Finite Group Databasebeta

← C3×#10596588 #13181388

Subgroups — C3×#10596588 #13181388

Order 1728.

840 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 106 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×#994C211yesmaximal, characteristic, complemented
2C3×#64267C213yesmaximal, complemented
3C3×#64266C213yesmaximal, complemented
4#10596588C311yesmaximal, characteristic, direct product factor
5C3×#5925—33nomaximal
6C3×#5835—31nomaximal
7C3×#3832—41nomaximal
8C32×#41⋊C3C2213yescomplemented
9C3×C3⋊GL(2,𝔽3)C2213yescomplemented
10C3×C3⋊#226C2211yescharacteristic, complemented
11C32×H(5,𝔽2)D311yescharacteristic, complemented
12C3×#1009—31no—
13C3×#994—21no—
14C3×#994—23no—
15C3×#994D313yescomplemented
16C3×#994C2×C311yescharacteristic, complemented
17#64267C2×C313yescomplemented
18#64266C2×C313yescomplemented
19C3×#986—39no—
20C3×#985—39no—
21C3×#950—33no—
22C3×#949—33no—
23C3×#948—31no—
24C3×#932—33no—
25C2×C3×#213—33no—
26C33×Q8—41no—
27C3×C4⋊(C3⋊D3)—43no—
28C3×C4×C3⋊D3—43no—
29C32×SL(2,𝔽3)C2311yescharacteristic
30#5925—33no—
31C3×#576—91no—
32#5835—31no—
33C32×#41—31no—
34C32×#41C2×D313yes—
35C2×C32×D4—33no—
36#3832—41no—
37C3×#239—61no—
38C3×#239—123no—
39C3×D3×D4—63no—
40C3×#235—33no—
41C2×C3×D12—33no—
42C3×#41⋊C3—23no—
43C3×#41⋊C3—29no—
44C3×#41⋊C3C2×D319yescomplemented
45C3×#41⋊C3C22×C313yescomplemented
46C3⋊GL(2,𝔽3)C22×C313yescomplemented
47C3⋊#226C22×C311yescharacteristic, complemented
48C3×GL(2,𝔽3)—39no—
49C3×#226—33no—
50C3×#216—31no—
51C3×#215—33no—
52C3×#214—33no—
53C3×#213—33no—
54C3×#213—36no—
55C3×#208—33no—
56C2×C3×C3⋊C8—33no—
57C2×C3×C3⋊D3—43no—
58C33×C4—43no—
59C3×C3⋊Dic3—41no—
60C3×H(5,𝔽2)—21no—
61C3×H(5,𝔽2)—91no—
62C3×H(5,𝔽2)C3×D311yescharacteristic, complemented
63C3×H(5,𝔽2)C3⋊D311yescharacteristic, complemented
64#1009—31no—
65#994—23no—
66#994C3×D313yescomplemented
67#986—39no—
68#985—39no—
69C3×AGL(1,ℤ/8)—93no—
70C3×AGL(1,ℤ/8)—93no—
71C3×#135—93no—
72C2×C3×D8—93no—
73C3×#131—91no—
74#950—33no—
75#949—33no—
76#948—31no—
77#932—33no—
78C2×#213—33no—
79C23×C32—61no—
80C22×C3×D3—63no—
81C32×Q8—41no—
82C32×Q8—43no—
83C32×Q8—83no—
84C32×Q8—81no—
85C32×Q8S411yescharacteristic
86C32×Q8C22×D311yescharacteristic
87C32×D4—33no—
88C32×D4—33no—
89C2×C32×C4—33no—
90C4⋊(C3⋊D3)—43no—
91C4×C3⋊D3—43no—
92C3×#67—63no—
93C3×D12—33no—
94C3×D12—63no—
95C3×D12—129no—
96C3×C4×D3—63no—
97C3×C4×D3—129no—
98C3×Dic6—31no—
99C3×SL(2,𝔽3)—21no—
100C3×SL(2,𝔽3)—23no—
101C3×SL(2,𝔽3)C22×D313yes—
102C3×SL(2,𝔽3)C23×C311yescharacteristic
103C3×C3⋊C8—31no—
104C3×C3⋊C8—33no—
105#576—91noSylow
106C2×C33—41no—
107C3×C3⋊D3—83no—
108C3×#41—23no—
109C3×#41—31no—
110C3×#41—31no—
111C3×#41—61no—
112C3×#41—93no—
113C3×#41—181no—
114C3×#41C2×C3×D313yes—
115C3×#41C2×C3⋊D313yes—
116C2×C3×D4—33no—
117C2×C3×D4—33no—
118C2×C3×D4—63no—
119C2×C3×D4—93no—
120C2×C3×D4—183no—
121#239—61no—
122#239—123no—
123D3×D4—63no—
124#235—33no—
125C2×D12—33no—
126#41⋊C3—29no—
127#41⋊C3C2×C3×D319yescomplemented
128GL(2,𝔽3)—39no—
129#226—33no—
130C3×Q16—91no—
131C3×SD16—93no—
132C3×SD16—93no—
133C3×D8—93no—
134C3×D8—96no—
135C3×M16—93no—
136C2×C3×C8—93no—
137#216—31no—
138#215—33no—
139#214—33no—
140#213—33no—
141#213—36no—
142#208—33no—
143C2×C3⋊C8—33no—
144C22×C32—33no—
145C22×C32—241no—
146C2×C3⋊D3—43no—
147C2×C3×D3—63no—
148C2×C3×D3—129no—
149C2×C3×D3—243no—
150C32×C4—31no—
151C32×C4—43no—
152C32×C4—49no—
153C32×C4—89no—
154C32×C4—83no—
155C32×C4C2×S413yes—
156C3⋊Dic3—41no—
157C3×Dic3—61no—
158C3×Dic3—123no—
