Finite Group Databasebeta

← C3×#10593220 #13165896

Subgroups — C3×#10593220 #13165896

Order 1728.

330 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 46 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×#64266C211yesmaximal, characteristic
2C3×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
3C32×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#5404—31nomaximal
5#10593220C311yesmaximal, characteristic, direct product factor
6C3×#4824—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3×C3⋊GL(2,𝔽3)C411yescharacteristic, complemented
9C3×C3⋊#226C411yescharacteristic, complemented
10C3×#3710—41nomaximal
11#64266C2×C311yescharacteristic
12C3×#985—31no—
13C3×#985—61no—
14C3×#950—31no—
15C3×#948D311yescharacteristic, complemented
16C3×#907—31no—
17C3×#131⋊C3—31no—
18C3⋊(#131⋊C3)—21no—
19C3⋊(#131⋊C3)C2×C311yescharacteristic, complemented
20C3×GU(2,𝔽3)—21no—
21C3×GU(2,𝔽3)D311yescharacteristic, complemented
22C3×GU(2,𝔽3)C2×C311yescharacteristic, complemented
23C32×#104—31no—
24C3×#822—31no—
25C3×#805—31no—
26C3×C4×C3⋊C8—31no—
27C3×C4×C3⋊D3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic
29C32×C3⋊C8—41no—
30C32×C3⋊C8—41no—
31#5404—31no—
32C3×#443—91no—
33#4824—31no—
34C32×#41C2×D311yescharacteristic
35C3×#235—31no—
36C3×#41⋊C3—21no—
37C3×#41⋊C3—21no—
38C3×#41⋊C3—21no—
39C3×#41⋊C3—41no—
40C3×#41⋊C3C2×D311yescharacteristic
41C3×#41⋊C3C22×C311yescharacteristic
42C3⋊GL(2,𝔽3)C3×C411yescharacteristic, complemented
43C3⋊#226C3×C411yescharacteristic, complemented
44C3×GL(2,𝔽3)—31no—
45C3×GL(2,𝔽3)—61no—
46C3×#226—31no—
47C3×#226—61no—
48C32×M16—31no—
49C32×C42—31no—
50C3×#216—31no—
51C3×#215—31no—
52C3×#214—31no—
53C3×#213—31no—
54C3×#208—31no—
55C2×C3×C3⋊C8—31no—
56C2×C3×C3⋊C8—31no—
57C3×ΓL(1,𝔽25)—61no—
58C3×D3×C8—61no—
59C3×D3×C8—121no—
60#3710—41no—
61C2×C3×C3⋊D3—41no—
62C33×C4—41no—
63C3×C3⋊Dic3—41no—
64#985—31no—
65#985—61no—
66C3×#135—91no—
67C3×#131—31no—
68C3×#131—91no—
69#950—31no—
70#948C3×D311yescharacteristic, complemented
71#907—31no—
72#131⋊C3—31no—
73#131⋊C3—61no—
74GU(2,𝔽3)C3×D311yescharacteristic, complemented
75C3×#108—91no—
76C3×#104—31no—
77C3×#104—31no—
78C3×#104—61no—
79C3×#104—91no—
80C3×C4×C8—91no—
81#822—31no—
82#805—31no—
83C4×C3⋊C8—31no—
84C32×Q8C4×D311yescharacteristic
85C32×D4—31no—
86C2×C32×C4—31no—
87C2×C32×C4—61no—
88C4×C3⋊D3—41no—
89C3×#67—61no—
90C3×D12—31no—
91C3×C4×D3—61no—
92C3×C4×D3—121no—
93C3×C4×D3—241no—
94C3×Dic6—31no—
95C3×SL(2,𝔽3)—21no—
96C3×SL(2,𝔽3)—21no—
97C3×SL(2,𝔽3)—21no—
98C3×SL(2,𝔽3)—41no—
99C3×SL(2,𝔽3)C4×D311yescharacteristic
100C3×SL(2,𝔽3)C2×C3×C411yescharacteristic
101C32×C8—61no—
102C32×C8—121no—
103C3×C3⋊C8—31no—
104C3×C3⋊C8—31no—
105C3×C3⋊C8—31no—
106C3×C3⋊C8—41no—
107C3×C3⋊C8—41no—
108C3×C3⋊C8—41no—
109C3×C3⋊C8—81no—
110C3×C3⋊C8—81no—
111C3×C3⋊C8S411yescharacteristic
112#443—91noSylow
113C2×C33—41no—
114C3×C3⋊D3—42no—
115C3×#41—21no—
116C3×#41—91no—
117C3×#41D3211yescharacteristic
118C3×#41C2×C3×D311yescharacteristic
119#235—31no—
120#41⋊C3—21no—
121#41⋊C3—21no—
122#41⋊C3—41no—
123#41⋊C3C2×C3×D311yescharacteristic
124GL(2,𝔽3)—31no—
125GL(2,𝔽3)—61no—
126#226—31no—
127#226—61no—
128C3×Q16—91no—
129C3×SD16—91no—
130C3×SD16—91no—
131C3×D8—91no—
132C3×M16—31no—
133C3×M16—31no—
134C3×M16—61no—
135C3×M16—91no—
136C3×M16—181no—
137C2×C3×C8—91no—
138C2×C3×C8—91no—
139C2×C3×C8—181no—
140C3×C42—31no—
141C3×C42—31no—
142C3×C42—61no—
143#216—31no—
144#215—31no—
145#214—31no—
146#213—31no—
147#208—31no—
148C2×C3⋊C8—31no—
149C2×C3⋊C8—31no—
150ΓL(1,𝔽25)—61no—
151D3×C8—61no—
152D3×C8—121no—
153C22×C32—31no—
154C2×C3⋊D3—41no—
155C2×C3×D3—61no—
156C2×C3×D3—121no—
157C2×C3×D3—241no—
