Finite Group Databasebeta

← #68112

Subgroups — #68112

Order 384.

288 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 53 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
2#4949C211yesmaximal, characteristic, complemented
3#4735C212yesmaximal, complemented
4C3×#323C211yesmaximal, characteristic, complemented
5#4470C212yesmaximal, complemented
6#1613—31nomaximal, Sylow
7C22×#67—21no—
8C2×C3×C22⋊C4C2211yescharacteristic
9C2×#217—21no—
10#930—21no—
11#930C2212yes—
12C2×C4⋊Dic3C2211yescharacteristic, complemented
13C2×Dic3⋊C4—21no—
14#886—22no—
15#882—22no—
16C3×C22⋊C8C2212yescomplemented
17C3⋊#95C2211yescharacteristic, complemented
18#821—22no—
19#817—22no—
20#534—31no—
21#481—32no—
22#399—31no—
23#327—32no—
24#323D311yescharacteristic, complemented
25C24×C3D411yescharacteristic
26C23×D3—21no—
27C22×C3×C4D411yescharacteristic
28C22×C3×C4C2311yescharacteristic
29C2×#67—22no—
30C2×#67—22no—
31C2×#67—42no—
32C2×#67—42no—
33C22×Dic3—21no—
34C22×Dic3—21no—
35C22×Dic3—21no—
36C2×D12—21no—
37C2×D12D412yes—
38C2×C3×C8—22no—
39C3×C22⋊C4—22no—
40#217—42no—
41(C2×D3)⋊C4—22no—
42(C2×D3)⋊C4—22no—
43C4⋊Dic3—21no—
44C4⋊Dic3—22no—
45C4⋊Dic3D412yes—
46Dic3⋊C4—42no—
47C22×D4—61no—
48#123—62no—
49#121—32no—
50#121—61no—
51#120—62no—
52C2×C4⋊C4—31no—
53C2×C4⋊C4—61no—
54C2×C22⋊C4—61no—
55C2×C22⋊C4C2×D311yescharacteristic
56#106—62no—
57#102—62no—
58C22⋊C8C2×D312yescomplemented
59#95—31no—
60C23×C3—21no—
61C23×C3—21no—
62C23×C3—41no—
63C23×C3—41no—
64C23×C3SD1612yes—
65C23×C3C2×D411yescharacteristic
66C22×D3—22no—
67C22×D3—42no—
68C22×D3—81no—
69C22×D3—81no—
70C2×C3×C4—21no—
71C2×C3×C4—41no—
72C2×C3×C4—41no—
73C2×C3×C4C2×D412yes—
74#67—44no—
75#67—42no—
76#67—42no—
77C2×Dic3—22no—
78C2×Dic3—22no—
79C2×Dic3—41no—
80C2×Dic3—42no—
81C2×Dic3—41no—
82C2×Dic3—41no—
83C2×Dic3—41no—
84C2×Dic3—41no—
85D12—42no—
86D12—81no—
87C3×C8—42no—
88C24—61no—
89C24D1211yescharacteristic
90C2×D4—32no—
91C2×D4—61no—
92C2×D4—62no—
93C2×D4—62no—
94C2×D4—122no—
95C2×D4—122no—
96C22×C4—61no—
97C22×C4—61no—
98C22×C4—61no—
99C22×C4D1211yescharacteristic
100C22×C4C22×D311yescharacteristic
101C2×C8—22no—
102C4⋊C4—32no—
103C4⋊C4—61no—
104C4⋊C4—62no—
105C4⋊C4—122no—
106C22⋊C4—22no—
107C22⋊C4—62no—
108C22⋊C4—62no—
109C22⋊C4—122no—
110C22×C3—21no—
111C22×C3—21no—
112C22×C3—22no—
113C22×C3—42no—
114C22×C3—41no—
115C22×C3—41no—
116C22×C3—81no—
117C22×C3#12011yescharacteristic
118C22×C3C2×SD1611yescharacteristic
119C22×C3AGL(1,ℤ/8)11yescharacteristic
120C2×D3—42no—
121C2×D3—42no—
122C2×D3—42no—
123C2×D3—84no—
124C3×C4—22no—
125C3×C4—41no—
126Dic3—42no—
127Dic3—42no—
128Dic3—81no—
129C23—21no—
130C23—21no—
131C23—41no—
132C23—41no—
133C23—62no—
134C23—122no—
135C23—241no—
136C23—241no—
137C23#20412yes—
138C23C2×D1211yescharacteristic
139D4—122no—
140D4—124no—
141D4—122no—
142D4—122no—
143D4—241no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—62no—
148C2×C4—62no—
149C2×C4—122no—
150C2×C4—121no—
151C2×C4—121no—
152C2×C4—121no—
153C2×C4—121no—
154C2×C4—121no—
155C2×C4D3×D412yes—
156C8—42no—
157C2×C3—21no—
158C2×C3—21no—
159C2×C3—42no—
160C2×C3#45011yescharacteristic
161C2×C3#45311yescharacteristic
162C2×C3#45811yescharacteristic
163D3—82no—
164C22—21no—
165C22—21no—
166C22—22no—
167C22—42no—
168C22—41no—
169C22—41no—
170C22—81no—
171C22—122no—
172C22—122no—
173C22—122no—
174C22—244no—
175C22#88211yescharacteristic
176C22C2×#20411yescharacteristic
177C22#90811yescharacteristic
178C4—22no—
179C4—41no—
180C4—122no—
181C4—122no—
182C4—241no—
183C3#161311yescharacteristic, complemented, Sylow
184C2—21no—
185C2—21no—
186C2—42no—
187C2—242no—
188C2#473111yescharacteristic
189C2#474211yescharacteristic
190C2#474711yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#323C2
socleC22×C3#120
prime radical#5493C2
hypercenter#323D3
hypocenterC3#1613
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #4949C2semidirect
3. #4735C2semidirect
4. C3×#323C2semidirect
5. #4470C2semidirect
12. C2×C4⋊Dic3C22semidirect
16. C3×C22⋊C8C22semidirect
17. C3⋊#95C22semidirect
24. #323D3semidirect
58. C22⋊C8C2×D3semidirect
183. C3#1613semidirect

Maximal subgroups

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

subgroupindex
#54932
#49492
#47352
C3×#3232
#44702
#16133