Finite Group Databasebeta

← #13048775

Subgroups — #13048775

Order 1296.

511 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 59 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#10619977C212yesmaximal, complemented
2C2×C3×#1084C211yesmaximal, characteristic, complemented
3#10619902C211yesmaximal, characteristic, complemented
4#10619901C212yesmaximal, complemented
5#10619900C211yesmaximal, characteristic, complemented
6#89031—32nomaximal
7#88672—31nomaximal
8C22×C3×H(3,𝔽3)C2211yescharacteristic
9C2×C3⋊((C3⋊D3)⋊C3)C2212yes—
10C3×#1084C2211yescharacteristic
11C3×#1084C2211yescharacteristic
12C3⋊((C3⋊Dic3)⋊C3)C2212yes—
13#6312D312yescomplemented
14C3×#638—32no—
15C2×C32×Dic3—32no—
16C2×C32×Dic3—61no—
17#6293—32no—
18#6292—32no—
19#6289—32no—
20Dic3×C3⋊D3—32no—
21C2×#1084D311yescharacteristic, complemented
22#638⋊C3—32no—
23#638⋊C3—62no—
24#6198—31no—
25((C3⋊D3)⋊C3)⋊C4—32no—
26#6196—31no—
27C2×C3×H(3,𝔽3)—21no—
28C2×C3×H(3,𝔽3)C2311yescharacteristic
29C3⋊((C3⋊D3)⋊C3)—22no—
30#3828—91no—
31#3804—92no—
32#3804—96no—
33C22×C33—21no—
34C22×C33C2×D312yes—
35C2×C3⋊(C3⋊D3)C2×D312yes—
36C2×C3×C3⋊D3—32no—
37C3⋊(C3⋊Dic3)C2×D312yes—
38C3×C3⋊Dic3—32no—
39C32×Dic3—32no—
40C32×Dic3—32no—
41C32×Dic3—61no—
42C32×Dic3—61no—
43C22×H(3,𝔽3)—22no—
44C22×H(3,𝔽3)—41no—
45C22×H(3,𝔽3)C2×D311yescharacteristic
46C2×(C3⋊D3)⋊C3—32no—
47C2×(C3⋊D3)⋊C3—62no—
48#1084C2×D311yescharacteristic
49#1084C2×D311yescharacteristic
50(C3⋊Dic3)⋊C3—32no—
51(C3⋊Dic3)⋊C3—62no—
52C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
53C2×C32×C4—91no—
54#638—32no—
55#638—32no—
56#638—62no—
57#638—96no—
58C4⋊(C3⋊D3)—91no—
59C4×C3⋊D3—92no—
60#634—91no—
61C3×#67—96no—
62C3×#67—92no—
63C2×C3×Dic3—36no—
64C2×C3×Dic3—32no—
65C2×C3×Dic3—61no—
66C2×C3×Dic3—63no—
67#627—92no—
68#627—96no—
69#626—92no—
70#626—96no—
71#624—92no—
72#624—96no—
73D3×Dic3—92no—
74D3×Dic3—96no—
75C2×C33—22no—
76C2×C33—22no—
77C2×C33—21no—
78C2×C33C22×D312yes—
79C3⋊(C3⋊D3)—22no—
80C3×C3⋊D3—62no—
81C2×H(3,𝔽3)—21no—
82C2×H(3,𝔽3)—22no—
83C2×H(3,𝔽3)—41no—
84C2×H(3,𝔽3)—42no—
85C2×H(3,𝔽3)—42no—
86C2×H(3,𝔽3)C22×D311yescharacteristic
87(C3⋊D3)⋊C3—62no—
88(C3⋊D3)⋊C3—122no—
89#237—272no—
90#235—273no—
91#235—271no—
92C22×C32—21no—
93C22×C32—22no—
94C22×C32—36no—
95C22×C32—41no—
96C22×C32—63no—
97C22×C32D3212yes—
98C22×C32D3211yescharacteristic
99C2×C3⋊D3—32no—
100C2×C3⋊D3—32no—
101C2×C3⋊D3—62no—
102C2×C3⋊D3—96no—
103C2×C3×D3—96no—
104C2×C3×D3—92no—
105C32×C4—91no—
106C32×C4—91no—
107C3⋊Dic3—32no—
108C3⋊Dic3—32no—
109C3⋊Dic3—62no—
110C3⋊Dic3—96no—
111C3×Dic3—36no—
112C3×Dic3—36no—
113C3×Dic3—32no—
114C3×Dic3—32no—
115C3×Dic3—61no—
116C3×Dic3—61no—
117C3×Dic3—63no—
118C3×Dic3—63no—
