Finite Group Databasebeta

← #37575837

Subgroups — #37575837

Order 24448.

1056 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15690287C211yesmaximal, characteristic, complemented
2C2×#20259292C211yesmaximal, characteristic, complemented
3C2×#20259290C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D191C211yesmaximal, characteristic, complemented
5#27132576C211yesmaximal, characteristic, complemented
6#27132575C211yesmaximal, characteristic, complemented
7#27132572C211yesmaximal, characteristic, complemented
8#27132571C211yesmaximal, characteristic, complemented
9#27132542C211yesmaximal, characteristic, complemented
10#27132541C211yesmaximal, characteristic, complemented
11#27132541C211yesmaximal, characteristic, complemented
12#27132539C211yesmaximal, characteristic, complemented
13#27132539C211yesmaximal, characteristic, complemented
14#27132525C211yesmaximal, characteristic, complemented
15#386×C191C211yesmaximal, characteristic, complemented
16C24×D191C2211yescharacteristic
17C2×#15690287—24no—
18C2×#15690287—24no—
19C2×#15690287—24no—
20C22×Dic382C2211yescharacteristic, complemented
21C22×D764C2211yescharacteristic
22C22×D764C2211yescharacteristic
23C22×C22⋊D191C2211yescharacteristic
24C22×C22⋊D191C2211yescharacteristic
25C22×C22⋊D191C2211yescharacteristic
26C22×C22⋊D191C2211yescharacteristic, complemented
27C22×C4×D191C2211yescharacteristic
28C22×C4×D191C2211yescharacteristic
29C22×C4×D191C2211yescharacteristic, complemented
30#20259292—24no—
31#20259290—24no—
32C2×C22⋊Dic191C2211yescharacteristic
33C2×C22⋊Dic191C2211yescharacteristic
34C2×C22⋊Dic191C2211yescharacteristic, complemented
35C2×(C2×D191)⋊C4C2211yescharacteristic
36C2×(C2×D191)⋊C4C2211yescharacteristic
37C2×(C2×D191)⋊C4C2211yescharacteristic
38C2×(C2×D191)⋊C4C2211yescharacteristic
39C2×(C2×D191)⋊C4C2211yescharacteristic
40C2×(C2×D191)⋊C4C2211yescharacteristic
41C2×(C2×D191)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D191)⋊C4C2211yescharacteristic, complemented
43C22⋊C4×D191—24no—
44C191⋊#95C2211yescharacteristic, complemented
45C191⋊#95C2211yescharacteristic, complemented
46C191⋊#95C2211yescharacteristic, complemented
47C191⋊#95C2211yescharacteristic, complemented
48C191⋊#95C2211yescharacteristic, complemented
49C191⋊#95C2211yescharacteristic, complemented
50C22×D4×C191C2211yescharacteristic
51C23×C4×C191C2211yescharacteristic, complemented
52C2×C22⋊C4×C191C2211yescharacteristic
53C2×C22⋊C4×C191C2211yescharacteristic
54C2×C22⋊C4×C191C2211yescharacteristic, complemented
55#95×C191C2211yescharacteristic, complemented
56#95×C191C2211yescharacteristic, complemented
57C23×D191—22no—
58C23×D191—22no—
59C23×D191—22no—
60C23×D191—22no—
61C23×D191—22no—
62C23×D191—41no—
63C23×D191—41no—
64C23×D191C2311yescharacteristic
