Finite Group Databasebeta

← #67649

Subgroups — #67649

Order 384.

146 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 37 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×#397C211yesmaximal, characteristic
2#4526C211yesmaximal, characteristic
3#4525C211yesmaximal, characteristic
4#1344—31nomaximal, Sylow
5C2×C3×C4⋊C4C411yescharacteristic, complemented
6C2×C3×C22⋊C4—21no—
7C2×C3×C22⋊C4C2211yescharacteristic
8C2×#217—21no—
9C3×#95—21no—
10C3×#95C411yescharacteristic, complemented
11#830—21no—
12#397D311yescharacteristic
13#342—31no—
14#323—31no—
15C24×C3D411yescharacteristic
16C22×C3×C4—21no—
17C22×C3×C4—21no—
18C22×C3×C4C2×C411yescharacteristic
19C22×C3×C4D411yescharacteristic
20C22×Dic3—41no—
21C3×C4⋊C4—41no—
22C3×C22⋊C4—41no—
23C3×C22⋊C4—41no—
24C3×C22⋊C4—41no—
25#217—22no—
26#217—41no—
27C2×C3⋊C8—41no—
28C2×C4⋊C4Dic311yescharacteristic, complemented
29C2×C22⋊C4—21no—
30C2×C22⋊C4—61no—
31C2×C22⋊C4C2×D311yescharacteristic
32C22⋊C8—61no—
33#95—21no—
34#95Dic311yescharacteristic, complemented
35C23×C3—21no—
36C23×C3—21no—
37C23×C3—41no—
38C23×C3—41no—
39C23×C3C22⋊C411yescharacteristic
40C23×C3D811yescharacteristic
41C23×C3SD1611yescharacteristic
42C2×C3×C4—21no—
43C2×C3×C4—21no—
44C2×C3×C4—41no—
45C2×C3×C4—41no—
46C2×C3×C4—41no—
47C2×C3×C4—41no—
48C2×C3×C4—41no—
49C2×C3×C4—41no—
50C2×C3×C4—41no—
51C2×C3×C4—41no—
52C2×Dic3—42no—
53C2×Dic3—81no—
54C2×Dic3—81no—
55C3⋊C8—81no—
56C24#6711yescharacteristic
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—121no—
60C22×C4C2×Dic311yescharacteristic
61C22×C4#6711yescharacteristic
62C2×C8—121no—
63C4⋊C4—41no—
64C22⋊C4—41no—
65C22⋊C4—41no—
66C22⋊C4—41no—
67C22⋊C4—62no—
68C22⋊C4—121no—
69C22×C3—21no—
70C22×C3—21no—
71C22×C3—21no—
72C22×C3—21no—
73C22×C3—41no—
74C22×C3—41no—
75C22×C3—41no—
76C22×C3—41no—
77C22×C3—81no—
78C22×C3#9911yescharacteristic
79C22×C3#10211yescharacteristic
80C22×C3#10411yescharacteristic
81C3×C4—41no—
82C3×C4—41no—
83C3×C4—81no—
84C3×C4—81no—
85Dic3—82no—
86C23—21no—
87C23—21no—
88C23—41no—
89C23—41no—
90C23#21311yescharacteristic
91C23#21411yescharacteristic
92C23#21711yescharacteristic
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—122no—
104C2×C4—241no—
105C2×C4—241no—
106C8—241no—
107C2×C3—21no—
108C2×C3—21no—
109C2×C3—41no—
110C2×C3—41no—
111C2×C3#32711yescharacteristic
112C2×C3#35211yescharacteristic
113C2×C3#35411yescharacteristic
114C22—21no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—41no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22—81no—
123C22#83211yescharacteristic
124C22#83411yescharacteristic
125C22#83711yescharacteristic
126C4—41no—
127C4—41no—
128C4—81no—
129C4—81no—
130C4—242no—
131C3#134411yescharacteristic, complemented, Sylow
132C2—21no—
133C2—21no—
134C2—41no—
135C2—41no—
136C2#453711yescharacteristic
137C2#453911yescharacteristic
138C2#454111yescharacteristic
subgroupG/H
centerC22#834
commutatorC22×C3×C4C2×C4
FrattiniC2×C22⋊C4C2×D3
FittingC3×#397C2
socleC22×C3#99
prime radicalC24×C3D4
cosocleC2×C3×C22⋊C4C22
hypercenter#397D3
hypocenterC3#1344
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #397

Complements

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

NH ≅ G/Ntype
5. C2×C3×C4⋊C4C4semidirect
10. C3×#95C4semidirect
28. C2×C4⋊C4Dic3semidirect
34. #95Dic3semidirect
131. C3#1344semidirect

Maximal subgroups

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

subgroupindex
C3×#3972
#45262
#45252
#13443