Finite Group Databasebeta

← #27186393

Subgroups — #27186393

Order 12928.

300 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 64 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#20266197C212yesmaximal, complemented
2#20266021C214yesmaximal, complemented
3C2×#15692217C211yesmaximal, characteristic
4#15692322—22no—
5C22⋊#11717581C2214yes—
6#15692310—22no—
7C22×D404C411yescharacteristic
8C22×C4×D101C412yes—
9#15692217—22no—
10#15692217—22no—
11C2×#13108051C2212yes—
12C2×C4×Dic101C2211yescharacteristic
13C2×C4⋊C4×C101C411yescharacteristic
14C2×#11717581—24no—
15C23×D101C2×C412yes—
16C2×D404—24no—
17C2×D404—22no—
18C22×Dic101C2×C411yescharacteristic
19C22×Dic101C2311yescharacteristic
20C2×C4×D101—22no—
21C2×C4×D101—22no—
22C2×C4×D101—44no—
23#13108051—24no—
24C4×Dic101D412yes—
25C4×Dic101D412yes—
26C22×C4×C101C2×C412yes—
27C22×C4×C101C2×C411yescharacteristic
28C4⋊C4×C101—22no—
29C23×C101C22×C411yescharacteristic
30C22×D101—22no—
31C22×D101—22no—
32C22×D101—42no—
33C22×D101—42no—
34C22×D101M1614yes—
35C2×C4×C101—22no—
36C2×C4×C101—22no—
37C2×C4×C101—42no—
38C2×C4×C101C22⋊C412yes—
39C2×Dic101—22no—
40C2×Dic101—41no—
41C2×Dic101C22⋊C411yescharacteristic
42C2×Dic101C22⋊C411yescharacteristic
43C2×Dic101C2×D412yes—
44D404—28no—
45C4×D101—44no—
46C4×D101—44no—
47#11717581—44no—
48C22×C101—21no—
49C22×C101—21no—
50C22×C101C2×C22⋊C411yescharacteristic
51C22×C101C2×M1612yes—
52C2×D101—24no—
53C2×D101—44no—
54C2×D101—42no—
55C2×D101—42no—
56C2×D101—44no—
57C4×C101—22no—
58C4×C101—42no—
59Dic101—22no—
60Dic101—21no—
61Dic101#10012yes—
62C2×C101—22no—
63C2×C101#40711yescharacteristic
64C2×C101#41011yescharacteristic
65C2×C101C2×#10011yescharacteristic
66D101—44no—
67#1466—1011nomaximal, Sylow
68C101#146611yescharacteristic, complemented, Sylow
69C2×C4×D4—1011no—
70#432—1012no—
71#323—1014no—
72C22×D4—1011no—
73C23×C4—1012no—
74C4×D4—2022no—
75C4×D4—2022no—
76C2×C4⋊C4#8793811yescharacteristic
77C2×C22⋊C4—1012no—
78C2×C42—1011no—
79C4⋊C8—2022no—
80C22⋊C8—1014no—
81#97—2022no—
82C24—1012no—
83C2×D4—2024no—
84C2×D4—2022no—
85C22×C4—1011no—
86C22×C4—1011no—
87C22×C4—4044no—
88C22×C4—2022no—
89C22×C4—2022no—
90C22×C4C2×#8793812yes—
91C22×C4C2×#8793811yescharacteristic
92C2×C8—2024no—
93C4⋊C4—22no—
94C22⋊C4—2024no—
95C42—1012no—
96C42—1012no—
97C23—1014no—
98C23—4042no—
99C23—4042no—
100C23—2022no—
101C23—2022no—
102C23C22×#8793811yescharacteristic
103D4—2028no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—42no—
107C2×C4—1011no—
108C2×C4—1012no—
109C2×C4—1011no—
110C2×C4—4044no—
111C2×C4—4044no—
112C2×C4—4041no—
113C2×C4—2022no—
114C2×C4C22⋊#8793812yes—
115C8—4044no—
116C22—21no—
117C22—21no—
118C22—4044no—
119C22—4044no—
120C22—4042no—
121C22—4042no—
122C22—2024no—
123C22C2×C22⋊#8793811yescharacteristic
124C22D101⋊M1612yes—
125C4—22no—
126C4—42no—
127C4—1012no—
128C4—2022no—
129C4—2021no—
130C2—22no—
131C2—4044no—
132C2#2026614911yescharacteristic
133C2#2026615811yescharacteristic
134C2D101⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC2×C4⋊C4×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC22×D404C4
cosocleC22×Dic101C23
hypercenterC2×C4⋊C4#87938
hypocenterC101#1466
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20266197C2semidirect
2. #20266021C2semidirect
68. C101#1466semidirect

Maximal subgroups

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

subgroupindex
#202661972
#202660212
C2×#156922172
#1466101