Finite Group Databasebeta

← #10592766⋊C3 #13142955

Subgroups — #10592766⋊C3 #13142955

Order 1728.

441 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 92 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#11721385C211yesmaximal, characteristic
3#63636⋊C3C214yesmaximal, complemented
4#63630⋊C3C211yesmaximal, characteristic, complemented
5#10592766C311yesmaximal, characteristic, complemented
6C3×#5067—31nomaximal
7C2×C4⋊((C3⋊D3)⋊C3)—22no—
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C4⋊C4×H(3,𝔽3)—22no—
10C4⋊C4×H(3,𝔽3)C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13C32×#125—21no—
14C32×#125D311yescharacteristic, complemented
15C32×#125C2×C311yescharacteristic, complemented
16#64083C2×C311yescharacteristic
17C3×#874—31no—
18#63636C2×C314yescomplemented
19#63630C2×C311yescharacteristic, complemented
20C3×#809—34no—
21C3×#803—31no—
22C22×(C3⋊D3)⋊C3—22no—
23C2×C4×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)D412yes—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C4⋊((C3⋊D3)⋊C3)—24no—
27C4⋊((C3⋊D3)⋊C3)—44no—
28(C3⋊(C3⋊C8))⋊C3—22no—
29C3×#490—91no—
30#5067—31no—
31#5067—91no—
32C2×C4⋊(C3⋊D3)—22no—
33C2×C4⋊(C3⋊D3)C22×C312yes—
34C2×C3×D12—32no—
35C2×C3×D12—62no—
36C32×C4⋊C4—24no—
37C32×C4⋊C4—22no—
38C32×C4⋊C4—22no—
39C32×C4⋊C4—22no—
40C32×C4⋊C4C2×D312yes—
41C32×C4⋊C4C22×C312yes—
42C32×C42—21no—
43C32×C42C2×D311yescharacteristic
44C32×C42C22×C311yescharacteristic
45C2×C3⋊(C3⋊C8)C22×C312yes—
46C2×C3×C3⋊C8—32no—
47C22×H(3,𝔽3)C2×D411yescharacteristic
48C2×(C3⋊D3)⋊C3—42no—
49C2×(C3⋊D3)⋊C3—44no—
50C4×H(3,𝔽3)—22no—
51C4×H(3,𝔽3)—42no—
52C4×H(3,𝔽3)#4112yes—
53C3×#127—91no—
54C3×#125—31no—
55C3×#125—31no—
56C3×#125—61no—
57C3×#125C3×D311yescharacteristic
58#874—31no—
59#874—91no—
60C3×#102—94no—
61C3×#97—91no—
62#809—34no—
63#809—94no—
64#803—31no—
65#803—91no—
66C22×C3⋊D3—22no—
67C22×C3×D3—62no—
68C2×C32×C4—24no—
69C2×C32×C4—22no—
70C2×C32×C4—22no—
71C2×C32×C4—22no—
72C2×C32×C4—21no—
73C2×C32×C4#6712yes—
74C2×C32×C4C3×D412yes—
75C2×C32×C4C22×D311yescharacteristic
76C2×C32×C4C23×C311yescharacteristic
77C4⋊(C3⋊D3)—24no—
78C4⋊(C3⋊D3)—44no—
79C3×D12—64no—
80C3×D12—124no—
81C3⋊(C3⋊C8)—22no—
82C3×C3⋊C8—62no—
83#490—91noSylow
84C2×H(3,𝔽3)#12411yescharacteristic
85C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
86(C3⋊D3)⋊C3—82no—
87C2×C3×D4—92no—
88C2×C3×D4—182no—
89C2×D12—32no—
90C2×D12—62no—
91C2×D12—92no—
92C2×D12—182no—
93C2×C3×C8—92no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—32no—
96C3×C4⋊C4—32no—
97C3×C4⋊C4—62no—
98C3×C4⋊C4—62no—
99C3×C4⋊C4—64no—
100C3×C4⋊C4—62no—
101C3×C4⋊C4C2×C3×D312yes—
102C3×C42—31no—
103C3×C42—31no—
104C3×C42—61no—
105C3×C42C2×C3×D311yescharacteristic
106C2×C3⋊C8—32no—
107C2×C3⋊C8—92no—
108C22×C32—21no—
109C22×C32C2×#6711yescharacteristic
110C22×C32C2×C3×D411yescharacteristic
