Finite Group Databasebeta

← #71955

Subgroups — #71955

Order 384.

206 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 49 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×#563C211yesmaximal, characteristic, complemented
2#5067C212yesmaximal, complemented
3#5058C211yesmaximal, characteristic
4#4543C212yesmaximal, complemented
5#4540C211yesmaximal, characteristic, complemented
6#1657—31nomaximal, Sylow
7C3×C4⋊Q8—21no—
8C3×#126—22no—
9C3×#125C2212yes—
10C3×#124C2211yescharacteristic
11C3×#123—21no—
12C3×C22⋊Q8—22no—
13C2×#215—21no—
14C2×#213—21no—
15#874C2211yescharacteristic
16#809—22no—
17#809—22no—
18#803C2212yes—
19#803C2211yescharacteristic
20#563D311yescharacteristic, complemented
21#496—31no—
22#490—32no—
23#330—32no—
24#329—31no—
25C2×C3×Q8—21no—
26C2×C3×Q8D411yescharacteristic
27C2×C3×D4D411yescharacteristic
28C22×C3×C4—21no—
29C2×D12—22no—
30C2×D12—21no—
31C3×C4⋊C4—22no—
32C3×C4⋊C4—22no—
33C3×C4⋊C4—22no—
34C3×C4⋊C4D412yes—
35C3×C4⋊C4D412yes—
36C3×C22⋊C4—22no—
37C3×C22⋊C4—21no—
38C3×C22⋊C4—22no—
39C3×C42—21no—
40C3×C42C2311yescharacteristic
41#215—42no—
42#213—42no—
43C2×C3⋊C8—21no—
44C2×C3⋊C8—22no—
45C2×SD16—61no—
46C2×D8—61no—
47C4⋊Q8—21no—
48#127—31no—
49#126—22no—
50#125C2×D312yes—
51#124C2×D311yescharacteristic
52#123—21no—
53C22⋊Q8—22no—
54#102—62no—
55#102—62no—
56#97—32no—
57#97—31no—
58C23×C3—21no—
59C22×D3—41no—
60C3×Q8—41no—
61C3×Q8—81no—
62C3×D4—41no—
63C2×C3×C4—22no—
64C2×C3×C4—21no—
65C2×C3×C4—22no—
66C2×C3×C4—81no—
67C2×C3×C4C2×D412yes—
68C2×C3×C4C2×D411yescharacteristic
69D12—41no—
70D12—41no—
71D12—44no—
72C3⋊C8—22no—
73C3⋊C8—42no—
74C2×Q8—21no—
75C2×Q8#6711yescharacteristic
76C2×D4—62no—
77C2×D4—61no—
78C2×D4#6711yescharacteristic
79C22×C4—21no—
80SD16—122no—
81D8—122no—
82C2×C8—61no—
83C2×C8—62no—
84C4⋊C4—22no—
85C4⋊C4—22no—
86C4⋊C4—22no—
87C4⋊C4#6712yes—
88C4⋊C4#6712yes—
89C22⋊C4—22no—
90C22⋊C4—21no—
91C22⋊C4—22no—
92C42—21no—
93C42C22×D311yescharacteristic
94C22×C3—41no—
95C22×C3—42no—
96C22×C3#12011yescharacteristic
97C2×D3—81no—
98C2×D3—82no—
99C3×C4—21no—
100C3×C4—22no—
101C3×C4—42no—
102C3×C4—41no—
103C3×C4—42no—
104C23—21no—
105C23—121no—
106Q8—41no—
107Q8—81no—
108D4—41no—
109D4—121no—
110D4—121no—
111D4—124no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4C2×#6712yes—
117C2×C4C2×#6711yescharacteristic
118C8—62no—
119C8—122no—
120C2×C3—81no—
121C2×C3#45311yescharacteristic
122C2×C3#45412yes—
123D3—161no—
124C22—41no—
125C22—42no—
126C22—241no—
127C22—242no—
128C22#95311yescharacteristic
129C4—21no—
130C4—22no—
131C4—42no—
132C4—41no—
133C4—42no—
134C3#165711yescharacteristic, complemented, Sylow
135C2—81no—
136C2—481no—
137C2#524111yescharacteristic
138C2#524512yes—
subgroupG/H
centerC22#953
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#563C2
socleC22×C3#120
prime radical#5058C2
hypercenter#563D3
hypocenterC3#1657
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. C3×#563C2semidirect
2. #5067C2semidirect
4. #4543C2semidirect
5. #4540C2semidirect
20. #563D3semidirect
134. C3#1657semidirect

Maximal subgroups

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

subgroupindex
C3×#5632
#50672
#50582
#45432
#45402
#16573