Finite Group Databasebeta

← C3×#10593221 #13165897

Subgroups — C3×#10593221 #13165897

Order 1728.

325 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 50 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×#985C211yesmaximal, characteristic
2C3×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
3C3×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#5405—31nomaximal
5#10593221C311yesmaximal, characteristic, direct product factor
6C3×#5197—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C32×GL(2,𝔽3)C411yescharacteristic, complemented
9C32×#226C411yescharacteristic, complemented
10C3×#3711—41nomaximal
11C3×#985—21no—
12C3×#985D311yescharacteristic
13C3×#985C2×C311yescharacteristic
14C32×#135—31no—
15C3×#948—31no—
16C3×#948D311yescharacteristic, complemented
17C3×#131⋊C3—31no—
18C3⋊(#131⋊C3)—21no—
19C3⋊(#131⋊C3)C2×C311yescharacteristic, complemented
20C3⋊GU(2,𝔽3)C2×C311yescharacteristic, complemented
21C3×GU(2,𝔽3)—31no—
22C3×GU(2,𝔽3)—61no—
23C3×#837—31no—
24C3×#837—31no—
25C3×#819—31no—
26C3×#815—31no—
27C32×C4×D3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic
29C3×C3⋊(C3⋊C8)—41no—
30C32×C3⋊C8—41no—
31#5405—31no—
32C3×#444—91no—
33#5197—31no—
34C32×#41—31no—
35C32×#41C2×D311yescharacteristic
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)—21no—
43C3×GL(2,𝔽3)Dic311yescharacteristic, complemented
44C3×GL(2,𝔽3)C3×C411yescharacteristic, complemented
45C3×#226—21no—
46C3×#226Dic311yescharacteristic, complemented
47C3×#226C3×C411yescharacteristic, complemented
48C32×Q16—31no—
49C32×SD16—31no—
50C32×SD16—31no—
51C32×D8—31no—
52C2×C32×C8—31no—
53C3×C4×Dic3—31no—
54C3×#208—31no—
55C3×#208—31no—
56C3×#208—61no—
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8—61no—
59C3×ΓL(1,𝔽25)—121no—
60#3711—41no—
61C2×C32×D3—41no—
62C33×C4—41no—
63C32×Dic3—41no—
64#985C3×D311yescharacteristic
65C3×#135—31no—
66C3×#135—31no—
67C3×#135—61no—
68C3×#131—31no—
69C3×#131—91no—
70#948—31no—
71#948C3×D311yescharacteristic, complemented
72#131⋊C3—31no—
73#131⋊C3—61no—
74GU(2,𝔽3)—31no—
75GU(2,𝔽3)—61no—
76C3×#108—91no—
77C3×#104—91no—
78C3×#104—91no—
79C3×#97—91no—
80#837—31no—
81#837—31no—
82#819—31no—
83#815—31no—
84C32×Q8—31no—
85C32×Q8C4×D311yescharacteristic
86C32×D4—31no—
87C32×D4—31no—
88C32×D4—61no—
89C2×C32×C4—31no—
90C2×C32×C4—61no—
91C2×C3×Dic3—61no—
92C3×C4×D3—41no—
93C3×C4×D3—41no—
94C3×C4×D3—81no—
95C3×SL(2,𝔽3)—21no—
96C3×SL(2,𝔽3)—21no—
97C3×SL(2,𝔽3)—21no—
98C3×SL(2,𝔽3)—41no—
99C3×SL(2,𝔽3)C2×Dic311yescharacteristic
100C3×SL(2,𝔽3)C2×C3×C411yescharacteristic
101C32×C8—31no—
102C32×C8—31no—
103C32×C8—121no—
104C3⋊(C3⋊C8)—41no—
105C3×C3⋊C8—31no—
106C3×C3⋊C8—41no—
107C3×C3⋊C8—61no—
108C3×C3⋊C8—81no—
109C3×C3⋊C8—121no—
110C3×C3⋊C8—241no—
111C3×C3⋊C8S411yescharacteristic
112#444—91noSylow
113C2×C33—41no—
114C32×D3—81no—
115C3×#41—21no—
116C3×#41—31no—
117C3×#41—31no—
118C3×#41—61no—
119C3×#41D3211yescharacteristic
120C3×#41C2×C3×D311yescharacteristic
121#41⋊C3—21no—
122#41⋊C3—21no—
123#41⋊C3—41no—
124#41⋊C3C2×C3×D311yescharacteristic
125GL(2,𝔽3)C3×Dic311yescharacteristic, complemented
126#226C3×Dic311yescharacteristic, complemented
127C3×Q16—31no—
128C3×Q16—31no—
129C3×Q16—61no—
130C3×SD16—31no—
131C3×SD16—31no—
132C3×SD16—31no—
133C3×SD16—31no—
134C3×SD16—61no—
135C3×SD16—61no—
136C3×D8—31no—
137C3×D8—31no—
138C3×D8—61no—
139C3×M16—91no—
140C3×M16—91no—
141C3×M16—181no—
142C2×C3×C8—31no—
143C2×C3×C8—31no—
144C2×C3×C8—61no—
145C2×C3×C8—91no—
146C2×C3×C8—181no—
147C3×C42—91no—
148C4×Dic3—31no—
149#208—31no—
150#208—31no—
151#208—61no—
152C2×C3⋊C8—31no—
153C2×C3⋊C8—61no—
154ΓL(1,𝔽25)—121no—
155C22×C32—31no—
156C22×C32—61no—
157C2×C3×D3—41no—
158C2×C3×D3—41no—
159C2×C3×D3—81no—
160C32×C4—31no—
161C32×C4—41no—
162C32×C4—41no—
