Finite Group Databasebeta

← #28401330

Subgroups — #28401330

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×#16813597C211yesmaximal, characteristic
3#20298928C211yesmaximal, characteristic, complemented
4#20298918C212yesmaximal, complemented
5#20298830C212yesmaximal, complemented
6#20298828C212yesmaximal, complemented
7#20298751C211yesmaximal, characteristic, complemented
8#20298748C211yesmaximal, characteristic, complemented
9#20298738C212yesmaximal, complemented
10C22⋊C8×D113C211yesmaximal, characteristic, complemented
11#483×C113C211yesmaximal, characteristic, complemented
12C2×Q8×D113C2211yescharacteristic, complemented
13C22×D452—21no—
14C22×D452C2211yescharacteristic
15C22×C4×D113C2211yescharacteristic
16C2×#14268636C2212yescomplemented
17C2×#14268634C2212yescomplemented
18C2×C8×D113C2212yescomplemented
19#16813602C2211yescharacteristic, complemented
20#16813601C2211yescharacteristic
21#16813600—21no—
22#16813599—21no—
23#16813598C2211yescharacteristic, complemented
24#16813597—21no—
25#16813597—21no—
26#16813597C2212yes—
27#16813597C2212yes—
28C4⋊C4×D113—21no—
29C4⋊C4×D113C2211yescharacteristic, complemented
30C2×#14268624—21no—
31C22⋊C4×D113—21no—
32#16813553C2212yescomplemented
33#16813545C2212yescomplemented
34#16813544C2212yescomplemented
35#16813542C2212yescomplemented
36C22⋊#11744751C2211yescharacteristic
37#16813534C2212yescomplemented
38C22⋊Q8×C113C2211yescharacteristic
39C2×C4⋊C4×C113C2211yescharacteristic
40#106×C113C2212yescomplemented
41#103×C113C2212yescomplemented
42C22⋊C8×C113C2211yescharacteristic
43C23×D113—21no—
44C23×D113D411yescharacteristic
45C2×Dic226—21no—
46C2×Dic226C2311yescharacteristic
47Q8×D113—42no—
48C2×D452—22no—
49C2×D452—22no—
50C2×D452—21no—
51C2×D452—42no—
52C2×D452—44no—
53C2×D452C2312yes—
54C22×Dic113D411yescharacteristic
55C2×C4×D113—21no—
56C2×C4×D113—21no—
57C2×C4×D113—42no—
58C2×C4×D113D411yescharacteristic
59C2×C4×D113D411yescharacteristic
60C2×C4×D113C2311yescharacteristic
61C2×C4×D113C2311yescharacteristic
62#14268636—42no—
63#14268634—42no—
64C2×#11744751C2312yes—
65C8×D113—42no—
66C4⋊Dic113—21no—
67C4⋊Dic113C2311yescharacteristic
68#14268626—21no—
69#14268626—21no—
70#14268626—21no—
71C22⋊Dic113—21no—
72#14268624—21no—
73#14268624—21no—
74#14268624—22no—
75#14268624—22no—
76C2×Q8×C113C2311yescharacteristic
77C22×C4×C113—21no—
78C22×C4×C113C2311yescharacteristic
79C2×C8×C113C2312yes—
80C4⋊C4×C113—21no—
81C4⋊C4×C113—21no—
82C4⋊C4×C113C2311yescharacteristic
83C4⋊C4×C113C2312yes—
84C22⋊C4×C113—21no—
85C23×C113C2×D411yescharacteristic
86C22×D113—22no—
87C22×D113—41no—
88C22×D113—41no—
89C22×D113—42no—
90C22×D113—42no—
91C22×D113—42no—
92C22×D113SD1614yescomplemented
93C22×D113C2×D411yescharacteristic
94C22×D113C2×D411yescharacteristic
95Q8×C113—22no—
96C2×C4×C113—22no—
97C2×C4×C113—21no—
98C2×C4×C113—21no—
99C2×C4×C113—22no—
100C2×C4×C113—42no—
101C2×C4×C113C2×D411yescharacteristic
102C2×C4×C113C2411yescharacteristic
103C2×Dic113—21no—
104C2×Dic113—21no—
105C2×Dic113—41no—
106C2×Dic113C2×D411yescharacteristic
107C2×Dic113C2×D411yescharacteristic
108D452—22no—
109D452—22no—
110D452—42no—
111D452—44no—
112D452—44no—
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×C113C2×SD1612yes—
126C22×C113C22×D411yescharacteristic
127C2×D113—21no—
128C2×D113—21no—
129C2×D113—21no—
130C2×D113—21no—
131C2×D113—44no—
132C2×D113—42no—
133C2×D113—42no—
134C2×D113—42no—
135C2×D113—84no—
136C2×D113C2×SD1612yes—
137C2×D113C2×SD1612yes—
138C4×C113—21no—
139C4×C113—41no—
140C4×C113—42no—
141C4×C113—41no—
142C4×C113H(5,𝔽2)12yes—
143Dic113—21no—
144Dic113—21no—
145Dic113—41no—
146Dic113—41no—
147C2×C113—22no—
148C2×C113#53411yescharacteristic
149C2×C113C22×SD1611yescharacteristic
150C2×C113#57511yescharacteristic
151D113—22no—
152D113—22no—
153D113—82no—
154#3187—1131nomaximal, Sylow
