Finite Group Databasebeta

← #71849

Subgroups — #71849

Order 384.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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
2C2×#935C212yesmaximal, complemented
3C2×#834C212yesmaximal, complemented
4#4943C211yesmaximal, characteristic, complemented
5#4526C211yesmaximal, characteristic, complemented
6#2022—31nomaximal, Sylow
7C22×C3×D4C2212yes—
8C23×Dic3C2211yescharacteristic
9C3×#123—22no—
10C3×#121—22no—
11C3×#120—22no—
12C2×C3×C22⋊C4C2211yescharacteristic
13C2×#217—22no—
14C2×#217C2211yescharacteristic
15C2×#217C2212yes—
16#935—24no—
17#935—24no—
18C2×Dic3⋊C4—22no—
19#834—24no—
20C3⋊#95—22no—
21#534D311yescharacteristic
22C2×#123—32no—
23C2×#99—32no—
24#381—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—22no—
35C22×C3×C4D412yes—
36C22×C3×C4D412yes—
37C22×Dic3—21no—
38C22×Dic3—22no—
39C22×Dic3—22no—
40C22×Dic3—22no—
41C22×Dic3—21no—
42C22×Dic3—21no—
43C22×Dic3D412yescomplemented
44C3×C4⋊C4—22no—
45C3×C22⋊C4—24no—
46C3×C22⋊C4—22no—
47#217—28no—
48#217—22no—
49#217—22no—
50#217D414yescomplemented
51Dic3⋊C4—28no—
52C22×D4C2×D312yes—
53C23×C4—31no—
54#123—22no—
55#123—64no—
56#123—64no—
57#121—22no—
58#120—22no—
59C2×C4⋊C4—62no—
60C2×C22⋊C4—32no—
61C2×C22⋊C4—31no—
62C2×C22⋊C4—62no—
63C2×C22⋊C4C2×D311yescharacteristic
64#99—64no—
65#95—62no—
66C23×C3—22no—
67C23×C3—22no—
68C23×C3—22no—
69C23×C3—42no—
70C23×C3—44no—
71C23×C3C2×D411yescharacteristic
72C23×C3C2×D412yes—
73C23×C3C2×D412yes—
74C23×C3C2×D411yescharacteristic
75C23×C3#4111yescharacteristic
76C3×D4—42no—
77C3×D4—42no—
78C3×D4—42no—
79C3×D4—42no—
80C3×D4—82no—
81C2×C3×C4—22no—
82C2×C3×C4—22no—
83C2×C3×C4—22no—
84C2×C3×C4—22no—
85C2×C3×C4—42no—
86C2×Dic3—24no—
87C2×Dic3—24no—
88C2×Dic3—22no—
89C2×Dic3—44no—
90C2×Dic3—41no—
91C2×Dic3—42no—
92C2×Dic3—41no—
93C2×Dic3—42no—
94C2×Dic3—41no—
95C24#6712yes—
96C24C22×D311yescharacteristic
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C22×C4—32no—
105C22×C4—62no—
106C22×C4—61no—
107C22×C4—61no—
108C22×C4—62no—
109C22×C4—62no—
110C22×C4—61no—
111C22×C4#6712yes—
112C22×C4#6712yes—
113C4⋊C4—22no—
114C4⋊C4—68no—
115C22⋊C4—24no—
116C22⋊C4—22no—
117C22⋊C4—34no—
118C22⋊C4—68no—
119C22⋊C4—62no—
120C22⋊C4—62no—
121C22×C3—22no—
122C22×C3—24no—
123C22×C3—41no—
124C22×C3—42no—
125C22×C3—41no—
126C22×C3—44no—
127C22×C3—42no—
128C22×C3—44no—
129C22×C3—82no—
130C22×C3#12012yes—
131C22×C3#12011yescharacteristic
132C22×C3#12111yescharacteristic
133C22×C3#12112yes—
134C22×C3#12711yescharacteristic
135C3×C4—42no—
136C3×C4—42no—
137Dic3—44no—
138Dic3—42no—
139Dic3—81no—
140C23—22no—
141C23—22no—
142C23—22no—
143C23—42no—
144C23—44no—
145C23D3×D411yescharacteristic
146C23D3×D412yes—
147C23#23711yescharacteristic
148C23C2×#6712yes—
149C23C2×#6711yescharacteristic
150D4—42no—
151D4—42no—
152D4—42no—
153D4—42no—
154D4—82no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—42no—
160C2×C4—64no—
161C2×C4—64no—
162C2×C4—62no—
163C2×C4—124no—
164C2×C4—122no—
165C2×C4—121no—
166C2×C4—121no—
167C2×C4—122no—
168C2×C4—121no—
169C2×C3—22no—
170C2×C3—21no—
171C2×C3—22no—
172C2×C3—21no—
173C2×C3—82no—
174C2×C3#39211yescharacteristic
175C2×C3#45812yes—
176C22—22no—
177C22—24no—
178C22—41no—
179C22—42no—
180C22—41no—
181C22—44no—
182C22—42no—
183C22—44no—
184C22—82no—
185C22#93712yes—
186C22#93811yescharacteristic
187C22#93912yes—
188C22#94011yescharacteristic
189C22#95311yescharacteristic
190C4—42no—
191C4—42no—
192C4—124no—
193C4—122no—
194C4—241no—
195C3#202211yescharacteristic, complemented, Sylow
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—82no—
201C2#522311yescharacteristic
202C2#524312yes—
subgroupG/H
centerC22#953
commutator, cosocleC24×C3C23
FrattiniC24C22×D3
Fitting, prime radicalC3×#534C2
socleC22×C3#120
hypercenter#534D3
hypocenterC3#2022
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. C2×#935C2semidirect
3. C2×#834C2semidirect
4. #4943C2semidirect
5. #4526C2semidirect
43. C22×Dic3D4semidirect
50. #217D4semidirect
195. C3#2022semidirect

Maximal subgroups

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

subgroupindex
C3×#5342
C2×#9352
C2×#8342
#49432
#45262
#20223