Finite Group Databasebeta

← #28402195

Subgroups — #28402195

Order 14464.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#16813649C212yesmaximal, complemented
2#20299270C211yesmaximal, characteristic, complemented
3D4×D452C211yesmaximal, characteristic, complemented
4#20298975C211yesmaximal, characteristic, complemented
5#20298852C212yesmaximal, complemented
6#20298798C211yesmaximal, characteristic, complemented
7#20298793C211yesmaximal, characteristic, complemented
8#20298746C212yesmaximal, complemented
9#20298733C212yesmaximal, complemented
10#20298677C211yesmaximal, characteristic, complemented
11C4×SD16×C113C211yesmaximal, characteristic, complemented
12C2×#14268648C2212yescomplemented
13C22×D452—21no—
14C2×D4×D113C2212yes—
15#16813649—28no—
16C2×#14268634C2212yescomplemented
17C2×D904C2212yescomplemented
18C2×D904C2212yescomplemented
19C2×#14268632C2212yes—
20C2×#14268630C2212yescomplemented
21#16813619—21no—
22#16813619C2211yescharacteristic
23#16813619C2211yescharacteristic, complemented
24#16813597—22no—
25#16813597C2212yes—
26#16813597C2212yescomplemented
27#16813596—21no—
28#16813592—22no—
29C4×D452—21no—
30C4×D452C2211yescharacteristic
31C4⋊#11744751C2211yescharacteristic, complemented
32#16813542C2212yescomplemented
33#16813542C2212yescomplemented
34#16813534C2212yescomplemented
35C2×SD16×C113C2211yescharacteristic
36C4×Q8×C113C2211yescharacteristic, complemented
37C4×D4×C113C2211yescharacteristic
38#106×C113C2211yescharacteristic, complemented
39#103×C113C2211yescharacteristic, complemented
40#102×C113C2211yescharacteristic
41C4×C8×C113C2211yescharacteristic, complemented
42C23×D113—22no—
43#14268648—24no—
44#14268648—42no—
45C2×D452—22no—
46C2×D452—22no—
47C2×D452—21no—
48C2×D452—21no—
49C2×D452—22no—
50C2×D452—21no—
51C2×D452—21no—
52C2×D452—44no—
53C2×D452D411yescharacteristic
54C2×D452C2312yes—
55C2×D452C2312yes—
56C2×C22⋊D113—22no—
57D4×D113—24no—
58D4×D113—42no—
59C2×C4×D113—22no—
60C2×C4×D113C2312yes—
61#14268634—24no—
62D904—24no—
63D904—42no—
64#14268632—24no—
65#14268630—24no—
66C2×#11744751C2312yes—
67C4⋊Dic113D411yescharacteristic
68#14268624—22no—
69#14268624—22no—
70#14268624D412yes—
71C2×Q8×C113C2311yescharacteristic
72C2×D4×C113C2311yescharacteristic
73C22×C4×C113—21no—
74SD16×C113D414yescomplemented
75C2×C8×C113C2311yescharacteristic
76C2×C8×C113C2311yescharacteristic
77C4⋊C4×C113—21no—
78C4⋊C4×C113C2311yescharacteristic
79C4⋊C4×C113C2311yescharacteristic
80C22⋊C4×C113—21no—
81C42×C113—21no—
82C42×C113C2311yescharacteristic
83C23×C113—21no—
84C22×D113—22no—
85C22×D113—22no—
86C22×D113—44no—
87C22×D113—44no—
88C22×D113—82no—
89C22×D113C2×D412yes—
90Q8×C113—21no—
91Q8×C113C2×D412yescomplemented
92D4×C113—21no—
93D4×C113C2×D412yes—
94C2×C4×C113—21no—
95C2×C4×C113—21no—
96C2×C4×C113—21no—
97C2×C4×C113—42no—
98C2×C4×C113C2×D411yescharacteristic
99C2×C4×C113C2×D411yescharacteristic
100C2×C4×C113C2411yescharacteristic
101C22⋊D113—44no—
102C2×Dic113C2×D412yes—
103D452—22no—
104D452—22no—
105D452—22no—
106D452—22no—
107D452—22no—
108D452—44no—
109D452—44no—
110D452—42no—
111D452—42no—
112D452—81no—
113D452—81no—
114D452—82no—
115C4×D113—22no—
116C4×D113—22no—
117C4×D113—44no—
118C8×C113—21no—
119C8×C113C2×D412yescomplemented
120#11744751—22no—
121C22×C113—22no—
122C22×C113—41no—
123C22×C113—41no—
124C22×C113C22×D411yescharacteristic
125C2×D113—22no—
126C2×D113—22no—
127C2×D113—22no—
128C2×D113—42no—
129C2×D113—42no—
130C2×D113—42no—
131C2×D113—42no—
132C2×D113—42no—
133C2×D113—42no—
134C2×D113—88no—
135C4×C113—22no—
136C4×C113—21no—
137C4×C113—41no—
138C4×C113—41no—
139C4×C113AGL(1,ℤ/8)12yescomplemented
140C4×C113C22×D411yescharacteristic
141C4×C113H(5,𝔽2)11yescharacteristic
142Dic113—22no—
143C2×C113—42no—
144C2×C113D4211yescharacteristic
145C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C113#57611yescharacteristic
147D113—42no—
148D113—82no—
149D113—82no—
150#3274—1131nomaximal, Sylow
151C113#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1132no—
153#550—1131no—
154D42—1131no—
155#493—1131no—
156#469—1132no—
157#460—1131no—
158#459—1131no—
159#449—1132no—
160#447—1132no—
161C4×SD16D11311yescharacteristic, complemented
