Finite Group Databasebeta

← #11797723

Subgroups — #11797723

Order 1152.

260 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 51 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
1C2×#63736C211yesmaximal, characteristic
2#10590790C212yesmaximal
3#10590174C211yesmaximal, characteristic
4#10590068C211yesmaximal, characteristic
5#10590062C212yesmaximal
6#63843—22no—
7C2×#3718C2212yes—
8C2×#3708C2211yescharacteristic
9C2×#3708C2211yescharacteristic
10#63736C414yescomplemented
11C8×C3⋊Dic3C2211yescharacteristic
12C3×#816—22no—
13#63576C2212yes—
14#4790—62no—
15C2×C3×Dic6—22no—
16C2×C32×C8D411yescharacteristic
17C4×C3⋊Dic3D411yescharacteristic
18C2×C3⋊(C3⋊C8)C2311yescharacteristic
19C3×C4⋊Dic3—22no—
20Dic3⋊Dic3—42no—
21Dic32—42no—
22#3718—22no—
23#3718C2×C414yes—
24#3708—21no—
25#3708C2×C412yes—
26C4×Q32—91nomaximal, Sylow
27Q8×Dic3—62no—
28C2×#216—62no—
29#887—62no—
30C3×#103—62no—
31#816—22no—
32C8×Dic3—62no—
33#807—62no—
34C2×C32×C4C2×D411yescharacteristic
35C2×C3⋊Dic3D811yescharacteristic
36C2×C3⋊Dic3D811yescharacteristic
37C2×C3×Dic3—42no—
38C2×C3×Dic3—42no—
39C3×Dic6—24no—
40C3×Dic6—42no—
41C32×C8—21no—
42C3⋊(C3⋊C8)C22×C411yescharacteristic
43C3⋊(C3⋊C8)#4111yescharacteristic
44C2×Q32—91no—
45C4×Q16—92no—
46#366—91no—
47#358—92no—
48C4×C16—91no—
49C2×C3×Q8—62no—
50C2×Dic6—22no—
51C2×C3×C8—22no—
52C3×C4⋊C4—62no—
53#216—64no—
54#216—64no—
55C4⋊Dic3—22no—
56C4⋊Dic3—122no—
57Dic3⋊C4—122no—
58C4×Dic3—62no—
59C4×Dic3—122no—
60C4×Dic3—122no—
61C2×C3⋊C8—62no—
62C22×C32C2×D811yescharacteristic
63C32×C4C4×D411yescharacteristic
64C32×C4#13511yescharacteristic
65C3⋊Dic3—21no—
66C3⋊Dic3Q3212yescomplemented
67C3×Dic3—44no—
68C3×Dic3—82no—
69C2×Q16—92no—
70C4×Q8—182no—
71Q32—94no—
72C2×C16—91no—
73C2×C16—91no—
74#107—92no—
75#103—62no—
76C4×C8—91no—
77C3×Q8—64no—
78C3×Q8—122no—
79C2×C3×C4—22no—
80C2×C3×C4—122no—
81C2×C3×C4—122no—
82C2×Dic3—42no—
83C2×Dic3—42no—
84C2×Dic3—62no—
85C2×Dic3—62no—
86Dic6—24no—
87Dic6—42no—
88C3×C8—42no—
89C3⋊C8—62no—
90C3⋊C8—62no—
91C2×C32C4×D811yescharacteristic
92C2×C32C2×Q3211yescharacteristic
93C2×C32#50811yescharacteristic
94C2×Q8—62no—
95Q16—94no—
96Q16—182no—
97C2×C8—91no—
98C2×C8GO+(4,𝔽2)11yescharacteristic
99C4⋊C4—62no—
100C4⋊C4—362no—
101C42—91no—
102C42—362no—
103C16—92no—
104C16—181no—
105C22×C3—22no—
106C3×C4—22no—
107C3×C4—22no—
108C3×C4—124no—
109C3×C4—242no—
110Dic3—44no—
111Dic3—64no—
112Dic3—82no—
113Dic3—122no—
114C32C4×Q3211yescharacteristic, complemented, Sylow
115Q8—64no—
116Q8—122no—
117C2×C4—91no—
118C2×C4—91no—
119C2×C4—122no—
120C2×C4—122no—
121C2×C4C2×GO+(4,𝔽2)11yescharacteristic
122C8—21no—
123C8—91no—
124C8—91no—
125C2×C3—22no—
126C2×C3—22no—
127C2×C3—22no—
128C22#6422511yescharacteristic
129C4—92no—
130C4—124no—
131C4—181no—
132C4—242no—
133C4C4×GO+(4,𝔽2)11yescharacteristic
134C4#6423711yescharacteristic
135C3—22no—
136C2#1059327511yescharacteristic
137C2#1059328011yescharacteristic
138C2#1059342011yescharacteristic
subgroupG/H
centerC22#64225
commutatorC3⋊(C3⋊C8)C22×C4
FrattiniC2×C4C2×GO+(4,𝔽2)
FittingC2×C32×C8D4
socle, prime radicalC22×C32C2×D8
cosocleC2×C3⋊(C3⋊C8)C23
hypercenterC2×C8GO+(4,𝔽2)
hypocenterC32C4×Q32
LCSG ▷ C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
10. #63736C4semidirect
66. C3⋊Dic3Q32semidirect
114. C32C4×Q32semidirect

Maximal subgroups

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

subgroupindex
C2×#637362
#105907902
#105901742
#105900682
#105900622
C4×Q329