Finite Group Databasebeta

← #30638665

Subgroups — #30638665

Order 16256.

592 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#16861133C214yesmaximal, complemented
2C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic
3#22549736C211yesmaximal, characteristic
4C23×Q8×C127C211yesmaximal, characteristic
5C22×C127⋊C8C2212yes—
6C2×C4⋊Dic127C2212yes—
7C2×C22⋊Dic127—21no—
8#16861133—28no—
9C22⋊(C127⋊C8)—22no—
10C127⋊#95—21no—
11C22×Q8×C127—28no—
12C22×Q8×C127—22no—
13C22×Q8×C127—21no—
14C22×Q8×C127C414yescomplemented
15C22×Q8×C127C2212yes—
16C23×C4×C127—21no—
17C23×C4×C127C2211yescharacteristic
18C22×Dic127—22no—
19C2×C127⋊C8—22no—
20C2×C127⋊C8—24no—
21C4⋊Dic127—24no—
22C22⋊Dic127—42no—
23C24×C127D411yescharacteristic
24C2×Q8×C127—216no—
25C2×Q8×C127—28no—
26C2×Q8×C127—216no—
27C2×Q8×C127—24no—
28C2×Q8×C127—22no—
29C2×Q8×C127—24no—
30C2×Q8×C127C2×C414yes—
31C2×Q8×C127D418yes—
32C22×C4×C127—28no—
33C22×C4×C127—24no—
34C22×C4×C127—22no—
35C22×C4×C127—24no—
36C22×C4×C127C2×C412yes—
37C22×C4×C127D411yescharacteristic
38C22×C4×C127D412yes—
39C22×C4×C127C2311yescharacteristic
40C2×Dic127—42no—
41C2×Dic127—42no—
42C2×Dic127—42no—
43C127⋊C8—42no—
44C23×C127—22no—
45C23×C127—22no—
46C23×C127C22⋊C412yes—
47C23×C127SD1612yes—
48C23×C127Q1612yes—
49C23×C127C2×D411yescharacteristic
50Q8×C127—216no—
51Q8×C127—24no—
52Q8×C127—28no—
53Q8×C127C22⋊C418yescomplemented
54C2×C4×C127—216no—
55C2×C4×C127—24no—
56C2×C4×C127—28no—
57C2×C4×C127—22no—
58C2×C4×C127—24no—
59C2×C4×C127—24no—
60C2×C4×C127C22⋊C412yes—
61C2×C4×C127C22×C411yescharacteristic
62C2×C4×C127C2×D414yes—
63C2×C4×C127C2×D411yescharacteristic
64C22×C127—22no—
65C22×C127—22no—
66C22×C127—24no—
67C22×C127—24no—
68C22×C127#10314yescomplemented
69C22×C127C2×C22⋊C411yescharacteristic
70C22×C127#12012yes—
71C22×C127C2×SD1611yescharacteristic
72C22×C127C2×Q1611yescharacteristic
73C22×C127#13712yes—
74C4×C127—28no—
75C4×C127—22no—
76C4×C127C2×C22⋊C412yes—
77C4×C127#12012yes—
78Dic127—82no—
79C2×C127—24no—
80C2×C127#37911yescharacteristic
81C2×C127C2×#10311yescharacteristic
82C2×C127#41911yescharacteristic
83C2×C127#44812yes—
84C2×C127#45112yes—
85#1787—1271nomaximal, Sylow
86C127#178711yescharacteristic, complemented, Sylow
87C23×Q8D12711yescharacteristic
88C2×#103—1274no—
89C2×C22⋊C8—1271no—
90#380—1271no—
91C22×Q8—28no—
92C22×Q8—22no—
93C22×Q8—21no—
94C22×Q8Dic12714yescomplemented
95C22×Q8C2×D12712yes—
96C23×C4—21no—
97C23×C4C2×D12711yescharacteristic
98C22×C8—1272no—
99C2×C4⋊C4—1272no—
100C2×C22⋊C4—2541no—
101#103—2548no—
102C22⋊C8—2542no—
103#95—2541no—
104C24C22⋊D12711yescharacteristic
105C2×Q8—216no—
106C2×Q8—28no—
107C2×Q8—216no—
108C2×Q8—24no—
109C2×Q8—22no—
110C2×Q8—24no—
111C2×Q8C2×Dic12714yes—
112C2×Q8C22⋊D12718yes—
113C22×C4—28no—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—24no—
117C22×C4—2542no—
118C22×C4C2×Dic12712yes—
119C22×C4C22⋊D12711yescharacteristic
120C22×C4C22⋊D12712yes—
121C22×C4C22×D12711yescharacteristic
122C2×C8—2542no—
123C2×C8—2544no—
124C4⋊C4—2544no—
125C22⋊C4—5082no—
126C23—22no—
127C23—22no—
128C23C22⋊Dic12712yes—
129C23#1452323512yes—
130C23#1452323812yes—
131C23C2×C22⋊D12711yescharacteristic
132Q8—216no—
133Q8—24no—
134Q8—28no—
135Q8C22⋊Dic12718yescomplemented
136C2×C4—216no—
137C2×C4—24no—
138C2×C4—28no—
139C2×C4—22no—
140C2×C4—24no—
141C2×C4—24no—
142C2×C4—5082no—
143C2×C4—5082no—
144C2×C4—5082no—
145C2×C4C22⋊Dic12712yes—
146C2×C4C22×Dic12711yescharacteristic
147C2×C4C2×C22⋊D12714yes—
148C2×C4C2×C22⋊D12711yescharacteristic
149C8—5082no—
150C22—24no—
151C22—22no—
152C22—22no—
153C22—24no—
154C22#1686113314yescomplemented
155C22C2×C22⋊Dic12711yescharacteristic
156C22#1686117812yes—
157C22C2×#1452323511yescharacteristic
158C22C2×#1452323811yescharacteristic
159C22#1686124012yes—
160C4—28no—
161C4—22no—
162C4—10162no—
163C4C2×C22⋊Dic12712yes—
164C4#1686117812yes—
165C2—24no—
166C2#2254973211yescharacteristic
167C2C2×#1686113311yescharacteristic
168C2#2254988511yescharacteristic
169C2#2255001512yes—
170C2#2255003212yes—
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC23×Q8×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
cosocleC22×C4×C127C23
hypercenterC23×Q8D127
hypocenterC127#1787
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16861133C2semidirect
14. C22×Q8×C127C4semidirect
53. Q8×C127C22⋊C4semidirect
68. C22×C127#103semidirect
86. C127#1787semidirect
94. C22×Q8Dic127semidirect
135. Q8C22⋊Dic127semidirect
154. C22#16861133semidirect

Maximal subgroups

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

subgroupindex
C2×#168611332
C2×C22⋊(C127⋊C8)2
#225497362
C23×Q8×C1272
#1787127