Finite Group Databasebeta

← #67701

Subgroups — #67701

Order 384.

162 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 53 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×#413C211yesmaximal, characteristic
2#5240C211yesmaximal, characteristic
3#5234C211yesmaximal, characteristic
4#1395—31nomaximal, Sylow
5C2×C3×#41C2211yescharacteristic
6C2×C3×M16—21no—
7#926—21no—
8C2×#208C411yescharacteristic, complemented
9C22×C3⋊C8C411yescharacteristic, complemented
10C3×#101C412yescomplemented
11C3×#100—21no—
12#834—21no—
13#834C412yes—
14#830—21no—
15#830—21no—
16#413D311yescharacteristic
17#410—31no—
18#408—31no—
19C3×#41—41no—
20C3×#41—41no—
21C2×C3×Q8C2×C411yescharacteristic
22C2×C3×D4C2×C411yescharacteristic
23C2×C3×D4D411yescharacteristic
24C2×C3×D4Q811yescharacteristic
25C22×C3×C4C2×C411yescharacteristic
26C22×C3×C4D411yescharacteristic
27C22×C3×C4D411yescharacteristic
28C3×M16—22no—
29C3×M16—41no—
30C2×C3×C8—41no—
31#217—22no—
32C4⋊Dic3—41no—
33C4×Dic3—41no—
34#208—41no—
35C2×C3⋊C8—21no—
36C2×C3⋊C8—21no—
37C2×C3⋊C8—41no—
38C2×#41C2×D311yescharacteristic
39C2×M16—21no—
40C2×M16—31no—
41C22×C8—31no—
42#117—61no—
43#101Dic312yescomplemented
44#100—21no—
45#99—32no—
46#99—61no—
47C22⋊C8—61no—
48C22⋊C8—61no—
49C23×C3C22⋊C411yescharacteristic
50C23×C3C4⋊C411yescharacteristic
51C23×C3C4⋊C411yescharacteristic
52C3×Q8—41no—
53C3×D4—41no—
54C3×D4—41no—
55C3×D4—41no—
56C2×C3×C4—21no—
57C2×C3×C4—21no—
58C2×C3×C4—21no—
59C2×C3×C4—21no—
60C2×C3×C4—21no—
61C2×C3×C4—21no—
62C2×C3×C4C4211yescharacteristic
63C2×C3×C4C22⋊C411yescharacteristic
64C2×C3×C4C22⋊C411yescharacteristic
65C2×C3×C4C4⋊C411yescharacteristic
66C2×Dic3—42no—
67C3×C8—42no—
68C3⋊C8—41no—
69C3⋊C8—41no—
70#41—41no—
71#41—41no—
72C2×Q8C2×Dic311yescharacteristic
73C2×D4Dic611yescharacteristic
74C2×D4C4×D311yescharacteristic
75C2×D4#6711yescharacteristic
76C22×C4C4×D311yescharacteristic
77C22×C4D1211yescharacteristic
78C22×C4#6711yescharacteristic
79M16—22no—
80M16—41no—
81M16—121no—
82C2×C8—41no—
83C2×C8—61no—
84C2×C8—61no—
85C2×C8—121no—
86C4⋊C4—121no—
87C22⋊C4—62no—
88C42—121no—
89C22×C3—21no—
90C22×C3—21no—
91C22×C3—21no—
92C22×C3—41no—
93C22×C3—41no—
94C22×C3—41no—
95C22×C3#9511yescharacteristic
96C3×C4—21no—
97C3×C4—21no—
98C3×C4—21no—
99C3×C4#9911yescharacteristic
100C3×C4#9911yescharacteristic
101Dic3—82no—
102C23Dic3⋊C411yescharacteristic
103C23Dic3⋊C411yescharacteristic
104C23(C2×D3)⋊C411yescharacteristic
105Q8—41no—
106D4—41no—
107D4—41no—
108D4—41no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—122no—
116C2×C4C4×Dic311yescharacteristic
117C2×C4C4⋊Dic311yescharacteristic
118C2×C4(C2×D3)⋊C411yescharacteristic
119C2×C4#21711yescharacteristic
120C8—42no—
121C8—121no—
122C8—121no—
123C2×C3—21no—
124C2×C3—41no—
125C2×C3—41no—
126C2×C3—41no—
127C2×C3#34211yescharacteristic
128C22—21no—
129C22—21no—
130C22—21no—
131C22—41no—
132C22—41no—
133C22—41no—
134C22C3⋊#9511yescharacteristic
135C4—21no—
136C4—21no—
137C4—21no—
138C4—242no—
139C4#80611yescharacteristic
140C4#80611yescharacteristic
141C3#139511yescharacteristic, complemented, Sylow
142C2—21no—
143C2—41no—
144C2—41no—
145C2—41no—
146C2#452711yescharacteristic
subgroupG/H
centerC4#806
commutatorC2×C3×C4C42
FrattiniC2×#41C2×D3
FittingC3×#413C2
socleC2×C3#342
prime radical, cosocleC2×C3×#41C22
hypercenter#413D3
hypocenterC3#1395
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
8. C2×#208C4semidirect
9. C22×C3⋊C8C4semidirect
10. C3×#101C4semidirect
43. #101Dic3semidirect
141. C3#1395semidirect

Maximal subgroups

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

subgroupindex
C3×#4132
#52402
#52342
#13953