Finite Group Databasebeta

← #69500

Subgroups — #69500

Order 384.

638 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 73 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×#953C212yesmaximal, complemented
2#5493C211yesmaximal, characteristic, complemented
3C3×#379C211yesmaximal, characteristic, complemented
4C2×#806C212yesmaximal, complemented
5#4526C211yesmaximal, characteristic, complemented
6#2021—31nomaximal, Sylow
7C25×C3C2211yescharacteristic
8C22×#67—22no—
9C22×#67C2212yes—
10C2×C3×C22⋊C4—22no—
11C2×C3×C22⋊C4C2211yescharacteristic
12C2×C3×C22⋊C4C2211yescharacteristic
13#953—24no—
14#953—24no—
15C2×#217—22no—
16C2×#217C2212yes—
17#930—22no—
18#886—22no—
19#882—22no—
20#806—24no—
21#534—31no—
22C2×#120—32no—
23C2×#99—32no—
24#379D311yescharacteristic, complemented
25#342—31no—
26C24×C3—22no—
27C24×C3—21no—
28C24×C3—22no—
29C24×C3—21no—
30C24×C3—44no—
31C24×C3D412yescomplemented
32C24×C3C2311yescharacteristic
33C23×D3D412yescomplemented
34C22×C3×C4—21no—
35C22×C3×C4D412yescomplemented
36C2×#67—28no—
37C2×#67—22no—
38C2×#67—22no—
39C2×#67—22no—
40C2×#67—22no—
41C2×#67—22no—
42C2×#67—22no—
43C2×#67—48no—
44C22×Dic3—22no—
45C22×Dic3D412yescomplemented
46C2×D12—22no—
47C3×C22⋊C4—22no—
48C3×C22⋊C4—22no—
49C3×C22⋊C4—44no—
50#217—28no—
51#217—22no—
52#217D414yescomplemented
53(C2×D3)⋊C4—24no—
54C4⋊Dic3—22no—
55C25C2×D311yescharacteristic
56C22×D4—32no—
57C22×D4—62no—
58#123—62no—
59#121—62no—
60#120—62no—
61#120—64no—
62#120—64no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—32no—
65C2×C22⋊C4—62no—
66C2×C22⋊C4C2×D311yescharacteristic
67C2×C22⋊C4C2×D311yescharacteristic
68#99—64no—
69C23×C3—24no—
70C23×C3—28no—
71C23×C3—24no—
72C23×C3—22no—
73C23×C3—44no—
74C23×C3—42no—
75C23×C3—44no—
76C23×C3—42no—
77C23×C3—48no—
78C23×C3—44no—
79C23×C3—41no—
80C23×C3—42no—
81C23×C3—41no—
82C23×C3C2×D411yescharacteristic
83C23×C3C2×D412yes—
84C23×C3C2×D412yes—
85C23×C3C2×D411yescharacteristic
86C23×C3#4111yescharacteristic
87C22×D3—22no—
88C22×D3—22no—
89C22×D3—22no—
90C22×D3—44no—
91C2×C3×C4—22no—
92C2×C3×C4—42no—
93C2×C3×C4—42no—
94C2×C3×C4—41no—
95#67—416no—
96#67—42no—
97#67—42no—
98#67—42no—
99#67—42no—
100C2×Dic3—22no—
101C2×Dic3—22no—
102C2×Dic3—24no—
103C2×Dic3—22no—
104C2×Dic3—44no—
105D12—82no—
106C24—22no—
107C24—21no—
108C24—21no—
109C24—22no—
110C24—32no—
111C24—44no—
112C24D1212yescomplemented
113C24C22×D311yescharacteristic
114C2×D4—62no—
115C2×D4—68no—
116C2×D4—62no—
117C2×D4—62no—
118C2×D4—62no—
119C2×D4—62no—
120C2×D4—62no—
121C2×D4—62no—
122C2×D4—128no—
123C22×C4—21no—
124C22×C4—32no—
125C22×C4—62no—
126C22×C4#6712yescomplemented
127C4⋊C4—62no—
128C22⋊C4—22no—
129C22⋊C4—22no—
130C22⋊C4—34no—
131C22⋊C4—44no—
132C22⋊C4—64no—
133C22⋊C4—68no—
134C22⋊C4—62no—
135C22×C3—24no—
136C22×C3—22no—
137C22×C3—28no—
138C22×C3—24no—
139C22×C3—41no—
140C22×C3—42no—
141C22×C3—41no—
142C22×C3—44no—
143C22×C3—44no—
144C22×C3—48no—
145C22×C3—42no—
146C22×C3—42no—
147C22×C3—44no—
148C22×C3#12012yes—
149C22×C3#12011yescharacteristic
150C22×C3#12112yes—
151C22×C3#12111yescharacteristic
152C22×C3#12711yescharacteristic
153C2×D3—44no—
154C2×D3—42no—
155C2×D3—44no—
156C2×D3—82no—
157C3×C4—42no—
158C3×C4—81no—
159Dic3—44no—
160Dic3—42no—
161C23—24no—
162C23—28no—
163C23—24no—
164C23—22no—
165C23—44no—
166C23—42no—
167C23—42no—
168C23—44no—
169C23—48no—
170C23—44no—
171C23—42no—
172C23—41no—
173C23—41no—
174C23—62no—
175C23—62no—
176C23—62no—
177C23—124no—
178C23C2×D1211yescharacteristic
179C23#23511yescharacteristic
180C23D3×D412yes—
181C23D3×D412yes—
182C23C2×#6711yescharacteristic
183D4—1216no—
184D4—122no—
185D4—122no—
186D4—122no—
187D4—122no—
188D4—242no—
189C2×C4—22no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—64no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—124no—
198C2×C3—22no—
199C2×C3—21no—
200C2×C3—22no—
201C2×C3—21no—
202C2×C3—44no—
203C2×C3#39211yescharacteristic
204C2×C3B(4,𝔽2)12yes—
205D3—82no—
206C22—22no—
207C22—24no—
208C22—28no—
209C22—24no—
210C22—41no—
211C22—41no—
212C22—42no—
213C22—44no—
214C22—44no—
215C22—42no—
216C22—48no—
217C22—42no—
218C22—44no—
219C22—124no—
220C22—122no—
221C22—124no—
222C22—242no—
223C22#88212yes—
224C22#88412yes—
225C22#93011yescharacteristic
226C22#93711yescharacteristic
227C22#94011yescharacteristic
228C4—42no—
229C4—81no—
230C4—124no—
231C4—122no—
232C3#202111yescharacteristic, complemented, Sylow
233C2—21no—
234C2—22no—
235C2—21no—
236C2—22no—
237C2—44no—
238C2—242no—
239C2#496011yescharacteristic
240C2#503212yes—
subgroupG/H
centerC22#937
commutator, cosocleC24×C3C23
FrattiniC24C22×D3
FittingC3×#379C2
socleC22×C3#120
hypercenter#379D3
hypocenterC3#2021
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#953C2semidirect
2. #5493C2semidirect
3. C3×#379C2semidirect
4. C2×#806C2semidirect
5. #4526C2semidirect
24. #379D3semidirect
31. C24×C3D4semidirect
33. C23×D3D4semidirect
35. C22×C3×C4D4semidirect
45. C22×Dic3D4semidirect
52. #217D4semidirect
112. C24D12semidirect
126. C22×C4#67semidirect
232. C3#2021semidirect

Maximal subgroups

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

subgroupindex
C2×#9532
#54932
C3×#3792
C2×#8062
#45262
#20213