Finite Group Databasebeta

← #37583480

Subgroups — #37583480

Order 24448.

410 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133192C211yesmaximal, characteristic, complemented
2#124×D191C211yesmaximal, characteristic, complemented
3#27133045C211yesmaximal, characteristic, complemented
4#27133018C211yesmaximal, characteristic, complemented
5#27133017C211yesmaximal, characteristic, complemented
6#27132887C211yesmaximal, characteristic, complemented
7#27132883C211yesmaximal, characteristic, complemented
8#27132880C211yesmaximal, characteristic, complemented
9#27132876C211yesmaximal, characteristic, complemented
10#27132817C211yesmaximal, characteristic, complemented
11#27132812C211yesmaximal, characteristic, complemented
12#27132800C211yesmaximal, characteristic, complemented
13#27132795C211yesmaximal, characteristic
14#27132714C211yesmaximal, characteristic, complemented
15#495×C191C211yesmaximal, characteristic, complemented
16C2×Q8×D191C2211yescharacteristic
17C2×D4×D191C2211yescharacteristic
18C2×#15690277C2211yescharacteristic, complemented
19C2×#15690275C2211yescharacteristic, complemented
20C2×#15690274C2211yescharacteristic
21C2×#15690272C2211yescharacteristic
22#20259315—21no—
23#20259314C2211yescharacteristic, complemented
24#20259311C2211yescharacteristic
25#20259310C2211yescharacteristic
26#20259309C2211yescharacteristic
27#20259308—21no—
28#20259308—21no—
29#20259304—21no—
30#20259298C2211yescharacteristic
31#20259293C2211yescharacteristic
32Q8×Dic191C2211yescharacteristic, complemented
33D4×Dic191C2211yescharacteristic, complemented
34(C22⋊D191)⋊C4—21no—
35#20259278—21no—
36C22⋊C4×D191—21no—
37C22⋊C4×D191—21no—
38C42×D191C2211yescharacteristic
39Dic191⋊C8C2211yescharacteristic, complemented
40Dic191⋊C8C2211yescharacteristic, complemented
41#20259247C2211yescharacteristic, complemented
42#20259247C2211yescharacteristic, complemented
43#20259246C2211yescharacteristic, complemented
44#20259246C2211yescharacteristic, complemented
45#20259244C2211yescharacteristic
46#20259244C2211yescharacteristic
47#20259243C2211yescharacteristic
48#20259243C2211yescharacteristic
49(C2×D191)⋊C8C2211yescharacteristic
50(C2×D191)⋊C8C2211yescharacteristic
51C4×C191⋊C8C2211yescharacteristic, complemented
52C2×Q16×C191C2211yescharacteristic, complemented
53C2×SD16×C191C2211yescharacteristic
54C2×SD16×C191C2211yescharacteristic, complemented
55C2×D8×C191C2211yescharacteristic
56#124×C191C2211yescharacteristic
57#124×C191C2211yescharacteristic, complemented
58C4×C8×C191C2211yescharacteristic, complemented
59C23×D191—21no—
60C2×Dic382—21no—
61C2×Dic382C2311yescharacteristic
62Q8×D191—41no—
63Q8×D191—41no—
64C2×D764C2311yescharacteristic
65C2×C22⋊D191—21no—
66C2×C22⋊D191—21no—
67D4×D191—41no—
68D4×D191—41no—
69C22×Dic191—21no—
70C2×C4×D191—21no—
71C2×C4×D191D411yescharacteristic
72C2×C4×D191D411yescharacteristic
73C2×C4×D191C2311yescharacteristic
74#15690277—41no—
75#15690275—41no—
76#15690274—41no—
77#15690272—41no—
78C2×C191⋊C8C2311yescharacteristic
79C2×C191⋊C8C2311yescharacteristic
80C4⋊Dic191—21no—
81C4⋊Dic191C2311yescharacteristic
82C4⋊Dic191C2311yescharacteristic
83Dic191⋊C4—21no—
84Dic191⋊C4—21no—
85Dic191⋊C4—21no—
86C22⋊Dic191—21no—
87C22⋊Dic191—21no—
88C22⋊Dic191—21no—
89C22⋊Dic191—21no—
90(C2×D191)⋊C4—21no—
91(C2×D191)⋊C4—21no—
92(C2×D191)⋊C4—21no—
93(C2×D191)⋊C4—21no—
94(C2×D191)⋊C4—21no—
95C4×Dic191—21no—
96C4×Dic191D411yescharacteristic
97C4×Dic191D411yescharacteristic