159H(5,𝔽2)—91no—
160H(5,𝔽2)C3×C3⋊D311yescharacteristic, complemented
161AGL(1,ℤ/8)—93no—
162AGL(1,ℤ/8)—93no—
163#135—93no—
164C2×D8—93no—
165#131—91no—
166C33—41noSylow
167C23×C3—61no—
168C23×C3—61no—
169C23×C3—62no—
170C23×C3—183no—
171C22×D3—63no—
172C3×Q8—21no—
173C3×Q8—21no—
174C3×Q8—43no—
175C3×Q8—83no—
176C3×Q8—91no—
177C3×Q8C3×S411yescharacteristic
178C3×Q8C3⋊S411yescharacteristic
179C3×Q8C22×C3×D311yescharacteristic
180C3×Q8C22×C3⋊D311yescharacteristic
181C3×D4—33no—
182C3×D4—33no—
183C3×D4—33no—
184C3×D4—33no—
185C3×D4—63no—
186C3×D4—63no—
187C3×D4—93no—
188C3×D4—183no—
189C3×D4—183no—
190C2×C3×C4—33no—
191C2×C3×C4—33no—
192C2×C3×C4—63no—
193C2×C3×C4—183no—
194#67—63no—
195D12—33no—
196D12—63no—
197D12—129no—
198C4×D3—63no—
199C4×D3—129no—
200Dic6—31no—
201SL(2,𝔽3)—23no—
202SL(2,𝔽3)C22×C3×D313yes—
203C3×C8—91no—
204C3×C8—93no—
205C3⋊C8—31no—
206C3⋊C8—33no—
207C2×C32—41no—
208C2×C32—43no—
209C2×C32—63no—
210C2×C32—83no—
211C2×C32—81no—
212C2×C32C22×S411yescharacteristic
213C3⋊D3—83no—
214C3×D3—123no—
215C3×D3—249no—
216#41—31no—
217#41—93no—
218#41—181no—
219#41C2×C3×C3⋊D313yes—
220C2×D4—33no—
221C2×D4—93no—
222C2×D4—183no—
223Q16—91no—
224SD16—93no—
225SD16—93no—
226D8—93no—
227D8—96no—
228M16—93no—
229C2×C8—93no—
230C22×C3—33no—
231C22×C3—33no—
232C22×C3—63no—
233C22×C3—183no—
234C22×C3—241no—
235C22×C3—241no—
236C22×C3—242no—
237C22×C3—723no—
238C2×D3—63no—
239C2×D3—129no—
240C2×D3—243no—
241C3×C4—23no—
242C3×C4—31no—
243C3×C4—31no—
244C3×C4—49no—
245C3×C4—61no—
246C3×C4—89no—
247C3×C4—181no—
248C3×C4C2×C3×S413yes—
249C3×C4C2×C3⋊S413yes—
250Dic3—61no—
251Dic3—123no—
252C32—41no—
253C32—43no—
254C32—83no—
255C32—81no—
256C32#592511yescharacteristic, complemented
257C23—61no—
258C23—183no—
259Q8—91no—
260Q8C3×C3⋊S411yescharacteristic
261Q8C22×C3×C3⋊D311yescharacteristic
262D4—33no—
263D4—33no—
264D4—93no—
265D4—183no—
266D4—183no—
267C2×C4—33no—
268C2×C4—183no—
269C8—91no—
270C8—93no—
271C2×C3—21no—
272C2×C3—43no—
273C2×C3—63no—
274C2×C3—63no—
275C2×C3—83no—
276C2×C3—123no—
277C2×C3—363no—
278C2×C3C22×C3×S411yescharacteristic
279C2×C3C22×C3⋊S411yescharacteristic
280D3—123no—
281D3—249no—
282C22—33no—
283C22—183no—
284C22—241no—
285C22—723no—
286C4—31no—
287C4—181no—
288C4C2×C3×C3⋊S413yes—
289C3—21no—
290C3—43no—
291C3—83no—
292C3C3×#592511yescharacteristic, complemented
293C3#1059658811yescharacteristic, direct product factor
294C2—63no—
295C2—363no—
296C2C22×C3×C3⋊S411yescharacteristic
subgroupG/H
centerC2×C3C22×C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2C22×C3×C3⋊S4
FittingC32×H(5,𝔽2)D3
socleC2×C32C22×S4
hypercenterC3×Q8C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
1. C32×#994C2semidirect
2. C3×#64267C2semidirect
3. C3×#64266C2semidirect
4. #10596588C3direct (N × H)
8. C32×#41⋊C3C22semidirect
9. C3×C3⋊GL(2,𝔽3)C22semidirect
10. C3×C3⋊#226C22semidirect
11. C32×H(5,𝔽2)D3semidirect
15. C3×#994D3semidirect
16. C3×#994C2×C3semidirect
17. #64267C2×C3semidirect
18. #64266C2×C3semidirect
44. C3×#41⋊C3C2×D3semidirect
45. C3×#41⋊C3C22×C3semidirect
46. C3⋊GL(2,𝔽3)C22×C3semidirect
47. C3⋊#226C22×C3semidirect
62. C3×H(5,𝔽2)C3×D3semidirect
63. C3×H(5,𝔽2)C3⋊D3semidirect
66. #994C3×D3semidirect
127. #41⋊C3C2×C3×D3semidirect
160. H(5,𝔽2)C3×C3⋊D3semidirect
256. C32#5925semidirect
292. C3C3×#5925semidirect
293. C3#10596588direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#9942
C3×#642672
C3×#642662
#105965883
C3×#59253
C3×#58353
C3×#38324