Finite Group Databasebeta

← #37576182

Subgroups — #37576182

Order 24448.

780 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259302C211yesmaximal, characteristic, complemented
2C2×#20259300C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D191C211yesmaximal, characteristic
4C2×C22⋊C4×D191C211yesmaximal, characteristic
5#27132599C211yesmaximal, characteristic, complemented
6#27132586C211yesmaximal, characteristic, complemented
7#27132575C211yesmaximal, characteristic, complemented
8#27132537C211yesmaximal, characteristic, complemented
9#27132534C211yesmaximal, characteristic, complemented
10#27132534C211yesmaximal, characteristic, complemented
11#27132534C211yesmaximal, characteristic, complemented
12#27132522C211yesmaximal, characteristic, complemented
13#27132520C211yesmaximal, characteristic, complemented
14#27132518C211yesmaximal, characteristic
15#396×C191C211yesmaximal, characteristic, complemented
16C24×D191C2211yescharacteristic
17C22×Dic382C2211yescharacteristic, complemented
18C22×C4×D191C2211yescharacteristic
19C22×C4×D191C2211yescharacteristic
20C22×C4×D191C2211yescharacteristic
21C22×C4×D191C2211yescharacteristic
22#20259302—24no—
23#20259300—24no—
24C2×C4⋊Dic191C2211yescharacteristic, complemented
25C2×C4⋊Dic191C2211yescharacteristic, complemented
26C2×Dic191⋊C4C2211yescharacteristic, complemented
27C2×Dic191⋊C4C2211yescharacteristic, complemented
28C2×Dic191⋊C4C2211yescharacteristic, complemented
29C2×Dic191⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic191C2211yescharacteristic
31C2×C22⋊Dic191C2211yescharacteristic
32C2×C22⋊Dic191C2211yescharacteristic
33C2×(C2×D191)⋊C4C2211yescharacteristic
34C2×(C2×D191)⋊C4C2211yescharacteristic
35C2×(C2×D191)⋊C4C2211yescharacteristic
36C2×(C2×D191)⋊C4C2211yescharacteristic
37C2×(C2×D191)⋊C4C2211yescharacteristic
38C2×(C2×D191)⋊C4C2211yescharacteristic
39C2×(C2×D191)⋊C4C2211yescharacteristic
40C2×(C2×D191)⋊C4C2211yescharacteristic
41C22⋊C4×D191—24no—
42C22⋊C4×D191—24no—
43C191⋊#95C2211yescharacteristic, complemented
44C191⋊#95C2211yescharacteristic, complemented
45C191⋊#95C2211yescharacteristic, complemented
46C191⋊#95C2211yescharacteristic, complemented
47C191⋊#95C2211yescharacteristic, complemented
48C2×C4⋊C4×C191C2211yescharacteristic, complemented
49C2×C4⋊C4×C191C2211yescharacteristic, complemented
50C2×C4⋊C4×C191C2211yescharacteristic, complemented
51C2×C22⋊C4×C191C2211yescharacteristic
52C2×C22⋊C4×C191C2211yescharacteristic
53C2×C22⋊C4×C191C2211yescharacteristic
54#95×C191C2211yescharacteristic, complemented
55C23×D191—22no—
56C23×D191—22no—
57C23×D191—22no—
58C23×D191—22no—
59C23×D191—41no—
60C23×D191—41no—
61C23×D191Q814yescomplemented
62C23×D191C2311yescharacteristic
63C23×D191C2311yescharacteristic
64C2×Dic382—22no—
65C2×Dic382—22no—
66C2×Dic382—41no—
67C22×Dic191C2311yescharacteristic
68C22×Dic191C2311yescharacteristic
69C22×Dic191C2311yescharacteristic
70C22×Dic191C2311yescharacteristic
71C22×Dic191C2311yescharacteristic
72C22×Dic191C2311yescharacteristic
73C2×C4×D191—22no—
74C2×C4×D191—22no—
75C2×C4×D191—22no—
