Finite Group Databasebeta

← #68966

Subgroups — #68966

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
1#5493C211yesmaximal, characteristic
2C2×#886C212yesmaximal, complemented
3C2×#806C212yesmaximal, complemented
4#4943C211yesmaximal, characteristic, complemented
5C3×#342C211yesmaximal, characteristic, complemented
6#2022—31nomaximal, Sylow
7C22×#67C2212yes—
8C23×Dic3C2211yescharacteristic
9C2×C3×C22⋊C4C2211yescharacteristic
10C2×C3×C22⋊C4C2212yes—
11C2×#217C2211yescharacteristic
12#930—22no—
13C2×(C2×D3)⋊C4—22no—
14C2×C4⋊Dic3—22no—
15#886—22no—
16#886—24no—
17#886—24no—
18#882—22no—
19C3⋊#95—22no—
20#806—24no—
21#534—31no—
22C2×#123—32no—
23C2×#99—32no—
24#381—31no—
25#342D311yescharacteristic, complemented
26C24×C3C2311yescharacteristic
27C23×D3D412yes—
28C22×C3×C4—22no—
29C22×C3×C4D412yes—
30C2×#67—22no—
31C2×#67—22no—
32C2×#67—22no—
33C2×#67—22no—
34C2×#67—22no—
35C2×#67—22no—
36C22×Dic3—21no—
37C22×Dic3—22no—
38C22×Dic3—22no—
39C22×Dic3—21no—
40C22×Dic3—21no—
41C22×Dic3D412yes—
42C22×Dic3D412yescomplemented
43C2×D12—22no—
44C3×C22⋊C4—22no—
45C3×C22⋊C4—22no—
46C3×C22⋊C4D414yescomplemented
47#217—22no—
48(C2×D3)⋊C4—24no—
49(C2×D3)⋊C4—28no—
50C4⋊Dic3—28no—
51C4⋊Dic3—22no—
52C22×D4—32no—
53C23×C4—31no—
54#123—64no—
55#123—62no—
56#123—64no—
57#121—62no—
58#120—62no—
59C2×C4⋊C4—62no—
60C2×C22⋊C4—31no—
61C2×C22⋊C4—62no—
62C2×C22⋊C4C2×D311yescharacteristic
63C2×C22⋊C4C2×D312yes—
64#99—64no—
65#95—62no—
66C23×C3—42no—
67C23×C3C2×D411yescharacteristic
68C23×C3C2×D412yes—
69C23×C3C2×D412yes—
70C23×C3C2×D411yescharacteristic
71C23×C3#4111yescharacteristic
72C22×D3—22no—
73C22×D3—22no—
74C22×D3—22no—
75C22×D3—44no—
76C2×C3×C4—24no—
77C2×C3×C4—22no—
78C2×C3×C4—44no—
79C2×C3×C4—42no—
80#67—42no—
81#67—42no—
82#67—42no—
83#67—42no—
84C2×Dic3—22no—
85C2×Dic3—24no—
86C2×Dic3—22no—
87C2×Dic3—22no—
88C2×Dic3—22no—
89C2×Dic3—41no—
90C2×Dic3—42no—
91C2×Dic3—41no—
92C2×Dic3—42no—
93C2×Dic3—41no—
94D12—82no—
95C24—32no—
96C24C22×D311yescharacteristic
97C2×D4—62no—
98C2×D4—62no—
99C2×D4—62no—
100C2×D4—62no—
101C2×D4—62no—
102C2×D4—62no—
103C2×D4—62no—
104C22×C4—22no—
105C22×C4—32no—
106C22×C4—32no—
107C22×C4—61no—
108C22×C4—62no—
109C22×C4—62no—
110C22×C4—61no—
111C22×C4—61no—
112C22×C4D1212yes—
113C4⋊C4—68no—
114C4⋊C4—62no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—68no—
118C22⋊C4—64no—
119C22⋊C4—62no—
120C22⋊C4D1214yescomplemented
121C22×C3—22no—
122C22×C3—24no—
123C22×C3—42no—
124C22×C3—41no—
125C22×C3—41no—
126C22×C3#12012yes—
127C22×C3#12011yescharacteristic
128C22×C3#12111yescharacteristic
129C22×C3#12112yes—
130C22×C3#12711yescharacteristic
131C2×D3—44no—
132C2×D3—42no—
133C2×D3—44no—
134C2×D3—82no—
135C3×C4—44no—
136C3×C4—42no—
137Dic3—42no—
138Dic3—42no—
139Dic3—81no—
140C23—42no—
141C23—62no—
142C23—62no—
143C23—62no—
144C23—124no—
145C23C2×D1211yescharacteristic
146C23C2×D1212yes—
147C23D3×D412yes—
148C23D3×D411yescharacteristic
149C23#23911yescharacteristic
150D4—122no—
151D4—122no—
152D4—122no—
153D4—122no—
154D4—242no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—64no—
160C2×C4—62no—
161C2×C4—62no—
162C2×C4—62no—
163C2×C4—62no—
164C2×C4—121no—
165C2×C4—121no—
166C2×C4—122no—
167C2×C4—122no—
168C2×C4—121no—
169C2×C3—21no—
170C2×C3—22no—
171C2×C3—22no—
172C2×C3—21no—
173C2×C3#39211yescharacteristic
174C2×C3#45812yes—
175D3—82no—
176C22—22no—
177C22—24no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—124no—
182C22—122no—
183C22—124no—
184C22—242no—
185C22#87411yescharacteristic
186C22#88212yes—
187C22#88211yescharacteristic
188C22#89511yescharacteristic
189C22#89512yes—
190C4—44no—
191C4—42no—
192C4—122no—
193C4—122no—
194C4—241no—
195C3#202211yescharacteristic, complemented, Sylow
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—242no—
201C2#467111yescharacteristic
202C2#474212yes—
subgroupG/H
centerC22#882
commutator, cosocleC24×C3C23
FrattiniC24C22×D3
FittingC3×#342C2
socleC22×C3#120
prime radical#5493C2
hypercenter#342D3
hypocenterC3#2022
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#886C2semidirect
3. C2×#806C2semidirect
4. #4943C2semidirect
5. C3×#342C2semidirect
25. #342D3semidirect
42. C22×Dic3D4semidirect
46. C3×C22⋊C4D4semidirect
120. C22⋊C4D12semidirect
195. C3#2022semidirect

Maximal subgroups

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

subgroupindex
#54932
C2×#8862
C2×#8062
#49432
C3×#3422
#20223