Finite Group Databasebeta

← #71940

Subgroups — #71940

Order 384.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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×Q82C211yesmaximal, characteristic
2#5092C211yesmaximal, characteristic
3#5021C212yesmaximal
4#4546C211yesmaximal, characteristic
5#4535C212yesmaximal
6#1633—31nomaximal, Sylow
7C3×C4⋊Q8—22no—
8C3×C4⋊Q8—22no—
9C3×C4⋊Q8C2211yescharacteristic
10C3×C4×Q8—22no—
11C3×C4×Q8C2212yes—
12C2×#216—22no—
13#869C2211yescharacteristic
14#835—22no—
15#807—22no—
16C4⋊(C3⋊C8)C2212yes—
17C4×C3⋊C8C2211yescharacteristic
18Q82D311yescharacteristic
19#494—31no—
20#477—32no—
21#333—31no—
22Q8⋊C8—32no—
23C2×C3×Q8—22no—
24C2×C3×Q8D412yes—
25C2×C3×Q8D412yes—
26C2×Dic6—21no—
27C3×C4⋊C4—24no—
28C3×C4⋊C4—22no—
29C3×C4⋊C4—22no—
30C3×C4⋊C4D412yes—
31C3×C42—22no—
32C3×C42—22no—
33C3×C42C2311yescharacteristic
34#216—44no—
35C4⋊Dic3—22no—
36C2×C3⋊C8—21no—
37C2×C3⋊C8—22no—
38C2×Q16—62no—
39C4⋊Q8—22no—
40C4⋊Q8—22no—
41C4⋊Q8—31no—
42C4⋊Q8C2×D311yescharacteristic
43C4×Q8—22no—
44C4×Q8C2×D312yes—
45#107—62no—
46C4⋊C8—32no—
47#103—62no—
48C4×C8—31no—
49C3×Q8—22no—
50C3×Q8—42no—
51C3×Q8—82no—
52C3×Q8Q1614yescomplemented
53C2×C3×C4—22no—
54C2×C3×C4—22no—
55C2×C3×C4—22no—
56C2×C3×C4C2×D411yescharacteristic
57C2×C3×C4C2×D412yes—
58C2×Dic3—41no—
59Dic6—42no—
60C3⋊C8—22no—
61C3⋊C8—42no—
62C2×Q8—22no—
63C2×Q8—61no—
64C2×Q8#6712yes—
65C2×Q8#6712yes—
66Q16—124no—
67C2×C8—61no—
68C2×C8—62no—
69C4⋊C4—24no—
70C4⋊C4—22no—
71C4⋊C4—22no—
72C4⋊C4—62no—
73C4⋊C4#6712yes—
74C42—22no—
75C42—22no—
76C42C22×D311yescharacteristic
77C22×C3#12011yescharacteristic
78C3×C4—24no—
79C3×C4—21no—
80C3×C4—42no—
81C3×C4—42no—
82C3×C4C2×Q1612yes—
83C3×C4#13712yes—
84Dic3—81no—
85Q8—22no—
86Q8—42no—
87Q8—82no—
88Q8—122no—
89Q8#21614yescomplemented
90C2×C4—22no—
91C2×C4—22no—
92C2×C4—22no—
93C2×C4—121no—
94C2×C4C2×#6711yescharacteristic
95C2×C4C2×#6712yes—
96C8—62no—
97C8—122no—
98C2×C3#45112yes—
99C2×C3#45311yescharacteristic
100C22#95311yescharacteristic
101C4—24no—
102C4—21no—
103C4—42no—
104C4—42no—
105C4—241no—
106C4#94212yes—
107C4C2×#21612yes—
108C3#163311yescharacteristic, complemented, Sylow
109C2#522812yes—
110C2#524111yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×Q82C2
socle, prime radicalC22×C3#120
hypercenterQ82D3
hypocenterC3#1633
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ Q82

Complements

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

NH ≅ G/Ntype
52. C3×Q8Q16semidirect
89. Q8#216semidirect
108. C3#1633semidirect

Maximal subgroups

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

subgroupindex
C3×Q822
#50922
#50212
#45462
#45352
#16333