Finite Group Databasebeta

← #28400368

Subgroups — #28400368

Order 14464.

440 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 143 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#20299151C211yesmaximal, characteristic, complemented
2C2×D4×Dic113C211yesmaximal, characteristic
3#20298900C212yesmaximal, complemented
4#20298896C212yesmaximal, complemented
5#20298886C212yesmaximal, complemented
6#20298876C212yesmaximal, complemented
7#20298641C211yesmaximal, characteristic, complemented
8#20298640C211yesmaximal, characteristic, complemented
9#20298621C211yesmaximal, characteristic, complemented
10#20298603C211yesmaximal, characteristic
11#418×C113C211yesmaximal, characteristic
12C23×Dic113—21no—
13C2×#14268631C2211yescharacteristic
14#16813621C2211yescharacteristic, complemented
15#16813621C2211yescharacteristic, complemented
16#16813613C2211yescharacteristic, complemented
17#16813613C2211yescharacteristic, complemented
18#16813599—21no—
19#16813599—21no—
20C4×Dic226—21no—
21C4×Dic226—21no—
22D4×Dic113—21no—
23D4×Dic113—21no—
24D4×Dic113C2212yes—
25D4×Dic113C2212yes—
26#16813581C2211yescharacteristic
27C2×C4⋊Dic113C2211yescharacteristic
28C2×C22⋊Dic113—21no—
29C2×C4×Dic113C2211yescharacteristic
30#16813556C2212yescomplemented
31#16813555C2212yescomplemented
32#16813554C2212yescomplemented
33#16813553C2212yescomplemented
34#16813551C2212yescomplemented
35#16813551C2212yescomplemented
36#16813543C2212yes—
37#16813543C2212yes—
38C22×D4×C113C2211yescharacteristic
39C2×M16×C113C2211yescharacteristic
40#117×C113C2211yescharacteristic
41#102×C113C2212yes—
42#102×C113C2212yes—
43C2×Dic226—21no—
44C2×Dic226—21no—
45C22×Dic113—21no—
46C22×Dic113—22no—
47C22×Dic113—22no—
48C22×Dic113—44no—
49C22×Dic113D411yescharacteristic
50C22×Dic113D411yescharacteristic
51#14268631—22no—
52C2×#11744751C2312yes—
53C4⋊Dic113—21no—
54C4⋊Dic113—21no—
55C4⋊Dic113—21no—
56C4⋊Dic113C2312yes—
57C4⋊Dic113C2311yescharacteristic
58C4⋊Dic113C2311yescharacteristic
59#14268626—21no—
60#14268626—21no—
61#14268626—21no—
62#14268626—21no—
63C22⋊Dic113—21no—
64C22⋊Dic113—22no—
65C22⋊Dic113—22no—
66C4×Dic113—21no—
67C4×Dic113D411yescharacteristic
68C4×Dic113D411yescharacteristic
69C4×Dic113C2311yescharacteristic
70C4×Dic113C2311yescharacteristic
71C24×C113—21no—
72C2×D4×C113—21no—
73C2×D4×C113—22no—
74C2×D4×C113Q814yescomplemented
75C2×D4×C113C2312yes—
76C22×C4×C113C2311yescharacteristic
77M16×C113—22no—
78C2×C8×C113C2312yes—
79C4⋊C4×C113C2311yescharacteristic
80C4⋊C4×C113C2311yescharacteristic
81C22⋊C4×C113—21no—
82C42×C113—21no—
83C23×C113—24no—
84C23×C113—22no—
85C23×C113—42no—
86C23×C113—42no—
87C23×C113C2×D411yescharacteristic
88D4×C113—24no—
89D4×C113—22no—
90D4×C113C2×Q814yes—
91C2×C4×C113—21no—
92C2×C4×C113—21no—
93C2×C4×C113C2×D411yescharacteristic
94C2×C4×C113C2×Q812yes—
95C2×C4×C113#4112yes—
96C2×C4×C113C2411yescharacteristic
97C2×Dic113—22no—
98C2×Dic113—21no—
99C2×Dic113—21no—
100C2×Dic113—22no—
101C2×Dic113—41no—
102C2×Dic113—44no—
103C2×Dic113—42no—
104C2×Dic113—44no—
105C2×Dic113C2×D411yescharacteristic
106C2×Dic113C2×D411yescharacteristic
107C2×Dic113C2×D411yescharacteristic
108C2×Dic113C2×D411yescharacteristic
109Dic226—42no—
110Dic226—42no—
111C8×C113—22no—
112#11744751—22no—
113C22×C113—21no—
114C22×C113—21no—
115C22×C113—24no—
116C22×C113—48no—
117C22×C113—42no—
118C22×C113—42no—
119C22×C113C22⋊Q812yes—
120C22×C113C22×D411yescharacteristic
121C4×C113—41no—
122C4×C113—41no—
123C4×C113C22⋊Q812yes—
124C4×C113C22×Q811yescharacteristic
125C4×C113C2×#4111yescharacteristic
126Dic113—21no—
127Dic113—21no—
128Dic113—21no—
129Dic113—41no—
130Dic113—42no—
131Dic113—41no—
132Dic113AGL(1,ℤ/8)12yes—
133C2×C113—22no—
134C2×C113—44no—
135C2×C113C2×C22⋊Q811yescharacteristic