65C23×D191C2311yescharacteristic
66C23×D191C2311yescharacteristic
67#15690287—232no—
68C2×Dic382—22no—
69C2×Dic382—22no—
70C2×Dic382—22no—
71C2×D764—22no—
72C2×D764—22no—
73C2×D764—22no—
74C2×D764—22no—
75C2×D764—41no—
76C2×D764D414yescomplemented
77C2×C22⋊D191—22no—
78C2×C22⋊D191—22no—
79C2×C22⋊D191—22no—
80C2×C22⋊D191—22no—
81C2×C22⋊D191—22no—
82C2×C22⋊D191—22no—
83C2×C22⋊D191—22no—
84C2×C22⋊D191—22no—
85C2×C22⋊D191—22no—
86C2×C22⋊D191—41no—
87C2×C22⋊D191—41no—
88C2×C22⋊D191D414yescomplemented
89C22×Dic191C2311yescharacteristic
90C22×Dic191C2311yescharacteristic
91C22×Dic191C2311yescharacteristic
92C22×Dic191C2311yescharacteristic
93C22×Dic191C2311yescharacteristic
94C2×C4×D191—22no—
95C2×C4×D191—22no—
96C2×C4×D191—22no—
97C2×C4×D191—22no—
98C2×C4×D191—22no—
99C2×C4×D191—22no—
100C2×C4×D191—41no—
101C2×C4×D191—41no—
102C2×C4×D191D414yescomplemented
103C22⋊Dic191—22no—
104C22⋊Dic191—22no—
105C22⋊Dic191—41no—
106(C2×D191)⋊C4—22no—
107(C2×D191)⋊C4—22no—
108(C2×D191)⋊C4—22no—
109(C2×D191)⋊C4—22no—
110(C2×D191)⋊C4—22no—
111(C2×D191)⋊C4—22no—
112(C2×D191)⋊C4—41no—
113(C2×D191)⋊C4—41no—
114C24×C191C2311yescharacteristic
115C24×C191C2311yescharacteristic
116C2×D4×C191—22no—
117C2×D4×C191—22no—
118C2×D4×C191—41no—
119C22×C4×C191—22no—
120C22×C4×C191—22no—
121C22×C4×C191—22no—
122C22×C4×C191C2311yescharacteristic
123C22×C4×C191C2311yescharacteristic
124C22×C4×C191C2311yescharacteristic
125C22×C4×C191C2311yescharacteristic
126C22×C4×C191C2311yescharacteristic
127C22⋊C4×C191—22no—
128C22⋊C4×C191—22no—
129C22⋊C4×C191—41no—
130C22×D191—21no—
131C22×D191—21no—
132C22×D191—21no—
133C22×D191—21no—
134C22×D191—28no—
135C22×D191—21no—
136C22×D191—21no—
137C22×D191—21no—
138C22×D191—21no—
139C22×D191—21no—
140C22×D191—21no—
141C22×D191—21no—
142C22×D191—21no—
143C22×D191—21no—
144C22×D191—21no—
145C22×D191—21no—
146C22×D191—21no—
147C22×D191—44no—
148C22×D191—44no—
149C22×D191—44no—
150C22×D191—44no—
151C22×D191—44no—
152C22×D191—44no—
153C22×D191C2×D412yes—
154C22×D191C2×D412yes—
155C22×D191C2×D412yes—
156C22×D191C2×D412yes—
157C22×D191#4112yescomplemented
158Dic382—28no—
159D764—28no—
160D764—44no—
161C22⋊D191—28no—
162C22⋊D191—28no—
163C22⋊D191—44no—
164C22⋊D191—44no—
165C2×Dic191—21no—
166C2×Dic191—21no—
167C2×Dic191—21no—
168C2×Dic191—21no—
169C2×Dic191—21no—
170C2×Dic191—21no—
171C2×Dic191—21no—
172C2×Dic191—21no—
173C2×Dic191—21no—
174C2×Dic191—21no—
175C2×Dic191—21no—
176C2×Dic191—21no—
177C2×Dic191—21no—
178C2×Dic191C2×D412yes—
179C2×Dic191C2×D412yes—
180C4×D191—28no—
181C4×D191—28no—
182C4×D191—44no—
183C23×C191—21no—
184C23×C191—21no—
185C23×C191—21no—
186C23×C191—21no—
187C23×C191—21no—
188C23×C191—21no—
189C23×C191—21no—
190C23×C191—21no—
191C23×C191—21no—
192C23×C191—21no—
193C23×C191—21no—
