Finite Group Databasebeta

← #85432

Subgroups — #85432

Order 384.

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

normal subgroups only

This group has 199 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
3C4×GL(2,𝔽3)C211yesmaximal, characteristic
4C4⋊#226C211yesmaximal, characteristic, complemented
5#5388C212yesmaximal, complemented
6#3369—31nomaximal, Sylow
7C2×GL(2,𝔽3)C2211yescharacteristic
8C2×#226C2212yes—
9#946—41nomaximal
10C4×SL(2,𝔽3)C2212yes—
11C4×SL(2,𝔽3)C2211yescharacteristic
12#859C2211yescharacteristic
13Q82D311yescharacteristic, complemented
14#556—31no—
15#489—32no—
16#487—31no—
17#476—31no—
18#475—32no—
19#464—32no—
20#462—31no—
21#446—31no—
22#441—32no—
23C4×SD16—31no—
24C2×C3×Q8—41no—
25C2×C4×D3—41no—
26C2×SL(2,𝔽3)C2311yescharacteristic
27GL(2,𝔽3)—21no—
28#226—22no—
29(C2×D3)⋊C4—42no—
30C4⋊Dic3—41no—
31Dic3⋊C4—42no—
32C2×Q16—32no—
33C2×SD16—31no—
34C4⋊Q8—32no—
35C4⋊Q8—31no—
36C4⋊Q8—31no—
37#126—62no—
38#124—32no—
39C22⋊Q8—61no—
40C4×Q8—31no—
41C4×Q8—32no—
42C4×Q8C2×D311yescharacteristic
43C4×Q8C2×D312yes—
44C4×D4—31no—
45#117—61no—
46#106—31no—
47#106—32no—
48C4⋊C8—31no—
49C4⋊C8—32no—
50#103—32no—
51#103—32no—
52#103—32no—
53#103—31no—
54#102—31no—
55#102—32no—
56#97—32no—
57C4×C8—31no—
58C22×D3—41no—
59C3×Q8—82no—
60C2×C3×C4—41no—
61C2×C3×C4—42no—
62C2×Dic3—41no—
63C2×Dic3—42no—
64C4×D3—82no—
65SL(2,𝔽3)#4111yescharacteristic
66C2×Q8—32no—
67C2×Q8—61no—
68C2×Q8S411yescharacteristic
69C2×Q8C22×D311yescharacteristic
70C2×D4—31no—
71C22×C4—61no—
72Q16—62no—
73Q16—62no—
74SD16—61no—
75SD16—61no—
76C2×C8—31no—
77C2×C8—31no—
78C2×C8—32no—
79C4⋊C4—32no—
80C4⋊C4—31no—
81C4⋊C4—31no—
82C4⋊C4—32no—
83C4⋊C4—31no—
84C4⋊C4—61no—
85C4⋊C4—61no—
86C4⋊C4—62no—
87C22⋊C4—62no—
88C22⋊C4—61no—
89C22⋊C4—62no—
90C42—31no—
91C42—32no—
92C42—62no—
93C22×C3—41no—
94C2×D3—42no—
95C2×D3—81no—
96C2×D3—81no—
97C3×C4—42no—
98C3×C4—82no—
99Dic3—81no—
100Dic3—82no—
101C23—61no—
102Q8—22no—
103Q8—31no—
104Q8—62no—
105Q8—62no—
106Q8—241no—
107Q8#23911yescharacteristic
108D4—61no—
109D4—61no—
110C2×C4—31no—
111C2×C4—32no—
112C2×C4—31no—
113C2×C4—61no—
114C2×C4—62no—
115C2×C4—122no—
116C2×C4C2×S412yes—
117C2×C4C2×S411yescharacteristic
118C8—61no—
119C8—61no—
120C8—62no—
121C2×C3—41no—
122C2×C3—41no—
123C2×C3—41no—
124D3—82no—
125C22—121no—
126C22—121no—
127C22—121no—
128C22C22×S411yescharacteristic
129C4—22no—
130C4—31no—
131C4—31no—
132C4—62no—
133C4—61no—
134C4—121no—
135C4—122no—
136C4#98312yes—
137C3—41noSylow
138C2—241no—
139C2C2×#98311yescharacteristic
140C2#591911yescharacteristic
141C2#592411yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingQ82D3
prime radicalC2×GL(2,𝔽3)C22
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
4. C4⋊#226C2semidirect
5. #5388C2semidirect
13. 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
C4×GL(2,𝔽3)2
C4⋊#2262
#53882
#33693
#9464