Finite Group Databasebeta

← #30648511

Subgroups — #30648511

Order 16256.

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

normal subgroups only

This group has 388 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×#14523247C211yesmaximal, characteristic
3#22550182C212yesmaximal, complemented
4#22550174C212yesmaximal, complemented
5#22550172C211yesmaximal, characteristic, complemented
6#22550166C211yesmaximal, characteristic, complemented
7#22550158C211yesmaximal, characteristic, complemented
8#22549913C212yesmaximal, complemented
9C4⋊C8×D127C211yesmaximal, characteristic, complemented
10#22549869C212yesmaximal
11#474×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2211yescharacteristic
13C2×C8×D127C2212yes—
14#16861209C2211yescharacteristic
15#16861200C2212yes—
16#16861189—21no—
17#16861185—21no—
18Q8×Dic127—21no—
19C4×Dic254C2211yescharacteristic
20#16861174—21no—
21#16861174C2211yescharacteristic
22D4×Dic127C2211yescharacteristic
23C4×C22⋊D127—21no—
24(C22⋊D127)⋊C4—21no—
25#16861167C2211yescharacteristic
26#16861167C2212yes—
27#16861165—21no—
28C2×C4×Dic127—21no—
29C42×D127C2211yescharacteristic
30#16861143C2212yes—
31#16861140C2212yes—
32#16861140C2212yes—
33C4⋊(C127⋊C8)C2211yescharacteristic
34Dic127⋊C8C2212yes—
35#16861134C2212yes—
36#16861132C2212yes—
37#16861130C2212yes—
38C4⋊Q8×C127C2211yescharacteristic
39C4×D4×C127C2211yescharacteristic
40#107×C127C2212yes—
41C4⋊C8×C127C2211yescharacteristic
42#102×C127C2212yes—
43#14523247—22no—
44#14523247Q814yescomplemented
45C2×Dic254—21no—
46C2×Dic254C2311yescharacteristic
47C2×C22⋊D127—21no—
48C22×Dic127—21no—
49C22×Dic127—21no—
50C2×C4×D127D411yescharacteristic
51C2×C4×D127D411yescharacteristic
52C2×C4×D127C2311yescharacteristic
53C2×C127⋊C8C2312yes—
54C8×D127—24no—
55C4⋊Dic127—22no—
56C4⋊Dic127—21no—
57C4⋊Dic127C2312yes—
58C4⋊Dic127C2311yescharacteristic
59Dic127⋊C4—22no—
60Dic127⋊C4—21no—
61Dic127⋊C4—21no—
62C22⋊Dic127—21no—
63(C2×D127)⋊C4—22no—
64(C2×D127)⋊C4—21no—
65C4×Dic127—21no—
66C4×Dic127—21no—
67C4×Dic127—22no—
68C4×Dic127—21no—
69C4×Dic127D411yescharacteristic
70C4×Dic127D411yescharacteristic
71C4×Dic127C2311yescharacteristic
72C2×Q8×C127—21no—
73C2×D4×C127C2311yescharacteristic
74C22×C4×C127—21no—
75C2×C8×C127C2312yes—
76C4⋊C4×C127—21no—
77C4⋊C4×C127C2312yes—
78C4⋊C4×C127C2311yescharacteristic
79C22⋊C4×C127—21no—
80C42×C127C2311yescharacteristic
81C22×D127C2×D411yescharacteristic
82Dic254—21no—
83Dic254—42no—
84Dic254C2×Q812yes—
85C22⋊D127—24no—
86C2×Dic127—21no—
87C2×Dic127—24no—
88C2×Dic127—21no—
89C2×Dic127—22no—
90C2×Dic127—42no—
91C2×Dic127C2×D411yescharacteristic
92C2×Dic127C2×D411yescharacteristic
93C2×Dic127C2×D411yescharacteristic
94C4×D127—22no—
95C4×D127—41no—
96C4×D127C2×Q812yes—
97C4×D127#4112yes—
98C127⋊C8—22no—
99C23×C127—21no—
100Q8×C127—42no—
101D4×C127—21no—
102D4×C127C2×Q812yes—
103C2×C4×C127—21no—
104C2×C4×C127—22no—
105C2×C4×C127—42no—
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2411yescharacteristic
109C8×C127—22no—
110C22×C127—22no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127C22×D411yescharacteristic
114C2×D127—21no—
115C2×D127—21no—
116C2×D127C22⋊Q812yes—
117C4×C127—21no—
118C4×C127—41no—
119C4×C127—42no—
120C4×C127#13512yes—
121C4×C127C22×Q811yescharacteristic
122C4×C127C2×#4111yescharacteristic
123Dic127—22no—
124Dic127—21no—
125Dic127—21no—
126Dic127—41no—