76C2×C4×D191—22no—
77C2×C4×D191—22no—
78C2×C4×D191—22no—
79C2×C4×D191—22no—
80C2×C4×D191—22no—
81C2×C4×D191—41no—
82C2×C4×D191—41no—
83C2×C4×D191—41no—
84C2×C4×D191—41no—
85C4⋊Dic191—22no—
86C4⋊Dic191—22no—
87Dic191⋊C4—22no—
88Dic191⋊C4—22no—
89Dic191⋊C4—41no—
90Dic191⋊C4—41no—
91C22⋊Dic191—22no—
92C22⋊Dic191—22no—
93C22⋊Dic191—41no—
94(C2×D191)⋊C4—22no—
95(C2×D191)⋊C4—22no—
96(C2×D191)⋊C4—22no—
97(C2×D191)⋊C4—22no—
98(C2×D191)⋊C4—22no—
99(C2×D191)⋊C4—22no—
100(C2×D191)⋊C4D414yes—
101(C2×D191)⋊C4D414yes—
102C24×C191C2311yescharacteristic
103C22×C4×C191C2311yescharacteristic
104C22×C4×C191C2311yescharacteristic
105C22×C4×C191C2311yescharacteristic
106C22×C4×C191C2311yescharacteristic
107C22×C4×C191C2311yescharacteristic
108C22×C4×C191C2311yescharacteristic
109C4⋊C4×C191—22no—
110C4⋊C4×C191—22no—
111C4⋊C4×C191—41no—
112C22⋊C4×C191—22no—
113C22⋊C4×C191—22no—
114C22⋊C4×C191—41no—
115C22×D191—21no—
116C22×D191—21no—
117C22×D191—21no—
118C22×D191—21no—
119C22×D191—21no—
120C22×D191—21no—
121C22×D191—21no—
122C22×D191—21no—
123C22×D191—28no—
124C22×D191—28no—
125C22×D191—44no—
126C22×D191—44no—
127C22×D191—44no—
128C22×D191—44no—
129C22×D191—44no—
130C22×D191C2×D412yes—
131C22×D191C2×D412yes—
132C22×D191C2×Q812yes—
133C22×D191C2×Q812yes—
134C22×D191#4112yes—
135C22×D191#4112yes—
136Dic382—44no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C2×Dic191—21no—
140C2×Dic191—21no—
141C2×Dic191—21no—
142C2×Dic191—21no—
143C2×Dic191—21no—
144C2×Dic191—21no—
145C2×Dic191—21no—
146C2×Dic191—21no—
147C2×Dic191—21no—
148C2×Dic191—21no—
149C2×Dic191—21no—
150C2×Dic191—21no—
151C2×Dic191—21no—
152C2×Dic191—21no—
153C2×Dic191C2×D412yes—
154C2×Dic191C2×D412yes—
155C4×D191—44no—
156C4×D191—44no—
157C4×D191—44no—
158C4×D191—44no—
159C23×C191—21no—
160C23×C191—21no—
161C23×C191—21no—
162C23×C191—21no—
163C23×C191—21no—
164C23×C191—21no—
165C23×C191C2×Q812yes—
166C23×C191C2411yescharacteristic
167C2×C4×C191—21no—
168C2×C4×C191—21no—
169C2×C4×C191—21no—
170C2×C4×C191—21no—
171C2×C4×C191—21no—
172C2×C4×C191—21no—
173C2×C4×C191—21no—
174C2×C4×C191—21no—
175C2×C4×C191—21no—
176C2×C4×C191—21no—
177C2×C4×C191—21no—
178C2×C4×C191—21no—
179C2×C4×C191—21no—
180C2×C4×C191—21no—
181C2×C4×C191—21no—
182C2×C4×C191—21no—
183C2×C4×C191C2×D412yes—
184C2×C4×C191C2×D412yes—
185C22×C191—22no—
186C22×C191—22no—
187C22×C191—22no—
188C22×C191—41no—
189C22×C191—41no—
190C22×C191—41no—
191C22×C191—41no—
192C22×C191C22×D411yescharacteristic
193C22×C191C22×D411yescharacteristic
194C22×C191C22×Q811yescharacteristic
195C22×C191C2×#4111yescharacteristic
196C22×C191C2×#4111yescharacteristic
197C22×C191H(5,𝔽2)11yescharacteristic
198C22×C191H(5,𝔽2)11yescharacteristic
199C2×D191—22no—
200C2×D191—22no—
201C2×D191—22no—
202C2×D191—22no—
203C2×D191—22no—
204C2×D191—22no—
205C2×D191—22no—
206C2×D191—22no—
207C2×D191—22no—
208C2×D191—22no—
209C2×D191—22no—
210C2×D191—22no—
211C2×D191—416no—
212C2×D191C22⋊Q814yescomplemented
213C2×D191C22⋊Q814yescomplemented
214C4×C191—22no—
215C4×C191—22no—
216C4×C191—41no—
217C4×C191—41no—
218C4×C191—41no—
