Finite Group Databasebeta

← #10588680⋊C3 #13142705

Subgroups — #10588680⋊C3 #13142705

Order 1728.

326 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 92 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
13−1+2×#117C211yesmaximal, characteristic
2#63486⋊C3C211yesmaximal, characteristic
3#63389⋊C3C211yesmaximal, characteristic
4#63368⋊C3C211yesmaximal, characteristic, complemented
5#63368⋊C3C211yesmaximal, characteristic, complemented
6#63367⋊C3C211yesmaximal, characteristic, complemented
7#63366⋊C3C211yesmaximal, characteristic, complemented
8C3×#4996—31nomaximal
9#10588680C311yesmaximal, characteristic, complemented
10C22×C4×3−1+2C2211yescharacteristic
11C2×3−1+2⋊Q8C2211yescharacteristic
12C4⋊C4×3−1+2C2211yescharacteristic
13C4⋊C4×3−1+2C2211yescharacteristic
14C22⋊C4×3−1+2—21no—
15C42×3−1+2—21no—
16#3676⋊C3—21no—
17C4⋊(3−1+2⋊C4)C2211yescharacteristic
18(3−1+2⋊C4)⋊C4—21no—
19C2×3−1+2⋊C8C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C32×#117D311yescharacteristic
22C3×#924—31no—
23C3×#830—31no—
24C3×#810—31no—
25C3×#810—31no—
26C3×#808—31no—
27C3×#807—31no—
28C9×#117—21no—
29C9×#117C2×C311yescharacteristic
30#63486C2×C311yescharacteristic
31#63389C2×C311yescharacteristic
32#63368C2×C311yescharacteristic, complemented
33#63368C2×C311yescharacteristic, complemented
34#63367C2×C311yescharacteristic, complemented
35#63366C2×C311yescharacteristic, complemented
36C23×3−1+2D411yescharacteristic
37C2×C4×3−1+2—21no—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2—21no—
41C2×C4×3−1+2D411yescharacteristic
42C2×C4×3−1+2C2311yescharacteristic
43C2×3−1+2⋊C4—21no—
44C2×3−1+2⋊C4—21no—
453−1+2⋊Q8—21no—
463−1+2⋊Q8—21no—
473−1+2⋊C8—21no—
483−1+2⋊C8—21no—
49C3×#485—91no—
50#4996—31no—
51C22×C32×C4C2×D311yescharacteristic
52C2×C3×Dic6—31no—
53C32×C4⋊C4C2×D311yescharacteristic
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C22⋊C4—21no—
56C32×C42—21no—
57C3×#217—61no—
58C3×C4⋊Dic3—31no—
59C3×Dic3⋊C4—61no—
60C2×C3×C3⋊C8—31no—
61C2×C3×C3⋊C8—31no—
62C22×C4×C9—21no—
63C22×C4×C9C22×C311yescharacteristic
64C2×Dic18C22×C311yescharacteristic
65C9×C4⋊C4—21no—
66C9×C4⋊C4—21no—
67C9×C4⋊C4C22×C311yescharacteristic
68C9×C4⋊C4C22×C311yescharacteristic
69C9×C22⋊C4—22no—
70C9×C22⋊C4—21no—
71C42×C9—22no—
72C42×C9—21no—
73#3676—21no—
74C4⋊Dic9C22×C311yescharacteristic
75Dic9⋊C4—21no—
76C2×C9⋊C8C22×C311yescharacteristic
77C2×C9⋊C8C22×C311yescharacteristic
78C22×3−1+2—21no—
79C22×3−1+2—21no—
80C22×3−1+2—21no—
81C22×3−1+2C2×D411yescharacteristic
82C4×3−1+2—21no—
83C4×3−1+2—41no—
84C4×3−1+2—41no—
85C4×3−1+2#4111yescharacteristic
86C4×3−1+2#4111yescharacteristic
873−1+2⋊C4—41no—
883−1+2⋊C4—41no—
89C3×C22⋊Q8—91no—
90C3×#117—31no—
91C3×#117C3×D311yescharacteristic
92#924—31no—
93C3×#107—91no—
94C3×#106—91no—
95C3×#103—91no—
96C3×#103—91no—
97C3×C22⋊C8—91no—
98#830—31no—
99#810—31no—
100#810—31no—
101#808—31no—
102#807—31no—
103C23×C32#6711yescharacteristic
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4#6711yescharacteristic
109C2×C32×C4C22×D311yescharacteristic
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C3×Dic6—61no—
113C3×Dic6—61no—
114C23×C9—21no—
115C23×C9C3×D411yescharacteristic
116C3×C3⋊C8—61no—
117C3×C3⋊C8—61no—
118C2×C4×C9—22no—
119C2×C4×C9—22no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—22no—
123C2×C4×C9—22no—
124C2×C4×C9—21no—
125C2×C4×C9—21no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9C3×D411yescharacteristic
129C2×C4×C9C23×C311yescharacteristic
130C2×Dic9—21no—
131C2×Dic9—21no—
132Dic18—21no—
133Dic18—21no—
134C9⋊C8—21no—
135C9⋊C8—21no—
136#485—91noSylow
137C2×3−1+2—41no—
138C2×3−1+2#12311yescharacteristic
139C2×3−1+2#13511yescharacteristic
140C2×3−1+2#13711yescharacteristic
141C2×C3×Q8—91no—
142C22×C3×C4—31no—
143C22×C3×C4C2×C3×D311yescharacteristic
144C2×Dic6—31no—
145C2×C3×C8—91no—
146C2×C3×C8—91no—
147C3×C4⋊C4—31no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—181no—
