Finite Group Databasebeta

← #30648831

Subgroups — #30648831

Order 16256.

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

normal subgroups only

This group has 444 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
1#22550431C211yesmaximal, characteristic, complemented
2C4×D4×D127C211yesmaximal, characteristic
3#22550213C212yesmaximal, complemented
4#22550153C211yesmaximal, characteristic, complemented
5#22550082C211yesmaximal, characteristic, complemented
6#22550081C211yesmaximal, characteristic, complemented
7#22550028C212yesmaximal
8#22549975C212yesmaximal, complemented
9#22549973C212yesmaximal, complemented
10#22549932C211yesmaximal, characteristic, complemented
11#474×C127C211yesmaximal, characteristic, complemented
12C2×D4×D127C2211yescharacteristic
13C22×C4×D127—21no—
14C2×#14523237C2212yes—
15C2×#14523236C2212yes—
16#16861209C2211yescharacteristic, complemented
17#16861200—21no—
18#16861189C2212yes—
19#16861185—21no—
20Q8×Dic127—21no—
21#16861174C2212yescomplemented
22C4×D508C2211yescharacteristic
23D4×Dic127C2211yescharacteristic
24#16861171C2211yescharacteristic
25C4×C22⋊D127—21no—
26(C22⋊D127)⋊C4—21no—
27#16861167—21no—
28C4⋊C4×D127C2211yescharacteristic
29C22⋊C4×D127—21no—
30C42×D127C2211yescharacteristic
31#16861142C2212yescomplemented
32C4⋊(C127⋊C8)C2211yescharacteristic, complemented
33#16861134C2212yescomplemented
34#16861132C2212yescomplemented
35#16861129C2212yes—
36(C2×D127)⋊C8C2212yes—
37#16861119C2212yescomplemented
38C4⋊Q8×C127C2211yescharacteristic, complemented
39C4×D4×C127C2211yescharacteristic
40#107×C127C2212yescomplemented
41C4⋊C8×C127C2211yescharacteristic, complemented
42#102×C127C2212yes—
43C23×D127—21no—
44C2×Dic254C2312yes—
45C2×D508C2311yescharacteristic
46C2×C22⋊D127—21no—
47D4×D127—22no—
48D4×D127—22no—
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#14523237—24no—
58#14523236—24no—
59C2×C127⋊C8C2312yes—
60C4⋊Dic127—22no—
61C4⋊Dic127—21no—
62C4⋊Dic127—21no—
63C4⋊Dic127C2311yescharacteristic
64Dic127⋊C4—22no—
65Dic127⋊C4—21no—
66C22⋊Dic127—21no—
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4—21no—
69(C2×D127)⋊C4—21no—
70C4×Dic127—22no—
71C4×Dic127D411yescharacteristic
72C4×Dic127D411yescharacteristic
73C4×Dic127C2311yescharacteristic
74C2×Q8×C127—21no—
75C2×D4×C127C2311yescharacteristic
76C22×C4×C127—21no—
77C2×C8×C127C2312yes—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2312yes—
80C4⋊C4×C127C2311yescharacteristic
81C22⋊C4×C127—21no—
82C42×C127C2311yescharacteristic
83C22×D127—24no—
84C22×D127—21no—
85C22×D127—42no—
86C22×D127—42no—
87C22×D127C2×D411yescharacteristic
88Dic254—22no—
89Dic254—22no—
90D508—21no—
91D508#4112yes—
92C22⋊D127—24no—
93C2×Dic127—22no—
94C2×Dic127—21no—
95C2×Dic127—42no—
96C2×Dic127C2×D411yescharacteristic
97C2×Dic127C2×D411yescharacteristic
98C2×Dic127C2×D411yescharacteristic
99C4×D127—22no—
100C4×D127—22no—
101C4×D127—21no—
102C4×D127—21no—
103C4×D127—42no—
104C4×D127—42no—
105C4×D127—42no—
106C127⋊C8—22no—
107C23×C127—21no—
108Q8×C127—42no—
109D4×C127—21no—
110D4×C127#4112yes—
111C2×C4×C127—21no—
112C2×C4×C127—22no—
113C2×C4×C127—42no—
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2411yescharacteristic
117C8×C127—22no—
118C22×C127—22no—
119C22×C127—41no—
120C22×C127—41no—
121C22×C127C22×D411yescharacteristic
122C2×D127—21no—
123C2×D127—21no—
124C2×D127—22no—
125C2×D127—48no—
126C2×D127—41no—
127C2×D127—41no—
128C2×D127#13712yes—
129C4×C127—21no—
130C4×C127—41no—
131C4×C127—42no—
132C4×C127AGL(1,ℤ/8)12yes—
133C4×C127C2×#4111yescharacteristic
134C4×C127C2×#4111yescharacteristic
135Dic127—21no—
136Dic127—21no—
137Dic127—21no—
138Dic127—42no—
139Dic127—41no—
140C2×C127—42no—
141C2×C127#52511yescharacteristic
