Finite Group Databasebeta

← #44402717

Subgroups — #44402717

Order 29056.

928 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#20300256C211yesmaximal, characteristic
2C2×C4×C22⋊D227C212yesmaximal, complemented
3C2×C22⋊C4×D227C212yesmaximal, complemented
4#28408935C211yesmaximal, characteristic, complemented
5#28408926C212yesmaximal, complemented
6#28408908C212yesmaximal, complemented
7#28408906C212yesmaximal, complemented
8#28408879C212yesmaximal, complemented
9#386×C227C211yesmaximal, characteristic, complemented
10C24×D227C2211yescharacteristic
11C22×C22⋊D227C2212yes—
12C22×C4×D227C2212yes—
13C22×C4×D227C2212yes—
14#20300256C418yescomplemented
15C4×C22⋊D227—28no—
16C2×C4⋊Dic227C2211yescharacteristic, complemented
17C2×Dic227⋊C4C2212yescomplemented
18C2×C22⋊Dic227C2212yes—
19C2×C22⋊Dic227C2211yescharacteristic
20C2×C22⋊Dic227C2212yescomplemented
21C2×(C2×D227)⋊C4C2212yes—
22C2×(C2×D227)⋊C4C2212yes—
23C2×(C2×D227)⋊C4C2212yes—
24C22⋊C4×D227—28no—
25C2×C4×Dic227C2212yescomplemented
26C2×C4×Dic227C2211yescharacteristic, complemented
27C227⋊#95C2212yescomplemented
28C227⋊#95C2212yescomplemented
29C22×D4×C227C2211yescharacteristic
30C23×C4×C227C2211yescharacteristic, complemented
31C2×C22⋊C4×C227C2212yes—
32C2×C22⋊C4×C227C2211yescharacteristic, complemented
33#95×C227C2212yescomplemented
34C23×D227—24no—
35C23×D227—24no—
36C23×D227—41no—
37C23×D227—41no—
38C23×D227C2×C414yescomplemented
39C23×D227C2312yes—
40C2×C22⋊D227—24no—
41C2×C22⋊D227—24no—
42C2×C22⋊D227C2×C418yescomplemented
43C22×Dic227C2312yes—
44C22×Dic227C2312yes—
45C22×Dic227C2312yes—
46C2×C4×D227—24no—
47C2×C4×D227—24no—
48C2×C4×D227—24no—
49C2×C4×D227—24no—
50C2×C4×D227—42no—
51C2×C4×D227—42no—
52C4⋊Dic227—41no—
53Dic227⋊C4—24no—
54C22⋊Dic227—24no—
55C22⋊Dic227—24no—
56C22⋊Dic227C2×C414yescomplemented
57(C2×D227)⋊C4—24no—
58(C2×D227)⋊C4C2×C418yescomplemented
59(C2×D227)⋊C4D418yescomplemented
60C4×Dic227—24no—
61C4×Dic227—41no—
62C24×C227C2311yescharacteristic
63C24×C227C2311yescharacteristic
64C2×D4×C227—41no—
65C2×D4×C227—41no—
66C2×D4×C227C2×C414yescomplemented
67C22×C4×C227—24no—
68C22×C4×C227—41no—
69C22×C4×C227C2312yes—
70C22×C4×C227C2311yescharacteristic
71C22×C4×C227C2312yes—
72C22⋊C4×C227—24no—
73C22⋊C4×C227—41no—
74C22×D227—22no—
75C22×D227—22no—
76C22×D227—22no—
77C22×D227—22no—
78C22×D227—216no—
79C22×D227—44no—
80C22×D227—44no—
81C22×D227—48no—
82C22×D227—44no—
83C22×D227C22×C414yes—
84C22×D227C2×D414yes—
85C22×D227#4114yes—
86C22⋊D227—216no—
87C2×Dic227—22no—
88C2×Dic227—22no—
89C2×Dic227—22no—
90C2×Dic227—22no—
91C2×Dic227—22no—
92C2×Dic227—22no—
93C2×Dic227—22no—
94C2×Dic227C22×C414yes—
95C2×Dic227C2×D414yes—
96C4×D227—48no—
97C4×D227—48no—
