Finite Group Databasebeta

← #85361

Subgroups — #85361

Order 384.

199 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 38 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#520⋊C3C211yesmaximal, characteristic
2#5426C211yesmaximal, characteristic
3#5424C211yesmaximal, characteristic
4#5389C212yesmaximal, complemented
5#5386C212yesmaximal, complemented
6#3084—31nomaximal, Sylow
7C2×#41⋊C3C2211yescharacteristic
8#888—41nomaximal
9C4×SL(2,𝔽3)C2212yes—
10#859C2211yescharacteristic
11#859C2212yes—
12#859C2211yescharacteristic
13#544—31no—
14#520D311yescharacteristic
15#479—32no—
16#478—32no—
17#476—32no—
18#475—32no—
19#428—31no—
20C2×#106—31no—
21#424—31no—
22#417—31no—
23#416—31no—
24C2×Dic6—42no—
25#41⋊C3Q812yescomplemented
26C2×SL(2,𝔽3)C2311yescharacteristic
27C3×C4⋊C4—41no—
28C4⋊Dic3—41no—
29Dic3⋊C4—42no—
30C4×Dic3—41no—
31C2×#41C2×D311yescharacteristic
32C2×M16—31no—
33C22×C8—31no—
34C4⋊Q8—32no—
35C4⋊Q8—61no—
36#125—32no—
37#125—61no—
38C22⋊Q8—62no—
39C4×Q8C2×D312yes—
40C4×D4—32no—
41#117—31no—
42#117—31no—
43C2×C4⋊C4—31no—
44C2×C4⋊C4—31no—
45#107—32no—
46#106—32no—
47#106—32no—
48#106—32no—
49C4⋊C8—32no—
50C4⋊C8—32no—
51#103—31no—
52#103—31no—
53#103—32no—
54#102—31no—
55#102—31no—
56#102—32no—
57C2×C3×C4—41no—
58C2×C3×C4—42no—
59C2×Dic3—41no—
60C2×Dic3—41no—
61C2×Dic3—42no—
62Dic6—84no—
63SL(2,𝔽3)C2×Q811yescharacteristic
64#41—32no—
65#41Dic612yescomplemented
66C2×Q8—62no—
67C2×Q8C22×D311yescharacteristic
68C2×D4—31no—
69C22×C4—31no—
70C22×C4—61no—
71C22×C4—61no—
72M16—62no—
73C2×C8—31no—
74C2×C8—31no—
75C2×C8—32no—
76C2×C8—62no—
77C4⋊C4—32no—
78C4⋊C4—31no—
79C4⋊C4—32no—
80C4⋊C4—31no—
81C4⋊C4—31no—
82C4⋊C4—62no—
83C4⋊C4—62no—
84C4⋊C4—61no—
85C4⋊C4—62no—
86C4⋊C4S411yescharacteristic
87C22⋊C4—32no—
88C22⋊C4—61no—
89C42—32no—
90C42—61no—
91C22×C3—41no—
92C3×C4—42no—
93C3×C4—82no—
94Dic3—42no—
95Dic3—42no—
96Dic3—82no—
97C23—31no—
98Q8—31no—
99Q8—242no—
100Q8C2×Dic611yescharacteristic
101D4—32no—
102D4—61no—
103C2×C4—31no—
104C2×C4—32no—
105C2×C4—32no—
106C2×C4—32no—
107C2×C4—61no—
108C2×C4—61no—
109C2×C4—62no—
110C2×C4—241no—
111C2×C4—241no—
112C2×C4C2×S412yes—
113C2×C4C2×S411yescharacteristic
114C8—61no—
115C8—61no—
116C8—62no—
117C2×C3—41no—
118C2×C3—41no—
119C2×C3—41no—
120C22—32no—
121C22—61no—
122C22—61no—
123C22C22×S411yescharacteristic
124C4—22no—
125C4—31no—
126C4—31no—
127C4—62no—
128C4—121no—
129C4—122no—
130C4—121no—
131C4A4⋊Q812yes—
132C3—41noSylow
133C2—62no—
134C2C2×A4⋊Q811yescharacteristic
135C2#592411yescharacteristic
136C2#592611yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#520D3
prime radicalC2×#41⋊C3C22
hypercenterC4⋊C4S4
hypocenterSL(2,𝔽3)C2×Q8
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
4. #5389C2semidirect
5. #5386C2semidirect
25. #41⋊C3Q8semidirect
65. #41Dic6semidirect

Maximal subgroups

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

subgroupindex
#520⋊C32
#54262
#54242
#53892
#53862
#30843
#8884