158C32×C4—31no—
159C32×C4—41no—
160C32×C4—41no—
161C32×C4—62no—
162C32×C4—81no—
163C32×C4—81no—
164C32×C4—81no—
165C32×C4—161no—
166C32×C4C2×S411yescharacteristic
167C3⋊Dic3—41no—
168C3×Dic3—61no—
169C3×Dic3—121no—
170C3×Dic3—241no—
171#135—91no—
172#131—31no—
173#131—91no—
174#108—91no—
175#104—31no—
176#104—91no—
177C4×C8—91no—
178C33—41noSylow
179C3×Q8—21no—
180C3×Q8—91no—
181C3×Q8#62411yescharacteristic
182C3×Q8C3×C4×D311yescharacteristic
183C3×D4—31no—
184C3×D4—31no—
185C3×D4—61no—
186C3×D4—91no—
187C3×D4—181no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—121no—
194C2×C3×C4—181no—
195#67—61no—
196D12—31no—
197C4×D3—61no—
198C4×D3—121no—
199C4×D3—241no—
200Dic6—31no—
201SL(2,𝔽3)—21no—
202SL(2,𝔽3)—21no—
203SL(2,𝔽3)—41no—
204SL(2,𝔽3)C3×C4×D311yescharacteristic
205C3×C8—31no—
206C3×C8—61no—
207C3×C8—61no—
208C3×C8—91no—
209C3×C8—91no—
210C3×C8—91no—
211C3×C8—121no—
212C3×C8—121no—
213C3×C8—241no—
214C3⋊C8—31no—
215C3⋊C8—31no—
216C3⋊C8—31no—
217C3⋊C8—41no—
218C3⋊C8C3×S411yescharacteristic
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—61no—
222C2×C32—81no—
223C2×C32—81no—
224C2×C32—81no—
225C2×C32—161no—
226C2×C32C4×S411yescharacteristic
227C3⋊D3—42no—
228C3×D3—122no—
229C3×D3—121no—
230C3×D3—242no—
231#41—91no—
232#41C3×D3211yescharacteristic
233Q16—91no—
234SD16—91no—
235SD16—91no—
236D8—91no—
237M16—31no—
238M16—91no—
239M16—181no—
240C2×C8—91no—
241C2×C8—91no—
242C2×C8—181no—
243C42—31no—
244C22×C3—31no—
245C22×C3—31no—
246C22×C3—61no—
247C22×C3—181no—
248C2×D3—61no—
249C2×D3—121no—
250C2×D3—241no—
251C3×C4—21no—
252C3×C4—31no—
253C3×C4—31no—
254C3×C4—41no—
255C3×C4—61no—
256C3×C4—62no—
257C3×C4—62no—
258C3×C4—81no—
259C3×C4—81no—
260C3×C4—122no—
261C3×C4—161no—
262C3×C4—181no—
263C3×C4D3×S411yescharacteristic
264C3×C4C2×C3×S411yescharacteristic
265Dic3—61no—
266Dic3—121no—
267Dic3—241no—
268C32—41no—
269C32—41no—
270C32—81no—
271C32—81no—
272C32—81no—
273C32—161no—
274C32#540411yescharacteristic, complemented
275Q8—91no—
276Q8C3×#62411yescharacteristic
277D4—31no—
278D4—91no—
279D4—181no—
280C2×C4—31no—
281C2×C4—61no—
282C2×C4—181no—
283C8—31no—
284C8—61no—
285C8—91no—
286C8—91no—
287C8—91no—
288C2×C3—21no—
289C2×C3—41no—
290C2×C3—61no—
291C2×C3—61no—
292C2×C3—81no—
293C2×C3—81no—
294C2×C3—121no—
295C2×C3—161no—
296C2×C3—361no—
297C2×C3(C3⋊S4)⋊C411yescharacteristic
298C2×C3C3×C4×S411yescharacteristic
299D3—122no—
300D3—121no—
301D3—242no—
302C22—31no—
303C22—181no—
304C4—31no—
305C4—62no—
306C4—181no—
307C4C3×D3×S411yescharacteristic
308C3—21no—
309C3—41no—
310C3—81no—
311C3—81no—
312C3—161no—
313C3#1059322011yescharacteristic, direct product factor
314C3C3×#540411yescharacteristic, complemented
315C2—61no—
316C2—361no—
317C2C3×(C3⋊S4)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC3×C4D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
FrattiniC4C3×D3×S4
FittingC32×#41C2×D3
socleC2×C32C4×S4
prime radicalC3×#64266C2
cosocleC3×#41⋊C3C22×C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊(#131⋊C3)C2semidirect
3. C32×GU(2,𝔽3)C2semidirect
5. #10593220C3direct (N × H)
8. C3×C3⋊GL(2,𝔽3)C4semidirect
9. C3×C3⋊#226C4semidirect
15. C3×#948D3semidirect
19. C3⋊(#131⋊C3)C2×C3semidirect
21. C3×GU(2,𝔽3)D3semidirect
22. C3×GU(2,𝔽3)C2×C3semidirect
42. C3⋊GL(2,𝔽3)C3×C4semidirect
43. C3⋊#226C3×C4semidirect
70. #948C3×D3semidirect
74. GU(2,𝔽3)C3×D3semidirect
274. C32#5404semidirect
313. C3#10593220direct (N × H)
314. C3C3×#5404semidirect

Maximal subgroups

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

subgroupindex
C3×#642662
C3×C3⋊(#131⋊C3)2
C32×GU(2,𝔽3)2
C3×#54043
#105932203
C3×#48243
C3×#37104