Finite Group Databasebeta

← #28400370

Subgroups — #28400370

Order 14464.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16813584C211yesmaximal, characteristic
3#20298901C212yesmaximal, complemented
4#20298897C212yesmaximal, complemented
5#20298887C212yesmaximal, complemented
6#20298877C212yesmaximal, complemented
7#20298638C211yesmaximal, characteristic, complemented
8#20298621C211yesmaximal, characteristic, complemented
9#20298604C211yesmaximal, characteristic
10#20298602C211yesmaximal, characteristic
11#428×C113C211yesmaximal, characteristic, complemented
12C22×D452C2211yescharacteristic
13C22×C4×D113—21no—
14C2×#14268631C2211yescharacteristic
15#16813621C2211yescharacteristic, complemented
16#16813621C2211yescharacteristic, complemented
17#16813613C2211yescharacteristic, complemented
18#16813613C2211yescharacteristic, complemented
19#16813599—21no—
20#16813599—21no—
21C4×Dic226—21no—
22C4×Dic226—21no—
23#16813584—21no—
24#16813584—21no—
25#16813584C2212yes—
26#16813584C2212yes—
27#16813581C2211yescharacteristic
28C2×#14268624—21no—
29C2×C4×Dic113C2211yescharacteristic
30#16813555C2212yescomplemented
31#16813553C2212yescomplemented
32#16813551C2212yescomplemented
33#16813551C2212yescomplemented
34#16813544C2212yes—
35#16813544C2212yes—
36#16813542C2212yes—
37#16813542C2212yes—
38C2×M16×C113C2211yescharacteristic
39#117×C113C2211yescharacteristic
40C2×C4⋊C4×C113C2211yescharacteristic
41#107×C113C2212yescomplemented
42#106×C113C2212yescomplemented
43C23×D113—21no—
44C2×Dic226—21no—
45C2×Dic226—21no—
46C2×D452—21no—
47C2×D452—22no—
48C2×D452Q814yescomplemented
49C2×D452C2312yes—
50C22×Dic113D411yescharacteristic
51C22×Dic113D411yescharacteristic
52C2×C4×D113—22no—
53C2×C4×D113—22no—
54C2×C4×D113—44no—
55#14268631—22no—
56C2×#11744751C2312yes—
57C4⋊Dic113—21no—
58C4⋊Dic113—21no—
59C4⋊Dic113C2311yescharacteristic
60C4⋊Dic113C2311yescharacteristic
61#14268626—21no—
62#14268626—21no—
63#14268626—21no—
64#14268626—21no—
65C22⋊Dic113—21no—
66#14268624—22no—
67#14268624—22no—
68C4×Dic113—21no—
69C4×Dic113D411yescharacteristic
70C4×Dic113D411yescharacteristic
71C4×Dic113C2311yescharacteristic
72C4×Dic113C2311yescharacteristic
73C22×C4×C113—21no—
74C22×C4×C113C2311yescharacteristic
75M16×C113—22no—
76C2×C8×C113C2312yes—
77C4⋊C4×C113—21no—
78C4⋊C4×C113C2312yes—
79C4⋊C4×C113C2311yescharacteristic
80C4⋊C4×C113C2311yescharacteristic
81C22⋊C4×C113—21no—
82C42×C113—21no—
83C23×C113C2×D411yescharacteristic
84C22×D113—24no—
85C22×D113—22no—
86C22×D113—42no—
87C22×D113—42no—
88C2×C4×C113—21no—
89C2×C4×C113—21no—
90C2×C4×C113—22no—
91C2×C4×C113—42no—
92C2×C4×C113C2×D411yescharacteristic
93C2×C4×C113C2×Q812yes—
94C2×C4×C113#4112yes—
95C2×C4×C113C2411yescharacteristic
96C2×Dic113—22no—
97C2×Dic113—21no—
98C2×Dic113—21no—
99C2×Dic113—41no—
100C2×Dic113C2×D411yescharacteristic
101C2×Dic113C2×D411yescharacteristic
102C2×Dic113C2×D411yescharacteristic
103C2×Dic113C2×D411yescharacteristic
104D452—24no—
105D452—22no—
106D452C2×Q814yes—
107C4×D113—44no—
108C4×D113—44no—
109Dic226—42no—
110Dic226—42no—
111C8×C113—22no—
112#11744751—22no—
113C22×C113—21no—
114C22×C113—21no—
115C22×C113C22⋊Q812yes—
116C22×C113C22×D411yescharacteristic
117C2×D113—24no—
118C2×D113—48no—
119C2×D113—42no—
120C2×D113—42no—
121C4×C113—41no—
122C4×C113—42no—
123C4×C113—41no—
124C4×C113C22⋊Q812yes—
125C4×C113C22×Q811yescharacteristic
126C4×C113C2×#4111yescharacteristic