119C3×Dic3—96no—
120C3×Dic3—92no—
121C33—21no—
122C33#23712yescomplemented
123H(3,𝔽3)—22no—
124H(3,𝔽3)—41no—
125H(3,𝔽3)#23511yescharacteristic, complemented
126C3×D4—272no—
127C2×C3×C4—93no—
128C2×C3×C4—91no—
129#67—96no—
130#67—92no—
131#67—272no—
132#67—272no—
133#67—542no—
134C2×Dic3—32no—
135C2×Dic3—61no—
136C2×Dic3—272no—
137D12—273no—
138D12—271no—
139C4×D3—272no—
140C4×D3—276no—
141C4×D3—272no—
142Dic6—272no—
143Dic6—273no—
144Dic6—271no—
145C2×C32—21no—
146C2×C32—22no—
147C2×C32—22no—
148C2×C32—22no—
149C2×C32—21no—
150C2×C32—36no—
151C2×C32—42no—
152C2×C32—42no—
153C2×C32—41no—
154C2×C32—66no—
155C2×C32—66no—
156C2×C32—63no—
157C2×C32C2×D3212yes—
158C2×C32C2×D3211yescharacteristic
159C3⋊D3—62no—
160C3⋊D3—62no—
161C3⋊D3—122no—
162C3⋊D3—186no—
163C3×D3—186no—
164C3×D3—182no—
165#41—811noSylow
166C22×C3—32no—
167C22×C3—61no—
168C22×C3—62no—
169C22×C3—121no—
170C22×C3—272no—
171C22×C3((C3⋊D3)⋊C3)⋊C213yes—
172C22×C3#111311yescharacteristic
173C2×D3—96no—
174C2×D3—92no—
175C2×D3—272no—
176C2×D3—542no—
177C3×C4—93no—
178C3×C4—93no—
179C3×C4—91no—
180C3×C4—91no—
181C3×C4—272no—
182Dic3—32no—
183Dic3—32no—
184Dic3—61no—
185Dic3—61no—
186Dic3—96no—
187Dic3—92no—
188Dic3—272no—
189Dic3—542no—
190C32—22no—
191C32—21no—
192C32—36no—
193C32—41no—
194C32—63no—
195C32#380412yescomplemented
196C32#380511yescharacteristic
197Q8—811no—
198D4—272no—
199D4—811no—
200C2×C4—91no—
201C2×C4—812no—
202C2×C3—23no—
203C2×C3—21no—
204C2×C3—32no—
205C2×C3—62no—
206C2×C3—62no—
207C2×C3—61no—
208C2×C3—62no—
209C2×C3—122no—
210C2×C3—122no—
211C2×C3—121no—
212C2×C3—542no—
213C2×C3C2×((C3⋊D3)⋊C3)⋊C213yes—
214C2×C3C2×#111311yescharacteristic
215D3—186no—
216D3—182no—
217D3—542no—
218D3—1082no—
219C22—272no—
220C22#6643211yescharacteristic
221C4—91no—
222C4—91no—
223C4—272no—
224C3—32no—
225C3—62no—
226C3—61no—
227C3—121no—
228C3#8867213yescomplemented
229C3#8904611yescharacteristic
230C2—21no—
231C2—542no—
232C2C2×#6643211yescharacteristic
subgroupG/H
centerC2C2×#66432
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×#1113
FittingC22×C3×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC22#66432
hypocenterC3×H(3,𝔽3)#41
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #10619977C2semidirect
2. C2×C3×#1084C2semidirect
3. #10619902C2semidirect
4. #10619901C2semidirect
5. #10619900C2semidirect
13. #6312D3semidirect
21. C2×#1084D3semidirect
52. C3×H(3,𝔽3)#41semidirect
122. C33#237semidirect
125. H(3,𝔽3)#235semidirect
195. C32#3804semidirect
228. C3#88672semidirect

Maximal subgroups

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

subgroupindex
#106199772
C2×C3×#10842
#106199022
#106199012
#106199002
#890313
#886723