Finite Group Databasebeta

← #26238373

Subgroups — #26238373

Order 12032.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252337C211yesmaximal, characteristic, complemented
2#20251956C211yesmaximal, characteristic
3#20244370C211yesmaximal, characteristic, complemented
4#20244164C211yesmaximal, characteristic, complemented
5#20244142C211yesmaximal, characteristic, complemented
6#20244137C211yesmaximal, characteristic, complemented
7#20243989C211yesmaximal, characteristic, complemented
8#20243989C211yesmaximal, characteristic, complemented
9#20243665C211yesmaximal, characteristic
10#20243665C211yesmaximal, characteristic
11#20243247C211yesmaximal, characteristic, complemented
12#20243235C211yesmaximal, characteristic
13#20243228C211yesmaximal, characteristic
14#20243198C211yesmaximal, characteristic
15C47×#3076C211yesmaximal, characteristic
16C2×#13094031C2211yescharacteristic
17C2×#13094031C2211yescharacteristic
18#15689254—21no—
19#15689254—21no—
20#15689254—21no—
21Q8×C47⋊C8—21no—
22C2×#13093960C2211yescharacteristic
23C2×#13093960C2211yescharacteristic
24#15689150—21no—
25#15689150—21no—
26#15689150C2211yescharacteristic
27#15689150C2211yescharacteristic
28#15689150C2211yescharacteristic
29#15689150C2211yescharacteristic
30#15689147—21no—
31#15689147C2211yescharacteristic
32C2×C4⋊(C47⋊C8)—21no—
33C2×C4⋊(C47⋊C8)C2211yescharacteristic
34C2×#13093951C2211yescharacteristic
35C2×#13093951C2211yescharacteristic
36#15689079C2211yescharacteristic
37#15689079C2211yescharacteristic
38C4×#10625329C2211yescharacteristic
39C2×#13093934C2211yescharacteristic
40C2×C8×Dic47C2211yescharacteristic
41#15688969C2211yescharacteristic
42#15688888C2211yescharacteristic
43#15688888C2211yescharacteristic
44#15688882C2211yescharacteristic
45#15688882C2211yescharacteristic
46#15688882C2211yescharacteristic
47#15688882C2211yescharacteristic
48#15688882C2211yescharacteristic
49#15688882C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52C47×#533C2211yescharacteristic
53C4×#41×C47C2211yescharacteristic
54C47×#479C411yescharacteristic, complemented
55C47×#479C411yescharacteristic, complemented
56C47×#478C411yescharacteristic, complemented
57C47×#478C411yescharacteristic, complemented
58C47×#477C411yescharacteristic, complemented
59C47×#476C411yescharacteristic, complemented
60C47×#475C411yescharacteristic, complemented
61C47×#474C411yescharacteristic, complemented
62C47×#427C2211yescharacteristic
63C47×#427C2211yescharacteristic
64C2×C4⋊C8×C47C2211yescharacteristic
65C47×#416C2211yescharacteristic
66C47×#416C2211yescharacteristic
67#13094031—22no—
68#13094031—22no—
69#13094031—41no—
70#13094031—41no—
71C2×#10625329—21no—
72C2×#10625329—21no—
73C2×#10625329C2311yescharacteristic
74C2×#10625329C2311yescharacteristic
75C2×#10625329C2311yescharacteristic
76C2×#10625329C2311yescharacteristic
77C22×C47⋊C8—21no—
78C22×C47⋊C8—21no—
79C22×C47⋊C8C2311yescharacteristic
80C22×C47⋊C8C2311yescharacteristic
81C2×C4×Dic47C2311yescharacteristic
82C2×C4×Dic47C2311yescharacteristic
83#13093960—41no—
84#13093960—41no—
85C4⋊(C47⋊C8)—21no—
86C4⋊(C47⋊C8)—21no—
87C4⋊(C47⋊C8)—21no—
88C4⋊(C47⋊C8)—21no—
89C4⋊(C47⋊C8)—21no—
90C4⋊(C47⋊C8)—21no—
91C4⋊(C47⋊C8)—21no—
92C4⋊(C47⋊C8)—21no—
