Finite Group Databasebeta

← C4×#120⋊C3 #73171

Subgroups — C4×#120⋊C3 #73171

Order 384.

254 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 27 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
1C4×C22⋊A4C211yesmaximal, characteristic, complemented
2C2×#120⋊C3C211yesmaximal, characteristic
3#379⋊C3C211yesmaximal, characteristic, complemented
4C4×#120C311yesmaximal, characteristic, complemented, Sylow
5C2×C22⋊A4C2211yescharacteristic
6C2×C4×A4—41nomaximal
7#120⋊C3C412yesdirect product factor
8C24×C4C2×C311yescharacteristic, complemented
9C2×#120C2×C311yescharacteristic
10C2×C4×D4—31no—
11#386—31no—
12#385—31no—
13#379C2×C311yescharacteristic, complemented
14C4×C22⋊C4—31no—
15C22⋊A4C2×C411yescharacteristic, complemented
16C22×A4—41no—
17C4×A4—41no—
18C4×A4—41no—
19C4×A4—82no—
20C25C22×C311yescharacteristic
21C22×D4—31no—
22C23×C4—31no—
23C23×C4—31no—
24C23×C4—62no—
25C23×C4—62no—
26C23×C4A411yescharacteristic, complemented
27#120—32no—
28#120C3×C412yescomplemented
29C4×D4—62no—
30C4×D4—62no—
31C2×C4⋊C4—31no—
32C2×C22⋊C4—31no—
33C2×C22⋊C4—31no—
34C2×C22⋊C4—31no—
35C2×C22⋊C4—31no—
36C2×C42—31no—
37#95—32no—
38C2×A4—42no—
39C2×A4—41no—
40C2×A4—82no—
41C2×C3×C4—161no—
42C24—31no—
43C24—31no—
44C24—62no—
45C24—62no—
46C24C2×C3×C411yescharacteristic, complemented
47C24C2×A411yescharacteristic
48C2×D4—32no—
49C2×D4—32no—
50C2×D4—62no—
51C2×D4—62no—
52C22×C4—22no—
53C22×C4—32no—
54C22×C4—32no—
55C22×C4—31no—
56C22×C4—31no—
57C22×C4—31no—
58C22×C4—61no—
59C22×C4—62no—
60C22×C4—62no—
61C22×C4—64no—
62C22×C4—61no—
63C22×C4—62no—
64C22×C4—62no—
65C22×C4—64no—
66C22×C4—62no—
67C22×C4—62no—
68C22×C4C2×A411yescharacteristic
69C22×C4C2×A411yescharacteristic
70C4⋊C4—62no—
71C22⋊C4—32no—
72C22⋊C4—32no—
73C22⋊C4—62no—
74C22⋊C4—62no—
75C22⋊C4—62no—
76C42—61no—
77C42—61no—
78C22×C3—161no—
79A4—41no—
80A4—82no—
81C3×C4—161no—
82C3×C4—161no—
83C23—22no—
84C23—32no—
85C23—32no—
86C23—31no—
87C23—31no—
88C23—62no—
89C23—62no—
90C23—61no—
91C23—62no—
92C23—62no—
93C23—62no—
94C23—64no—
95C23—61no—
96C23—62no—
97C23—62no—
98C23—64no—
99C23C4×A412yescomplemented
100C23C22×A411yescharacteristic
101D4—64no—
102D4—64no—
103C2×C4—31no—
104C2×C4—31no—
105C2×C4—32no—
106C2×C4—32no—
107C2×C4—41no—
108C2×C4—62no—
109C2×C4—64no—
110C2×C4—61no—
111C2×C4—61no—
112C2×C4—62no—
113C2×C4—61no—
114C2×C4—64no—
115C2×C4—62no—
116C2×C4—62no—
117C2×C4—62no—
118C2×C4—61no—
119C2×C4—62no—
120C2×C4—121no—
121C2×C3—162no—
122C2×C3—161no—
123C22—22no—
124C22—31no—
125C22—31no—
126C22—32no—
127C22—32no—
128C22—41no—
129C22—62no—
130C22—62no—
131C22—62no—
132C22—62no—
133C22—64no—
134C22—62no—
135C22—62no—
136C22—62no—
137C22—64no—
138C22—62no—
139C22—121no—
140C22C2×C4×A411yescharacteristic
141C4—31no—
142C4—41no—
143C4—62no—
144C4—62no—
145C4—121no—
146C4#120⋊C311yescharacteristic, direct product factor
147C3—161noSylow
148C2—31no—
149C2—31no—
150C2—42no—
151C2—62no—
152C2—62no—
153C2C2×#120⋊C311yescharacteristic
subgroupG/H
center, hypercenterC4#120⋊C3
commutator, hypocenterC24C2×C3×C4
Frattini, socleC23C22×A4
FittingC4×#120C3
prime radicalC2×#120⋊C3C2
cosocleC25C22×C3
LCSG ▷ C24
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4×C22⋊A4C2semidirect
3. #379⋊C3C2semidirect
4. C4×#120C3semidirect
7. #120⋊C3C4direct (N × H)
8. C24×C4C2×C3semidirect
13. #379C2×C3semidirect
15. C22⋊A4C2×C4semidirect
26. C23×C4A4semidirect
28. #120C3×C4semidirect
46. C24C2×C3×C4semidirect
99. C23C4×A4semidirect
146. C4#120⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C22⋊A42
C2×#120⋊C32
#379⋊C32
C4×#1203
C2×C4×A44