Finite Group Databasebeta

← #86571

Subgroups — #86571

Order 384.

996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 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
1C3×#543C211yesmaximal, characteristic
2#5623C2112yesmaximal, complemented
3#5622C2112yesmaximal, complemented
4#5483C213yesmaximal
5C2×#877C213yesmaximal
6#3458—31nomaximal, Sylow
7C23×Dic3C2211yescharacteristic
8C3×C22⋊Q8C22112yes—
9C2×C3×C22⋊C4C2213yes—
10C2×#217C2213yes—
11Q8×Dic3C2214yescomplemented
12#944C22112yescomplemented
13#924C22112yes—
14C2×Dic3⋊C4C2216yes—
15C2×C4×Dic3C2216yes—
16#889C22124yescomplemented
17#888C22112yescomplemented
18#887C22112yescomplemented
19#878C22124yes—
20#877C22112yes—
21#877C22112yes—
22#544—312no—
23#543—33no—
24#543D311yescharacteristic
25#533—312no—
26C2×#117—33no—
27C24×C3C2311yescharacteristic
28C2×C3×Q8C2314yes—
29C22×C3×C4C2316yes—
30C22×Dic3Q818yescomplemented
31C22×Dic3C2316yes—
32C22×Dic3C2313yes—
33C22×Dic3C2313yes—
34C2×Dic6C23112yes—
35C3×C4⋊C4C23112yes—
36C3×C22⋊C4C23112yes—
37#217C23112yes—
38C4⋊Dic3C23112yes—
39Dic3⋊C4C23124yes—
40Dic3⋊C4C23124yes—
41C4×Dic3C23112yes—
42C4×Dic3C23112yes—
43C23×C4—31no—
44C4⋊Q8—312no—
45C4⋊Q8—312no—
46#125—324no—
47C22⋊Q8—312no—
48C22⋊Q8—324no—
49C22⋊Q8C2×D3112yes—
50C4×Q8—34no—
51C4×Q8—312no—
52#117—312no—
53#117—312no—
54C2×C4⋊C4—36no—
55C2×C22⋊C4—33no—
56C2×C22⋊C4C2×D313yes—
57C2×C42—36no—
58C23×C3—42no—
59C23×C3C2×Q814yes—
60C23×C3C2413yes—
61C3×Q8—44no—
62C2×C3×C4—46no—
63C2×C3×C4C24112yes—
64C2×Dic3—46no—
65C2×Dic3C2×Q8112yes—
66C2×Dic3C2×Q8112yes—
67C2×Dic3C24112yes—
68C2×Dic3C2411yescharacteristic
69C2×Dic3C2413yes—
70Dic6—412no—
71C24C22×D311yescharacteristic
72C2×Q8—312no—
73C2×Q8C22×D314yes—
74C22×C4—38no—
75C22×C4—33no—
76C22×C4—33no—
77C22×C4—36no—
78C22×C4C22×D316yes—
79C4⋊C4—324no—
80C4⋊C4—324no—
81C4⋊C4—312no—
82C4⋊C4C22×D3112yes—
83C22⋊C4—312no—
84C22⋊C4C22×D3112yes—
85C42—312no—
86C42—312no—
87C22×C3—26no—
88C22×C3—44no—
89C22×C3C22×Q816yes—
90C22×C3C2511yescharacteristic
91C3×C4—212no—
92Dic3—212no—
93Dic3C22×Q812yes—
94Dic3C22×Q816yes—
95C23—42no—
96C23D3×Q814yes—
97C23C23×D313yes—
98Q8—44no—
99Q8—1212no—
100C2×C4—312no—
101C2×C4—312no—
102C2×C4—31no—
103C2×C4—33no—
104C2×C4—312no—
105C2×C4—46no—
106C2×C4—126no—
107C2×C4C23×D3112yes—
108C2×C3—26no—
109C2×C3C23×Q811yescharacteristic
110C2×C3#58512yes—
111C22—26no—
112C22—44no—
113C22C2×D3×Q816yes—
114C22C24×D311yescharacteristic
115C4—212no—
116C4—32no—
117C4—36no—
118C4—612no—
119C3#345811yescharacteristic, complemented, Sylow
120C2—26no—
121C2C22×D3×Q811yescharacteristic
122C2#596612yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#543C2
prime radicalC24×C3C23
hypercenter#543D3
hypocenterC3#3458
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
2. #5623C2semidirect
3. #5622C2semidirect
11. Q8×Dic3C22semidirect
12. #944C22semidirect
16. #889C22semidirect
17. #888C22semidirect
18. #887C22semidirect
30. C22×Dic3Q8semidirect
119. C3#3458semidirect

Maximal subgroups

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

subgroupindex
C3×#5432
#56232
#56222
#54832
C2×#8772
#34583