Finite Group Databasebeta

← #30648522

Subgroups — #30648522

Order 16256.

440 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 131 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#22550431C211yesmaximal, characteristic, complemented
2C4×D4×D127C211yesmaximal, characteristic
3#22550179C212yesmaximal, complemented
4#22550178C212yesmaximal, complemented
5#22550167C211yesmaximal, characteristic, complemented
6#22550166C211yesmaximal, characteristic, complemented
7#22550153C211yesmaximal, characteristic, complemented
8#22549917C212yesmaximal, complemented
9#22549898C211yesmaximal, characteristic, complemented
10#22549879C212yesmaximal
11#474×C127C211yesmaximal, characteristic, complemented
12C2×D4×D127C2211yescharacteristic
13C22×C4×D127—21no—
14C2×#14523231C2212yes—
15#16861209C2211yescharacteristic, complemented
16#16861200C2212yescomplemented
17#16861189—21no—
18#16861185—21no—
19Q8×Dic127—21no—
20#16861174—21no—
21C4×D508C2211yescharacteristic
22D4×Dic127C2211yescharacteristic
23#16861171C2211yescharacteristic
24C4×C22⋊D127—21no—
25(C22⋊D127)⋊C4—21no—
26#16861167C2212yes—
27C4⋊C4×D127C2211yescharacteristic
28C22⋊C4×D127—21no—
29C42×D127C2211yescharacteristic
30#16861142C2212yescomplemented
31#16861141C2212yescomplemented
32#16861140C2212yescomplemented
33C4⋊(C127⋊C8)C2211yescharacteristic, complemented
34Dic127⋊C8C2212yescomplemented
35#16861131C2212yes—
36#16861130C2212yes—
37#16861129C2212yes—
38C4⋊Q8×C127C2211yescharacteristic, complemented
39C4×D4×C127C2211yescharacteristic
40#107×C127C2212yescomplemented
41C4⋊C8×C127C2211yescharacteristic, complemented
42#102×C127C2212yes—
43C23×D127—21no—
44C2×Dic254—21no—
45C2×D508C2311yescharacteristic
46C2×C22⋊D127—21no—
47D4×D127—22no—
48D4×D127Q814yescomplemented
49C22×Dic127—21no—
50C2×C4×D127—21no—
51C2×C4×D127—21no—
52C2×C4×D127—21no—
53C2×C4×D127—44no—
54C2×C4×D127D411yescharacteristic
55C2×C4×D127D411yescharacteristic
56C2×C4×D127C2311yescharacteristic
57#14523231—24no—
58C2×C127⋊C8C2312yes—
59C4⋊Dic127—22no—
60C4⋊Dic127—21no—
61C4⋊Dic127C2312yes—
62C4⋊Dic127C2311yescharacteristic
63Dic127⋊C4—22no—
64Dic127⋊C4—21no—
65C22⋊Dic127—21no—
66(C2×D127)⋊C4—22no—
67(C2×D127)⋊C4—21no—
68(C2×D127)⋊C4—21no—
69C4×Dic127—22no—
70C4×Dic127D411yescharacteristic
71C4×Dic127D411yescharacteristic
72C4×Dic127C2311yescharacteristic
73C2×Q8×C127—21no—
74C2×D4×C127C2311yescharacteristic
75C22×C4×C127—21no—
76C2×C8×C127C2312yes—
77C4⋊C4×C127—21no—
78C4⋊C4×C127C2312yes—
79C4⋊C4×C127C2311yescharacteristic
80C22⋊C4×C127—21no—
81C42×C127C2311yescharacteristic
82C22×D127—21no—
83C22×D127—24no—
84C22×D127—42no—
85C22×D127—42no—
86C22×D127C2×D411yescharacteristic
87Dic254—42no—
88D508—21no—
89D508C2×Q812yes—
90C22⋊D127—24no—
91C2×Dic127—22no—
92C2×Dic127—21no—
93C2×Dic127—42no—
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C2×Dic127C2×D411yescharacteristic
97C4×D127—22no—
98C4×D127—41no—
99C4×D127—42no—
100C4×D127—42no—
101C4×D127—42no—
102C4×D127C2×Q812yes—
103C4×D127#4112yes—
104C127⋊C8—22no—
105C23×C127—21no—
106Q8×C127—42no—
107D4×C127—21no—
108D4×C127C2×Q812yes—
109C2×C4×C127—21no—
110C2×C4×C127—22no—
111C2×C4×C127—42no—
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2411yescharacteristic
115C8×C127—22no—
116C22×C127—22no—
117C22×C127—41no—
118C22×C127—41no—
119C22×C127C22×D411yescharacteristic
120C2×D127—21no—
121C2×D127—21no—
122C2×D127—22no—
123C2×D127—48no—
124C2×D127—41no—
125C2×D127—41no—
126C2×D127C22⋊Q812yes—
127C4×C127—21no—
128C4×C127—41no—
129C4×C127—42no—
130C4×C127AGL(1,ℤ/8)12yes—
131C4×C127C22×Q811yescharacteristic
132C4×C127C2×#4111yescharacteristic
133Dic127—21no—
134Dic127—21no—
135Dic127—41no—
136Dic127—42no—
137Dic127C22⋊Q812yes—
138C2×C127—42no—
139C2×C127C2×C22⋊Q811yescharacteristic
140C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
141C2×C127#57511yescharacteristic