142C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C127C2×#13711yescharacteristic
144D127—22no—
145D127—42no—
146#3110—1271nomaximal, Sylow
147C127#311011yescharacteristic, complemented, Sylow
148#533—1271no—
149C2×C4×D4—1271no—
150#484—1272no—
151#474—1271no—
152#474D12711yescharacteristic, complemented
153#461—1271no—
154#461—1271no—
155#450—1272no—
156#440—1272no—
157#440—1272no—
158#432—1271no—
159C22×D4—1271no—
160C23×C4—2541no—
161C2×SD16—1272no—
162C2×SD16—1272no—
163C4⋊Q8—1271no—
164C4⋊Q8C2×D12711yescharacteristic, complemented
165#125—2541no—
166C22⋊Q8—1272no—
167C22⋊Q8—2541no—
168C4×Q8—1272no—
169C4×Q8—2541no—
170C4×D4—1271no—
171C4×D4—1271no—
172C4×D4—1271no—
173C4×D4—2541no—
174C4×D4—2541no—
175C4×D4C2×D12711yescharacteristic
176#117—2541no—
177C2×C4⋊C4—1271no—
178C2×C22⋊C4—2541no—
179C2×C42—1271no—
180#107—1272no—
181#107C2×D12712yescomplemented
182C4⋊C8—1271no—
183C4⋊C8C2×D12711yescharacteristic, complemented
184#103—1272no—
185#103—1272no—
186#102—1272no—
187#102C2×D12712yes—
188C22⋊C8—1272no—
189#97—1272no—
190C24—2541no—
191C2×Q8—21no—
192C2×Q8—1272no—
193C2×D4—1271no—
194C2×D4—2541no—
195C2×D4—2542no—
196C2×D4—2542no—
197C2×D4C22×D12711yescharacteristic
198C22×C4—21no—
199C22×C4—1271no—
200C22×C4—1271no—
201C22×C4—1271no—
202C22×C4—5084no—
203C22×C4—2541no—
204C22×C4—2541no—
205C22×C4—2541no—
206C22×C4—2541no—
207SD16—2544no—
208SD16—2544no—
209C2×C8—1272no—
210C2×C8C22×D12712yes—
211C4⋊C4—21no—
212C4⋊C4—1271no—
213C4⋊C4—2541no—
214C4⋊C4—2542no—
215C4⋊C4—2542no—
216C4⋊C4—2541no—
217C4⋊C4—2541no—
218C4⋊C4C22×D12712yes—
219C4⋊C4C22×D12711yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—2541no—
222C22⋊C4—2542no—
223C22⋊C4—2541no—
224C22⋊C4—2541no—
225C42—1271no—
226C42—1271no—
227C42—1271no—
228C42—2542no—
229C42C22×D12711yescharacteristic
230C23—21no—
231C23—1271no—
232C23—5082no—
233C23—5082no—
234C23—2544no—
235C23—2541no—
236Q8—42no—
237Q8—2542no—
238Q8—2542no—
239D4—21no—
240D4—1272no—
241D4—2544no—
242D4—2541no—
243D4#1452324912yes—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—1271no—
248C2×C4—1271no—
249C2×C4—1271no—
250C2×C4—5082no—
251C2×C4—5082no—
252C2×C4—5082no—
253C2×C4—5082no—
254C2×C4—2542no—
255C2×C4—2542no—
256C2×C4—2541no—
257C2×C4—2542no—
258C2×C4—2541no—
259C2×C4—2541no—
260C2×C4D4×D12711yescharacteristic
261C2×C4D4×D12711yescharacteristic
262C2×C4C23×D12711yescharacteristic
263C8—22no—
264C8—2542no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—1272no—
269C22—5088no—
270C22—5081no—
271C22—5081no—
272C22—2542no—
273C22—2541no—
274C22—2541no—
275C22C2×D4×D12711yescharacteristic
276C4—21no—
277C4—41no—
278C4—42no—
279C4—5082no—
280C4—5081no—
281C4—2541no—
282C4—2541no—
283C4—2541no—
284C4#1686123312yes—
285C4#41×D12711yescharacteristic
286C4C2×#1452324911yescharacteristic
287C2—42no—
288C2—5082no—
289C2—2542no—
290C2#2255038911yescharacteristic
291C2C2×#1686123311yescharacteristic
292C2#137×D12711yescharacteristic
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#3110
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. #22550213C2semidirect
4. #22550153C2semidirect
5. #22550082C2semidirect
6. #22550081C2semidirect
8. #22549975C2semidirect
9. #22549973C2semidirect
10. #22549932C2semidirect
11. #474×C127C2semidirect
16. #16861209C22semidirect
21. #16861174C22semidirect
31. #16861142C22semidirect
32. C4⋊(C127⋊C8)C22semidirect
33. #16861134C22semidirect
34. #16861132C22semidirect
37. #16861119C22semidirect
38. C4⋊Q8×C127C22semidirect
40. #107×C127C22semidirect
41. C4⋊C8×C127C22semidirect
147. C127#3110semidirect
152. #474D127semidirect
164. C4⋊Q8C2×D127semidirect
181. #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
#225502132
#225501532
#225500822
#225500812
#225500282
#225499752
#225499732
#225499322
#474×C1272
#3110127