Finite Group Databasebeta

← #84031

Subgroups — #84031

Order 384.

1068 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 237 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
1C24×C3×C4C211yesmaximal, characteristic
2C2×#924C213yesmaximal, complemented
3#5247C212yesmaximal
4#5212C213yesmaximal, complemented
5#5210C216yesmaximal, complemented
6#2848—31nomaximal, Sylow
7C25×C3C2211yescharacteristic
8C23×C3×C4—212no—
9C23×C3×C4C2213yes—
10C23×C3×C4C2213yes—
11C22×Dic6C2211yescharacteristic, complemented
12C2×#217C2216yes—
13C2×#217C2216yes—
14C2×C4⋊Dic3C2213yescomplemented
15#924—212no—
16C2×Dic3⋊C4C2216yescomplemented
17C3⋊#95C2216yescomplemented
18C24×C4D311yescharacteristic
19C2×C22⋊Q8—33no—
20#385—36no—
21#380—33no—
22#379—32no—
23C24×C3—26no—
24C24×C3—26no—
25C24×C3Q814yescomplemented
26C24×C3C2313yes—
27C22×C3×C4—232no—
28C22×C3×C4—26no—
29C22×C3×C4—212no—
30C22×C3×C4—212no—
31C22×C3×C4D4112yes—
32C22×C3×C4C2313yes—
33C22×C3×C4C2311yescharacteristic
34C22×Dic3C2316yes—
35C22×Dic3C2312yes—
36C2×Dic6—26no—
37#217—212no—
38#217—212no—
39C4⋊Dic3—26no—
40Dic3⋊C4—212no—
41C25C2×D311yescharacteristic
42C22×Q8—31no—
43C23×C4—212no—
44C23×C4C2×D313yes—
45C23×C4C2×D313yes—
46C22⋊Q8—612no—
47C2×C4⋊C4—33no—
48C2×C4⋊C4—36no—
49C2×C22⋊C4—36no—
50C2×C22⋊C4—36no—
51#95—36no—
52C23×C3—28no—
53C23×C3—224no—
54C23×C3—224no—
55C23×C3—26no—
56C23×C3—26no—
57C23×C3C2×D416yes—
58C23×C3C2×Q816yes—
59C23×C3#4116yes—
60C23×C3C2411yescharacteristic
61C2×C3×C4—224no—
62C2×C3×C4—26no—
63C2×C3×C4—224no—
64C2×C3×C4C2×D416yes—
65C2×C3×C4C2×D416yes—
66C2×Dic3—26no—
67C2×Dic3—26no—
68C2×Dic3—212no—
69Dic6—44no—
70C24—26no—
71C24—26no—
72C24Dic614yescomplemented
73C24C22×D313yes—
74C2×Q8—66no—
75C22×C4—232no—
76C22×C4—26no—
77C22×C4—212no—
78C22×C4—212no—
79C22×C4—36no—
80C22×C4—32no—
81C22×C4#67112yes—
82C22×C4C22×D313yes—
83C22×C4C22×D311yescharacteristic
84C4⋊C4—612no—
85C4⋊C4—66no—
86C22⋊C4—612no—
87C22⋊C4—612no—
88C22×C3—232no—
89C22×C3—26no—
90C22×C3—212no—
91C22×C3—26no—
92C22×C3—212no—
93C22×C3C22⋊Q8112yescomplemented
94C22×C3C22×D413yes—
95C22×C3C22×Q811yescharacteristic
96C22×C3C2×#4113yes—
97C3×C4—26no—
98C3×C4#12014yes—
99Dic3—42no—
100Dic3—46no—
101C23—28no—
102C23—224no—
103C23—224no—
104C23—26no—
105C23—26no—
106C23C2×Dic616yes—
107C23#23516yes—
108C23C2×#6716yes—
109C23C23×D311yescharacteristic
110Q8—124no—
111C2×C4—224no—
112C2×C4—26no—
113C2×C4—224no—
114C2×C4—612no—
115C2×C4—66no—
116C2×C4—66no—
117C2×C4C2×#6716yes—
118C2×C4C2×#6716yes—
119C2×C3—212no—
120C2×C3C2×#12011yescharacteristic
121C2×C3C2×C22⋊Q813yes—
122C2×C3#52513yes—
123C22—232no—
124C22—212no—
125C22—212no—
126C22—26no—
127C22—26no—
128C22#924112yescomplemented
129C22C22×Dic611yescharacteristic
130C22C2×#23513yes—
131C22C22×#6713yes—
132C4—26no—
133C4—126no—
134C4—122no—
135C4#95314yes—
136C3#284811yescharacteristic, complemented, Sylow
137C2—212no—
138C2C2×#92413yes—
139C2#579113yes—
140C2C2×#95311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC24×C3×C4C2
prime radicalC25×C3C22
hypercenterC24×C4D3
hypocenterC3#2848
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
2. C2×#924C2semidirect
4. #5212C2semidirect
5. #5210C2semidirect
11. C22×Dic6C22semidirect
14. C2×C4⋊Dic3C22semidirect
16. C2×Dic3⋊C4C22semidirect
17. C3⋊#95C22semidirect
25. C24×C3Q8semidirect
72. C24Dic6semidirect
93. C22×C3C22⋊Q8semidirect
128. C22#924semidirect
136. C3#2848semidirect

Maximal subgroups

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

subgroupindex
C24×C3×C42
C2×#9242
#52472
#52122
#52102
#28483