Finite Group Databasebeta

← #85393

Subgroups — #85393

Order 384.

226 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 42 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
1D4×SL(2,𝔽3)C211yesmaximal, characteristic
2#5419C212yesmaximal, complemented
3C2×#859C212yesmaximal, complemented
4#5391C211yesmaximal, characteristic, complemented
5C4×#226C211yesmaximal, characteristic, complemented
6#3299—31nomaximal, Sylow
7C22×SL(2,𝔽3)C2212yes—
8C2×#226C2211yescharacteristic
9D4×Dic3—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11#859C2211yescharacteristic
12#859C2212yes—
13#554—31no—
14D4×Q8D311yescharacteristic
15#487—31no—
16#484—32no—
17#477—31no—
18#467—32no—
19#462—31no—
20#451—32no—
21C4×Q16—31no—
22C8×D4—31no—
23C2×#103—32no—
24C2×C3×D4—41no—
25C22×Dic3—42no—
26C2×SL(2,𝔽3)—22no—
27C2×SL(2,𝔽3)C2311yescharacteristic
28#226—21no—
29#217—42no—
30C4⋊Dic3—41no—
31C4×Dic3—41no—
32C22×Q8C2×D312yes—
33C2×Q16—31no—
34C2×Q16—32no—
35C22×C8—32no—
36C4⋊Q8—31no—
37C4⋊Q8—31no—
38#125—61no—
39C22⋊Q8—32no—
40C22⋊Q8—32no—
41C22⋊Q8—62no—
42C4×Q8—31no—
43C4×Q8C2×D311yescharacteristic
44C4×D4—31no—
45C4×D4—61no—
46C2×C4⋊C4—32no—
47#107—31no—
48#107—32no—
49C4⋊C8—31no—
50#103—32no—
51#103—32no—
52#103—31no—
53#103—32no—
54#103—31no—
55#103—32no—
56C22⋊C8—32no—
57C4×C8—31no—
58C23×C3—42no—
59C3×D4—44no—
60C2×C3×C4—41no—
61C2×Dic3—41no—
62C2×Dic3—42no—
63C2×Dic3—41no—
64C2×Dic3—84no—
65SL(2,𝔽3)#4111yescharacteristic
66C2×Q8—22no—
67C2×Q8—32no—
68C2×Q8—31no—
69C2×Q8—61no—
70C2×Q8—62no—
71C2×Q8C22×D311yescharacteristic
72C2×D4S411yescharacteristic
73C22×C4—32no—
74C22×C4—62no—
75Q16—61no—
76Q16—61no—
77Q16—122no—
78C2×C8—31no—
79C2×C8—31no—
80C2×C8—32no—
81C2×C8—122no—
82C4⋊C4—31no—
83C4⋊C4—31no—
84C4⋊C4—31no—
85C4⋊C4—32no—
86C4⋊C4—61no—
87C4⋊C4—62no—
88C4⋊C4—61no—
89C4⋊C4—62no—
90C4⋊C4—61no—
91C22⋊C4—32no—
92C22⋊C4—62no—
93C42—31no—
94C42—61no—
95C22×C3—44no—
96C22×C3—41no—
97C22×C3—82no—
98C22×C3—82no—
99C3×C4—42no—
100Dic3—42no—
101Dic3—82no—
102Dic3—81no—
103C23C2×S412yes—
104Q8—31no—
105Q8—62no—
106Q8—62no—
107Q8—61no—
108Q8—61no—
109Q8—241no—
110Q8#23711yescharacteristic
111D4#22614yescomplemented
112C2×C4—31no—
113C2×C4—31no—
114C2×C4—32no—
115C2×C4—62no—
116C2×C4—62no—
117C2×C4—61no—
118C2×C4—61no—
119C2×C4—62no—
120C2×C4—242no—
121C2×C4C2×S411yescharacteristic
122C8—61no—
123C8—62no—
124C8—61no—
125C2×C3—41no—
126C2×C3—41no—
127C2×C3—41no—
128C2×C3—84no—
129C22—22no—
130C22—22no—
131C22C2×#22614yes—
132C22C22×S411yescharacteristic
133C4—31no—
134C4—31no—
135C4—62no—
136C4—61no—
137C4—121no—
138C4—121no—
139C4—122no—
140C4C2×#22612yes—
141C3—41noSylow
142C2—24no—
143C2#591411yescharacteristic
144C2C22×#22611yescharacteristic
145C2#592611yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingD4×Q8D3
prime radicalD4×SL(2,𝔽3)C2
hypercenterC2×D4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #5419C2semidirect
3. C2×#859C2semidirect
4. #5391C2semidirect
5. C4×#226C2semidirect
111. D4#226semidirect

Maximal subgroups

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

subgroupindex
D4×SL(2,𝔽3)2
#54192
C2×#8592
#53912
C4×#2262
#32993
D4×Dic34