Finite Group Databasebeta

← #67647

Subgroups — #67647

Order 384.

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

normal subgroups only

This group has 168 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×#394C211yesmaximal, characteristic
2#4530C211yesmaximal, characteristic, complemented
3#4528C211yesmaximal, characteristic, complemented
4#1345—31nomaximal, Sylow
5C22×C3×D4C411yescharacteristic
6C2×C3×C4⋊C4C2211yescharacteristic
7C2×C3×C22⋊C4—21no—
8C2×C3×C22⋊C4—21no—
9C2×#208—21no—
10C3×#95C411yescharacteristic
11#830—21no—
12#394D311yescharacteristic
13#343—31no—
14#324—31no—
15C24×C3—21no—
16C2×C3×D4—41no—
17C2×C3×D4—42no—
18C22×C3×C4—21no—
19C22×C3×C4C2×C411yescharacteristic
20C22×C3×C4D411yescharacteristic
21C22×C3×C4D411yescharacteristic
22C3×C4⋊C4—41no—
23C3×C22⋊C4—42no—
24C3×C22⋊C4—41no—
25C3×C22⋊C4—41no—
26#208—22no—
27#208—41no—
28C2×C3⋊C8—41no—
29C2×C3⋊C8—41no—
30C22×D4Dic311yescharacteristic
31C2×M16—61no—
32C2×C4⋊C4C2×D311yescharacteristic
33C2×C22⋊C4—21no—
34C2×C22⋊C4—21no—
35C22⋊C8—61no—
36#95Dic311yescharacteristic
37C23×C3—22no—
38C23×C3—41no—
39C23×C3—41no—
40C23×C3—41no—
41C23×C3—41no—
42C23×C3—41no—
43C23×C3—41no—
44C23×C3C22⋊C411yescharacteristic
45C3×D4—42no—
46C3×D4—42no—
47C2×C3×C4—21no—
48C2×C3×C4—21no—
49C2×C3×C4—21no—
50C2×C3×C4—21no—
51C2×C3×C4—41no—
52C2×C3×C4—41no—
53C2×C3×C4—41no—
54C2×C3×C4—41no—
55C2×C3×C4—41no—
56C2×C3×C4D811yescharacteristic
57C2×C3×C4SD1611yescharacteristic
58C3⋊C8—42no—
59C3⋊C8—81no—
60C24—21no—
61C2×D4—41no—
62C2×D4—42no—
63C22×C4—21no—
64C22×C4C2×Dic311yescharacteristic
65C22×C4#6711yescharacteristic
66C22×C4#6711yescharacteristic
67M16—62no—
68M16—121no—
69C2×C8—121no—
70C2×C8—121no—
71C4⋊C4—41no—
72C22⋊C4—42no—
73C22⋊C4—41no—
74C22⋊C4—41no—
75C22×C3—21no—
76C22×C3—21no—
77C22×C3—42no—
78C22×C3—42no—
79C22×C3—42no—
80C22×C3—81no—
81C22×C3—81no—
82C22×C3—81no—
83C22×C3—81no—
84C22×C3#9911yescharacteristic
85C22×C3#10211yescharacteristic
86C22×C3#10411yescharacteristic
87C3×C4—21no—
88C3×C4—21no—
89C3×C4—41no—
90C3×C4—41no—
91C3×C4—81no—
92C23—22no—
93C23—41no—
94C23—41no—
95C23—41no—
96C23—41no—
97C23—41no—
98C23—41no—
99C23#21711yescharacteristic
100D4—42no—
101D4—42no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4#21311yescharacteristic
112C2×C4#21411yescharacteristic
113C8—122no—
114C8—241no—
115C2×C3—21no—
116C2×C3—21no—
117C2×C3—82no—
118C2×C3#32711yescharacteristic
119C2×C3AΣL(1,𝔽16)11yescharacteristic
120C2×C3#35511yescharacteristic
121C22—21no—
122C22—21no—
123C22—42no—
124C22—42no—
125C22—42no—
126C22—81no—
127C22—81no—
128C22—81no—
129C22—81no—
130C22#83211yescharacteristic
131C22#83411yescharacteristic
132C22#83711yescharacteristic
133C4—21no—
134C4—21no—
135C4—41no—
136C4—41no—
137C4—81no—
138C3#134511yescharacteristic, complemented, Sylow
139C2—21no—
140C2—21no—
141C2—82no—
142C2#453611yescharacteristic
143C2#453711yescharacteristic
144C2#454211yescharacteristic
subgroupG/H
centerC22#834
commutatorC22×C3×C4C2×C4
FrattiniC2×C4⋊C4C2×D3
FittingC3×#394C2
socleC22×C3#99
prime radicalC22×C3×D4C4
cosocleC2×C3×C4⋊C4C22
hypercenter#394D3
hypocenterC3#1345
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #394

Complements

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

NH ≅ G/Ntype
2. #4530C2semidirect
3. #4528C2semidirect
138. C3#1345semidirect

Maximal subgroups

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

subgroupindex
C3×#3942
#45302
#45282
#13453