93C4⋊(C47⋊C8)—21no—
94C4⋊(C47⋊C8)—21no—
95C4⋊(C47⋊C8)D411yescharacteristic
96C4⋊(C47⋊C8)D411yescharacteristic
97C4⋊(C47⋊C8)D411yescharacteristic
98C4⋊(C47⋊C8)D411yescharacteristic
99#13093951—22no—
100#13093951Q814yescomplemented
101#13093935—21no—
102#13093935—21no—
103#13093935—21no—
104#13093934—41no—
105C4×C47⋊C8—21no—
106C4×C47⋊C8—21no—
107C4×C47⋊C8—21no—
108C8×Dic47—41no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C2×#41×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C4⋊Q8×C47C2×C411yescharacteristic
115C4⋊Q8×C47C2×C411yescharacteristic
116#125×C47C2×C411yescharacteristic
117#125×C47C2×C411yescharacteristic
118C22⋊Q8×C47—21no—
119C22⋊Q8×C47—21no—
120C4×Q8×C47—21no—
121C4×Q8×C47—21no—
122C4×Q8×C47C2×C411yescharacteristic
123C4×Q8×C47C2×C411yescharacteristic
124C4×D4×C47—21no—
125C4×D4×C47—21no—
126C4×D4×C47C2×C411yescharacteristic
127C4×D4×C47C2×C411yescharacteristic
128#117×C47—21no—
129#117×C47C2311yescharacteristic
130#117×C47C2311yescharacteristic
131#117×C47C2311yescharacteristic
132C2×C42×C47—21no—
133C2×C42×C47C2311yescharacteristic
134#107×C47C2×C411yescharacteristic
135#107×C47C2×C411yescharacteristic
136#107×C47C2×C411yescharacteristic
137#107×C47C2×C411yescharacteristic
138#106×C47C2×C411yescharacteristic
139#106×C47C2×C411yescharacteristic
140#106×C47C2×C411yescharacteristic
141#106×C47C2×C411yescharacteristic
142C4⋊C8×C47C2×C411yescharacteristic
143C4⋊C8×C47C2×C411yescharacteristic
144C4⋊C8×C47C2×C411yescharacteristic
145C4⋊C8×C47C2×C411yescharacteristic
146#103×C47C2×C411yescharacteristic
147#103×C47C2×C411yescharacteristic
148#103×C47C2×C411yescharacteristic
149#103×C47C2×C411yescharacteristic
150#102×C47C2×C411yescharacteristic
151#102×C47C2×C411yescharacteristic
152#102×C47C2×C411yescharacteristic
153#102×C47C2×C411yescharacteristic
154#41×C47—21no—
155#41×C47—21no—
156#41×C47C2×Q812yes—
157#41×C47#4112yes—
158C2×Q8×C47—21no—
159C2×Q8×C47—21no—
160C2×Q8×C47C22×C411yescharacteristic
161C2×D4×C47—21no—
162C2×D4×C47C22×C411yescharacteristic
163C22×C4×C47—21no—
164C22×C4×C47—21no—
165C22×C4×C47—21no—
166C22×C4×C47#4111yescharacteristic
167C22×C4×C47#4111yescharacteristic
168C22×C4×C47C2411yescharacteristic
169C22×Dic47—21no—
170C22×Dic47—21no—
171C2×C8×C47—21no—
172C2×C8×C47—21no—
173C2×C8×C47—21no—
174C2×C8×C47—21no—
175C2×C8×C47C22×C411yescharacteristic
176C2×C8×C47C22×C411yescharacteristic
177C2×C8×C47C22×C411yescharacteristic
178C2×C8×C47C22×C411yescharacteristic
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—21no—
185C4⋊C4×C47—21no—
186C4⋊C4×C47—21no—
187C4⋊C4×C47C22×C411yescharacteristic
188C4⋊C4×C47C22×C411yescharacteristic
189C4⋊C4×C47C22×C411yescharacteristic
190C4⋊C4×C47C22×C411yescharacteristic
191C4⋊C4×C47C22×C411yescharacteristic
192C4⋊C4×C47C22×C411yescharacteristic
193C22⋊C4×C47—21no—
194C22⋊C4×C47—21no—
195C22⋊C4×C47—21no—
196C22⋊C4×C47—21no—
197C22⋊C4×C47—21no—
198C42×C47—21no—
199C42×C47—21no—
200C42×C47—21no—
201C42×C47—21no—
202C42×C47—21no—
203C42×C47C22×C411yescharacteristic
204C42×C47C22×C411yescharacteristic
205C42×C47C2×D411yescharacteristic
206C42×C47C2×D411yescharacteristic
207C4×Dic47—21no—
208C4×Dic47—21no—
209C4×Dic47—21no—
210C4×Dic47C2×Q812yes—
211#10625329—21no—
212#10625329—21no—
