Finite Group Databasebeta

← C3×#10593224 #13165900

Subgroups — C3×#10593224 #13165900

Order 1728.

334 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C3×#64267C211yesmaximal, characteristic
2C32×#984C211yesmaximal, characteristic, complemented
3C3×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4C3×#5406—31nomaximal
5#10593224C311yesmaximal, characteristic, direct product factor
6C3×#5203—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3×C3⋊GL(2,𝔽3)—21no—
9C32×#226—21no—
10C3×#3717—41nomaximal
11#64267C2×C311yescharacteristic
12C3×#986—31no—
13C3×#986—61no—
14C3×#984—21no—
15C3×#984D311yescharacteristic, complemented
16C3×#984C2×C311yescharacteristic, complemented
17C32×#137—31no—
18C3×#949—31no—
19C3×#948D311yescharacteristic, complemented
20C2×C3×#215—31no—
21C3×#131⋊C3—31no—
22C3⋊(#131⋊C3)—21no—
23C3⋊(#131⋊C3)C2×C311yescharacteristic, complemented
24C3×#836—31no—
25C3×#823—31no—
26C3×#822—31no—
27C3×C4⋊(C3⋊D3)—41no—
28C32×Dic6—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30C32×C3⋊C8—41no—
31#5406—31no—
32C3×#471—91no—
33#5203—31no—
34C32×#41C2×D311yescharacteristic
35C2×C32×Q8—31no—
36C2×C3×D12—31no—
37C3×#41⋊C3—21no—
38C3×#41⋊C3—21no—
39C3×#41⋊C3—21no—
40C3×#41⋊C3—41no—
41C3×#41⋊C3C2×D311yescharacteristic
42C3×#41⋊C3C22×C311yescharacteristic
43C3⋊GL(2,𝔽3)—21no—
44C3×GL(2,𝔽3)—62no—
45C3×GL(2,𝔽3)—61no—
46C3×#226—22no—
47C3×#226—21no—
48C3×#226—21no—
49C32×Q16—61no—
50C32×SD16—61no—
51C32×M16—31no—
52C3×#215—61no—
53C3×#215—61no—
54C3×#215—61no—
55C3×#213—61no—
56C3×#208—31no—
57C3×#208—31no—
58C2×C3×C3⋊C8—31no—
59C3×#204—121no—
60#3717—41no—
61C2×C3×C3⋊D3—81no—
62C33×C4—41no—
63C32×Dic3—81no—
64#986—31no—
65#986—61no—
66#984C3×D311yescharacteristic, complemented
67C3×#137—31no—
68C3×#137—31no—
69C3×#137—61no—
70C3×AGL(1,ℤ/8)—91no—
71C2×C3×SD16—91no—
72C3×#131—31no—
73#949—31no—
74#948C3×D311yescharacteristic, complemented
75C2×#215—31no—
76#131⋊C3—31no—
77#131⋊C3—61no—
78C3×#108—91no—
79C3×#101—91no—
80C3×#100—91no—
81#836—31no—
82#823—31no—
83#822—31no—
84C22×C3×D3—61no—
85C32×Q8—61no—
86C32×Q8—61no—
87C32×Q8D1211yescharacteristic
88C32×D4—31no—
89C2×C32×C4—31no—
90C2×C32×C4—61no—
91C4⋊(C3⋊D3)—41no—
92C3×D12—61no—
93C3×D12—61no—
94C3×D12—121no—
95C3×D12—241no—
96C3×Dic6—41no—
97C3×Dic6—41no—
98C3×Dic6—81no—
99C3×SL(2,𝔽3)—21no—
100C3×SL(2,𝔽3)—21no—
101C3×SL(2,𝔽3)—21no—
102C3×SL(2,𝔽3)—41no—
103C3×SL(2,𝔽3)#6711yescharacteristic
104C3×SL(2,𝔽3)C3×D411yescharacteristic
105C32×C8—61no—
106C32×C8—121no—
107C3×C3⋊C8—31no—
108C3×C3⋊C8—41no—
109C3×C3⋊C8—61no—
110C3×C3⋊C8—81no—
111C3×C3⋊C8S411yescharacteristic
112#471—91noSylow
113C2×C33—41no—
114C3×C3⋊D3—161no—
115C3×#41—21no—
116C3×#41D3211yescharacteristic
117C3×#41C2×C3×D311yescharacteristic
118C2×C3×Q8—31no—
119C2×C3×Q8—31no—
120C2×C3×Q8—61no—
121C2×C3×D4—91no—
122C2×D12—31no—
123#41⋊C3—21no—
124#41⋊C3—21no—
125#41⋊C3—41no—
126#41⋊C3C2×C3×D311yescharacteristic
127GL(2,𝔽3)—62no—
128GL(2,𝔽3)—61no—
129#226—21no—
130C3×Q16—62no—
131C3×Q16—61no—
132C3×Q16—61no—
133C3×SD16—61no—
134C3×SD16—61no—
135C3×SD16—62no—
136C3×SD16—181no—
137C3×SD16—181no—
138C3×SD16—181no—
139C3×D8—181no—
140C3×M16—31no—
141C3×M16—31no—
142C3×M16—61no—
143C3×M16—91no—
144C3×M16—91no—
145C2×C3×C8—91no—
146#215—61no—
147#215—61no—
148#215—61no—
149#213—61no—
150#208—31no—
151#208—31no—
152C2×C3⋊C8—31no—
153#204—121no—
154C22×C32—31no—
155C2×C3⋊D3—81no—
156C2×C3×D3—121no—
157C2×C3×D3—241no—
158C2×C3×D3—242no—
159C2×C3×D3—241no—
160C32×C4—31no—
161C32×C4—41no—
