Finite Group Databasebeta

← #37582436

Subgroups — #37582436

Order 24448.

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

normal subgroups only

This group has 420 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
1#27133176C211yesmaximal, characteristic
2C2×#20259280C211yesmaximal, characteristic, complemented
3#27133003C212yesmaximal, complemented
4#27132991C211yesmaximal, characteristic, complemented
5#27132990C212yesmaximal, complemented
6#27132981C211yesmaximal, characteristic, complemented
7#27132977C211yesmaximal, characteristic, complemented
8#27132700C212yesmaximal, complemented
9#27132658C212yesmaximal, complemented
10C22⋊C8×D191C211yesmaximal, characteristic, complemented
11#484×C191C211yesmaximal, characteristic, complemented
12C2×#15690288C2211yescharacteristic
13C22×C4×D191C2211yescharacteristic
14C2×C8×D191C2212yes—
15#20259307—21no—
16#20259298C2211yescharacteristic
17#20259297C2211yescharacteristic
18#20259296C2211yescharacteristic
19#20259292—21no—
20#20259284—21no—
21(C22⋊D191)⋊C4—21no—
22#20259280—21no—
23#20259280—21no—
24#20259280C2211yescharacteristic
25#20259280C2212yes—
26#20259280C2212yes—
27C2×C4⋊Dic191C2211yescharacteristic
28C2×(C2×D191)⋊C4—21no—
29C2×C4×Dic191—21no—
30#20259256C2212yescomplemented
31#20259253C2212yescomplemented
32#20259253C2212yescomplemented
33#20259246C2212yescomplemented
34#20259244C2212yescomplemented
35#20259242C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic
37(C2×D191)⋊C8C2212yes—
38C22⋊Q8×C191C2211yescharacteristic
39C2×C4⋊C4×C191C2211yescharacteristic
40#107×C191C2212yescomplemented
41#103×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic
43C23×D191D411yescharacteristic
44#15690288—24no—
45C2×D764—21no—
46C2×D764—21no—
47C2×D764C2311yescharacteristic
48C2×C22⋊D191—21no—
49C22×Dic191—21no—
50C22×Dic191D411yescharacteristic
51C2×C4×D191—24no—
52C2×C4×D191—21no—
53C2×C4×D191D411yescharacteristic
54C2×C4×D191D411yescharacteristic
55C2×C4×D191C2311yescharacteristic
56C2×C4×D191C2311yescharacteristic
57C2×C191⋊C8C2312yes—
58C8×D191—24no—
59C4⋊Dic191—21no—
60C4⋊Dic191C2312yes—
61C4⋊Dic191C2311yescharacteristic
62Dic191⋊C4—21no—
63(C2×D191)⋊C4—22no—
64(C2×D191)⋊C4—22no—
65(C2×D191)⋊C4—21no—
66(C2×D191)⋊C4—21no—
67(C2×D191)⋊C4—21no—
68(C2×D191)⋊C4—21no—
69C4×Dic191—22no—
70C4×Dic191—22no—
71C4×Dic191—21no—
72C2×Q8×C191C2311yescharacteristic
73C22×C4×C191—21no—
74C22×C4×C191C2311yescharacteristic
75C2×C8×C191C2312yes—
76C4⋊C4×C191—21no—
77C4⋊C4×C191—21no—
78C4⋊C4×C191C2312yes—
79C4⋊C4×C191C2311yescharacteristic
80C22⋊C4×C191—21no—
81C22×D191—22no—
82C22×D191—22no—
83C22×D191—21no—
84C22×D191—41no—
85C22×D191C2×D411yescharacteristic
86C22×D191C2×D411yescharacteristic
87D764—22no—
88D764—42no—
89D764—42no—
90C22⋊D191—42no—
91C2×Dic191—22no—
92C2×Dic191—22no—
93C2×Dic191—21no—
94C2×Dic191—42no—
95C2×Dic191C2×D411yescharacteristic
96C2×Dic191C2×D411yescharacteristic
97C4×D191—22no—
98C4×D191—22no—
99C4×D191—22no—
100C4×D191—42no—
101C4×D191#4114yes—
102C191⋊C8—22no—
103C23×C191C2×D411yescharacteristic
104Q8×C191—22no—
105C2×C4×C191—22no—
106C2×C4×C191—21no—
107C2×C4×C191—21no—
108C2×C4×C191—22no—
109C2×C4×C191—42no—
110C2×C4×C191C2×D411yescharacteristic
111C2×C4×C191C2411yescharacteristic
112C8×C191—22no—
113C22×C191—21no—
114C22×C191—21no—
115C22×C191#13512yes—
116C22×C191C22×D411yescharacteristic
117C2×D191—21no—
118C2×D191—21no—
119C2×D191—21no—
120C2×D191—21no—
121C2×D191—21no—
122C2×D191—44no—
123C2×D191—41no—
124C2×D191—41no—
125C2×D191—41no—
126C2×D191#12312yes—
127C4×C191—21no—
128C4×C191—41no—
129C4×C191—42no—
130C4×C191—41no—
131C4×C191C2×#4112yes—
132Dic191—21no—
133Dic191—42no—
134Dic191—41no—
135Dic191#12312yes—
136C2×C191—22no—
137C2×C191C2×#12311yescharacteristic
