Finite Group Databasebeta

← #30649281

Subgroups — #30649281

Order 16256.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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
2#125×D127C211yesmaximal, characteristic
3#22550286C211yesmaximal, characteristic, complemented
4#22550285C211yesmaximal, characteristic, complemented
5#22550282C211yesmaximal, characteristic, complemented
6#22550277C212yesmaximal, complemented
7#22550276C212yesmaximal, complemented
8#22549917C212yesmaximal
9#22549915C212yesmaximal
10#22549828C211yesmaximal, characteristic
11#501×C127C211yesmaximal, characteristic, complemented
12C2×#14523231C2212yes—
13#16861209C2211yescharacteristic
14#16861201C2211yescharacteristic
15#16861200C2211yescharacteristic
16#16861200C2212yes—
17#16861200C2211yescharacteristic
18#16861199—21no—
19#16861199—21no—
20#16861189—21no—
21#16861185—21no—
22Q8×Dic127—21no—
23#16861174—21no—
24#16861167C2212yes—
25C4⋊C4×D127—21no—
26C4⋊C4×D127—21no—
27C4⋊C4×D127C2211yescharacteristic
28C4⋊C4×D127C2211yescharacteristic
29C42×D127C2211yescharacteristic
30#16861143C2212yes—
31#16861142C2212yes—
32#16861142C2212yes—
33#16861141C2212yes—
34#16861140C2212yes—
35#16861140C2212yes—
36#16861120C2211yescharacteristic
37C8×Dic127C2212yes—
38C4⋊Q8×C127C2211yescharacteristic
39#125×C127C2211yescharacteristic
40#107×C127C2212yes—
41#106×C127C2212yes—
42#97×C127C2211yescharacteristic
43C2×Dic254—21no—
44C2×C4×D127—21no—
45C2×C4×D127—21no—
46C2×C4×D127D411yescharacteristic
47C2×C4×D127D411yescharacteristic
48C2×C4×D127C2311yescharacteristic
49#14523231Q818yescomplemented
50C2×C127⋊C8C2312yes—
51C4⋊Dic127—22no—
52C4⋊Dic127—21no—
53C4⋊Dic127—21no—
54C4⋊Dic127—21no—
55C4⋊Dic127C2311yescharacteristic
56C4⋊Dic127C2311yescharacteristic
57C4⋊Dic127C2312yes—
58Dic127⋊C4—22no—
59Dic127⋊C4—21no—
60Dic127⋊C4—21no—
61Dic127⋊C4—21no—
62Dic127⋊C4—21no—
63Dic127⋊C4—21no—
64Dic127⋊C4—21no—
65(C2×D127)⋊C4—22no—
66C4×Dic127—22no—
67C4×Dic127D411yescharacteristic
68C4×Dic127D411yescharacteristic
69C4×Dic127C2311yescharacteristic
70C2×Q8×C127—21no—
71C2×C8×C127C2312yes—
72C4⋊C4×C127—21no—
73C4⋊C4×C127—21no—
74C4⋊C4×C127—21no—
75C4⋊C4×C127C2311yescharacteristic
76C4⋊C4×C127C2312yes—
77C4⋊C4×C127C2311yescharacteristic
78C42×C127C2311yescharacteristic
79C22×D127C2×D411yescharacteristic
80Dic254—42no—
81C2×Dic127—22no—
82C2×Dic127—21no—
83C2×Dic127—21no—
84C2×Dic127C2×D411yescharacteristic
85C2×Dic127C2×D411yescharacteristic
86C2×Dic127C2×D411yescharacteristic
87C4×D127—41no—
88C4×D127—41no—
89C4×D127—42no—
90C4×D127—42no—
91C4×D127C2×Q814yes—
92C127⋊C8C2×Q814yes—
93Q8×C127—42no—
94C2×C4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127—22no—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2411yescharacteristic
100C8×C127C2×Q814yes—
101C22×C127C22×D411yescharacteristic
102C2×D127—21no—
103C2×D127—21no—
104C2×D127C4⋊Q812yes—
105C4×C127—21no—
106C4×C127—21no—
107C4×C127—42no—
108C4×C127—41no—
109C4×C127—41no—
110C4×C127C22×Q812yes—
111Dic127—21no—
112Dic127—21no—
113Dic127—41no—
114Dic127—41no—
115Dic127—42no—