98C23×C227—22no—
99C23×C227—22no—
100C23×C227—22no—
101C23×C227—22no—
102C23×C227—21no—
103C23×C227—21no—
104C23×C227—21no—
105C23×C227—21no—
106C23×C227C22×C412yes—
107C23×C227C22×C412yes—
108C23×C227C2411yescharacteristic
109D4×C227—44no—
110C2×C4×C227—22no—
111C2×C4×C227—22no—
112C2×C4×C227—21no—
113C2×C4×C227—21no—
114C2×C4×C227—22no—
115C2×C4×C227—22no—
116C2×C4×C227—22no—
117C2×C4×C227—44no—
118C2×C4×C227C22×C412yes—
119C2×C4×C227C2×D414yes—
120C22×C227—24no—
121C22×C227—24no—
122C22×C227—22no—
123C22×C227—22no—
124C22×C227—41no—
125C22×C227—41no—
126C22×C227—42no—
127C22×C227—42no—
128C22×C227—41no—
129C22×C227—41no—
130C22×C227C23×C411yescharacteristic
131C22×C227C22×D412yes—
132C22×C227C2×#4112yes—
133C22×C227H(5,𝔽2)11yescharacteristic
134C22×C227H(5,𝔽2)11yescharacteristic
135C2×D227—24no—
136C2×D227—24no—
137C2×D227—24no—
138C2×D227—24no—
139C2×D227—24no—
140C2×D227—24no—
141C2×D227—416no—
142C2×D227C4×D418yescomplemented
143C4×C227—22no—
144C4×C227—24no—
145C4×C227—42no—
146Dic227—24no—
147Dic227—24no—
148Dic227—42no—
149C2×C227—42no—
150C2×C227—42no—
151C2×C227C2×C4×D412yes—
152C2×C227#51811yescharacteristic
153C2×C227D4211yescharacteristic
154C2×C227#54612yes—
155C2×C227#55111yescharacteristic
156D227—28no—
157C227#235811yescharacteristic, complemented, Sylow
158#2358—2271nomaximal, Sylow
159C2×#120—2271no—
160C2×C4×D4—2272no—
161C22×C22⋊C4—2272no—
162#390—2271no—
163#388—2272no—
164#386—2272no—
165#386D22711yescharacteristic, complemented
166#385—2272no—
167C4×C22⋊C4—2272no—
168C25—2271no—
169C22×D4—2272no—
170C22×D4C2×D22711yescharacteristic
171C23×C4—2272no—
172C23×C4—2272no—
173C23×C4C2×D22711yescharacteristic, complemented
174#120—2278no—
175C4×D4—4548no—
176C2×C4⋊C4—2271no—
177C2×C4⋊C4—2272no—
178C2×C22⋊C4—4548no—
179C2×C22⋊C4—2272no—
180C2×C22⋊C4—2272no—
181C2×C22⋊C4—2272no—
182C2×C22⋊C4—2271no—
183C2×C22⋊C4—2272no—
184C2×C22⋊C4—2272no—
185C2×C22⋊C4C2×D22712yes—
186C2×C22⋊C4C2×D22711yescharacteristic, complemented
187C2×C42—2271no—
188C2×C42—2272no—
189#95—2272no—
190#95—2272no—
191#95C2×D22712yescomplemented
192C24—9081no—
193C24—9081no—
194C24—4544no—
195C24—4544no—
196C24—2274no—
197C24—2272no—
198C24C22×D22711yescharacteristic
199C24C22×D22711yescharacteristic
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—4544no—
203C2×D4—4544no—
204C2×D4—2278no—
205C2×D4C4×D22714yescomplemented
206C22×C4—24no—
207C22×C4—41no—
208C22×C4—9082no—
209C22×C4—9082no—
210C22×C4—4544no—
211C22×C4—4544no—
212C22×C4—4544no—
213C22×C4—4544no—
214C22×C4—2272no—
215C22×C4—2272no—
216C22×C4—2272no—
217C22×C4C22×D22712yes—
218C22×C4C22×D22711yescharacteristic
219C22×C4C22×D22712yes—
220C4⋊C4—9081no—
221C4⋊C4—4544no—
