Finite Group Databasebeta

← C9×#4891 #13142181

Subgroups — C9×#4891 #13142181

Order 1728.

342 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#893C211yesmaximal, characteristic
2C9×#890C211yesmaximal, characteristic, complemented
3C3×C9×#107C211yesmaximal, characteristic, complemented
4C9×#821C211yesmaximal, characteristic
5C9×Dic3⋊C8C211yesmaximal, characteristic, complemented
6C9×#809C211yesmaximal, characteristic
7C9×#808C211yesmaximal, characteristic, complemented
8C3×#4891C311yesmaximal, characteristic
9C9×#478—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×C4×D3×C9—21no—
12C2×C3×C8×C9C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C9×(C2×D3)⋊C4—21no—
16C9×C4⋊Dic3—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C9×Dic3⋊C4—21no—
19C4×C9×Dic3C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C3×#893C2×C311yescharacteristic
22C3×#890C2×C311yescharacteristic
23C32×#107C2×C311yescharacteristic
24C3×#821C2×C311yescharacteristic
25C3×Dic3⋊C8C2×C311yescharacteristic
26C3×#809C2×C311yescharacteristic
27C3×#808C2×C311yescharacteristic
28C9×#125—31no—
29C4×D4×C9—31no—
30C9×#107—21no—
31C9×#107D311yescharacteristic, complemented
32C9×#106—31no—
33C9×C4⋊C8—31no—
34C9×#102—31no—
35C9×#102—31no—
36C22×D3×C9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C9×Dic3—21no—
41C2×C9×Dic3D411yescharacteristic
42C2×C9×Dic3D411yescharacteristic
43C9×D12—21no—
44C9×D12Q812yes—
45C4×D3×C9—42no—
46C3×C8×C9—21no—
47C9×C3⋊C8—21no—
48C3×#478—31no—
49#4891C911yescharacteristic, direct product factor
50C2×C3×D12C22×C311yescharacteristic
51C2×C3×C4×D3—21no—
52C2×C32×C8C22×C311yescharacteristic
53C32×C4⋊C4C22×C311yescharacteristic
54C32×C4⋊C4C22×C311yescharacteristic
55C3×(C2×D3)⋊C4—21no—
56C3×C4⋊Dic3—21no—
57C3×C4⋊Dic3C22×C311yescharacteristic
58C3×Dic3⋊C4—21no—
59C3×C4×Dic3C22×C311yescharacteristic
60C2×C3×C3⋊C8C22×C311yescharacteristic
61C2×D4×C9—31no—
62C22×C4×C9—61no—
63C2×C8×C9—21no—
64C2×C8×C9—31no—
65C2×C8×C9C2×D311yescharacteristic
66C9×C4⋊C4—21no—
67C9×C4⋊C4—21no—
68C9×C4⋊C4—31no—
69C9×C4⋊C4—61no—
70C9×C4⋊C4—61no—
71C9×C4⋊C4C2×D311yescharacteristic
72C9×C4⋊C4C2×D311yescharacteristic
73C9×C22⋊C4—61no—
74C42×C9—31no—
75C22×C3×C9C2×D411yescharacteristic
76C2×D3×C9—22no—
77C2×D3×C9—41no—
78C2×D3×C9—41no—
79C3×C4×C9—41no—
80C3×C4×C9—41no—
81C3×C4×C9C2×Q811yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C9×Dic3—21no—
84C9×Dic3—21no—
85C9×Dic3—41no—
86C3×#125—31no—
87C3×C4×D4—31no—
88#893C2×C911yescharacteristic
89#890C2×C911yescharacteristic, complemented
90C3×#107—21no—
91C3×#107C2×C911yescharacteristic, complemented
92C3×#107C3×D311yescharacteristic
93C3×#106—31no—
94C3×C4⋊C8—31no—
95C3×#102—31no—
96C3×#102—31no—
97#821C2×C911yescharacteristic
98Dic3⋊C8C2×C911yescharacteristic, complemented
99#809C2×C911yescharacteristic
100#808C2×C911yescharacteristic, complemented
101C22×C3×D3—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4C23×C311yescharacteristic
105C2×C3×Dic3—21no—
106C2×C3×Dic3C3×D411yescharacteristic
107C2×C3×Dic3C3×D411yescharacteristic
108C3×D12—21no—
109C3×D12C3×Q812yes—
110C3×C4×D3—42no—
111C23×C9—61no—
112C32×C8—21no—
113C3×C3⋊C8—21no—
114D4×C9—32no—
115D4×C9—61no—
116C2×C4×C9—21no—
117C2×C4×C9—21no—
118C2×C4×C9—21no—
119C2×C4×C9—31no—
120C2×C4×C9—31no—
121C2×C4×C9—41no—
122C2×C4×C9—41no—
123C2×C4×C9—61no—
124C2×C4×C9—122no—
125C2×C4×C9C22×D311yescharacteristic
126C8×C9—22no—
127C8×C9—21no—
128C8×C9—61no—
129#478—31noSylow
130C2×C3×C9C22⋊Q811yescharacteristic
131C2×C3×C9#13511yescharacteristic
132C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
133D3×C9—42no—
134C2×C3×D4—31no—
135C22×C3×C4—61no—
136C2×D12C22×C911yescharacteristic
137C2×C4×D3—21no—
138C2×C3×C8—21no—
139C2×C3×C8—31no—
140C2×C3×C8C22×C911yescharacteristic
141C2×C3×C8C2×C3×D311yescharacteristic
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—61no—
146C3×C4⋊C4—61no—
147C3×C4⋊C4C22×C911yescharacteristic
148C3×C4⋊C4C22×C911yescharacteristic
149C3×C4⋊C4C2×C3×D311yescharacteristic
150C3×C4⋊C4C2×C3×D311yescharacteristic
151C3×C22⋊C4—61no—
152C3×C42—31no—
153(C2×D3)⋊C4—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3C22×C911yescharacteristic
156Dic3⋊C4—21no—
