Finite Group Databasebeta

← #26238365

Subgroups — #26238365

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#20253176C211yesmaximal, characteristic
2#20252335C211yesmaximal, characteristic, complemented
3#20251956C211yesmaximal, characteristic
4#20244369C211yesmaximal, characteristic, complemented
5#20244163C211yesmaximal, characteristic, complemented
6#20244140C211yesmaximal, characteristic, complemented
7#20244135C211yesmaximal, characteristic, complemented
8#20243990C211yesmaximal, characteristic, complemented
9#20243990C211yesmaximal, characteristic, complemented
10#20243661C211yesmaximal, characteristic
11#20243661C211yesmaximal, characteristic
12#20243245C211yesmaximal, characteristic, complemented
13#20243233C211yesmaximal, characteristic
14#20243226C211yesmaximal, characteristic
15C47×#1759C211yesmaximal, characteristic
16C2×#13094031C2211yescharacteristic
17C2×#13094031C2211yescharacteristic
18#15689680C2211yescharacteristic
19#15689430C411yescharacteristic, complemented
20#15689430C411yescharacteristic, complemented
21#15689423C411yescharacteristic, complemented
22#15689423C411yescharacteristic, complemented
23#15689418C411yescharacteristic, complemented
24#15689413C411yescharacteristic, complemented
25#15689408C411yescharacteristic, complemented
26#15689403C411yescharacteristic, complemented
27#15689253—21no—
28#15689251—21no—
29#15689251—21no—
30C8×Dic94—21no—
31C2×#13093959C2211yescharacteristic
32C2×#13093959C2211yescharacteristic
33#15689159C2211yescharacteristic
34#15689159C2211yescharacteristic
35#15689151C2211yescharacteristic
36#15689151C2211yescharacteristic
37#15689150—21no—
38#15689150—21no—
39#15689150C2211yescharacteristic
40#15689150C2211yescharacteristic
41#15689147—21no—
42#15689147C2211yescharacteristic
43C2×C4⋊(C47⋊C8)—21no—
44C2×C4⋊(C47⋊C8)C2211yescharacteristic
45C2×C4⋊(C47⋊C8)C2211yescharacteristic
46#15689119C2211yescharacteristic
47#15689119C2211yescharacteristic
48#15689081C2211yescharacteristic
49#15689081C2211yescharacteristic
50C2×#13093935C2211yescharacteristic
51C2×C4×C47⋊C8C2211yescharacteristic
52#15688887C2211yescharacteristic
53#15688887C2211yescharacteristic
54#15688883C2211yescharacteristic
55#15688883C2211yescharacteristic
56#15688883C2211yescharacteristic
57#15688883C2211yescharacteristic
58#15688878C2211yescharacteristic
59#15688878C2211yescharacteristic
60C4×#41×C47C2211yescharacteristic
61C2×#104×C47C2211yescharacteristic
62C2×#104×C47C2211yescharacteristic
63C4×M16×C47C2211yescharacteristic
64C47×#376C2211yescharacteristic
65C47×#339C2211yescharacteristic
66C47×#339C2211yescharacteristic
67#13094031—22no—
68#13094031—22no—
69#13094031—41no—
70#13094031—41no—
71C2×#10625329—21no—
72C2×#10625329—21no—
73C2×#10625329C2311yescharacteristic
74C2×#10625329C2311yescharacteristic
75C22×C47⋊C8—21no—
76C22×C47⋊C8—21no—
77C22×C47⋊C8C2311yescharacteristic
78C22×C47⋊C8C2311yescharacteristic
79C22×C47⋊C8C2311yescharacteristic
80C22×C47⋊C8C2311yescharacteristic
81#13094024C2×C411yescharacteristic
82#13094024C2×C411yescharacteristic
83#13094016C2×C411yescharacteristic
84#13094016C2×C411yescharacteristic
85#13094003—21no—
86#13094003—21no—
87C4×Dic94—21no—
88C4×Dic94—21no—
89#13093983C2311yescharacteristic
90#13093983C2311yescharacteristic
91#13093959—41no—
92#13093959—41no—
93#13093957C2×C411yescharacteristic
94#13093957C2×C411yescharacteristic
95#13093957C2×C411yescharacteristic
96#13093957C2×C411yescharacteristic
97#13093955C2×C411yescharacteristic
98#13093955C2×C411yescharacteristic
