Finite Group Databasebeta

← #37583302

Subgroups — #37583302

Order 24448.

420 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133176C211yesmaximal, characteristic
2C22⋊Q8×D191C211yesmaximal, characteristic, complemented
3#27133006C211yesmaximal, characteristic, complemented
4#27133002C211yesmaximal, characteristic, complemented
5#27132992C211yesmaximal, characteristic, complemented
6#27132978C211yesmaximal, characteristic, complemented
7#27132902C211yesmaximal, characteristic, complemented
8#27132894C211yesmaximal, characteristic, complemented
9#27132892C211yesmaximal, characteristic, complemented
10#27132828C211yesmaximal, characteristic, complemented
11#27132824C211yesmaximal, characteristic, complemented
12#27132817C211yesmaximal, characteristic, complemented
13#27132803C211yesmaximal, characteristic, complemented
14C2×(C2×D191)⋊C8C211yesmaximal, characteristic, complemented
15#467×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18C22×C4×D191C2211yescharacteristic
19C2×#15690277C2211yescharacteristic, complemented
20C2×#15690276C2211yescharacteristic, complemented
21C2×#15690274C2211yescharacteristic, complemented
22C22×C191⋊C8C2211yescharacteristic
23#20259307—21no—
24#20259302C2211yescharacteristic
25#20259301C2211yescharacteristic
26#20259300—21no—
27#20259299—21no—
28#20259298C2211yescharacteristic
29#20259298C2211yescharacteristic
30#20259297C2211yescharacteristic
31#20259296C2211yescharacteristic
32#20259292—21no—
33#20259284—21no—
34(C22⋊D191)⋊C4—21no—
35#20259280C2211yescharacteristic
36C4⋊C4×D191—21no—
37C4⋊C4×D191C2211yescharacteristic
38C22⋊C4×D191—21no—
39#20259255C2211yescharacteristic, complemented
40#20259254C2211yescharacteristic, complemented
41#20259253C2211yescharacteristic, complemented
42#20259247C2211yescharacteristic, complemented
43#20259246C2211yescharacteristic, complemented
44#20259246C2211yescharacteristic, complemented
45#20259245C2211yescharacteristic, complemented
46#20259244C2211yescharacteristic, complemented
47#20259242C2211yescharacteristic, complemented
48(C2×D191)⋊C8C2211yescharacteristic
49(C2×D191)⋊C8C2211yescharacteristic
50(C2×D191)⋊C8C2211yescharacteristic
51(C2×D191)⋊C8C2211yescharacteristic
52C2×Q16×C191C2211yescharacteristic, complemented
53C22×C8×C191C2211yescharacteristic
54C22⋊Q8×C191C2211yescharacteristic
55C22⋊Q8×C191C2211yescharacteristic
56#107×C191C2211yescharacteristic, complemented
57#103×C191C2211yescharacteristic, complemented
58#103×C191C2211yescharacteristic, complemented
59C23×D191D411yescharacteristic
60#15690288—41no—
61#15690288—41no—
62C2×Dic382—21no—
63C2×Dic382C2311yescharacteristic
64Q8×D191—41no—
65Q8×D191—41no—
66C2×D764—21no—
67C2×D764—21no—
68C2×D764C2311yescharacteristic
69C2×C22⋊D191—21no—
70C22×Dic191D411yescharacteristic
71C2×C4×D191—21no—
72C2×C4×D191—21no—
73C2×C4×D191—21no—
74C2×C4×D191—41no—
75C2×C4×D191—41no—
76C2×C4×D191D411yescharacteristic
77C2×C4×D191D411yescharacteristic
78C2×C4×D191C2311yescharacteristic
79C2×C4×D191C2311yescharacteristic
80#15690277—41no—
81#15690276—41no—
82#15690274—41no—
83C2×C191⋊C8—41no—
84C2×C191⋊C8C2311yescharacteristic
85C2×C191⋊C8C2311yescharacteristic
86C4⋊Dic191—21no—
87C4⋊Dic191C2311yescharacteristic
88C4⋊Dic191C2311yescharacteristic
89Dic191⋊C4—21no—
90Dic191⋊C4—21no—
91Dic191⋊C4—21no—
92Dic191⋊C4—21no—
93C22⋊Dic191—21no—
94(C2×D191)⋊C4—21no—
95(C2×D191)⋊C4—21no—
96(C2×D191)⋊C4—21no—
97(C2×D191)⋊C4—21no—
98(C2×D191)⋊C4—21no—
99(C2×D191)⋊C4—21no—
100C4×Dic191—21no—
