Finite Group Databasebeta

← C3×#10593225 #13165901

Subgroups — C3×#10593225 #13165901

Order 1728.

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

normal subgroups only

This group has 352 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, complemented
2C32×#986C211yesmaximal, characteristic, complemented
3C3×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4C3×#5407—31nomaximal
5#10593225C311yesmaximal, characteristic, direct product factor
6C3×#5199—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3×C3⋊GL(2,𝔽3)—21no—
9C32×GL(2,𝔽3)—21no—
10C3×#3714—41nomaximal
11#64267C2×C311yescharacteristic, complemented
12C3×#986—21no—
13C3×#986—31no—
14C3×#986—61no—
15C3×#986D311yescharacteristic, complemented
16C3×#986C2×C311yescharacteristic, complemented
17C32×AGL(1,ℤ/8)—31no—
18C3×#949—31no—
19C3×#948D311yescharacteristic, complemented
20C2×C3×#213—31no—
21C3×#131⋊C3—31no—
22C3⋊(#131⋊C3)—21no—
23C3⋊(#131⋊C3)C2×C311yescharacteristic, complemented
24C3×#833—31no—
25C3×#823—31no—
26C3×#822—31no—
27C3×C4⋊(C3⋊D3)—41no—
28C32×D12—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30C32×C3⋊C8—41no—
31#5407—31no—
32C3×#472—91no—
33#5199—31no—
34C32×#41C2×D311yescharacteristic
35C2×C32×D4—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)—22no—
45C3×GL(2,𝔽3)—21no—
46C3×GL(2,𝔽3)—21no—
47C3×GL(2,𝔽3)—62no—
48C3×GL(2,𝔽3)—61no—
49C32×SD16—61no—
50C32×D8—61no—
51C32×M16—31no—
52C3×#215—61no—
53C3×#213—61no—
54C3×#213—61no—
55C3×#213—61no—
56C3×#208—31no—
57C3×#208—31no—
58C2×C3×C3⋊C8—31no—
59C3×D24—121no—
60#3714—41no—
61C2×C3×C3⋊D3—81no—
62C2×C32×D3—81no—
63C33×C4—41no—
64#986—31no—
65#986—61no—
66#986C3×D311yescharacteristic, complemented
67C3×AGL(1,ℤ/8)—31no—
68C3×AGL(1,ℤ/8)—31no—
69C3×AGL(1,ℤ/8)—61no—
70C3×AGL(1,ℤ/8)—91no—
71C2×C3×D8—91no—
72C3×#131—31no—
73#949—31no—
74#948C3×D311yescharacteristic, complemented
75C2×#213—31no—
76#131⋊C3—31no—
77#131⋊C3—61no—
78C3×#108—91no—
79C3×#100—91no—
80C3×#100—91no—
81#833—31no—
82#823—31no—
83#822—31no—
84C23×C32—61no—
85C22×C3×D3—61no—
86C32×Q8D1211yescharacteristic
87C32×D4—31no—
88C32×D4—61no—
89C32×D4—61no—
90C2×C32×C4—31no—
91C4⋊(C3⋊D3)—41no—
92C3×D12—41no—
93C3×D12—41no—
94C3×D12—61no—
95C3×D12—61no—
96C3×D12—81no—
97C3×D12—121no—
98C3×D12—241no—
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#472—91noSylow
113C2×C33—41no—
114C3×C3⋊D3—161no—
115C32×D3—161no—
116C3×#41—21no—
117C3×#41D3211yescharacteristic
118C3×#41C2×C3×D311yescharacteristic
119C2×C3×D4—31no—
120C2×C3×D4—31no—
121C2×C3×D4—61no—
122C2×C3×D4—91no—
123C2×D12—31no—
124#41⋊C3—21no—
125#41⋊C3—21no—
126#41⋊C3—41no—
127#41⋊C3C2×C3×D311yescharacteristic
128GL(2,𝔽3)—21no—
129GL(2,𝔽3)—62no—
130GL(2,𝔽3)—61no—
131C3×SD16—62no—
132C3×SD16—61no—
133C3×SD16—61no—
134C3×SD16—181no—
135C3×D8—61no—
136C3×D8—61no—
137C3×D8—62no—
138C3×D8—181no—
139C3×D8—181no—
140C3×D8—181no—
141C3×M16—31no—
142C3×M16—31no—
143C3×M16—61no—
144C3×M16—91no—
145C3×M16—91no—
146C2×C3×C8—91no—
147#215—61no—
148#213—61no—
149#213—61no—
150#213—61no—
151#208—31no—
152#208—31no—
153C2×C3⋊C8—31no—
154D24—121no—
155C22×C32—31no—
156C22×C32—121no—
157C22×C32—241no—
158C2×C3⋊D3—81no—
159C2×C3×D3—81no—
160C2×C3×D3—81no—
161C2×C3×D3—82no—
162C2×C3×D3—121no—
163C2×C3×D3—241no—
164C2×C3×D3—242no—
165C2×C3×D3—241no—
166C32×C4—31no—
167C32×C4—41no—
168C32×C4—41no—
169C32×C4—81no—
170C32×C4—81no—
171C32×C4—81no—
172C32×C4—161no—
173C32×C4C2×S411yescharacteristic
174AGL(1,ℤ/8)—31no—
175AGL(1,ℤ/8)—91no—
