Finite Group Databasebeta

← #37582465

Subgroups — #37582465

Order 24448.

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

normal subgroups only

This group has 410 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
2C2×#20259280C211yesmaximal, characteristic
3#27133006C212yesmaximal, complemented
4#27133002C212yesmaximal, complemented
5#27132998C211yesmaximal, characteristic, complemented
6#27132994C211yesmaximal, characteristic, complemented
7#27132991C211yesmaximal, characteristic, complemented
8#27132704C212yesmaximal, complemented
9#27132670C212yesmaximal, complemented
10#27132625C211yesmaximal, characteristic
11#484×C191C211yesmaximal, characteristic, complemented
12C2×Q8×D191C2211yescharacteristic
13C22×C4×D191C2211yescharacteristic
14C2×#15690270C2212yes—
15#20259302C2211yescharacteristic
16#20259301C2211yescharacteristic
17#20259300—21no—
18#20259299—21no—
19#20259298C2211yescharacteristic
20#20259280—21no—
21#20259280—21no—
22#20259280C2212yes—
23#20259280C2212yes—
24C2×C4⋊Dic191C2211yescharacteristic
25C4⋊C4×D191—21no—
26C4⋊C4×D191C2211yescharacteristic
27C2×(C2×D191)⋊C4—21no—
28C22⋊C4×D191—21no—
29C2×C4×Dic191—21no—
30#20259255C2212yescomplemented
31#20259254C2212yescomplemented
32#20259253C2212yescomplemented
33#20259247C2212yescomplemented
34#20259246C2212yescomplemented
35#20259245C2212yescomplemented
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
44C2×Dic382—21no—
45C2×Dic382C2311yescharacteristic
46Q8×D191—24no—
47C22×Dic191—21no—
48C22×Dic191D411yescharacteristic
49C2×C4×D191—24no—
50C2×C4×D191—21no—
51C2×C4×D191—21no—
52C2×C4×D191D411yescharacteristic
53C2×C4×D191D411yescharacteristic
54C2×C4×D191C2311yescharacteristic
55C2×C4×D191C2311yescharacteristic
56#15690270—24no—
57C2×C191⋊C8C2312yes—
58C4⋊Dic191—21no—
59C4⋊Dic191—21no—
60C4⋊Dic191C2312yes—
61C4⋊Dic191C2311yescharacteristic
62Dic191⋊C4—21no—
63Dic191⋊C4—21no—
64Dic191⋊C4—21no—
65C22⋊Dic191—21no—
66(C2×D191)⋊C4—21no—
67(C2×D191)⋊C4—21no—
68(C2×D191)⋊C4—22no—
69(C2×D191)⋊C4—22no—
70C4×Dic191—22no—
71C4×Dic191—22no—
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—41no—
84C22×D191C2×D411yescharacteristic
85C22×D191C2×D411yescharacteristic
86Dic382—22no—
87Dic382—42no—
88C2×Dic191—22no—
89C2×Dic191—22no—
90C2×Dic191—21no—
91C2×Dic191—21no—
92C2×Dic191—42no—
93C2×Dic191C2×D411yescharacteristic
94C2×Dic191C2×D411yescharacteristic
95C4×D191—22no—
96C4×D191—22no—
97C4×D191—22no—
98C4×D191—42no—
99C4×D191—42no—
100C4×D191#4114yes—
101C191⋊C8—22no—
102C23×C191C2×D411yescharacteristic
103Q8×C191—22no—
104C2×C4×C191—22no—
105C2×C4×C191—21no—
106C2×C4×C191—21no—
107C2×C4×C191—22no—
108C2×C4×C191—42no—
109C2×C4×C191C2×D411yescharacteristic
110C2×C4×C191C2411yescharacteristic
111C8×C191—22no—
112C22×C191—21no—
113C22×C191—21no—
114C22×C191#13712yes—
115C22×C191C22×D411yescharacteristic
116C2×D191—21no—
117C2×D191—21no—
118C2×D191—21no—
119C2×D191—21no—
120C2×D191—21no—
121C2×D191—44no—
122C2×D191#12312yes—
123C4×C191—21no—
124C4×C191—41no—
125C4×C191—42no—
126C4×C191—41no—
127C4×C191C2×#4112yes—
128Dic191—21no—
129Dic191—41no—
130Dic191—42no—
131Dic191—41no—
132Dic191#12312yes—
133C2×C191—22no—
134C2×C191C2×#12311yescharacteristic
135C2×C191C2×#13711yescharacteristic
136C2×C191#57511yescharacteristic
137D191—22no—
138D191—41no—
139C191#309511yescharacteristic, complemented, Sylow
140#3095—1911nomaximal, Sylow
141C2×C22⋊Q8—1911no—
