Finite Group Databasebeta

← #37583276

Subgroups — #37583276

Order 24448.

468 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D191C211yesmaximal, characteristic, complemented
2C2×#20259297C211yesmaximal, characteristic
3#27132993C211yesmaximal, characteristic, complemented
4#27132983C212yesmaximal, complemented
5#27132895C212yesmaximal, complemented
6#27132893C212yesmaximal, complemented
7#27132816C211yesmaximal, characteristic, complemented
8#27132813C211yesmaximal, characteristic, complemented
9#27132803C212yesmaximal, complemented
10C22⋊C8×D191C211yesmaximal, characteristic, complemented
11#483×C191C211yesmaximal, characteristic, complemented
12C2×Q8×D191C2211yescharacteristic, complemented
13C22×D764—21no—
14C22×D764C2211yescharacteristic
15C22×C4×D191C2211yescharacteristic
16C2×#15690276C2212yescomplemented
17C2×#15690274C2212yescomplemented
18C2×C8×D191C2212yescomplemented
19#20259302C2211yescharacteristic, complemented
20#20259301C2211yescharacteristic
21#20259300—21no—
22#20259299—21no—
23#20259298C2211yescharacteristic, complemented
24#20259297—21no—
25#20259297—21no—
26#20259297C2212yes—
27#20259297C2212yes—
28C4⋊C4×D191—21no—
29C4⋊C4×D191C2211yescharacteristic, complemented
30C2×(C2×D191)⋊C4—21no—
31C22⋊C4×D191—21no—
32#20259253C2212yescomplemented
33#20259245C2212yescomplemented
34#20259244C2212yescomplemented
35#20259242C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic
37(C2×D191)⋊C8C2212yescomplemented
38C22⋊Q8×C191C2211yescharacteristic
39C2×C4⋊C4×C191C2211yescharacteristic
40#106×C191C2212yescomplemented
41#103×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic
43C23×D191—21no—
44C23×D191D411yescharacteristic
45C2×Dic382—21no—
46C2×Dic382C2311yescharacteristic
47Q8×D191—42no—
48C2×D764—22no—
49C2×D764—22no—
50C2×D764—21no—
51C2×D764—42no—
52C2×D764—44no—
53C2×D764C2312yes—
54C22×Dic191D411yescharacteristic
55C2×C4×D191—21no—
56C2×C4×D191—21no—
57C2×C4×D191—42no—
58C2×C4×D191D411yescharacteristic
59C2×C4×D191D411yescharacteristic
60C2×C4×D191C2311yescharacteristic
61C2×C4×D191C2311yescharacteristic
62#15690276—42no—
63#15690274—42no—
64C2×C191⋊C8C2312yes—
65C8×D191—42no—
66C4⋊Dic191—21no—
67C4⋊Dic191C2311yescharacteristic
68Dic191⋊C4—21no—
69Dic191⋊C4—21no—
70Dic191⋊C4—21no—
71C22⋊Dic191—21no—
72(C2×D191)⋊C4—21no—
73(C2×D191)⋊C4—21no—
74(C2×D191)⋊C4—22no—
75(C2×D191)⋊C4—22no—
76C2×Q8×C191C2311yescharacteristic
77C22×C4×C191—21no—
78C22×C4×C191C2311yescharacteristic
79C2×C8×C191C2312yes—
80C4⋊C4×C191—21no—
81C4⋊C4×C191—21no—
82C4⋊C4×C191C2312yes—
83C4⋊C4×C191C2311yescharacteristic
84C22⋊C4×C191—21no—
85C22×D191—22no—
86C22×D191—41no—
87C22×D191—41no—
88C22×D191—42no—
89C22×D191—42no—
90C22×D191—42no—
91C22×D191SD1614yescomplemented
92C22×D191C2×D411yescharacteristic
93C22×D191C2×D411yescharacteristic
94Dic382—22no—
95Dic382—42no—
96D764—22no—
97D764—22no—
98D764—42no—
99D764—44no—
100D764—44no—
101C2×Dic191—21no—
102C2×Dic191—21no—
103C2×Dic191—41no—
104C2×Dic191C2×D411yescharacteristic
105C2×Dic191C2×D411yescharacteristic
106C4×D191—22no—
107C4×D191—22no—
108C4×D191—41no—
109C4×D191—41no—
110C4×D191—42no—
111C4×D191—42no—
112C191⋊C8—22no—
113C23×C191C2×D411yescharacteristic
114Q8×C191—22no—
115C2×C4×C191—22no—
116C2×C4×C191—21no—
117C2×C4×C191—21no—
118C2×C4×C191—22no—
119C2×C4×C191—42no—
120C2×C4×C191C2×D411yescharacteristic
121C2×C4×C191C2411yescharacteristic
122C8×C191—22no—
123C22×C191—21no—
124C22×C191—21no—
125C22×C191C2×SD1612yes—
126C22×C191C22×D411yescharacteristic
127C2×D191—21no—
128C2×D191—21no—
129C2×D191—21no—
130C2×D191—21no—
131C2×D191—44no—
132C2×D191—42no—
133C2×D191—42no—
134C2×D191—42no—
135C2×D191—84no—
136C2×D191C2×SD1612yes—
137C2×D191C2×SD1612yes—
138C4×C191—21no—
139C4×C191—41no—
140C4×C191—42no—
141C4×C191—41no—
142C4×C191H(5,𝔽2)12yes—
143Dic191—21no—
144Dic191—21no—
145Dic191—41no—
146Dic191—41no—
147C2×C191—22no—
148C2×C191#53411yescharacteristic
149C2×C191C22×SD1611yescharacteristic
150C2×C191#57511yescharacteristic
151D191—22no—
152D191—22no—