157C4×Dic3C22×C911yescharacteristic
158C2×C3⋊C8C22×C911yescharacteristic
159C22×C32C2×C3×D411yescharacteristic
160C2×C3×D3—22no—
161C2×C3×D3—41no—
162C2×C3×D3—41no—
163C32×C4—41no—
164C32×C4—41no—
165C32×C4C2×C3×Q811yescharacteristic
166C32×C4C3×#4111yescharacteristic
167C3×Dic3—21no—
168C3×Dic3—21no—
169C3×Dic3—41no—
170C22×C9—21no—
171C22×C9—62no—
172C22×C9—121no—
173C22×C9—121no—
174C22×C9D3×D411yescharacteristic
175C4×C9—21no—
176C4×C9—21no—
177C4×C9—41no—
178C4×C9—41no—
179C4×C9—61no—
180C4×C9—61no—
181C4×C9—81no—
182C4×C9—81no—
183C4×C9—121no—
184C4×C9#23511yescharacteristic
185C4×C9D3×Q811yescharacteristic
186#125—31no—
187C4×D4—31no—
188#107D3×C911yescharacteristic, complemented
189#106—31no—
190C4⋊C8—31no—
191#102—31no—
192#102—31no—
193C3×C9#47811yescharacteristic, complemented, Sylow
194C23×C3—61no—
195C22×D3—21no—
196C3×D4—32no—
197C3×D4—61no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—61no—
208C2×C3×C4—122no—
209C2×C3×C4C23×C911yescharacteristic
210C2×C3×C4C22×C3×D311yescharacteristic
211C2×Dic3—21no—
212C2×Dic3D4×C911yescharacteristic
213C2×Dic3D4×C911yescharacteristic
214D12—21no—
215D12Q8×C912yes—
216C4×D3—42no—
217C3×C8—22no—
218C3×C8—21no—
219C3×C8—21no—
220C3×C8—61no—
221C3⋊C8—21no—
222C2×C32C3×C22⋊Q811yescharacteristic
223C2×C32C3×#13511yescharacteristic
224C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
225C3×D3—42no—
226C2×C9—21no—
227C2×C9—21no—
228C2×C9—21no—
229C2×C9—122no—
230C2×C9#89611yescharacteristic
231C2×C9#91111yescharacteristic
232C2×C9#91911yescharacteristic
233C2×D4—31no—
234C22×C4—61no—
235C2×C8—31no—
236C2×C8C2×D3×C911yescharacteristic
237C4⋊C4—31no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4C2×D3×C911yescharacteristic
241C4⋊C4C2×D3×C911yescharacteristic
242C22⋊C4—61no—
243C42—31no—
244C22×C3—21no—
245C22×C3—62no—
246C22×C3—121no—
247C22×C3—121no—
248C22×C3C2×D4×C911yescharacteristic
249C22×C3C3×D3×D411yescharacteristic
250C2×D3—22no—
251C2×D3—41no—
252C2×D3—41no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—61no—
260C3×C4—61no—
261C3×C4—81no—
262C3×C4—81no—
263C3×C4—121no—
264C3×C4C2×Q8×C911yescharacteristic
265C3×C4C9×#4111yescharacteristic
266C3×C4C3×#23511yescharacteristic
267C3×C4C3×D3×Q811yescharacteristic
268Dic3—21no—
269Dic3—21no—
270Dic3—41no—
271C32C3×#47811yescharacteristic
272C9—21no—
273C9#489111yescharacteristic, direct product factor
274C23—61no—
275D4—32no—
276D4—61no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—61no—
282C2×C4—122no—
283C2×C4C22×D3×C911yescharacteristic
284C8—21no—
285C8—61no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—122no—
290C2×C3C9×C22⋊Q811yescharacteristic
291C2×C3C9×#13511yescharacteristic
292C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
293C2×C3C3×#89611yescharacteristic
294C2×C3C3×#91111yescharacteristic
295C2×C3C3×#91911yescharacteristic
296D3—42no—
297C22—62no—
298C22—121no—
299C22—121no—
300C22D3×D4×C911yescharacteristic
301C4—41no—
302C4—41no—
303C4—61no—
304C4—61no—
305C4—121no—
306C4C9×#23511yescharacteristic
307C4D3×Q8×C911yescharacteristic
308C3—21no—
309C3C9×#47811yescharacteristic, complemented
310C3C3×#489111yescharacteristic
311C2—122no—
312C2C9×#89611yescharacteristic
313C2C9×#91111yescharacteristic
314C2C9×#91911yescharacteristic
subgroupG/H
centerC22×C9D3×D4
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#107C2
socleC22×C32C2×C3×D4
prime radicalC2×C3×D12C22×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#107D3
hypocenterC3C9×#478
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C9×#107

Complements

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

NH ≅ G/Ntype
2. C9×#890C2semidirect
3. C3×C9×#107C2semidirect
5. C9×Dic3⋊C8C2semidirect
7. C9×#808C2semidirect
31. C9×#107D3semidirect
49. #4891C9direct (N × H)
89. #890C2×C9semidirect
91. C3×#107C2×C9semidirect
98. Dic3⋊C8C2×C9semidirect
100. #808C2×C9semidirect
188. #107D3×C9semidirect
193. C3×C9#478semidirect
273. C9#4891direct (N × H)
309. C3C9×#478semidirect

Maximal subgroups

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

subgroupindex
C9×#8932
C9×#8902
C3×C9×#1072
C9×#8212
C9×Dic3⋊C82
C9×#8092
C9×#8082
C3×#48913
C9×#4783