Finite Group Databasebeta

← #13048722

Subgroups — #13048722

Order 1296.

576 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 77 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#10619994C211yesmaximal, characteristic, complemented
2C3×#6235C211yesmaximal, characteristic, complemented
3C2×Dic3×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#10619894C211yesmaximal, characteristic, complemented
5#10619893C211yesmaximal, characteristic, complemented
6#10619890C211yesmaximal, characteristic, complemented
7Dic3×#272C211yesmaximal, characteristic, complemented
8C3×#3804—34nomaximal
9#88740—31nomaximal
10C22×C3×H(3,𝔽3)C2211yescharacteristic
11C2×C3⋊#272C2211yescharacteristic
12C2×C3×#272C2211yescharacteristic
13C3⋊#1084C2211yescharacteristic
14C3×#1084C2211yescharacteristic
15Dic3×H(3,𝔽3)C2211yescharacteristic
16Dic3×H(3,𝔽3)C2211yescharacteristic
17C3×#638—34no—
18C32×#67—34no—
19C2×C32×Dic3D314yescomplemented
20C3×#627—34no—
21C3×#626—34no—
22C3×#624—34no—
23C3×D3×Dic3—34no—
24C2×C4×H(3,𝔽3)—31no—
25#6235—31no—
26#6235—64no—
27#6235D311yescharacteristic, complemented
28C4⋊#272—31no—
29C4×#272—31no—
30C4×#272—31no—
31#6229—31no—
32C2×C3×H(3,𝔽3)—21no—
33C2×C3×H(3,𝔽3)C2311yescharacteristic
34C3⋊#272—21no—
35C3×#272—21no—
36C3×#237—91no—
37C3×#235—94no—
38#3804—34no—
39C22×C33C2×D314yes—
40C2×C3×C3⋊D3—34no—
41C2×C32×D3—34no—
42C3×C3⋊Dic3—34no—
43C32×Dic3—34no—
44C32×Dic3C2×D314yes—
45C32×Dic3C2×D314yes—
46C22×H(3,𝔽3)—24no—
47C22×H(3,𝔽3)C2×D311yescharacteristic
48C2×#272—31no—
49C2×#272—64no—
50C2×#272C2×D311yescharacteristic
51C4×H(3,𝔽3)—31no—
52C4×H(3,𝔽3)—31no—
53#1084—31no—
54#1084—64no—
55#1084C2×D311yescharacteristic
56C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
57C32×D4—91no—
58C2×C32×C4—34no—
59#638—34no—
60C3×#67—34no—
61C3×#67—34no—
62C3×#67—64no—
63C3×#67—94no—
64C3×#67—91no—
65C3×#67—91no—
66C3×#67—184no—
67C2×C3×Dic3—34no—
68C2×C3×Dic3—91no—
69C2×C3×Dic3C3⋊D311yescharacteristic
70C3×D12—94no—
71C3×C4×D3—94no—
72C3×C4×D3—94no—
73C3×C4×D3—91no—
74C3×Dic6—94no—
75C3×Dic6—91no—
76#627—34no—
77#626—34no—
78#624—34no—
79D3×Dic3—34no—
80C2×C33—24no—
81C2×C33C22×D314yes—
82C3×C3⋊D3—64no—
83C32×D3—64no—
84C2×H(3,𝔽3)—21no—
85C2×H(3,𝔽3)—24no—
86C2×H(3,𝔽3)—44no—
87C2×H(3,𝔽3)C22×D311yescharacteristic
88#272—21no—
89#272—61no—
90#272—124no—
91C3×#41—271no—
92#237—91no—
93#235—94no—
94C22×C32—24no—
95C22×C32—34no—
96C22×C32—64no—
97C22×C32—91no—
98C22×C32D3214yes—
99C22×C32C2×C3⋊D311yescharacteristic
100C2×C3⋊D3—34no—
101C2×C3×D3—34no—
102C2×C3×D3—34no—
103C2×C3×D3—64no—
104C2×C3×D3—94no—
105C2×C3×D3—91no—
106C2×C3×D3—184no—
107C32×C4—34no—
108C32×C4—34no—
109C32×C4—91no—
110C3⋊Dic3—34no—
111C3×Dic3—34no—
112C3×Dic3—34no—
113C3×Dic3—34no—
114C3×Dic3—34no—
115C3×Dic3—64no—
116C3×Dic3—94no—
117C3×Dic3—91no—
118C3×Dic3—184no—
119C3×Dic3C2×C3⋊D311yescharacteristic
120C3×Dic3C2×C3⋊D311yescharacteristic
121C33#23514yescomplemented
