Finite Group Databasebeta

← #10588564⋊C3 #13142245

Subgroups — #10588564⋊C3 #13142245

Order 1728.

330 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 90 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
2C4⋊C4×AGL(1,ℤ/9)C211yesmaximal, 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#63366⋊C3C211yesmaximal, characteristic, complemented
8C3×#4884—31nomaximal
9#10588564C311yesmaximal, characteristic, complemented
10C2×C4×AGL(1,ℤ/9)—21no—
11C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
12C2×3−1+2⋊Q8C2211yescharacteristic
13C2×C8×3−1+2C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15C4⋊C4×3−1+2C2211yescharacteristic
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17C4⋊(3−1+2⋊C4)—21no—
18C4⋊(3−1+2⋊C4)C2211yescharacteristic
19(3−1+2⋊C4)⋊C4—21no—
20C2×3−1+2⋊C8C2211yescharacteristic
21C3×#897—31no—
22C3×D3×C4⋊C4—31no—
23C32×#107D311yescharacteristic
24C3×(C2×D3)⋊C8—31no—
25C3×#816—31no—
26C3×#810—31no—
27C3×#807—31no—
28#63459C2×C311yescharacteristic
29C4⋊C4×D9C2×C311yescharacteristic
30C9×#107—21no—
31C9×#107C2×C311yescharacteristic, complemented
32(C2×D9)⋊C8C2×C311yescharacteristic
33#63376C2×C311yescharacteristic, complemented
34#63368C2×C311yescharacteristic, complemented
35#63366C2×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—
45C4×AGL(1,ℤ/9)—42no—
463−1+2⋊Q8—21no—
473−1+2⋊Q8—21no—
48C8×3−1+2—21no—
493−1+2⋊C8—21no—
50C3×#484—91no—
51#4884—31no—
52C2×C3×C4×D3—31no—
53C2×C3×C4×D3—61no—
54C2×C3×Dic6—31no—
55C2×C32×C8C2×D311yescharacteristic
56C32×C4⋊C4C2×D311yescharacteristic
57C32×C4⋊C4C2×D311yescharacteristic
58C3×(C2×D3)⋊C4—61no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—61no—
61C3×Dic3⋊C4—61no—
62C2×C3×C3⋊C8—31no—
63C2×C4×D9—21no—
64C2×C4×D9C22×C311yescharacteristic
65C2×Dic18C22×C311yescharacteristic
66C2×C8×C9—21no—
67C2×C8×C9C22×C311yescharacteristic
68C9×C4⋊C4—21no—
69C9×C4⋊C4—21no—
70C9×C4⋊C4C22×C311yescharacteristic
71C9×C4⋊C4C22×C311yescharacteristic
72(C2×D9)⋊C4—21no—
73C4⋊Dic9—21no—
74C4⋊Dic9C22×C311yescharacteristic
75Dic9⋊C4—21no—
76C2×C9⋊C8C22×C311yescharacteristic
77C22×3−1+2C2×D411yescharacteristic
78C2×AGL(1,ℤ/9)—21no—
79C2×AGL(1,ℤ/9)—21no—
80C2×AGL(1,ℤ/9)Q1612yes—
81C4×3−1+2—41no—
82C4×3−1+2—41no—
83C4×3−1+2#4111yescharacteristic
84C4×3−1+2#4111yescharacteristic
853−1+2⋊C4—21no—
863−1+2⋊C4—41no—
873−1+2⋊C4—41no—
88C3×C22⋊Q8—91no—
89C2×C3×C4⋊C4—91no—
90#897—31no—
91D3×C4⋊C4—31no—
92C3×#107—31no—
93C3×#107—91no—
94C3×#107C3×D311yescharacteristic
95C3×#103—91no—
96C3×#103—91no—
97C3×C22⋊C8—91no—
98(C2×D3)⋊C8—31no—
99#816—31no—
100#810—31no—
101#807—31no—
102C22×C3×D3—31no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3×Dic3—31no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C3×C4×D3—61no—
110C3×C4×D3—61no—
111C3×C4×D3—122no—
112C3×Dic6—61no—
113C3×Dic6—61no—
114C22×D9C3×D411yescharacteristic
115C32×C8—21no—
116C3×C3⋊C8—61no—
117C2×C4×C9—21no—
118C2×C4×C9—21no—
119C2×C4×C9—21no—
120C2×C4×C9—41no—
121C2×C4×C9—41no—
122C2×C4×C9C23×C311yescharacteristic
123C2×Dic9—21no—
124C2×Dic9—21no—
125C2×Dic9C3×D411yescharacteristic
126C4×D9—21no—
127C4×D9—21no—
128C4×D9—42no—
129Dic18—21no—
130Dic18—21no—
131C8×C9—22no—
132C8×C9—21no—
133C9⋊C8—21no—
134#484—91noSylow
135C2×3−1+2#12311yescharacteristic
136C2×3−1+2C2×Q1611yescharacteristic
137C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
138AGL(1,ℤ/9)—22no—
139C2×C3×Q8—91no—
140C22×C3×C4—91no—
141C22×C3×C4—181no—
142C2×C4×D3—31no—
143C2×C4×D3—61no—
144C2×Dic6—31no—
145C2×C3×C8—31no—
146C2×C3×C8—91no—
147C2×C3×C8C2×C3×D311yescharacteristic
148C3×C4⋊C4—31no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—181no—
152C3×C4⋊C4—181no—
153C3×C4⋊C4C2×C3×D311yescharacteristic
154C3×C4⋊C4C2×C3×D311yescharacteristic
155C3×C22⋊C4—181no—
156(C2×D3)⋊C4—61no—
157C4⋊Dic3—31no—