99#13093955C2×C411yescharacteristic
100#13093955C2×C411yescharacteristic
101C4⋊(C47⋊C8)—21no—
102C4⋊(C47⋊C8)—21no—
103C4⋊(C47⋊C8)—21no—
104C4⋊(C47⋊C8)—21no—
105C4⋊(C47⋊C8)—21no—
106C4⋊(C47⋊C8)—21no—
107C4⋊(C47⋊C8)C2×C411yescharacteristic
108C4⋊(C47⋊C8)C2×C411yescharacteristic
109C4⋊(C47⋊C8)C2×C411yescharacteristic
110C4⋊(C47⋊C8)C2×C411yescharacteristic
111C4⋊(C47⋊C8)D411yescharacteristic
112C4⋊(C47⋊C8)D411yescharacteristic
113C4⋊(C47⋊C8)D411yescharacteristic
114C4⋊(C47⋊C8)D411yescharacteristic
115Dic47⋊C8—21no—
116Dic47⋊C8—21no—
117Dic47⋊C8—21no—
118Dic47⋊C8—21no—
119#13093948C2×C411yescharacteristic
120#13093948C2×C411yescharacteristic
121#13093948C2×C411yescharacteristic
122#13093948C2×C411yescharacteristic
123#13093945C2×C411yescharacteristic
124#13093945C2×C411yescharacteristic
125#13093945C2×C411yescharacteristic
126#13093945C2×C411yescharacteristic
127#13093935—41no—
128#13093934—21no—
129#13093934—21no—
130C4×C47⋊C8—41no—
131C8×Dic47—21no—
132C8×Dic47—21no—
133C2×#41×C47C2311yescharacteristic
134C2×M16×C47C2311yescharacteristic
135C2×M16×C47C2311yescharacteristic
136C4×Q8×C47C2×C411yescharacteristic
137C4×Q8×C47C2×C411yescharacteristic
138C4×D4×C47—21no—
139C4×D4×C47—21no—
140C4×D4×C47C2×C411yescharacteristic
141C4×D4×C47C2×C411yescharacteristic
142#117×C47—21no—
143#117×C47C2311yescharacteristic
144C2×C42×C47—21no—
145C2×C42×C47C2311yescharacteristic
146C2×C42×C47C2311yescharacteristic
147C2×C42×C47C2311yescharacteristic
148#104×C47—22no—
149#104×C47Q814yescomplemented
150#97×C47—21no—
151C4×C8×C47—21no—
152#95×C47—21no—
153#95×C47—21no—
154#41×C47—21no—
155#41×C47—21no—
156#41×C47C2×Q812yes—
157#41×C47#4112yes—
158C2×Q8×C47C22×C411yescharacteristic
159C2×D4×C47—21no—
160C2×D4×C47C22×C411yescharacteristic
161C22×C4×C47—21no—
162C22×C4×C47—21no—
163C22×C4×C47—21no—
164C22×C4×C47—21no—
165C22×C4×C47—21no—
166C22×C4×C47#4111yescharacteristic
167C22×C4×C47#4111yescharacteristic
168C22×C4×C47C2411yescharacteristic
169C2×Dic94—21no—
170C2×Dic94—21no—
171M16×C47—21no—
172M16×C47—21no—
173M16×C47—21no—
174M16×C47C2×Q812yes—
175C2×C8×C47—21no—
176C2×C8×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47C22×C411yescharacteristic
180C4⋊C4×C47C22×C411yescharacteristic
181C22⋊C4×C47—21no—
182C22⋊C4×C47—21no—
183C22⋊C4×C47—21no—
184C42×C47—21no—
185C42×C47—21no—
186C42×C47—21no—
187C42×C47—21no—
188C42×C47—21no—
189C42×C47—21no—
190C42×C47C22×C411yescharacteristic
191C42×C47C22×C411yescharacteristic
192C42×C47C2×D411yescharacteristic
193C42×C47C2×D411yescharacteristic
194C42×C47C2×Q812yes—
195C22⋊Dic47—21no—
196C22⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47C22×C411yescharacteristic
200C4⋊Dic47C22×C411yescharacteristic
201C4⋊Dic47C22×C411yescharacteristic
202C4⋊Dic47C22×C411yescharacteristic
203Dic47⋊C4—21no—
204Dic47⋊C4—21no—
205Dic47⋊C4—21no—
206Dic47⋊C4—21no—
207C4×Dic47—21no—
208C4×Dic47—21no—
209#10625329—21no—
210#10625329—21no—
211#10625329—21no—
212#10625329—21no—
213#10625329—42no—
214#10625329—42no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8—21no—
221C2×C47⋊C8—21no—
222C2×C47⋊C8—21no—
223C2×C47⋊C8—21no—
224C2×C47⋊C8—21no—
225C2×C47⋊C8—42no—
226C2×C47⋊C8—42no—
227C2×C47⋊C8C22×C411yescharacteristic
228C2×C47⋊C8C22×C411yescharacteristic
229C2×C47⋊C8C22×C411yescharacteristic
230C2×C47⋊C8C22×C411yescharacteristic
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×C47C4×Q812yes—
239D4×C47—22no—
240D4×C47—22no—
241D4×C47—21no—
242D4×C47C4×Q812yes—