98C4×Dic191C2311yescharacteristic
99C2×Q8×C191C2311yescharacteristic
100C2×Q8×C191C2311yescharacteristic
101C2×D4×C191C2311yescharacteristic
102C2×D4×C191C2311yescharacteristic
103Q16×C191—41no—
104SD16×C191—41no—
105SD16×C191—41no—
106D8×C191—41no—
107C2×C8×C191C2311yescharacteristic
108C2×C8×C191C2311yescharacteristic
109C22⋊C4×C191—21no—
110C22⋊C4×C191—21no—
111C22⋊C4×C191—21no—
112C22⋊C4×C191—21no—
113C42×C191C2311yescharacteristic
114C22×D191—21no—
115C22×D191—42no—
116C22×D191—42no—
117C22×D191—42no—
118C22×D191C2×D411yescharacteristic
119Dic382—21no—
120Dic382—21no—
121Dic382—42no—
122D764—21no—
123D764—21no—
124C22⋊D191—42no—
125C22⋊D191—42no—
126C2×Dic191—21no—
127C2×Dic191—21no—
128C2×Dic191—21no—
129C2×Dic191—42no—
130C2×Dic191C2×D411yescharacteristic
131C2×Dic191C2×D411yescharacteristic
132C2×Dic191C2×D411yescharacteristic
133C4×D191—22no—
134C4×D191—21no—
135C4×D191—21no—
136C4×D191—41no—
137C4×D191—42no—
138C191⋊C8—21no—
139C191⋊C8—21no—
140C23×C191—21no—
141C23×C191—21no—
142Q8×C191—21no—
143Q8×C191—21no—
144Q8×C191—21no—
145Q8×C191—21no—
146D4×C191—21no—
147D4×C191—21no—
148D4×C191—21no—
149D4×C191—21no—
150C2×C4×C191—21no—
151C2×C4×C191—21no—
152C2×C4×C191C2×D411yescharacteristic
153C2×C4×C191C2×D411yescharacteristic
154C2×C4×C191C2411yescharacteristic
155C8×C191—21no—
156C8×C191—21no—
157C22×C191—41no—
158C22×C191—41no—
159C22×C191—41no—
160C22×C191—41no—
161C22×C191—41no—
162C22×C191—41no—
163C22×C191C22×D411yescharacteristic
164C2×D191—21no—
165C2×D191—21no—
166C2×D191—41no—
167C2×D191—41no—
168C2×D191—41no—
169C2×D191—84no—
170C2×D191#13512yes—
171C4×C191—21no—
172C4×C191—21no—
173C4×C191—41no—
174C4×C191—41no—
175C4×C191H(5,𝔽2)11yescharacteristic
176C4×C191H(5,𝔽2)11yescharacteristic
177Dic191—21no—
178Dic191—21no—
179Dic191—21no—
180Dic191—41no—
181Dic191—41no—
182Dic191—41no—
183C2×C191—81no—
184C2×C191—81no—
185C2×C191#53511yescharacteristic
186C2×C191C2×#13511yescharacteristic
187C2×C191#57511yescharacteristic
188D191—22no—
189D191—81no—
190C191#321111yescharacteristic, complemented, Sylow
191#3211—1911nomaximal, Sylow
192#529—1911no—
193C2×#124—1911no—
194#495—1911no—
195#495D19111yescharacteristic, complemented
196#488—1911no—
197#488—1911no—
198#464—1911no—
199#464—1911no—
200#463—1911no—
201#463—1911no—
202#451—1911no—
203#450—1911no—
204#448—1911no—
205#447—1911no—
206#431—1911no—
207C22×Q8—1911no—
208C22×D4—1911no—
209C2×Q16—1911no—
210C2×Q16C2×D19111yescharacteristic, complemented
211C2×SD16—1911no—
212C2×SD16—1911no—
213C2×SD16C2×D19111yescharacteristic
214C2×SD16C2×D19111yescharacteristic, complemented
215C2×D8—1911no—
216C2×D8C2×D19111yescharacteristic
217#126—3821no—
218#125—1911no—
219#124—3821no—
220#124—3821no—
221#124—1911no—
222#124—1911no—
223#124—1911no—
224#124C2×D19111yescharacteristic
225#124C2×D19111yescharacteristic, complemented
226#123—3821no—
227C22⋊Q8—1911no—
228#121—1911no—
229C4×Q8—1911no—
230C4×D4—3821no—
231C4×D4—1911no—
232#117—3821no—
233C2×C22⋊C4—3821no—
234C2×C22⋊C4—3821no—
235C2×C42—1911no—
236C4⋊C8—1911no—
237C4⋊C8—1911no—
238#103—1911no—
239#103—1911no—
240#103—1911no—
241#103—1911no—
242#102—1911no—
243#102—1911no—
244#102—1911no—
245#102—1911no—
246C22⋊C8—1911no—
247C22⋊C8—1911no—
248C4×C8—1911no—
249C4×C8C2×D19111yescharacteristic, complemented
250C24—3821no—
251C2×Q8—3821no—
252C2×Q8—1911no—
253C2×Q8—7641no—
254C2×Q8—7641no—
255C2×Q8C22×D19111yescharacteristic
256C2×Q8C22×D19111yescharacteristic
257C2×D4—3821no—
