Finite Group Databasebeta

← #13154257

Subgroups — #13154257

Order 1728.

1386 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 191 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
1H(3,𝔽3)×#126C211yesmaximal, characteristic, complemented
2#11721763C213yesmaximal, complemented
3#11721762C213yesmaximal, complemented
4#11721740C213yesmaximal, complemented
5#11721738C213yesmaximal, complemented
6#11721722C211yesmaximal, characteristic, complemented
7#11721720C211yesmaximal, characteristic, complemented
8C3×#5711—34nomaximal
9C23×#272C2211yescharacteristic
10C2×#6235C2213yes—
11C2×C4⋊#272C2213yes—
12C2×C4⋊#272C2213yes—
13C2×C4⋊#272C2213yescomplemented
14C2×C4×#272C2213yescomplemented
15C4⋊C4×H(3,𝔽3)C2213yescomplemented
16C22⋊C4×H(3,𝔽3)C2213yes—
17C42×H(3,𝔽3)C2211yescharacteristic, complemented
18#88542C2213yes—
19#88542C2213yes—
20#88542C2213yescomplemented
21#88540C2213yescomplemented
22C32×#126D314yescomplemented
23C3×#895—312no—
24C3×#894—312no—
25C3×#884—312no—
26C3×#882—312no—
27C3×#876—34no—
28C3×#874—34no—
29C23×H(3,𝔽3)C2311yescharacteristic
30C22×#272—43no—
31C22×#272C2311yescharacteristic
32C22×#272C2311yescharacteristic
33C22×#272C2313yes—
34C2×C4×H(3,𝔽3)C2313yes—
35C2×C4×H(3,𝔽3)C2313yes—
36#6235—43no—
37C2×#1084C2313yes—
38C4⋊#272—43no—
39C4⋊#272—43no—
40C4⋊#272—43no—
41C4×#272—43no—
42C3×#560—91no—
43#5711—34no—
44C23×C3×D3—34no—
45C2×C3×#67—312no—
46C2×C3×D12—312no—
47C2×C3×D12—312no—
48C2×C3×D12—312no—
49C2×C3×C4×D3—312no—
50C32×C4⋊C4C2×D3112yescomplemented
51C32×C22⋊C4C2×D3112yes—
52C32×C42C2×D314yescomplemented
53C3×(C2×D3)⋊C4—312no—
54C3×(C2×D3)⋊C4—312no—
55C3×(C2×D3)⋊C4—312no—
56C3×Dic3⋊C4—312no—
57C22×H(3,𝔽3)—23no—
58C22×H(3,𝔽3)C2411yescharacteristic
59C2×#272—23no—
60C2×#272—23no—
61C2×#272—23no—
62C2×#272—23no—
63C2×#272—23no—
64C2×#272—44no—
65C4×H(3,𝔽3)—23no—
66C4×H(3,𝔽3)—23no—
67#1084—23no—
68C3×#127—91no—
69C3×#126—34no—
70C3×#126—91no—
71C3×#126C3⋊D311yescharacteristic
72C3×#123—93no—
73C3×#121—93no—
74C3×#121—93no—
75C3×#120—93no—
76#895—312no—
77#894—312no—
78#884—312no—
79#882—312no—
80#876—34no—
81#874—34no—
82C23×C32C22×D314yes—
83C22×C3×D3—34no—
84C22×C3×D3—34no—
85C22×C3×D3—312no—
86C22×C3×D3—1212no—
87C2×C32×C4C22×D3112yes—
88C2×C32×C4C22×D3112yes—
89C3×#67—1212no—
90C2×C3×Dic3—312no—
91C3×D12—1212no—
92C3×D12—1212no—
93C3×D12—1212no—
94C3×C4×D3—1212no—
95#560—91noSylow
96C2×H(3,𝔽3)—41no—
97C2×H(3,𝔽3)H(5,𝔽2)13yes—
98#272—43no—
99#272—41no—
100#272—41no—
101C24×C3—91no—
102C23×D3—34no—
103C2×C3×D4—93no—
104C2×C3×D4—93no—
105C2×C3×D4—93no—
106C2×C3×D4—93no—
107C22×C3×C4—93no—
108C2×#67—312no—
109C2×D12—312no—
110C2×D12—312no—
111C2×D12—312no—
112C2×C4×D3—312no—
113C3×C4⋊C4—312no—
114C3×C4⋊C4—93no—
115C3×C4⋊C4C2×C3⋊D313yes—
116C3×C22⋊C4—312no—
117C3×C22⋊C4—93no—
118C3×C22⋊C4—93no—
119C3×C22⋊C4—93no—
120C3×C22⋊C4C2×C3⋊D313yes—
121C3×C42—34no—
122C3×C42C2×C3⋊D311yescharacteristic
123(C2×D3)⋊C4—312no—
124(C2×D3)⋊C4—312no—
125(C2×D3)⋊C4—312no—
126Dic3⋊C4—312no—
127C22×C32—212no—
128C22×C32C23×D314yes—
129C2×C3×D3—612no—
130C2×C3×D3—612no—
131C2×C3×D3—612no—
