Finite Group Databasebeta

← #27187416

Subgroups — #27187416

Order 12928.

238 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 54 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#20266156C212yesmaximal
2#20266084C214yesmaximal, complemented
3#20265753C211yesmaximal, characteristic
4#15692334—22no—
5#15692314C2214yes—
6#15692313—22no—
7C2×#13108074C2211yescharacteristic
8C22×Dic202C411yescharacteristic, complemented
9#15692255C412yescomplemented
10#15692243C2212yes—
11#15692239—22no—
12#15692235—22no—
13#117×C101C411yescharacteristic, complemented
14C4⋊#87938—42no—
15C22⋊#87938—24no—
16C4×#87938—42no—
17#13108074—42no—
18C2×Dic202—21no—
19C2×Dic202—42no—
20C2×Dic202C2×C412yes—
21C2×Dic202C2×C411yescharacteristic
22C2×D404C2311yescharacteristic
23C2×C22⋊D101D412yes—
24C22×Dic101—21no—
25C2×C4×D101D412yes—
26C4⋊Dic101—24no—
27#13108051—24no—
28C22×C4×C101C2×C411yescharacteristic
29C4⋊C4×C101—21no—
30C22⋊C4×C101—21no—
31C42×C101C2×C412yes—
32C23×C101C22⋊C411yescharacteristic
33C22×D101C2×D412yes—
34C2×#87938—44no—
35C2×C4×C101—22no—
36C2×C4×C101—22no—
37C2×C4×C101C22⋊C411yescharacteristic
38C2×C4×C101C22×C411yescharacteristic
39C22⋊D101—42no—
40C2×Dic101—22no—
41C2×Dic101—42no—
42C2×Dic101C22⋊C412yes—
43D404—41no—
44C4×D101—22no—
45C4×D101—22no—
46Dic202—41no—
47Dic202—42no—
48Dic202—42no—
49C22×C101—21no—
50C22×C101—41no—
51C22×C101C2×C22⋊C411yescharacteristic
52C2×D101—22no—
53C2×D101—42no—
54#87938—84no—
55C4×C101—21no—
56C4×C101—42no—
57C4×C101#9912yes—
58Dic101—22no—
59Dic101—42no—
60C2×C101—21no—
61C2×C101—41no—
62C2×C101C2×#9911yescharacteristic
63D101—42no—
64#2129—1011nomaximal, Sylow
65C101#212911yescharacteristic, complemented, Sylow
66#536—1011no—
67#410—1012no—
68#354—1014no—
69C2×#41—1011no—
70C22×Q8—1011no—
71C4⋊Q8—1012no—
72#124—1012no—
73#123—2022no—
74C22⋊Q8—2022no—
75#117—2022no—
76#117#8793811yescharacteristic, complemented
77#99—1014no—
78#99—2022no—
79#41—4042no—
80C2×Q8—1011no—
81C2×Q8—1012no—
82C2×Q8—4042no—
83C2×Q8—2021no—
84C2×D4—1012no—
85C2×D4—1011no—
86C22×C4—1012no—
87C22×C4—2021no—
88C22×C4C2×#8793811yescharacteristic
89C4⋊C4—21no—
90C4⋊C4—4042no—
91C4⋊C4—2024no—
92C22⋊C4—21no—
93C22⋊C4—2024no—
94C22⋊C4—2024no—
95C42—4042no—
96C42C2×#8793812yes—
97C23—1012no—
98C23C22⋊#8793811yescharacteristic
99Q8—4042no—
100Q8—4041no—
101Q8—4042no—
102D4—4041no—
103D4—4042no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—1012no—
107C2×C4—4044no—
108C2×C4—4042no—
109C2×C4—2022no—
110C2×C4—2022no—
111C2×C4—2022no—
112C2×C4C22⋊#8793811yescharacteristic
113C2×C4C22×#8793811yescharacteristic
114C22—21no—
115C22—41no—
116C22—4042no—
117C22—2022no—
118C22C2×C22⋊#8793811yescharacteristic
119C4—21no—
120C4—42no—
121C4—8084no—
122C4—4042no—
123C4—2022no—
124C4#1569231412yes—
125C2—21no—
126C2—41no—
127C2—4042no—
128C2C2×#1569231411yescharacteristic
subgroupG/H
centerC2C2×#15692314
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#117×C101C4
socleC2×C101C2×#99
prime radicalC2×#13108074C22
cosocleC2×D404C23
hypercenter#117#87938
hypocenterC101#2129
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #20266084C2semidirect
8. C22×Dic202C4semidirect
9. #15692255C4semidirect
13. #117×C101C4semidirect
65. C101#2129semidirect
76. #117#87938semidirect

Maximal subgroups

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

subgroupindex
#202661562
#202660842
#202657532
#2129101