Finite Group Databasebeta

← #73032

Subgroups — #73032

Order 384.

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

normal subgroups only

This group has 135 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#5935C211yesmaximal, characteristic, complemented
2#5934C211yesmaximal, characteristic, complemented
3#458⋊C3C211yesmaximal, characteristic, complemented
4#2200—31nomaximal, Sylow
5#997C2211yescharacteristic
6C4⋊S4—41nomaximal
7#560—31no—
8#458D311yescharacteristic, complemented
9B(4,𝔽2)—31no—
10#453—31no—
11#353—31no—
12#352—31no—
13AΣL(1,𝔽16)—31no—
14C2×S4—41no—
15C2×S4—41no—
16C2×S4—81no—
17C4×A4—41no—
18GL(2,𝔽3)—81no—
19H(5,𝔽2)C2×D311yescharacteristic
20AGL(1,ℤ/8)—61no—
21#127—31no—
22#126—61no—
23#123—31no—
24#123—61no—
25#121—61no—
26#121—61no—
27#121—61no—
28#120—31no—
29#120—61no—
30#120—61no—
31#104—61no—
32#100—31no—
33#99—31no—
34#99—31no—
35#99—61no—
36C2×A4—41no—
37C2×A4—81no—
38S4—81no—
39S4—81no—
40S4—82no—
41D12—161no—
42SL(2,𝔽3)—81no—
43C24—31no—
44#41—61no—
45C2×D4—31no—
46C2×D4—31no—
47C2×D4—61no—
48C2×D4—61no—
49C2×D4—61no—
50C2×D4—61no—
51C2×D4—61no—
52C2×D4—61no—
53C2×D4—121no—
54C22×C4—61no—
55C22×C4S411yescharacteristic, complemented
56SD16—121no—
57D8—121no—
58M16—61no—
59C4⋊C4—61no—
60C4⋊C4—61no—
61C4⋊C4—61no—
62C22⋊C4—31no—
63C22⋊C4—31no—
64C22⋊C4—61no—
65C22⋊C4—61no—
66C22⋊C4—61no—
67C22⋊C4—61no—
68C22⋊C4—61no—
69C22⋊C4—61no—
70C22⋊C4—121no—
71C42—31no—
72C2×D3—161no—
73C2×D3—161no—
74A4—41no—
75A4—81no—
76C3×C4—161no—
77C23—21no—
78C23—31no—
79C23—61no—
80C23—61no—
81C23—121no—
82C23—241no—
83C23C2×S411yescharacteristic
84Q8—21no—
85D4—61no—
86D4—121no—
87D4—121no—
88D4—241no—
89D4—241no—
90D4—241no—
91D4—241no—
92D4—241no—
93D4—242no—
94C2×C4—31no—
95C2×C4—61no—
96C2×C4—61no—
97C2×C4—61no—
98C2×C4—61no—
99C2×C4—61no—
100C2×C4—121no—
101C2×C4—241no—
102C8—121no—
103C2×C3—161no—
104D3—321no—
105D3—321no—
106C22—31no—
107C22—41no—
108C22—61no—
109C22—81no—
110C22—121no—
111C22—121no—
112C22—121no—
113C22—121no—
114C22—121no—
115C22—122no—
116C22—121no—
117C22—121no—
118C22—241no—
119C4—41no—
120C4—61no—
121C4—121no—
122C4—121no—
123C4—121no—
124C4—241no—
125C3—161noSylow
126C2—61no—
127C2—121no—
128C2—241no—
129C2—241no—
130C2#1020⋊C211yescharacteristic
subgroupG/H
center, socle, hypercenterC2#1020⋊C2
commutator, cosocle, hypocenter#997C22
FrattiniC23C2×S4
Fitting#458D3
LCSG ▷ #997
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #5935C2semidirect
2. #5934C2semidirect
3. #458⋊C3C2semidirect
8. #458D3semidirect
55. C22×C4S4semidirect

Maximal subgroups

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

subgroupindex
#59352
#59342
#458⋊C32
#22003
C4⋊S44