153D191—82no—
154C191#318711yescharacteristic, complemented, Sylow
155#3187—1911nomaximal, Sylow
156C2×C22⋊Q8—1911no—
157C2×#121—1911no—
158#483—1911no—
159#483D19111yescharacteristic, complemented
160#481—1912no—
161#465—1912no—
162#465—1912no—
163#450—1911no—
164#450—1911no—
165#448—1912no—
166C2×C22⋊C8—1911no—
167C22×Q8—1911no—
168C22×D4—3821no—
169C22×D4—1911no—
170C23×C4—1911no—
171C2×SD16—1912no—
172C2×SD16—1912no—
173C22×C8—1912no—
174C22⋊Q8—3821no—
175C22⋊Q8—3821no—
176C22⋊Q8—1911no—
177C22⋊Q8—1911no—
178C22⋊Q8—1911no—
179C22⋊Q8C2×D19111yescharacteristic
180#121—3821no—
181#121—3821no—
182#121—1912no—
183#121—1912no—
184C2×C4⋊C4—3821no—
185C2×C4⋊C4—1911no—
186C2×C4⋊C4C2×D19111yescharacteristic
187C2×C22⋊C4—3821no—
188C2×C22⋊C4—3821no—
189#106—1912no—
190#106C2×D19112yescomplemented
191#103—1912no—
192#103C2×D19112yescomplemented
193#102—1912no—
194#102—1912no—
195C22⋊C8—1911no—
196C22⋊C8—1912no—
197C22⋊C8C2×D19111yescharacteristic
198C24—3821no—
199C24—1911no—
200C2×Q8—3821no—
201C2×Q8—1911no—
202C2×Q8—7642no—
203C2×Q8C22×D19111yescharacteristic
204C2×D4—3822no—
205C2×D4—3822no—
206C2×D4—3821no—
207C2×D4—1912no—
208C2×D4—7642no—
209C2×D4—7644no—
210C22×C4—21no—
211C22×C4—3821no—
212C22×C4—3821no—
213C22×C4—1911no—
214C22×C4—1911no—
215C22×C4—1911no—
216C22×C4—1911no—
217C22×C4—1911no—
218C22×C4—7642no—
219C22×C4C22×D19111yescharacteristic
220SD16—7642no—
221SD16—7642no—
222C2×C8—1912no—
223C2×C8—7642no—
224C2×C8C22×D19112yes—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—3821no—
228C4⋊C4—3821no—
229C4⋊C4—3821no—
230C4⋊C4—3821no—
231C4⋊C4—1911no—
232C4⋊C4C22×D19111yescharacteristic
233C4⋊C4C22×D19112yes—
234C22⋊C4—21no—
235C22⋊C4—3821no—
236C22⋊C4—3821no—
237C22⋊C4—3822no—
238C22⋊C4—3822no—
239C22⋊C4—3821no—
240C23—3822no—
241C23—1911no—
242C23—1911no—
243C23—1914no—
244C23—7641no—
245C23—7641no—
246C23—7642no—
247C23—7642no—
248C23—7642no—
249C23D4×D19111yescharacteristic
250Q8—22no—
251Q8—3822no—
252Q8—7642no—
253D4—3822no—
254D4—3822no—
255D4—7642no—
256D4—7644no—
257D4—7644no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—3822no—
264C2×C4—3822no—
265C2×C4—3821no—
266C2×C4—3821no—
267C2×C4—1911no—
268C2×C4—1911no—
269C2×C4—7641no—
270C2×C4—7641no—
271C2×C4—7642no—
272C2×C4—7642no—
273C2×C4—7641no—
274C2×C4D4×D19111yescharacteristic
275C2×C4C23×D19111yescharacteristic
276C8—22no—
277C8—3822no—
278C22—21no—
279C22—21no—
280C22—3821no—
281C22—3821no—
282C22—3821no—
283C22—3821no—
284C22—1912no—
285C22—1912no—
286C22—15284no—
287C22—7644no—
288C22—7642no—
289C22—7642no—
290C22—7642no—
291C22SD16×D19112yes—
292C22C2×D4×D19111yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—42no—
297C4—3821no—
298C4—3821no—
299C4—7641no—
300C4—7641no—
301C4#2025936812yes—
302C2—22no—
303C2—3822no—
304C2—3822no—
305C2—15282no—
306C2#2713322411yescharacteristic
307C2C2×SD16×D19111yescharacteristic
308C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#483×C191C2
socleC22×C191C22×D4
prime radicalC2×#20259297C2
hypercenter#483D191
hypocenterC191#3187
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D191C2semidirect
3. #27132993C2semidirect
4. #27132983C2semidirect
5. #27132895C2semidirect
6. #27132893C2semidirect
7. #27132816C2semidirect
8. #27132813C2semidirect
9. #27132803C2semidirect
10. C22⋊C8×D191C2semidirect
11. #483×C191C2semidirect
12. C2×Q8×D191C22semidirect
16. C2×#15690276C22semidirect
17. C2×#15690274C22semidirect
18. C2×C8×D191C22semidirect
19. #20259302C22semidirect
23. #20259298C22semidirect
29. C4⋊C4×D191C22semidirect
32. #20259253C22semidirect
33. #20259245C22semidirect
34. #20259244C22semidirect
35. #20259242C22semidirect
37. (C2×D191)⋊C8C22semidirect
40. #106×C191C22semidirect
41. #103×C191C22semidirect
91. C22×D191SD16semidirect
154. C191#3187semidirect
159. #483D191semidirect
190. #106C2×D191semidirect
192. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1912
C2×#202592972
#271329932
#271329832
#271328952
#271328932
#271328162
#271328132
#271328032
C22⋊C8×D1912
#483×C1912
#3187191