Finite Group Databasebeta

← #13173077

Subgroups — #13173077

Order 1728.

898 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 123 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×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4#10593661—312nomaximal
5#10592937—31nomaximal
6C32×#41⋊C3—21no—
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3⋊(C3⋊GL(2,𝔽3))—21no—
9#88859—41nomaximal
10C2×C32×#41D311yescharacteristic, complemented
11C2×C3×#41⋊C3D3112yescomplemented
12C2×C3⋊GL(2,𝔽3)—312no—
13C2×#3755—31no—
14C2×#3753—31no—
15#64139—31no—
16C3⋊#859—312no—
17#63662—31no—
18#63659—31no—
19#63657—31no—
20C22×C3⋊(C3⋊D3)—41no—
21C2×C33×C4—41no—
22C2×C3⋊(C3⋊Dic3)—41no—
23C32×SL(2,𝔽3)D411yescharacteristic, complemented
24#5427—99no—
25#5238—94no—
26C32×#41—21no—
27C32×#41—61no—
28C2×C32×Q8C2×D311yescharacteristic
29C2×C32×D4—31no—
30C22×C32×C4—31no—
31C2×#638—61no—
32C2×C4⋊(C3⋊D3)—31no—
33C3×#41⋊C3—212no—
34C2×C3×SL(2,𝔽3)C2×D3112yes—
35C3⋊GL(2,𝔽3)—612no—
36#3755—61no—
37#3755—61no—
38#3753—62no—
39#3752—61no—
40#3752—1212no—
41C4⋊(C3⋊Dic3)—31no—
42C2×C3⋊(C3⋊C8)—31no—
43C2×C3⋊(C3⋊C8)—31no—
44C22×C33—41no—
45C2×C3⋊(C3⋊D3)—42no—
46C2×C3⋊(C3⋊D3)—81no—
47C2×C3⋊(C3⋊D3)—81no—
48C33×C4—81no—
49C3⋊(C3⋊Dic3)—81no—
50C2×C3×#41C3⋊D314yescomplemented
51C2×#41⋊C3C3⋊D319yescomplemented
52C2×GL(2,𝔽3)—99no—
53C2×#215—94no—
54C2×#213—94no—
55#930—94no—
56#859—99no—
57#835—94no—
58#832—94no—
59#830—94no—
60C23×C32—31no—
61C22×C3⋊D3—61no—
62C22×C3⋊D3—1212no—
63C32×Q8—31no—
64C32×Q8#6711yescharacteristic
65C32×D4—32no—
66C32×D4—61no—
67C2×C32×C4—31no—
68C2×C32×C4—412no—
69C2×C32×C4—61no—
70C2×C32×C4—61no—
71C2×C32×C4S411yescharacteristic
72#638—122no—
73C2×C3⋊Dic3—61no—
74C2×C3⋊Dic3—1212no—
75C4⋊(C3⋊D3)—61no—
76C4⋊(C3⋊D3)—61no—
77C3×SL(2,𝔽3)#67112yescomplemented
78C3⋊(C3⋊C8)—61no—
79C3⋊(C3⋊C8)—61no—
80#449—271noSylow
81C2×C33—41no—
82C2×C33—41no—
83C2×C33—41no—
84C3⋊(C3⋊D3)—82no—
85C3×#41—24no—
86C3×#41—64no—
87C2×C3×Q8C2×C3⋊D314yes—
88C2×C3×D4—34no—
89C22×C3×C4—34no—
90C2×#67—184no—
91C2×D12—94no—
92#41⋊C3—29no—
93C2×SL(2,𝔽3)C2×C3⋊D319yes—
94GL(2,𝔽3)—189no—
95#215—184no—
96#215—184no—
97#213—188no—
98(C2×D3)⋊C4—184no—
99(C2×D3)⋊C4—369no—
100C4⋊Dic3—94no—
101C2×C3⋊C8—94no—
102C2×C3⋊C8—94no—
103C22×C32—412no—
104C22×C32—61no—
105C22×C32—61no—
106C22×C32—61no—
107C22×C32C2×S411yescharacteristic
108C2×C3⋊D3—1224no—
109C2×C3⋊D3—121no—
110C2×C3⋊D3—121no—
111C2×C3⋊D3—121no—
112C2×C3⋊D3—2412no—
113C2×C3⋊D3—2412no—
114C32×C4—21no—
115C32×C4—31no—
116C32×C4—31no—
117C32×C4—812no—
118C3⋊Dic3—121no—
119C3⋊Dic3—2412no—
120C2×#41C3⋊(C3⋊D3)11yescharacteristic, complemented
121C2×SD16—271no—
122C2×D8—271no—
123#121—271no—