136C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C113#57511yescharacteristic
138#3077—1131nomaximal, Sylow
139C113#307711yescharacteristic, complemented, Sylow
140#533—1131no—
141C2×C4×D4—1131no—
142#478—1132no—
143#478—1132no—
144#476—1132no—
145#474—1132no—
146#428—1131no—
147#428—1131no—
148#423—1131no—
149#418—1131no—
150#418D11311yescharacteristic
151C22×D4C2×D11311yescharacteristic
152C23×C4—2261no—
153C2×M16—1131no—
154C2×M16C2×D11311yescharacteristic
155C4⋊Q8—1131no—
156C4⋊Q8—1131no—
157#125—1131no—
158#125—1131no—
159C22⋊Q8—2261no—
160C22⋊Q8—2261no—
161C4×Q8—2261no—
162C4×Q8—2261no—
163C4×D4—1132no—
164C4×D4—1132no—
165C4×D4—2261no—
166C4×D4—2261no—
167#117—1131no—
168#117C2×D11311yescharacteristic
169C2×C4⋊C4—1131no—
170C2×C22⋊C4—2261no—
171C2×C42—1131no—
172#107—1132no—
173#107—1132no—
174#106—1132no—
175#106—1132no—
176C4⋊C8—1132no—
177C4⋊C8—1132no—
178#102—1132no—
179#102—1132no—
180#102C2×D11312yes—
181#102C2×D11312yes—
182C24—21no—
183C2×Q8—2261no—
184C2×Q8—2261no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4Dic22614yescomplemented
188C2×D4C22×D11312yes—
189C22×C4—1131no—
190C22×C4—1131no—
191C22×C4—4524no—
192C22×C4—2262no—
193C22×C4—2262no—
194C22×C4—2261no—
195C22×C4C22×D11311yescharacteristic
196M16—22no—
197M16—2262no—
198C2×C8—1132no—
199C2×C8C22×D11312yes—
200C4⋊C4—1131no—
201C4⋊C4—1131no—
202C4⋊C4—1132no—
203C4⋊C4—2261no—
204C4⋊C4—2261no—
205C4⋊C4—2261no—
206C4⋊C4—2261no—
207C4⋊C4—2261no—
208C4⋊C4—2261no—
209C4⋊C4—2261no—
210C4⋊C4C22×D11311yescharacteristic
211C4⋊C4C22×D11311yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—2262no—
214C22⋊C4—2262no—
215C22⋊C4—2261no—
216C42—21no—
217C42—1131no—
218C42—1131no—
219C42—1131no—
220C42—1131no—
221C42—2261no—
222C23—24no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23D4×D11311yescharacteristic
227Q8—4522no—
228Q8—4522no—
229D4—24no—
230D4—22no—
231D4C2×Dic22614yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—1131no—
235C2×C4—1131no—
236C2×C4—1131no—
237C2×C4—1131no—
238C2×C4—4524no—
239C2×C4—4522no—
240C2×C4—4524no—
241C2×C4—4521no—
242C2×C4—2262no—
243C2×C4—2261no—
244C2×C4—2261no—
245C2×C4—2262no—
246C2×C4D4×D11311yescharacteristic
247C2×C4C2×Dic22612yes—
248C2×C4#1426864612yes—
249C2×C4C23×D11311yescharacteristic
250C8—22no—
251C8—2262no—
252C22—21no—
253C22—21no—
254C22—24no—
255C22—48no—
256C22—42no—
257C22—42no—
258C22#1681359912yes—
259C22C2×D4×D11311yescharacteristic
260C4—41no—
261C4—41no—
262C4—1132no—
263C4—4522no—
264C4—4521no—
265C4—4521no—
266C4—2261no—
267C4—2261no—
268C4—2261no—
269C4#1681359912yes—
270C4C22×Dic22611yescharacteristic
271C4C2×#1426864611yescharacteristic
272C2—22no—
273C2—44no—
274C2C2×#1681359911yescharacteristic
275C2AGL(1,ℤ/8)×D11311yescharacteristic
276C2#2029939411yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#418×C113C2
socleC22×C113C22×D4
prime radicalC22×D4×C113C22
hypercenter#418D113
hypocenterC113#3077
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #20299151C2semidirect
3. #20298900C2semidirect
4. #20298896C2semidirect
5. #20298886C2semidirect
6. #20298876C2semidirect
7. #20298641C2semidirect
8. #20298640C2semidirect
9. #20298621C2semidirect
14. #16813621C22semidirect
15. #16813621C22semidirect
16. #16813613C22semidirect
17. #16813613C22semidirect
30. #16813556C22semidirect
31. #16813555C22semidirect
32. #16813554C22semidirect
33. #16813553C22semidirect
34. #16813551C22semidirect
35. #16813551C22semidirect
74. C2×D4×C113Q8semidirect
139. C113#3077semidirect
187. C2×D4Dic226semidirect

Maximal subgroups

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

subgroupindex
#202991512
C2×D4×Dic1132
#202989002
#202988962
#202988862
#202988762
#202986412
#202986402
#202986212
#202986032
#418×C1132
#3077113