Finite Group Databasebeta

← #71602

Subgroups — #71602

Order 384.

316 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×C3×C4×D4C211yesmaximal, characteristic
2#5207C212yesmaximal, complemented
3#4528C212yesmaximal, complemented
4#4525C212yesmaximal, complemented
5#1465—31nomaximal, Sylow
6C22×C3×D4C411yescharacteristic
7C23×C3×C4C411yescharacteristic
8C23×C3×C4C2211yescharacteristic
9C3×C4×D4—22no—
10C3×C4×D4—22no—
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×C3×C22⋊C4C411yescharacteristic
13C2×C3×C22⋊C4C2211yescharacteristic
14C2×C3×C42C411yescharacteristic
15C2×#208—22no—
16#830—22no—
17#830C2214yes—
18C2×C4×D4D311yescharacteristic
19#407—32no—
20#324—32no—
21#323—32no—
22C24×C3C2×C411yescharacteristic
23C24×C3C2×C411yescharacteristic
24C2×C3×D4—22no—
25C2×C3×D4—22no—
26C2×C3×D4—22no—
27C22×C3×C4—21no—
28C22×C3×C4—21no—
29C22×C3×C4—42no—
30C22×C3×C4—42no—
31C22×C3×C4C2×C411yescharacteristic
32C22×C3×C4C2×C411yescharacteristic
33C22×C3×C4C2×C411yescharacteristic
34C22×C3×C4C2×C411yescharacteristic
35C22×C3×C4D412yes—
36C22×C3×C4D412yes—
37C22×C3×C4C2311yescharacteristic
38C3×C4⋊C4—22no—
39C3×C22⋊C4—22no—
40C3×C22⋊C4—22no—
41C3×C42—21no—
42C3×C42—21no—
43#208—44no—
44C2×C3⋊C8—24no—
45C22×D4Dic311yescharacteristic
46C23×C4Dic311yescharacteristic
47C23×C4C2×D311yescharacteristic
48C2×M16—62no—
49C4×D4—22no—
50C4×D4—22no—
51C2×C4⋊C4C2×D311yescharacteristic
52C2×C22⋊C4Dic311yescharacteristic
53C2×C22⋊C4C2×D311yescharacteristic
54C2×C42Dic311yescharacteristic
55C22⋊C8—34no—
56C22⋊C8—62no—
57C23×C3—21no—
58C23×C3—21no—
59C23×C3—21no—
60C23×C3—21no—
61C23×C3—41no—
62C23×C3—41no—
63C23×C3—41no—
64C23×C3—41no—
65C23×C3C22⋊C412yes—
66C23×C3M1612yescomplemented
67C23×C3C22×C411yescharacteristic
68C3×D4—28no—
69C2×C3×C4—21no—
70C2×C3×C4—21no—
71C2×C3×C4—24no—
72C2×C3×C4—22no—
73C2×C3×C4—21no—
74C2×C3×C4—21no—
75C2×C3×C4—42no—
76C2×C3×C4—41no—
77C2×C3×C4—42no—
78C2×C3×C4—42no—
79C2×C3×C4—41no—
80C2×C3×C4—41no—
81C2×C3×C4C22⋊C411yescharacteristic
82C2×C3×C4C22⋊C411yescharacteristic
83C2×C3×C4M1612yescomplemented
84C2×C3×C4C2×D412yes—
85C3⋊C8—44no—
86C24C2×Dic311yescharacteristic
87C24C2×Dic311yescharacteristic
88C2×D4—22no—
89C2×D4—22no—
90C2×D4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4C2×Dic311yescharacteristic
96C22×C4C2×Dic311yescharacteristic
97C22×C4C2×Dic311yescharacteristic
98C22×C4C2×Dic311yescharacteristic
99C22×C4#6712yes—
100C22×C4#6712yes—
101C22×C4C22×D311yescharacteristic
102M16—124no—
103C2×C8—64no—
104C4⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C42—21no—
108C42—21no—
109C22×C3—21no—
110C22×C3—21no—
111C22×C3—22no—
112C22×C3—22no—
113C22×C3—22no—
114C22×C3—22no—
115C22×C3—42no—
116C22×C3—42no—
117C22×C3—42no—
118C22×C3—42no—
119C22×C3—41no—
120C22×C3—41no—
121C22×C3C2×C22⋊C411yescharacteristic
122C22×C3C2×M1611yescharacteristic
123C22×C3C2×M1611yescharacteristic
124C3×C4—22no—
125C3×C4—22no—
126C3×C4—21no—
127C3×C4—41no—
128C3×C4—41no—
129C3×C4#9912yes—
130C23—21no—
131C23—21no—
132C23—21no—
133C23—21no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23#20812yescomplemented
139C23#21712yes—
140C23C22×Dic311yescharacteristic
141D4—28no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4#20812yescomplemented
155C2×C4#21711yescharacteristic
156C2×C4#21711yescharacteristic
157C2×C4C2×#6712yes—
158C8—124no—
159C2×C3—21no—
160C2×C3—21no—
161C2×C3—42no—
162C2×C3—42no—
163C2×C3#40711yescharacteristic
164C2×C3C2×#9911yescharacteristic
165C2×C3#41311yescharacteristic
166C22—21no—
167C22—21no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—42no—
176C22—41no—
177C22—41no—
178C22C2×#20811yescharacteristic
179C22C2×#20811yescharacteristic
180C22C2×#21711yescharacteristic
181C4—22no—
182C4—22no—
183C4—21no—
184C4—41no—
185C4—41no—
186C4#83412yes—
187C3#146511yescharacteristic, complemented, Sylow
188C2—21no—
189C2—21no—
190C2—42no—
191C2—42no—
192C2#520711yescharacteristic
193C2C2×#83411yescharacteristic
194C2#523711yescharacteristic
subgroupG/H
centerC22C2×#217
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC2×C3×C4×D4C2
socleC22×C3C2×C22⋊C4
prime radicalC22×C3×D4C4
cosocleC22×C3×C4C23
hypercenterC2×C4×D4D3
hypocenterC3#1465
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #5207C2semidirect
3. #4528C2semidirect
4. #4525C2semidirect
66. C23×C3M16semidirect
83. C2×C3×C4M16semidirect
138. C23#208semidirect
154. C2×C4#208semidirect
187. C3#1465semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
#52072
#45282
#45252
#14653