194C23×C191—21no—
195C23×C191C2×D412yes—
196C23×C191#4112yescomplemented
197C23×C191C2411yescharacteristic
198D4×C191—44no—
199C2×C4×C191—28no—
200C2×C4×C191—21no—
201C2×C4×C191—21no—
202C2×C4×C191—21no—
203C2×C4×C191—21no—
204C2×C4×C191—21no—
205C2×C4×C191—21no—
206C2×C4×C191—21no—
207C2×C4×C191—21no—
208C2×C4×C191—21no—
209C2×C4×C191—21no—
210C2×C4×C191—21no—
211C2×C4×C191—21no—
212C2×C4×C191—21no—
213C2×C4×C191C2×D412yes—
214C2×C4×C191C2×D412yes—
215C22×C191—22no—
216C22×C191—22no—
217C22×C191—22no—
218C22×C191—22no—
219C22×C191—22no—
220C22×C191—22no—
221C22×C191—41no—
222C22×C191—41no—
223C22×C191—41no—
224C22×C191—41no—
225C22×C191—41no—
226C22×C191—41no—
227C22×C191—41no—
228C22×C191—41no—
229C22×C191C22×D411yescharacteristic
230C22×C191C22×D411yescharacteristic
231C22×C191C22×D411yescharacteristic
232C22×C191C2×#4111yescharacteristic
233C22×C191C2×#4111yescharacteristic
234C22×C191H(5,𝔽2)11yescharacteristic
235C22×C191H(5,𝔽2)11yescharacteristic
236C2×D191—22no—
237C2×D191—22no—
238C2×D191—22no—
239C2×D191—22no—
240C2×D191—22no—
241C2×D191—22no—
242C2×D191—22no—
243C2×D191—22no—
244C2×D191—22no—
245C2×D191—22no—
246C2×D191—22no—
247C2×D191—22no—
248C2×D191—416no—
249C2×D191—41no—
250C2×D191—41no—
251C2×D191—41no—
252C2×D191—41no—
253C2×D191—41no—
254C2×D191—41no—
255C2×D191—41no—
256C2×D191—41no—
257C2×D191#12014yes—
258C4×C191—22no—
259C4×C191—22no—
260C4×C191—41no—
261C4×C191—41no—
262C4×C191—41no—
263Dic191—22no—
264Dic191—22no—
265Dic191—41no—
266Dic191—41no—
267Dic191—41no—
268C2×C191—42no—
269C2×C191—42no—
270C2×C191C2×#12011yescharacteristic
271C2×C191#52511yescharacteristic
272C2×C191#53511yescharacteristic
273C2×C191#54611yescharacteristic
274C2×C191#54611yescharacteristic
275C2×C191#54611yescharacteristic
276C2×C191#54611yescharacteristic
277D191—24no—
278D191—42no—
279D191—42no—
280C191#240411yescharacteristic, complemented, Sylow
281#2404—1911nomaximal, Sylow
282C22×#41—1911no—
283C2×#120—1911no—
284C2×#120—1911no—
285C22×C22⋊C4—1911no—
286#393—1911no—
287#393—1911no—
288#392—1911no—
289#392—1911no—
290#386—1911no—
291#386—1911no—
292#386—1911no—
293#386—1911no—
294#386—1911no—
295#386D19111yescharacteristic, complemented
296#381—1911no—
297C25—1911no—
298C2×#41—3824no—
299C2×#41—3824no—
300C2×#41—3824no—
301C22×Q8—1911no—
302C22×D4—1911no—
303C22×D4—1911no—
304C22×D4—1911no—
305C22×D4—1911no—
306C22×D4—1911no—
307C22×D4—1911no—
308C22×D4C2×D19111yescharacteristic
309C23×C4—1911no—
310C23×C4—1911no—
311C23×C4—1911no—
312C23×C4C2×D19111yescharacteristic, complemented
313#120—3824no—
314#120—3824no—
315C2×C22⋊C4—3824no—
316C2×C22⋊C4—1911no—
317C2×C22⋊C4—1911no—
318C2×C22⋊C4—1911no—
319C2×C22⋊C4—1911no—
320C2×C22⋊C4—1911no—
321C2×C22⋊C4—1911no—
322C2×C22⋊C4—1911no—