124#103—271no—
125#102—271no—
126C22⋊C8—271no—
127C33—41noSylow
128C23×C3—34no—
129C22×D3—184no—
130C22×D3—369no—
131C3×Q8—34no—
132C3×Q8#63814yes—
133C3×D4—38no—
134C3×D4—64no—
135C2×C3×C4—34no—
136C2×C3×C4—49no—
137C2×C3×C4—64no—
138C2×C3×C4—64no—
139C2×C3×C4C3⋊S414yes—
140#67—368no—
141C2×Dic3—184no—
142C2×Dic3—369no—
143D12—184no—
144D12—184no—
145SL(2,𝔽3)#63819yescomplemented
146C3⋊C8—184no—
147C3⋊C8—184no—
148C2×C32—412no—
149C2×C32—412no—
150C2×C32—412no—
151C2×C32—121no—
152C2×C32#98511yescharacteristic
153C2×C32#98611yescharacteristic
154C2×C32GL(2,ℤ/4)11yescharacteristic
155C3⋊D3—2424no—
156C3⋊D3—241no—
157#41—21no—
158#41—61no—
159C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
160C2×D4—31no—
161C2×D4—271no—
162C2×D4—541no—
163C22×C4—31no—
164SD16—541no—
165SD16—541no—
166D8—542no—
167C2×C8—271no—
168C2×C8—271no—
169C4⋊C4—271no—
170C22⋊C4—541no—
171C22×C3—49no—
172C22×C3—64no—
173C22×C3—64no—
174C22×C3—64no—
175C22×C3C2×C3⋊S414yes—
176C2×D3—3618no—
177C2×D3—364no—
178C2×D3—364no—
179C2×D3—364no—
180C2×D3—729no—
181C2×D3—729no—
182C3×C4—24no—
183C3×C4—34no—
184C3×C4—34no—
185C3×C4—89no—
186Dic3—364no—
187Dic3—729no—
188C32—412no—
189C32#542711yescharacteristic, complemented
190C23—31no—
191C23—541no—
192Q8—31no—
193Q8#631211yescharacteristic
194D4—32no—
195D4—61no—
196D4—541no—
197D4—541no—
198D4—1082no—
199C2×C4—31no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—541no—
203C2×C4C3⋊(C3⋊S4)11yescharacteristic
204C8—541no—
205C8—541no—
206C2×C3—49no—
207C2×C3—49no—
208C2×C3—49no—
209C2×C3—124no—
210C2×C3#6426614yes—
211C2×C3#6426714yes—
212C2×C3#6426914yes—
213D3—7218no—
214D3—724no—
215C22—61no—
216C22—61no—
217C22—61no—
218C22—1081no—
219C22—1081no—
220C22—1081no—
221C22C2×C3⋊(C3⋊S4)11yescharacteristic
222C4—21no—
223C4—31no—
224C4—31no—
225C4—1081no—
226C3—49no—
227C3#1059366114yescomplemented
228C2—121no—
229C2—2161no—
230C2#1172447011yescharacteristic
231C2#1172447111yescharacteristic
232C2#1172447311yescharacteristic
subgroupG/H
center, FrattiniC22C2×C3⋊(C3⋊S4)
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×#41D3
socleC22×C32C2×S4
hypercenterC2×C4C3⋊(C3⋊S4)
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C2×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
10. C2×C32×#41D3semidirect
11. C2×C3×#41⋊C3D3semidirect
23. C32×SL(2,𝔽3)D4semidirect
50. C2×C3×#41C3⋊D3semidirect
51. C2×#41⋊C3C3⋊D3semidirect
77. C3×SL(2,𝔽3)#67semidirect
120. C2×#41C3⋊(C3⋊D3)semidirect
145. SL(2,𝔽3)#638semidirect
189. C32#5427semidirect
227. C3#10593661semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C2×C3⋊(C3⋊GL(2,𝔽3))2
C3⋊(C3⋊#859)2
#105936613
#105929373
#888594