Finite Group Databasebeta

← #71861

Subgroups — #71861

Order 384.

270 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 73 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×#535C211yesmaximal, characteristic
2#5240C212yesmaximal, complemented
3#4545C214yesmaximal, complemented
4#2128—31nomaximal, Sylow
5C2×C3×#41C2211yescharacteristic
6C22×C3×D4C411yescharacteristic, complemented
7C3×#127C2212yes—
8C3×#124C412yescomplemented
9C3×#121—22no—
10C3×#120—22no—
11C3×#117C411yescharacteristic, complemented
12#926—22no—
13#834—22no—
14#834C2214yescomplemented
15#535D311yescharacteristic
16#410—32no—
17#353—34no—
18C24×C3—21no—
19C3×#41—42no—
20C2×C3×Q8C2×C411yescharacteristic
21C2×C3×D4—24no—
22C2×C3×D4—21no—
23C2×C3×D4—42no—
24C2×C3×D4C2×C412yes—
25C2×C3×D4D412yes—
26C2×C3×D4C2311yescharacteristic
27C22×C3×C4C2×C411yescharacteristic
28C22×C3×C4D412yes—
29C3×C4⋊C4—21no—
30C3×C22⋊C4—21no—
31C3×C22⋊C4—24no—
32C3×C42C2×C412yes—
33#217—24no—
34C4⋊Dic3—42no—
35C4×Dic3—42no—
36C2×#41C2×D311yescharacteristic
37C22×D4Dic311yescharacteristic, complemented
38#127C2×D312yes—
39#124Dic312yescomplemented
40#121—22no—
41#120—22no—
42#117—62no—
43#117Dic311yescharacteristic, complemented
44#99—34no—
45#99—62no—
46C23×C3—22no—
47C23×C3—42no—
48C23×C3—82no—
49C23×C3C22⋊C411yescharacteristic
50C23×C3C2×D412yes—
51C3×Q8—41no—
52C3×D4—42no—
53C3×D4—41no—
54C3×D4—42no—
55C3×D4—42no—
56C3×D4—84no—
57C2×C3×C4—22no—
58C2×C3×C4—22no—
59C2×C3×C4—22no—
60C2×C3×C4—22no—
61C2×C3×C4C22⋊C412yes—
62C2×C3×C4C22⋊C411yescharacteristic
63C2×C3×C4C22×C411yescharacteristic
64C2×Dic3—44no—
65C24—21no—
66#41—42no—
67C2×Q8C2×Dic311yescharacteristic
68C2×D4—24no—
69C2×D4—21no—
70C2×D4—42no—
71C2×D4C2×Dic312yes—
72C2×D4#6712yes—
73C2×D4C22×D311yescharacteristic
74C22×C4C2×Dic311yescharacteristic
75C22×C4#6712yes—
76C4⋊C4—21no—
77C4⋊C4—122no—
78C22⋊C4—21no—
79C22⋊C4—24no—
80C22⋊C4—64no—
81C42—122no—
82C42C2×Dic312yes—
83C22×C3—22no—
84C22×C3—21no—
85C22×C3—42no—
86C22×C3—41no—
87C22×C3—44no—
88C22×C3—42no—
89C22×C3—84no—
90C22×C3C2×C22⋊C411yescharacteristic
91C3×C4—22no—
92C3×C4—21no—
93C3×C4—42no—
94C3×C4#9912yes—
95Dic3—84no—
96C23—22no—
97C23—42no—
98C23—82no—
99C23#21711yescharacteristic
100C23C2×#6712yes—
101Q8—41no—
102D4—42no—
103D4—41no—
104D4—42no—
105D4—42no—
106D4—84no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—124no—
112C2×C4#21712yes—
113C2×C4#21711yescharacteristic
114C2×C4C22×Dic311yescharacteristic
115C2×C3—21no—
116C2×C3—42no—
117C2×C3—41no—
118C2×C3—82no—
119C2×C3C2×#9911yescharacteristic
120C22—22no—
121C22—21no—
122C22—42no—
123C22—41no—
124C22—44no—
125C22—42no—
126C22—84no—
127C22C2×#21711yescharacteristic
128C4—22no—
129C4—21no—
130C4—42no—
131C4—244no—
132C4#83412yes—
133C3#212811yescharacteristic, complemented, Sylow
134C2—21no—
135C2—42no—
136C2—41no—
137C2—82no—
138C2C2×#83411yescharacteristic
subgroupG/H
centerC2C2×#834
commutatorC2×C3×C4C22×C4
FrattiniC2×D4C22×D3
Fitting, prime radicalC3×#535C2
socleC2×C3C2×#99
cosocleC2×C3×D4C23
hypercenter#535D3
hypocenterC3#2128
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
2. #5240C2semidirect
3. #4545C2semidirect
6. C22×C3×D4C4semidirect
8. C3×#124C4semidirect
11. C3×#117C4semidirect
14. #834C22semidirect
37. C22×D4Dic3semidirect
39. #124Dic3semidirect
43. #117Dic3semidirect
133. C3#2128semidirect

Maximal subgroups

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

subgroupindex
C3×#5352
#52402
#45452
#21283