213#10625329—21no—
214#10625329—21no—
215#10625329—21no—
216#10625329—21no—
217#10625329—21no—
218#10625329—42no—
219#10625329—42no—
220#10625329C2×Q812yes—
221C2×C47⋊C8—21no—
222C2×C47⋊C8—21no—
223C2×C47⋊C8—21no—
224C2×C47⋊C8—21no—
225C2×C47⋊C8—21no—
226C2×C47⋊C8—21no—
227C2×C47⋊C8—21no—
228C2×C47⋊C8—21no—
229C2×C47⋊C8—42no—
230C2×C47⋊C8—42no—
231C2×C47⋊C8C2×D411yescharacteristic
232C2×C47⋊C8C2×D411yescharacteristic
233C2×C47⋊C8C2×D411yescharacteristic
234C2×C47⋊C8C2×D411yescharacteristic
235C23×C47—21no—
236C23×C47C2×#4111yescharacteristic
237Q8×C47—21no—
238Q8×C47—42no—
239Q8×C47—81no—
240Q8×C47C4×Q812yes—
241D4×C47—22no—
242D4×C47—22no—
243D4×C47—21no—
244D4×C47C4×Q812yes—
245C2×C4×C47—21no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—21no—
249C2×C4×C47—21no—
250C2×C4×C47—21no—
251C2×C4×C47—21no—
252C2×C4×C47—21no—
253C2×C4×C47—21no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—42no—
257C2×C4×C47—81no—
258C2×C4×C47C4×D411yescharacteristic
259C2×C4×C47C4×D411yescharacteristic
260C2×C4×C47C4×D411yescharacteristic
261C2×C4×C47C4×D411yescharacteristic
262C2×C4×C47C23×C411yescharacteristic
263C2×C4×C47C22×D411yescharacteristic
264C2×C4×C47C22×Q811yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×C4×C47H(5,𝔽2)11yescharacteristic
267C2×C4×C472−1+411yescharacteristic
268C2×Dic47—22no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C8×C47—21no—
275C8×C47—21no—
276C8×C47—21no—
277C8×C47—21no—
278C47⋊C8—21no—
279C47⋊C8—21no—
280C47⋊C8—21no—
281C47⋊C8—21no—
282C47⋊C8—22no—
283C47⋊C8—41no—
284C47⋊C8—41no—
285C47⋊C8—41no—
286#22984—471nomaximal, Sylow
287C22×C47—21no—
288C22×C47—21no—
289C22×C47—22no—
290C22×C47—41no—
291C22×C47—41no—
292C22×C47C2×C4×D411yescharacteristic
293C22×C47#55211yescharacteristic
294C22×C47#55411yescharacteristic
295C4×C47—22no—
296C4×C47—21no—
297C4×C47—21no—
298C4×C47—21no—
299C4×C47—21no—
300C4×C47—41no—
301C4×C47—41no—
302C4×C47—41no—
303C4×C47—41no—
304C4×C47—41no—
305C4×C47—41no—
306C4×C47C2×C4×Q811yescharacteristic
307C4×C47#52011yescharacteristic
308C4×C47#54611yescharacteristic
309C4×C47D4×Q811yescharacteristic
310Dic47—42no—
311Dic47—81no—
312#3076D4711yescharacteristic
313#2962—471no—
314#2906—471no—
315#1995—471no—
316#1950—471no—
317#1946—471no—
318#1945—471no—
319#1922—471no—
320#1922—471no—
321#1866—471no—
322#1866—471no—
323#1768—471no—
324#1765—471no—
325#1764—471no—
326#1759—471no—
327C2×C47—21no—
328C2×C47—21no—
329C2×C47—42no—
330C2×C47#236011yescharacteristic
331C2×C47#295511yescharacteristic
332C2×C47#295711yescharacteristic
333C2×#131—471no—
334C2×#131—471no—
335#533C2×D4711yescharacteristic
336C4×#41C2×D4711yescharacteristic
337#479Dic4711yescharacteristic, complemented
338#479Dic4711yescharacteristic, complemented
339#478Dic4711yescharacteristic, complemented
340#478Dic4711yescharacteristic, complemented
341#477Dic4711yescharacteristic, complemented
342#476Dic4711yescharacteristic, complemented
343#475Dic4711yescharacteristic, complemented
344#474Dic4711yescharacteristic, complemented
345#446—941no—
346#446—941no—
347#446—941no—
348C8×Q8—941no—
349C2×#108—471no—
350C2×#108—471no—
351#427C2×D4711yescharacteristic
352#427C2×D4711yescharacteristic
353#424—471no—
354#424—471no—