243C2×C4×C47—21no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—22no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—42no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
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—21no—
269C2×Dic47—21no—
270C2×Dic47—21no—
271C2×Dic47—21no—
272Dic94—42no—
273Dic94—81no—
274C8×C47—22no—
275C8×C47—41no—
276C47⋊C8—21no—
277C47⋊C8—21no—
278C47⋊C8—21no—
279C47⋊C8—21no—
280C47⋊C8—21no—
281C47⋊C8—21no—
282C47⋊C8—21no—
283C47⋊C8—21no—
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—42no—
301C4×C47—41no—
302C4×C47—41no—
303C4×C47—81no—
304C4×C47C2×C4×Q811yescharacteristic
305C4×C47#52011yescharacteristic
306C4×C47#54611yescharacteristic
307C4×C47D4×Q811yescharacteristic
308Dic47—41no—
309Dic47—41no—
310Dic47—41no—
311Dic47—41no—
312#3076—471no—
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#1759D4711yescharacteristic
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#533—471no—
336C4×#41C2×D4711yescharacteristic
337#479—471no—
338#479—471no—
339#478—471no—
340#478—471no—
341#477—471no—
342#476—471no—
343#475—471no—
344#474—471no—
345#446—941no—
346#446—941no—
347#446—941no—
348C8×Q8—941no—
349C2×#108—471no—
350C2×#108—471no—
351#427—471no—
352#427—471no—
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—471no—
363C2×C4⋊C8—941no—
364C2×#104C2×D4711yescharacteristic
365C2×#104C2×D4711yescharacteristic
366#416—471no—
367#416—471no—
368#405—471no—
369#405—471no—
370C4×M16C2×D4711yescharacteristic
371C2×#97—471no—
372C2×C4×C8—471no—
373#376C2×D4711yescharacteristic
374#341—471no—
375#341—471no—
376#339—471no—
377#339—471no—
378#339—471no—
379#339—471no—
380#339C2×D4711yescharacteristic
381#339C2×D4711yescharacteristic
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—941no—
393C2×M16—941no—
394C2×M16C22×D4711yescharacteristic
395C2×M16C22×D4711yescharacteristic
396C22×C8—471no—
397C22×C8—471no—
398C22×C8—471no—
399C22×C8—471no—
400C22×C8—941no—
401C22×C8—941no—
402C4⋊Q8—471no—
403C4⋊Q8—471no—
404#125—471no—
405#125—471no—
406C22⋊Q8—941no—
407C22⋊Q8—941no—
408C4×Q8—941no—
409C4×Q8—941no—
410C4×Q8C4×D4711yescharacteristic
411C4×Q8C4×D4711yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4C4×D4711yescharacteristic
415C4×D4C4×D4711yescharacteristic
416#117—21no—
417#117—471no—
418#117—471no—
419#117C22×D4711yescharacteristic
420C2×C42—21no—
421C2×C42C22×D4711yescharacteristic
422C2×C42C22×D4711yescharacteristic
423C2×C42C22×D4711yescharacteristic
424#108—1881no—
425#108—1881no—
426#107—471no—
427#107—471no—
428#107—471no—
429#107—471no—
430#106—471no—
431#106—471no—
432#106—471no—
433#106—471no—
434C4⋊C8—471no—
435C4⋊C8—471no—
436C4⋊C8—471no—
437C4⋊C8—471no—
438C4⋊C8—471no—
439C4⋊C8—471no—
440C4⋊C8—471no—
441C4⋊C8—471no—
442C4⋊C8—941no—
443C4⋊C8—941no—
444C4⋊C8—941no—
445C4⋊C8—941no—
446C4⋊C8—941no—
447C4⋊C8—941no—
448C4⋊C8—941no—
449C4⋊C8—941no—
450C4⋊C8—941no—
451C4⋊C8—941no—
452#104—22no—
453#104Dic9414yescomplemented
454#103—471no—
455#103—471no—
456#103—471no—
457#103—471no—
458#102—471no—
459#102—471no—
460#102—471no—
461#102—471no—
462#97—21no—
463#97—941no—
464#97—941no—
465#97—1881no—
466C4×C8—21no—
467C4×C8—941no—
468C4×C8—941no—
469C4×C8—1881no—
470#95—21no—
471#95—21no—
472#41—21no—
473#41—21no—
474#41C2×Dic9412yes—
475#41#1062535012yes—
476C2×Q8—941no—
477C2×Q8—941no—
478C2×Q8C2×C4×D4711yescharacteristic
479C2×D4—21no—
480C2×D4C2×C4×D4711yescharacteristic
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4#1062535211yescharacteristic
487C22×C4#1062535211yescharacteristic
488C22×C4C23×D4711yescharacteristic