101C2×Q8×C191C2311yescharacteristic
102C2×Q8×C191C2311yescharacteristic
103C22×C4×C191C2311yescharacteristic
104Q16×C191—41no—
105C2×C8×C191—41no—
106C2×C8×C191C2311yescharacteristic
107C2×C8×C191C2311yescharacteristic
108C4⋊C4×C191—21no—
109C4⋊C4×C191—21no—
110C4⋊C4×C191C2311yescharacteristic
111C4⋊C4×C191C2311yescharacteristic
112C22⋊C4×C191—21no—
113C22⋊C4×C191—21no—
114C22×D191—22no—
115C22×D191—22no—
116C22×D191—21no—
117C22×D191—41no—
118C22×D191C2×D411yescharacteristic
119C22×D191C2×D411yescharacteristic
120Dic382—21no—
121Dic382—21no—
122Dic382—42no—
123D764—21no—
124D764—21no—
125D764—42no—
126D764—42no—
127C22⋊D191—42no—
128C2×Dic191—21no—
129C2×Dic191—21no—
130C2×Dic191—21no—
131C2×Dic191—41no—
132C2×Dic191C2×D411yescharacteristic
133C2×Dic191C2×D411yescharacteristic
134C4×D191—21no—
135C4×D191—21no—
136C4×D191—21no—
137C4×D191—21no—
138C4×D191—41no—
139C4×D191—41no—
140C4×D191—42no—
141C4×D191—42no—
142C4×D191—42no—
143C191⋊C8—21no—
144C191⋊C8—21no—
145C23×C191C2×D411yescharacteristic
146Q8×C191—21no—
147Q8×C191—21no—
148Q8×C191—21no—
149Q8×C191—21no—
150C2×C4×C191—21no—
151C2×C4×C191—21no—
152C2×C4×C191—21no—
153C2×C4×C191—21no—
154C2×C4×C191—21no—
155C2×C4×C191—21no—
156C2×C4×C191C2×D411yescharacteristic
157C2×C4×C191C2411yescharacteristic
158C8×C191—21no—
159C8×C191—21no—
160C22×C191—21no—
161C22×C191—21no—
162C22×C191#13712yes—
163C22×C191C22×D411yescharacteristic
164C2×D191—21no—
165C2×D191—21no—
166C2×D191—21no—
167C2×D191—21no—
168C2×D191—21no—
169C2×D191—44no—
170C2×D191—41no—
171C2×D191—41no—
172C2×D191—41no—
173C2×D191#13512yes—
174C4×C191—21no—
175C4×C191—41no—
176C4×C191—41no—
177C4×C191—41no—
178C4×C191—41no—
179C4×C191H(5,𝔽2)11yescharacteristic
180C4×C191H(5,𝔽2)11yescharacteristic
181Dic191—21no—
182Dic191—21no—
183Dic191—41no—
184Dic191—41no—
185Dic191—41no—
186C2×C191—22no—
187C2×C191#53411yescharacteristic
188C2×C191C2×#13511yescharacteristic
189C2×C191C2×#13711yescharacteristic
190D191—22no—
191D191—41no—
192D191—81no—
193C191#319011yescharacteristic, complemented, Sylow
194#3190—1911nomaximal, Sylow
195#525—1911no—
196C2×C22⋊Q8—1911no—
197#485—1911no—
198#485—1911no—
199#483—1911no—
200#480—1911no—
201#467—1911no—
202#467D19111yescharacteristic, complemented
203#465—1911no—
204#465—1911no—
205#452—1911no—
206#452—1911no—
207#451—1911no—
208#448—1911no—
209C2×C22⋊C8—1911no—
210C2×#41—1911no—
211C22×Q8—1911no—
212C23×C4—1911no—
213C2×Q16—1911no—
214C2×Q16C2×D19111yescharacteristic, complemented
215C2×SD16—1911no—
216C2×SD16—1911no—
217C22×C8—1911no—
218C22×C8C2×D19111yescharacteristic
219#123—3821no—
220C22⋊Q8—3821no—
221C22⋊Q8—3821no—
222C22⋊Q8—1911no—
223C22⋊Q8—1911no—
224C22⋊Q8—1911no—
225C22⋊Q8—1911no—
226C22⋊Q8C2×D19111yescharacteristic
227C22⋊Q8C2×D19111yescharacteristic
228#121—1911no—
229#121—1911no—
230#120—3821no—
231C4×D4—3821no—
232C4×D4—3821no—
233#117—1911no—
234C2×C4⋊C4—3821no—
235C2×C4⋊C4—1911no—
236C2×C22⋊C4—3821no—
237#107—1911no—
238#107C2×D19111yescharacteristic, complemented
239#106—1911no—
240#106—1911no—
241#103—1911no—
242#103—1911no—
243#103—1911no—
244#103—1911no—
245#103C2×D19111yescharacteristic, complemented
246#103C2×D19111yescharacteristic, complemented
247#102—1911no—
248#102—1911no—
249C22⋊C8—1911no—
250C22⋊C8—1911no—
251C22⋊C8—1911no—
252C22⋊C8—1911no—
253C24—1911no—
254#41—7641no—
255#41—7641no—
256C2×Q8—3821no—
257C2×Q8—1911no—
258C2×Q8—7641no—
259C2×Q8—7641no—
260C2×Q8C22×D19111yescharacteristic
261C2×Q8C22×D19111yescharacteristic
262C2×D4—3821no—
263C2×D4—3821no—
