Finite Group Databasebeta

← #85334

Subgroups — #85334

Order 384.

204 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 40 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
1C4×#41⋊C3C211yesmaximal, characteristic
2#5424C212yesmaximal
3#5391C211yesmaximal, characteristic, complemented
4#5389C212yesmaximal, complemented
5#5386C211yesmaximal, characteristic, complemented
6#3076—31nomaximal, Sylow
7C2×#41⋊C3C2211yescharacteristic
8C4×Dic6—41nomaximal
9C4×SL(2,𝔽3)C2211yescharacteristic
10C4×SL(2,𝔽3)C2211yescharacteristic
11#859C2212yes—
12#859C2212yes—
13#533—31no—
14C4×#41D311yescharacteristic
15#479—32no—
16#478—32no—
17#477—31no—
18#476—31no—
19#475—31no—
20#474—31no—
21#427—32no—
22C2×C4⋊C8—31no—
23#416—32no—
24C2×Dic6—41no—
25#41⋊C3Q812yescomplemented
26C2×SL(2,𝔽3)C2311yescharacteristic
27C3×C42—41no—
28C4⋊Dic3—41no—
29Dic3⋊C4—42no—
30C4×Dic3—42no—
31C2×#41C2×D311yescharacteristic
32C22×C8—32no—
33C4⋊Q8—31no—
34C4⋊Q8—31no—
35#125—32no—
36C22⋊Q8—61no—
37C22⋊Q8—61no—
38C4×Q8—61no—
39C4×Q8—61no—
40C4×Q8C2×D311yescharacteristic
41C4×Q8C2×D311yescharacteristic
42C4×D4—31no—
43C4×D4—31no—
44#117—31no—
45#117—32no—
46C2×C42—31no—
47#107—32no—
48#107—32no—
49#106—32no—
50#106—32no—
51C4⋊C8—31no—
52C4⋊C8—31no—
53C4⋊C8—32no—
54#103—32no—
55#103—32no—
56#102—32no—
57#102—32no—
58C2×C3×C4—41no—
59C2×C3×C4—41no—
60C2×C3×C4—41no—
61C2×Dic3—42no—
62C2×Dic3—42no—
63Dic6—44no—
64SL(2,𝔽3)C2×Q811yescharacteristic
65#41—32no—
66#41Dic612yescomplemented
67C2×Q8—61no—
68C2×Q8—61no—
69C2×Q8C22×D311yescharacteristic
70C2×D4—31no—
71C22×C4—31no—
72C22×C4—31no—
73C22×C4—31no—
74C2×C8—32no—
75C2×C8—32no—
76C2×C8—64no—
77C4⋊C4—31no—
78C4⋊C4—31no—
79C4⋊C4—32no—
80C4⋊C4—32no—
81C4⋊C4—62no—
82C4⋊C4—61no—
83C4⋊C4—62no—
84C4⋊C4—61no—
85C22⋊C4—61no—
86C22⋊C4—62no—
87C42—31no—
88C42—31no—
89C42—31no—
90C42—62no—
91C42S411yescharacteristic
92C22×C3—41no—
93C3×C4—42no—
94C3×C4—42no—
95C3×C4—81no—
96Dic3—44no—
97Dic3—82no—
98C23—31no—
99Q8—31no—
100Q8—122no—
101Q8—122no—
102Q8C2×Dic611yescharacteristic
103D4—32no—
104D4—61no—
105C2×C4—31no—
106C2×C4—32no—
107C2×C4—32no—
108C2×C4—31no—
109C2×C4—31no—
110C2×C4—62no—
111C2×C4—62no—
112C2×C4—62no—
113C2×C4—121no—
114C2×C4C2×S411yescharacteristic
115C2×C4C2×S411yescharacteristic
116C2×C4C2×S411yescharacteristic
117C8—62no—
118C8—62no—
119C2×C3—41no—
120C2×C3—41no—
121C2×C3—41no—
122C22—32no—
123C22—61no—
124C22—61no—
125C22C22×S411yescharacteristic
126C4—21no—
127C4—31no—
128C4—31no—
129C4—61no—
130C4—61no—
131C4—122no—
132C4—122no—
133C4A4⋊Q812yes—
134C4#98512yes—
135C3—41noSylow
136C2—62no—
137C2C2×A4⋊Q811yescharacteristic
138C2C2×#98511yescharacteristic
139C2#592311yescharacteristic
subgroupG/H
centerC2×C4C2×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Frattini, socleC22C22×S4
FittingC4×#41D3
prime radicalC2×#41⋊C3C22
hypercenterC42S4
hypocenterSL(2,𝔽3)C2×Q8
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
3. #5391C2semidirect
4. #5389C2semidirect
5. #5386C2semidirect
25. #41⋊C3Q8semidirect
66. #41Dic6semidirect

Maximal subgroups

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

subgroupindex
C4×#41⋊C32
#54242
#53912
#53892
#53862
#30763
C4×Dic64