Finite Group Databasebeta

← #28400500

Subgroups — #28400500

Order 14464.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#20299110C211yesmaximal, characteristic
2C2×C4⋊C4×D113C211yesmaximal, characteristic, complemented
3#20298935C212yesmaximal, complemented
4#20298930C212yesmaximal, complemented
5#20298918C212yesmaximal, complemented
6#20298913C212yesmaximal, complemented
7#20298641C211yesmaximal, characteristic, complemented
8#20298600C211yesmaximal, characteristic, complemented
9#20298597C211yesmaximal, characteristic, complemented
10#20298561C211yesmaximal, characteristic, complemented
11#428×C113C211yesmaximal, characteristic, complemented
12C2×#14268647C2211yescharacteristic
13C22×C4×D113—21no—
14C22×C4×D113C2211yescharacteristic
15C2×#14268631C2211yescharacteristic
16#16813606—21no—
17#16813602C2211yescharacteristic
18#16813602C2211yescharacteristic
19#16813597C2211yescharacteristic
20#16813597C2211yescharacteristic
21#16813592—21no—
22C4×D452—21no—
23C4×D452—21no—
24C2×C4⋊Dic113C2211yescharacteristic
25C2×#14268626—21no—
26C4⋊C4×D113—21no—
27C4⋊C4×D113—21no—
28C4⋊C4×D113C2212yes—
29C4⋊C4×D113C2212yes—
30#16813555C2212yescomplemented
31#16813553C2212yescomplemented
32#16813547C2212yescomplemented
33#16813545C2212yescomplemented
34#16813544C2212yescomplemented
35#16813542C2212yescomplemented
36#16813534C2212yes—
37#16813534C2212yes—
38C2×M16×C113C2211yescharacteristic
39#117×C113C2211yescharacteristic
40C2×C4⋊C4×C113C2211yescharacteristic
41#107×C113C2212yescomplemented
42#106×C113C2212yescomplemented
43C23×D113D411yescharacteristic
44#14268647—22no—
45#14268647—22no—
46C2×Dic226C2311yescharacteristic
47C2×D452—21no—
48C2×D452—21no—
49C2×D452C2311yescharacteristic
50C2×C22⋊D113—21no—
51C22×Dic113—21no—
52C22×Dic113D411yescharacteristic
53C2×C4×D113—22no—
54C2×C4×D113—22no—
55C2×C4×D113—22no—
56C2×C4×D113—22no—
57C2×C4×D113—21no—
58C2×C4×D113—44no—
59C2×C4×D113D411yescharacteristic
60C2×C4×D113D411yescharacteristic
61C2×C4×D113C2311yescharacteristic
62C2×C4×D113C2311yescharacteristic
63#14268631—22no—
64C2×#11744751C2312yes—
65C4⋊Dic113—21no—
66C4⋊Dic113—21no—
67C4⋊Dic113—21no—
68C4⋊Dic113C2312yes—
69#14268626—22no—
70#14268626—22no—
71#14268624—21no—
72#14268624—21no—
73#14268624—21no—
74#14268624—21no—
75C22×C4×C113—21no—
76C22×C4×C113C2311yescharacteristic
77M16×C113—22no—
78C2×C8×C113C2312yes—
79C4⋊C4×C113—21no—
80C4⋊C4×C113C2312yes—
81C4⋊C4×C113C2311yescharacteristic
82C4⋊C4×C113C2311yescharacteristic
83C22⋊C4×C113—21no—
84C42×C113—21no—
85C23×C113C2×D411yescharacteristic
86C22×D113—22no—
87C22×D113—22no—
88C22×D113—22no—
89C22×D113—21no—
90C22×D113C2×D411yescharacteristic
91C22×D113C2×D411yescharacteristic
92C2×C4×C113—21no—
93C2×C4×C113—21no—
94C2×C4×C113—22no—
95C2×C4×C113—42no—
96C2×C4×C113C2×D411yescharacteristic
97C2×C4×C113#4112yes—
98C2×C4×C113#4112yes—
99C2×C4×C113C2411yescharacteristic
100C22⋊D113—42no—
101C2×Dic113—22no—
102C2×Dic113—21no—
103C2×Dic113—22no—
104C2×Dic113—42no—
105C2×Dic113C2×D411yescharacteristic
106C2×Dic113C2×D411yescharacteristic
107D452—21no—
108D452—21no—
109D452—42no—
110D452—42no—
111C4×D113—22no—
112C4×D113—22no—
113C4×D113—21no—
114C4×D113—21no—
115C4×D113—21no—
116C4×D113—21no—
117C4×D113—44no—
118C4×D113—44no—
119C4×D113—42no—
120Dic226—21no—
121Dic226—21no—
122C8×C113—22no—
123#11744751—22no—
124C22×C113—21no—
125C22×C113—21no—
126C22×C113#12312yes—
127C22×C113C22×D411yescharacteristic
128C2×D113—28no—
129C2×D113—21no—
130C2×D113—21no—
131C2×D113—21no—
132C2×D113—21no—
133C2×D113—41no—
134C2×D113—41no—
135C2×D113—41no—
136C2×D113AGL(1,ℤ/8)12yes—
137C2×D113#13712yes—
138C4×C113—42no—
139C4×C113—41no—
140C4×C113—41no—
141C4×C113#12312yes—
142C4×C113C2×#4111yescharacteristic
143C4×C113C2×#4111yescharacteristic
144Dic113—21no—
145Dic113—21no—
146Dic113—41no—
147Dic113—42no—
148C2×C113—22no—
149C2×C113C2×#12311yescharacteristic
150C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C113C2×#13711yescharacteristic
152D113—22no—
153D113—22no—
