Finite Group Databasebeta

← C3×#10596587 #13181387

Subgroups — C3×#10596587 #13181387

Order 1728.

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

normal subgroups only

This group has 812 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×#64266C213yesmaximal, complemented
3C3×#64265C213yesmaximal, complemented
4#10596587C311yesmaximal, characteristic, direct product factor
5C3×#5924—33nomaximal
6C3×#5836—31nomaximal
7C3×Q8×C3⋊D3—41nomaximal
8C32×#41⋊C3C2213yes—
9C3×C3⋊GL(2,𝔽3)C2211yescharacteristic, complemented
10C3×C3⋊#226C2213yescomplemented
11C32×H(5,𝔽2)D311yescharacteristic, complemented
12C3×#1010—31no—
13C3×#994—21no—
14C3×#994—23no—
15C3×#994D313yescomplemented
16C3×#994C2×C311yescharacteristic, complemented
17#64266C2×C313yescomplemented
18#64265C2×C313yescomplemented
19C3×#985—39no—
20C3×#984—39no—
21C3×#951—33no—
22C3×#950—33no—
23C3×#948—31no—
24C2×C3×#214—33no—
25C3×#932—33no—
26C33×Q8—41no—
27C3×C4×C3⋊D3—43no—
28C3×#634—43no—
29C32×SL(2,𝔽3)C2311yescharacteristic
30#5924—33no—
31C3×#577—91no—
32#5836—31no—
33C32×#41—31no—
34C32×#41C2×D313yes—
35C2×C32×D4—33no—
36Q8×C3⋊D3—41no—
37C3×D3×Q8—61no—
38C3×D3×Q8—123no—
39C3×#237—63no—
40C3×#235—33no—
41C2×C3×Dic6—33no—
42C3×#41⋊C3—23no—
43C3×#41⋊C3—29no—
44C3×#41⋊C3C2×D319yes—
45C3×#41⋊C3C22×C313yes—
46C3⋊GL(2,𝔽3)C22×C311yescharacteristic, complemented
47C3⋊#226C22×C313yescomplemented
48C3×GL(2,𝔽3)—33no—
49C3×#226—39no—
50C3×#216—33no—
51C3×#215—31no—
52C3×#214—36no—
53C3×#214—33no—
54C3×#213—33no—
55C3×#208—33no—
56C2×C3×C3⋊C8—33no—
57C2×C3×C3⋊D3—41no—
58C33×C4—43no—
59C3×C3⋊Dic3—43no—
60C3×2−1+4—91no—
61C3×H(5,𝔽2)—21no—
62C3×H(5,𝔽2)C3×D311yescharacteristic, complemented
63C3×H(5,𝔽2)C3⋊D311yescharacteristic, complemented
64#1010—31no—
65#994—23no—
66#994C3×D313yescomplemented
67#985—39no—
68#984—39no—
69C3×#137—93no—
70C3×AGL(1,ℤ/8)—93no—
71C3×#135—93no—
72C2×C3×SD16—93no—
73C3×#131—91no—
74#951—33no—
75#950—33no—
76#948—31no—
77C2×#214—33no—
78#932—33no—
79C23×C32—61no—
80C32×Q8—41no—
81C32×Q8—43no—
82C32×Q8—83no—
83C32×Q8—81no—
84C32×Q8S411yescharacteristic
85C32×Q8C22×D311yescharacteristic
86C32×D4—33no—
87C32×D4—33no—
88C2×C32×C4—33no—
89C4×C3⋊D3—43no—
90#634—43no—
91C3×#67—63no—
92C2×C3×Dic3—63no—
93C3×D12—31no—
94C3×C4×D3—63no—
95C3×C4×D3—129no—
96C3×Dic6—33no—
97C3×Dic6—63no—
98C3×Dic6—129no—
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#577—91noSylow
106C2×C33—41no—
107C3×C3⋊D3—42no—
108C3×#41—23no—
109C3×#41—31no—
110C3×#41—31no—
111C3×#41—61no—
112C3×#41—93no—
113C3×#41—183no—
114C3×#41C2×C3×D313yes—
115C3×#41C2×C3⋊D313yes—
116C2×C3×Q8—93no—
117C2×C3×Q8—181no—
118C2×C3×D4—33no—
119C2×C3×D4—33no—
120C2×C3×D4—63no—
121D3×Q8—61no—
122D3×Q8—123no—
123#237—63no—
124#235—33no—
125C2×Dic6—33no—
126#41⋊C3—29no—
127#41⋊C3C2×C3×D319yes—
128GL(2,𝔽3)—33no—
129#226—39no—
130C3×Q16—93no—
131C3×SD16—93no—
132C3×SD16—96no—
133C3×SD16—91no—
134C3×D8—93no—
135C3×M16—93no—
136C2×C3×C8—93no—
137#216—33no—
138#215—31no—
139#214—36no—
140#214—33no—
141#213—33no—
142#208—33no—
143C2×C3⋊C8—33no—
144C22×C32—33no—
145C22×C32—241no—
