Finite Group Databasebeta

← #85420

Subgroups — #85420

Order 384.

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

normal subgroups only

This group has 214 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
1#557⋊C3C211yesmaximal, characteristic, complemented
2#5395C212yesmaximal, complemented
3C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
4C4×GL(2,𝔽3)C212yesmaximal, complemented
5C4×#226C211yesmaximal, characteristic, complemented
6#3365—31nomaximal, Sylow
7C2×GL(2,𝔽3)C2212yes—
8C2×#226C2211yescharacteristic
9#893—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11C4×SL(2,𝔽3)C2212yes—
12#859C2211yescharacteristic
13#559—31no—
14#557D311yescharacteristic, complemented
15#488—32no—
16#486—31no—
17#478—32no—
18#475—31no—
19#464—32no—
20#460—31no—
21C8×Q8—31no—
22C4×Q16—31no—
23C4×SD16—32no—
24C2×D12—41no—
25C2×C4×D3—42no—
26C2×SL(2,𝔽3)C2311yescharacteristic
27GL(2,𝔽3)—22no—
28#226—21no—
29C3×C4⋊C4—41no—
30(C2×D3)⋊C4—42no—
31C4×Dic3—41no—
32C2×Q16—31no—
33C2×SD16—32no—
34C4⋊Q8—31no—
35#127—31no—
36#125—32no—
37#124—32no—
38#124—61no—
39#123—62no—
40C4×Q8—31no—
41C4×Q8—31no—
42C4×Q8C2×D312yes—
43C4×Q8C2×D311yescharacteristic
44C4×D4—32no—
45C4×D4—61no—
46#107—31no—
47#106—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—
56C4×C8—32no—
57C4×C8—31no—
58C22×D3—42no—
59C2×C3×C4—41no—
60C2×C3×C4—42no—
61C2×Dic3—41no—
62C2×Dic3—41no—
63D12—44no—
64C4×D3—84no—
65SL(2,𝔽3)#4111yescharacteristic
66C2×Q8—31no—
67C2×Q8—31no—
68C2×Q8C22×D311yescharacteristic
69C2×D4—32no—
70C2×D4—61no—
71C2×D4—61no—
72C22×C4—62no—
73Q16—61no—
74Q16—61no—
75SD16—62no—
76SD16—62no—
77C2×C8—31no—
78C2×C8—31no—
79C2×C8—32no—
80C4⋊C4—32no—
81C4⋊C4—31no—
82C4⋊C4—32no—
83C4⋊C4—31no—
84C4⋊C4—61no—
85C4⋊C4—61no—
86C4⋊C4S411yescharacteristic
87C22⋊C4—62no—
88C22⋊C4—62no—
89C22⋊C4—62no—
90C42—32no—
91C42—31no—
92C42—61no—
93C22×C3—41no—
94C2×D3—44no—
95C2×D3—82no—
96C2×D3—82no—
97C3×C4—42no—
98C3×C4—82no—
99Dic3—42no—
100Dic3—81no—
101C23—62no—
102Q8—31no—
103Q8—61no—
104Q8—61no—
105Q8—62no—
106Q8#23911yescharacteristic
107D4—62no—
108D4—62no—
109D4—122no—
110D4—241no—
111C2×C4—31no—
112C2×C4—32no—
113C2×C4—31no—
114C2×C4—61no—
115C2×C4—61no—
116C2×C4—242no—
117C2×C4C2×S412yes—
118C2×C4C2×S411yescharacteristic
119C8—61no—
120C8—62no—
121C8—61no—
122C2×C3—41no—
123C2×C3—41no—
124C2×C3—41no—
125D3—84no—
126C22—122no—
127C22—122no—
128C22—122no—
129C22C22×S411yescharacteristic
130C4—22no—
131C4—31no—
132C4—31no—
133C4—62no—
134C4—61no—
135C4—121no—
136C4—121no—
137C4#98512yes—
138C3—41noSylow
139C2—242no—
140C2#591911yescharacteristic
141C2C2×#98511yescharacteristic
142C2#592611yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#557D3
prime radicalC4⋊GL(2,𝔽3)C2
hypercenterC4⋊C4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #557⋊C3C2semidirect
2. #5395C2semidirect
4. C4×GL(2,𝔽3)C2semidirect
5. C4×#226C2semidirect
14. #557D3semidirect

Maximal subgroups

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

subgroupindex
#557⋊C32
#53952
C4⋊GL(2,𝔽3)2
C4×GL(2,𝔽3)2
C4×#2262
#33653
#8934