176C2×D8—91no—
177#131—31no—
178#108—91no—
179#100—91no—
180#100—91no—
181C33—41noSylow
182C23×C3—61no—
183C23×C3—61no—
184C23×C3—121no—
185C23×C3—181no—
186C22×D3—61no—
187C3×Q8—21no—
188C3×Q8#62611yescharacteristic
189C3×Q8C3×D1211yescharacteristic
190C3×D4—31no—
191C3×D4—31no—
192C3×D4—61no—
193C3×D4—61no—
194C3×D4—61no—
195C3×D4—62no—
196C3×D4—61no—
197C3×D4—61no—
198C3×D4—121no—
199C3×D4—181no—
200C3×D4—181no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—61no—
204D12—41no—
205D12—61no—
206D12—61no—
207D12—121no—
208D12—241no—
209SL(2,𝔽3)—21no—
210SL(2,𝔽3)—21no—
211SL(2,𝔽3)—41no—
212SL(2,𝔽3)C3×#6711yescharacteristic
213C3×C8—31no—
214C3×C8—61no—
215C3×C8—61no—
216C3×C8—62no—
217C3×C8—91no—
218C3×C8—121no—
219C3×C8—181no—
220C3×C8—241no—
221C3⋊C8—31no—
222C3⋊C8—61no—
223C3⋊C8C3×S411yescharacteristic
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—61no—
227C2×C32—81no—
228C2×C32—81no—
229C2×C32—81no—
230C2×C32—161no—
231C2×C32—241no—
232C2×C32C4⋊S411yescharacteristic
233C3⋊D3—161no—
234C3×D3—161no—
235C3×D3—161no—
236C3×D3—162no—
237C3×D3—241no—
238C3×D3—481no—
239C3×D3—482no—
240#41C3×D3211yescharacteristic
241C2×D4—31no—
242C2×D4—91no—
243SD16—61no—
244SD16—181no—
245D8—61no—
246D8—181no—
247D8—181no—
248D8—181no—
249M16—31no—
250M16—91no—
251M16—91no—
252C2×C8—91no—
253C22×C3—31no—
254C22×C3—31no—
255C22×C3—61no—
256C22×C3—122no—
257C22×C3—121no—
258C22×C3—121no—
259C22×C3—241no—
260C22×C3—241no—
261C22×C3—242no—
262C22×C3—361no—
263C22×C3—721no—
264C2×D3—81no—
265C2×D3—121no—
266C2×D3—241no—
267C2×D3—242no—
268C2×D3—241no—
269C3×C4—21no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—41no—
273C3×C4—61no—
274C3×C4—81no—
275C3×C4—81no—
276C3×C4—161no—
277C3×C4D3×S411yescharacteristic
278C3×C4C2×C3×S411yescharacteristic
279C32—41no—
280C32—41no—
281C32—81no—
282C32—81no—
283C32—81no—
284C32—161no—
285C32#540711yescharacteristic, complemented
286C23—61no—
287C23—181no—
288Q8C3×#62611yescharacteristic
289D4—31no—
290D4—61no—
291D4—61no—
292D4—181no—
293D4—181no—
294C2×C4—31no—
295C8—31no—
296C8—61no—
297C8—91no—
298C8—181no—
299C2×C3—21no—
300C2×C3—41no—
301C2×C3—61no—
302C2×C3—61no—
303C2×C3—81no—
304C2×C3—81no—
305C2×C3—121no—
306C2×C3—161no—
307C2×C3—242no—
308C2×C3—241no—
309C2×C3—241no—
310C2×C3—721no—
311C2×C3#6420711yescharacteristic
312C2×C3C3×C4⋊S411yescharacteristic
313D3—161no—
314D3—241no—
315D3—481no—
316D3—482no—
317C22—31no—
318C22—121no—
319C22—241no—
320C22—361no—
321C22—721no—
322C4—31no—
323C4C3×D3×S411yescharacteristic
324C3—21no—
325C3—41no—
326C3—81no—
327C3—81no—
328C3—161no—
329C3#1059322511yescharacteristic, direct product factor
330C3C3×#540711yescharacteristic, complemented
331C2—61no—
332C2—241no—
333C2—721no—
334C2C3×#6420711yescharacteristic
subgroupG/H
centerC2×C3#64207
commutator, cosocleC3×#41⋊C3C22×C3
FrattiniC4C3×D3×S4
FittingC32×#41C2×D3
socleC2×C32C4⋊S4
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
1. C3×#64267C2semidirect
2. C32×#986C2semidirect
3. C3×C3⋊(#131⋊C3)C2semidirect
5. #10593225C3direct (N × H)
11. #64267C2×C3semidirect
15. C3×#986D3semidirect
16. C3×#986C2×C3semidirect
19. C3×#948D3semidirect
23. C3⋊(#131⋊C3)C2×C3semidirect
66. #986C3×D3semidirect
74. #948C3×D3semidirect
285. C32#5407semidirect
329. C3#10593225direct (N × H)
330. C3C3×#5407semidirect

Maximal subgroups

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

subgroupindex
C3×#642672
C32×#9862
C3×C3⋊(#131⋊C3)2
C3×#54073
#105932253
C3×#51993
C3×#37144