264C2×D4—3821no—
265C2×D4—1911no—
266C22×C4—3821no—
267C22×C4—3821no—
268C22×C4—3821no—
269C22×C4—1911no—
270C22×C4—1911no—
271C22×C4—1911no—
272C22×C4—1911no—
273C22×C4—1911no—
274C22×C4—7641no—
275C22×C4—7641no—
276C22×C4C22×D19111yescharacteristic
277Q16—41no—
278Q16—7641no—
279SD16—7641no—
280SD16—7641no—
281C2×C8—41no—
282C2×C8—1911no—
283C2×C8—1911no—
284C2×C8—7641no—
285C2×C8C22×D19111yescharacteristic
286C2×C8C22×D19111yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—3821no—
290C4⋊C4—3821no—
291C4⋊C4—3821no—
292C4⋊C4—3821no—
293C4⋊C4—3821no—
294C4⋊C4—1911no—
295C4⋊C4—1911no—
296C4⋊C4C22×D19111yescharacteristic
297C4⋊C4C22×D19111yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3821no—
301C22⋊C4—3821no—
302C22⋊C4—3821no—
303C22⋊C4—3821no—
304C22⋊C4—3821no—
305C22⋊C4—3821no—
306C22⋊C4—3821no—
307C42—3821no—
308C23—3822no—
309C23—3822no—
310C23—3821no—
311C23—1911no—
312C23—1911no—
313C23—7641no—
314C23D4×D19111yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—21no—
318Q8—21no—
319Q8—3821no—
320Q8—3821no—
321Q8—7642no—
322D4—3821no—
323D4—3821no—
324D4—7642no—
325D4—7642no—
326D4—7642no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—3821no—
334C2×C4—3821no—
335C2×C4—3821no—
336C2×C4—3821no—
337C2×C4—3821no—
338C2×C4—3821no—
339C2×C4—3821no—
340C2×C4—1911no—
341C2×C4—1911no—
342C2×C4—7641no—
343C2×C4—7641no—
344C2×C4—7642no—
345C2×C4—7642no—
346C2×C4—7642no—
347C2×C4—7641no—
348C2×C4D4×D19111yescharacteristic
349C2×C4C23×D19111yescharacteristic
350C8—21no—
351C8—21no—
352C8—3821no—
353C8—3821no—
354C22—21no—
355C22—21no—
356C22—3821no—
357C22—3821no—
358C22—3821no—
359C22—3821no—
360C22—3821no—
361C22—1912no—
362C22—7644no—
363C22—7641no—
364C22—7641no—
365C22—7641no—
366C22#2025935412yes—
367C22C2×D4×D19111yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—3821no—
374C4—3821no—
375C4—7641no—
376C4—7641no—
377C4—7641no—
378C4#2025936711yescharacteristic
379C4#2025936811yescharacteristic
380C2—22no—
381C2—3822no—
382C2—15281no—
383C2—7641no—
384C2#2713322111yescharacteristic
385C2#135×D19111yescharacteristic
386C2C2×#2025935411yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#467×C191C2
socleC22×C191C22×D4
prime radical#27133176C2
hypercenter#467D191
hypocenterC191#3190
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. C22⋊Q8×D191C2semidirect
3. #27133006C2semidirect
4. #27133002C2semidirect
5. #27132992C2semidirect
6. #27132978C2semidirect
7. #27132902C2semidirect
8. #27132894C2semidirect
9. #27132892C2semidirect
10. #27132828C2semidirect
11. #27132824C2semidirect
12. #27132817C2semidirect
13. #27132803C2semidirect
14. C2×(C2×D191)⋊C8C2semidirect
15. #467×C191C2semidirect
19. C2×#15690277C22semidirect
20. C2×#15690276C22semidirect
21. C2×#15690274C22semidirect
39. #20259255C22semidirect
40. #20259254C22semidirect
41. #20259253C22semidirect
42. #20259247C22semidirect
43. #20259246C22semidirect
44. #20259246C22semidirect
45. #20259245C22semidirect
46. #20259244C22semidirect
47. #20259242C22semidirect
52. C2×Q16×C191C22semidirect
56. #107×C191C22semidirect
57. #103×C191C22semidirect
58. #103×C191C22semidirect
193. C191#3190semidirect
202. #467D191semidirect
214. C2×Q16C2×D191semidirect
238. #107C2×D191semidirect
245. #103C2×D191semidirect
246. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331762
C22⋊Q8×D1912
#271330062
#271330022
#271329922
#271329782
#271329022
#271328942
#271328922
#271328282
#271328242
#271328172
#271328032
C2×(C2×D191)⋊C82
#467×C1912
#3190191