154D113—81no—
155#3092—1131nomaximal, Sylow
156C113#309211yescharacteristic, complemented, Sylow
157#525—1131no—
158C22×C4⋊C4—1131no—
159#484—1132no—
160#483—1132no—
161#481—1132no—
162#480—1132no—
163#428—1131no—
164#428D11311yescharacteristic, complemented
165#417—1131no—
166#417—1131no—
167#407—1131no—
168C2×#41—1131no—
169C23×C4—1131no—
170C23×C4—2261no—
171C2×M16—1131no—
172C2×M16C2×D11311yescharacteristic
173#123—2261no—
174C22⋊Q8—1131no—
175C22⋊Q8—1131no—
176#121—1131no—
177#121—1131no—
178#120—2261no—
179C4×D4—2261no—
180C4×D4—2261no—
181#117C2×D11311yescharacteristic
182C2×C4⋊C4—1131no—
183C2×C4⋊C4—1132no—
184C2×C4⋊C4—1132no—
185C2×C4⋊C4—2261no—
186C2×C4⋊C4—2261no—
187C2×C4⋊C4—2261no—
188C2×C4⋊C4C2×D11311yescharacteristic
189#107—1132no—
190#107C2×D11312yescomplemented
191#106—1132no—
192#106C2×D11312yescomplemented
193#103—1132no—
194#103—1132no—
195#102—1132no—
196#102—1132no—
197C22⋊C8—1132no—
198C22⋊C8—1132no—
199C24—1131no—
200#41—2262no—
201#41—2262no—
202C2×Q8—1131no—
203C2×D4—1131no—
204C2×D4—2261no—
205C2×D4—2261no—
206C2×D4—2261no—
207C22×C4—21no—
208C22×C4—1131no—
209C22×C4—1131no—
210C22×C4—1131no—
211C22×C4—1131no—
212C22×C4—1131no—
213C22×C4—4524no—
214C22×C4—2262no—
215C22×C4—2262no—
216C22×C4—2261no—
217C22×C4—2261no—
218C22×C4—2262no—
219C22×C4—2262no—
220C22×C4C22×D11311yescharacteristic
221M16—22no—
222M16—2262no—
223C2×C8—1132no—
224C2×C8C22×D11312yes—
225C4⋊C4—21no—
226C4⋊C4—1132no—
227C4⋊C4—2262no—
228C4⋊C4—2262no—
229C4⋊C4—2261no—
230C4⋊C4—2261no—
231C4⋊C4—2261no—
232C4⋊C4C22×D11312yes—
233C4⋊C4C22×D11311yescharacteristic
234C4⋊C4C22×D11311yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—2261no—
237C22⋊C4—2261no—
238C22⋊C4—2261no—
239C22⋊C4—2261no—
240C42—21no—
241C23—1131no—
242C23—1131no—
243C23—2262no—
244C23—2262no—
245C23—2261no—
246C23—2262no—
247C23D4×D11311yescharacteristic
248Q8—2261no—
249Q8—2261no—
250D4—4522no—
251D4—4522no—
252D4—4522no—
253D4—2261no—
254D4—2261no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—1131no—
260C2×C4—1131no—
261C2×C4—4524no—
262C2×C4—4524no—
263C2×C4—4522no—
264C2×C4—4522no—
265C2×C4—2261no—
266C2×C4—2262no—
267C2×C4—2262no—
268C2×C4—2262no—
269C2×C4—2262no—
270C2×C4—2261no—
271C2×C4—2261no—
272C2×C4—2261no—
273C2×C4—2261no—
274C2×C4D4×D11311yescharacteristic
275C2×C4#1426864712yes—
276C2×C4#1426864812yes—
277C2×C4C23×D11311yescharacteristic
278C8—22no—
279C8—2262no—
280C22—21no—
281C22—21no—
282C22—1132no—
283C22—1132no—
284C22—4521no—
285C22—4521no—
286C22—4521no—
287C22—2268no—
288C22—2261no—
289C22—2261no—
290C22—2261no—
291C22—2261no—
292C22#1681360712yes—
293C22C2×D4×D11311yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—4521no—
298C4—4522no—
299C4—2261no—
300C4—2261no—
301C4#1681360712yes—
302C4C2×#1426864711yescharacteristic
303C4C2×#1426864811yescharacteristic
304C2—22no—
305C2—9041no—
306C2—2262no—
307C2—2262no—
308C2C2×#1681360711yescharacteristic
309C2AGL(1,ℤ/8)×D11311yescharacteristic
310C2#137×D11311yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#428×C113C2
socleC22×C113C22×D4
prime radical#20299110C2
hypercenter#428D113
hypocenterC113#3092
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
2. C2×C4⋊C4×D113C2semidirect
3. #20298935C2semidirect
4. #20298930C2semidirect
5. #20298918C2semidirect
6. #20298913C2semidirect
7. #20298641C2semidirect
8. #20298600C2semidirect
9. #20298597C2semidirect
10. #20298561C2semidirect
11. #428×C113C2semidirect
30. #16813555C22semidirect
31. #16813553C22semidirect
32. #16813547C22semidirect
33. #16813545C22semidirect
34. #16813544C22semidirect
35. #16813542C22semidirect
41. #107×C113C22semidirect
42. #106×C113C22semidirect
156. C113#3092semidirect
164. #428D113semidirect
190. #107C2×D113semidirect
192. #106C2×D113semidirect

Maximal subgroups

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

subgroupindex
#202991102
C2×C4⋊C4×D1132
#202989352
#202989302
#202989182
#202989132
#202986412
#202986002
#202985972
#202985612
#428×C1132
#3092113