142C2×#117—1911no—
143#485—1912no—
144#485—1912no—
145#484—1911no—
146#484D19111yescharacteristic, complemented
147#483—1911no—
148#483—1911no—
149#428—1912no—
150#419—1912no—
151#407—1911no—
152C22×Q8—1911no—
153C23×C4—1911no—
154C2×M16—1912no—
155C22⋊Q8—3821no—
156C22⋊Q8—3821no—
157C22⋊Q8—1911no—
158C22⋊Q8—1911no—
159C22⋊Q8—1911no—
160C22⋊Q8C2×D19111yescharacteristic
161#117—3821no—
162#117—3821no—
163#117—1912no—
164#117—1912no—
165C2×C4⋊C4—3821no—
166C2×C4⋊C4—1911no—
167C2×C4⋊C4—1911no—
168C2×C4⋊C4C2×D19111yescharacteristic
169C2×C22⋊C4—3821no—
170C2×C22⋊C4—3821no—
171C2×C42—3821no—
172#107—1912no—
173#107C2×D19112yescomplemented
174#106—1912no—
175#106—1912no—
176#103—1912no—
177#103—1912no—
178#103—1912no—
179#103C2×D19112yescomplemented
180C22⋊C8—1911no—
181C22⋊C8—1912no—
182C22⋊C8C2×D19111yescharacteristic
183C24—1911no—
184C2×Q8—3821no—
185C2×Q8—3824no—
186C2×Q8—1911no—
187C2×Q8C22×D19111yescharacteristic
188C22×C4—21no—
189C22×C4—3824no—
190C22×C4—3821no—
191C22×C4—3821no—
192C22×C4—3821no—
193C22×C4—1911no—
194C22×C4—1911no—
195C22×C4—1911no—
196C22×C4—1911no—
197C22×C4—1911no—
198C22×C4C22×D19111yescharacteristic
199M16—3824no—
200C2×C8—1912no—
201C2×C8C22×D19112yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—3821no—
205C4⋊C4—3821no—
206C4⋊C4—3821no—
207C4⋊C4—3821no—
208C4⋊C4—3821no—
209C4⋊C4—1912no—
210C4⋊C4—1911no—
211C4⋊C4C22×D19112yes—
212C4⋊C4C22×D19111yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—3822no—
215C22⋊C4—3822no—
216C22⋊C4—3821no—
217C22⋊C4—3821no—
218C22⋊C4—3821no—
219C42—3822no—
220C42—3822no—
221C23—3822no—
222C23—3822no—
223C23—1911no—
224C23—1911no—
225C23—7641no—
226C23D4×D19111yescharacteristic
227Q8—22no—
228Q8—3822no—
229Q8—7642no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—3822no—
236C2×C4—3822no—
237C2×C4—3822no—
238C2×C4—3822no—
239C2×C4—3822no—
240C2×C4—3821no—
241C2×C4—3821no—
242C2×C4—1911no—
243C2×C4—1911no—
244C2×C4—1914no—
245C2×C4—7642no—
246C2×C4—7642no—
247C2×C4—7642no—
248C2×C4D4×D19111yescharacteristic
249C2×C4C23×D19111yescharacteristic
250C8—22no—
251C8—3822no—
252C22—21no—
253C22—21no—
254C22—3821no—
255C22—3821no—
256C22—3821no—
257C22—3821no—
258C22—3821no—
259C22—1912no—
260C22—7644no—
261C22#2025935412yes—
262C22C2×D4×D19111yescharacteristic
263C4—21no—
264C4—42no—
265C4—41no—
266C4—41no—
267C4—3821no—
268C4—1912no—
269C4—7642no—
270C4—7641no—
271C4—7641no—
272C4#41×D19112yes—
273C2—22no—
274C2—3822no—
275C2—7641no—
276C2#123×D19111yescharacteristic
277C2C2×#2025935411yescharacteristic
278C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#484×C191C2
socleC22×C191C22×D4
prime radicalC23×D191D4
hypercenter#484D191
hypocenterC191#3095
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
3. #27133006C2semidirect
4. #27133002C2semidirect
5. #27132998C2semidirect
6. #27132994C2semidirect
7. #27132991C2semidirect
8. #27132704C2semidirect
9. #27132670C2semidirect
11. #484×C191C2semidirect
30. #20259255C22semidirect
31. #20259254C22semidirect
32. #20259253C22semidirect
33. #20259247C22semidirect
34. #20259246C22semidirect
35. #20259245C22semidirect
40. #107×C191C22semidirect
41. #103×C191C22semidirect
139. C191#3095semidirect
146. #484D191semidirect
173. #107C2×D191semidirect
179. #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×#202592802
#271330062
#271330022
#271329982
#271329942
#271329912
#271327042
#271326702
#271326252
#484×C1912
#3095191