Finite Group Databasebeta

← #73025

Subgroups — #73025

Order 384.

127 conjugacy classes of subgroups; only the 11 normal subgroups are shown.

normal subgroups only

This group has 127 conjugacy classes of subgroups — too many to draw. Only its 11 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#5933C211yesmaximal, characteristic, complemented
2#5932C211yesmaximal, characteristic, complemented
3B(4,𝔽2)⋊C3C211yesmaximal, characteristic
4#2197—31nomaximal, Sylow
5#997C2211yescharacteristic
6C2×SL(2,ℤ/4)—41nomaximal
7#551—31no—
8#458—31no—
9B(4,𝔽2)D311yescharacteristic
10#455—31no—
11#354—31no—
12#352—31no—
13AΣL(1,𝔽16)—31no—
14C22×A4—41no—
15SL(2,ℤ/4)—41no—
16SL(2,ℤ/4)—41no—
17SL(2,ℤ/4)—81no—
18#226—81no—
19H(5,𝔽2)C2×D311yescharacteristic
20#137—61no—
21#126—61no—
22#124—31no—
23#123—31no—
24#123—61no—
25#123—61no—
26C22⋊Q8—61no—
27#120—31no—
28C4×D4—61no—
29#117—61no—
30C2×C22⋊C4—61no—
31#104—61no—
32#100—31no—
33#99—31no—
34#99—31no—
35#99—61no—
36C2×A4—42no—
37C2×A4—41no—
38C2×A4—81no—
39C2×Dic3—161no—
40SL(2,𝔽3)—81no—
41C24S411yescharacteristic
42#41—61no—
43C2×Q8—31no—
44C2×D4—31no—
45C2×D4—61no—
46C2×D4—61no—
47C22×C4—31no—
48C22×C4—61no—
49C22×C4—61no—
50Q16—121no—
51SD16—121no—
52M16—61no—
53C4⋊C4—61no—
54C4⋊C4—61no—
55C4⋊C4—61no—
56C4⋊C4—61no—
57C4⋊C4—121no—
58C22⋊C4—31no—
59C22⋊C4—31no—
60C22⋊C4—61no—
61C22⋊C4—61no—
62C22⋊C4—61no—
63C22⋊C4—61no—
64C22⋊C4—61no—
65C22⋊C4—61no—
66C22⋊C4—61no—
67C22⋊C4—121no—
68C42—31no—
69C42—61no—
70C22×C3—161no—
71A4—41no—
72A4—81no—
73Dic3—161no—
74Dic3—161no—
75C23—21no—
76C23—31no—
77C23—42no—
78C23—61no—
79C23C2×S411yescharacteristic
80Q8—21no—
81Q8—121no—
82D4—61no—
83D4—121no—
84D4—122no—
85C2×C4—31no—
86C2×C4—61no—
87C2×C4—61no—
88C2×C4—61no—
89C2×C4—61no—
90C2×C4—61no—
91C2×C4—61no—
92C2×C4—122no—
93C2×C4—121no—
94C2×C4—121no—
95C2×C4—241no—
96C8—121no—
97C2×C3—162no—
98C2×C3—161no—
99C22—31no—
100C22—41no—
101C22—41no—
102C22—62no—
103C22—61no—
104C22—81no—
105C22—121no—
106C22—121no—
107C4—61no—
108C4—121no—
109C4—121no—
110C4—121no—
111C4—121no—
112C4—121no—
113C4—241no—
114C3—161noSylow
115C2—42no—
116C2—61no—
117C2—121no—
118C2#1020⋊C211yescharacteristic
subgroupG/H
center, socle, hypercenterC2#1020⋊C2
commutator, cosocle, hypocenter#997C22
FrattiniC23C2×S4
FittingB(4,𝔽2)D3
prime radicalB(4,𝔽2)⋊C3C2
LCSG ▷ #997
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #5933C2semidirect
2. #5932C2semidirect

Maximal subgroups

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

subgroupindex
#59332
#59322
B(4,𝔽2)⋊C32
#21973
C2×SL(2,ℤ/4)4