Finite Group Databasebeta

← #13181410

Subgroups — #13181410

Order 1728.

1600 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1600 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C32×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2#11724471C211yesmaximal, characteristic, complemented
3#11724470C211yesmaximal, characteristic, complemented
4#11724468C212yesmaximal, complemented
5C2×C3⋊(C3⋊GL(2,𝔽3))C212yesmaximal, complemented
6#10596589—312nomaximal
7#10596308—31nomaximal
8D4×C3⋊(C3⋊D3)—41nomaximal
9C32×#41⋊C3C2211yescharacteristic, complemented
10C2×C32×SL(2,𝔽3)C2212yescomplemented
11C3⋊(C3⋊GL(2,𝔽3))C2211yescharacteristic, complemented
12C3⋊(C3⋊GL(2,𝔽3))C2212yescomplemented
13C3⋊(C3⋊#226)C2211yescharacteristic, complemented
14C32×2−1+4D311yescharacteristic, complemented
15#64366—31no—
16C3×2−1+4⋊C3D3112yescomplemented
17#64267—312no—
18#64266—312no—
19#64264—324no—
20C2×C3⋊GL(2,𝔽3)—324no—
21#64159—32no—
22#64159—31no—
23#64158—32no—
24#64158—31no—
25#64157—31no—
26#64151—31no—
27#64151—32no—
28C2×#3755—31no—
29C2×#3755—32no—
30C22×C3⋊(C3⋊D3)—42no—
31C33×D4—41no—
32#6312—42no—
33C4⋊(C3⋊(C3⋊D3))—41no—
34C4×C3⋊(C3⋊D3)—41no—
35C32×SL(2,𝔽3)C2311yescharacteristic
36#5926—99no—
37#5837—94no—
38C32×#41—31no—
39C32×#41—32no—
40C32×#41C2×D311yescharacteristic, complemented
41C2×C32×Q8—31no—
42C2×C32×Q8C2×D312yescomplemented
43#3832—61no—
44D4×C3⋊D3—62no—
45D4×C3⋊D3—61no—
46D4×C3⋊D3—1212no—
47#3828—32no—
48#3828—31no—
49C2×C4⋊(C3⋊D3)—32no—
50C2×C4⋊(C3⋊D3)—31no—
51C3×#41⋊C3C2×D3112yescomplemented
52C2×C3×SL(2,𝔽3)C2×D3124yescomplemented
53C3⋊GL(2,𝔽3)—312no—
54C3⋊GL(2,𝔽3)—324no—
55C3⋊#226—312no—
56#3756—32no—
57#3756—31no—
58#3755—32no—
59#3755—32no—
60#3755—32no—
61#3755—31no—
62#3755—32no—
63#3754—31no—
64#3753—31no—
65#3753—32no—
66#3748—32no—
67#3748—31no—
68C2×C3⋊(C3⋊C8)—32no—
69C2×C3⋊(C3⋊C8)—31no—
70C22×C33—42no—
71C2×C3⋊(C3⋊D3)—41no—
72C2×C3⋊(C3⋊D3)—42no—
73C2×C3⋊(C3⋊D3)—84no—
74C33×C4—41no—
75C3⋊(C3⋊Dic3)—41no—
76C3×2−1+4C3⋊D314yescomplemented
77#1009—94no—
782−1+4⋊C3C3⋊D319yescomplemented
79#986—99no—
80#985—99no—
81#983—918no—
82C2×GL(2,𝔽3)—918no—
83#950—98no—
84#950—94no—
85#949—98no—
86#949—94no—
87#948—94no—
88#942—94no—
89#942—98no—
90C2×#215—94no—
91C2×#215—98no—
92C22×C3⋊D3—61no—
93C22×C3⋊D3—62no—
94C22×C3⋊D3—1224no—
95C32×Q8—32no—
96C32×Q8—31no—
97C32×Q8C22×D311yescharacteristic
98C32×D4—31no—
99C32×D4—32no—
100C32×D4—412no—
101C32×D4S411yescharacteristic
102C2×C32×C4—32no—
103C2×C32×C4—31no—
104C2×C32×C4—61no—
105#638—61no—
