Finite Group Databasebeta

← #87519

Subgroups — #87519

Order 384.

293 conjugacy classes of subgroups; only the 16 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 16 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#5950C211yesmaximal, characteristic, complemented
2#5934C211yesmaximal, characteristic, complemented
3#5932C211yesmaximal, characteristic, complemented
4#3025—31nomaximal, Sylow
5#997C2211yescharacteristic
6C4×S4—43nomaximal
7#585D311yescharacteristic, complemented
8#525—33no—
9#458—33no—
10#458—31no—
11B(4,𝔽2)—33no—
12B(4,𝔽2)—31no—
13#410—33no—
14C2×S4—43no—
15#41⋊C3—81no—
16C4×A4—43no—
17SL(2,ℤ/4)—43no—
182−1+4—61no—
19H(5,𝔽2)—33no—
20H(5,𝔽2)C2×D311yescharacteristic
21C2×#41—33no—
22C2×#41—61no—
23C23×C4—31no—
24#123—33no—
25#123—33no—
26C22⋊Q8—63no—
27#121—63no—
28#120—33no—
29#120—33no—
30C4×D4—63no—
31C4×D4—63no—
32#117—33no—
33#99—36no—
34#99—33no—
35#99—33no—
36C2×A4—43no—
37S4—46no—
38C4×D3—161no—
39SL(2,𝔽3)—81no—
40C24—31no—
41#41—21no—
42#41—61no—
43#41—63no—
44#41—66no—
45#41—63no—
46C2×Q8—33no—
47C2×Q8—61no—
48C2×D4—33no—
49C2×D4—33no—
50C2×D4—33no—
51C2×D4—63no—
52C2×D4—63no—
53C2×D4—63no—
54C22×C4—31no—
55C22×C4—33no—
56C22×C4—31no—
57C22×C4—63no—
58C22×C4—63no—
59C22×C4S413yescomplemented
60C4⋊C4—63no—
61C4⋊C4—63no—
62C4⋊C4—63no—
63C22⋊C4—33no—
64C22⋊C4—33no—
65C22⋊C4—63no—
66C22⋊C4—63no—
67C22⋊C4—63no—
68C22⋊C4—63no—
69C42—63no—
70C2×D3—161no—
71A4—43no—
72C3×C4—161no—
73Dic3—161no—
74C23—31no—
75C23—33no—
76C23—63no—
77C23—122no—
78C23C2×S413yes—
79Q8—21no—
80Q8—63no—
81D4—66no—
82D4—63no—
83D4—61no—
84D4—126no—
85D4—126no—
86C2×C4—33no—
87C2×C4—33no—
88C2×C4—33no—
89C2×C4—33no—
90C2×C4—63no—
91C2×C4—61no—
92C2×C4—63no—
93C2×C4—63no—
94C2×C4—63no—
95C2×C4—123no—
96C2×C4—121no—
97C2×C3—161no—
98D3—162no—
99C22—33no—
100C22—43no—
101C22—66no—
102C22—61no—
103C22—121no—
104C22—123no—
105C22—123no—
106C22—121no—
107C4—33no—
108C4—61no—
109C4—123no—
110C4—123no—
111C4—121no—
112C4#102011yescharacteristic
113C3—161noSylow
114C2—63no—
115C2—122no—
116C2—121no—
117C2C2×#102011yescharacteristic
subgroupG/H
center, hypercenterC4#1020
commutator, cosocle, hypocenter#997C22
Frattini, socleC2C2×#1020
Fitting#585D3
LCSG ▷ #997
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #5950C2semidirect
2. #5934C2semidirect
3. #5932C2semidirect
7. #585D3semidirect
59. C22×C4S4semidirect

Maximal subgroups

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

subgroupindex
#59502
#59342
#59322
#30253
C4×S44