Finite Group Databasebeta

← #13154273

Subgroups — #13154273

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)×#127C211yesmaximal, characteristic, complemented
2#11721849C214yesmaximal, complemented
3#11721848C212yesmaximal, complemented
4#11721845C214yesmaximal, complemented
5D4×#1084C212yesmaximal, complemented
6#11721718C211yesmaximal, characteristic, complemented
7#11721716C211yesmaximal, characteristic, complemented
8C3×#5719—34nomaximal
9C2×D4×H(3,𝔽3)C2212yes—
10C2×D4×H(3,𝔽3)C2214yes—
11C2×#6235C2214yes—
12C22×#1084C2214yes—
13C2×C4×#272C2211yescharacteristic
14C42×H(3,𝔽3)C2211yescharacteristic
15#88552C2214yes—
16#88552C2214yes—
17#88542C2212yes—
18C4⋊#1084C2212yes—
19#88540C2212yes—
20#88540C2214yes—
21C4×#1084C2211yescharacteristic
22C32×#127D314yescomplemented
23C3×#939—316no—
24C3×#938—38no—
25C3×#935—316no—
26C3×D4×Dic3—38no—
27C3×#872—34no—
28C3×#870—34no—
29C23×H(3,𝔽3)C2314yes—
30C22×#272C2311yescharacteristic
31D4×H(3,𝔽3)—24no—
32D4×H(3,𝔽3)—44no—
33C2×C4×H(3,𝔽3)C2312yes—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35#6235—44no—
36C2×#1084—44no—
37C2×#1084C2311yescharacteristic
38C2×#1084C2312yes—
39C2×#1084C2314yes—
40C4×#272—22no—
41C3×#540—91no—
42#5719—34no—
43C2×C32×D4C2×D318yes—
44C2×C32×D4C2×D3116yes—
45C2×C3×#67—316no—
46C22×C3×Dic3—316no—
47C2×C3×C4×D3—34no—
48C32×C42C2×D314yes—
49C3×#217—316no—
50C3×#217—316no—
51C3×(C2×D3)⋊C4—38no—
52C3×C4⋊Dic3—38no—
53C3×Dic3⋊C4—38no—
54C3×Dic3⋊C4—316no—
55C3×C4×Dic3—34no—
56C22×H(3,𝔽3)—24no—
57C22×H(3,𝔽3)—24no—
58C22×H(3,𝔽3)—24no—
59C22×H(3,𝔽3)C2411yescharacteristic
60C2×#272—21no—
61C2×#272—22no—
62C4×H(3,𝔽3)—22no—
63C4×H(3,𝔽3)#4112yes—
64#1084—21no—
65#1084—22no—
66#1084—24no—
67C3×#127—34no—
68C3×#127C3⋊D311yescharacteristic
69C3×#125—91no—
70C3×#123—94no—
71C3×#121—92no—
72C3×#121—94no—
73C3×C4×D4—92no—
74C3×#117—91no—
75#939—316no—
76#938—38no—
77#935—316no—
78D4×Dic3—38no—
79#872—34no—
80#870—34no—
81C23×C32C22×D3116yes—
82C22×C3×D3—34no—
83C32×D4—216no—
84C32×D4—416no—
85C2×C32×C4C22×D318yes—
86C2×C32×C4C22×D314yes—
87C3×#67—1216no—
88C2×C3×Dic3—38no—
89C2×C3×Dic3—316no—
90C2×C3×Dic3—34no—
91C2×C3×Dic3—1216no—
92C3×C4×D3—68no—
93#540—91noSylow
94C2×H(3,𝔽3)—44no—
95C2×H(3,𝔽3)C2×#4111yescharacteristic
96C2×H(3,𝔽3)H(5,𝔽2)12yes—
97#272—41no—
98C2×C3×D4—38no—
99C2×C3×D4—316no—
100C2×C3×D4—94no—
101C2×C3×D4C2×C3⋊D312yes—
102C2×C3×D4C2×C3⋊D314yes—
103C22×C3×C4—94no—
104C22×C3×C4—91no—
105C2×#67—316no—
106C22×Dic3—316no—
107C2×C4×D3—34no—
108C3×C4⋊C4—92no—
109C3×C4⋊C4—94no—
110C3×C4⋊C4—92no—
111C3×C22⋊C4—94no—
112C3×C22⋊C4—94no—
113C3×C22⋊C4—92no—
114C3×C42—34no—
115C3×C42—91no—
116C3×C42C2×C3⋊D311yescharacteristic
117#217—316no—
118#217—316no—
119(C2×D3)⋊C4—38no—
120C4⋊Dic3—38no—
121Dic3⋊C4—316no—
122Dic3⋊C4—38no—
123C4×Dic3—34no—
124C22×C32—216no—