132C2×C3×D3—612no—
133C2×C3×D3—612no—
134C2×C3×D3—1216no—
135C32×C4—212no—
136C32×C4—212no—
137C3×Dic3—612no—
138#127—91no—
139#126—91no—
140#126#27211yescharacteristic, complemented
141#123—93no—
142#121—93no—
143#121—93no—
144#120—93no—
145H(3,𝔽3)#56011yescharacteristic, complemented, Sylow
146C23×C3—34no—
147C23×C3—91no—
148C23×C3—91no—
149C23×C3—93no—
150C23×C3—363no—
151C23×C3C22×C3⋊D311yescharacteristic
152C22×D3—34no—
153C22×D3—34no—
154C22×D3—312no—
155C22×D3—1212no—
156C3×D4—363no—
157C3×D4—363no—
158C3×D4—363no—
159C3×D4—363no—
160C2×C3×C4—312no—
161C2×C3×C4—312no—
162C2×C3×C4—93no—
163C2×C3×C4—363no—
164C2×C3×C4C22×C3⋊D313yes—
165C2×C3×C4C22×C3⋊D313yes—
166#67—1212no—
167C2×Dic3—312no—
168D12—1212no—
169D12—1212no—
170D12—1212no—
171C4×D3—1212no—
172C2×C32—44no—
173C2×C32#1009112yes—
174C3×D3—1212no—
175C3×D3—124no—
176C3×D3—124no—
177C24—91no—
178C2×D4—93no—
179C2×D4—93no—
180C2×D4—93no—
181C2×D4—93no—
182C22×C4—93no—
183C4⋊C4—93no—
184C4⋊C4C2×#27213yescomplemented
185C22⋊C4—93no—
186C22⋊C4—93no—
187C22⋊C4—93no—
188C22⋊C4C2×#27213yes—
189C42C2×#27211yescharacteristic, complemented
190C22×C3—23no—
191C22×C3—34no—
192C22×C3—612no—
193C22×C3—183no—
194C22×C3—183no—
195C22×C3—183no—
196C22×C3—183no—
197C22×C3—183no—
198C22×C3—364no—
199C22×C3C23×C3⋊D311yescharacteristic
200C2×D3—612no—
201C2×D3—612no—
202C2×D3—612no—
203C2×D3—612no—
204C2×D3—612no—
205C2×D3—1216no—
206C3×C4—23no—
207C3×C4—23no—
208C3×C4—612no—
209C3×C4—612no—
210C3×C4—183no—
211Dic3—612no—
212C32#571114yescomplemented
213C23—91no—
214C23—91no—
215C23—93no—
216C23—363no—
217C23C22×#27211yescharacteristic
218D4—363no—
219D4—363no—
220D4—363no—
221D4—363no—
222C2×C4—93no—
223C2×C4—363no—
224C2×C4C22×#27213yes—
225C2×C4C22×#27213yes—
226C2×C3—312no—
227C2×C3—41no—
228C2×C3—124no—
229C2×C3—361no—
230C2×C3—361no—
231C2×C3—363no—
232C2×C3#6436613yes—
233D3—124no—
234D3—124no—
235D3—1212no—
236C22—23no—
237C22—183no—
238C22—183no—
239C22—183no—
240C22—183no—
241C22—183no—
242C22—364no—
243C22C23×#27211yescharacteristic
244C4—23no—
245C4—23no—
246C4—183no—
247C3—34no—
248C3#1059618211yescharacteristic
249C2—41no—
250C2—361no—
251C2—361no—
252C2—363no—
253C2#1172267513yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#126C2
hypercenterC3×#126C3⋊D3
hypocenterH(3,𝔽3)#560
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#126

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#126C2semidirect
2. #11721763C2semidirect
3. #11721762C2semidirect
4. #11721740C2semidirect
5. #11721738C2semidirect
6. #11721722C2semidirect
7. #11721720C2semidirect
13. C2×C4⋊#272C22semidirect
14. C2×C4×#272C22semidirect
15. C4⋊C4×H(3,𝔽3)C22semidirect
17. C42×H(3,𝔽3)C22semidirect
20. #88542C22semidirect
21. #88540C22semidirect
22. C32×#126D3semidirect
50. C32×C4⋊C4C2×D3semidirect
52. C32×C42C2×D3semidirect
140. #126#272semidirect
145. H(3,𝔽3)#560semidirect
184. C4⋊C4C2×#272semidirect
189. C42C2×#272semidirect
212. C32#5711semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1262
#117217632
#117217622
#117217402
#117217382
#117217222
#117217202
C3×#57113