Finite Group Databasebeta

← #13154231

Subgroups — #13154231

Order 1728.

1320 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 205 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)×#125C211yesmaximal, characteristic, complemented
2#11721763C212yesmaximal, complemented
3#11721763C214yesmaximal, complemented
4#11721762C214yesmaximal, complemented
5#11721761C212yesmaximal, complemented
6#11721720C211yesmaximal, characteristic, complemented
7#11721718C211yesmaximal, characteristic, complemented
8C3×#5697—34nomaximal
9C2×C4⋊#272C2212yes—
10C2×C4⋊#272C2214yes—
11C2×C4⋊#272C2214yes—
12C2×C4×#272C2211yescharacteristic
13C2×C4×#272C2214yes—
14C4⋊C4×H(3,𝔽3)C2212yes—
15C4⋊C4×H(3,𝔽3)C2214yes—
16C42×H(3,𝔽3)C2211yescharacteristic
17#88542C2214yes—
18#88542C2214yes—
19#88542C2212yes—
20#88540C2212yes—
21C4×#1084C2211yescharacteristic
22C32×#125D314yescomplemented
23C3×#895—316no—
24C3×#895—38no—
25C3×#894—316no—
26C3×#893—38no—
27C3×#874—34no—
28C3×#872—34no—
29C22×#272C2311yescharacteristic
30C22×#272C2314yes—
31C2×C4×H(3,𝔽3)C2314yes—
32C2×C4×H(3,𝔽3)C2312yes—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084C2312yes—
35C2×#1084C2311yescharacteristic
36C4⋊#272—24no—
37C4⋊#272—44no—
38C4⋊#272—44no—
39C4×#272—22no—
40C4×#272—44no—
41C3×#540—91no—
42#5697—34no—
43C2×C3×D12—38no—
44C2×C3×D12—316no—
45C2×C3×D12—316no—
46C2×C3×C4×D3—34no—
47C2×C3×C4×D3—316no—
48C32×C4⋊C4C2×D318yes—
49C32×C4⋊C4C2×D3116yes—
50C32×C42C2×D314yes—
51C3×(C2×D3)⋊C4—316no—
52C3×(C2×D3)⋊C4—316no—
53C3×(C2×D3)⋊C4—38no—
54C3×Dic3⋊C4—38no—
55C3×C4×Dic3—34no—
56C22×H(3,𝔽3)C2411yescharacteristic
57C2×#272—21no—
58C2×#272—24no—
59C2×#272—24no—
60C2×#272—22no—
61C2×#272—24no—
62C4×H(3,𝔽3)—24no—
63C4×H(3,𝔽3)—22no—
64C4×H(3,𝔽3)#4112yes—
65#1084—21no—
66#1084—22no—
67C3×#127—91no—
68C3×#125—34no—
69C3×#125C3⋊D311yescharacteristic
70C3×#123—94no—
71C3×#121—94no—
72C3×#121—92no—
73C3×C4×D4—92no—
74C3×#117—91no—
75#895—316no—
76#895—38no—
77#894—316no—
78#893—38no—
79#874—34no—
80#872—34no—
81C22×C3×D3—34no—
82C22×C3×D3—316no—
83C2×C32×C4C22×D318yes—
84C2×C32×C4C22×D3116yes—
85C2×C32×C4C22×D314yes—
86C2×C3×Dic3—38no—
87C2×C3×Dic3—34no—
88C3×D12—616no—
89C3×D12—1216no—
90C3×D12—1216no—
91C3×C4×D3—68no—
92C3×C4×D3—1216no—
93#540—91noSylow
94C2×H(3,𝔽3)C2×#4111yescharacteristic
95C2×H(3,𝔽3)H(5,𝔽2)12yes—
96#272—41no—
97#272—44no—
98C2×C3×D4—92no—
99C2×C3×D4—94no—
100C2×C3×D4—94no—
101C22×C3×C4—91no—
102C22×C3×C4—94no—
103C2×D12—38no—
104C2×D12—316no—
105C2×D12—316no—
106C2×C4×D3—34no—
107C2×C4×D3—316no—
108C3×C4⋊C4—38no—
109C3×C4⋊C4—316no—
110C3×C4⋊C4—92no—
111C3×C4⋊C4C2×C3⋊D312yes—
112C3×C4⋊C4C2×C3⋊D314yes—
113C3×C22⋊C4—92no—
114C3×C22⋊C4—94no—
115C3×C22⋊C4—94no—
116C3×C42—34no—
117C3×C42—91no—
118C3×C42C2×C3⋊D311yescharacteristic
119(C2×D3)⋊C4—38no—
120(C2×D3)⋊C4—316no—
