Finite Group Databasebeta

← #28401328

Subgroups — #28401328

Order 14464.

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

normal subgroups only

This group has 468 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×D113C211yesmaximal, characteristic, complemented
2C2×#16813593C211yesmaximal, characteristic
3#20298929C211yesmaximal, characteristic, complemented
4#20298926C211yesmaximal, characteristic, complemented
5#20298915C212yesmaximal, complemented
6#20298826C212yesmaximal, complemented
7#20298824C212yesmaximal, complemented
8#20298749C211yesmaximal, characteristic, complemented
9#20298735C212yesmaximal, complemented
10C22⋊C8×D113C211yesmaximal, characteristic, complemented
11#450×C113C211yesmaximal, characteristic, complemented
12C2×Q8×D113C2211yescharacteristic, complemented
13C22×C22⋊D113—21no—
14C22×C4×D113C2211yescharacteristic
15C2×#14268635C2212yescomplemented
16C2×C8×D113C2212yescomplemented
17#16813602C2211yescharacteristic, complemented
18#16813601C2211yescharacteristic
19#16813600—21no—
20#16813599—21no—
21#16813598C2211yescharacteristic, complemented
22#16813593—21no—
23#16813593—21no—
24#16813593C2212yes—
25#16813593C2212yes—
26C2×C4⋊Dic113C2211yescharacteristic
27C4⋊C4×D113—21no—
28C4⋊C4×D113C2211yescharacteristic, complemented
29C2×C22⋊Dic113—21no—
30C22⋊C4×D113—21no—
31#16813554C2212yescomplemented
32#16813553C2212yescomplemented
33#16813547C2212yescomplemented
34#16813546C2212yescomplemented
35#16813543C2212yescomplemented
36C22⋊#11744751C2211yescharacteristic
37#16813534C2212yescomplemented
38C22×D4×C113C2211yescharacteristic
39C2×SD16×C113C2212yescomplemented
40C22⋊Q8×C113C2211yescharacteristic
41#102×C113C2212yescomplemented
42C22⋊C8×C113C2211yescharacteristic
43C23×D113D411yescharacteristic
44C2×Dic226—21no—
45C2×Dic226C2311yescharacteristic
46Q8×D113—42no—
47C2×C22⋊D113—22no—
48C2×C22⋊D113—22no—
49C2×C22⋊D113—44no—
50C22×Dic113—21no—
51C22×Dic113D411yescharacteristic
52C2×C4×D113—21no—
53C2×C4×D113—21no—
54C2×C4×D113—42no—
55C2×C4×D113D411yescharacteristic
56C2×C4×D113D411yescharacteristic
57C2×C4×D113C2311yescharacteristic
58C2×C4×D113C2311yescharacteristic
59#14268635—42no—
60C2×#11744751C2312yes—
61C8×D113—42no—
62C4⋊Dic113—21no—
63C4⋊Dic113—21no—
64C4⋊Dic113C2311yescharacteristic
65C4⋊Dic113C2312yes—
66#14268626—21no—
67#14268626—21no—
68#14268626—21no—
69C22⋊Dic113—21no—
70C22⋊Dic113—22no—
71C22⋊Dic113—22no—
72#14268624—21no—
73#14268624—21no—
74C24×C113—21no—
75C2×Q8×C113C2311yescharacteristic
76C2×D4×C113—21no—
77C2×D4×C113—42no—
78C2×D4×C113C2312yes—
79C22×C4×C113C2311yescharacteristic
80SD16×C113—42no—
81C2×C8×C113C2312yes—
82C4⋊C4×C113—21no—
83C4⋊C4×C113C2311yescharacteristic
84C22⋊C4×C113—21no—
85C23×C113—22no—
86C23×C113—42no—
87C23×C113—42no—
88C23×C113—42no—
89C23×C113C2×D411yescharacteristic
90C22×D113—41no—
91C22×D113—41no—
92C22×D113SD1614yescomplemented
93C22×D113C2×D411yescharacteristic
94C22×D113C2×D411yescharacteristic
95Q8×C113—22no—
96D4×C113—22no—
97D4×C113—22no—
98D4×C113—42no—
99C2×C4×C113—22no—
100C2×C4×C113—21no—
101C2×C4×C113—21no—
102C2×C4×C113C2×D411yescharacteristic
103C2×C4×C113C2411yescharacteristic
104C22⋊D113—44no—
105C22⋊D113—44no—
106C2×Dic113—21no—
107C2×Dic113—21no—
108C2×Dic113—22no—
109C2×Dic113—41no—
110C2×Dic113—42no—
111C2×Dic113C2×D411yescharacteristic
112C2×Dic113C2×D411yescharacteristic
113C4×D113—22no—
114C4×D113—22no—
115C4×D113—41no—
116C4×D113—41no—
117C4×D113—42no—
118C4×D113—42no—
119Dic226—22no—
120Dic226—42no—
121C8×C113—22no—
122#11744751—22no—
123C22×C113—21no—
124C22×C113—21no—
125C22×C113—42no—
126C22×C113—42no—
127C22×C113—42no—
128C22×C113—84no—
129C22×C113C2×SD1612yes—
130C22×C113C22×D411yescharacteristic
131C2×D113—21no—
132C2×D113—21no—
133C2×D113—21no—
134C2×D113—21no—
135C2×D113—44no—
136C2×D113C2×SD1612yes—
137C2×D113C2×SD1612yes—
138C4×C113—21no—
139C4×C113—41no—
140C4×C113—41no—
141C4×C113H(5,𝔽2)12yes—
142Dic113—21no—
143Dic113—21no—
144Dic113—41no—
145Dic113—42no—
146Dic113—41no—
147C2×C113—22no—
148C2×C113—82no—
149C2×C113#53411yescharacteristic
150C2×C113C22×SD1611yescharacteristic
