Finite Group Databasebeta

← #11711232

Subgroups — #11711232

Order 768.

202 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 19 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
2C8×C42⋊C3C211yesmaximal, characteristic
3#9845—31nomaximal, Sylow
4#5398—41nomaximal
5C4×C42⋊C3C2211yescharacteristic
6#4623—22no—
7#3170—31no—
8#1994—32no—
9C42×C8D311yescharacteristic
10#1567—31no—
11#1493—32no—
12A4⋊Q8—42no—
13C8×A4—41no—
14C2×C42⋊C3D411yescharacteristic
15#533—32no—
16#500—31no—
17#494—31no—
18#487—32no—
19#467—61no—
20#467—61no—
21C4×Q16—61no—
22C4×Q16—61no—
23#423—32no—
24C2×C4×C8—31no—
25C2×C4×C8—31no—
26C43C2×D311yescharacteristic
27#339—62no—
28#339—62no—
29#335—61no—
30#335—61no—
31Q8⋊C8—64no—
32C4×A4—41no—
33SL(2,ℤ/4)—82no—
34Dic12—161no—
35C42⋊C3Q1611yescharacteristic, complemented
36C2×Q16—61no—
37C2×Q16—61no—
38C2×M16—62no—
39C22×C8—31no—
40C22×C8S411yescharacteristic
41C4⋊Q8—32no—
42C4⋊Q8—32no—
43#125—62no—
44C22⋊Q8—62no—
45C22⋊Q8—62no—
46C4×Q8—62no—
47C4×Q8—62no—
48#117—62no—
49C2×C42—31no—
50C2×C42—31no—
51C2×C42D1211yescharacteristic
52#107—61no—
53#107—61no—
54C4⋊C8—62no—
55C4⋊C8—62no—
56#103—62no—
57#103—62no—
58C4×C8—31no—
59C4×C8—31no—
60C4×C8—31no—
61C4×C8—31no—
62C4×C8—32no—
63C4×C8—61no—
64C2×A4—41no—
65Dic6—162no—
66C3×C8—161no—
67C2×Q8—62no—
68C2×Q8—62no—
69C22×C4—31no—
70C22×C4—31no—
71C22×C4C2×S411yescharacteristic
72Q16—122no—
73Q16—121no—
74Q16—121no—
75M16—122no—
76M16—122no—
77C2×C8—31no—
78C2×C8—31no—
79C2×C8—31no—
80C2×C8—31no—
81C2×C8—62no—
82C2×C8—122no—
83C4⋊C4—62no—
84C4⋊C4—62no—
85C4⋊C4—62no—
86C4⋊C4—62no—
87C4⋊C4—122no—
88C4⋊C4—122no—
89C22⋊C4—122no—
90C42—31no—
91C42—31no—
92C42—31no—
93C42—31no—
94C42—31no—
95C42—32no—
96C42—61no—
97C42—122no—
98C42Dic1211yescharacteristic, complemented
99A4—41no—
100C3×C4—161no—
101Dic3—322no—
102C23C4⋊S411yescharacteristic
103Q8—122no—
104Q8—124no—
105Q8—122no—
106C2×C4—31no—
107C2×C4—31no—
108C2×C4—31no—
109C2×C4—31no—
110C2×C4—31no—
111C2×C4—31no—
112C2×C4—31no—
113C2×C4—31no—
114C2×C4—31no—
115C2×C4—31no—
116C2×C4—62no—
117C2×C4—122no—
118C2×C4—122no—
119C8—32no—
120C8—31no—
121C8—61no—
122C8—242no—
123C8#85711yescharacteristic
124C2×C3—161no—
125C22—31no—
126C22—31no—
127C22#539811yescharacteristic
128C4—31no—
129C4—31no—
130C4—31no—
131C4—31no—
132C4—32no—
133C4—31no—
134C4—61no—
135C4—242no—
136C4—242no—
137C4C2×#85711yescharacteristic
138C3—161noSylow
139C2—31no—
140C2—31no—
141C2C4⋊#85711yescharacteristic
subgroupG/H
centerC2C4⋊#857
commutator, cosocleC4×C42⋊C3C22
FrattiniC22×C4C2×S4
FittingC42×C8D3
socleC23C4⋊S4
prime radicalC2×C42⋊C3D4
hypercenterC8#857
hypocenterC42⋊C3Q16
LCSG ▷ C4×C42⋊C3 ▷ C2×C42⋊C3 ▷ C42⋊C3
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
35. C42⋊C3Q16semidirect
98. C42Dic12semidirect

Maximal subgroups

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

subgroupindex
#729802
C8×C42⋊C32
#98453
#53984