Finite Group Databasebeta

← #27187411

Subgroups — #27187411

Order 12928.

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

normal subgroups only

This group has 270 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, complemented
2#20266081C214yesmaximal, complemented
3#20265744C211yesmaximal, characteristic
4#15692334—22no—
5#15692314C2214yescomplemented
6#15692313—22no—
7C2×#13108074C2211yescharacteristic
8#15692251C2212yes—
9#15692242C412yescomplemented
10#15692226—22no—
11#15692224—22no—
12#15692214C411yescharacteristic, complemented
13C22×D4×C101C411yescharacteristic, complemented
14C4⋊#87938—42no—
15C22⋊#87938—24no—
16C4×#87938—42no—
17#13108074—42no—
18C2×Dic202C2×C411yescharacteristic
19C2×D404C2311yescharacteristic
20C2×C22⋊D101—24no—
21C2×C22⋊D101D412yes—
22C2×C4×D101D412yes—
23C4⋊Dic101—21no—
24C22⋊Dic101—21no—
25C22⋊Dic101—24no—
26C4×Dic101C2×C412yes—
27C24×C101—21no—
28C2×D4×C101—21no—
29C2×D4×C101—42no—
30C2×D4×C101C2×C412yes—
31C22×C4×C101C2×C411yescharacteristic
32C23×C101—22no—
33C23×C101—42no—
34C23×C101—82no—
35C23×C101C22⋊C411yescharacteristic
36C22×D101C2×D412yes—
37C2×#87938—44no—
38D4×C101—42no—
39D4×C101—42no—
40C2×C4×C101—22no—
41C2×C4×C101C22⋊C411yescharacteristic
42C2×C4×C101C22×C411yescharacteristic
43C22⋊D101—42no—
44C22⋊D101—84no—
45C2×Dic101—22no—
46C2×Dic101C22⋊C412yes—
47D404—41no—
48C4×D101—22no—
49C4×D101—22no—
50Dic202—41no—
51C22×C101—21no—
52C22×C101—41no—
53C22×C101—44no—
54C22×C101—42no—
55C22×C101—84no—
56C22×C101C2×C22⋊C411yescharacteristic
57C2×D101—22no—
58C2×D101—42no—
59#87938—84no—
60C4×C101—21no—
61C4×C101#9912yes—
62Dic101—22no—
63Dic101—42no—
64C2×C101—21no—
65C2×C101—41no—
66C2×C101—82no—
67C2×C101C2×#9911yescharacteristic
68D101—42no—
69#2128—1011nomaximal, Sylow
70C101#212811yescharacteristic, complemented, Sylow
71#535—1011no—
72#410—1012no—
73#353—1014no—
74C2×#41—1011no—
75C22×D4#8793811yescharacteristic, complemented
76#127—1012no—
77#124—1012no—
78#121—2022no—
79#120—2022no—
80#117—1011no—
81#117—2022no—
82#99—1014no—
83#99—2022no—
84C24—21no—
85#41—4042no—
86C2×Q8—1011no—
87C2×D4—21no—
88C2×D4—42no—
89C2×D4—1012no—
90C2×D4—1011no—
91C2×D4—2024no—
92C2×D4C2×#8793812yes—
93C22×C4—1012no—
94C22×C4C2×#8793811yescharacteristic
95C4⋊C4—4042no—
96C4⋊C4—2021no—
97C22⋊C4—2024no—
98C22⋊C4—2024no—
99C22⋊C4—2021no—
100C42—1012no—
101C42—4042no—
102C23—22no—
103C23—42no—
104C23—82no—
105C23—1012no—
106C23C22⋊#8793811yescharacteristic
107Q8—4041no—
108D4—42no—
109D4—42no—
110D4—8084no—
111D4—4041no—
112D4—4042no—
113C2×C4—22no—
114C2×C4—1012no—
115C2×C4—4044no—
116C2×C4—2022no—
117C2×C4—2022no—
118C2×C4—2022no—
119C2×C4C22⋊#8793811yescharacteristic
120C2×C4C22×#8793811yescharacteristic
121C22—21no—
122C22—41no—
123C22—44no—
124C22—42no—
125C22—84no—
126C22—4042no—
127C22—2022no—
128C22C2×C22⋊#8793811yescharacteristic
129C4—21no—
130C4—8084no—
131C4—4042no—
132C4—2022no—
133C4#1569231312yes—
134C2—21no—
135C2—41no—
136C2—82no—
137C2—4042no—
138C2C2×#1569231311yescharacteristic
subgroupG/H
centerC2C2×#15692313
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC22×D4×C101C4
socleC2×C101C2×#99
prime radical#20265744C2
cosocleC2×D404C23
hypercenterC22×D4#87938
hypocenterC101#2128
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #20266156C2semidirect
2. #20266081C2semidirect
5. #15692314C22semidirect
9. #15692242C4semidirect
12. #15692214C4semidirect
13. C22×D4×C101C4semidirect
70. C101#2128semidirect
75. C22×D4#87938semidirect

Maximal subgroups

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

subgroupindex
#202661562
#202660812
#202657442
#2128101