106#638—62no—
107#638—1224no—
108C4⋊(C3⋊D3)—32no—
109C4⋊(C3⋊D3)—31no—
110C4⋊(C3⋊D3)—62no—
111C4⋊(C3⋊D3)—61no—
112C4⋊(C3⋊D3)—1212no—
113C4×C3⋊D3—62no—
114C4×C3⋊D3—61no—
115C4×C3⋊D3—1212no—
116#634—31no—
117C3×SL(2,𝔽3)C22×D3112yes—
118C3⋊(C3⋊C8)—31no—
119C3⋊(C3⋊C8)—31no—
120C3⋊(C3⋊C8)—32no—
121#577—271noSylow
122C2×C33—41no—
123C2×C33—82no—
124C3⋊(C3⋊D3)—42no—
125C3⋊(C3⋊D3)—82no—
126C3×#41—34no—
127C3×#41—38no—
128C3×#41C2×C3⋊D314yescomplemented
129C2×C3×Q8—34no—
130C2×C3×Q8C2×C3⋊D318yescomplemented
131#239—184no—
132D3×D4—188no—
133D3×D4—184no—
134D3×D4—369no—
135#235—98no—
136#235—94no—
137C2×D12—98no—
138C2×D12—94no—
139#41⋊C3C2×C3⋊D319yescomplemented
140C2×SL(2,𝔽3)C2×C3⋊D3118yescomplemented
141GL(2,𝔽3)—99no—
142GL(2,𝔽3)—918no—
143#226—99no—
144#216—98no—
145#216—94no—
146#215—98no—
147#215—98no—
148#215—98no—
149#215—94no—
150#215—98no—
151#214—94no—
152#213—94no—
153#213—98no—
154#208—98no—
155#208—94no—
156C2×C3⋊C8—98no—
157C2×C3⋊C8—94no—
158C22×C32—31no—
159C22×C32—424no—
160C22×C32C2×S412yes—
161C2×C3⋊D3—61no—
162C2×C3⋊D3—62no—
163C2×C3⋊D3—1212no—
164C2×C3⋊D3—1224no—
165C2×C3⋊D3—242no—
166C2×C3⋊D3—241no—
167C2×C3⋊D3—2448no—
168C32×C4—31no—
169C32×C4—32no—
170C32×C4—412no—
171C32×C4C2×S411yescharacteristic
172C3⋊Dic3—61no—
173C3⋊Dic3—1212no—
1742−1+4C3⋊(C3⋊D3)11yescharacteristic, complemented
175H(5,𝔽2)—271no—
176#137—271no—
177#137—272no—
178AGL(1,ℤ/8)—272no—
179AGL(1,ℤ/8)—271no—
180#135—272no—
181#135—271no—
182C2×SD16—271no—
183C2×SD16—272no—
184#131—271no—
185C33—41noSylow
186C22×D3—184no—
187C22×D3—188no—
188C22×D3—3618no—
189C3×Q8—38no—
190C3×Q8—34no—
191C3×Q8C22×C3⋊D314yes—
192C3×D4—34no—
193C3×D4—38no—
194C3×D4—49no—
195C3×D4C3⋊S414yes—
196C2×C3×C4—38no—
197C2×C3×C4—34no—
198C2×C3×C4—64no—
199#67—184no—
200#67—188no—
201#67—3618no—
202D12—98no—
203D12—94no—
204D12—188no—
205D12—184no—
206D12—369no—
207C4×D3—188no—
208C4×D3—184no—
209C4×D3—369no—
210Dic6—94no—
211SL(2,𝔽3)C22×C3⋊D319yes—
212C3⋊C8—94no—
213C3⋊C8—94no—
214C3⋊C8—98no—
215C2×C32—22no—
216C2×C32—412no—
217C2×C32—61no—
218C2×C32—824no—
219C2×C32C22×S411yescharacteristic
220C3⋊D3—1224no—
221C3⋊D3—121no—
222C3⋊D3—122no—
223C3⋊D3—2424no—
224#41—31no—
225#41—32no—
226#41—272no—
227#41—271no—
228#41—541no—
229#41C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
230C2×Q8—31no—
231C2×Q8C2×C3⋊(C3⋊D3)12yescomplemented
232C2×D4—272no—
233C2×D4—271no—
234C2×D4—542no—
235C2×D4—541no—
236Q16—272no—
237Q16—271no—
238SD16—271no—
239SD16—272no—
240SD16—272no—
241SD16—272no—
