Finite Group Databasebeta

← #26219759

Subgroups — #26219759

Order 12032.

1022 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689164C211yesmaximal, characteristic
2C2×#15689162C211yesmaximal, characteristic
3C2×#15689145C211yesmaximal, characteristic
4#382×D47C211yesmaximal, characteristic
5C4×#13093982C211yesmaximal, characteristic
6#20244159C211yesmaximal, characteristic, complemented
7#20244158C211yesmaximal, characteristic, complemented
8#20244157C211yesmaximal, characteristic, complemented
9#20244156C211yesmaximal, characteristic, complemented
10#20244155C211yesmaximal, characteristic, complemented
11#20244154C211yesmaximal, characteristic, complemented
12#20244153C211yesmaximal, characteristic, complemented
13#20243106C211yesmaximal, characteristic
14#20243106C211yesmaximal, characteristic
15C47×#1949C211yesmaximal, characteristic, complemented
16C22×#10625324C2211yescharacteristic
17C22×#10625324C2211yescharacteristic
18C2×#13093982C2211yescharacteristic
19C2×#13093982C2211yescharacteristic
20C2×C4⋊C4×D47C2211yescharacteristic
21C2×C4⋊C4×D47C2211yescharacteristic
22C2×C42×D47C2211yescharacteristic
23#15689164—24no—
24#15689162—24no—
25C2×#13093958C2211yescharacteristic
26C2×#13093957C2211yescharacteristic
27C2×#13093957C2211yescharacteristic
28C2×#13093956C2211yescharacteristic
29C2×#13093955C2211yescharacteristic
30C2×#13093955C2211yescharacteristic
31#15689145C418yescomplemented
32C2×C4⋊(C47⋊C8)C2211yescharacteristic
33C2×Dic47⋊C8C2211yescharacteristic
34C2×Dic47⋊C8C2211yescharacteristic
35#15688989C2211yescharacteristic
36#15688988C2211yescharacteristic
37#15688988C2211yescharacteristic
38#15688987—21no—
39#15688987—21no—
40C4⋊C4×Dic47C2211yescharacteristic
41C4⋊C4×Dic47C2211yescharacteristic
42C4×C4⋊Dic47C2211yescharacteristic
43C4×(C2×D47)⋊C4—21no—
44#15688970—21no—
45#15688968—21no—
46#95×D47—21no—
47#95×D47—21no—
48C42×Dic47—21no—
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×#107×C47C2211yescharacteristic
56C2×#106×C47C2211yescharacteristic
57C2×C4⋊C8×C47C2211yescharacteristic
58C47×#382C2211yescharacteristic
59C4×C4⋊C4×C47C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×C4×D47—21no—
63C22×C4×D47—21no—
64C22×C4×D47D411yescharacteristic
65C22×C4×D47D411yescharacteristic
66C22×C4×D47C2311yescharacteristic
67C2×#10625324—22no—
68C2×#10625324—22no—
69C2×#10625324—22no—
70C2×#10625324—22no—
71C2×#10625324C2×C414yes—
72C2×#10625324C2×C414yes—
73C22×C47⋊C8C2311yescharacteristic
74C22×C47⋊C8C2311yescharacteristic
75#13093982—22no—
76#13093982—22no—
77#13093982D414yes—
78#13093982Q814yes—
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C4⋊Dic47C2311yescharacteristic
81C2×C4⋊Dic47C2311yescharacteristic
82C2×C4⋊Dic47C2311yescharacteristic
83C2×Dic47⋊C4—21no—
84C2×Dic47⋊C4—21no—
85C4⋊C4×D47—22no—
86C4⋊C4×D47—22no—
87C4⋊C4×D47—22no—
88C4⋊C4×D47—22no—
89C2×(C2×D47)⋊C4—21no—
90C2×(C2×D47)⋊C4—21no—
91C2×C4×Dic47—21no—
92C2×C4×Dic47—21no—
93C2×C4×Dic47—21no—
94C2×C4×Dic47—21no—
95C2×C4×Dic47—21no—
96C2×C4×Dic47D411yescharacteristic
97C2×C4×Dic47D411yescharacteristic
98C2×C4×Dic47C2311yescharacteristic
99C42×D47C2×C414yes—
100C42×D47Q814yes—
101#13093958—22no—
102#13093957—22no—
103#13093957—22no—
104#13093956—22no—
105#13093955—22no—
106#13093955—22no—
107C4⋊(C47⋊C8)C2×C414yescomplemented
108Dic47⋊C8C2×C414yescomplemented
109Dic47⋊C8C2×C414yescomplemented
110C47⋊#95—21no—
111C47⋊#95—21no—
112C47⋊#95—21no—