142D127—22no—
143D127—42no—
144#3077—1271nomaximal, Sylow
145C127#307711yescharacteristic, complemented, Sylow
146#533—1271no—
147C2×C4×D4—1271no—
148#478—1272no—
149#478—1272no—
150#476—1271no—
151#476—1271no—
152#474—1271no—
153#474D12711yescharacteristic, complemented
154#428—1272no—
155#423—1271no—
156#418—1272no—
157C22×D4—1271no—
158C23×C4—2541no—
159C2×M16—1272no—
160C4⋊Q8—1271no—
161C4⋊Q8C2×D12711yescharacteristic, complemented
162#125—1272no—
163C22⋊Q8—2541no—
164C22⋊Q8—2541no—
165C4×Q8—2541no—
166C4×Q8—2541no—
167C4×D4—1271no—
168C4×D4—1271no—
169C4×D4—1271no—
170C4×D4—2541no—
171C4×D4—2541no—
172C4×D4C2×D12711yescharacteristic
173#117—1272no—
174C2×C4⋊C4—1271no—
175C2×C22⋊C4—2541no—
176C2×C42—1271no—
177#107—1272no—
178#107C2×D12712yescomplemented
179#106—1272no—
180#106—1272no—
181C4⋊C8—1272no—
182C4⋊C8—1271no—
183C4⋊C8C2×D12711yescharacteristic, complemented
184#102—1272no—
185#102—1272no—
186#102—1272no—
187#102C2×D12712yes—
188C24—2541no—
189C2×Q8—21no—
190C2×Q8—2541no—
191C2×D4—1274no—
192C2×D4—1271no—
193C2×D4—2542no—
194C2×D4—2541no—
195C2×D4C22×D12711yescharacteristic
196C22×C4—21no—
197C22×C4—1271no—
198C22×C4—1271no—
199C22×C4—1271no—
200C22×C4—5084no—
201C22×C4—2541no—
202C22×C4—2541no—
203C22×C4—2541no—
204C22×C4—2541no—
205M16—2544no—
206C2×C8—1272no—
207C2×C8C22×D12712yes—
208C4⋊C4—21no—
209C4⋊C4—1272no—
210C4⋊C4—1271no—
211C4⋊C4—2541no—
212C4⋊C4—2542no—
213C4⋊C4—2542no—
214C4⋊C4—2541no—
215C4⋊C4C22×D12712yes—
216C4⋊C4C22×D12711yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—2541no—
219C22⋊C4—2542no—
220C22⋊C4—2541no—
221C22⋊C4—2541no—
222C42—1271no—
223C42—1271no—
224C42—1271no—
225C42—2542no—
226C42C22×D12711yescharacteristic
227C23—21no—
228C23—1271no—
229C23—5082no—
230C23—5082no—
231C23—2544no—
232C23—2541no—
233Q8—42no—
234Q8—5082no—
235D4—21no—
236D4—1272no—
237D4—2544no—
238D4—2541no—
239D4Q8×D12712yes—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—1271no—
244C2×C4—1272no—
245C2×C4—1272no—
246C2×C4—1271no—
247C2×C4—1271no—
248C2×C4—5081no—
249C2×C4—5082no—
250C2×C4—5082no—
251C2×C4—5082no—
252C2×C4—5082no—
253C2×C4—2542no—
254C2×C4—2541no—
255C2×C4—2542no—
256C2×C4D4×D12711yescharacteristic
257C2×C4D4×D12711yescharacteristic
258C2×C4C23×D12711yescharacteristic
259C8—22no—
260C8—2542no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—1272no—
265C22—5088no—
266C22—5081no—
267C22—5081no—
268C22—2542no—
269C22—2541no—
270C22—2541no—
271C22C2×D4×D12711yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—1272no—
276C4—5081no—
277C4—5082no—
278C4—2541no—
279C4—2541no—
280C4#1686123312yes—
281C4C2×Q8×D12711yescharacteristic
282C4#41×D12711yescharacteristic
283C2—42no—
284C2—5082no—
285C2—2542no—
286C2C22⋊Q8×D12711yescharacteristic
287C2C2×#1686123311yescharacteristic
288C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#474×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×D127C22
hypercenter#474D127
hypocenterC127#3077
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #22550431C2semidirect
3. #22550179C2semidirect
4. #22550178C2semidirect
5. #22550167C2semidirect
6. #22550166C2semidirect
7. #22550153C2semidirect
8. #22549917C2semidirect
9. #22549898C2semidirect
11. #474×C127C2semidirect
15. #16861209C22semidirect
16. #16861200C22semidirect
30. #16861142C22semidirect
31. #16861141C22semidirect
32. #16861140C22semidirect
33. C4⋊(C127⋊C8)C22semidirect
34. Dic127⋊C8C22semidirect
38. C4⋊Q8×C127C22semidirect
40. #107×C127C22semidirect
41. C4⋊C8×C127C22semidirect
48. D4×D127Q8semidirect
145. C127#3077semidirect
153. #474D127semidirect
161. C4⋊Q8C2×D127semidirect
178. #107C2×D127semidirect
183. C4⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
C4×D4×D1272
#225501792
#225501782
#225501672
#225501662
#225501532
#225499172
#225498982
#225498792
#474×C1272
#3077127