258C2×D4—3821no—
259C2×D4—1911no—
260C2×D4—7641no—
261C2×D4—7641no—
262C2×D4C22×D19111yescharacteristic
263C2×D4C22×D19111yescharacteristic
264C22×C4—3821no—
265C22×C4—3821no—
266C22×C4—1911no—
267C22×C4—1911no—
268C22×C4—1911no—
269Q16—41no—
270Q16—7641no—
271SD16—41no—
272SD16—41no—
273SD16—7641no—
274SD16—7641no—
275D8—41no—
276D8—7641no—
277C2×C8—1911no—
278C2×C8—1911no—
279C2×C8C22×D19111yescharacteristic
280C2×C8C22×D19111yescharacteristic
281C4⋊C4—3821no—
282C4⋊C4—3821no—
283C4⋊C4—3821no—
284C4⋊C4—3821no—
285C4⋊C4—1911no—
286C4⋊C4—1911no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—3821no—
292C22⋊C4—3821no—
293C22⋊C4—3821no—
294C22⋊C4—3821no—
295C22⋊C4—3821no—
296C22⋊C4—3821no—
297C22⋊C4—3821no—
298C22⋊C4—3821no—
299C22⋊C4—3821no—
300C42—3821no—
301C42—1911no—
302C42—1911no—
303C42—1911no—
304C42C22×D19111yescharacteristic
305C23—21no—
306C23—21no—
307C23—3821no—
308C23—1911no—
309C23—7642no—
310C23—7642no—
311C23—7642no—
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—21no—
316Q8—3821no—
317Q8—3821no—
318Q8—7642no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—3821no—
324D4—3821no—
325D4—7642no—
326D4—7642no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—3822no—
330C2×C4—3821no—
331C2×C4—3821no—
332C2×C4—3821no—
333C2×C4—3821no—
334C2×C4—3821no—
335C2×C4—1911no—
336C2×C4—1911no—
337C2×C4—1911no—
338C2×C4—7641no—
339C2×C4—7642no—
340C2×C4—7642no—
341C2×C4D4×D19111yescharacteristic
342C2×C4D4×D19111yescharacteristic
343C2×C4C23×D19111yescharacteristic
344C8—21no—
345C8—21no—
346C8—3821no—
347C8—3821no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—3821no—
355C22—3821no—
356C22—1912no—
357C22—15284no—
358C22—7641no—
359C22—7641no—
360C22—7641no—
361C22C2×D4×D19111yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—3821no—
367C4—3821no—
368C4—3821no—
369C4—7641no—
370C4—7641no—
371C4—7641no—
372C4#2025936711yescharacteristic
373C4#2025936711yescharacteristic
374C2—81no—
375C2—81no—
376C2—3822no—
377C2—15281no—
378C2#2713322611yescharacteristic
379C2#135×D19111yescharacteristic
380C2#2713346011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#495×C191C2
socleC22×C191C22×D4
prime radical#27132795C2
hypercenter#495D191
hypocenterC191#3211
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #27133192C2semidirect
2. #124×D191C2semidirect
3. #27133045C2semidirect
4. #27133018C2semidirect
5. #27133017C2semidirect
6. #27132887C2semidirect
7. #27132883C2semidirect
8. #27132880C2semidirect
9. #27132876C2semidirect
10. #27132817C2semidirect
11. #27132812C2semidirect
12. #27132800C2semidirect
14. #27132714C2semidirect
15. #495×C191C2semidirect
18. C2×#15690277C22semidirect
19. C2×#15690275C22semidirect
23. #20259314C22semidirect
32. Q8×Dic191C22semidirect
33. D4×Dic191C22semidirect
39. Dic191⋊C8C22semidirect
40. Dic191⋊C8C22semidirect
41. #20259247C22semidirect
42. #20259247C22semidirect
43. #20259246C22semidirect
44. #20259246C22semidirect
51. C4×C191⋊C8C22semidirect
52. C2×Q16×C191C22semidirect
54. C2×SD16×C191C22semidirect
57. #124×C191C22semidirect
58. C4×C8×C191C22semidirect
190. C191#3211semidirect
195. #495D191semidirect
210. C2×Q16C2×D191semidirect
214. C2×SD16C2×D191semidirect
225. #124C2×D191semidirect
249. C4×C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331922
#124×D1912
#271330452
#271330182
#271330172
#271328872
#271328832
#271328802
#271328762
#271328172
#271328122
#271328002
#271327952
#271327142
#495×C1912
#3211191