125C22×C32—216no—
126C22×C32—216no—
127C22×C32C23×D314yes—
128C2×C3×D3—64no—
129C2×C3×D3—68no—
130C32×C4—28no—
131C32×C4#23718yes—
132C3×Dic3—64no—
133C3×Dic3—68no—
134C3×Dic3—616no—
135#127#27211yescharacteristic, complemented
136#125—91no—
137#123—94no—
138#121—92no—
139#121—94no—
140C4×D4—92no—
141#117—91no—
142H(3,𝔽3)#54011yescharacteristic, complemented, Sylow
143C23×C3—316no—
144C23×C3—91no—
145C23×C3C22×C3⋊D314yes—
146C22×D3—34no—
147C3×D4—24no—
148C3×D4—44no—
149C3×D4—616no—
150C3×D4—1216no—
151C3×D4—364no—
152C2×C3×C4—34no—
153C2×C3×C4—38no—
154C2×C3×C4—92no—
155C2×C3×C4—91no—
156C2×C3×C4—94no—
157C2×C3×C4—182no—
158C2×C3×C4—364no—
159C2×C3×C4C22×C3⋊D312yes—
160C2×C3×C4C22×C3⋊D311yescharacteristic
161#67—1216no—
162C2×Dic3—38no—
163C2×Dic3—316no—
164C2×Dic3—34no—
165C2×Dic3—1216no—
166C4×D3—68no—
167C2×C32—416no—
168C2×C32C2×#23714yes—
169C2×C32#100418yes—
170C3×D3—124no—
171C2×D4—94no—
172C2×D4C2×#27212yes—
173C2×D4C2×#27214yes—
174C22×C4—94no—
175C22×C4—91no—
176C4⋊C4—92no—
177C4⋊C4—92no—
178C4⋊C4—94no—
179C22⋊C4—94no—
180C22⋊C4—94no—
181C22⋊C4—92no—
182C42—91no—
183C42C2×#27211yescharacteristic
184C22×C3—24no—
185C22×C3—24no—
186C22×C3—24no—
187C22×C3—34no—
188C22×C3—616no—
189C22×C3—616no—
190C22×C3—616no—
191C22×C3—181no—
192C22×C3—182no—
193C22×C3C23×C3⋊D311yescharacteristic
194C2×D3—64no—
195C2×D3—68no—
196C3×C4—22no—
197C3×C4—38no—
198C3×C4—68no—
199C3×C4—181no—
200C3×C4—182no—
201C3×C4—184no—
202C3×C4#383012yes—
203Dic3—64no—
204Dic3—68no—
205Dic3—616no—
206C32#571914yescomplemented
207C23—91no—
208C23C22×#27214yes—
209D4—24no—
210D4—44no—
211D4—364no—
212C2×C4—92no—
213C2×C4—91no—
214C2×C4—94no—
215C2×C4—182no—
216C2×C4—364no—
217C2×C4C22×#27212yes—
218C2×C4C22×#27211yescharacteristic
219C2×C3—34no—
220C2×C3—38no—
221C2×C3—44no—
222C2×C3—1216no—
223C2×C3—361no—
224C2×C3C2×#383011yescharacteristic
225C2×C3#6436112yes—
226D3—124no—
227C22—24no—
228C22—24no—
229C22—24no—
230C22—181no—
231C22—182no—
232C22C23×#27211yescharacteristic
233C4—22no—
234C4—181no—
235C4—182no—
236C4—184no—
237C4#8874412yes—
238C3—34no—
239C3#1059619011yescharacteristic
240C2—44no—
241C2—361no—
242C2C2×#8874411yescharacteristic
243C2#1172266412yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#127C2
hypercenterC3×#127C3⋊D3
hypocenterH(3,𝔽3)#540
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#127

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#127C2semidirect
2. #11721849C2semidirect
3. #11721848C2semidirect
4. #11721845C2semidirect
5. D4×#1084C2semidirect
6. #11721718C2semidirect
7. #11721716C2semidirect
22. C32×#127D3semidirect
135. #127#272semidirect
142. H(3,𝔽3)#540semidirect
206. C32#5719semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1272
#117218492
#117218482
#117218452
D4×#10842
#117217182
#117217162
C3×#57193