Finite Group Databasebeta

← #85319

Subgroups — #85319

Order 384.

209 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 36 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
1C2×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2#5419C212yesmaximal
3C4⋊#226C211yesmaximal, characteristic, complemented
4#5388C212yesmaximal, complemented
5#5386C211yesmaximal, characteristic, complemented
6#3113—31nomaximal, Sylow
7C22×SL(2,𝔽3)C2211yescharacteristic
8C2×#226C2212yes—
9#924—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11C4×SL(2,𝔽3)C2211yescharacteristic
12#859C2212yes—
13#533—31no—
14C2×C4×Q8D311yescharacteristic
15#483—32no—
16#475—31no—
17#475—31no—
18#462—31no—
19#462—31no—
20#451—32no—
21#441—32no—
22#441—32no—
23#432—31no—
24C22×C3×C4—41no—
25C2×Dic6—41no—
26C2×SL(2,𝔽3)—21no—
27C2×SL(2,𝔽3)C2311yescharacteristic
28#226—22no—
29#217—42no—
30C4⋊Dic3—41no—
31Dic3⋊C4—42no—
32C22×Q8C2×D311yescharacteristic
33C2×Q16—32no—
34C2×Q16—32no—
35C4⋊Q8—31no—
36C4⋊Q8—31no—
37#125—61no—
38C22⋊Q8—32no—
39C22⋊Q8—61no—
40C4×Q8—31no—
41C4×Q8—31no—
42C4×Q8—32no—
43C4×Q8—61no—
44C4×Q8C2×D311yescharacteristic
45C4×Q8C2×D311yescharacteristic
46#117—61no—
47C2×C4⋊C4—31no—
48C2×C42—31no—
49#106—32no—
50#106—32no—
51C4⋊C8—31no—
52C4⋊C8—31no—
53#103—32no—
54#103—32no—
55#103—32no—
56#103—32no—
57C22⋊C8—32no—
58#97—32no—
59C23×C3—41no—
60C2×C3×C4—41no—
61C2×C3×C4—41no—
62C2×C3×C4—82no—
63C2×Dic3—42no—
64C2×Dic3—42no—
65Dic6—82no—
66SL(2,𝔽3)#4111yescharacteristic
67C2×Q8—21no—
68C2×Q8—31no—
69C2×Q8—32no—
70C2×Q8—61no—
71C2×Q8—61no—
72C2×Q8C22×D311yescharacteristic
73C22×C4—31no—
74C22×C4—31no—
75C22×C4S411yescharacteristic
76Q16—62no—
77Q16—64no—
78Q16—62no—
79C2×C8—32no—
80C2×C8—32no—
81C4⋊C4—31no—
82C4⋊C4—31no—
83C4⋊C4—32no—
84C4⋊C4—61no—
85C4⋊C4—61no—
86C4⋊C4—61no—
87C4⋊C4—62no—
88C4⋊C4—62no—
89C4⋊C4—61no—
90C22⋊C4—62no—
91C42—31no—
92C42—31no—
93C42—31no—
94C42—31no—
95C42—62no—
96C22×C3—42no—
97C22×C3—41no—
98C22×C3—81no—
99C22×C3—81no—
100C3×C4—42no—
101C3×C4—81no—
102Dic3—82no—
103Dic3—82no—
104C23C2×S411yescharacteristic
105Q8—31no—
106Q8—32no—
107Q8—61no—
108Q8—62no—
109Q8—62no—
110Q8—122no—
111Q8#23511yescharacteristic
112C2×C4—22no—
113C2×C4—31no—
114C2×C4—31no—
115C2×C4—31no—
116C2×C4—31no—
117C2×C4—61no—
118C2×C4—61no—
119C2×C4—62no—
120C2×C4—62no—
121C2×C4—62no—
122C2×C4C2×S411yescharacteristic
123C2×C4C2×S411yescharacteristic
124C8—62no—
125C8—62no—
126C2×C3—41no—
127C2×C3—41no—
128C2×C3—41no—
129C2×C3—82no—
130C22—21no—
131C22—21no—
132C22#98412yes—
133C22C22×S411yescharacteristic
134C4—21no—
135C4—31no—
136C4—31no—
137C4—61no—
138C4—61no—
139C4—61no—
140C4—122no—
141C4—122no—
142C4#98312yes—
143C3—41noSylow
144C2—22no—
145C2#591211yescharacteristic
146C2C2×#98311yescharacteristic
147C2C2×#98411yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingC2×C4×Q8D3
prime radicalC22×SL(2,𝔽3)C22
hypercenterC22×C4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C4⋊#226C2semidirect
4. #5388C2semidirect
5. #5386C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×SL(2,𝔽3)2
#54192
C4⋊#2262
#53882
#53862
#31133
#9244