Finite Group Databasebeta

← #11711228

Subgroups — #11711228

Order 768.

210 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 17 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#72988C212yesmaximal, complemented
2#1726⋊C3C211yesmaximal, characteristic, complemented
3#9762—31nomaximal, Sylow
4#5402—41nomaximal
5#376⋊C3C2211yescharacteristic
6#4622—22no—
7#3169—31no—
8#1726D311yescharacteristic, complemented
9#1568—31no—
10#1530—32no—
11#1505—32no—
12C4⋊S4—42no—
13C8×A4—41no—
14#95⋊C3D411yescharacteristic
15#532—32no—
16#500—31no—
17#493—31no—
18#486—32no—
19#466—61no—
20#466—61no—
21C4×D8—61no—
22C4×D8—61no—
23#432—32no—
24C2×#97—31no—
25C2×C4×C8—31no—
26#376C2×D311yescharacteristic
27#335—61no—
28#334—61no—
29#329—64no—
30#327—62no—
31#327—62no—
32C2×S4—82no—
33C4×A4—41no—
34D24—161no—
35C2×#41—62no—
36C2×D8—61no—
37C2×D8—61no—
38C22×C8—31no—
39C22×C8S411yescharacteristic
40C4⋊Q8—32no—
41#127—32no—
42#124—62no—
43#121—62no—
44#121—62no—
45C4×D4—62no—
46C4×D4—62no—
47C2×C42—31no—
48#107—61no—
49#107—61no—
50C4⋊C8—62no—
51#102—62no—
52#102—62no—
53C22⋊C8—62no—
54#97—32no—
55#97—61no—
56#97—62no—
57C4×C8—31no—
58C4×C8—31no—
59C4×C8—31no—
60C4×C8—31no—
61#95—31no—
62#95D1211yescharacteristic
63C2×A4—41no—
64S4—162no—
65D12—162no—
66C3×C8—161no—
67#41—242no—
68#41—242no—
69C2×Q8—62no—
70C2×D4—62no—
71C2×D4—62no—
72C2×D4—62no—
73C2×D4—122no—
74C22×C4—31no—
75C22×C4—31no—
76C22×C4—122no—
77C22×C4C2×S411yescharacteristic
78D8—122no—
79D8—121no—
80D8—121no—
81C2×C8—31no—
82C2×C8—31no—
83C2×C8—31no—
84C2×C8—31no—
85C2×C8—62no—
86C2×C8—122no—
87C4⋊C4—62no—
88C4⋊C4—62no—
89C22⋊C4—122no—
90C22⋊C4—122no—
91C42—31no—
92C42—31no—
93C42—31no—
94C42—31no—
95C2×D3—322no—
96A4—41no—
97C3×C4—161no—
98C23—122no—
99C23C4⋊S411yescharacteristic
100Q8—242no—
101D4—122no—
102D4—122no—
103D4—242no—
104D4—242no—
105D4—242no—
106D4—242no—
107C2×C4—31no—
108C2×C4—31no—
109C2×C4—31no—
110C2×C4—31no—
111C2×C4—31no—
112C2×C4—31no—
113C2×C4—61no—
114C2×C4—61no—
115C2×C4—121no—
116C2×C4—122no—
117C2×C4—244no—
118C8—32no—
119C8—31no—
120C8—61no—
121C8—242no—
122C8#85711yescharacteristic
123C2×C3—161no—
124D3—642no—
125C22—31no—
126C22—31no—
127C22—242no—
128C22—242no—
129C22—242no—
130C22#540711yescharacteristic
131C4—31no—
132C4—61no—
133C4—61no—
134C4—61no—
135C4—61no—
136C4—242no—
137C4C2×#85711yescharacteristic
138C3—161noSylow
139C2—31no—
140C2—31no—
141C2—482no—
142C2C4⋊#85711yescharacteristic
subgroupG/H
centerC2C4⋊#857
commutator, cosocle#376⋊C3C22
FrattiniC22×C4C2×S4
Fitting#1726D3
socleC23C4⋊S4
hypercenterC8#857
hypocenter#95⋊C3D4
LCSG ▷ #376⋊C3 ▷ #95⋊C3
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #72988C2semidirect
2. #1726⋊C3C2semidirect
8. #1726D3semidirect

Maximal subgroups

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

subgroupindex
#729882
#1726⋊C32
#97623
#54024