Finite Group Databasebeta

← #13173071

Subgroups — #13173071

Order 1728.

1130 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4#10593655—312nomaximal
5#10592926—31nomaximal
6C2×#6312—41nomaximal
7C2×C32×SL(2,𝔽3)—21no—
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C3⋊(C3⋊GL(2,𝔽3))—21no—
10C22×C32×Q8D311yescharacteristic, complemented
11C22×C3×SL(2,𝔽3)D3112yescomplemented
12C2×C3⋊GL(2,𝔽3)—312no—
13C2×#3755—31no—
14C2×#3755—31no—
15#64139—31no—
16C3⋊#859—312no—
17#63662—31no—
18#63662—31no—
19#63657—31no—
20C23×C33—41no—
21C22×C3⋊(C3⋊D3)—41no—
22#6312—44no—
23C2×C3⋊(C3⋊Dic3)—41no—
24C32×SL(2,𝔽3)D411yescharacteristic, complemented
25#5421—99no—
26#5227—94no—
27C2×C32×Q8—21no—
28C2×C32×Q8—31no—
29C2×C32×Q8—61no—
30C2×C32×Q8C2×D311yescharacteristic
31C22×C32×C4—31no—
32C2×#638—61no—
33C2×#638—1212no—
34C2×C4⋊(C3⋊D3)—31no—
35C2×C3×SL(2,𝔽3)—212no—
36C2×C3×SL(2,𝔽3)C2×D3112yes—
37C3⋊GL(2,𝔽3)—612no—
38#3755—61no—
39#3755—62no—
40#3755—61no—
41#3752—61no—
42C4⋊(C3⋊Dic3)—31no—
43C2×C3⋊(C3⋊C8)—31no—
44C2×C3⋊(C3⋊C8)—31no—
45C22×C33—41no—
46C22×C33—42no—
47C22×C33—81no—
48C22×C33—81no—
49C2×C3⋊(C3⋊D3)—42no—
50C2×C3⋊(C3⋊D3)—81no—
51C2×C3⋊(C3⋊D3)—81no—
52C3⋊(C3⋊Dic3)—42no—
53C22×C3×Q8C3⋊D314yescomplemented
54C22×SL(2,𝔽3)C3⋊D319yescomplemented
55C2×GL(2,𝔽3)—99no—
56C2×#215—94no—
57C2×#215—94no—
58#930—94no—
59#859—99no—
60#835—94no—
61#835—94no—
62#830—94no—
63C23×C32—412no—
64C23×C32S411yescharacteristic
65C22×C3⋊D3—61no—
66C22×C3⋊D3—1212no—
67C32×Q8—31no—
68C32×Q8—32no—
69C32×Q8—61no—
70C32×Q8#6711yescharacteristic, complemented
71C2×C32×C4—31no—
72C2×C32×C4—31no—
73C2×C32×C4—61no—
74C2×C32×C4—61no—
75#638—1248no—
76#638—122no—
77C2×C3⋊Dic3—61no—
78C2×C3⋊Dic3—1212no—
79C4⋊(C3⋊D3)—61no—
80C4⋊(C3⋊D3)—61no—
81C3×SL(2,𝔽3)#67112yescomplemented
82C3⋊(C3⋊C8)—61no—
83C3⋊(C3⋊C8)—61no—
84#448—271noSylow
85C2×C33—41no—
86C2×C33—41no—
87C2×C33—41no—
88C2×C33—82no—
89C3⋊(C3⋊D3)—82no—
90C2×C3×Q8—24no—
91C2×C3×Q8—34no—
92C2×C3×Q8—64no—
93C2×C3×Q8C2×C3⋊D314yes—
94C22×C3×C4—34no—
95C2×#67—184no—
96C2×#67—369no—
97C2×D12—94no—
98C2×SL(2,𝔽3)—29no—
99C2×SL(2,𝔽3)C2×C3⋊D319yes—
100GL(2,𝔽3)—189no—
101#215—184no—
102#215—188no—
103#215—184no—
104(C2×D3)⋊C4—184no—
105C4⋊Dic3—94no—
106C2×C3⋊C8—94no—
107C2×C3⋊C8—94no—
108C22×C32—21no—
109C22×C32—21no—
110C22×C32—412no—
111C22×C32—424no—
112C22×C32—812no—
113C22×C32—812no—
114C22×C32GL(2,𝔽3)12yescomplemented
115C22×C32C2×S411yescharacteristic
116C2×C3⋊D3—1224no—
117C2×C3⋊D3—121no—
118C2×C3⋊D3—121no—
119C2×C3⋊D3—121no—
120C2×C3⋊D3—2412no—
121C2×C3⋊D3—2412no—
122C32×C4—31no—
123C32×C4—31no—
124C32×C4—61no—
125C3⋊Dic3—1224no—
126C3⋊Dic3—121no—
127C22×Q8C3⋊(C3⋊D3)11yescharacteristic, complemented
128C2×SD16—271no—
129C2×SD16—271no—
130#121—271no—
131#103—271no—
132#103—271no—
133C22⋊C8—271no—
134C33—41noSylow
135C23×C3—49no—
136C23×C3C3⋊S414yes—
137C22×D3—184no—
138C22×D3—369no—
139C3×Q8—34no—
140C3×Q8—38no—