116Dic127C4⋊Q812yes—
117C2×C127C2×C4⋊Q811yescharacteristic
118C2×C127#57511yescharacteristic
119C2×C127#57511yescharacteristic
120D127—22no—
121#3166—1271nomaximal, Sylow
122C127#316611yescharacteristic, complemented, Sylow
123#533—1271no—
124C2×#125—1271no—
125#501—1271no—
126#501—1271no—
127#501—1271no—
128#501D12711yescharacteristic, complemented
129#499—1272no—
130#499—1272no—
131#428—1272no—
132#428—1272no—
133C4×M16—1271no—
134C2×M16—1272no—
135C4⋊Q8—1271no—
136C4⋊Q8C2×D12711yescharacteristic
137#125—1271no—
138#125—1271no—
139#125—1272no—
140#125—1271no—
141#125—2541no—
142#125—2541no—
143#125C2×D12711yescharacteristic
144C22⋊Q8—2541no—
145C22⋊Q8—2541no—
146C4×Q8—2541no—
147C4×Q8—2541no—
148#117—1272no—
149C2×C4⋊C4—1271no—
150C2×C4⋊C4—1271no—
151C2×C4⋊C4—2541no—
152C2×C4⋊C4—2541no—
153C2×C42—1271no—
154#107—1272no—
155#107—1272no—
156#107—1272no—
157#107C2×D12712yes—
158#106—1272no—
159#106—1272no—
160#106—1272no—
161#106C2×D12712yes—
162#97—1271no—
163#97C2×D12711yescharacteristic
164C4×C8—1272no—
165C2×Q8—21no—
166C2×Q8—2541no—
167C22×C4—1271no—
168C22×C4—1271no—
169C22×C4—1271no—
170C22×C4—2541no—
171C22×C4—2541no—
172M16—1278no—
173C2×C8—1272no—
174C2×C8C22×D12712yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—1271no—
179C4⋊C4—1271no—
180C4⋊C4—1272no—
181C4⋊C4—2541no—
182C4⋊C4—2541no—
183C4⋊C4—2541no—
184C4⋊C4—2541no—
185C4⋊C4—2542no—
186C4⋊C4—2541no—
187C4⋊C4—2541no—
188C4⋊C4—2542no—
189C4⋊C4—2541no—
190C4⋊C4—2541no—
191C4⋊C4—2541no—
192C4⋊C4C22×D12711yescharacteristic
193C4⋊C4C22×D12712yes—
194C4⋊C4C22×D12711yescharacteristic
195C22⋊C4—2542no—
196C42—1271no—
197C42—1271no—
198C42—1271no—
199C42—2542no—
200C42C22×D12711yescharacteristic
201C23—1271no—
202Q8—42no—
203Q8—5082no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—1271no—
208C2×C4—1274no—
209C2×C4—1271no—
210C2×C4—1271no—
211C2×C4—5081no—
212C2×C4—5081no—
213C2×C4—5082no—
214C2×C4—5082no—
215C2×C4—2542no—
216C2×C4—2541no—
217C2×C4—2541no—
218C2×C4D4×D12711yescharacteristic
219C2×C4D4×D12711yescharacteristic
220C2×C4C23×D12711yescharacteristic
221C8—1274no—
222C8Q8×D12714yes—
223C22—1272no—
224C22—2541no—
225C22—2541no—
226C22C2×D4×D12711yescharacteristic
227C4—21no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—41no—
232C4—1272no—
233C4—5081no—
234C4—5081no—
235C4—5082no—
236C4—2541no—
237C4—2541no—
238C4C2×Q8×D12712yes—
239C2—2542no—
240C2C4⋊Q8×D12711yescharacteristic
241C2#2255067211yescharacteristic
242C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#501×C127C2
socleC22×C127C22×D4
prime radicalC22×D127C2×D4
hypercenter#501D127
hypocenterC127#3166
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #22550286C2semidirect
4. #22550285C2semidirect
5. #22550282C2semidirect
6. #22550277C2semidirect
7. #22550276C2semidirect
11. #501×C127C2semidirect
49. #14523231Q8semidirect
122. C127#3166semidirect
128. #501D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
#125×D1272
#225502862
#225502852
#225502822
#225502772
#225502762
#225499172
#225499152
#225498282
#501×C1272
#3166127