162#436—1131no—
163C2×#41—1132no—
164C22×D4—1132no—
165C22×D4—2261no—
166AGL(1,ℤ/8)—2268no—
167C2×SD16—1132no—
168C2×SD16C2×D11311yescharacteristic
169C2×D8—1132no—
170C2×D8—1132no—
171C2×D8—1132no—
172C2×M16—1132no—
173#127—1131no—
174#127—1131no—
175#127—2261no—
176#121—1132no—
177#121—1132no—
178#121—2262no—
179#121—2261no—
180#120—2262no—
181C4×Q8C2×D11311yescharacteristic, complemented
182C4×D4—1131no—
183C4×D4—2261no—
184C4×D4C2×D11311yescharacteristic
185#106C2×D11311yescharacteristic, complemented
186C4⋊C8—1131no—
187#103C2×D11311yescharacteristic, complemented
188#102—1132no—
189#102—1132no—
190#102C2×D11311yescharacteristic
191C22⋊C8—1132no—
192C4×C8C2×D11311yescharacteristic, complemented
193C24—2262no—
194#41—4522no—
195#41—2264no—
196C2×Q8C22×D11311yescharacteristic
197C2×D4—1131no—
198C2×D4—1132no—
199C2×D4—1132no—
200C2×D4—4524no—
201C2×D4—4522no—
202C2×D4—2262no—
203C2×D4—2262no—
204C2×D4—2262no—
205C2×D4—2262no—
206C2×D4—2264no—
207C2×D4—2261no—
208C2×D4—2261no—
209C2×D4—2261no—
210C2×D4—2261no—
211C2×D4C22×D11311yescharacteristic
212C22×C4—21no—
213C22×C4—1132no—
214C22×C4—2262no—
215SD16—2264no—
216SD16D45214yescomplemented
217D8—4522no—
218D8—2264no—
219D8—2264no—
220M16—2264no—
221C2×C8—1132no—
222C2×C8C22×D11311yescharacteristic
223C2×C8C22×D11311yescharacteristic
224C4⋊C4—21no—
225C4⋊C4—1131no—
226C4⋊C4C22×D11311yescharacteristic
227C4⋊C4C22×D11311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—1132no—
230C22⋊C4—2262no—
231C22⋊C4—2262no—
232C42—21no—
233C42C22×D11311yescharacteristic
234C23—21no—
235C23—1132no—
236C23—9042no—
237C23—4524no—
238C23—4524no—
239C23—2262no—
240C23—2262no—
241Q8—21no—
242Q8C2×D45212yescomplemented
243D4—21no—
244D4—9041no—
245D4—9041no—
246D4—9042no—
247D4—4524no—
248D4—4524no—
249D4—4524no—
250D4—4522no—
251D4—4522no—
252D4—2262no—
253D4—2262no—
254D4—2262no—
255D4—2262no—
256D4—2262no—
257D4C2×D45212yes—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—1132no—
263C2×C4—4524no—
264C2×C4—2262no—
265C2×C4—2262no—
266C2×C4D4×D11311yescharacteristic
267C2×C4D4×D11311yescharacteristic
268C2×C4C23×D11311yescharacteristic
269C8—21no—
270C8—2262no—
271C8C2×D45212yescomplemented
272C22—22no—
273C22—41no—
274C22—41no—
275C22—9048no—
276C22—4522no—
277C22—4522no—
278C22—4522no—
279C22—4522no—
280C22—4522no—
281C22—4522no—
282C22—2262no—
283C22—2262no—
284C22—2262no—
285C22C2×D4×D11311yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—2262no—
291C4#1681364912yescomplemented
292C4C22×D45211yescharacteristic
293C4#1681366811yescharacteristic
294C2—42no—
295C2—9042no—
296C2—9042no—
297C2—4522no—
298C2D4×D45211yescharacteristic
299C2C2×#1681364911yescharacteristic
300C2#2029940011yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC4×SD16×C113C2
socleC22×C113C22×D4
hypercenterC4×SD16D113
hypocenterC113#3274
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16813649C2semidirect
2. #20299270C2semidirect
3. D4×D452C2semidirect
4. #20298975C2semidirect
5. #20298852C2semidirect
6. #20298798C2semidirect
7. #20298793C2semidirect
8. #20298746C2semidirect
9. #20298733C2semidirect
10. #20298677C2semidirect
11. C4×SD16×C113C2semidirect
12. C2×#14268648C22semidirect
16. C2×#14268634C22semidirect
17. C2×D904C22semidirect
18. C2×D904C22semidirect
20. C2×#14268630C22semidirect
23. #16813619C22semidirect
26. #16813597C22semidirect
31. C4⋊#11744751C22semidirect
32. #16813542C22semidirect
33. #16813542C22semidirect
34. #16813534C22semidirect
36. C4×Q8×C113C22semidirect
38. #106×C113C22semidirect
39. #103×C113C22semidirect
41. C4×C8×C113C22semidirect
74. SD16×C113D4semidirect
91. Q8×C113C2×D4semidirect
119. C8×C113C2×D4semidirect
139. C4×C113AGL(1,ℤ/8)semidirect
151. C113#3274semidirect
161. C4×SD16D113semidirect
181. C4×Q8C2×D113semidirect
185. #106C2×D113semidirect
187. #103C2×D113semidirect
192. C4×C8C2×D113semidirect
216. SD16D452semidirect
242. Q8C2×D452semidirect
271. C8C2×D452semidirect
291. C4#16813649semidirect

Maximal subgroups

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

subgroupindex
C2×#168136492
#202992702
D4×D4522
#202989752
#202988522
#202987982
#202987932
#202987462
#202987332
#202986772
C4×SD16×C1132
#3274113