Finite Group Databasebeta

← #10594022

Subgroups — #10594022

Order 576.

534 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 120 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#64085C211yesmaximal, characteristic, complemented
2C3×#876C211yesmaximal, characteristic, complemented
3C3×#874C211yesmaximal, characteristic, complemented
4#63922C213yesmaximal, complemented
5#63886C213yesmaximal, complemented
6#63885C213yesmaximal, complemented
7#63884C213yesmaximal, complemented
8#5720—31nomaximal
9#5711—31nomaximal
10C22×D32C2211yescharacteristic
11C2×C3×D12C2213yes—
12C2×C3×D12C2213yescomplemented
13C2×#626C2213yes—
14C2×#625C2213yes—
15C2×D3×Dic3C2213yescomplemented
16C32×C42C2211yescharacteristic, complemented
17#3752C2213yes—
18#3750C2213yescomplemented
19C3×(C2×D3)⋊C4C2213yes—
20C3×Dic3⋊C4C2213yescomplemented
21(C2×D3)⋊Dic3C2213yes—
22(C2×D3)⋊Dic3C2213yescomplemented
23C3×#127—31no—
24C3×#126—31no—
25#939—33no—
26#939—33no—
27#937—33no—
28#935—33no—
29#895—33no—
30#894—33no—
31#884—33no—
32#882—33no—
33#876—31no—
34#876—31no—
35#876—61no—
36#876D311yescharacteristic, complemented
37#874D311yescharacteristic, complemented
38C22×C3⋊D3C2311yescharacteristic
39C22×C3×D3C2311yescharacteristic
40C22×C3×D3C2311yescharacteristic
41C22×C3×D3C2313yes—
42C2×D32—43no—
43C2×C32×C4C2313yes—
44C2×C3⋊Dic3C2313yes—
45C2×C3×Dic3C2313yes—
46C3×D12—43no—
47C3×D12—43no—
48#626—43no—
49#625—43no—
50D3×Dic3—43no—
51#560—91noSylow
52C23×D3—31no—
53C23×D3—31no—
54C2×C3×D4—33no—
55C2×C3×D4—33no—
56C2×#67—33no—
57C2×#67—33no—
58C2×#67—33no—
59C22×Dic3—33no—
60C2×D12—33no—
61C2×D12C2×D313yes—
62C2×D12C2×D313yescomplemented
63C2×C4×D3—33no—
64C3×C4⋊C4—33no—
65C3×C22⋊C4—33no—
66C3×C42—21no—
67C3×C42C2×D311yescharacteristic, complemented
68C3×C42C2×D311yescharacteristic, complemented
69#217—33no—
70#217—33no—
71(C2×D3)⋊C4—33no—
72(C2×D3)⋊C4—33no—
73(C2×D3)⋊C4—33no—
74(C2×D3)⋊C4—33no—
75(C2×D3)⋊C4—63no—
76(C2×D3)⋊C4C2×D313yes—
77Dic3⋊C4—33no—
78Dic3⋊C4—33no—
79Dic3⋊C4—63no—
80Dic3⋊C4C2×D313yescomplemented
81C22×C32C2411yescharacteristic
82C2×C3⋊D3—23no—
83C2×C3×D3—23no—
84C2×C3×D3—23no—
85C2×C3×D3—23no—
86C2×C3×D3—23no—
87C2×C3×D3—23no—
88D32—44no—
89C32×C4—23no—
90C3⋊Dic3—23no—
91C3×Dic3—23no—
92#127—31no—
93#126—31no—
94#126—91no—
95#123—93no—
96#121—93no—
97#121—93no—
98#120—93no—
99C23×C3—31no—
100C23×C3—33no—
101C23×C3—31no—
102C22×D3—31no—
103C22×D3—31no—
104C22×D3—61no—
105C22×D3—123no—
106C22×D3—123no—
107C22×D3C22×D311yescharacteristic
108C22×D3C22×D313yes—
109C22×D3C22×D311yescharacteristic
110C3×D4—123no—
111C3×D4—123no—
112C2×C3×C4—23no—
113C2×C3×C4—33no—