138C2×C191C2×#13511yescharacteristic
139C2×C191#57511yescharacteristic
140D191—22no—
141D191—41no—
142D191—81no—
143C191#309111yescharacteristic, complemented, Sylow
144#3091—1911nomaximal, Sylow
145#525—1911no—
146C2×#117—1911no—
147#485—1912no—
148#484D19111yescharacteristic, complemented
149#483—1911no—
150#482—1912no—
151#481—1911no—
152#480—1911no—
153#427—1912no—
154#416—1912no—
155C2×C22⋊C8—1911no—
156C2×#41—1911no—
157C23×C4—1911no—
158C22×C8—1912no—
159#123—3821no—
160C22⋊Q8—1911no—
161C22⋊Q8C2×D19111yescharacteristic
162#121—1911no—
163#121—1911no—
164#120—3821no—
165C4×D4—3821no—
166C4×D4—3821no—
167#117—3821no—
168#117—3821no—
169#117—1912no—
170#117—1912no—
171#117—1911no—
172C2×C4⋊C4—1911no—
173C2×C4⋊C4C2×D19111yescharacteristic
174C2×C22⋊C4—3821no—
175C2×C42—3821no—
176#107—1912no—
177#107C2×D19112yescomplemented
178#106—1912no—
179#106—1912no—
180#103—1912no—
181#103C2×D19112yescomplemented
182#102—1912no—
183#102—1912no—
184C22⋊C8—1911no—
185C22⋊C8—1912no—
186C22⋊C8C2×D19111yescharacteristic
187C24—1911no—
188#41—3824no—
189C2×Q8C22×D19111yescharacteristic
190C2×D4—3821no—
191C2×D4—3821no—
192C2×D4—3821no—
193C2×D4—1911no—
194C22×C4—21no—
195C22×C4—3824no—
196C22×C4—3821no—
197C22×C4—3821no—
198C22×C4—1911no—
199C22×C4—1911no—
200C22×C4—1911no—
201C22×C4—1911no—
202C22×C4—1911no—
203C22×C4C22×D19111yescharacteristic
204C2×C8—3824no—
205C2×C8—1912no—
206C2×C8C22×D19112yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—3821no—
210C4⋊C4—3821no—
211C4⋊C4—1912no—
212C4⋊C4—1911no—
213C4⋊C4C22×D19112yes—
214C4⋊C4C22×D19111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—3822no—
217C22⋊C4—3822no—
218C22⋊C4—3821no—
219C22⋊C4—3821no—
220C22⋊C4—3821no—
221C22⋊C4—3821no—
222C42—3822no—
223C42—3822no—
224C42—3821no—
225C23—3822no—
226C23—3822no—
227C23—3821no—
228C23—1911no—
229C23—1911no—
230C23—7641no—
231C23D4×D19111yescharacteristic
232Q8—22no—
233D4—3822no—
234D4—7642no—
235D4—7642no—
236D4—7642no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—3822no—
243C2×C4—3822no—
244C2×C4—3822no—
245C2×C4—3822no—
246C2×C4—3822no—
247C2×C4—3821no—
248C2×C4—1911no—
249C2×C4—1911no—
250C2×C4—1914no—
251C2×C4—7642no—
252C2×C4—7642no—
253C2×C4D4×D19111yescharacteristic
254C2×C4C23×D19111yescharacteristic
255C8—22no—
256C8—3822no—
257C22—21no—
258C22—21no—
259C22—3821no—
260C22—3821no—
261C22—3821no—
262C22—3821no—
263C22—3821no—
264C22—1912no—
265C22—7644no—
266C22—7641no—
267C22—7641no—
268C22—7641no—
269C22#2025934512yes—
270C22C2×D4×D19111yescharacteristic
271C4—21no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—3821no—
276C4—1912no—
277C4—7642no—
278C4—7641no—
279C4#41×D19112yes—
280C2—22no—
281C2—3822no—
282C2—15281no—
283C2—7641no—
284C2#123×D19111yescharacteristic
285C2C2×#2025934511yescharacteristic
286C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#484×C191C2
socleC22×C191C22×D4
prime radical#27133176C2
hypercenter#484D191
hypocenterC191#3091
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#20259280C2semidirect
3. #27133003C2semidirect
4. #27132991C2semidirect
5. #27132990C2semidirect
6. #27132981C2semidirect
7. #27132977C2semidirect
8. #27132700C2semidirect
9. #27132658C2semidirect
10. C22⋊C8×D191C2semidirect
11. #484×C191C2semidirect
30. #20259256C22semidirect
31. #20259253C22semidirect
32. #20259253C22semidirect
33. #20259246C22semidirect
34. #20259244C22semidirect
35. #20259242C22semidirect
40. #107×C191C22semidirect
41. #103×C191C22semidirect
143. C191#3091semidirect
148. #484D191semidirect
177. #107C2×D191semidirect
181. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331762
C2×#202592802
#271330032
#271329912
#271329902
#271329812
#271329772
#271327002
#271326582
C22⋊C8×D1912
#484×C1912
#3091191