121(C2×D3)⋊C4—316no—
122Dic3⋊C4—38no—
123C4×Dic3—34no—
124C22×C32C23×D314yes—
125C2×C3×D3—64no—
126C2×C3×D3—616no—
127C2×C3×D3—616no—
128C2×C3×D3—68no—
129C2×C3×D3—616no—
130C32×C4—216no—
131C32×C4—28no—
132C32×C4#23918yes—
133C3×Dic3—64no—
134C3×Dic3—68no—
135#127—91no—
136#125#27211yescharacteristic, complemented
137#123—94no—
138#121—94no—
139#121—92no—
140C4×D4—92no—
141#117—91no—
142H(3,𝔽3)#54011yescharacteristic, complemented, Sylow
143C23×C3—91no—
144C23×C3—94no—
145C22×D3—34no—
146C22×D3—316no—
147C3×D4—184no—
148C3×D4—364no—
149C3×D4—364no—
150C2×C3×C4—34no—
151C2×C3×C4—38no—
152C2×C3×C4—316no—
153C2×C3×C4—92no—
154C2×C3×C4—91no—
155C2×C3×C4—182no—
156C2×C3×C4—364no—
157C2×C3×C4C22×C3⋊D312yes—
158C2×C3×C4C22×C3⋊D314yes—
159C2×C3×C4C22×C3⋊D311yescharacteristic
160C2×Dic3—38no—
161C2×Dic3—34no—
162D12—616no—
163D12—1216no—
164D12—1216no—
165C4×D3—68no—
166C4×D3—1216no—
167C2×C32C2×#23914yes—
168C2×C32#100918yes—
169C3×D3—124no—
170C3×D3—1216no—
171C2×D4—92no—
172C2×D4—94no—
173C2×D4—94no—
174C22×C4—91no—
175C22×C4—94no—
176C4⋊C4—92no—
177C4⋊C4C2×#27212yes—
178C4⋊C4C2×#27214yes—
179C22⋊C4—92no—
180C22⋊C4—94no—
181C22⋊C4—94no—
182C42—91no—
183C42C2×#27211yescharacteristic
184C22×C3—34no—
185C22×C3—181no—
186C22×C3—184no—
187C22×C3—184no—
188C22×C3—182no—
189C22×C3—184no—
190C22×C3C23×C3⋊D311yescharacteristic
191C2×D3—64no—
192C2×D3—616no—
193C2×D3—616no—
194C2×D3—68no—
195C2×D3—616no—
196C3×C4—24no—
197C3×C4—22no—
198C3×C4—38no—
199C3×C4—616no—
200C3×C4—68no—
201C3×C4—181no—
202C3×C4—182no—
203C3×C4#383212yes—
204Dic3—64no—
205Dic3—68no—
206C32#569714yescomplemented
207C23—91no—
208C23—94no—
209D4—184no—
210D4—364no—
211D4—364no—
212C2×C4—92no—
213C2×C4—91no—
214C2×C4—182no—
215C2×C4—364no—
216C2×C4C22×#27212yes—
217C2×C4C22×#27214yes—
218C2×C4C22×#27211yescharacteristic
219C2×C3—34no—
220C2×C3—38no—
221C2×C3—361no—
222C2×C3—364no—
223C2×C3C2×#383211yescharacteristic
224C2×C3#6436612yes—
225D3—124no—
226D3—1216no—
227C22—181no—
228C22—184no—
229C22—184no—
230C22—182no—
231C22—184no—
232C22C23×#27211yescharacteristic
233C4—24no—
234C4—22no—
235C4—181no—
236C4—182no—
237C4#8874812yes—
238C3—34no—
239C3#1059616811yescharacteristic
240C2—361no—
241C2—364no—
242C2C2×#8874811yescharacteristic
243C2#1172267512yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#125C2
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#540
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
2. #11721763C2semidirect
3. #11721763C2semidirect
4. #11721762C2semidirect
5. #11721761C2semidirect
6. #11721720C2semidirect
7. #11721718C2semidirect
22. C32×#125D3semidirect
136. #125#272semidirect
142. H(3,𝔽3)#540semidirect
206. C32#5697semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117217632
#117217622
#117217612
#117217202
#117217182
C3×#56973