Finite Group Databasebeta

← #11711280

Subgroups — #11711280

Order 768.

182 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 18 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#72980C212yesmaximal
2#1842⋊C3C211yesmaximal, characteristic
3#10522—31nomaximal, Sylow
4#5398—41nomaximal
5C4×C42⋊C3C2211yescharacteristic
6#4623—22no—
7#3249—31no—
8#1994—32no—
9#1842D311yescharacteristic
10#1550—32no—
11#1292—31no—
12A4⋊Q8—42no—
13C8×A4—41no—
14C2×C42⋊C3D411yescharacteristic
15#533—32no—
16#477—62no—
17#467—61no—
18#467—61no—
19#462—62no—
20#423—32no—
21C2×C4⋊C8—31no—
22C2×C4⋊C8—31no—
23C43C2×D311yescharacteristic
24#339—62no—
25#339—62no—
26#333—62no—
27#332—62no—
28C4×A4—41no—
29SL(2,ℤ/4)—82no—
30Dic12—161no—
31C42⋊C3Q1611yescharacteristic, complemented
32C2×Q16—61no—
33C2×Q16—61no—
34C2×M16—62no—
35C22×C8—31no—
36C22×C8S411yescharacteristic
37C4⋊Q8—32no—
38C4⋊Q8—32no—
39#125—62no—
40C22⋊Q8—62no—
41C22⋊Q8—62no—
42C4×Q8—62no—
43C4×Q8—62no—
44#117—62no—
45C2×C42—31no—
46C2×C42—31no—
47C2×C42D1211yescharacteristic
48#107—61no—
49#107—61no—
50C4⋊C8—62no—
51C4⋊C8—62no—
52C4⋊C8—62no—
53C4⋊C8—62no—
54#103—62no—
55#103—62no—
56C2×A4—41no—
57Dic6—162no—
58C3×C8—161no—
59C2×Q8—62no—
60C2×Q8—62no—
61C22×C4—31no—
62C22×C4—31no—
63C22×C4C2×S411yescharacteristic
64Q16—241no—
65Q16—241no—
66M16—122no—
67M16—122no—
68C2×C8—61no—
69C2×C8—61no—
70C2×C8—121no—
71C2×C8—122no—
72C4⋊C4—62no—
73C4⋊C4—62no—
74C4⋊C4—62no—
75C4⋊C4—62no—
76C4⋊C4—122no—
77C4⋊C4—122no—
78C22⋊C4—122no—
79C42—31no—
80C42—32no—
81C42—32no—
82C42—62no—
83C42—122no—
84C42Dic1211yescharacteristic, complemented
85A4—41no—
86C3×C4—161no—
87Dic3—322no—
88C23C4⋊S411yescharacteristic
89Q8—122no—
90Q8—122no—
91Q8—242no—
92C2×C4—32no—
93C2×C4—32no—
94C2×C4—31no—
95C2×C4—31no—
96C2×C4—31no—
97C2×C4—31no—
98C2×C4—32no—
99C2×C4—121no—
100C2×C4—122no—
101C2×C4—122no—
102C8—41no—
103C8—121no—
104C8—242no—
105C2×C3—161no—
106C22—31no—
107C22—31no—
108C22#539811yescharacteristic
109C4—32no—
110C4—32no—
111C4—31no—
112C4—62no—
113C4—242no—
114C4—242no—
115C4#857⋊C211yescharacteristic
116C3—161noSylow
117C2—31no—
118C2—31no—
119C2#7299411yescharacteristic
subgroupG/H
centerC2#72994
commutator, cosocleC4×C42⋊C3C22
FrattiniC22×C4C2×S4
Fitting#1842D3
socleC23C4⋊S4
prime radicalC2×C42⋊C3D4
hypercenterC4#857⋊C2
hypocenterC42⋊C3Q16
LCSG ▷ C4×C42⋊C3 ▷ C2×C42⋊C3 ▷ C42⋊C3
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
31. C42⋊C3Q16semidirect
84. C42Dic12semidirect

Maximal subgroups

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

subgroupindex
#729802
#1842⋊C32
#105223
#53984