323C2×C22⋊C4—1911no—
324C2×C22⋊C4—1911no—
325C2×C22⋊C4—1911no—
326C2×C22⋊C4—1911no—
327C2×C22⋊C4C2×D19111yescharacteristic
328C2×C22⋊C4C2×D19111yescharacteristic
329C2×C22⋊C4C2×D19111yescharacteristic, complemented
330#95—1911no—
331#95—1911no—
332#95—1911no—
333#95—1911no—
334#95—1911no—
335#95—1911no—
336#95C2×D19111yescharacteristic, complemented
337#95C2×D19111yescharacteristic, complemented
338C24—3822no—
339C24—3822no—
340C24—3822no—
341C24—3822no—
342C24—3822no—
343C24—1911no—
344C24—1911no—
345C24—1911no—
346C24—7641no—
347C24—7641no—
348C24C22×D19111yescharacteristic
349C24C22×D19111yescharacteristic
350#41—38232no—
351C2×Q8—3822no—
352C2×Q8—3822no—
353C2×Q8—3822no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—41no—
357C2×D4—3822no—
358C2×D4—3822no—
359C2×D4—3822no—
360C2×D4—3822no—
361C2×D4—3822no—
362C2×D4—3822no—
363C2×D4—3822no—
364C2×D4—3822no—
365C2×D4—3822no—
366C2×D4—3822no—
367C2×D4—3822no—
368C2×D4—3822no—
369C2×D4—3822no—
370C2×D4—1914no—
371C2×D4—1914no—
372C2×D4—7641no—
373C2×D4—7641no—
374C2×D4—7641no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—3822no—
379C22×C4—3822no—
380C22×C4—3822no—
381C22×C4—3822no—
382C22×C4—3822no—
383C22×C4—3822no—
384C22×C4—1911no—
385C22×C4—1911no—
386C22×C4—1911no—
387C22×C4—1911no—
388C22×C4—1911no—
389C22×C4—1914no—
390C22×C4—7641no—
391C22×C4—7641no—
392C22×C4C22×D19111yescharacteristic
393C22×C4C22×D19111yescharacteristic
394C22×C4C22×D19111yescharacteristic
395C22×C4C22×D19111yescharacteristic
396C22×C4C22×D19111yescharacteristic
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—41no—
400C22⋊C4—3822no—
401C22⋊C4—3822no—
402C22⋊C4—3822no—
403C22⋊C4—3822no—
404C22⋊C4—3822no—
405C22⋊C4—3822no—
406C22⋊C4—3822no—
407C22⋊C4—3822no—
408C22⋊C4—7641no—
409C22⋊C4—7641no—
410C22⋊C4—7641no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—3828no—
424C23—3821no—
425C23—3821no—
426C23—3821no—
427C23—3821no—
428C23—3821no—
429C23—3821no—
430C23—3821no—
431C23—3821no—
432C23—3821no—
433C23—3821no—
434C23—3821no—
435C23—3821no—
436C23—3821no—
437C23—3821no—
438C23—3821no—
439C23—3821no—
440C23—1912no—
441C23—1912no—
442C23—1912no—
443C23—1912no—
444C23—1912no—
445C23—7644no—
446C23—7644no—
447C23—7644no—
448C23—7644no—
449C23—7644no—
450C23—7644no—
451C23D4×D19112yes—
452C23#1569028712yescomplemented
453C23C23×D19111yescharacteristic
454Q8—3828no—
455D4—44no—
456D4—3828no—
457D4—3828no—
458D4—3828no—
459D4—7644no—
460D4—7644no—
461D4—7644no—
462C2×C4—28no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—3828no—
477C2×C4—3828no—
478C2×C4—3821no—
479C2×C4—3821no—
480C2×C4—3821no—
481C2×C4—3821no—
482C2×C4—3821no—
483C2×C4—3821no—
484C2×C4—3821no—
485C2×C4—3821no—
486C2×C4—3821no—
487C2×C4—3821no—
488C2×C4—3821no—
489C2×C4—3821no—