151C2×C113#57511yescharacteristic
152D113—22no—
153D113—22no—
154#3187—1131nomaximal, Sylow
155C113#318711yescharacteristic, complemented, Sylow
156C2×C22⋊Q8—1131no—
157C2×#121—1131no—
158#483—1131no—
159#483—1131no—
160#481—1132no—
161#465—1132no—
162#465—1132no—
163#450—1131no—
164#450D11311yescharacteristic, complemented
165#448—1132no—
166C2×C22⋊C8—1131no—
167C22×Q8—1131no—
168C22×D4—2261no—
169C22×D4C2×D11311yescharacteristic
170C23×C4—1131no—
171C2×SD16—1132no—
172C2×SD16C2×D11312yescomplemented
173C22×C8—1132no—
174C22⋊Q8—1131no—
175C22⋊Q8—1131no—
176C22⋊Q8—1131no—
177C22⋊Q8—2261no—
178C22⋊Q8—2261no—
179C22⋊Q8C2×D11311yescharacteristic
180#121—1132no—
181#121—1132no—
182#121—2261no—
183#121—2261no—
184C2×C4⋊C4—1131no—
185C2×C4⋊C4—1131no—
186C2×C4⋊C4—2261no—
187C2×C22⋊C4—2261no—
188C2×C22⋊C4—2261no—
189#106—1132no—
190#106—1132no—
191#103—1132no—
192#103—1132no—
193#102—1132no—
194#102C2×D11312yescomplemented
195C22⋊C8—1131no—
196C22⋊C8—1132no—
197C22⋊C8C2×D11311yescharacteristic
198C24—21no—
199C24—1131no—
200C2×Q8—1131no—
201C2×Q8—4522no—
202C2×Q8—2261no—
203C2×Q8C22×D11311yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—4524no—
207C2×D4—2262no—
208C2×D4—2262no—
209C2×D4C22×D11312yes—
210C22×C4—1131no—
211C22×C4—1131no—
212C22×C4—1131no—
213C22×C4—1131no—
214C22×C4—1131no—
215C22×C4—4522no—
216C22×C4—2261no—
217C22×C4—2261no—
218C22×C4—2261no—
219C22×C4C22×D11311yescharacteristic
220SD16—42no—
221SD16—4522no—
222C2×C8—1132no—
223C2×C8—4522no—
224C2×C8C22×D11312yes—
225C4⋊C4—21no—
226C4⋊C4—1131no—
227C4⋊C4—1132no—
228C4⋊C4—2261no—
229C4⋊C4—2261no—
230C4⋊C4—2261no—
231C4⋊C4—2261no—
232C4⋊C4—2261no—
233C4⋊C4C22×D11311yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—2261no—
236C22⋊C4—2261no—
237C22⋊C4—2262no—
238C22⋊C4—2262no—
239C22⋊C4—2261no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—1134no—
245C23—1131no—
246C23—1131no—
247C23—4521no—
248C23—4521no—
249C23D4×D11311yescharacteristic
250Q8—22no—
251Q8—4522no—
252Q8—2262no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—4524no—
257D4—4524no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—1131no—
262C2×C4—1131no—
263C2×C4—4521no—
264C2×C4—4521no—
265C2×C4—4522no—
266C2×C4—4522no—
267C2×C4—4522no—
268C2×C4—4521no—
269C2×C4—2261no—
270C2×C4—2262no—
271C2×C4—2261no—
272C2×C4—2262no—
273C2×C4—2262no—
274C2×C4D4×D11311yescharacteristic
275C2×C4C23×D11311yescharacteristic
276C8—22no—
277C8—2262no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—1132no—
285C22—1132no—
286C22—4524no—
287C22—2261no—
288C22—2261no—
289C22—2261no—
290C22—2261no—
291C22SD16×D11312yes—
292C22C2×D4×D11311yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—4522no—
297C4—4521no—
298C4—4521no—
299C4—2261no—
300C4—2261no—
301C4#1681366712yes—
302C2—22no—
303C2—82no—
304C2—2262no—
305C2—2262no—
306C2#2029915811yescharacteristic
307C2C2×SD16×D11311yescharacteristic
308C2#2029939411yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#450×C113C2
socleC22×C113C22×D4
prime radicalC2×#16813593C2
hypercenter#450D113
hypocenterC113#3187
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D113C2semidirect
3. #20298929C2semidirect
4. #20298926C2semidirect
5. #20298915C2semidirect
6. #20298826C2semidirect
7. #20298824C2semidirect
8. #20298749C2semidirect
9. #20298735C2semidirect
10. C22⋊C8×D113C2semidirect
11. #450×C113C2semidirect
12. C2×Q8×D113C22semidirect
15. C2×#14268635C22semidirect
16. C2×C8×D113C22semidirect
17. #16813602C22semidirect
21. #16813598C22semidirect
28. C4⋊C4×D113C22semidirect
31. #16813554C22semidirect
32. #16813553C22semidirect
33. #16813547C22semidirect
34. #16813546C22semidirect
35. #16813543C22semidirect
37. #16813534C22semidirect
39. C2×SD16×C113C22semidirect
41. #102×C113C22semidirect
92. C22×D113SD16semidirect
155. C113#3187semidirect
164. #450D113semidirect
172. C2×SD16C2×D113semidirect
194. #102C2×D113semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1132
C2×#168135932
#202989292
#202989262
#202989152
#202988262
#202988242
#202987492
#202987352
C22⋊C8×D1132
#450×C1132
#3187113