219C4×C191—41no—
220Dic191—22no—
221Dic191—22no—
222Dic191—41no—
223Dic191—41no—
224Dic191—41no—
225Dic191—41no—
226C2×C191—42no—
227C2×C191C2×C22⋊Q811yescharacteristic
228C2×C191C2×C22⋊Q811yescharacteristic
229C2×C191#54311yescharacteristic
230C2×C191D4211yescharacteristic
231C2×C191#54611yescharacteristic
232C2×C191#54611yescharacteristic
233C2×C191#55111yescharacteristic
234D191—24no—
235D191—24no—
236C191#243711yescharacteristic, complemented, Sylow
237#2437—1911nomaximal, Sylow
238C2×C22⋊Q8—1911no—
239C2×C22⋊Q8—1911no—
240C22×C22⋊C4—1911no—
241C22×C22⋊C4—1911no—
242#399—1911no—
243#396—1911no—
244#396D19111yescharacteristic, complemented
245#393—1911no—
246#385—1911no—
247#385—1911no—
248#385—1911no—
249#385—1911no—
250#380—1911no—
251#380—1911no—
252#379—1911no—
253C25—1911no—
254C22×Q8—1911no—
255C23×C4—1911no—
256C23×C4—1911no—
257C23×C4—1911no—
258C23×C4—1911no—
259C22⋊Q8—3824no—
260C22⋊Q8—3824no—
261C2×C4⋊C4—1911no—
262C2×C4⋊C4—1911no—
263C2×C4⋊C4—1911no—
264C2×C4⋊C4—1911no—
265C2×C4⋊C4—1911no—
266C2×C4⋊C4—1911no—
267C2×C4⋊C4C2×D19111yescharacteristic, complemented
268C2×C4⋊C4C2×D19111yescharacteristic, complemented
269C2×C4⋊C4C2×D19111yescharacteristic, complemented
270C2×C22⋊C4—3824no—
271C2×C22⋊C4—3824no—
272C2×C22⋊C4—1911no—
273C2×C22⋊C4—1911no—
274C2×C22⋊C4—1911no—
275C2×C22⋊C4—1911no—
276C2×C22⋊C4—1911no—
277C2×C22⋊C4—1911no—
278C2×C22⋊C4—1911no—
279C2×C22⋊C4—1911no—
280C2×C22⋊C4—1911no—
281C2×C22⋊C4—1911no—
282C2×C22⋊C4—1911no—
283C2×C22⋊C4C2×D19111yescharacteristic
284C2×C22⋊C4C2×D19111yescharacteristic
285C2×C22⋊C4C2×D19111yescharacteristic
286#95—1911no—
287#95—1911no—
288#95—1911no—
289#95—1911no—
290#95—1911no—
291#95C2×D19111yescharacteristic, complemented
292C24—3822no—
293C24—3822no—
294C24—3822no—
295C24—3822no—
296C24—1911no—
297C24—1911no—
298C24—1914no—
299C24—7641no—
300C24—7641no—
301C24C22×D19111yescharacteristic
302C2×Q8—3822no—
303C2×Q8—3822no—
304C2×Q8—7641no—
305C22×C4—3822no—
306C22×C4—3822no—
307C22×C4—3822no—
308C22×C4—3822no—
309C22×C4—3822no—
310C22×C4—3822no—
311C22×C4—3822no—
312C22×C4—3822no—
313C22×C4—1911no—
314C22×C4—1911no—
315C22×C4—1911no—
316C22×C4—1911no—
317C22×C4—1911no—
318C22×C4—1911no—
319C22×C4—7641no—
320C22×C4—7641no—
321C22×C4—7641no—
322C22×C4—7641no—
323C22×C4C22×D19111yescharacteristic
324C22×C4C22×D19111yescharacteristic
325C22×C4C22×D19111yescharacteristic
326C22×C4C22×D19111yescharacteristic
327C22×C4C22×D19111yescharacteristic
328C22×C4C22×D19111yescharacteristic
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—41no—
332C4⋊C4—3822no—
333C4⋊C4—3822no—
334C4⋊C4—3822no—
335C4⋊C4—3822no—
336C4⋊C4—7641no—
337C4⋊C4—7641no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—41no—
341C22⋊C4—3822no—
342C22⋊C4—3822no—
343C22⋊C4—3822no—
344C22⋊C4—3822no—
345C22⋊C4—3822no—
346C22⋊C4—3822no—
347C22⋊C4—3822no—
348C22⋊C4—3822no—
349C22⋊C4—1914no—
350C22⋊C4—1914no—
351C22⋊C4—7641no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—3828no—
359C23—3828no—
360C23—3821no—
361C23—3821no—
362C23—3821no—
363C23—3821no—
364C23—3821no—