162C32×C4—41no—
163C32×C4—81no—
164C32×C4—81no—
165C32×C4—81no—
166C32×C4—121no—
167C32×C4—161no—
168C32×C4C2×S411yescharacteristic
169C3×Dic3—81no—
170C3×Dic3—81no—
171C3×Dic3—82no—
172#137—31no—
173AGL(1,ℤ/8)—91no—
174C2×SD16—91no—
175#131—31no—
176#108—91no—
177#101—91no—
178#100—91no—
179C33—41noSylow
180C23×C3—181no—
181C22×D3—61no—
182C3×Q8—21no—
183C3×Q8—61no—
184C3×Q8—61no—
185C3×Q8—62no—
186C3×Q8—61no—
187C3×Q8—61no—
188C3×Q8—121no—
189C3×Q8#62611yescharacteristic
190C3×Q8C3×D1211yescharacteristic
191C3×D4—31no—
192C3×D4—31no—
193C3×D4—61no—
194C3×D4—181no—
195C3×D4—181no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—61no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—121no—
202D12—61no—
203D12—61no—
204D12—121no—
205D12—241no—
206Dic6—41no—
207SL(2,𝔽3)—21no—
208SL(2,𝔽3)—21no—
209SL(2,𝔽3)—41no—
210SL(2,𝔽3)C3×#6711yescharacteristic
211C3×C8—31no—
212C3×C8—61no—
213C3×C8—61no—
214C3×C8—62no—
215C3×C8—91no—
216C3×C8—121no—
217C3×C8—181no—
218C3×C8—241no—
219C3⋊C8—31no—
220C3⋊C8—61no—
221C3⋊C8C3×S411yescharacteristic
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—61no—
225C2×C32—81no—
226C2×C32—81no—
227C2×C32—81no—
228C2×C32—161no—
229C2×C32C4⋊S411yescharacteristic
230C3⋊D3—161no—
231C3×D3—241no—
232C3×D3—481no—
233C3×D3—482no—
234#41C3×D3211yescharacteristic
235C2×Q8—31no—
236C2×D4—91no—
237Q16—61no—
238SD16—61no—
239SD16—181no—
240SD16—181no—
241SD16—181no—
242D8—181no—
243M16—31no—
244M16—91no—
245M16—91no—
246C2×C8—91no—
247C22×C3—31no—
248C22×C3—31no—
249C22×C3—61no—
250C22×C3—361no—
251C22×C3—721no—
252C2×D3—121no—
253C2×D3—241no—
254C2×D3—242no—
255C2×D3—241no—
256C3×C4—21no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—41no—
260C3×C4—61no—
261C3×C4—81no—
262C3×C4—81no—
263C3×C4—122no—
264C3×C4—121no—
265C3×C4—121no—
266C3×C4—161no—
267C3×C4D3×S411yescharacteristic
268C3×C4C2×C3×S411yescharacteristic
269Dic3—81no—
270C32—41no—
271C32—41no—
272C32—81no—
273C32—81no—
274C32—81no—
275C32—161no—
276C32#540611yescharacteristic, complemented
277C23—181no—
278Q8—61no—
279Q8—61no—
280Q8C3×#62611yescharacteristic
281D4—31no—
282D4—181no—
283D4—181no—
284C2×C4—31no—
285C2×C4—61no—
286C8—31no—
287C8—61no—
288C8—91no—
289C8—181no—
290C2×C3—21no—
291C2×C3—41no—
292C2×C3—61no—
293C2×C3—61no—
294C2×C3—81no—
295C2×C3—81no—
296C2×C3—121no—
297C2×C3—161no—
298C2×C3—721no—
299C2×C3#6420711yescharacteristic
300C2×C3C3×C4⋊S411yescharacteristic
301D3—241no—
302D3—481no—
303D3—482no—
304C22—31no—
305C22—361no—
306C22—721no—
307C4—31no—
308C4—121no—
309C4C3×D3×S411yescharacteristic
310C3—21no—
311C3—41no—
312C3—81no—
313C3—81no—
314C3—161no—
315C3#1059322411yescharacteristic, direct product factor
316C3C3×#540611yescharacteristic, complemented
317C2—61no—
318C2—721no—
319C2C3×#6420711yescharacteristic
subgroupG/H
centerC2×C3#64207
commutator, cosocleC3×#41⋊C3C22×C3
FrattiniC4C3×D3×S4
FittingC32×#41C2×D3
socleC2×C32C4⋊S4
prime radicalC3×#64267C2
hypercenterC3×C4D3×S4
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
2. C32×#984C2semidirect
3. C3×C3⋊(#131⋊C3)C2semidirect
5. #10593224C3direct (N × H)
15. C3×#984D3semidirect
16. C3×#984C2×C3semidirect
19. C3×#948D3semidirect
23. C3⋊(#131⋊C3)C2×C3semidirect
66. #984C3×D3semidirect
74. #948C3×D3semidirect
276. C32#5406semidirect
315. C3#10593224direct (N × H)
316. C3C3×#5406semidirect

Maximal subgroups

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

subgroupindex
C3×#642672
C32×#9842
C3×C3⋊(#131⋊C3)2
C3×#54063
#105932243
C3×#52033
C3×#37174