Finite Group Databasebeta

← #85332

Subgroups — #85332

Order 384.

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

normal subgroups only

This group has 241 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, complemented
2#5421C212yesmaximal, complemented
3#5394C212yesmaximal, complemented
4C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5#5391C211yesmaximal, characteristic, complemented
6#3112—31nomaximal, Sylow
7C22×SL(2,𝔽3)C2211yescharacteristic
8C2×GL(2,𝔽3)C2212yes—
9#930—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11C4×SL(2,𝔽3)C2211yescharacteristic
12#859C2212yes—
13#532—31no—
14C2×C4×Q8D311yescharacteristic, complemented
15#480—32no—
16#477—31no—
17#477—31no—
18#460—31no—
19#460—31no—
20#448—32no—
21#440—32no—
22#440—32no—
23#432—31no—
24C22×C3×C4—41no—
25C2×#67—42no—
26C2×D12—41no—
27C2×SL(2,𝔽3)—21no—
28C2×SL(2,𝔽3)C2311yescharacteristic
29GL(2,𝔽3)—22no—
30(C2×D3)⋊C4—42no—
31C4⋊Dic3—41no—
32C2×#41—61no—
33C22×Q8C2×D311yescharacteristic
34C2×SD16—32no—
35C2×SD16—32no—
36C4⋊Q8—31no—
37#127—31no—
38#124—61no—
39#121—32no—
40#121—61no—
41C4×Q8—31no—
42C4×Q8—31no—
43C4×Q8C2×D311yescharacteristic
44C4×Q8C2×D311yescharacteristic
45C4×D4—32no—
46C4×D4—61no—
47C2×C4⋊C4—31no—
48C2×C42—31no—
49#107—32no—
50#107—32no—
51C4⋊C8—31no—
52C4⋊C8—31no—
53#103—32no—
54#103—32no—
55#102—32no—
56#102—32no—
57C22⋊C8—32no—
58#97—32no—
59C23×C3—41no—
60C22×D3—42no—
61C2×C3×C4—41no—
62C2×C3×C4—41no—
63C2×C3×C4—82no—
64#67—84no—
65C2×Dic3—42no—
66D12—82no—
67SL(2,𝔽3)#4111yescharacteristic, complemented
68#41—124no—
69C2×Q8—21no—
70C2×Q8—31no—
71C2×Q8—61no—
72C2×Q8—61no—
73C2×Q8C22×D311yescharacteristic
74C2×D4—32no—
75C2×D4—61no—
76C2×D4—62no—
77C2×D4—61no—
78C22×C4—31no—
79C22×C4—31no—
80C22×C4—62no—
81C22×C4S411yescharacteristic
82SD16—62no—
83SD16—64no—
84SD16—62no—
85C2×C8—32no—
86C2×C8—32no—
87C4⋊C4—31no—
88C4⋊C4—31no—
89C4⋊C4—32no—
90C4⋊C4—61no—
91C4⋊C4—61no—
92C22⋊C4—62no—
93C22⋊C4—62no—
94C42—31no—
95C42—31no—
96C42—31no—
97C42—31no—
98C22×C3—42no—
99C22×C3—41no—
100C22×C3—81no—
101C22×C3—81no—
102C2×D3—82no—
103C2×D3—82no—
104C2×D3—82no—
105C3×C4—42no—
106C3×C4—81no—
107Dic3—82no—
108C23—62no—
109C23C2×S411yescharacteristic
110Q8—31no—
111Q8—32no—
112Q8—61no—
113Q8—122no—
114Q8#23511yescharacteristic
115D4—62no—
116D4—62no—
117D4—122no—
118D4—124no—
119D4—122no—
120C2×C4—22no—
121C2×C4—31no—
122C2×C4—31no—
123C2×C4—31no—
124C2×C4—31no—
125C2×C4—61no—
126C2×C4—61no—
127C2×C4—62no—
128C2×C4—62no—
129C2×C4—124no—
130C2×C4C2×S411yescharacteristic
131C2×C4C2×S411yescharacteristic
132C8—62no—
133C8—62no—
134C2×C3—41no—
135C2×C3—41no—
136C2×C3—41no—
137C2×C3—82no—
138D3—162no—
139C22—21no—
140C22—21no—
141C22—122no—
142C22—122no—
143C22—122no—
144C22#98612yes—
145C22C22×S411yescharacteristic
146C4—21no—
147C4—31no—
148C4—31no—
149C4—61no—
150C4—61no—
151C4—61no—
152C4—122no—
153C4#98312yes—
154C3—41noSylow
155C2—22no—
156C2—242no—
157C2#591211yescharacteristic
158C2C2×#98311yescharacteristic
159C2C2×#98611yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingC2×C4×Q8D3
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
1. C2×C4×SL(2,𝔽3)C2semidirect
2. #5421C2semidirect
3. #5394C2semidirect
4. C4⋊GL(2,𝔽3)C2semidirect
5. #5391C2semidirect
14. C2×C4×Q8D3semidirect
67. SL(2,𝔽3)#41semidirect

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
#54212
#53942
C4⋊GL(2,𝔽3)2
#53912
#31123
#9304