365C23—3821no—
366C23—3821no—
367C23—3821no—
368C23—1912no—
369C23—1912no—
370C23—1912no—
371C23—1912no—
372C23—1912no—
373C23—1912no—
374C23—7644no—
375C23—7644no—
376C23—7644no—
377C23—7644no—
378C23—7644no—
379C23Q8×D19112yes—
380C23C23×D19111yescharacteristic
381Q8—7644no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—3821no—
399C2×C4—3821no—
400C2×C4—3821no—
401C2×C4—3821no—
402C2×C4—3821no—
403C2×C4—3821no—
404C2×C4—3821no—
405C2×C4—3821no—
406C2×C4—3821no—
407C2×C4—3821no—
408C2×C4—3821no—
409C2×C4—3821no—
410C2×C4—3821no—
411C2×C4—3821no—
412C2×C4—3821no—
413C2×C4—3821no—
414C2×C4—1912no—
415C2×C4—1912no—
416C2×C4—7644no—
417C2×C4—7644no—
418C2×C4—7644no—
419C2×C4—7644no—
420C2×C4D4×D19112yes—
421C2×C4D4×D19112yes—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—3822no—
430C22—3822no—
431C22—3822no—
432C22—3822no—
433C22—3822no—
434C22—3822no—
435C22—3822no—
436C22—3822no—
437C22—3822no—
438C22—3822no—
439C22—3822no—
440C22—3822no—
441C22—1914no—
442C22—1914no—
443C22—76416no—
444C22C2×D4×D19111yescharacteristic
445C22C2×D4×D19111yescharacteristic
446C22C2×Q8×D19111yescharacteristic
447C22#41×D19111yescharacteristic
448C22#41×D19111yescharacteristic
449C22#2025936711yescharacteristic
450C22#2025936811yescharacteristic
451C4—22no—
452C4—22no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—3822no—
458C4—3822no—
459C4—7641no—
460C4—7641no—
461C4—7641no—
462C4—7641no—
463C2—42no—
464C2—3824no—
465C2—3824no—
466C2C22⋊Q8×D19111yescharacteristic
467C2C22⋊Q8×D19111yescharacteristic
468C2#2713328811yescharacteristic
469C2#2713329711yescharacteristic
470C2#2713330211yescharacteristic
471C2#2713331011yescharacteristic
472C2#2713333811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#396×C191C2
prime radicalC24×D191C22
hypercenter#396D191
hypocenterC191#2437
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#20259302C2semidirect
2. C2×#20259300C2semidirect
5. #27132599C2semidirect
6. #27132586C2semidirect
7. #27132575C2semidirect
8. #27132537C2semidirect
9. #27132534C2semidirect
10. #27132534C2semidirect
11. #27132534C2semidirect
12. #27132522C2semidirect
13. #27132520C2semidirect
15. #396×C191C2semidirect
17. C22×Dic382C22semidirect
24. C2×C4⋊Dic191C22semidirect
25. C2×C4⋊Dic191C22semidirect
26. C2×Dic191⋊C4C22semidirect
27. C2×Dic191⋊C4C22semidirect
28. C2×Dic191⋊C4C22semidirect
29. C2×Dic191⋊C4C22semidirect
43. C191⋊#95C22semidirect
44. C191⋊#95C22semidirect
45. C191⋊#95C22semidirect
46. C191⋊#95C22semidirect
47. C191⋊#95C22semidirect
48. C2×C4⋊C4×C191C22semidirect
49. C2×C4⋊C4×C191C22semidirect
50. C2×C4⋊C4×C191C22semidirect
54. #95×C191C22semidirect
61. C23×D191Q8semidirect
212. C2×D191C22⋊Q8semidirect
213. C2×D191C22⋊Q8semidirect
236. C191#2437semidirect
244. #396D191semidirect
267. C2×C4⋊C4C2×D191semidirect
268. C2×C4⋊C4C2×D191semidirect
269. C2×C4⋊C4C2×D191semidirect
291. #95C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593022
C2×#202593002
C2×C22⋊C4×D1912
#271325992
#271325862
#271325752
#271325372
#271325342
#271325222
#271325202
#271325182
#396×C1912
#2437191