111C2×C3⋊D3—42no—
112C2×C3⋊D3—44no—
113C2×C3×D3—122no—
114C2×C3×D3—124no—
115C32×C4—22no—
116C32×C4—22no—
117C32×C4—22no—
118C32×C4—42no—
119C32×C4—44no—
120C32×C4—42no—
121C32×C4—42no—
122C32×C4#23912yes—
123C32×C4C3×#4112yes—
124#127—91no—
125#125(C3⋊D3)⋊C311yescharacteristic, complemented
126#102—94no—
127#97—91no—
128H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
129C23×C3—182no—
130C22×D3—62no—
131C22×D3—182no—
132C3×D4—184no—
133C3×D4—364no—
134C2×C3×C4—22no—
135C2×C3×C4—32no—
136C2×C3×C4—32no—
137C2×C3×C4—31no—
138C2×C3×C4—31no—
139C2×C3×C4—62no—
140C2×C3×C4—62no—
141C2×C3×C4—64no—
142C2×C3×C4—62no—
143C2×C3×C4—61no—
144C2×C3×C4C3×#6712yes—
145C2×C3×C4C22×C3×D311yescharacteristic
146D12—64no—
147D12—124no—
148D12—184no—
149D12—364no—
150C3×C8—182no—
151C3⋊C8—62no—
152C3⋊C8—182no—
153C2×C32—21no—
154C2×C32—22no—
155C2×C32#94711yescharacteristic
156C2×C32#94912yes—
157C2×C32C3×#12411yescharacteristic
158C2×C32C3×AGL(1,ℤ/8)12yes—
159C3⋊D3—82no—
160C3×D3—242no—
161C2×D4—92no—
162C2×D4—182no—
163C2×C8—92no—
164C4⋊C4—22no—
165C4⋊C4C2×(C3⋊D3)⋊C312yes—
166C42C2×(C3⋊D3)⋊C311yescharacteristic
167C22×C3—31no—
168C22×C3—31no—
169C22×C3—61no—
170C22×C3—362no—
171C22×C3—364no—
172C22×C3C2×C3×#6711yescharacteristic
173C2×D3—122no—
174C2×D3—124no—
175C2×D3—362no—
176C2×D3—364no—
177C3×C4—22no—
178C3×C4—32no—
179C3×C4—32no—
180C3×C4—42no—
181C3×C4—62no—
182C3×C4—62no—
183C3×C4—62no—
184C3×C4—122no—
185C3×C4—122no—
186C3×C4—122no—
187C3×C4—124no—
188C3×C4C3×#23912yes—
189C32—21no—
190C32#506711yescharacteristic, complemented
191C32C3×#49011yescharacteristic, complemented
192C23—182no—
193D4—184no—
194D4—364no—
195C2×C4—22no—
196C2×C4#638⋊C312yes—
197C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
198C8—182no—
199C2×C3—31no—
200C2×C3—31no—
201C2×C3—32no—
202C2×C3—32no—
203C2×C3—61no—
204C2×C3—62no—
205C2×C3—722no—
206C2×C3C3×#94711yescharacteristic
207C2×C3C3×#94912yes—
208D3—242no—
209D3—722no—
210C22—362no—
211C22—364no—
212C22C2×#638⋊C311yescharacteristic
213C4—22no—
214C4—42no—
215C4#8872212yes—
216C3—31no—
217C3—31no—
218C3—61no—
219C3C3×#506711yescharacteristic
220C2—722no—
221C2#64156⋊C311yescharacteristic
222C2#64158⋊C312yes—
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#125C2
socleC22×C3C2×C3×#67
prime radical#11721385C2
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#490
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
3. #63636⋊C3C2semidirect
4. #63630⋊C3C2semidirect
5. #10592766C3semidirect
14. C32×#125D3semidirect
15. C32×#125C2×C3semidirect
18. #63636C2×C3semidirect
19. #63630C2×C3semidirect
125. #125(C3⋊D3)⋊C3semidirect
128. H(3,𝔽3)#490semidirect
190. C32#5067semidirect
191. C32C3×#490semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117213852
#63636⋊C32
#63630⋊C32
#105927663
C3×#50673