222C22⋊C4—24no—
223C22⋊C4—41no—
224C22⋊C4—4544no—
225C22⋊C4—4544no—
226C22⋊C4—4544no—
227C22⋊C4—2274no—
228C22⋊C4—2278no—
229C22⋊C4—2278no—
230C42—9081no—
231C42—4544no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—9084no—
241C23—9084no—
242C23—9088no—
243C23—9084no—
244C23—45416no—
245C23—4542no—
246C23—4542no—
247C23—4542no—
248C23—4542no—
249C23—2274no—
250C23—2274no—
251C23—2274no—
252C23C2×C4×D22712yes—
253C23C2×C4×D22712yes—
254C23C23×D22711yescharacteristic
255D4—44no—
256D4—45416no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—44no—
265C2×C4—9088no—
266C2×C4—9088no—
267C2×C4—4542no—
268C2×C4—4542no—
269C2×C4—4542no—
270C2×C4—4542no—
271C2×C4—4542no—
272C2×C4—4542no—
273C2×C4—4542no—
274C2×C4—2274no—
275C2×C4—2274no—
276C2×C4C2×C4×D22712yes—
277C2×C4D4×D22714yes—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—90816no—
289C22—4544no—
290C22—4544no—
291C22—4544no—
292C22—4544no—
293C22—4544no—
294C22—4544no—
295C22—2278no—
296C22C22×C4×D22711yescharacteristic
297C22C2×D4×D22712yes—
298C22#41×D22712yes—
299C22#2030033311yescharacteristic
300C22#2030033311yescharacteristic
301C4—24no—
302C4—22no—
303C4—42no—
304C4—9082no—
305C4—4544no—
306C4—4544no—
307C2—42no—
308C2—42no—
309C2—4548no—
310C2C4×D4×D22712yes—
311C2#2840950511yescharacteristic
312C2#2840966411yescharacteristic
313C2#2840967912yes—
314C2#2840971011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutatorC22×C227C23×C4
Fitting#386×C227C2
socle, cosocleC23×C227C24
prime radicalC2×#20300256C2
hypercenter#386D227
hypocenterC227#2358
LCSG ▷ C22×C227 ▷ C227
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D227C2semidirect
3. C2×C22⋊C4×D227C2semidirect
4. #28408935C2semidirect
5. #28408926C2semidirect
6. #28408908C2semidirect
7. #28408906C2semidirect
8. #28408879C2semidirect
9. #386×C227C2semidirect
14. #20300256C4semidirect
16. C2×C4⋊Dic227C22semidirect
17. C2×Dic227⋊C4C22semidirect
20. C2×C22⋊Dic227C22semidirect
25. C2×C4×Dic227C22semidirect
26. C2×C4×Dic227C22semidirect
27. C227⋊#95C22semidirect
28. C227⋊#95C22semidirect
30. C23×C4×C227C22semidirect
32. C2×C22⋊C4×C227C22semidirect
33. #95×C227C22semidirect
38. C23×D227C2×C4semidirect
42. C2×C22⋊D227C2×C4semidirect
56. C22⋊Dic227C2×C4semidirect
58. (C2×D227)⋊C4C2×C4semidirect
59. (C2×D227)⋊C4D4semidirect
66. C2×D4×C227C2×C4semidirect
142. C2×D227C4×D4semidirect
157. C227#2358semidirect
165. #386D227semidirect
173. C23×C4C2×D227semidirect
186. C2×C22⋊C4C2×D227semidirect
191. #95C2×D227semidirect
205. C2×D4C4×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203002562
C2×C4×C22⋊D2272
C2×C22⋊C4×D2272
#284089352
#284089262
#284089082
#284089062
#284088792
#386×C2272
#2358227