146C2×C3⋊D3—41no—
147C2×C3×D3—61no—
148C2×C3×D3—123no—
149C32×C4—31no—
150C32×C4—43no—
151C32×C4—49no—
152C32×C4—89no—
153C32×C4—83no—
154C32×C4C2×S413yes—
155C3⋊Dic3—43no—
156C3×Dic3—63no—
157C3×Dic3—129no—
1582−1+4—91no—
159H(5,𝔽2)C3×C3⋊D311yescharacteristic, complemented
160#137—93no—
161AGL(1,ℤ/8)—93no—
162#135—93no—
163C2×SD16—93no—
164#131—91no—
165C33—41noSylow
166C23×C3—61no—
167C23×C3—61no—
168C23×C3—62no—
169C3×Q8—21no—
170C3×Q8—21no—
171C3×Q8—43no—
172C3×Q8—83no—
173C3×Q8—93no—
174C3×Q8—183no—
175C3×Q8C3×S411yescharacteristic
176C3×Q8C3⋊S411yescharacteristic
177C3×Q8C22×C3×D311yescharacteristic
178C3×Q8C22×C3⋊D311yescharacteristic
179C3×D4—33no—
180C3×D4—33no—
181C3×D4—33no—
182C3×D4—33no—
183C3×D4—63no—
184C3×D4—63no—
185C3×D4—91no—
186C3×D4—183no—
187C2×C3×C4—33no—
188C2×C3×C4—33no—
189C2×C3×C4—63no—
190C2×C3×C4—183no—
191C2×C3×C4—183no—
192#67—63no—
193C2×Dic3—63no—
194D12—31no—
195C4×D3—63no—
196C4×D3—129no—
197Dic6—33no—
198Dic6—63no—
199Dic6—129no—
200SL(2,𝔽3)—23no—
201SL(2,𝔽3)C22×C3×D313yes—
202C3×C8—91no—
203C3×C8—93no—
204C3⋊C8—31no—
205C3⋊C8—33no—
206C2×C32—41no—
207C2×C32—43no—
208C2×C32—63no—
209C2×C32—83no—
210C2×C32—81no—
211C2×C32C22×S411yescharacteristic
212C3⋊D3—42no—
213C3×D3—126no—
214C3×D3—121no—
215#41—31no—
216#41—93no—
217#41—183no—
218#41C2×C3×C3⋊D313yes—
219C2×Q8—93no—
220C2×Q8—181no—
221C2×D4—33no—
222Q16—93no—
223SD16—93no—
224SD16—96no—
225SD16—91no—
226D8—93no—
227M16—93no—
228C2×C8—93no—
229C22×C3—33no—
230C22×C3—33no—
231C22×C3—63no—
232C22×C3—181no—
233C22×C3—241no—
234C22×C3—241no—
235C22×C3—242no—
236C2×D3—61no—
237C2×D3—123no—
238C3×C4—23no—
239C3×C4—31no—
240C3×C4—31no—
241C3×C4—49no—
242C3×C4—61no—
243C3×C4—89no—
244C3×C4—183no—
245C3×C4C2×C3×S413yes—
246C3×C4C2×C3⋊S413yes—
247Dic3—63no—
248Dic3—129no—
249C32—41no—
250C32—43no—
251C32—83no—
252C32—81no—
253C32#592411yescharacteristic, complemented
254C23—61no—
255Q8—93no—
256Q8—183no—
257Q8C3×C3⋊S411yescharacteristic
258Q8C22×C3×C3⋊D311yescharacteristic
259D4—33no—
260D4—33no—
261D4—91no—
262D4—183no—
263C2×C4—33no—
264C2×C4—183no—
265C2×C4—183no—
266C8—91no—
267C8—93no—
268C2×C3—21no—
269C2×C3—43no—
270C2×C3—63no—
271C2×C3—63no—
272C2×C3—83no—
273C2×C3—123no—
274C2×C3—361no—
275C2×C3C22×C3×S411yescharacteristic
276C2×C3C22×C3⋊S411yescharacteristic
277D3—126no—
278D3—121no—
279C22—33no—
280C22—181no—
281C22—241no—
282C4—31no—
283C4—183no—
284C4C2×C3×C3⋊S413yes—
285C3—21no—
286C3—43no—
287C3—83no—
288C3C3×#592411yescharacteristic, complemented
289C3#1059658711yescharacteristic, direct product factor
290C2—63no—
291C2—361no—
292C2C22×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×#64266C2semidirect
3. C3×#64265C2semidirect
4. #10596587C3direct (N × H)
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. #64266C2×C3semidirect
18. #64265C2×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
159. H(5,𝔽2)C3×C3⋊D3semidirect
253. C32#5924semidirect
288. C3C3×#5924semidirect
289. C3#10596587direct (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×#642662
C3×#642652
#105965873
C3×#59243
C3×#58363
C3×Q8×C3⋊D34