Finite Group Databasebeta

← #11711368

Subgroups — #11711368

Order 768.

155 conjugacy classes of subgroups; only the 14 normal subgroups are shown.

normal subgroups only

This group has 155 conjugacy classes of subgroups — too many to draw. Only its 14 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#1747⋊C3C211yesmaximal, characteristic, complemented
2#67994C212yesmaximal, complemented
3#11770—31nomaximal, Sylow
4#5402—41nomaximal
5#338⋊C3C2211yescharacteristic
6#3176—31no—
7#2166—31no—
8#1747D311yescharacteristic, complemented
9#1341—32no—
10#1339—32no—
11C4⋊S4—42no—
12C8×A4—41no—
13#540—61no—
14#499—31no—
15#493—31no—
16#490—61no—
17#469—32no—
18#466—61no—
19#442—32no—
20C4×D8—61no—
21#405—31no—
22#364—61no—
23#349—32no—
24#347—61no—
25#338C2×D311yescharacteristic
26H(3,ℤ/4)—31no—
27C2×S4—82no—
28C4×A4—41no—
29D24—161no—
30C2×D8—32no—
31C2×D8—61no—
32C2×M16—31no—
33C22×C8S411yescharacteristic
34#127—61no—
35#125—61no—
36#123—122no—
37#121—62no—
38#121—122no—
39C4×D4—62no—
40#117—31no—
41M32—62no—
42#107—61no—
43#106—32no—
44#102—62no—
45#102—62no—
46#97—31no—
47#97—31no—
48C4×C8—31no—
49C4×C8—31no—
50C2×A4—41no—
51S4—162no—
52D12—162no—
53C3×C8—161no—
54C2×D4—62no—
55C2×D4—122no—
56C2×D4—122no—
57C22×C4—122no—
58C22×C4C2×S411yescharacteristic
59D8—121no—
60D8—122no—
61D8—121no—
62M16—121no—
63C2×C8—31no—
64C2×C8—31no—
65C2×C8—31no—
66C2×C8—31no—
67C4⋊C4—61no—
68C4⋊C4—62no—
69C4⋊C4—122no—
70C22⋊C4—61no—
71C22⋊C4—122no—
72C22⋊C4—122no—
73C42—31no—
74C42—31no—
75C16—122no—
76C2×D3—322no—
77A4—41no—
78C3×C4—161no—
79C23—122no—
80C23#98611yescharacteristic
81D4—122no—
82D4—122no—
83D4—482no—
84D4—482no—
85C2×C4—31no—
86C2×C4—31no—
87C2×C4—61no—
88C2×C4—61no—
89C2×C4—122no—
90C2×C4—482no—
91C8—31no—
92C8—61no—
93C8—61no—
94C8#85711yescharacteristic
95C2×C3—161no—
96D3—642no—
97C22—31no—
98C22—41no—
99C22—242no—
100C22—242no—
101C22—242no—
102C4—31no—
103C4—121no—
104C4—121no—
105C4—242no—
106C4C2×#85711yescharacteristic
107C3—161noSylow
108C2—61no—
109C2—482no—
110C2#7298811yescharacteristic
subgroupG/H
center, socleC2#72988
commutator, cosocle, hypocenter#338⋊C3C22
FrattiniC22×C4C2×S4
Fitting#1747D3
hypercenterC8#857
LCSG ▷ #338⋊C3
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #1747⋊C3C2semidirect
2. #67994C2semidirect
8. #1747D3semidirect

Maximal subgroups

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

subgroupindex
#1747⋊C32
#679942
#117703
#54024