Finite Group Databasebeta

← #11772412

Subgroups — #11772412

Order 1152.

170 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 48 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
1C22×#3771C211yesmaximal, characteristic
2C2×#63748C211yesmaximal, characteristic
3#10593511C211yesmaximal, characteristic
4C2×#3792—21no—
5C22×#622C2211yescharacteristic
6C2×#3771—21no—
7C2×#3771—21no—
8C2×#3771C412yescomplemented
9#63957—21no—
10#63748—21no—
11#63748C412yescomplemented
12#63743C412yes—
13#63742—21no—
14C22×C3⋊Dic3D411yescharacteristic
15#3792—22no—
16#3792—41no—
17C2×#622—41no—
18C2×#622C2×C411yescharacteristic
19C2×#622C2×C411yescharacteristic
20C2×#622C2×C411yescharacteristic
21C2×#622D411yescharacteristic
22C2×#622D411yescharacteristic
23C2×#622Q811yescharacteristic
24#3771—22no—
25C3×#217—41no—
26#3724—22no—
27Dic32—41no—
28#1382—91nomaximal, Sylow
29#877—121no—
30C23×C32SD1611yescharacteristic
31C2×C3⋊Dic3C22⋊C411yescharacteristic
32C2×C3⋊Dic3C22⋊C411yescharacteristic
33C2×C3⋊Dic3C4⋊C411yescharacteristic
34C2×C3⋊Dic3D811yescharacteristic
35C2×C3⋊Dic3SD1611yescharacteristic
36C2×C3⋊Dic3Q1611yescharacteristic
37C2×C3×Dic3—42no—
38#622—42no—
39#622C4211yescharacteristic
40#622C22⋊C411yescharacteristic
41#622C4⋊C411yescharacteristic
42#622C4⋊C411yescharacteristic
43C22×C16—91no—
44C2×#108—91no—
45#427—91no—
46C22×Dic3—121no—
47C3×C22⋊C4—121no—
48#217—41no—
49Dic3⋊C4—122no—
50C4×Dic3—122no—
51C22×C32—22no—
52C22×C32#10211yescharacteristic
53C22×C32#10311yescharacteristic
54C22×C32#10611yescharacteristic
55C3⋊Dic3#9511yescharacteristic
56C3⋊Dic3#10211yescharacteristic
57C3⋊Dic3#10311yescharacteristic
58C3⋊Dic3#10711yescharacteristic
59C3×Dic3—82no—
60C2×M16—181no—
61C22×C8—91no—
62#117—181no—
63C2×C16—92no—
64C2×C16—181no—
65C2×C16—181no—
66#108—92no—
67#108—181no—
68#107—92no—
69#106—181no—
70C23×C3—41no—
71C2×C3×C4—122no—
72C2×Dic3—42no—
73C2×Dic3—121no—
74C2×Dic3—121no—
75C2×Dic3—121no—
76C2×Dic3—121no—
77C2×Dic3—121no—
78C2×Dic3—121no—
79C2×C32—21no—
80C2×C32—21no—
81C2×C32#34011yescharacteristic
82C2×C32#35912yes—
83C22×C4—91no—
84M16—182no—
85M16—361no—
86C2×C8—91no—
87C2×C8—91no—
88C2×C8—91no—
89C2×C8—91no—
90C2×C8—91no—
91C2×C8—91no—
92C2×C8—361no—
93C4⋊C4—182no—
94C22⋊C4—121no—
95C42—361no—
96C16—182no—
97C22×C3—41no—
98C22×C3—41no—
99C22×C3—41no—
100C22×C3—82no—
101C3×C4—242no—
102Dic3—82no—
103Dic3—121no—
104Dic3—121no—
105Dic3—121no—
106Dic3—121no—
107C32#138211yescharacteristic, complemented, Sylow
108C23AΓL(1,𝔽9)11yescharacteristic
109C2×C4—91no—
110C2×C4—91no—
111C2×C4—91no—
112C2×C4—91no—
113C2×C4—91no—
114C2×C4—91no—
115C2×C4—122no—
116C8—91no—
117C8—91no—
118C8—91no—
119C8—91no—
120C8—362no—
121C2×C3—41no—
122C2×C3—42no—
123C2×C3—81no—
124C2×C3—81no—
125C22—22no—
126C22GO+(4,𝔽2)⋊C411yescharacteristic
127C22PSU(3,𝔽2)⋊C411yescharacteristic
128C22AGL(1,𝔽9)⋊C411yescharacteristic
129C4—91no—
130C4—91no—
131C4—91no—
132C4—91no—
133C4—242no—
134C3—41no—
135C2—21no—
136C2—21no—
137C2#1059316611yescharacteristic
138C2#1059316712yes—
subgroupG/H
centerC22GO+(4,𝔽2)⋊C4
commutator#622C42
Frattini, hypercenterC23AΓL(1,𝔽9)
Fitting, prime radicalC23×C32SD16
socleC22×C32#102
cosocleC22×#622C22
hypocenterC32#1382
LCSG ▷ #622 ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
8. C2×#3771C4semidirect
11. #63748C4semidirect
107. C32#1382semidirect

Maximal subgroups

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

subgroupindex
C22×#37712
C2×#637482
#105935112
#13829