Finite Group Databasebeta

← #13047428

Subgroups — #13047428

Order 1296.

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

normal subgroups only

This group has 400 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
1C2×C4×AΣL(1,𝔽27)C211yesmaximal, characteristic, complemented
2#10619679C212yesmaximal, complemented
3C4⋊#4183C211yesmaximal, characteristic, complemented
4C4×#4183C212yesmaximal, complemented
5#10619660C211yesmaximal, characteristic, complemented
6C3×#3828—31nomaximal
7#3697⋊C3—32nomaximal
8#3828⋊C3—31nomaximal
9C22×AΣL(1,𝔽27)C2211yescharacteristic
10C2×#4183C2212yes—
11C4×AΣL(1,𝔽27)C2211yescharacteristic
12C4×AΣL(1,𝔽27)C2211yescharacteristic
13#66330C2212yes—
14C2×C33×C4D311yescharacteristic, complemented
15C3×#638—32no—
16C3×C4⋊(C3⋊D3)—31no—
17C3×C4×C3⋊D3—32no—
18C3×#634—31no—
19C2×C4×3−1+2D312yescomplemented
20C2×C4×H(3,𝔽3)D311yescharacteristic, complemented
21#611⋊C3—34no—
22#638⋊C3—32no—
23C4⋊AGL(1,ℤ/9)—32no—
24C4×AGL(1,ℤ/9)—34no—
253−1+2⋊Q8—32no—
26C4⋊((C3⋊D3)⋊C3)—31no—
27C4×(C3⋊D3)⋊C3—32no—
28#634⋊C3—31no—
29C2×AΣL(1,𝔽27)—21no—
30C2×AΣL(1,𝔽27)C2311yescharacteristic
31#4183—22no—
32#3828—31no—
33#3828—31no—
34C3×#235—93no—
35C3×#235—91no—
36#3697—32no—
37C22×C33C2×D311yescharacteristic
38C2×C3×C3⋊D3—32no—
39C33×C4C2×D311yescharacteristic
40C33×C4C2×D311yescharacteristic
41C3×C3⋊Dic3—32no—
42C22×3−1+2C2×D312yes—
43C22×H(3,𝔽3)C2×D311yescharacteristic
44C2×AGL(1,ℤ/9)—34no—
45C2×(C3⋊D3)⋊C3—32no—
46C4×3−1+2C2×D312yes—
47C4×3−1+2C2×D312yes—
48C4×H(3,𝔽3)C2×D311yescharacteristic
49C4×H(3,𝔽3)C2×D311yescharacteristic
503−1+2⋊C4—34no—
51(C3⋊Dic3)⋊C3—32no—
52AΣL(1,𝔽27)#4111yescharacteristic, complemented, Sylow
53C2×C32×C4—31no—
54C2×C32×C4—33no—
55C2×C32×C4—31no—
56C2×C32×C4C3⋊D311yescharacteristic
57#638—32no—
58#638—32no—
59C4⋊(C3⋊D3)—31no—
60C4⋊(C3⋊D3)—31no—
61C4×C3⋊D3—32no—
62C4×C3⋊D3—32no—
63#634—31no—
64#634—31no—
65C3×#67—96no—
66C3×#67—92no—
67C3×D12—93no—
68C3×D12—91no—
69C3×C4×D3—96no—
70C3×C4×D3—92no—
71C3×Dic6—93no—
72C3×Dic6—91no—
73C2×C4×C9—32no—
74#611—34no—
75D36—32no—
76C4×D9—34no—
77Dic18—32no—
78C2×C33—21no—
79C2×C33C22×D311yescharacteristic
80C3×C3⋊D3—62no—
81C2×3−1+2—22no—
82C2×3−1+2C22×D312yes—
83C2×H(3,𝔽3)—21no—
84C2×H(3,𝔽3)C22×D311yescharacteristic
85AGL(1,ℤ/9)—64no—
86(C3⋊D3)⋊C3—62no—
87C3×#41—271no—
88#235—93no—
89#235—91no—
90#235—271no—
91C22×C32—31no—
92C22×C32—33no—
93C22×C32—31no—
94C22×C32C2×C3⋊D311yescharacteristic
95C2×C3⋊D3—32no—
96C2×C3⋊D3—32no—
97C2×C3×D3—96no—
98C2×C3×D3—92no—
99C32×C4—31no—
100C32×C4—31no—
101C32×C4—33no—
102C32×C4—33no—
103C32×C4—31no—
104C32×C4—31no—
105C32×C4C2×C3⋊D311yescharacteristic
106C32×C4C2×C3⋊D311yescharacteristic
