Finite Group Databasebeta

← #10590782

Subgroups — #10590782

Order 576.

223 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 62 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#63898C211yesmaximal, characteristic
2#63872C211yesmaximal, characteristic, complemented
3#63641C211yesmaximal, characteristic, complemented
4C3×#821C211yesmaximal, characteristic
5C3×#817C211yesmaximal, characteristic, complemented
6#63576C211yesmaximal, characteristic, complemented
7#63562C211yesmaximal, characteristic
8#4871—31nomaximal
9#4763—31nomaximal
10C2×C3×D12C2211yescharacteristic
11C2×D3×Dic3—21no—
12C2×C32×C8C2211yescharacteristic
13C4×C3⋊Dic3C2211yescharacteristic
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic3C2211yescharacteristic
18Dic3⋊Dic3—21no—
19Dic3⋊Dic3—21no—
20(C2×D3)⋊Dic3—21no—
21D4×Dic3—31no—
22#893—31no—
23#890—31no—
24#890—31no—
25C3×#106—31no—
26C3×#102—31no—
27#832—31no—
28#821D311yescharacteristic
29#817D311yescharacteristic, complemented
30Dic3⋊C8—31no—
31Dic3⋊C8—31no—
32Dic3⋊C8—61no—
33#809—31no—
34#807—31no—
35#807—31no—
36C22×C3×D3—21no—
37C2×C32×C4C2311yescharacteristic
38C2×C3⋊Dic3D411yescharacteristic
39C2×C3⋊Dic3D411yescharacteristic
40C2×C3×Dic3—21no—
41C2×C3×Dic3—21no—
42C2×C3×Dic3—21no—
43C3×D12—21no—
44C3×D12Q812yes—
45D3×Dic3—42no—
46C32×C8—21no—
47C3⋊(C3⋊C8)—21no—
48#478—91noSylow
49C2×C3×D4—31no—
50C22×Dic3—61no—
51C2×D12C2×D311yescharacteristic
52C2×C4×D3—61no—
53C2×C3×C8—21no—
54C2×C3×C8C2×D311yescharacteristic
55C2×C3×C8C2×D311yescharacteristic
56C3×C4⋊C4—31no—
57C3×C4⋊C4—31no—
58C3×C4⋊C4—31no—
59#217—61no—
60(C2×D3)⋊C4—61no—
61C4⋊Dic3—61no—
62C4⋊Dic3—61no—
63C4⋊Dic3C2×D311yescharacteristic
64C4⋊Dic3C2×D311yescharacteristic
65C4⋊Dic3C2×D311yescharacteristic
66Dic3⋊C4—61no—
67Dic3⋊C4—61no—
68C4×Dic3—31no—
69C4×Dic3—31no—
70C4×Dic3—61no—
71C2×C3⋊C8—31no—
72C2×C3⋊C8—31no—
73C2×C3⋊C8—61no—
74C22×C32C2×D411yescharacteristic
75C2×C3×D3—22no—
76C2×C3×D3—41no—
77C2×C3×D3—41no—
78C32×C4C2×Q811yescharacteristic
79C32×C4#4111yescharacteristic
80C3⋊Dic3—21no—
81C3⋊Dic3—21no—
82C3×Dic3—41no—
83C3×Dic3—41no—
84C3×Dic3—41no—
85#125—91no—
86C4×D4—91no—
87#107—91no—
88#106—31no—
89C4⋊C8—91no—
90#102—31no—
91#102—91no—
92C23×C3—61no—
93C22×D3—21no—
94C3×D4—32no—
95C3×D4—61no—
96C2×C3×C4—21no—
97C2×C3×C4—61no—
98C2×C3×C4—61no—
99C2×C3×C4—61no—
100C2×C3×C4C22×D311yescharacteristic
101C2×C3×C4C22×D311yescharacteristic
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—21no—
105C2×Dic3—31no—
106C2×Dic3—31no—
107C2×Dic3—31no—
108C2×Dic3—31no—
109C2×Dic3—61no—
110C2×Dic3—61no—
111C2×Dic3—122no—
112D12—21no—
113D12Dic612yes—
114C4×D3—122no—
115C3×C8—21no—
116C3×C8—21no—
117C3×C8—41no—
118C3⋊C8—61no—
119C3⋊C8—61no—
120C3⋊C8—121no—
121C2×C32C22⋊Q811yescharacteristic
122C2×C32#13511yescharacteristic
123C2×C32AGL(1,ℤ/8)11yescharacteristic
124C3×D3—42no—
125C2×D4—31no—
126C22×C4—181no—
127C2×C8—91no—
128C2×C8D3211yescharacteristic
129C4⋊C4—31no—
130C4⋊C4—31no—
131C4⋊C4—31no—
132C4⋊C4—181no—
133C4⋊C4—181no—
134C22⋊C4—181no—
135C42—91no—
136C22×C3—21no—
137C22×C3—62no—
138C22×C3—121no—
139C22×C3—121no—
140C22×C3D3×D411yescharacteristic
141C22×C3D3×D411yescharacteristic
142C2×D3—22no—
143C2×D3—41no—
144C2×D3—41no—
145C3×C4—21no—
146C3×C4—21no—
147C3×C4—121no—
148C3×C4—121no—
149C3×C4—121no—
150C3×C4C2×Dic611yescharacteristic
151C3×C4#23511yescharacteristic
152C3×C4#23711yescharacteristic
153C3×C4D3×Q811yescharacteristic
154Dic3—41no—
155Dic3—41no—
156Dic3—41no—
157Dic3—62no—
158Dic3—61no—
159Dic3—61no—
160Dic3—61no—
161Dic3—61no—
162Dic3—121no—
163C32#47811yescharacteristic, complemented, Sylow
164C23—61no—
165D4—32no—
166D4—61no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—61no—
170C2×C4—91no—
171C2×C4—91no—
172C2×C4—362no—
173C2×C4C2×D3211yescharacteristic
174C8—21no—
175C8—181no—
176C2×C3—21no—
177C2×C3—21no—
178C2×C3—21no—
179C2×C3—122no—
180C2×C3#87811yescharacteristic
181C2×C3#89611yescharacteristic
182C2×C3#91111yescharacteristic
183C2×C3#91411yescharacteristic
184C2×C3#91611yescharacteristic
185C2×C3#91611yescharacteristic
186D3—42no—
187C22—62no—
188C22—121no—
189C22—121no—
190C22#380211yescharacteristic
191C4—121no—
192C4—121no—
193C4—121no—
194C4—181no—
195C4—181no—
196C4D3×Dic611yescharacteristic
197C4#380411yescharacteristic
198C3—21no—
199C3#476311yescharacteristic, complemented
200C3#487111yescharacteristic, complemented
201C2—122no—
202C2#6379811yescharacteristic
203C2#6380911yescharacteristic
204C2#6389911yescharacteristic
subgroupG/H
centerC22#3802
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×C8C22
socleC22×C32C2×D4
prime radicalC2×C3×D12C22
hypercenterC2×C8D32
hypocenterC32#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #63872C2semidirect
3. #63641C2semidirect
5. C3×#817C2semidirect
6. #63576C2semidirect
29. #817D3semidirect
163. C32#478semidirect
199. C3#4763semidirect
200. C3#4871semidirect

Maximal subgroups

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

subgroupindex
#638982
#638722
#636412
C3×#8212
C3×#8172
#635762
#635622
#48713
#47633