113C47⋊#95—21no—
114C47⋊#95—21no—
115C47⋊#95—21no—
116C47⋊#95—21no—
117C47⋊#95—21no—
118C47⋊#95—21no—
119C22×C8×C47C2311yescharacteristic
120C22×C8×C47C2311yescharacteristic
121C2×C4⋊C4×C47C2311yescharacteristic
122C2×C4⋊C4×C47C2311yescharacteristic
123C2×C4⋊C4×C47C2311yescharacteristic
124C2×C4⋊C4×C47C2311yescharacteristic
125C2×C42×C47—21no—
126C2×C42×C47C2311yescharacteristic
127#107×C47—22no—
128#106×C47—22no—
129C4⋊C8×C47C2×C414yescomplemented
130#95×C47—21no—
131#95×C47—21no—
132#95×C47—21no—
133C23×D47C2×D411yescharacteristic
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2411yescharacteristic
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47C2×D411yescharacteristic
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C2×C4×D47—41no—
149C2×C4×D47—41no—
150C2×C4×D47—41no—
151C2×C4×D47—41no—
152C2×C4×D47—42no—
153C2×C4×D47—42no—
154C2×C4×D47—42no—
155C2×C4×D47—42no—
156C2×C4×D47—42no—
157C2×C4×D47—42no—
158C2×C4×D47C4⋊C414yes—
159C2×C4×D47C4⋊C414yes—
160C2×C4×D47C22×C412yes—
161C2×C4×D47C2×D412yes—
162C2×C4×D47C2×Q812yes—
163C2×C4×D47C2×Q812yes—
164C2×C4×D47#4112yes—
165C2×C4×D47#4112yes—
166C2×C8×C47—21no—
167C2×C8×C47—21no—
168C2×C8×C47—21no—
169C2×C8×C47—21no—
170C2×C8×C47C22×C412yes—
171C2×C8×C47C22×C412yes—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47C2×D412yes—
179C4⋊C4×C47C2×Q812yes—
180C42×C47—42no—
181C42×C47C22×C412yes—
182C42×C47C2×Q812yes—
183(C2×D47)⋊C4—24no—
184(C2×D47)⋊C4—24no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47C2×D412yes—
192C4⋊Dic47C2×Q812yes—
193Dic47⋊C4—42no—
194Dic47⋊C4—42no—
195C4×Dic47—24no—
196C4×Dic47—24no—
197C4×Dic47—42no—
198C4×Dic47—42no—
199C4×Dic47—42no—
200C4×Dic47C4⋊C414yes—
201C4×Dic47C4⋊C414yes—
202C4×Dic47C22×C412yes—
203C4×Dic47C2×Q812yes—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8C22×C412yes—
209C2×C47⋊C8C22×C412yes—
210#10625324—28no—
211#10625324—28no—
212C23×C47C22×D411yescharacteristic
213C22×D47—21no—
214C22×D47—21no—
215C22×D47—21no—
216C22×D47—21no—
217C22×D47C2×C4⋊C412yes—
218C22×D47#12112yes—
219C22×D47C22⋊Q812yes—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47C2×C4⋊C412yes—
237C2×C4×C47C2×C4⋊C412yes—
238C2×C4×C47C23×C411yescharacteristic
239C2×C4×C47C22×D411yescharacteristic
240C2×C4×C47C22×Q811yescharacteristic
241C2×C4×C47C22×Q811yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47C2×C4⋊C412yes—
261C2×Dic47C2×C4⋊C412yes—
262C2×Dic47C2×C4⋊C412yes—
263C2×Dic47#12112yes—
264C2×Dic47C22⋊Q812yes—
265C4×D47—44no—
266C4×D47—44no—
267C4×D47—84no—
268C4×D47—84no—
269C4×D47C4×D414yes—
270C4×D47C4×Q814yes—
271C4×D47#12514yes—
272C4×D47C4⋊Q814yes—
273C8×C47—22no—
274C8×C47—22no—
275C47⋊C8—22no—
276C47⋊C8—22no—
277#21172—471nomaximal, Sylow
278C22×C47C22×C4⋊C411yescharacteristic
279C22×C47C2×#12111yescharacteristic
280C22×C47C2×C22⋊Q811yescharacteristic
281C22×C47#57511yescharacteristic
282C22×C47#57511yescharacteristic
283C22×C47#57511yescharacteristic
284C22×C47#57511yescharacteristic
285C2×D47—22no—
286C2×D47—22no—
287C2×D47—22no—
288C2×D47—22no—
289C2×D47—22no—
290C2×D47—22no—
291C2×D47#38914yes—
292C4×C47—22no—
293C4×C47—22no—
294C4×C47—42no—
295C4×C47—42no—
296C4×C47—81no—
297C4×C47—81no—
298C4×C47C2×C4×D411yescharacteristic
299C4×C47C2×C4×Q811yescharacteristic
300C4×C47C2×#12511yescharacteristic
301C4×C47C2×C4⋊Q811yescharacteristic
302Dic47—22no—
303Dic47—22no—
304Dic47—42no—
305Dic47—42no—
306Dic47—81no—
307Dic47—81no—