242SD16—271no—
243SD16—272no—
244D8—272no—
245D8—271no—
246M16—272no—
247M16—271no—
248C2×C8—272no—
249C2×C8—271no—
250C22×C3—34no—
251C22×C3—418no—
252C22×C3C2×C3⋊S418yes—
253C2×D3—184no—
254C2×D3—188no—
255C2×D3—369no—
256C2×D3—3618no—
257C2×D3—728no—
258C2×D3—724no—
259C2×D3—7236no—
260C3×C4—34no—
261C3×C4—38no—
262C3×C4—49no—
263C3×C4C2×C3⋊S414yes—
264Dic3—184no—
265Dic3—369no—
266C32—412no—
267C32#592611yescharacteristic, complemented
268C23—541no—
269C23—542no—
270Q8—32no—
271Q8—31no—
272Q8—271no—
273Q8C22×C3⋊(C3⋊D3)11yescharacteristic
274D4—31no—
275D4—32no—
276D4—272no—
277D4—271no—
278D4—542no—
279D4—541no—
280D4—542no—
281D4—541no—
282D4C3⋊(C3⋊S4)11yescharacteristic
283C2×C4—32no—
284C2×C4—31no—
285C2×C4—61no—
286C2×C4—542no—
287C2×C4—541no—
288C8—271no—
289C8—271no—
290C8—272no—
291C2×C3—28no—
292C2×C3—49no—
293C2×C3—64no—
294C2×C3—818no—
295C2×C3C22×C3⋊S414yes—
296D3—3618no—
297D3—364no—
298D3—368no—
299D3—7218no—
300C22—31no—
301C22—541no—
302C22—542no—
303C22—2162no—
304C22—2161no—
305C22C2×C3⋊(C3⋊S4)12yes—
306C4—31no—
307C4—32no—
308C4—541no—
309C4C2×C3⋊(C3⋊S4)11yescharacteristic
310C3—49no—
311C3#1059658914yescomplemented
312C2—22no—
313C2—61no—
314C2—1081no—
315C2—1082no—
316C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
center, FrattiniC2C22×C3⋊(C3⋊S4)
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FittingC32×2−1+4D3
socleC2×C32C22×S4
hypercenterD4C3⋊(C3⋊S4)
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C32×2−1+4⋊C3C2semidirect
2. #11724471C2semidirect
3. #11724470C2semidirect
4. #11724468C2semidirect
5. C2×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
9. C32×#41⋊C3C22semidirect
10. C2×C32×SL(2,𝔽3)C22semidirect
11. C3⋊(C3⋊GL(2,𝔽3))C22semidirect
12. C3⋊(C3⋊GL(2,𝔽3))C22semidirect
13. C3⋊(C3⋊#226)C22semidirect
14. C32×2−1+4D3semidirect
16. C3×2−1+4⋊C3D3semidirect
40. C32×#41C2×D3semidirect
42. C2×C32×Q8C2×D3semidirect
51. C3×#41⋊C3C2×D3semidirect
52. C2×C3×SL(2,𝔽3)C2×D3semidirect
76. C3×2−1+4C3⋊D3semidirect
78. 2−1+4⋊C3C3⋊D3semidirect
128. C3×#41C2×C3⋊D3semidirect
130. C2×C3×Q8C2×C3⋊D3semidirect
139. #41⋊C3C2×C3⋊D3semidirect
140. C2×SL(2,𝔽3)C2×C3⋊D3semidirect
174. 2−1+4C3⋊(C3⋊D3)semidirect
229. #41C2×C3⋊(C3⋊D3)semidirect
231. C2×Q8C2×C3⋊(C3⋊D3)semidirect
267. C32#5926semidirect
311. C3#10596589semidirect

Maximal subgroups

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

subgroupindex
C32×2−1+4⋊C32
#117244712
#117244702
#117244682
C2×C3⋊(C3⋊GL(2,𝔽3))2
#105965893
#105963083
D4×C3⋊(C3⋊D3)4