Finite Group Databasebeta

← #85422

Subgroups — #85422

Order 384.

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

normal subgroups only

This group has 231 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#5394C212yesmaximal, complemented
3C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C4⋊GL(2,𝔽3)C212yesmaximal, complemented
5C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#3368—31nomaximal, Sylow
7C2×GL(2,𝔽3)C2211yescharacteristic
8C2×GL(2,𝔽3)C2212yes—
9#895—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11C4×SL(2,𝔽3)C2212yes—
12#859C2211yescharacteristic
13#557D311yescharacteristic, complemented
14#550—31no—
15#490—32no—
16#486—31no—
17#478—32no—
18#474—31no—
19#460—32no—
20#460—31no—
21#446—31no—
22#440—32no—
23C4×SD16—31no—
24C2×D12—41no—
25C2×D12—42no—
26C2×C4×D3—41no—
27C2×SL(2,𝔽3)C2311yescharacteristic
28GL(2,𝔽3)—21no—
29GL(2,𝔽3)—22no—
30C3×C4⋊C4—41no—
31(C2×D3)⋊C4—42no—
32C2×#41—61no—
33C2×SD16—31no—
34C2×SD16—32no—
35C4⋊Q8—31no—
36#127—31no—
37#127—32no—
38#125—32no—
39#121—62no—
40#121—61no—
41C4×Q8—31no—
42C4×Q8C2×D311yescharacteristic
43C4×Q8C2×D312yes—
44C4×D4—31no—
45C4×D4—32no—
46#107—32no—
47#106—31no—
48C4⋊C8—31no—
49C4⋊C8—32no—
50#103—31no—
51#102—31no—
52#102—32no—
53#102—32no—
54#102—32no—
55#102—32no—
56#97—32no—
57C4×C8—31no—
58C22×D3—41no—
59C22×D3—42no—
60C2×C3×C4—41no—
61C2×C3×C4—42no—
62C2×Dic3—41no—
63D12—82no—
64D12—84no—
65C4×D3—82no—
66SL(2,𝔽3)#4111yescharacteristic, complemented
67#41—124no—
68C2×Q8—31no—
69C2×Q8C22×D311yescharacteristic
70C2×D4—31no—
71C2×D4—32no—
72C2×D4—62no—
73C2×D4—61no—
74C2×D4—61no—
75C2×D4—62no—
76C22×C4—62no—
77C22×C4—61no—
78SD16—61no—
79SD16—62no—
80SD16—61no—
81SD16—62no—
82C2×C8—31no—
83C2×C8—31no—
84C2×C8—32no—
85C4⋊C4—31no—
86C4⋊C4—32no—
87C4⋊C4—32no—
88C4⋊C4—31no—
89C4⋊C4—61no—
90C4⋊C4S411yescharacteristic
91C22⋊C4—61no—
92C22⋊C4—62no—
93C42—31no—
94C42—32no—
95C22×C3—41no—
96C2×D3—42no—
97C2×D3—81no—
98C2×D3—81no—
99C2×D3—82no—
100C2×D3—82no—
101C2×D3—82no—
102C3×C4—42no—
103C3×C4—82no—
104Dic3—81no—
105C23—61no—
106C23—62no—
107Q8—31no—
108Q8—62no—
109Q8#23911yescharacteristic
110D4—61no—
111D4—62no—
112D4—61no—
113D4—62no—
114D4—124no—
115D4—122no—
116D4—242no—
117D4—241no—
118C2×C4—31no—
119C2×C4—31no—
120C2×C4—32no—
121C2×C4—61no—
122C2×C4—124no—
123C2×C4—122no—
124C2×C4C2×S412yes—
125C2×C4C2×S411yescharacteristic
126C8—61no—
127C8—61no—
128C8—62no—
129C2×C3—41no—
130C2×C3—41no—
131C2×C3—41no—
132D3—82no—
133D3—162no—
134C22—121no—
135C22—122no—
136C22—121no—
137C22—122no—
138C22—121no—
139C22—122no—
140C22C22×S411yescharacteristic
141C4—22no—
142C4—31no—
143C4—31no—
144C4—62no—
145C4—61no—
146C4—121no—
147C4#98612yes—
148C3—41noSylow
149C2—241no—
150C2—242no—
151C2#591911yescharacteristic
152C2C2×#98611yescharacteristic
153C2#592611yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#557D3
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. #5394C2semidirect
3. C4⋊GL(2,𝔽3)C2semidirect
4. C4⋊GL(2,𝔽3)C2semidirect
5. C4×GL(2,𝔽3)C2semidirect
13. #557D3semidirect
66. SL(2,𝔽3)#41semidirect

Maximal subgroups

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

subgroupindex
#557⋊C32
#53942
C4⋊GL(2,𝔽3)2
C4×GL(2,𝔽3)2
#33683
#8954