308Dic47#38914yes—
309C2×#428—471no—
310C2×#428—471no—
311C2×#423—471no—
312C2×#382—471no—
313C4×#117—471no—
314#1949—471no—
315#1949—471no—
316#1949—471no—
317#1949—471no—
318#1949—471no—
319#1949—471no—
320#1949—471no—
321#1949D4711yescharacteristic, complemented
322#1736—471no—
323#1736—471no—
324C2×C47C2×#38911yescharacteristic
325C2×C47#291111yescharacteristic
326C2×C47#291111yescharacteristic
327C2×C47#304011yescharacteristic
328C2×C47#304011yescharacteristic
329C2×C47#308011yescharacteristic
330C2×C47#308011yescharacteristic
331D47—24no—
332C22×M16—471no—
333C22×M16—471no—
334C2×#117—471no—
335C2×#117—471no—
336C22×C4⋊C4—471no—
337C22×C4⋊C4—471no—
338C22×C42—471no—
339#428—944no—
340#428—944no—
341C2×#107—471no—
342C2×#107—471no—
343C2×#107—471no—
344C2×#107C2×D4711yescharacteristic
345C2×#106—471no—
346C2×#106—471no—
347C2×#106—471no—
348C2×#106C2×D4711yescharacteristic
349#423—478no—
350C2×C4⋊C8—471no—
351C2×C4⋊C8—471no—
352C2×C4⋊C8—471no—
353C2×C4⋊C8C2×D4711yescharacteristic
354#382—471no—
355#382—471no—
356#382—471no—
357#382—941no—
358#382—941no—
359#382C2×D4711yescharacteristic
360C4×C4⋊C4—471no—
361C4×C4⋊C4—471no—
362C4×C4⋊C4—471no—
363C4×C4⋊C4C2×D4711yescharacteristic
364C4×C22⋊C4—941no—
365C4×C22⋊C4—941no—
366#376—941no—
367C2×#95—941no—
368C2×#95—941no—
369C43—941no—
370#340—471no—
371#340—471no—
372#340—471no—
373#340—471no—
374#340—471no—
375#340—471no—
376#340C2×D4711yescharacteristic
377#340C2×D4711yescharacteristic
378C47#2117211yescharacteristic, complemented, Sylow
379C23×C4—471no—
380C23×C4—471no—
381C23×C4—471no—
382C23×C4—941no—
383C23×C4—941no—
384C2×M16—474no—
385C2×M16—474no—
386C2×M16—942no—
387C2×M16—942no—
388C2×M16—942no—
389C2×M16—942no—
390C22×C8—471no—
391C22×C8—471no—
392C22×C8C22×D4711yescharacteristic
393C22×C8C22×D4711yescharacteristic
394#117—474no—
395#117—474no—
396#117—942no—
397#117—942no—
398C2×C4⋊C4—471no—
399C2×C4⋊C4—471no—
400C2×C4⋊C4—471no—
401C2×C4⋊C4—471no—
402C2×C4⋊C4—942no—
403C2×C4⋊C4—942no—
404C2×C4⋊C4—942no—
405C2×C4⋊C4—942no—
406C2×C4⋊C4—941no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4C22×D4711yescharacteristic
409C2×C4⋊C4C22×D4711yescharacteristic
410C2×C4⋊C4C22×D4711yescharacteristic
411C2×C4⋊C4C22×D4711yescharacteristic
412C2×C22⋊C4—941no—
413C2×C22⋊C4—941no—
414C2×C42—21no—
415C2×C42—471no—
416C2×C42—471no—
417C2×C42—471no—
418C2×C42—474no—
419C2×C42—474no—
420C2×C42—941no—
421C2×C42—941no—
422C2×C42—941no—
423C2×C42—941no—
424C2×C42—941no—
425C2×C42C22×D4711yescharacteristic
426#107—22no—
427#107—942no—
428#107—942no—
429#107—942no—
430#106—22no—
431#106—942no—
432#106—942no—
433#106—942no—
434C4⋊C8—474no—
435C4⋊C8—474no—
436C4⋊C8—474no—
437C4⋊C8C4×D4714yescomplemented
438#95—21no—
439#95—21no—
440#95—21no—
441#95—941no—
442#95—941no—
443#95—941no—
444#95—941no—
445#95—941no—
446#95—941no—
447#95—941no—
448#95—941no—
449#95—941no—
450C24—471no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—471no—
456C22×C4—472no—
457C22×C4—472no—
458C22×C4—472no—
459C22×C4—472no—
460C22×C4—472no—
461C22×C4—472no—
462C22×C4—471no—
463C22×C4—471no—
464C22×C4—474no—
465C22×C4—474no—
466C22×C4—941no—
467C22×C4—941no—
468C22×C4—941no—
469C22×C4—941no—
470C22×C4—1881no—
471C22×C4—1881no—
472C22×C4—1881no—
473C22×C4—1881no—
474C22×C4—1882no—
475C22×C4—1882no—
476C22×C4—1882no—
477C22×C4—1882no—
478C22×C4—1882no—
479C22×C4—1882no—
480C22×C4D4×D4711yescharacteristic