489M16—21no—
490M16—21no—
491M16—21no—
492M16—941no—
493M16—941no—
494M16—941no—
495M16—941no—
496M16—1882no—
497M16—1882no—
498M16C2×Dic9412yes—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—471no—
502C2×C8—471no—
503C2×C8—471no—
504C2×C8—471no—
505C2×C8—471no—
506C2×C8—471no—
507C2×C8—471no—
508C2×C8—471no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—941no—
515C2×C8—941no—
516C2×C8—941no—
517C2×C8—941no—
518C2×C8—941no—
519C2×C8—1882no—
520C2×C8—1882no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—471no—
524C4⋊C4—471no—
525C4⋊C4—471no—
526C4⋊C4—471no—
527C4⋊C4—941no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4—941no—
531C4⋊C4—941no—
532C4⋊C4—941no—
533C4⋊C4C2×C4×D4711yescharacteristic
534C4⋊C4C2×C4×D4711yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—941no—
539C22⋊C4—941no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—941no—
547C42—941no—
548C42C2×Dic9412yes—
549C42C2×C4×D4711yescharacteristic
550C42C2×C4×D4711yescharacteristic
551C42D4×D4711yescharacteristic
552C42D4×D4711yescharacteristic
553C23—21no—
554C23C2×#1062535211yescharacteristic
555Q8—21no—
556Q8—3761no—
557Q8—1882no—
558Q8C4×Dic9412yes—
559D4—22no—
560D4—22no—
561D4—21no—
562D4C4×Dic9412yes—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—81no—
578C2×C4—941no—
579C2×C4—941no—
580C2×C4—941no—
581C2×C4—941no—
582C2×C4(C22⋊D47)⋊C411yescharacteristic
583C2×C4(C22⋊D47)⋊C411yescharacteristic
584C2×C4(C22⋊D47)⋊C411yescharacteristic
585C2×C4(C22⋊D47)⋊C411yescharacteristic
586C2×C4C22×C4×D4711yescharacteristic
587C2×C4C2×D4×D4711yescharacteristic
588C2×C4C22×Dic9411yescharacteristic
589C2×C4C2×#1062535011yescharacteristic
590C2×C4#1309407011yescharacteristic
591C2×C4#1309407111yescharacteristic
592C8—22no—
593C8—41no—
594C8—941no—
595C8—941no—
596C8—941no—
597C8—941no—
598C8—941no—
599C8—941no—
600C8—941no—
601C8—941no—
602C8—1881no—
603C8—1881no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22C2×(C22⋊D47)⋊C411yescharacteristic
610C22#1568981311yescharacteristic
611C22#1568983011yescharacteristic
612C4—22no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—42no—
618C4—41no—
619C4—41no—
620C4—81no—
621C4—1881no—
622C4—1881no—
623C4—1881no—
624C4—1881no—
625C4C2×C4×Dic9411yescharacteristic
626C4#1568960311yescharacteristic
627C4#1568976911yescharacteristic
628C4D4×Dic9411yescharacteristic
629C2—21no—
630C2—21no—
631C2—42no—
632C2#2024639111yescharacteristic
633C2#2025228711yescharacteristic
634C2#2025230111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1759C2
socleC22×C47C2×C4×D4
prime radicalC2×#41×C47C23
cosocleC22×C4×C47C24
hypercenter#1759D47
hypocenterC47#22984
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1759

Complements

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

NH ≅ G/Ntype
2. #20252335C2semidirect
4. #20244369C2semidirect
5. #20244163C2semidirect
6. #20244140C2semidirect
7. #20244135C2semidirect
8. #20243990C2semidirect
9. #20243990C2semidirect
12. #20243245C2semidirect
19. #15689430C4semidirect
20. #15689430C4semidirect
21. #15689423C4semidirect
22. #15689423C4semidirect
23. #15689418C4semidirect
24. #15689413C4semidirect
25. #15689408C4semidirect
26. #15689403C4semidirect
149. #104×C47Q8semidirect
384. C47#22984semidirect
453. #104Dic94semidirect

Maximal subgroups

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

subgroupindex
#202531762
#202523352
#202519562
#202443692
#202441632
#202441402
#202441352
#202439902
#202436612
#202432452
#202432332
#202432262
C47×#17592
#2298447