Finite Group Databasebeta

← #77401

Subgroups — #77401

Order 384.

420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 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×#396C211yesmaximal, characteristic, complemented
2#4990C213yesmaximal, complemented
3#4984C213yesmaximal, complemented
4#4961C213yesmaximal
5#4951C213yesmaximal, complemented
6#4677C211yesmaximal, characteristic, complemented
7C3⋊#401C211yesmaximal, characteristic, complemented
8#2801—31nomaximal, Sylow
9C22×#67C2211yescharacteristic
10C2×C3×C4⋊C4C2213yes—
11C2×C3×C22⋊C4C2213yes—
12C2×#217C2213yes—
13C2×(C2×D3)⋊C4C2213yes—
14C2×(C2×D3)⋊C4C2213yes—
15C2×C4⋊Dic3C2213yes—
16C2×C4⋊Dic3C2213yes—
17C2×C4×Dic3C2213yes—
18C3×#95C2211yescharacteristic
19C3⋊#95C2213yes—
20C3⋊#95C2213yes—
21C3⋊#95C2213yes—
22#401—31no—
23#399—31no—
24#396D311yescharacteristic, complemented
25#394—33no—
26#389—33no—
27#388—33no—
28#388—33no—
29C24×C3C2311yescharacteristic
30C23×D3C2311yescharacteristic
31C22×C3×C4C2313yes—
32C22×C3×C4C2313yes—
33C2×#67—43no—
34C22×Dic3C2313yes—
35C22×Dic3C2311yescharacteristic
36C22×Dic3C2313yes—
37C3×C4⋊C4—26no—
38C3×C22⋊C4—26no—
39#217—43no—
40(C2×D3)⋊C4—26no—
41(C2×D3)⋊C4—43no—
42C4⋊Dic3—43no—
43C4⋊Dic3—43no—
44C4×Dic3—43no—
45C22×D4—31no—
46C2×C4⋊C4—33no—
47C2×C4⋊C4—33no—
48C2×C4⋊C4C2×D313yes—
49C2×C22⋊C4—33no—
50C2×C22⋊C4—33no—
51C2×C22⋊C4—33no—
52C2×C22⋊C4C2×D313yes—
53C2×C42—33no—
54#95—33no—
55#95—33no—
56#95—33no—
57#95C2×D311yescharacteristic
58C23×C3—23no—
59C23×C3—23no—
60C23×C3—21no—
61C23×C3C2411yescharacteristic
62C22×D3—23no—
63C22×D3—23no—
64C22×D3—21no—
65C2×C3×C4—23no—
66C2×C3×C4—23no—
67C2×C3×C4—23no—
68C2×C3×C4—23no—
69C2×C3×C4—23no—
70C2×C3×C4#4116yes—
71#67—44no—
72C2×Dic3—23no—
73C2×Dic3—23no—
74C2×Dic3—23no—
75C2×Dic3—23no—
76C2×Dic3—23no—
77C2×Dic3—23no—
78C2×Dic3—23no—
79C24—31no—
80C24C22×D311yescharacteristic
81C2×D4—123no—
82C22×C4—33no—
83C22×C4—31no—
84C22×C4—33no—
85C22×C4C22×D313yes—
86C22×C4C22×D313yes—
87C4⋊C4—26no—
88C4⋊C4—123no—
89C4⋊C4—123no—
90C22⋊C4—26no—
91C22⋊C4—66no—
92C22⋊C4—123no—
93C22⋊C4—123no—
94C42—123no—
95C22×C3—43no—
96C22×C3—43no—
97C22×C3—41no—
98C22×C3C2×#4113yes—
99C22×C3H(5,𝔽2)13yes—
100C22×C32−1+411yescharacteristic
101C2×D3—43no—
102C2×D3—43no—
103C2×D3—41no—
104C3×C4—26no—
105C3×C4—43no—
106Dic3—43no—
107Dic3—43no—
108Dic3—41no—
109C23—23no—
110C23—21no—
111C23—23no—
112C23—63no—
113C23—61no—
114C23—63no—
115C23C23×D311yescharacteristic
116D4—124no—
117C2×C4—23no—
118C2×C4—23no—
119C2×C4—23no—
120C2×C4—23no—
121C2×C4—23no—
122C2×C4—63no—
123C2×C4—63no—
124C2×C4—63no—
125C2×C4—63no—
126C2×C4—63no—
127C2×C4—63no—
128C2×C4—63no—
129C2×C4#23716yes—
130C2×C3—81no—
131C2×C3#54213yes—
132C2×C3#55313yes—
133C2×C3#56011yescharacteristic
134D3—81no—
135C22—43no—
136C22—41no—
137C22—43no—
138C22—123no—
139C22—121no—
140C22—123no—
141C22C2×#23713yes—
142C22#100711yescharacteristic
143C22#100913yes—
144C4—26no—
145C4—43no—
146C4—123no—
147C4—121no—
148C4—123no—
149C3#280111yescharacteristic, complemented, Sylow
150C2—81no—
151C2—241no—
152C2#561813yes—
153C2#564313yes—
154C2#571111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#396C2
prime radicalC22×#67C22
hypercenter#396D3
hypocenterC3#2801
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C3×#396C2semidirect
2. #4990C2semidirect
3. #4984C2semidirect
5. #4951C2semidirect
6. #4677C2semidirect
7. C3⋊#401C2semidirect
24. #396D3semidirect
149. C3#2801semidirect

Maximal subgroups

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

subgroupindex
C3×#3962
#49902
#49842
#49612
#49512
#46772
C3⋊#4012
#28013