158C4⋊Dic3—61no—
159Dic3⋊C4—61no—
160C2×C3⋊C8—31no—
161C22×C32D3×D411yescharacteristic
162C2×C3×D3—32no—
163C2×C3×D3—61no—
164C2×C3×D3—61no—
165C32×C4—41no—
166C32×C4—41no—
167C32×C4#23511yescharacteristic
168C32×C4#23911yescharacteristic
169C3×Dic3—61no—
170C3×Dic3—121no—
171C3×Dic3—121no—
172C22×C9—21no—
173C22×C9C2×C3×D411yescharacteristic
174C2×D9—21no—
175C2×D9—21no—
176C2×D9C3×Q1612yes—
177C4×C9—21no—
178C4×C9—21no—
179C4×C9—41no—
180C4×C9—41no—
181C4×C9—81no—
182C4×C9—81no—
183C4×C9C3×#4111yescharacteristic
184C4×C9C3×#4111yescharacteristic
185Dic9—21no—
186Dic9—41no—
187Dic9—41no—
188C22⋊Q8—91no—
189C2×C4⋊C4—91no—
190#107—91no—
191#107AGL(1,ℤ/9)11yescharacteristic, complemented
192#103—91no—
193#103—91no—
194C22⋊C8—91no—
1953−1+2#48411yescharacteristic, complemented, Sylow
196C23×C3—91no—
197C22×D3—31no—
198C3×Q8—181no—
199C3×Q8—181no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—31no—
203C2×C3×C4—61no—
204C2×C3×C4—61no—
205C2×C3×C4—91no—
206C2×C3×C4—181no—
207C2×C3×C4—181no—
208C2×C3×C4—181no—
209C2×C3×C4—181no—
210C2×C3×C4—362no—
211C2×C3×C4C22×C3×D311yescharacteristic
212C2×Dic3—31no—
213C2×Dic3—61no—
214C2×Dic3—61no—
215C4×D3—61no—
216C4×D3—61no—
217C4×D3—122no—
218Dic6—61no—
219Dic6—61no—
220C3×C8—21no—
221C3×C8—61no—
222C3×C8—181no—
223C3⋊C8—61no—
224C2×C32#89411yescharacteristic
225C2×C32#91111yescharacteristic
226C2×C32D3×Q1611yescharacteristic
227C3×D3—62no—
228C2×C9—21no—
229C2×C9—21no—
230C2×C9—21no—
231C2×C9C3×#12311yescharacteristic
232C2×C9C2×C3×Q1611yescharacteristic
233C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
234D9—22no—
235C2×Q8—91no—
236C22×C4—91no—
237C22×C4—181no—
238C2×C8—91no—
239C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
240C4⋊C4—91no—
241C4⋊C4—181no—
242C4⋊C4—181no—
243C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
244C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
245C22⋊C4—181no—
246C22×C3—31no—
247C22×C3—92no—
248C22×C3—181no—
249C22×C3—181no—
250C22×C3C3×D3×D411yescharacteristic
251C2×D3—32no—
252C2×D3—61no—
253C2×D3—61no—
254C3×C4—31no—
255C3×C4—31no—
256C3×C4—41no—
257C3×C4—41no—
258C3×C4—121no—
259C3×C4—121no—
260C3×C4—181no—
261C3×C4—361no—
262C3×C4—361no—
263C3×C4C3×#23511yescharacteristic
264C3×C4C3×#23911yescharacteristic
265Dic3—61no—
266Dic3—121no—
267Dic3—121no—
268C32#488411yescharacteristic
269C9—21no—
270C9C3×#48411yescharacteristic, complemented
271C23—91no—
272Q8—181no—
273Q8—181no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—91no—
277C2×C4—181no—
278C2×C4—181no—
279C2×C4—181no—
280C2×C4—181no—
281C2×C4—362no—
282C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
283C8—21no—
284C8—181no—
285C2×C3—31no—
286C2×C3—31no—
287C2×C3—31no—
288C2×C3—182no—
289C2×C3C3×#89411yescharacteristic
290C2×C3C3×#91111yescharacteristic
291C2×C3C3×D3×Q1611yescharacteristic
292D3—62no—
293C22—92no—
294C22—181no—
295C22—181no—
296C22D4×AGL(1,ℤ/9)11yescharacteristic
297C4—41no—
298C4—41no—
299C4—181no—
300C4—361no—
301C4—361no—
302C4#3697⋊C311yescharacteristic
303C4#8872411yescharacteristic
304C3—31no—
305C3C3×#488411yescharacteristic
306C2—182no—
307C2#63456⋊C311yescharacteristic
308C2#63473⋊C311yescharacteristic
309C2Q16×AGL(1,ℤ/9)11yescharacteristic
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×#484
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. #63366⋊C3C2semidirect
9. #10588564C3semidirect
31. C9×#107C2×C3semidirect
33. #63376C2×C3semidirect
34. #63368C2×C3semidirect
35. #63366C2×C3semidirect
191. #107AGL(1,ℤ/9)semidirect
195. 3−1+2#484semidirect
270. C9C3×#484semidirect

Maximal subgroups

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

subgroupindex
#63459⋊C32
C4⋊C4×AGL(1,ℤ/9)2
3−1+2×#1072
(C2×AGL(1,ℤ/9))⋊C82
#63376⋊C32
#63368⋊C32
#63366⋊C32
C3×#48843
#105885643