Finite Group Databasebeta

← #85433

Subgroups — #85433

Order 384.

206 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 34 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
1Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#5395C212yesmaximal, complemented
3#5394C212yesmaximal, complemented
4C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
5C4×#226C211yesmaximal, characteristic, complemented
6#3375—31nomaximal, Sylow
7C2×GL(2,𝔽3)C2212yes—
8C2×#226C2211yescharacteristic
9#947—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11C4×SL(2,𝔽3)C2212yes—
12#859C2211yescharacteristic
13#559—31no—
14Q82D311yescharacteristic, complemented
15#489—32no—
16#486—31no—
17#477—31no—
18#474—32no—
19#464—32no—
20#460—31no—
21#446—31no—
22#440—32no—
23C4×Q16—31no—
24C2×C3×Q8—41no—
25C2×D12—41no—
26C2×SL(2,𝔽3)C2311yescharacteristic
27GL(2,𝔽3)—22no—
28#226—21no—
29(C2×D3)⋊C4—42no—
30(C2×D3)⋊C4—42no—
31C4×Dic3—41no—
32C2×Q16—31no—
33C2×SD16—32no—
34C4⋊Q8—32no—
35C4⋊Q8—31no—
36#127—31no—
37#124—32no—
38#124—61no—
39#123—62no—
40C4×Q8—31no—
41C4×Q8—31no—
42C4×Q8C2×D311yescharacteristic
43C4×Q8C2×D312yes—
44C4×D4—32no—
45C4×D4—61no—
46#107—31no—
47#107—32no—
48C4⋊C8—31no—
49C4⋊C8—32no—
50#103—31no—
51#103—32no—
52#103—31no—
53#102—32no—
54#102—32no—
55#102—32no—
56#97—32no—
57C4×C8—31no—
58C22×D3—42no—
59C3×Q8—82no—
60C2×C3×C4—41no—
61C2×C3×C4—42no—
62C2×Dic3—41no—
63C2×Dic3—41no—
64D12—82no—
65SL(2,𝔽3)#4111yescharacteristic
66C2×Q8—31no—
67C2×Q8—61no—
68C2×Q8S411yescharacteristic
69C2×Q8C22×D311yescharacteristic
70C2×D4—32no—
71C2×D4—61no—
72C2×D4—61no—
73C22×C4—62no—
74Q16—61no—
75Q16—61no—
76SD16—62no—
77SD16—62no—
78C2×C8—31no—
79C2×C8—31no—
80C2×C8—32no—
81C4⋊C4—32no—
82C4⋊C4—31no—
83C4⋊C4—31no—
84C4⋊C4—32no—
85C4⋊C4—31no—
86C4⋊C4—61no—
87C22⋊C4—62no—
88C22⋊C4—62no—
89C22⋊C4—62no—
90C42—31no—
91C42—32no—
92C42—61no—
93C22×C3—41no—
94C2×D3—82no—
95C2×D3—82no—
96C2×D3—82no—
97C3×C4—42no—
98C3×C4—82no—
99Dic3—81no—
100Dic3—81no—
101C23—62no—
102Q8—22no—
103Q8—31no—
104Q8—61no—
105Q8—61no—
106Q8—241no—
107Q8#23911yescharacteristic
108D4—62no—
109D4—62no—
110D4—122no—
111D4—241no—
112C2×C4—31no—
113C2×C4—32no—
114C2×C4—31no—
115C2×C4—61no—
116C2×C4—61no—
117C2×C4—242no—
118C2×C4C2×S412yes—
119C2×C4C2×S411yescharacteristic
120C8—61no—
121C8—62no—
122C8—61no—
123C2×C3—41no—
124C2×C3—41no—
125C2×C3—41no—
126D3—162no—
127C22—122no—
128C22—122no—
129C22—122no—
130C22C22×S411yescharacteristic
131C4—22no—
132C4—31no—
133C4—31no—
134C4—62no—
135C4—61no—
136C4—121no—
137C4—121no—
138C4#98312yes—
139C3—41noSylow
140C2—242no—
141C2C2×#98311yescharacteristic
142C2#591911yescharacteristic
143C2#592511yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingQ82D3
prime radicalC4⋊GL(2,𝔽3)C2
hypercenterC2×Q8S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×SL(2,𝔽3)C2semidirect
2. #5395C2semidirect
3. #5394C2semidirect
5. C4×#226C2semidirect
14. Q82D3semidirect

Maximal subgroups

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

subgroupindex
Q8×SL(2,𝔽3)2
#53952
#53942
C4⋊GL(2,𝔽3)2
C4×#2262
#33753
#9474