127Dic113—21no—
128Dic113—21no—
129Dic113—21no—
130Dic113—41no—
131Dic113—41no—
132Dic113AGL(1,ℤ/8)12yes—
133C2×C113—22no—
134C2×C113C2×C22⋊Q811yescharacteristic
135C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C113#57511yescharacteristic
137D113—44no—
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#428D11311yescharacteristic, complemented
148#423—1131no—
149#418—1131no—
150#418—1131no—
151C22×D4—1131no—
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⋊C4C2×D11311yescharacteristic
170C2×C22⋊C4—2261no—
171C2×C42—1131no—
172#107—1132no—
173#107C2×D11312yescomplemented
174#106—1132no—
175#106C2×D11312yescomplemented
176C4⋊C8—1132no—
177C4⋊C8—1132no—
178#102—1132no—
179#102—1132no—
180#102—1132no—
181#102—1132no—
182C24—2261no—
183C2×Q8—2261no—
184C2×Q8—2261no—
185C2×D4—1134no—
186C2×D4—1132no—
187C2×D4—2261no—
188C2×D4—2262no—
189C22×C4—21no—
190C22×C4—1131no—
191C22×C4—1131no—
192C22×C4—4524no—
193C22×C4—2262no—
194C22×C4—2262no—
195C22×C4C22×D11311yescharacteristic
196M16—22no—
197M16—2262no—
198C2×C8—1132no—
199C2×C8C22×D11312yes—
200C4⋊C4—21no—
201C4⋊C4—1131no—
202C4⋊C4—1131no—
203C4⋊C4—2261no—
204C4⋊C4—2261no—
205C4⋊C4—2261no—
206C4⋊C4—2261no—
207C4⋊C4—2261no—
208C4⋊C4—2261no—
209C4⋊C4C22×D11311yescharacteristic
210C4⋊C4C22×D11311yescharacteristic
211C4⋊C4C22×D11312yes—
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—4522no—
223C23—4522no—
224C23—2264no—
225C23—2262no—
226C23D4×D11311yescharacteristic
227Q8—4522no—
228Q8—4522no—
229D4—1134no—
230D4—2264no—
231D4—2262no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—1131no—
237C2×C4—1131no—
238C2×C4—1131no—
239C2×C4—1131no—
240C2×C4—4524no—
241C2×C4—4524no—
242C2×C4—4521no—
243C2×C4—2262no—
244C2×C4—2261no—
245C2×C4—2261no—
246C2×C4D4×D11311yescharacteristic
247C2×C4Q8×D11312yes—
248C2×C4#1426864712yes—
249C2×C4C23×D11311yescharacteristic
250C8—22no—
251C8—2262no—
252C22—21no—
253C22—21no—
254C22—4528no—
255C22—4522no—
256C22—4522no—
257C22—2264no—
258C22#1681360012yes—
259C22C2×D4×D11311yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—1132no—
264C4—4521no—
265C4—4521no—
266C4—2261no—
267C4—2261no—
268C4—2261no—
269C4#1681360012yes—
270C4C2×Q8×D11311yescharacteristic
271C4C2×#1426864711yescharacteristic
272C2—22no—
273C2—4524no—
274C2C2×#1681360011yescharacteristic
275C2AGL(1,ℤ/8)×D11311yescharacteristic
276C2#2029939711yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#428×C113C2
socleC22×C113C22×D4
prime radicalC22×D452C22
hypercenter#428D113
hypocenterC113#3077
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #20299151C2semidirect
3. #20298901C2semidirect
4. #20298897C2semidirect
5. #20298887C2semidirect
6. #20298877C2semidirect
7. #20298638C2semidirect
8. #20298621C2semidirect
11. #428×C113C2semidirect
15. #16813621C22semidirect
16. #16813621C22semidirect
17. #16813613C22semidirect
18. #16813613C22semidirect
30. #16813555C22semidirect
31. #16813553C22semidirect
32. #16813551C22semidirect
33. #16813551C22semidirect
41. #107×C113C22semidirect
42. #106×C113C22semidirect
48. C2×D452Q8semidirect
139. C113#3077semidirect
147. #428D113semidirect
173. #107C2×D113semidirect
175. #106C2×D113semidirect

Maximal subgroups

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

subgroupindex
#202991512
C2×#168135842
#202989012
#202988972
#202988872
#202988772
#202986382
#202986212
#202986042
#202986022
#428×C1132
#3077113