122H(3,𝔽3)—24no—
123H(3,𝔽3)#23711yescharacteristic, complemented
124C3×Q8—271no—
125C3×D4—91no—
126C3×D4—91no—
127C3×D4—181no—
128C3×D4—271no—
129C3×D4—271no—
130C2×C3×C4—31no—
131C2×C3×C4—94no—
132C2×C3×C4—271no—
133C2×C3×C4—271no—
134#67—34no—
135#67—94no—
136#67—91no—
137#67—91no—
138#67—184no—
139C2×Dic3—91no—
140C2×Dic3#27211yescharacteristic, complemented
141D12—94no—
142C4×D3—94no—
143C4×D3—94no—
144C4×D3—91no—
145Dic6—94no—
146Dic6—91no—
147C2×C32—21no—
148C2×C32—24no—
149C2×C32—24no—
150C2×C32—34no—
151C2×C32—44no—
152C2×C32—64no—
153C2×C32—64no—
154C2×C32—124no—
155C2×C32—181no—
156C2×C32C2×D3214yes—
157C2×C32C22×C3⋊D311yescharacteristic
158C3⋊D3—64no—
159C3×D3—64no—
160C3×D3—64no—
161C3×D3—124no—
162C3×D3—184no—
163C3×D3—181no—
164C3×D3—364no—
165#41—271noSylow
166C22×C3—21no—
167C22×C3—34no—
168C22×C3—64no—
169C22×C3—91no—
170C22×C3—91no—
171C22×C3—181no—
172C22×C3—271no—
173C22×C3C2×#27211yescharacteristic
174C22×C3D3×C3⋊D311yescharacteristic
175C2×D3—34no—
176C2×D3—94no—
177C2×D3—91no—
178C2×D3—184no—
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—94no—
182C3×C4—94no—
183C3×C4—91no—
184C3×C4—91no—
185C3×C4—181no—
186C3×C4—271no—
187Dic3—34no—
188Dic3—94no—
189Dic3—91no—
190Dic3—184no—
191Dic3C2×#27211yescharacteristic
192Dic3C2×#27211yescharacteristic
193C32—24no—
194C32—34no—
195C32—64no—
196C32#380414yescomplemented
197C32#382811yescharacteristic
198Q8—271no—
199D4—91no—
200D4—271no—
201D4—271no—
202C2×C4—31no—
203C2×C4—271no—
204C2×C4—271no—
205C2×C3—21no—
206C2×C3—21no—
207C2×C3—21no—
208C2×C3—34no—
209C2×C3—41no—
210C2×C3—64no—
211C2×C3—64no—
212C2×C3—124no—
213C2×C3—181no—
214C2×C3—181no—
215C2×C3—361no—
216C2×C3—541no—
217C2×C3C22×#27211yescharacteristic
218C2×C3C2×D3×C3⋊D311yescharacteristic
219D3—64no—
220D3—184no—
221D3—181no—
222D3—364no—
223C22—91no—
224C22—271no—
225C22D3×#27211yescharacteristic
226C4—31no—
227C4—31no—
228C4—91no—
229C4—271no—
230C3—21no—
231C3—34no—
232C3—64no—
233C3#8874011yescharacteristic, complemented
234C3#8903111yescharacteristic
235C2—21no—
236C2—181no—
237C2—541no—
238C2C2×D3×#27211yescharacteristic
subgroupG/H
center, FrattiniC2×C3C2×D3×C3⋊D3
commutator, cosocleC2×C3×H(3,𝔽3)C23
FittingC22×C3×H(3,𝔽3)C22
socleC2×C32C22×C3⋊D3
hypercenterC22×C3D3×C3⋊D3
hypocenterC3×H(3,𝔽3)#41
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. #10619994C2semidirect
2. C3×#6235C2semidirect
3. C2×Dic3×H(3,𝔽3)C2semidirect
4. #10619894C2semidirect
5. #10619893C2semidirect
6. #10619890C2semidirect
7. Dic3×#272C2semidirect
19. C2×C32×Dic3D3semidirect
27. #6235D3semidirect
56. C3×H(3,𝔽3)#41semidirect
121. C33#235semidirect
123. H(3,𝔽3)#237semidirect
140. C2×Dic3#272semidirect
196. C32#3804semidirect
233. C3#88740semidirect

Maximal subgroups

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

subgroupindex
#106199942
C3×#62352
C2×Dic3×H(3,𝔽3)2
#106198942
#106198932
#106198902
Dic3×#2722
C3×#38043
#887403