490C2×C4—3821no—
491C2×C4—1912no—
492C2×C4—1912no—
493C2×C4—7644no—
494C2×C4D4×D19112yes—
495C2×C4D4×D19112yes—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—3822no—
511C22—3822no—
512C22—3822no—
513C22—3822no—
514C22—3822no—
515C22—3822no—
516C22—3822no—
517C22—3822no—
518C22—3822no—
519C22—3822no—
520C22—3822no—
521C22—3822no—
522C22—1914no—
523C22—76416no—
524C22—7641no—
525C22—7641no—
526C22—7641no—
527C22—7641no—
528C22—7641no—
529C22—7641no—
530C22—7641no—
531C22—7641no—
532C22C2×D4×D19111yescharacteristic
533C22C2×D4×D19111yescharacteristic
534C22C2×D4×D19111yescharacteristic
535C22#41×D19111yescharacteristic
536C22C2×#1569028711yescharacteristic
537C22#2025936711yescharacteristic
538C22#2025936811yescharacteristic
539C4—22no—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—3822no—
545C4—3822no—
546C4—7641no—
547C4—7641no—
548C4—7641no—
549C2—42no—
550C2—42no—
551C2—3824no—
552C2—7642no—
553C2—7642no—
554C2#120×D19111yescharacteristic
555C2#2713317211yescharacteristic
556C2#2713322911yescharacteristic
557C2#2713330311yescharacteristic
558C2#2713330511yescharacteristic
559C2#2713330911yescharacteristic
560C2#2713331011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#386×C191C2
hypercenter#386D191
hypocenterC191#2404
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#15690287C2semidirect
2. C2×#20259292C2semidirect
3. C2×#20259290C2semidirect
4. C2×C22⋊C4×D191C2semidirect
5. #27132576C2semidirect
6. #27132575C2semidirect
7. #27132572C2semidirect
8. #27132571C2semidirect
9. #27132542C2semidirect
10. #27132541C2semidirect
11. #27132541C2semidirect
12. #27132539C2semidirect
13. #27132539C2semidirect
14. #27132525C2semidirect
15. #386×C191C2semidirect
20. C22×Dic382C22semidirect
26. C22×C22⋊D191C22semidirect
29. C22×C4×D191C22semidirect
34. C2×C22⋊Dic191C22semidirect
41. C2×(C2×D191)⋊C4C22semidirect
42. C2×(C2×D191)⋊C4C22semidirect
44. C191⋊#95C22semidirect
45. C191⋊#95C22semidirect
46. C191⋊#95C22semidirect
47. C191⋊#95C22semidirect
48. C191⋊#95C22semidirect
49. C191⋊#95C22semidirect
51. C23×C4×C191C22semidirect
54. C2×C22⋊C4×C191C22semidirect
55. #95×C191C22semidirect
56. #95×C191C22semidirect
76. C2×D764D4semidirect
88. C2×C22⋊D191D4semidirect
102. C2×C4×D191D4semidirect
157. C22×D191#41semidirect
196. C23×C191#41semidirect
280. C191#2404semidirect
295. #386D191semidirect
312. C23×C4C2×D191semidirect
329. C2×C22⋊C4C2×D191semidirect
336. #95C2×D191semidirect
337. #95C2×D191semidirect
452. C23#15690287semidirect

Maximal subgroups

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

subgroupindex
C22×#156902872
C2×#202592922
C2×#202592902
C2×C22⋊C4×D1912
#271325762
#271325752
#271325722
#271325712
#271325422
#271325412
#271325392
#271325252
#386×C1912
#2404191