Finite Group Databasebeta

← #27186622

Subgroups — #27186622

Order 12928.

238 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 56 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#20266200C212yesmaximal, complemented
2#20266033C214yesmaximal, complemented
3#20265728C211yesmaximal, characteristic
4#15692322—22no—
5#15692312—22no—
6#15692310C2214yes—
7C2×#13108075C412yes—
8#15692252C411yescharacteristic
9#15692244C2212yes—
10#15692230—22no—
11#15692217—22no—
12C42×D101C2211yescharacteristic
13C4⋊Q8×C101C411yescharacteristic
14C2×#11717581—24no—
15#13108075—44no—
16C2×D404—22no—
17C2×D404—22no—
18C2×D404C2×C412yes—
19C2×C4×D101—22no—
20C2×C4×D101C2×C411yescharacteristic
21C2×C4×D101D412yes—
22#13108051—24no—
23C4×Dic101D412yes—
24C4×Dic101C2311yescharacteristic
25C2×Q8×C101C2×C412yes—
26C4⋊C4×C101—22no—
27C42×C101C2×C411yescharacteristic
28C22×D101—22no—
29C22×D101C22⋊C411yescharacteristic
30Q8×C101—42no—
31C2×C4×C101—22no—
32C2×C4×C101C22⋊C412yes—
33C2×C4×C101C22×C411yescharacteristic
34C2×Dic101C22⋊C411yescharacteristic
35C2×Dic101C2×D412yes—
36D404—42no—
37D404—44no—
38D404—44no—
39C4×D101—22no—
40C4×D101—42no—
41C4×D101—44no—
42#11717581—44no—
43C22×C101C2×C22⋊C411yescharacteristic
44C2×D101—21no—
45C2×D101—22no—
46C2×D101—42no—
47C2×D101—44no—
48C4×C101—21no—
49C4×C101—22no—
50C4×C101—42no—
51Dic101—22no—
52Dic101#10012yes—
53C2×C101C2×#10011yescharacteristic
54C2×C101#42112yes—
55D101—41no—
56D101—82no—
57#1536—1011nomaximal, Sylow
58C101#153611yescharacteristic, complemented, Sylow
59#532—1011no—
60#432—1012no—
61#329—1014no—
62C2×#41—1012no—
63C4⋊Q8#8793811yescharacteristic
64#127—1011no—
65#124—1012no—
66#121—2022no—
67C4×D4—2022no—
68C2×C42—1011no—
69C4⋊C8—2022no—
70C22⋊C8—2022no—
71#97—1014no—
72#41—4044no—
73C2×Q8C2×#8793812yes—
74C2×D4—1012no—
75C2×D4—2022no—
76C2×D4—2022no—
77C22×C4—1012no—
78C22×C4—1011no—
79C22×C4—2022no—
80C2×C8—2024no—
81C4⋊C4—22no—
82C22⋊C4—2024no—
83C42—1012no—
84C42—1011no—
85C42C2×#8793811yescharacteristic
86C23—1011no—
87C23—2022no—
88Q8—42no—
89D4—4042no—
90D4—4044no—
91D4—4044no—
92C2×C4—22no—
93C2×C4—1011no—
94C2×C4—1012no—
95C2×C4—4042no—
96C2×C4—4044no—
97C2×C4—2022no—
98C2×C4C22⋊#8793812yes—
99C2×C4C22×#8793811yescharacteristic
100C8—4044no—
101C22—4042no—
102C22—4044no—
103C22—2021no—
104C22—2022no—
105C22C2×C22⋊#8793811yescharacteristic
106C4—21no—
107C4—22no—
108C4—42no—
109C4—1012no—
110C4—2022no—
111C2—8082no—
112C2—4041no—
113C2D101⋊#10011yescharacteristic
114C2#2026617612yes—
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC4⋊Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radical#20265728C2
cosocleC4×Dic101C23
hypercenterC4⋊Q8#87938
hypocenterC101#1536
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20266200C2semidirect
2. #20266033C2semidirect
58. C101#1536semidirect

Maximal subgroups

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

subgroupindex
#202662002
#202660332
#202657282
#1536101