Finite Group Databasebeta

← #10588568⋊C3 #13142249

Subgroups — #10588568⋊C3 #13142249

Order 1728.

306 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 86 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
1#63459⋊C3C211yesmaximal, characteristic
2#63454⋊C3C211yesmaximal, characteristic
33−1+2×#107C211yesmaximal, characteristic, complemented
4(C2×AGL(1,ℤ/9))⋊C8C211yesmaximal, characteristic
5#63376⋊C3C211yesmaximal, characteristic, complemented
6#63368⋊C3C211yesmaximal, characteristic, complemented
7#63367⋊C3C211yesmaximal, characteristic, complemented
8C3×#4888—31nomaximal
9#10588568C311yesmaximal, characteristic, complemented
10C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
11C2×3−1+2⋊Q8C2211yescharacteristic
12C2×C8×3−1+2C2211yescharacteristic
13C4⋊C4×3−1+2C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15(C2×AGL(1,ℤ/9))⋊C4—21no—
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17C4⋊(3−1+2⋊C4)—21no—
18C4⋊(3−1+2⋊C4)C2211yescharacteristic
19C4×3−1+2⋊C4—21no—
20C2×3−1+2⋊C8C2211yescharacteristic
21C3×#897—31no—
22C3×#892—31no—
23C32×#107D311yescharacteristic
24C3×(C2×D3)⋊C8—31no—
25C3×#816—31no—
26C3×#810—31no—
27C3×#808—31no—
28#63459C2×C311yescharacteristic
29#63454C2×C311yescharacteristic
30C9×#107—21no—
31C9×#107C2×C311yescharacteristic, complemented
32(C2×D9)⋊C8C2×C311yescharacteristic
33#63376C2×C311yescharacteristic, complemented
34#63368C2×C311yescharacteristic, complemented
35#63367C2×C311yescharacteristic, complemented
36C22×AGL(1,ℤ/9)D411yescharacteristic
37C2×C4×3−1+2—21no—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2C2311yescharacteristic
40C2×3−1+2⋊C4—21no—
41C2×3−1+2⋊C4—21no—
42C2×3−1+2⋊C4D411yescharacteristic
43C4×AGL(1,ℤ/9)—21no—
44C4×AGL(1,ℤ/9)—21no—
453−1+2⋊Q8—21no—
463−1+2⋊Q8—21no—
47C8×3−1+2—21no—
483−1+2⋊C8—21no—
49C3×#485—91no—
50#4888—31no—
51C2×C3×C4×D3—31no—
52C2×C3×Dic6—31no—
53C2×C32×C8C2×D311yescharacteristic
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C3×(C2×D3)⋊C4—61no—
57C3×(C2×D3)⋊C4—61no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—61no—
60C3×C4×Dic3—61no—
61C2×C3×C3⋊C8—31no—
62C2×C4×D9C22×C311yescharacteristic
63C2×Dic18C22×C311yescharacteristic
64C2×C8×C9—21no—
65C2×C8×C9C22×C311yescharacteristic
66C9×C4⋊C4—21no—
67C9×C4⋊C4—21no—
68C9×C4⋊C4C22×C311yescharacteristic
69C9×C4⋊C4C22×C311yescharacteristic
70(C2×D9)⋊C4—21no—
71(C2×D9)⋊C4—21no—
72C4⋊Dic9—21no—
73C4⋊Dic9C22×C311yescharacteristic
74C4×Dic9—21no—
75C2×C9⋊C8C22×C311yescharacteristic
76C22×3−1+2C2×D411yescharacteristic
77C2×AGL(1,ℤ/9)—21no—
78C2×AGL(1,ℤ/9)—21no—
79C2×AGL(1,ℤ/9)—21no—
80C4×3−1+2—41no—
81C4×3−1+2—41no—
82C4×3−1+2#4111yescharacteristic
83C4×3−1+2#4111yescharacteristic
843−1+2⋊C4—21no—
853−1+2⋊C4—41no—
863−1+2⋊C4—41no—
87C3×C22⋊Q8—91no—
88C3×#117—91no—
89#897—31no—
90#892—31no—
91C3×#107—31no—
92C3×#107C3×D311yescharacteristic
93C3×#106—91no—
94C3×#103—91no—
95C3×#103—91no—
96C3×C22⋊C8—91no—
97(C2×D3)⋊C8—31no—
98#816—31no—
99#810—31no—
100#808—31no—
101C22×C3×D3—31no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3×Dic3—31no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C3×C4×D3—61no—
109C3×C4×D3—61no—
110C3×Dic6—61no—
111C3×Dic6—61no—
112C22×D9C3×D411yescharacteristic
113C32×C8—21no—
114C3×C3⋊C8—61no—
115C2×C4×C9—21no—
116C2×C4×C9—21no—
117C2×C4×C9—21no—
118C2×C4×C9—41no—
119C2×C4×C9—41no—
120C2×C4×C9C23×C311yescharacteristic
121C2×Dic9—21no—
122C2×Dic9—21no—
123C2×Dic9C3×D411yescharacteristic
124C4×D9—21no—
125C4×D9—21no—
126Dic18—21no—
127Dic18—21no—
128C8×C9—22no—
129C8×C9—21no—
130C9⋊C8—21no—
131#485—91noSylow
132C2×3−1+2#12311yescharacteristic
133C2×3−1+2#13511yescharacteristic
134C2×3−1+2#13711yescharacteristic
135AGL(1,ℤ/9)—41no—
136C2×C3×Q8—91no—
137C22×C3×C4—91no—
138C2×C4×D3—31no—
139C2×Dic6—31no—
140C2×C3×C8—31no—
141C2×C3×C8—91no—
142C2×C3×C8C2×C3×D311yescharacteristic
143C3×C4⋊C4—31no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—91no—
146C3×C4⋊C4—181no—