355#424—471no—
356#424—471no—
357#424—941no—
358#424—941no—
359#423—471no—
360#423—941no—
361C2×C4⋊C8—471no—
362C2×C4⋊C8—941no—
363C2×C4⋊C8C2×D4711yescharacteristic
364C2×#104—471no—
365C2×#104—471no—
366#416C2×D4711yescharacteristic
367#416C2×D4711yescharacteristic
368#405—471no—
369#405—471no—
370C4×M16—471no—
371C2×#97—471no—
372C2×C4×C8—471no—
373#376—471no—
374#341—471no—
375#341—471no—
376#339—471no—
377#339—471no—
378#339—471no—
379#339—471no—
380#339—471no—
381#339—471no—
382#336—471no—
383#336—471no—
384C47#2298411yescharacteristic, complemented, Sylow
385C2×#41C22×D4711yescharacteristic
386#131—942no—
387#131—942no—
388#131—1881no—
389#131—1881no—
390C2×M16—471no—
391C2×M16—471no—
392C2×M16—471no—
393C2×M16—471no—
394C2×M16—941no—
395C2×M16—941no—
396C22×C8—471no—
397C22×C8—471no—
398C22×C8—941no—
399C22×C8—941no—
400C22×C8C22×D4711yescharacteristic
401C22×C8C22×D4711yescharacteristic
402C4⋊Q8C2×Dic4711yescharacteristic
403C4⋊Q8C2×Dic4711yescharacteristic
404#125C2×Dic4711yescharacteristic
405#125C2×Dic4711yescharacteristic
406C22⋊Q8—21no—
407C22⋊Q8—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8C2×Dic4711yescharacteristic
411C4×Q8C2×Dic4711yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4C2×Dic4711yescharacteristic
415C4×D4C2×Dic4711yescharacteristic
416#117—21no—
417#117C22×D4711yescharacteristic
418#117C22×D4711yescharacteristic
419#117C22×D4711yescharacteristic
420C2×C42—21no—
421C2×C42—471no—
422C2×C42—471no—
423C2×C42C22×D4711yescharacteristic
424#108—1881no—
425#108—1881no—
426#107C2×Dic4711yescharacteristic
427#107C2×Dic4711yescharacteristic
428#107C2×Dic4711yescharacteristic
429#107C2×Dic4711yescharacteristic
430#106C2×Dic4711yescharacteristic
431#106C2×Dic4711yescharacteristic
432#106C2×Dic4711yescharacteristic
433#106C2×Dic4711yescharacteristic
434C4⋊C8—471no—
435C4⋊C8—471no—
436C4⋊C8—471no—
437C4⋊C8—471no—
438C4⋊C8—941no—
439C4⋊C8—941no—
440C4⋊C8—941no—
441C4⋊C8—941no—
442C4⋊C8—941no—
443C4⋊C8—941no—
444C4⋊C8—941no—
445C4⋊C8—941no—
446C4⋊C8—941no—
447C4⋊C8—941no—
448C4⋊C8C2×Dic4711yescharacteristic
449C4⋊C8C2×Dic4711yescharacteristic
450C4⋊C8C2×Dic4711yescharacteristic
451C4⋊C8C2×Dic4711yescharacteristic
452#104—474no—
453#104—942no—
454#103C2×Dic4711yescharacteristic
455#103C2×Dic4711yescharacteristic
456#103C2×Dic4711yescharacteristic
457#103C2×Dic4711yescharacteristic
458#102C2×Dic4711yescharacteristic
459#102C2×Dic4711yescharacteristic
460#102C2×Dic4711yescharacteristic
461#102C2×Dic4711yescharacteristic
462#97—941no—
463#97—941no—
464#97—941no—
465#97—1881no—
466C4×C8—941no—
467C4×C8—941no—
468C4×C8—941no—
469C4×C8—1881no—
470#95—941no—
471#95—941no—
472#41—21no—
473#41—21no—
474#41Q8×D4712yes—
475#41#1062535412yes—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8C22×Dic4711yescharacteristic
479C2×D4—21no—
480C2×D4C22×Dic4711yescharacteristic
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—941no—
485C22×C4—941no—
486C22×C4#1062535211yescharacteristic
487C22×C4#1062535211yescharacteristic
488C22×C4C23×D4711yescharacteristic
489M16—472no—
490M16—941no—
491M16—941no—
492M16—941no—
493M16—941no—
494M16—941no—
495M16—941no—
496M16—941no—
497M16—1882no—
498M16—1882no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—471no—