141C3×Q8—64no—
142C3×Q8#63814yescomplemented
143C2×C3×C4—34no—
144C2×C3×C4—34no—
145C2×C3×C4—64no—
146C2×C3×C4—64no—
147#67—3636no—
148#67—368no—
149C2×Dic3—184no—
150C2×Dic3—369no—
151D12—184no—
152D12—184no—
153SL(2,𝔽3)#63819yescomplemented
154C3⋊C8—184no—
155C3⋊C8—184no—
156C2×C32—22no—
157C2×C32—412no—
158C2×C32—412no—
159C2×C32—412no—
160C2×C32—824no—
161C2×C32C2×GL(2,𝔽3)11yescharacteristic
162C2×C32#98311yescharacteristic
163C2×C32GL(2,ℤ/4)11yescharacteristic
164C3⋊D3—2424no—
165C3⋊D3—241no—
166C2×Q8—21no—
167C2×Q8—31no—
168C2×Q8—61no—
169C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
170C2×D4—271no—
171C2×D4—541no—
172C22×C4—31no—
173SD16—541no—
174SD16—542no—
175SD16—541no—
176C2×C8—271no—
177C2×C8—271no—
178C4⋊C4—271no—
179C22⋊C4—541no—
180C22×C3—24no—
181C22×C3—24no—
182C22×C3—49no—
183C22×C3—418no—
184C22×C3—89no—
185C22×C3—89no—
186C22×C3C3⋊GL(2,𝔽3)18yescomplemented
187C22×C3C2×C3⋊S414yes—
188C2×D3—3618no—
189C2×D3—364no—
190C2×D3—364no—
191C2×D3—364no—
192C2×D3—729no—
193C2×D3—729no—
194C3×C4—34no—
195C3×C4—34no—
196C3×C4—64no—
197Dic3—3618no—
198Dic3—364no—
199C32—412no—
200C32#542111yescharacteristic, complemented
201C23—541no—
202C23C3⋊(C3⋊S4)11yescharacteristic
203Q8—31no—
204Q8—32no—
205Q8—61no—
206Q8#631211yescharacteristic, complemented
207D4—541no—
208D4—541no—
209D4—1082no—
210C2×C4—31no—
211C2×C4—31no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—541no—
215C8—541no—
216C8—541no—
217C2×C3—28no—
218C2×C3—49no—
219C2×C3—49no—
220C2×C3—49no—
221C2×C3—818no—
222C2×C3C2×C3⋊GL(2,𝔽3)14yes—
223C2×C3#6426414yes—
224C2×C3#6426914yes—
225D3—7218no—
226D3—724no—
227C22—21no—
228C22—21no—
229C22—1081no—
230C22—1081no—
231C22—1081no—
232C22C3⋊(C3⋊GL(2,𝔽3))12yescomplemented
233C22C2×C3⋊(C3⋊S4)11yescharacteristic
234C4—31no—
235C4—31no—
236C4—61no—
237C4—1081no—
238C3—49no—
239C3#1059365514yescomplemented
240C2—22no—
241C2—2161no—
242C2C2×C3⋊(C3⋊GL(2,𝔽3))11yescharacteristic
243C2#1172446811yescharacteristic
244C2#1172447311yescharacteristic
subgroupG/H
center, FrattiniC22C2×C3⋊(C3⋊S4)
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC22×C32×Q8D3
socleC22×C32C2×S4
hypercenterC23C3⋊(C3⋊S4)
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C32×SL(2,𝔽3)C2semidirect
2. C2×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
10. C22×C32×Q8D3semidirect
11. C22×C3×SL(2,𝔽3)D3semidirect
24. C32×SL(2,𝔽3)D4semidirect
53. C22×C3×Q8C3⋊D3semidirect
54. C22×SL(2,𝔽3)C3⋊D3semidirect
70. C32×Q8#67semidirect
81. C3×SL(2,𝔽3)#67semidirect
114. C22×C32GL(2,𝔽3)semidirect
127. C22×Q8C3⋊(C3⋊D3)semidirect
142. C3×Q8#638semidirect
153. SL(2,𝔽3)#638semidirect
186. C22×C3C3⋊GL(2,𝔽3)semidirect
200. C32#5421semidirect
206. Q8#6312semidirect
232. C22C3⋊(C3⋊GL(2,𝔽3))semidirect
239. C3#10593655semidirect

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
C2×C3⋊(C3⋊GL(2,𝔽3))2
C3⋊(C3⋊#859)2
#105936553
#105929263
C2×#63124