Finite Group Databasebeta

← #72989

Subgroups — #72989

Order 384.

177 conjugacy classes of subgroups; only the 14 normal subgroups are shown.

normal subgroups only

This group has 177 conjugacy classes of subgroups — too many to draw. Only its 14 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#5936C212yesmaximal
2#379⋊C3C211yesmaximal, characteristic
3#2029—31nomaximal, Sylow
4C2×C22⋊A4C2211yescharacteristic
5A4⋊Q8—41nomaximal
6#543—31no—
7#379—32no—
8#379D311yescharacteristic
9#342—32no—
10#342—31no—
11C22⋊A4Q811yescharacteristic, complemented
12C4×A4—41no—
13SL(2,ℤ/4)—42no—
14SL(2,ℤ/4)—82no—
15C25C2×D311yescharacteristic
16C22⋊Q8—62no—
17C22⋊Q8—62no—
18C22⋊Q8—61no—
19C22⋊Q8—61no—
20C2×C22⋊C4—32no—
21C2×C22⋊C4—31no—
22C2×C22⋊C4—31no—
23C2×C22⋊C4—32no—
24C2×C22⋊C4—62no—
25C2×C22⋊C4—62no—
26C2×A4—41no—
27C2×A4—82no—
28Dic6—161no—
29C24—31no—
30C24—31no—
31C24—62no—
32C24—62no—
33C24Dic611yescharacteristic, complemented
34C2×Q8—61no—
35C2×Q8—61no—
36C22×C4—32no—
37C22×C4—32no—
38C22×C4—31no—
39C22×C4—62no—
40C22×C4S411yescharacteristic
41C4⋊C4—61no—
42C4⋊C4—62no—
43C4⋊C4—62no—
44C4⋊C4—61no—
45C22⋊C4—62no—
46C22⋊C4—62no—
47C22⋊C4—62no—
48C22⋊C4—62no—
49C22⋊C4—62no—
50C22⋊C4—62no—
51C22⋊C4—122no—
52C22⋊C4—122no—
53C22⋊C4—122no—
54C22⋊C4—122no—
55A4—41no—
56A4—82no—
57C3×C4—161no—
58Dic3—162no—
59C23—22no—
60C23—32no—
61C23—32no—
62C23—31no—
63C23—31no—
64C23—62no—
65C23—62no—
66C23—62no—
67C23—62no—
68C23—62no—
69C23—121no—
70C23—121no—
71C23—122no—
72C23—122no—
73C23C2×S411yescharacteristic
74Q8—241no—
75Q8—241no—
76C2×C4—61no—
77C2×C4—61no—
78C2×C4—62no—
79C2×C4—62no—
80C2×C4—121no—
81C2×C4—122no—
82C2×C4—122no—
83C2×C4—122no—
84C2×C4—122no—
85C2×C4—122no—
86C2×C3—161no—
87C22—22no—
88C22—31no—
89C22—31no—
90C22—32no—
91C22—32no—
92C22—62no—
93C22—62no—
94C22—62no—
95C22—62no—
96C22—62no—
97C22—122no—
98C22—121no—
99C22—121no—
100C22—122no—
101C22A4⋊Q811yescharacteristic
102C4—41no—
103C4—121no—
104C4—122no—
105C4—122no—
106C4—242no—
107C3—161noSylow
108C2—31no—
109C2—31no—
110C2—62no—
111C2—62no—
112C2#1020⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2#1020⋊C2
commutator, prime radical, cosocleC2×C22⋊A4C22
Frattini, socleC23C2×S4
Fitting#379D3
hypocenterC22⋊A4Q8
LCSG ▷ C2×C22⋊A4 ▷ C22⋊A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
11. C22⋊A4Q8semidirect
33. C24Dic6semidirect

Maximal subgroups

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

subgroupindex
#59362
#379⋊C32
#20293
A4⋊Q84