Finite Group Databasebeta

← #10590505

Subgroups — #10590505

Order 576.

230 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 60 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×#892C211yesmaximal, characteristic, complemented
2#63908C211yesmaximal, characteristic
3#63635C211yesmaximal, characteristic, complemented
4C3×#809C211yesmaximal, characteristic, complemented
5#63575C211yesmaximal, characteristic, complemented
6#63562C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#5003—31nomaximal
9#4784—31nomaximal
10C2×C3×D12C2211yescharacteristic
11C2×C3×C4×D3C2211yescharacteristic
12C2×#625—21no—
13C32×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×(C2×D3)⋊C4—21no—
17C3×C4⋊Dic3C2211yescharacteristic
18C3×C4×Dic3—21no—
19C2×C3×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic3—21no—
21C3×#117—31no—
22#938—31no—
23#930—31no—
24#892D311yescharacteristic, complemented
25C3×#102—31no—
26#832—31no—
27#830—31no—
28(C2×D3)⋊C8—31no—
29#818—31no—
30#809—31no—
31#809D311yescharacteristic, complemented
32#808—31no—
33#808—31no—
34#808—61no—
35#807—31no—
36C22×C3×D3—21no—
37C22×C3×D3D411yescharacteristic
38C2×C32×C4—21no—
39C2×C32×C4C2311yescharacteristic
40C2×C3⋊Dic3—21no—
41C2×C3×Dic3—21no—
42C2×C3×Dic3D411yescharacteristic
43C3×D12—21no—
44C3×D12—21no—
45C3×C4×D3—21no—
46C3×C4×D3—21no—
47#625—42no—
48C3⋊(C3⋊C8)—21no—
49C3×C3⋊C8—21no—
50#482—91noSylow
51C2×C3×D4—31no—
52C22×C3×C4—31no—
53C2×#67—61no—
54C2×#67—61no—
55C2×D12C2×D311yescharacteristic
56C2×C4×D3C2×D311yescharacteristic
57C2×C3×C8—31no—
58C3×C4⋊C4—21no—
59C3×C4⋊C4—31no—
60C3×C4⋊C4C2×D311yescharacteristic
61C3×C4⋊C4C2×D311yescharacteristic
62C3×C22⋊C4—61no—
63C3×C42—61no—
64#217—61no—
65(C2×D3)⋊C4—21no—
66(C2×D3)⋊C4—61no—
67C4⋊Dic3—31no—
68C4⋊Dic3—31no—
69C4⋊Dic3—61no—
70C4⋊Dic3C2×D311yescharacteristic
71C4×Dic3—21no—
72C2×C3⋊C8—31no—
73C2×C3⋊C8—31no—
74C2×C3⋊C8—61no—
75C2×C3⋊C8C2×D311yescharacteristic
76C22×C32C2×D411yescharacteristic
77C2×C3×D3—21no—
78C2×C3×D3—21no—
79C2×C3×D3—21no—
80C2×C3×D3—41no—
81C2×C3×D3—41no—
82C2×C3×D3—41no—
83C32×C4—41no—
84C32×C4#4111yescharacteristic
85C32×C4#4111yescharacteristic
86C3⋊Dic3—41no—
87C3×Dic3—21no—
88C3×Dic3—41no—
89#121—91no—
90#117—31no—
91#107—91no—
92#106—91no—
93#102—31no—
94#102—91no—
95C22⋊C8—91no—
96C23×C3—31no—
97C23×C3—61no—
98C22×D3—21no—
99C22×D3#6711yescharacteristic
100C3×D4—61no—
101C3×D4—61no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—31no—
106C2×C3×C4—41no—
107C2×C3×C4—61no—
108C2×C3×C4—61no—
109C2×C3×C4—61no—
110C2×C3×C4C22×D311yescharacteristic
111C2×C3×C4C22×D311yescharacteristic
112#67—122no—
113#67—122no—
114C2×Dic3—21no—
115C2×Dic3—61no—
116C2×Dic3—61no—
