Finite Group Databasebeta

← #71047

Subgroups — #71047

Order 384.

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

normal subgroups only

This group has 212 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#5563C211yesmaximal, characteristic
2C3×#461C211yesmaximal, characteristic, complemented
3#5093C211yesmaximal, characteristic, complemented
4#5087C211yesmaximal, characteristic, complemented
5D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#4483C211yesmaximal, characteristic, complemented
7#4481C211yesmaximal, characteristic, complemented
8#1646—31nomaximal, Sylow
9C2×C3×SD16—21no—
10C3×C4⋊Q8C2211yescharacteristic
11C3×C4×D4C2211yescharacteristic
12#938—21no—
13#930—21no—
14#926—21no—
15#892—21no—
16#885—22no—
17C4×D12C2211yescharacteristic
18#869C2211yescharacteristic
19C3×C4⋊C8C2211yescharacteristic
20C3×#103—21no—
21#810—22no—
22#809—21no—
23#807—21no—
24C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C3⋊C8C2211yescharacteristic
26#555—31no—
27#487—31no—
28#474—31no—
29#461D311yescharacteristic, complemented
30#333—31no—
31D4⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34C2×C3×D4D411yescharacteristic
35C22×C3×C4—21no—
36C2×#67—22no—
37C2×D12D411yescharacteristic
38C2×C4×D3—21no—
39C2×Dic6D412yes—
40C3×SD16—42no—
41C2×C3×C8—21no—
42C3×C4⋊C4—21no—
43C3×C4⋊C4—21no—
44C3×C4⋊C4D411yescharacteristic
45C3×C22⋊C4—21no—
46C3×C42C2311yescharacteristic
47#217—22no—
48(C2×D3)⋊C4—22no—
49(C2×D3)⋊C4—21no—
50C4⋊Dic3—21no—
51C4⋊Dic3—21no—
52C4⋊Dic3D411yescharacteristic
53C4×Dic3—22no—
54C2×C3⋊C8—21no—
55C2×C3⋊C8—21no—
56C2×SD16—21no—
57C4⋊Q8—31no—
58C4⋊Q8C2×D311yescharacteristic
59#124—62no—
60#121—61no—
61#121—61no—
62C4×D4—31no—
63C4×D4C2×D311yescharacteristic
64#117—61no—
65#117—61no—
66#107—61no—
67C4⋊C8—31no—
68C4⋊C8C2×D311yescharacteristic
69#103—21no—
70#103—62no—
71#102—61no—
72C4×C8—31no—
73C23×C3—21no—
74C22×D3—21no—
75C3×Q8—41no—
76C3×Q8—41no—
77C3×D4—21no—
78C3×D4—21no—
79C2×C3×C4—21no—
80C2×C3×C4—41no—
81C2×C3×C4—41no—
82C2×C3×C4—41no—
83C2×C3×C4C2×D411yescharacteristic
84C2×C3×C4C2×D411yescharacteristic
85C2×C3×C4C2×D411yescharacteristic
86#67—82no—
87C2×Dic3—21no—
88C2×Dic3—22no—
89D12—21no—
90D12—21no—
91C4×D3—41no—
92C4×D3—41no—
93Dic6—42no—
94C3×C8—41no—
95C3⋊C8—41no—
96C3⋊C8—41no—
97C2×Q8—21no—
98C2×Q8—32no—
99C2×D4—31no—
100C2×D4—62no—
101C2×D4#6711yescharacteristic
102C22×C4—21no—
103C22×C4—61no—
104SD16—42no—
105C2×C8—21no—
106C2×C8—61no—
107C2×C8—61no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—31no—
111C4⋊C4—61no—
112C4⋊C4—61no—
113C4⋊C4#6711yescharacteristic
114C22⋊C4—21no—
115C22⋊C4—62no—
116C22⋊C4—62no—
117C22⋊C4—61no—
118C42—62no—
119C42C22×D311yescharacteristic
120C22×C3—41no—
121C22×C3—41no—
122C22×C3—41no—
123C22×C3#12011yescharacteristic
124C2×D3—41no—
125C2×D3—41no—
126C2×D3—41no—
127C3×C4—21no—
128C3×C4—41no—
129C3×C4—81no—
130C3×C4#13511yescharacteristic
131C3×C4#13511yescharacteristic
132C3×C4AGL(1,ℤ/8)11yescharacteristic
133C3×C4#13711yescharacteristic
134Dic3—41no—
135Dic3—42no—
136C23—21no—
137C23—61no—
138Q8—41no—
139Q8—41no—
140Q8—122no—
141D4—21no—
142D4—21no—
143D4—61no—
144D4—61no—
145D4—242no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—61no—
151C2×C4—62no—
152C2×C4—121no—
153C2×C4—121no—
154C2×C4D3×D411yescharacteristic
155C2×C4D3×D411yescharacteristic
156C2×C4C2×#6711yescharacteristic
157C8—41no—
158C8—121no—
159C8—121no—
160C2×C3—81no—
161C2×C3#44911yescharacteristic
162C2×C3#45211yescharacteristic
163C2×C3#45511yescharacteristic
164D3—81no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—121no—
169C22—121no—
170C22—121no—
171C22#93711yescharacteristic
172C4—21no—
173C4—41no—
174C4—81no—
175C4—122no—
176C4—121no—
177C4#91511yescharacteristic
178C4#91611yescharacteristic
179C4#93211yescharacteristic
180C4#95011yescharacteristic
181C3#164611yescharacteristic, complemented, Sylow
182C2—81no—
183C2—241no—
184C2#503711yescharacteristic
185C2#517311yescharacteristic
186C2#520211yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#461C2
socleC22×C3#120
prime radical#5563C2
hypercenter#461D3
hypocenterC3#1646
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. C3×#461C2semidirect
3. #5093C2semidirect
4. #5087C2semidirect
5. D4⋊(C3⋊C8)C2semidirect
6. #4483C2semidirect
7. #4481C2semidirect
29. #461D3semidirect
181. C3#1646semidirect

Maximal subgroups

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

subgroupindex
#55632
C3×#4612
#50932
#50872
D4⋊(C3⋊C8)2
#44832
#44812
#16463