163C32×C4—61no—
164C32×C4—81no—
165C32×C4—81no—
166C32×C4—81no—
167C32×C4—161no—
168C32×C4C2×S411yescharacteristic
169C3×Dic3—41no—
170C3×Dic3—41no—
171C3×Dic3—81no—
172C3×Dic3—121no—
173#135—31no—
174#131—31no—
175#131—91no—
176#108—91no—
177#104—91no—
178#104—91no—
179#97—91no—
180C33—41noSylow
181C3×Q8—21no—
182C3×Q8—31no—
183C3×Q8—31no—
184C3×Q8—61no—
185C3×Q8D3×Dic311yescharacteristic
186C3×Q8C3×C4×D311yescharacteristic
187C3×D4—31no—
188C3×D4—31no—
189C3×D4—31no—
190C3×D4—31no—
191C3×D4—61no—
192C3×D4—61no—
193C3×D4—61no—
194C3×D4—61no—
195C3×D4—121no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—61no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—121no—
202C2×C3×C4—181no—
203C2×Dic3—61no—
204C4×D3—41no—
205SL(2,𝔽3)—21no—
206SL(2,𝔽3)—21no—
207SL(2,𝔽3)—41no—
208SL(2,𝔽3)C2×C3×Dic311yescharacteristic
209C3×C8—31no—
210C3×C8—31no—
211C3×C8—31no—
212C3×C8—31no—
213C3×C8—31no—
214C3×C8—61no—
215C3×C8—61no—
216C3×C8—91no—
217C3×C8—121no—
218C3×C8—181no—
219C3×C8—241no—
220C3⋊C8—31no—
221C3⋊C8—61no—
222C3⋊C8—121no—
223C3⋊C8—241no—
224C3⋊C8C3×S411yescharacteristic
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—61no—
228C2×C32—81no—
229C2×C32—81no—
230C2×C32—81no—
231C2×C32—121no—
232C2×C32—161no—
233C2×C32C4×S411yescharacteristic
234C3×D3—81no—
235C3×D3—81no—
236C3×D3—82no—
237#41—31no—
238#41C3×D3211yescharacteristic
239Q16—31no—
240SD16—31no—
241SD16—31no—
242D8—31no—
243M16—91no—
244M16—91no—
245M16—181no—
246C2×C8—31no—
247C2×C8—91no—
248C2×C8—181no—
249C42—91no—
250C22×C3—31no—
251C22×C3—31no—
252C22×C3—61no—
253C22×C3—61no—
254C22×C3—61no—
255C22×C3—121no—
256C2×D3—41no—
257C3×C4—21no—
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—41no—
261C3×C4—61no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—81no—
265C3×C4—81no—
266C3×C4—121no—
267C3×C4—161no—
268C3×C4—361no—
269C3×C4D3×S411yescharacteristic
270C3×C4C2×C3×S411yescharacteristic
271Dic3—41no—
272Dic3—121no—
273C32—41no—
274C32—41no—
275C32—81no—
276C32—81no—
277C32—81no—
278C32—161no—
279C32#540511yescharacteristic, complemented
280Q8—31no—
281Q8C3×D3×Dic311yescharacteristic
282D4—31no—
283D4—31no—
284D4—61no—
285C2×C4—31no—
286C2×C4—61no—
287C2×C4—181no—
288C8—31no—
289C8—31no—
290C8—31no—
291C8—91no—
292C8—181no—
293C2×C3—21no—
294C2×C3—41no—
295C2×C3—61no—
296C2×C3—61no—
297C2×C3—81no—
298C2×C3—81no—
299C2×C3—121no—
300C2×C3—121no—
301C2×C3—121no—
302C2×C3—161no—
303C2×C3—241no—
304C2×C3Dic3×S411yescharacteristic
305C2×C3C3×C4×S411yescharacteristic
306D3—81no—
307C22—31no—
308C22—61no—
309C4—31no—
310C4—61no—
311C4—361no—
312C4C3×D3×S411yescharacteristic
313C3—21no—
314C3—41no—
315C3—81no—
316C3—81no—
317C3—161no—
318C3#1059322111yescharacteristic, direct product factor
319C3C3×#540511yescharacteristic, complemented
320C2—61no—
321C2—121no—
322C2C3×Dic3×S411yescharacteristic
subgroupG/H
center, hypercenterC3×C4D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
FrattiniC4C3×D3×S4
FittingC32×#41C2×D3
socleC2×C32C4×S4
prime radicalC32×#985C2
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. C3×C3⋊GU(2,𝔽3)C2semidirect
5. #10593221C3direct (N × H)
8. C32×GL(2,𝔽3)C4semidirect
9. C32×#226C4semidirect
16. C3×#948D3semidirect
19. C3⋊(#131⋊C3)C2×C3semidirect
20. C3⋊GU(2,𝔽3)C2×C3semidirect
43. C3×GL(2,𝔽3)Dic3semidirect
44. C3×GL(2,𝔽3)C3×C4semidirect
46. C3×#226Dic3semidirect
47. C3×#226C3×C4semidirect
71. #948C3×D3semidirect
125. GL(2,𝔽3)C3×Dic3semidirect
126. #226C3×Dic3semidirect
279. C32#5405semidirect
318. C3#10593221direct (N × H)
319. C3C3×#5405semidirect

Maximal subgroups

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

subgroupindex
C32×#9852
C3×C3⋊(#131⋊C3)2
C3×C3⋊GU(2,𝔽3)2
C3×#54053
#105932213
C3×#51973
C3×#37114