114C2×C3×C4C22×D313yes—
115C2×C3×C4C22×D313yes—
116#67—123no—
117#67—123no—
118#67—123no—
119C2×Dic3—33no—
120C2×Dic3—33no—
121C2×Dic3—63no—
122C2×Dic3—123no—
123C2×Dic3C22×D313yes—
124D12—43no—
125D12—43no—
126D12—123no—
127C4×D3—123no—
128C2×C32H(5,𝔽2)13yes—
129C3⋊D3—41no—
130C3×D3—41no—
131C3×D3—41no—
132C3×D3—43no—
133C24—91no—
134C2×D4—33no—
135C2×D4—33no—
136C2×D4—93no—
137C2×D4—93no—
138C22×C4—93no—
139C4⋊C4—33no—
140C4⋊C4—93no—
141C22⋊C4—33no—
142C22⋊C4—93no—
143C22⋊C4—93no—
144C22⋊C4—93no—
145C42D3211yescharacteristic, complemented
146C22×C3—21no—
147C22×C3—63no—
148C22×C3—63no—
149C22×C3—63no—
150C22×C3—63no—
151C22×C3—63no—
152C22×C3C23×D311yescharacteristic
153C22×C3C23×D311yescharacteristic
154C2×D3—23no—
155C2×D3—23no—
156C2×D3—23no—
157C2×D3—23no—
158C2×D3—23no—
159C2×D3—63no—
160C2×D3—63no—
161C2×D3—123no—
162C2×D3—124no—
163C2×D3—124no—
164C3×C4—23no—
165C3×C4—23no—
166C3×C4—43no—
167C3×C4—63no—
168Dic3—23no—
169Dic3—63no—
170Dic3—63no—
171Dic3—123no—
172C32#56011yescharacteristic, complemented, Sylow
173C23—31no—
174C23—31no—
175C23—33no—
176C23—91no—
177C23—363no—
178D4—123no—
179D4—123no—
180D4—363no—
181D4—363no—
182C2×C4—33no—
183C2×C4—93no—
184C2×C4—363no—
185C2×C4C2×D3213yes—
186C2×C3—23no—
187C2×C3—121no—
188C2×C3—123no—
189C2×C3—121no—
190C2×C3#100413yes—
191C2×C3#100913yes—
192D3—41no—
193D3—41no—
194D3—43no—
195D3—121no—
196D3—121no—
197D3—241no—
198C22—63no—
199C22—63no—
200C22—63no—
201C22—63no—
202C22—63no—
203C22—183no—
204C22—364no—
205C22C22×D3211yescharacteristic
206C4—23no—
207C4—63no—
208C4—183no—
209C3—21no—
210C3#571111yescharacteristic, complemented
211C3#572011yescharacteristic, complemented
212C2—121no—
213C2—121no—
214C2—123no—
215C2—361no—
216C2#6430713yes—
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
hypercenterC42D32
hypocenterC32#560
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #64085C2semidirect
2. C3×#876C2semidirect
3. C3×#874C2semidirect
4. #63922C2semidirect
5. #63886C2semidirect
6. #63885C2semidirect
7. #63884C2semidirect
12. C2×C3×D12C22semidirect
15. C2×D3×Dic3C22semidirect
16. C32×C42C22semidirect
18. #3750C22semidirect
20. C3×Dic3⋊C4C22semidirect
22. (C2×D3)⋊Dic3C22semidirect
36. #876D3semidirect
37. #874D3semidirect
62. C2×D12C2×D3semidirect
67. C3×C42C2×D3semidirect
68. C3×C42C2×D3semidirect
80. Dic3⋊C4C2×D3semidirect
145. C42D32semidirect
172. C32#560semidirect
210. C3#5711semidirect
211. C3#5720semidirect

Maximal subgroups

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

subgroupindex
#640852
C3×#8762
C3×#8742
#639222
#638862
#638852
#638842
#57203
#57113