147C3×C4⋊C4C2×C3×D311yescharacteristic
148C3×C4⋊C4C2×C3×D311yescharacteristic
149C3×C22⋊C4—181no—
150C3×C22⋊C4—181no—
151C3×C42—181no—
152(C2×D3)⋊C4—61no—
153(C2×D3)⋊C4—61no—
154C4⋊Dic3—31no—
155C4⋊Dic3—61no—
156C4×Dic3—61no—
157C2×C3⋊C8—31no—
158C22×C32D3×D411yescharacteristic
159C2×C3×D3—61no—
160C2×C3×D3—61no—
161C2×C3×D3—61no—
162C32×C4—41no—
163C32×C4—41no—
164C32×C4#23511yescharacteristic
165C32×C4#23911yescharacteristic
166C3×Dic3—61no—
167C3×Dic3—121no—
168C3×Dic3—121no—
169C22×C9—21no—
170C22×C9C2×C3×D411yescharacteristic
171C2×D9—21no—
172C2×D9—21no—
173C2×D9—21no—
174C4×C9—21no—
175C4×C9—21no—
176C4×C9—41no—
177C4×C9—41no—
178C4×C9—81no—
179C4×C9—81no—
180C4×C9C3×#4111yescharacteristic
181C4×C9C3×#4111yescharacteristic
182Dic9—21no—
183Dic9—41no—
184Dic9—41no—
185C22⋊Q8—91no—
186#117—91no—
187#107AGL(1,ℤ/9)11yescharacteristic, complemented
188#106—91no—
189#103—91no—
190#103—91no—
191C22⋊C8—91no—
1923−1+2#48511yescharacteristic, complemented, Sylow
193C23×C3—91no—
194C22×D3—31no—
195C3×Q8—181no—
196C3×Q8—181no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—31no—
200C2×C3×C4—61no—
201C2×C3×C4—61no—
202C2×C3×C4—91no—
203C2×C3×C4—181no—
204C2×C3×C4—181no—
205C2×C3×C4—181no—
206C2×C3×C4—181no—
207C2×C3×C4C22×C3×D311yescharacteristic
208C2×Dic3—31no—
209C2×Dic3—61no—
210C2×Dic3—61no—
211C4×D3—61no—
212C4×D3—61no—
213Dic6—61no—
214Dic6—61no—
215C3×C8—21no—
216C3×C8—61no—
217C3×C8—181no—
218C3⋊C8—61no—
219C2×C32#89411yescharacteristic
220C2×C32#91211yescharacteristic
221C2×C32#91811yescharacteristic
222C3×D3—121no—
223C2×C9—21no—
224C2×C9—21no—
225C2×C9—21no—
226C2×C9C3×#12311yescharacteristic
227C2×C9C3×#13511yescharacteristic
228C2×C9C3×#13711yescharacteristic
229D9—41no—
230C2×Q8—91no—
231C22×C4—91no—
232C2×C8—91no—
233C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
234C4⋊C4—91no—
235C4⋊C4—181no—
236C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
237C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
238C22⋊C4—181no—
239C22⋊C4—181no—
240C42—181no—
241C22×C3—31no—
242C22×C3—181no—
243C22×C3—181no—
244C22×C3—181no—
245C22×C3C3×D3×D411yescharacteristic
246C2×D3—61no—
247C2×D3—61no—
248C2×D3—61no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—121no—
254C3×C4—121no—
255C3×C4—181no—
256C3×C4—361no—
257C3×C4—361no—
258C3×C4C3×#23511yescharacteristic
259C3×C4C3×#23911yescharacteristic
260Dic3—61no—
261Dic3—121no—
262Dic3—121no—
263C32#488811yescharacteristic
264C9—21no—
265C9C3×#48511yescharacteristic, complemented
266C23—91no—
267Q8—181no—
268Q8—181no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—91no—
272C2×C4—181no—
273C2×C4—181no—
274C2×C4—181no—
275C2×C4—181no—
276C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
277C8—21no—
278C8—181no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—361no—
283C2×C3C3×#89411yescharacteristic
284C2×C3C3×#91211yescharacteristic
285C2×C3C3×#91811yescharacteristic
286D3—121no—
287C22—181no—
288C22—181no—
289C22—181no—
290C22D4×AGL(1,ℤ/9)11yescharacteristic
291C4—41no—
292C4—41no—
293C4—181no—
294C4—361no—
295C4—361no—
296C4#3697⋊C311yescharacteristic
297C4#8872411yescharacteristic
298C3—31no—
299C3C3×#488811yescharacteristic
300C2—361no—
301C2#63456⋊C311yescharacteristic
302C2#63474⋊C311yescharacteristic
303C2#63480⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×AGL(1,ℤ/9)D4
hypercenter#107AGL(1,ℤ/9)
hypocenterC9C3×#485
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
3. 3−1+2×#107C2semidirect
5. #63376⋊C3C2semidirect
6. #63368⋊C3C2semidirect
7. #63367⋊C3C2semidirect
9. #10588568C3semidirect
31. C9×#107C2×C3semidirect
33. #63376C2×C3semidirect
34. #63368C2×C3semidirect
35. #63367C2×C3semidirect
187. #107AGL(1,ℤ/9)semidirect
192. 3−1+2#485semidirect
265. C9C3×#485semidirect

Maximal subgroups

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

subgroupindex
#63459⋊C32
#63454⋊C32
3−1+2×#1072
(C2×AGL(1,ℤ/9))⋊C82
#63376⋊C32
#63368⋊C32
#63367⋊C32
C3×#48883
#105885683