Finite Group Databasebeta

← #68110

Subgroups — #68110

Order 384.

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

normal subgroups only

This group has 280 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#4953C211yesmaximal, characteristic, complemented
3#4736C212yesmaximal, complemented
4C3×#323C211yesmaximal, characteristic, complemented
5#4470C212yesmaximal, complemented
6#1607—31nomaximal, Sylow
7C22×#67—21no—
8C2×C3×C22⋊C4C2211yescharacteristic
9C2×#217—21no—
10#930—21no—
11#930C2212yes—
12C2×C4⋊Dic3C2211yescharacteristic, complemented
13#886—22no—
14#882—22no—
15C3×C22⋊C8C2212yescomplemented
16C3⋊#95—21no—
17C3⋊#95C2211yescharacteristic, complemented
18#821—22no—
19#818—22no—
20#534—31no—
21#480—32no—
22#397—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—
46C22×D4—61no—
47#123—62no—
48#121—32no—
49#121—61no—
50#120—62no—
51C2×C4⋊C4—31no—
52C2×C22⋊C4—61no—
53C2×C22⋊C4C2×D311yescharacteristic
54#107—62no—
55#102—62no—
56C22⋊C8C2×D312yescomplemented
57#95—31no—
58#95—61no—
59C23×C3—21no—
60C23×C3—21no—
61C23×C3—41no—
62C23×C3—41no—
63C23×C3D812yes—
64C23×C3C2×D411yescharacteristic
65C22×D3—22no—
66C22×D3—42no—
67C22×D3—81no—
68C22×D3—81no—
69C2×C3×C4—21no—
70C2×C3×C4—41no—
71C2×C3×C4—41no—
72C2×C3×C4C2×D412yes—
73#67—44no—
74#67—42no—
75#67—42no—
76C2×Dic3—22no—
77C2×Dic3—42no—
78C2×Dic3—41no—
79C2×Dic3—41no—
80C2×Dic3—41no—
81C2×Dic3—41no—
82C2×Dic3—41no—
83C2×Dic3—41no—
84D12—42no—
85D12—81no—
86C3×C8—42no—
87C24—61no—
88C24D1211yescharacteristic
89C2×D4—32no—
90C2×D4—61no—
91C2×D4—62no—
92C2×D4—62no—
93C2×D4—122no—
94C2×D4—122no—
95C22×C4—61no—
96C22×C4—61no—
97C22×C4—61no—
98C22×C4D1211yescharacteristic
99C22×C4C22×D311yescharacteristic
100C2×C8—22no—
101C4⋊C4—32no—
102C4⋊C4—61no—
103C4⋊C4—62no—
104C22⋊C4—22no—
105C22⋊C4—62no—
106C22⋊C4—62no—
107C22⋊C4—122no—
108C22×C3—21no—
109C22×C3—21no—
110C22×C3—22no—
111C22×C3—42no—
112C22×C3—41no—
113C22×C3—41no—
114C22×C3—81no—
115C22×C3#12011yescharacteristic
116C22×C3C2×D811yescharacteristic
117C22×C3AGL(1,ℤ/8)11yescharacteristic
118C2×D3—42no—
119C2×D3—42no—
120C2×D3—42no—
121C2×D3—84no—
122C3×C4—22no—
123C3×C4—41no—
124Dic3—42no—
125Dic3—81no—
126Dic3—81no—
127C23—21no—
128C23—21no—
129C23—41no—
130C23—41no—
131C23—62no—
132C23—122no—
133C23—241no—
134C23—241no—
135C23D2412yes—
136C23C2×D1211yescharacteristic
137D4—122no—
138D4—124no—
139D4—122no—
140D4—122no—
141D4—241no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—62no—
146C2×C4—122no—
147C2×C4—121no—
148C2×C4—121no—
149C2×C4—121no—
150C2×C4—121no—
151C2×C4—121no—
152C2×C4—121no—
153C2×C4D3×D412yes—
154C8—42no—
155C2×C3—21no—
156C2×C3—21no—
157C2×C3—42no—
158C2×C3#44711yescharacteristic
159C2×C3#45511yescharacteristic
160C2×C3#45811yescharacteristic
161D3—82no—
162C22—21no—
163C22—21no—
164C22—22no—
165C22—42no—
166C22—41no—
167C22—41no—
168C22—81no—
169C22—122no—
170C22—122no—
171C22—122no—
172C22—244no—
173C22#88211yescharacteristic
174C22C2×D2411yescharacteristic
175C22#90811yescharacteristic
176C4—22no—
177C4—41no—
178C4—122no—
179C4—241no—
180C4—241no—
181C3#160711yescharacteristic, complemented, Sylow
182C2—21no—
183C2—21no—
184C2—42no—
185C2—242no—
186C2#473211yescharacteristic
187C2#474211yescharacteristic
188C2#474611yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#323C2
socleC22×C3#120
prime radical#5493C2
hypercenter#323D3
hypocenterC3#1607
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. #4953C2semidirect
3. #4736C2semidirect
4. C3×#323C2semidirect
5. #4470C2semidirect
12. C2×C4⋊Dic3C22semidirect
15. C3×C22⋊C8C22semidirect
17. C3⋊#95C22semidirect
24. #323D3semidirect
56. C22⋊C8C2×D3semidirect
181. C3#1607semidirect

Maximal subgroups

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

subgroupindex
#54932
#49532
#47362
C3×#3232
#44702
#16073