Finite Group Databasebeta

← #71850

Subgroups — #71850

Order 384.

382 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 71 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×#534C211yesmaximal, characteristic
2#5220C212yesmaximal, complemented
3C2×#834C212yesmaximal, complemented
4#4946C211yesmaximal, characteristic, complemented
5#4526C211yesmaximal, characteristic, complemented
6#2044—31nomaximal, Sylow
7C22×C3×D4C2212yes—
8C23×Dic3C2211yescharacteristic
9C3×#123—22no—
10C3×#121—21no—
11C3×#121—21no—
12C3×#120—22no—
13C2×C3×C22⋊C4C2211yescharacteristic
14C2×#217—22no—
15C2×#217C2212yes—
16C2×#217C2211yescharacteristic
17C2×Dic3⋊C4—21no—
18#834—24no—
19C3⋊#95—21no—
20C3⋊#95—22no—
21#534D311yescharacteristic
22C2×#99—32no—
23#386—32no—
24#385—31no—
25#342—31no—
26C24×C3D412yes—
27C24×C3C2311yescharacteristic
28C2×C3×D4—22no—
29C2×C3×D4—22no—
30C2×C3×D4—22no—
31C2×C3×D4—22no—
32C2×C3×D4—22no—
33C2×C3×D4—22no—
34C2×C3×D4—21no—
35C2×C3×D4—21no—
36C22×C3×C4D412yes—
37C22×C3×C4D411yescharacteristic
38C22×C3×C4D411yescharacteristic
39C22×Dic3—21no—
40C22×Dic3—21no—
41C22×Dic3—22no—
42C22×Dic3—22no—
43C22×Dic3—42no—
44C3×C4⋊C4—21no—
45C3×C4⋊C4—21no—
46C3×C22⋊C4—22no—
47C3×C22⋊C4—22no—
48C3×C22⋊C4—22no—
49#217—21no—
50#217—22no—
51#217—22no—
52#217—44no—
53#217D412yescomplemented
54Dic3⋊C4—42no—
55C22×D4C2×D312yes—
56C23×C4—31no—
57#123—22no—
58#121—21no—
59#121—21no—
60#120—22no—
61C2×C4⋊C4—61no—
62C2×C22⋊C4—31no—
63C2×C22⋊C4—32no—
64C2×C22⋊C4—62no—
65C2×C22⋊C4C2×D311yescharacteristic
66#99—64no—
67#95—61no—
68#95—62no—
69C23×C3—22no—
70C23×C3—22no—
71C23×C3—22no—
72C23×C3—41no—
73C23×C3—41no—
74C23×C3—42no—
75C23×C3—42no—
76C23×C3C2×D411yescharacteristic
77C23×C3C2×D412yes—
78C23×C3C2×D411yescharacteristic
79C23×C3#4111yescharacteristic
80C23×C3#4112yes—
81C3×D4—42no—
82C3×D4—42no—
83C3×D4—42no—
84C3×D4—42no—
85C3×D4—81no—
86C3×D4—81no—
87C2×C3×C4—21no—
88C2×C3×C4—21no—
89C2×C3×C4—22no—
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—41no—
93C2×C3×C4—41no—
94C2×Dic3—22no—
95C2×Dic3—24no—
96C2×Dic3—22no—
97C2×Dic3—41no—
98C2×Dic3—41no—
99C2×Dic3—41no—
100C2×Dic3—42no—
101C2×Dic3—41no—
102C2×Dic3—42no—
103C2×Dic3—42no—
104C2×Dic3—42no—
105C24#6712yes—
106C24C22×D311yescharacteristic
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—21no—
115C22×C4—61no—
116C22×C4—62no—
117C22×C4—61no—
118C22×C4—62no—
119C22×C4—122no—
120C22×C4#6712yes—
121C22×C4#6711yescharacteristic
122C22×C4#6711yescharacteristic
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—122no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—22no—
129C22⋊C4—32no—
130C22⋊C4—61no—
131C22⋊C4—62no—
132C22⋊C4—62no—
133C22⋊C4—124no—
134C22×C3—21no—
135C22×C3—22no—
136C22×C3—21no—
137C22×C3—22no—
138C22×C3—41no—
139C22×C3—41no—
140C22×C3—42no—
141C22×C3—44no—
142C22×C3—42no—
143C22×C3—42no—
144C22×C3—42no—
145C22×C3—82no—
146C22×C3#12011yescharacteristic
147C22×C3#12111yescharacteristic
148C22×C3#12112yes—
149C22×C3#12312yes—
150C22×C3#12411yescharacteristic
151C3×C4—42no—
152C3×C4—41no—
153C3×C4—41no—
154Dic3—42no—
155Dic3—42no—
156Dic3—82no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—41no—
161C23—41no—
162C23—42no—
163C23—42no—
164C23D3×D411yescharacteristic
165C23#23711yescharacteristic
166C23#23712yes—
167C23C2×#6712yes—
168C23C2×#6711yescharacteristic
169D4—42no—
170D4—42no—
171D4—42no—
172D4—42no—
173D4—81no—
174D4—81no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—64no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—121no—
186C2×C4—121no—
187C2×C4—122no—
188C2×C4—121no—
189C2×C4—121no—
190C2×C4—122no—
191C2×C4—122no—
192C2×C4—122no—
193C2×C3—21no—
194C2×C3—22no—
195C2×C3—22no—
196C2×C3—21no—
197C2×C3—82no—
198C2×C3#39411yescharacteristic
199C2×C3B(4,𝔽2)11yescharacteristic
200C2×C3#45811yescharacteristic
201C22—21no—
202C22—22no—
203C22—21no—
204C22—22no—
205C22—41no—
206C22—41no—
207C22—42no—
208C22—44no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—82no—
213C22#93512yes—
214C22#93611yescharacteristic
215C22#93811yescharacteristic
216C22#93912yes—
217C22#95311yescharacteristic
218C4—42no—
219C4—41no—
220C4—41no—
221C4—122no—
222C4—122no—
223C4—242no—
224C3#204411yescharacteristic, complemented, Sylow
225C2—21no—
226C2—22no—
227C2—22no—
228C2—21no—
229C2—82no—
230C2#522211yescharacteristic
231C2#524311yescharacteristic
232C2#524411yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC24×C3C23
FrattiniC24C22×D3
Fitting, prime radicalC3×#534C2
socleC22×C3#120
hypercenter#534D3
hypocenterC3#2044
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
2. #5220C2semidirect
3. C2×#834C2semidirect
4. #4946C2semidirect
5. #4526C2semidirect
53. #217D4semidirect
224. C3#2044semidirect

Maximal subgroups

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

subgroupindex
C3×#5342
#52202
C2×#8342
#49462
#45262
#20443