117C2×Dic3—121no—
118C2×Dic3#6711yescharacteristic
119D12—21no—
120D12—21no—
121C4×D3—21no—
122C4×D3—21no—
123C3×C8—61no—
124C3⋊C8—21no—
125C3⋊C8—62no—
126C3⋊C8—61no—
127C3⋊C8—61no—
128C2×C32#12311yescharacteristic
129C2×C32#13511yescharacteristic
130C2×C32AGL(1,ℤ/8)11yescharacteristic
131C3×D3—41no—
132C3×D3—81no—
133C2×D4—31no—
134C2×D4—181no—
135C22×C4—31no—
136C2×C8—31no—
137C2×C8—91no—
138C4⋊C4—31no—
139C4⋊C4—91no—
140C4⋊C4D3211yescharacteristic
141C22⋊C4—61no—
142C22⋊C4—181no—
143C42—61no—
144C22×C3—21no—
145C22×C3—61no—
146C22×C3—61no—
147C22×C3—61no—
148C22×C3—121no—
149C22×C3—121no—
150C22×C3—121no—
151C22×C3D3×D411yescharacteristic
152C22×C3C2×#6711yescharacteristic
153C2×D3—21no—
154C2×D3—21no—
155C2×D3—21no—
156C2×D3—41no—
157C2×D3—41no—
158C2×D3—41no—
159C3×C4—21no—
160C3×C4—21no—
161C3×C4—41no—
162C3×C4—41no—
163C3×C4—61no—
164C3×C4—81no—
165C3×C4—121no—
166C3×C4#23511yescharacteristic
167C3×C4#23511yescharacteristic
168C3×C4#23511yescharacteristic
169C3×C4#23711yescharacteristic
170Dic3—21no—
171Dic3—41no—
172Dic3—121no—
173Dic3—121no—
174Dic3—241no—
175C32#48211yescharacteristic, complemented, Sylow
176C23—31no—
177C23—61no—
178D4—61no—
179D4—61no—
180D4—362no—
181C2×C4—21no—
182C2×C4—31no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—61no—
186C2×C4—181no—
187C2×C4C2×D3211yescharacteristic
188C8—61no—
189C8—181no—
190C2×C3—21no—
191C2×C3—21no—
192C2×C3—21no—
193C2×C3—121no—
194C2×C3—241no—
195C2×C3#88311yescharacteristic
196C2×C3#91211yescharacteristic
197C2×C3#91411yescharacteristic
198C2×C3#92911yescharacteristic
199C2×C3#94911yescharacteristic
200C2×C3#95011yescharacteristic
201D3—41no—
202D3—81no—
203C22—61no—
204C22—61no—
205C22—61no—
206C22—121no—
207C22—121no—
208C22—121no—
209C22D3×#6711yescharacteristic
210C4—41no—
211C4—61no—
212C4—121no—
213C4—361no—
214C4#379511yescharacteristic
215C4#379811yescharacteristic
216C3—21no—
217C3#478411yescharacteristic, complemented
218C3#500311yescharacteristic, complemented
219C2—121no—
220C2—241no—
221C2#6390511yescharacteristic
222C2#6393311yescharacteristic
223C2#6393911yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
prime radical#63908C2
hypercenterC4⋊C4D32
hypocenterC32#482
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
1. C3×#892C2semidirect
3. #63635C2semidirect
4. C3×#809C2semidirect
5. #63575C2semidirect
6. #63562C2semidirect
7. (C2×D3)⋊(C3⋊C8)C2semidirect
24. #892D3semidirect
31. #809D3semidirect
175. C32#482semidirect
217. C3#4784semidirect
218. C3#5003semidirect

Maximal subgroups

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

subgroupindex
C3×#8922
#639082
#636352
C3×#8092
#635752
#635622
(C2×D3)⋊(C3⋊C8)2
#50033
#47843