Finite Group Databasebeta

← #10590545

Subgroups — #10590545

Order 576.

222 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C3×#893C211yesmaximal, characteristic
2#63869C211yesmaximal, characteristic, complemented
3#63634C211yesmaximal, characteristic, complemented
4C3×#808C211yesmaximal, characteristic, complemented
5Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#63563C211yesmaximal, characteristic
7#63562C211yesmaximal, characteristic
8#5009—31nomaximal
9#4871—31nomaximal
10C2×C3×D12C2211yescharacteristic
11C2×C3×C4×D3—21no—
12C32×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×(C2×D3)⋊C4—21no—
16C3×C4⋊Dic3C2211yescharacteristic
17C3×C4×Dic3C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19#3724—21no—
20Dic3⋊Dic3—21no—
21C3×C4×D4—31no—
22#893D311yescharacteristic
23#890—31no—
24#870—31no—
25C3×#106—31no—
26#832—31no—
27#832—31no—
28#821—31no—
29Dic3⋊C8—31no—
30#809—31no—
31#808D311yescharacteristic, complemented
32#807—31no—
33#807—31no—
34#807—61no—
35C4⋊(C3⋊C8)—31no—
36C22×C3×D3—21no—
37C2×C32×C4—21no—
38C2×C32×C4C2311yescharacteristic
39C2×C3⋊Dic3—21no—
40C2×C3×Dic3—21no—
41C2×C3×Dic3D411yescharacteristic
42C2×C3×Dic3D411yescharacteristic
43C3×D12—21no—
44C3×D12Q812yes—
45C3×C4×D3—42no—
46C3⋊(C3⋊C8)—21no—
47C3×C3⋊C8—21no—
48#478—91noSylow
49C2×C3×D4—31no—
50C22×C3×C4—61no—
51C2×D12C2×D311yescharacteristic
52C2×C4×D3—21no—
53C2×C3×C8—31no—
54C3×C4⋊C4—21no—
55C3×C4⋊C4—31no—
56C3×C4⋊C4C2×D311yescharacteristic
57C3×C4⋊C4C2×D311yescharacteristic
58C3×C22⋊C4—61no—
59C3×C42—31no—
60(C2×D3)⋊C4—21no—
61C4⋊Dic3—31no—
62C4⋊Dic3—31no—
63C4⋊Dic3—61no—
64C4⋊Dic3—61no—
65C4⋊Dic3C2×D311yescharacteristic
66Dic3⋊C4—61no—
67Dic3⋊C4—61no—
68Dic3⋊C4—61no—
69C4×Dic3C2×D311yescharacteristic
70C2×C3⋊C8—31no—
71C2×C3⋊C8—31no—
72C2×C3⋊C8—61no—
73C2×C3⋊C8C2×D311yescharacteristic
74C22×C32C2×D411yescharacteristic
75C2×C3×D3—22no—
76C2×C3×D3—41no—
77C2×C3×D3—41no—
78C32×C4—41no—
79C32×C4C2×Q811yescharacteristic
80C32×C4#4111yescharacteristic
81C3⋊Dic3—41no—
82C3×Dic3—21no—
83C3×Dic3—21no—
84C3×Dic3—41no—
85#125—91no—
86C4×D4—31no—
87#107—91no—
88#106—31no—
89C4⋊C8—91no—
90#102—91no—
91#102—91no—
92C23×C3—61no—
93C22×D3—21no—
94C3×D4—32no—
95C3×D4—61no—
96C2×C3×C4—21no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—31no—
100C2×C3×C4—31no—
101C2×C3×C4—41no—
102C2×C3×C4—61no—
103C2×C3×C4—122no—
104C2×C3×C4C22×D311yescharacteristic
105C2×C3×C4C22×D311yescharacteristic
106C2×Dic3—21no—
107C2×Dic3—61no—
108C2×Dic3—61no—
109C2×Dic3—121no—
110C2×Dic3#6711yescharacteristic
111C2×Dic3#6711yescharacteristic
112D12—21no—
113D12Dic612yes—
114C4×D3—42no—
115C3×C8—61no—
116C3⋊C8—21no—
117C3⋊C8—62no—
118C3⋊C8—61no—
119C3⋊C8—61no—
120C2×C32C22⋊Q811yescharacteristic
121C2×C32#13511yescharacteristic
122C2×C32AGL(1,ℤ/8)11yescharacteristic
123C3×D3—42no—
124C2×D4—31no—
125C22×C4—61no—
126C2×C8—31no—
127C2×C8—91no—
128C4⋊C4—31no—
129C4⋊C4—91no—
130C4⋊C4—181no—
131C4⋊C4—181no—
132C4⋊C4D3211yescharacteristic
133C22⋊C4—61no—
134C42—31no—
135C22×C3—21no—
136C22×C3—62no—
137C22×C3—121no—
138C22×C3—121no—
139C22×C3D3×D411yescharacteristic
140C22×C3C2×#6711yescharacteristic
141C2×D3—22no—
142C2×D3—41no—
143C2×D3—41no—
144C3×C4—21no—
145C3×C4—21no—
146C3×C4—41no—
147C3×C4—41no—
148C3×C4—61no—
149C3×C4—61no—
150C3×C4—81no—
151C3×C4—121no—
152C3×C4C2×Dic611yescharacteristic
153C3×C4#23511yescharacteristic
154C3×C4#23511yescharacteristic
155C3×C4D3×Q811yescharacteristic
156Dic3—21no—
157Dic3—21no—
158Dic3—41no—
159Dic3—121no—
160Dic3—121no—
161Dic3—241no—
162C32#47811yescharacteristic, complemented, Sylow
163C23—61no—
164D4—32no—
165D4—61no—
166C2×C4—21no—
167C2×C4—31no—
168C2×C4—31no—
169C2×C4—61no—
170C2×C4—122no—
171C2×C4—181no—
172C2×C4C2×D3211yescharacteristic
173C8—61no—
174C8—181no—
175C2×C3—21no—
176C2×C3—21no—
177C2×C3—21no—
178C2×C3—122no—
179C2×C3#89611yescharacteristic
180C2×C3#91411yescharacteristic
181C2×C3#91611yescharacteristic
182C2×C3#92411yescharacteristic
183C2×C3#93211yescharacteristic
184C2×C3#95011yescharacteristic
185D3—42no—
186C22—62no—
187C22—121no—
188C22—121no—
189C22D3×#6711yescharacteristic
190C4—41no—
191C4—61no—
192C4—61no—
193C4—121no—
194C4—361no—
195C4D3×Dic611yescharacteristic
196C4#379811yescharacteristic
197C3—21no—
198C3#487111yescharacteristic, complemented
199C3#500911yescharacteristic, complemented
200C2—122no—
201C2#6385111yescharacteristic
202C2#6393411yescharacteristic
203C2#6394711yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
prime radicalC2×C3×D12C22
hypercenterC4⋊C4D32
hypocenterC32#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #63869C2semidirect
3. #63634C2semidirect
4. C3×#808C2semidirect
5. Dic3⋊(C3⋊C8)C2semidirect
31. #808D3semidirect
162. C32#478semidirect
198. C3#4871semidirect
199. C3#5009semidirect

Maximal subgroups

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

subgroupindex
C3×#8932
#638692
#636342
C3×#8082
Dic3⋊(C3⋊C8)2
#635632
#635622
#50093
#48713