155C113#318711yescharacteristic, complemented, Sylow
156C2×C22⋊Q8—1131no—
157C2×#121—1131no—
158#483—1131no—
159#483D11311yescharacteristic, complemented
160#481—1132no—
161#465—1132no—
162#465—1132no—
163#450—1131no—
164#450—1131no—
165#448—1132no—
166C2×C22⋊C8—1131no—
167C22×Q8—1131no—
168C22×D4—1131no—
169C22×D4—2261no—
170C23×C4—1131no—
171C2×SD16—1132no—
172C2×SD16—1132no—
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—2261no—
186C2×C4⋊C4C2×D11311yescharacteristic
187C2×C22⋊C4—2261no—
188C2×C22⋊C4—2261no—
189#106—1132no—
190#106C2×D11312yescomplemented
191#103—1132no—
192#103C2×D11312yescomplemented
193#102—1132no—
194#102—1132no—
195C22⋊C8—1131no—
196C22⋊C8—1132no—
197C22⋊C8C2×D11311yescharacteristic
198C24—1131no—
199C24—2261no—
200C2×Q8—1131no—
201C2×Q8—4522no—
202C2×Q8—2261no—
203C2×Q8C22×D11311yescharacteristic
204C2×D4—1132no—
205C2×D4—4522no—
206C2×D4—4524no—
207C2×D4—2262no—
208C2×D4—2262no—
209C2×D4—2261no—
210C22×C4—21no—
211C22×C4—1131no—
212C22×C4—1131no—
213C22×C4—1131no—
214C22×C4—1131no—
215C22×C4—1131no—
216C22×C4—4522no—
217C22×C4—2261no—
218C22×C4—2261no—
219C22×C4C22×D11311yescharacteristic
220SD16—4522no—
221SD16—4522no—
222C2×C8—1132no—
223C2×C8—4522no—
224C2×C8C22×D11312yes—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—1131no—
228C4⋊C4—2261no—
229C4⋊C4—2261no—
230C4⋊C4—2261no—
231C4⋊C4—2261no—
232C4⋊C4C22×D11311yescharacteristic
233C4⋊C4C22×D11312yes—
234C22⋊C4—21no—
235C22⋊C4—2262no—
236C22⋊C4—2262no—
237C22⋊C4—2261no—
238C22⋊C4—2261no—
239C22⋊C4—2261no—
240C23—1134no—
241C23—1131no—
242C23—1131no—
243C23—4522no—
244C23—4522no—
245C23—4521no—
246C23—4521no—
247C23—4522no—
248C23—2262no—
249C23D4×D11311yescharacteristic
250Q8—22no—
251Q8—4522no—
252Q8—2262no—
253D4—4522no—
254D4—4524no—
255D4—4524no—
256D4—2262no—
257D4—2262no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—1131no—
264C2×C4—1131no—
265C2×C4—4521no—
266C2×C4—4521no—
267C2×C4—4522no—
268C2×C4—4522no—
269C2×C4—4521no—
270C2×C4—2261no—
271C2×C4—2261no—
272C2×C4—2262no—
273C2×C4—2262no—
274C2×C4D4×D11311yescharacteristic
275C2×C4C23×D11311yescharacteristic
276C8—22no—
277C8—2262no—
278C22—21no—
279C22—21no—
280C22—1132no—
281C22—1132no—
282C22—9044no—
283C22—4524no—
284C22—4522no—
285C22—4522no—
286C22—4522no—
287C22—2261no—
288C22—2261no—
289C22—2261no—
290C22—2261no—
291C22SD16×D11312yes—
292C22C2×D4×D11311yescharacteristic
293C4—21no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—4521no—
298C4—4521no—
299C4—2261no—
300C4—2261no—
301C4#1681366812yes—
302C2—22no—
303C2—9042no—
304C2—2262no—
305C2—2262no—
306C2#2029915911yescharacteristic
307C2C2×SD16×D11311yescharacteristic
308C2#2029939711yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#483×C113C2
socleC22×C113C22×D4
prime radicalC2×#16813597C2
hypercenter#483D113
hypocenterC113#3187
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D113C2semidirect
3. #20298928C2semidirect
4. #20298918C2semidirect
5. #20298830C2semidirect
6. #20298828C2semidirect
7. #20298751C2semidirect
8. #20298748C2semidirect
9. #20298738C2semidirect
10. C22⋊C8×D113C2semidirect
11. #483×C113C2semidirect
12. C2×Q8×D113C22semidirect
16. C2×#14268636C22semidirect
17. C2×#14268634C22semidirect
18. C2×C8×D113C22semidirect
19. #16813602C22semidirect
23. #16813598C22semidirect
29. C4⋊C4×D113C22semidirect
32. #16813553C22semidirect
33. #16813545C22semidirect
34. #16813544C22semidirect
35. #16813542C22semidirect
37. #16813534C22semidirect
40. #106×C113C22semidirect
41. #103×C113C22semidirect
92. C22×D113SD16semidirect
155. C113#3187semidirect
159. #483D113semidirect
190. #106C2×D113semidirect
192. #103C2×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×#168135972
#202989282
#202989182
#202988302
#202988282
#202987512
#202987482
#202987382
C22⋊C8×D1132
#483×C1132
#3187113