Finite Group Databasebeta

← #69967

Subgroups — #69967

Order 384.

440 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C22×(C2×D3)⋊C4C211yesmaximal, characteristic
2C3×#407C211yesmaximal, characteristic, complemented
3#5207C211yesmaximal, characteristic, complemented
4#4468C214yesmaximal, complemented
5#1462—31nomaximal, Sylow
6C24×D3C411yescharacteristic
7C23×C3×C4C2211yescharacteristic
8C23×Dic3C411yescharacteristic
9C2×C3×M16—21no—
10C2×(C2×D3)⋊C4—21no—
11C2×(C2×D3)⋊C4—21no—
12C2×(C2×D3)⋊C4—21no—
13C2×(C2×D3)⋊C4—21no—
14C2×(C2×D3)⋊C4C412yes—
15C2×(C2×D3)⋊C4C2212yes—
16C2×#208—21no—
17C3×C22⋊C8—21no—
18C3×C22⋊C8C2212yescomplemented
19#830—21no—
20#830C2212yescomplemented
21C22×C22⋊C4—31no—
22#407—31no—
23#407D311yescharacteristic, complemented
24#323—34no—
25C24×C3C2×C411yescharacteristic
26C23×D3—21no—
27C23×D3—21no—
28C23×D3—24no—
29C23×D3—42no—
30C23×D3—42no—
31C23×D3C2×C412yes—
32C22×C3×C4—42no—
33C22×C3×C4C2×C411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C22×C3×C4D411yescharacteristic
37C22×C3×C4D411yescharacteristic
38C22×C3×C4C2311yescharacteristic
39C22×Dic3—21no—
40C22×Dic3—21no—
41C22×Dic3—42no—
42C22×Dic3C2×C412yes—
43C3×M16—42no—
44C2×C3×C8—22no—
45(C2×D3)⋊C4—22no—
46(C2×D3)⋊C4—22no—
47(C2×D3)⋊C4—22no—
48(C2×D3)⋊C4—22no—
49#208—42no—
50C2×C3⋊C8—22no—
51C25—31no—
52C23×C4—31no—
53C23×C4C2×D311yescharacteristic
54C2×M16—21no—
55C2×M16—61no—
56C2×C22⋊C4—32no—
57C2×C22⋊C4—32no—
58C2×C22⋊C4—61no—
59C2×C22⋊C4—61no—
60C2×C22⋊C4—61no—
61C2×C22⋊C4—61no—
62C22⋊C8—21no—
63C22⋊C8—32no—
64C22⋊C8—61no—
65C22⋊C8C2×D312yescomplemented
66C23×C3—22no—
67C23×C3—22no—
68C23×C3—22no—
69C23×C3C22⋊C411yescharacteristic
70C23×C3C22⋊C411yescharacteristic
71C23×C3C22×C411yescharacteristic
72C22×D3—24no—
73C22×D3—24no—
74C22×D3—22no—
75C22×D3—22no—
76C22×D3—416no—
77C22×D3—42no—
78C22×D3—42no—
79C22×D3—42no—
80C22×D3—42no—
81C22×D3—42no—
82C22×D3—42no—
83C22×D3M1614yescomplemented
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4C22⋊C411yescharacteristic
91C2×C3×C4C22⋊C411yescharacteristic
92C2×C3×C4C2×D411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94C2×Dic3—22no—
95C2×Dic3—42no—
96C2×Dic3—42no—
97C2×Dic3—42no—
98C3×C8—42no—
99C3⋊C8—42no—
100C24—32no—
101C24—64no—
102C24—61no—
103C24—61no—
104C24—122no—
105C24—122no—
106C24C4×D311yescharacteristic
107C22×C4—32no—
108C22×C4—42no—
109C22×C4—61no—
110C22×C4—61no—
111C22×C4—122no—
112C22×C4C4×D311yescharacteristic
113C22×C4D1211yescharacteristic
114C22×C4D1211yescharacteristic
115C22×C4#6711yescharacteristic
116C22×C4#6711yescharacteristic
117C22×C4C22×D311yescharacteristic
118M16—42no—
119M16—122no—
120C2×C8—22no—
121C2×C8—62no—
122C22⋊C4—62no—
123C22⋊C4—62no—
124C22⋊C4—62no—
125C22⋊C4—62no—
126C22×C3—28no—
127C22×C3—21no—
128C22×C3—21no—
129C22×C3—21no—
130C22×C3—21no—
131C22×C3—21no—
132C22×C3—21no—
133C22×C3#9912yes—
134C22×C3#10012yes—
135C22×C3C2×C22⋊C411yescharacteristic
136C22×C3C2×M1612yes—
137C2×D3—24no—
138C2×D3—48no—
139C2×D3—48no—
140C2×D3—44no—
141C2×D3—42no—
142C2×D3—42no—
143C2×D3—44no—
144C3×C4—21no—
145C3×C4—21no—
146C3×C4—21no—
147C3×C4—21no—
148Dic3—42no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23—34no—
153C23—64no—
154C23—64no—
155C23—62no—
156C23—62no—
157C23—122no—
158C23—122no—
159C23—1216no—
160C23—122no—
161C23—122no—
162C23—122no—
163C23—122no—
164C23(C2×D3)⋊C411yescharacteristic
165C23(C2×D3)⋊C411yescharacteristic
166C23C2×C4×D311yescharacteristic
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—62no—
174C2×C4—122no—
175C2×C4—122no—
176C2×C4—122no—
177C2×C4(C2×D3)⋊C411yescharacteristic
178C2×C4(C2×D3)⋊C411yescharacteristic
179C2×C4C2×D1211yescharacteristic
180C2×C4C2×#6711yescharacteristic
181C8—42no—
182C8—122no—
183C2×C3—22no—
184C2×C3—22no—
185C2×C3—22no—
186C2×C3#40711yescharacteristic
187C2×C3C2×#9911yescharacteristic
188C2×C3C2×#10011yescharacteristic
189D3—44no—
190C22—28no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—64no—
198C22—124no—
199C22—128no—
200C22—128no—
201C22—124no—
202C22—122no—
203C22—122no—
204C22#80612yes—
205C22#82312yes—
206C22D3×M1612yes—
207C22C2×(C2×D3)⋊C411yescharacteristic
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—122no—
213C3#146211yescharacteristic, complemented, Sylow
214C2—22no—
215C2—22no—
216C2—22no—
217C2—124no—
218C2C2×#80611yescharacteristic
219C2#512611yescharacteristic
220C2C2×#82311yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#407C2
socleC22×C3C2×C22⋊C4
prime radicalC24×D3C4
cosocleC22×C3×C4C23
hypercenter#407D3
hypocenterC3#1462
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C3×#407C2semidirect
3. #5207C2semidirect
4. #4468C2semidirect
18. C3×C22⋊C8C22semidirect
20. #830C22semidirect
23. #407D3semidirect
65. C22⋊C8C2×D3semidirect
83. C22×D3M16semidirect
213. C3#1462semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D3)⋊C42
C3×#4072
#52072
#44682
#14623