Finite Group Databasebeta

← #30641108

Subgroups — #30641108

Order 16256.

1088 conjugacy classes of subgroups; only the 261 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 261 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
1C24×Dic127C211yesmaximal, characteristic
2C2×#16861165C212yesmaximal, complemented
3#22549737C212yesmaximal, complemented
4#22549734C214yesmaximal, complemented
5#22549732C211yesmaximal, characteristic
6C22⋊C4×Dic127C214yesmaximal, complemented
7#379×C127C211yesmaximal, characteristic
8C23×Dic127—28no—
9C23×Dic127C418yescomplemented
10C23×Dic127C2212yes—
11C23×Dic127C2212yes—
12C23×Dic127C2211yescharacteristic
13C23×Dic127C2211yescharacteristic
14#16861165—28no—
15C2×Dic127⋊C4C2214yescomplemented
16C2×C22⋊Dic127C2212yes—
17C2×C22⋊Dic127C2214yes—
18C2×C4×Dic127C2214yescomplemented
19C127⋊#95C2218yescomplemented
20C25×C127C2211yescharacteristic
21C2×C22⋊C4×C127C2212yes—
22C2×C22⋊C4×C127C2214yes—
23C22×Dic127—232no—
24C22×Dic127—28no—
25C22×Dic127—28no—
26C22×Dic127—24no—
27C22×Dic127C2×C414yes—
28C22×Dic127C2×C418yes—
29C22×Dic127C2×C418yes—
30C22×Dic127C2×C414yes—
31C22×Dic127D418yes—
32C22×Dic127C2314yes—
33C22×Dic127C2312yes—
34C22×Dic127C2311yescharacteristic
35C22×Dic127C2311yescharacteristic
36Dic127⋊C4—28no—
37C22⋊Dic127—24no—
38C22⋊Dic127—28no—
39C4×Dic127—28no—
40C24×C127—24no—
41C24×C127—28no—
42C24×C127C2×C414yes—
43C24×C127C2311yescharacteristic
44C24×C127C2312yes—
45C22×C4×C127C2314yes—
46C22⋊C4×C127—24no—
47C22⋊C4×C127—28no—
48C2×Dic127—28no—
49C2×Dic127—24no—
50C2×Dic127—216no—
51C2×Dic127—216no—
52C2×Dic127—24no—
53C2×Dic127—28no—
54C2×Dic127C22⋊C4116yescomplemented
55C2×Dic127C22×C414yes—
56C2×Dic127C22×C412yes—
57C2×Dic127C22×C412yes—
58C2×Dic127C2×D414yes—
59C2×Dic127C2×D414yes—
60C23×C127—232no—
61C23×C127—28no—
62C23×C127—216no—
63C23×C127—24no—
64C23×C127—24no—
65C23×C127—24no—
66C23×C127C22×C412yes—
67C23×C127C22×C414yes—
68C23×C127C2×D414yes—
69C23×C127#4118yes—
70C23×C127C2411yescharacteristic
71C2×C4×C127—24no—
72C2×C4×C127—28no—
73C22×C127—232no—
74C22×C127—24no—
75C22×C127—24no—
76C22×C127—24no—
77C22×C127—28no—
78C22×C127—216no—
79C22×C127C2×C22⋊C414yes—
80C22×C127#11718yescomplemented
81C22×C127C23×C411yescharacteristic
82C22×C127C22×D412yes—
83C22×C127C2×#4114yes—
84C4×C127—44no—
85Dic127—24no—
86Dic127—44no—
87Dic127C2×C22⋊C414yes—
88Dic127C2×C22⋊C414yes—
89C2×C127—24no—
90C2×C127—28no—
91C2×C127C22×C22⋊C411yescharacteristic
92C2×C127C2×#11712yes—
93C2×C127#52514yes—
94#2280—1271nomaximal, Sylow
95C127#228011yescharacteristic, complemented, Sylow
96C24×C4—1271no—
97C2×#117—1272no—
98#381—1272no—
99#380—1274no—
100#379—1271no—
101#379D12711yescharacteristic
102C4×C22⋊C4—1274no—
103C25C2×D12711yescharacteristic
104C23×C4—1271no—
105C23×C4—1278no—
106C23×C4—1272no—
107C23×C4—1272no—
108C23×C4—1271no—
109C23×C4—2548no—
110#117—2548no—
111C2×C4⋊C4—1274no—
112C2×C22⋊C4—1272no—
113C2×C22⋊C4—1274no—
114C2×C22⋊C4C2×D12712yes—
115C2×C22⋊C4C2×D12714yes—
116C2×C42—1274no—
117#95—1278no—
118C24—24no—
119C24—28no—
120C24C4×D12714yes—
121C24C22×D12711yescharacteristic
122C24C22×D12712yes—
123C22×C4—1274no—
124C22×C4—1278no—
125C22×C4—1271no—
126C22×C4—1271no—
127C22×C4—1278no—
128C22×C4—1278no—
129C22×C4—1274no—
130C22×C4—1272no—
131C22×C4—1274no—
132C22×C4—2548no—
133C22×C4—2548no—
134C22×C4—25432no—
135C22×C4—2544no—
136C22×C4C22×D12714yes—
137C4⋊C4—2548no—
138C22⋊C4—24no—
139C22⋊C4—28no—
140C22⋊C4—2544no—
141C22⋊C4—2548no—
142C42—2548no—
143C23—232no—
144C23—24no—
145C23—24no—
146C23—24no—
147C23—28no—
148C23—216no—
149C23C2×C4×D12712yes—
150C23C2×C4×D12714yes—
151C23D4×D12714yes—
152C23#1452324718yes—
153C23C23×D12711yescharacteristic
154C2×C4—24no—
155C2×C4—28no—
156C2×C4—1272no—
157C2×C4—1272no—
158C2×C4—1274no—
159C2×C4—1274no—
160C2×C4—12716no—
161C2×C4—1274no—
162C2×C4—25416no—
163C2×C4—25416no—
164C2×C4—2544no—
165C2×C4—2548no—
166C2×C4—2544no—
167C2×C4—2548no—
168C22—232no—
169C22—24no—
170C22—24no—
171C22—24no—
172C22—28no—
173C22—216no—
174C22C22⋊C4×D12714yes—
175C22#1686116518yescomplemented
176C22C22×C4×D12711yescharacteristic
177C22C2×D4×D12712yes—
178C22C2×#1452324714yes—
179C4—44no—
180C4—1274no—
181C4—1274no—
182C4—5084no—
183C4—2544no—
184C2—24no—
185C2—28no—
186C2C2×C22⋊C4×D12711yescharacteristic
187C2C2×#1686116512yes—
188C2#2255038214yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#379×C127C2
socle, cosocleC23×C127C24
prime radicalC25×C127C22
hypercenter#379D127
hypocenterC127#2280
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#16861165C2semidirect
3. #22549737C2semidirect
4. #22549734C2semidirect
6. C22⋊C4×Dic127C2semidirect
9. C23×Dic127C4semidirect
15. C2×Dic127⋊C4C22semidirect
18. C2×C4×Dic127C22semidirect
19. C127⋊#95C22semidirect
54. C2×Dic127C22⋊C4semidirect
80. C22×C127#117semidirect
95. C127#2280semidirect
175. C22#16861165semidirect

Maximal subgroups

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

subgroupindex
C24×Dic1272
C2×#168611652
#225497372
#225497342
#225497322
C22⋊C4×Dic1272
#379×C1272
#2280127