481C22×C4D4×D4711yescharacteristic
482C22×C4C23×D4711yescharacteristic
483M16—948no—
484M16—948no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—472no—
490C2×C8—472no—
491C2×C8—941no—
492C2×C8—941no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8C2×C4×D4712yes—
496C2×C8C2×C4×D4712yes—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—472no—
504C4⋊C4—472no—
505C4⋊C4—941no—
506C4⋊C4—941no—
507C4⋊C4—941no—
508C4⋊C4—941no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—1882no—
512C4⋊C4—1882no—
513C4⋊C4D4×D4712yes—
514C4⋊C4Q8×D4712yes—
515C22⋊C4—944no—
516C22⋊C4—944no—
517C42—42no—
518C42—474no—
519C42—474no—
520C42—472no—
521C42—472no—
522C42—944no—
523C42—944no—
524C42—1882no—
525C42—1882no—
526C42—1882no—
527C42C2×C4×D4712yes—
528C42Q8×D4712yes—
529C23—472no—
530C23—472no—
531C23—472no—
532C23—941no—
533C23—941no—
534C23—941no—
535C23—941no—
536C23C2×D4×D4711yescharacteristic
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—472no—
554C2×C4—472no—
555C2×C4—472no—
556C2×C4—474no—
557C2×C4—474no—
558C2×C4—474no—
559C2×C4—474no—
560C2×C4—472no—
561C2×C4—472no—
562C2×C4—941no—
563C2×C4—941no—
564C2×C4—941no—
565C2×C4—941no—
566C2×C4—942no—
567C2×C4—942no—
568C2×C4—3764no—
569C2×C4—3764no—
570C2×C4—1884no—
571C2×C4—1884no—
572C2×C4—1881no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4—1881no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4C4⋊C4×D4712yes—
583C2×C4C4⋊C4×D4712yes—
584C2×C4C22×C4×D4711yescharacteristic
585C2×C4C2×D4×D4711yescharacteristic
586C2×C4C2×Q8×D4711yescharacteristic
587C2×C4C2×Q8×D4711yescharacteristic
588C2×C4#41×D4711yescharacteristic
589C2×C4#41×D4711yescharacteristic
590C8—22no—
591C8—22no—
592C8—942no—
593C8—942no—
594C22—474no—
595C22—942no—
596C22—942no—
597C22—942no—
598C22—942no—
599C22—942no—
600C22—942no—
601C22C2×C4⋊C4×D4711yescharacteristic
602C22#121×D4711yescharacteristic
603C22C22⋊Q8×D4711yescharacteristic
604C22#1568991911yescharacteristic
605C22#1568992011yescharacteristic
606C22#1568992011yescharacteristic
607C22#1568992211yescharacteristic
608C4—22no—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—81no—
613C4—81no—
614C4—474no—
615C4—942no—
616C4—942no—
617C4—3761no—
618C4—3761no—
619C4—1882no—
620C4—1882no—
621C4C4×D4×D4711yescharacteristic
622C4C4×Q8×D4711yescharacteristic
623C4#125×D4711yescharacteristic
624C4C4⋊Q8×D4711yescharacteristic
625C2—944no—
626C2#389×D4711yescharacteristic
627C2#2025197511yescharacteristic
628C2#2025197611yescharacteristic
629C2#2025287711yescharacteristic
630C2#2025287911yescharacteristic
631C2#2025320511yescharacteristic
632C2#2025320711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1949C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1949D47
hypocenterC47#21172
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
6. #20244159C2semidirect
7. #20244158C2semidirect
8. #20244157C2semidirect
9. #20244156C2semidirect
10. #20244155C2semidirect
11. #20244154C2semidirect
12. #20244153C2semidirect
15. C47×#1949C2semidirect
31. #15689145C4semidirect
107. C4⋊(C47⋊C8)C2×C4semidirect
108. Dic47⋊C8C2×C4semidirect
109. Dic47⋊C8C2×C4semidirect
129. C4⋊C8×C47C2×C4semidirect
321. #1949D47semidirect
378. C47#21172semidirect
437. C4⋊C8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891642
C2×#156891622
C2×#156891452
#382×D472
C4×#130939822
#202441592
#202441582
#202441572
#202441562
#202441552
#202441542
#202441532
#202431062
C47×#19492
#2117247