151C3×C4⋊C4C2×C3×D311yescharacteristic
152C3×C4⋊C4C2×C3×D311yescharacteristic
153C3×C22⋊C4—21no—
154C3×C22⋊C4—61no—
155C3×C22⋊C4—181no—
156C3×C42—21no—
157C3×C42—61no—
158#217—61no—
159C4⋊Dic3—31no—
160Dic3⋊C4—61no—
161C2×C3⋊C8—31no—
162C2×C3⋊C8—31no—
163C22×C32—21no—
164C22×C32—21no—
165C22×C32—21no—
166C22×C32C2×#6711yescharacteristic
167C32×C4—21no—
168C32×C4—41no—
169C32×C4—41no—
170C32×C4#23511yescharacteristic
171C32×C4#23511yescharacteristic
172C3×Dic3—121no—
173C3×Dic3—121no—
174C22×C9—22no—
175C22×C9—21no—
176C22×C9—21no—
177C22×C9—21no—
178C22×C9—21no—
179C22×C9—41no—
180C22×C9—41no—
181C22×C9C2×C3×D411yescharacteristic
182C4×C9—22no—
183C4×C9—21no—
184C4×C9—21no—
185C4×C9—21no—
186C4×C9—42no—
187C4×C9—42no—
188C4×C9—41no—
189C4×C9—41no—
190C4×C9C3×#4111yescharacteristic
191C4×C9C3×#4111yescharacteristic
192Dic9—41no—
193Dic9—41no—
194C22⋊Q8—91no—
195#117AGL(1,ℤ/9)11yescharacteristic
196#107—91no—
197#106—91no—
198#103—91no—
199#103—91no—
200C22⋊C8—91no—
2013−1+2#48511yescharacteristic, complemented, Sylow
202C23×C3—31no—
203C23×C3C3×#6711yescharacteristic
204C3×Q8—181no—
205C3×Q8—181no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—61no—
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4C3×#6711yescharacteristic
219C2×C3×C4C22×C3×D311yescharacteristic
220C2×Dic3—61no—
221C2×Dic3—61no—
222Dic6—61no—
223Dic6—61no—
224C3×C8—181no—
225C3×C8—181no—
226C3⋊C8—61no—
227C3⋊C8—61no—
228C2×C32—41no—
229C2×C32#92911yescharacteristic
230C2×C32#95011yescharacteristic
231C2×C32#95111yescharacteristic
232C2×C9—21no—
233C2×C9—21no—
234C2×C9—21no—
235C2×C9—42no—
236C2×C9—41no—
237C2×C9C3×#12311yescharacteristic
238C2×C9C3×#13511yescharacteristic
239C2×C9C3×#13711yescharacteristic
240C2×Q8—91no—
241C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
242C2×C8—91no—
243C2×C8—91no—
244C4⋊C4—91no—
245C4⋊C4—181no—
246C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
247C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—181no—
250C42—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—31no—
255C22×C3—61no—
256C22×C3—61no—
257C22×C3—61no—
258C22×C3C2×C3×#6711yescharacteristic
259C3×C4—21no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—61no—
265C3×C4—121no—
266C3×C4—121no—
267C3×C4—361no—
268C3×C4—361no—
269C3×C4C3×#23511yescharacteristic
270C3×C4C3×#23511yescharacteristic
271Dic3—121no—
272Dic3—121no—
273C32#499611yescharacteristic
274C9—21no—
275C9C3×#48511yescharacteristic, complemented
276C23#611⋊C311yescharacteristic
277Q8—181no—
278Q8—181no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—181no—
284C2×C4—181no—
285C2×C4#611⋊C311yescharacteristic
286C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
287C8—181no—
288C8—181no—
289C2×C3—31no—
290C2×C3—31no—
291C2×C3—31no—
292C2×C3—41no—
293C2×C3—121no—
294C2×C3C3×#92911yescharacteristic
295C2×C3C3×#95011yescharacteristic
296C2×C3C3×#95111yescharacteristic
297C22—21no—
298C22—21no—
299C22—21no—
300C22C2×#611⋊C311yescharacteristic
301C4—21no—
302C4—41no—
303C4—41no—
304C4—361no—
305C4—361no—
306C4#3697⋊C311yescharacteristic
307C4#3697⋊C311yescharacteristic
308C3—31no—
309C3C3×#499611yescharacteristic
310C2—41no—
311C2#63491⋊C311yescharacteristic
312C2#63511⋊C311yescharacteristic
313C2#63513⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#117C2
socleC22×C3C2×C3×#67
prime radicalC23×C32#67
hypercenter#117AGL(1,ℤ/9)
hypocenterC9C3×#485
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
4. #63368⋊C3C2semidirect
5. #63368⋊C3C2semidirect
6. #63367⋊C3C2semidirect
7. #63366⋊C3C2semidirect
9. #10588680C3semidirect
32. #63368C2×C3semidirect
33. #63368C2×C3semidirect
34. #63367C2×C3semidirect
35. #63366C2×C3semidirect
201. 3−1+2#485semidirect
275. C9C3×#485semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1172
#63486⋊C32
#63389⋊C32
#63368⋊C32
#63367⋊C32
#63366⋊C32
C3×#49963
#105886803