Finite Group Databasebeta

← #73080

Subgroups — #73080

Order 384.

215 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 20 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, complemented
2C2×#120⋊C3C211yesmaximal, characteristic
3#1889—31nomaximal, Sylow
4C2×C22⋊A4C2211yescharacteristic
5C2×SL(2,ℤ/4)—41nomaximal
6#120⋊C3C412yescomplemented
7C2×#120D311yescharacteristic
8#518—31no—
9C2×#99—31no—
10#379—32no—
11#342—32no—
12C22⋊A4C2×C411yescharacteristic, complemented
13C22×A4—41no—
14SL(2,ℤ/4)—42no—
15SL(2,ℤ/4)—82no—
16C25C2×D311yescharacteristic
17C22×D4—31no—
18#120—32no—
19#120Dic312yescomplemented
20C4×D4—62no—
21C4×D4—62no—
22#117—61no—
23C2×C22⋊C4—31no—
24C2×C22⋊C4—32no—
25C2×C22⋊C4—32no—
26C2×C22⋊C4—61no—
27C2×C22⋊C4—62no—
28C2×C22⋊C4—62no—
29#99—62no—
30C2×A4—42no—
31C2×A4—41no—
32C2×A4—82no—
33C2×Dic3—161no—
34C24—31no—
35C24—31no—
36C24—62no—
37C24—62no—
38C24C2×Dic311yescharacteristic, complemented
39C24S411yescharacteristic
40C2×D4—32no—
41C2×D4—32no—
42C2×D4—62no—
43C2×D4—62no—
44C22×C4—31no—
45C22×C4—32no—
46C22×C4—32no—
47C22×C4—61no—
48C22×C4—61no—
49C22×C4—62no—
50C4⋊C4—61no—
51C4⋊C4—61no—
52C22⋊C4—32no—
53C22⋊C4—61no—
54C22⋊C4—62no—
55C22⋊C4—62no—
56C22⋊C4—62no—
57C22⋊C4—62no—
58C22⋊C4—62no—
59C22⋊C4—122no—
60C22⋊C4—122no—
61C22⋊C4—122no—
62C22⋊C4—122no—
63C42—62no—
64C22×C3—161no—
65A4—41no—
66A4—82no—
67Dic3—162no—
68C23—22no—
69C23—32no—
70C23—32no—
71C23—31no—
72C23—31no—
73C23—62no—
74C23—62no—
75C23—61no—
76C23—61no—
77C23—62no—
78C23—62no—
79C23—62no—
80C23—121no—
81C23—121no—
82C23—122no—
83C23—122no—
84C23SL(2,ℤ/4)12yescomplemented
85C23C2×S411yescharacteristic
86D4—64no—
87D4—64no—
88C2×C4—32no—
89C2×C4—61no—
90C2×C4—61no—
91C2×C4—62no—
92C2×C4—62no—
93C2×C4—122no—
94C2×C4—122no—
95C2×C4—122no—
96C2×C4—122no—
97C2×C4—122no—
98C2×C4—241no—
99C2×C4—241no—
100C2×C3—162no—
101C2×C3—161no—
102C22—22no—
103C22—31no—
104C22—31no—
105C22—32no—
106C22—32no—
107C22—41no—
108C22—62no—
109C22—62no—
110C22—62no—
111C22—62no—
112C22—62no—
113C22—62no—
114C22—121no—
115C22—121no—
116C22—121no—
117C22—122no—
118C22—122no—
119C22C2×SL(2,ℤ/4)11yescharacteristic
120C4—62no—
121C4—122no—
122C4—122no—
123C4—242no—
124C3—161noSylow
125C2—31no—
126C2—31no—
127C2—42no—
128C2—62no—
129C2—62no—
130C2#1020⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2#1020⋊C2
commutator, hypocenterC22⋊A4C2×C4
Frattini, socleC23C2×S4
FittingC2×#120D3
prime radicalC2×#120⋊C3C2
cosocleC2×C22⋊A4C22
LCSG ▷ C22⋊A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #5936C2semidirect
6. #120⋊C3C4semidirect
12. C22⋊A4C2×C4semidirect
19. #120Dic3semidirect
38. C24C2×Dic3semidirect
84. C23SL(2,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
#59362
C2×#120⋊C32
#18893
C2×SL(2,ℤ/4)4