127Dic127—42no—
128Dic127C22⋊Q812yes—
129C2×C127—42no—
130C2×C127C2×C22⋊Q811yescharacteristic
131C2×C127C2×#13511yescharacteristic
132C2×C127#57511yescharacteristic
133D127—22no—
134#3076—1271nomaximal, Sylow
135C127#307611yescharacteristic, complemented, Sylow
136#533—1271no—
137C4×#41—1271no—
138#479—1272no—
139#478—1272no—
140#477—1271no—
141#476—1271no—
142#475—1271no—
143#474D12711yescharacteristic, complemented
144#427—1272no—
145C2×C4⋊C8—1271no—
146#416—1272no—
147C2×#41—1271no—
148C22×C8—1272no—
149C4⋊Q8—1271no—
150C4⋊Q8C2×D12711yescharacteristic
151#125—1272no—
152C22⋊Q8—2541no—
153C22⋊Q8—2541no—
154C4×Q8—1271no—
155C4×Q8—1271no—
156C4×Q8—2541no—
157C4×Q8—2541no—
158C4×D4—1271no—
159C4×D4—2541no—
160C4×D4—2541no—
161C4×D4C2×D12711yescharacteristic
162#117—1271no—
163#117—1272no—
164#117—2541no—
165C2×C42—1271no—
166C2×C42—2541no—
167#107—1272no—
168#107C2×D12712yes—
169#106—1272no—
170#106—1272no—
171C4⋊C8—1272no—
172C4⋊C8—1271no—
173C4⋊C8C2×D12711yescharacteristic
174#103—1272no—
175#103—1272no—
176#102—1272no—
177#102C2×D12712yes—
178#41—1274no—
179#41—2542no—
180C2×Q8—21no—
181C2×Q8—1271no—
182C2×Q8—2541no—
183C2×D4—2541no—
184C2×D4C22×D12711yescharacteristic
185C22×C4—21no—
186C22×C4—1271no—
187C22×C4—1271no—
188C22×C4—1271no—
189C22×C4—2541no—
190C22×C4—2541no—
191C2×C8—1272no—
192C2×C8—2544no—
193C2×C8C22×D12712yes—
194C4⋊C4—21no—
195C4⋊C4—1272no—
196C4⋊C4—1271no—
197C4⋊C4—2541no—
198C4⋊C4—2542no—
199C4⋊C4—2542no—
200C4⋊C4—2541no—
201C4⋊C4—2541no—
202C4⋊C4C22×D12712yes—
203C4⋊C4C22×D12711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—2541no—
206C22⋊C4—2542no—
207C22⋊C4—2541no—
208C42—1271no—
209C42—1271no—
210C42—1271no—
211C42—2541no—
212C42—2541no—
213C42—2542no—
214C42—2541no—
215C42C22×D12711yescharacteristic
216C23—21no—
217C23—1271no—
218Q8—42no—
219Q8—1272no—
220Q8—5082no—
221Q8—2541no—
222D4—21no—
223D4—2544no—
224D4Q8×D12712yes—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—1271no—
229C2×C4—1272no—
230C2×C4—1272no—
231C2×C4—1271no—
232C2×C4—1271no—
233C2×C4—5081no—
234C2×C4—5082no—
235C2×C4—2544no—
236C2×C4—2541no—
237C2×C4—2542no—
238C2×C4—2541no—
239C2×C4—2542no—
240C2×C4D4×D12711yescharacteristic
241C2×C4D4×D12711yescharacteristic
242C2×C4C23×D12711yescharacteristic
243C8—22no—
244C8—2542no—
245C22—22no—
246C22—41no—
247C22—41no—
248C22—1272no—
249C22—2541no—
250C22—2541no—
251C22C2×D4×D12711yescharacteristic
252C4—21no—
253C4—41no—
254C4—42no—
255C4—1272no—
256C4—5081no—
257C4—5082no—
258C4—2542no—
259C4—2541no—
260C4—2541no—
261C4#1686122812yes—
262C4C2×Q8×D12711yescharacteristic
263C4#41×D12711yescharacteristic
264C2—42no—
265C2—2542no—
266C2C22⋊Q8×D12711yescharacteristic
267C2C2×#1686122811yescharacteristic
268C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#474×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523247C22
hypercenter#474D127
hypocenterC127#3076
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. #22550182C2semidirect
4. #22550174C2semidirect
5. #22550172C2semidirect
6. #22550166C2semidirect
7. #22550158C2semidirect
8. #22549913C2semidirect
9. C4⋊C8×D127C2semidirect
11. #474×C127C2semidirect
44. #14523247Q8semidirect
135. C127#3076semidirect
143. #474D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
C4×#145232472
#225501822
#225501742
#225501722
#225501662
#225501582
#225499132
C4⋊C8×D1272
#225498692
#474×C1272
#3076127