Finite Group Databasebeta

← #69596

Subgroups — #69596

Order 384.

382 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 69 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
1#5493C211yesmaximal, characteristic
2C3×#385C211yesmaximal, characteristic, complemented
3#5213C212yesmaximal, complemented
4C2×#806C212yesmaximal, complemented
5#4526C211yesmaximal, characteristic, complemented
6#2044—31nomaximal, Sylow
7C23×C3×C4C2211yescharacteristic
8C22×#67C2212yes—
9C2×C3×C4⋊C4—21no—
10C2×C3×C22⋊C4C2211yescharacteristic
11C2×C3×C22⋊C4C2211yescharacteristic
12C2×#217C2212yes—
13#930—21no—
14#930—21no—
15C2×(C2×D3)⋊C4—22no—
16#886—22no—
17#882—22no—
18C3×#95—21no—
19C3⋊#95—22no—
20#806—24no—
21#534—31no—
22C2×#99—32no—
23#386—32no—
24#385D311yescharacteristic, complemented
25#342—31no—
26C24×C3C2311yescharacteristic
27C23×D3D412yes—
28C22×C3×C4—22no—
29C22×C3×C4—21no—
30C22×C3×C4—21no—
31C22×C3×C4—42no—
32C22×C3×C4D411yescharacteristic
33C22×C3×C4D411yescharacteristic
34C2×#67—22no—
35C2×#67—22no—
36C2×#67—22no—
37C2×#67—22no—
38C2×#67—22no—
39C2×#67—22no—
40C22×Dic3—22no—
41C22×Dic3D412yes—
42C2×D12—21no—
43C2×D12—21no—
44C3×C4⋊C4—42no—
45C3×C22⋊C4—22no—
46C3×C22⋊C4—21no—
47C3×C22⋊C4D412yescomplemented
48#217—22no—
49#217—22no—
50(C2×D3)⋊C4—22no—
51(C2×D3)⋊C4—22no—
52(C2×D3)⋊C4—44no—
53C4⋊Dic3—21no—
54C4⋊Dic3—21no—
55C22×D4—32no—
56C23×C4C2×D311yescharacteristic
57#123—62no—
58#121—61no—
59#121—61no—
60#120—62no—
61C2×C4⋊C4—21no—
62C2×C22⋊C4—32no—
63C2×C22⋊C4—62no—
64C2×C22⋊C4C2×D311yescharacteristic
65C2×C22⋊C4C2×D311yescharacteristic
66#99—64no—
67#95—21no—
68#95—62no—
69C23×C3—41no—
70C23×C3—41no—
71C23×C3C2×D411yescharacteristic
72C23×C3C2×D412yes—
73C23×C3C2×D411yescharacteristic
74C23×C3#4111yescharacteristic
75C23×C3#4112yes—
76C22×D3—22no—
77C22×D3—22no—
78C22×D3—22no—
79C22×D3—42no—
80C22×D3—42no—
81C2×C3×C4—24no—
82C2×C3×C4—22no—
83C2×C3×C4—21no—
84C2×C3×C4—21no—
85C2×C3×C4—22no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4—42no—
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4—41no—
92C2×C3×C4—41no—
93#67—42no—
94#67—42no—
95#67—42no—
96#67—42no—
97C2×Dic3—22no—
98C2×Dic3—22no—
99C2×Dic3—22no—
100C2×Dic3—42no—
101C2×Dic3—42no—
102C2×Dic3—42no—
103D12—81no—
104D12—81no—
105C24—32no—
106C24C22×D311yescharacteristic
107C2×D4—61no—
108C2×D4—61no—
109C2×D4—62no—
110C2×D4—62no—
111C2×D4—62no—
112C2×D4—62no—
113C2×D4—62no—
114C2×D4—62no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—32no—
119C22×C4—42no—
120C22×C4—62no—
121C22×C4#6711yescharacteristic
122C22×C4#6711yescharacteristic
123C4⋊C4—42no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C22⋊C4—21no—
127C22⋊C4—22no—
128C22⋊C4—62no—
129C22⋊C4—62no—
130C22⋊C4—62no—
131C22⋊C4—62no—
132C22⋊C4—124no—
133C22⋊C4D1212yescomplemented
134C22×C3—22no—
135C22×C3—21no—
136C22×C3—22no—
137C22×C3—21no—
138C22×C3—41no—
139C22×C3—41no—
140C22×C3—42no—
141C22×C3#12011yescharacteristic
142C22×C3#12112yes—
143C22×C3#12111yescharacteristic
144C22×C3#12312yes—
145C22×C3#12411yescharacteristic
146C2×D3—44no—
147C2×D3—42no—
148C2×D3—42no—
149C2×D3—42no—
150C2×D3—82no—
151C3×C4—42no—
152C3×C4—41no—
153C3×C4—42no—
154C3×C4—41no—
155Dic3—42no—
156Dic3—82no—
157C23—41no—
158C23—41no—
159C23—62no—
160C23—62no—
161C23—62no—
162C23—122no—
163C23—122no—
164C23C2×D1211yescharacteristic
165C23#23511yescharacteristic
166C23D3×D412yes—
167C23#23912yes—
168C23C2×#6711yescharacteristic
169D4—122no—
170D4—122no—
171D4—122no—
172D4—122no—
173D4—241no—
174D4—241no—
175C2×C4—24no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—122no—
191C2×C4—122no—
192C2×C4—122no—
193C2×C3—22no—
194C2×C3—21no—
195C2×C3—22no—
196C2×C3—21no—
197C2×C3#39411yescharacteristic
198C2×C3B(4,𝔽2)11yescharacteristic
199C2×C3#45811yescharacteristic
200D3—82no—
201C22—21no—
202C22—22no—
203C22—21no—
204C22—22no—
205C22—41no—
206C22—41no—
207C22—42no—
208C22—124no—
209C22—122no—
210C22—122no—
211C22—122no—
212C22—242no—
213C22#89412yes—
214C22#89512yes—
215C22#93011yescharacteristic
216C22#93711yescharacteristic
217C22#94711yescharacteristic
218C4—42no—
219C4—42no—
220C4—41no—
221C4—41no—
222C4—122no—
223C4—242no—
224C3#204411yescharacteristic, complemented, Sylow
225C2—21no—
226C2—22no—
227C2—22no—
228C2—21no—
229C2—242no—
230C2#499111yescharacteristic
231C2#503211yescharacteristic
232C2#505311yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC24×C3C23
FrattiniC24C22×D3
FittingC3×#385C2
socleC22×C3#120
prime radical#5493C2
hypercenter#385D3
hypocenterC3#2044
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C3×#385C2semidirect
3. #5213C2semidirect
4. C2×#806C2semidirect
5. #4526C2semidirect
24. #385D3semidirect
47. C3×C22⋊C4D4semidirect
133. C22⋊C4D12semidirect
224. C3#2044semidirect

Maximal subgroups

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

subgroupindex
#54932
C3×#3852
#52132
C2×#8062
#45262
#20443