107C3⋊Dic3—32no—
108C3⋊Dic3—32no—
109C3×Dic3—96no—
110C3×Dic3—92no—
111C22×C9—32no—
112C2×D9—34no—
113C4×C9—32no—
114C4×C9—32no—
115Dic9—34no—
116C33#23511yescharacteristic, complemented
1173−1+2#23512yescomplemented
118H(3,𝔽3)#23511yescharacteristic, complemented
119C3×Q8—271no—
120C3×D4—271no—
121C3×D4—272no—
122C2×C3×C4—31no—
123C2×C3×C4—33no—
124C2×C3×C4—91no—
125C2×C3×C4—272no—
126C2×C3×C4#27211yescharacteristic
127#67—96no—
128#67—92no—
129#67—272no—
130D12—93no—
131D12—91no—
132D12—271no—
133C4×D3—96no—
134C4×D3—92no—
135C4×D3—272no—
136Dic6—93no—
137Dic6—91no—
138Dic6—271no—
139C2×C32—21no—
140C2×C32—31no—
141C2×C32—33no—
142C2×C32—31no—
143C2×C32—61no—
144C2×C32—61no—
145C2×C32—63no—
146C2×C32C22×C3⋊D311yescharacteristic
147C3⋊D3—62no—
148C3⋊D3—62no—
149C3×D3—186no—
150C3×D3—182no—
151C2×C9—32no—
152C2×C9—62no—
153D9—64no—
154#41—271noSylow
155C22×C3—33no—
156C22×C3—31no—
157C22×C3—91no—
158C22×C3—272no—
159C22×C3C2×#27211yescharacteristic
160C2×D3—96no—
161C2×D3—92no—
162C2×D3—272no—
163C3×C4—33no—
164C3×C4—31no—
165C3×C4—33no—
166C3×C4—31no—
167C3×C4—91no—
168C3×C4—91no—
169C3×C4—272no—
170C3×C4C2×#27211yescharacteristic
171C3×C4C2×#27211yescharacteristic
172Dic3—96no—
173Dic3—92no—
174Dic3—272no—
175C32—33no—
176C32—31no—
177C32—31no—
178C32#382811yescharacteristic
179C9—32no—
180Q8—271no—
181D4—272no—
182D4—271no—
183C2×C4—272no—
184C2×C4#418311yescharacteristic, complemented
185C2×C3—21no—
186C2×C3—33no—
187C2×C3—31no—
188C2×C3—61no—
189C2×C3—63no—
190C2×C3—91no—
191C2×C3—181no—
192C2×C3—542no—
193C2×C3C22×#27211yescharacteristic
194D3—186no—
195D3—182no—
196D3—542no—
197C22—272no—
198C22C2×#418311yescharacteristic
199C4—272no—
200C4C2×#418311yescharacteristic
201C4C2×#418311yescharacteristic
202C3—33no—
203C3—31no—
204C3—91no—
205C3#8874011yescharacteristic
206C2—21no—
207C2—542no—
208C2C22×#418311yescharacteristic
subgroupG/H
centerC4C2×#4183
commutator, cosocleC2×AΣL(1,𝔽27)C23
FrattiniC2×C32C22×C3⋊D3
FittingC2×C4×AΣL(1,𝔽27)C2
socleC2×C3C22×#272
hypercenterC2×C4#4183
hypocenterAΣL(1,𝔽27)#41
LCSG ▷ C2×AΣL(1,𝔽27) ▷ AΣL(1,𝔽27)
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×AΣL(1,𝔽27)C2semidirect
2. #10619679C2semidirect
3. C4⋊#4183C2semidirect
4. C4×#4183C2semidirect
5. #10619660C2semidirect
14. C2×C33×C4D3semidirect
19. C2×C4×3−1+2D3semidirect
20. C2×C4×H(3,𝔽3)D3semidirect
52. AΣL(1,𝔽27)#41semidirect
116. C33#235semidirect
117. 3−1+2#235semidirect
118. H(3,𝔽3)#235semidirect
184. C2×C4#4183semidirect

Maximal subgroups

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

subgroupindex
C2×C4×AΣL(1,𝔽27)2
#106196792
C4⋊#41832
C4×#41832
#106196602
C3×#38283
#3697⋊C33
#3828⋊C33