504C2×C8—471no—
505C2×C8—471no—
506C2×C8—471no—
507C2×C8—941no—
508C2×C8—941no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—941no—
515C2×C8—1882no—
516C2×C8—1882no—
517C2×C8C22×Dic4711yescharacteristic
518C2×C8C22×Dic4711yescharacteristic
519C2×C8C22×Dic4711yescharacteristic
520C2×C8C22×Dic4711yescharacteristic
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4C22×Dic4711yescharacteristic
530C4⋊C4C22×Dic4711yescharacteristic
531C4⋊C4C22×Dic4711yescharacteristic
532C4⋊C4C22×Dic4711yescharacteristic
533C4⋊C4C22×Dic4711yescharacteristic
534C4⋊C4C22×Dic4711yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—472no—
546C42—941no—
547C42—941no—
548C42—941no—
549C42D4×D4711yescharacteristic
550C42D4×D4711yescharacteristic
551C42C22×Dic4711yescharacteristic
552C42C22×Dic4711yescharacteristic
553C23—21no—
554C23C2×#1062535211yescharacteristic
555Q8—21no—
556Q8—42no—
557Q8—81no—
558Q8Q8×Dic4712yes—
559D4—22no—
560D4—22no—
561D4—21no—
562D4Q8×Dic4712yes—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—81no—
576C2×C4—942no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4D4×Dic4711yescharacteristic
583C2×C4D4×Dic4711yescharacteristic
584C2×C4D4×Dic4711yescharacteristic
585C2×C4D4×Dic4711yescharacteristic
586C2×C4C23×Dic4711yescharacteristic
587C2×C4C2×D4×D4711yescharacteristic
588C2×C4C2×Q8×D4711yescharacteristic
589C2×C4C2×#1062535411yescharacteristic
590C2×C4#1309407011yescharacteristic
591C2×C4#1309407111yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—941no—
597C8—941no—
598C8—941no—
599C8—941no—
600C8—942no—
601C8—1881no—
602C8—1881no—
603C8—1881no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22C2×D4×Dic4711yescharacteristic
610C22#1568981511yescharacteristic
611C22#1568983111yescharacteristic
612C4—22no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—3761no—
624C4—1882no—
625C4C2×Q8×Dic4711yescharacteristic
626C4#1568960611yescharacteristic
627C4#1568976411yescharacteristic
628C4#1568978611yescharacteristic
629C2—21no—
630C2—21no—
631C2—42no—
632C2#2024639711yescharacteristic
633C2#2025229111yescharacteristic
634C2#2025230511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#3076C2
socleC22×C47C2×C4×D4
prime radicalC2×#41×C47C23
cosocleC22×C4×C47C24
hypercenter#3076D47
hypocenterC47#22984
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20252337C2semidirect
3. #20244370C2semidirect
4. #20244164C2semidirect
5. #20244142C2semidirect
6. #20244137C2semidirect
7. #20243989C2semidirect
8. #20243989C2semidirect
11. #20243247C2semidirect
54. C47×#479C4semidirect
55. C47×#479C4semidirect
56. C47×#478C4semidirect
57. C47×#478C4semidirect
58. C47×#477C4semidirect
59. C47×#476C4semidirect
60. C47×#475C4semidirect
61. C47×#474C4semidirect
100. #13093951Q8semidirect
337. #479Dic47semidirect
338. #479Dic47semidirect
339. #478Dic47semidirect
340. #478Dic47semidirect
341. #477Dic47semidirect
342. #476Dic47semidirect
343. #475Dic47semidirect
344. #474Dic47semidirect
384. C47#22984semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202523372
#202519562
#202443702
#202441642
#202441422
#202441372
#202439892
#202436652
#202432472
#202432352
#202432282
#202431982
C47×#30762
#2298447