Finite Group Databasebeta

← C9×#4768 #13141731

Subgroups — C9×#4768 #13141731

Order 1728.

338 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#936C211yesmaximal, characteristic
2C9×#890C211yesmaximal, characteristic, complemented
3C3×C9×#102C211yesmaximal, characteristic
4C9×#832C211yesmaximal, characteristic
5C9×#816C211yesmaximal, characteristic, complemented
6C8×C9×Dic3C211yesmaximal, characteristic, complemented
7C9×#810C211yesmaximal, characteristic, complemented
8C3×#4768C311yesmaximal, characteristic
9C9×#488—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C2×C3×C8×C9C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C9×#217—21no—
15C9×#217—21no—
16C9×C4⋊Dic3—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C9×Dic3⋊C4—21no—
19C4×C9×Dic3C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C3×#936C2×C311yescharacteristic
22C3×#890C2×C311yescharacteristic
23C32×#102C2×C311yescharacteristic
24C3×#832C2×C311yescharacteristic
25C3×#816C2×C311yescharacteristic
26C3×C8×Dic3C2×C311yescharacteristic
27C3×#810C2×C311yescharacteristic
28C9×#125—31no—
29C9×#124—31no—
30C9×#103—31no—
31C9×#103—31no—
32C9×#102—21no—
33C9×#102—31no—
34C9×#102D311yescharacteristic
35C4×C8×C9—31no—
36C23×C3×C9—21no—
37C3×D4×C9—21no—
38C3×D4×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C9×Dic3—21no—
42C2×C9×Dic3—21no—
43C2×C9×Dic3D411yescharacteristic
44C2×C9×Dic3D411yescharacteristic
45C9×Dic6—21no—
46C9×Dic6—21no—
47C3×C8×C9—21no—
48C9×C3⋊C8—21no—
49C3×#488—31no—
50#4768C911yescharacteristic, direct product factor
51C2×C32×D4C22×C311yescharacteristic
52C2×C3×Dic6C22×C311yescharacteristic
53C2×C32×C8C22×C311yescharacteristic
54C32×C4⋊C4C22×C311yescharacteristic
55C3×#217—21no—
56C3×#217—21no—
57C3×C4⋊Dic3—21no—
58C3×C4⋊Dic3C22×C311yescharacteristic
59C3×Dic3⋊C4—21no—
60C3×C4×Dic3C22×C311yescharacteristic
61C2×C3×C3⋊C8C22×C311yescharacteristic
62C2×Q8×C9—31no—
63C2×D4×C9—21no—
64C2×D4×C9C2×D311yescharacteristic
65C2×C8×C9—21no—
66C2×C8×C9—31no—
67C2×C8×C9C2×D311yescharacteristic
68C9×C4⋊C4—21no—
69C9×C4⋊C4—31no—
70C9×C4⋊C4—61no—
71C9×C4⋊C4—61no—
72C9×C4⋊C4C2×D311yescharacteristic
73C9×C22⋊C4—61no—
74C9×C22⋊C4—61no—
75C42×C9—31no—
76C22×C3×C9—41no—
77C22×C3×C9—41no—
78C22×C3×C9—41no—
79C22×C3×C9C2×D411yescharacteristic
80C3×C4×C9—41no—
81C3×C4×C9#4111yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C9×Dic3—21no—
84C9×Dic3—21no—
85C9×Dic3—41no—
86C9×Dic3—41no—
87C3×#125—31no—
88C3×#124—31no—
89#936C2×C911yescharacteristic
90#890C2×C911yescharacteristic, complemented
91C3×#103—31no—
92C3×#103—31no—
93C3×#102—21no—
94C3×#102—31no—
95C3×#102C2×C911yescharacteristic
96C3×#102C3×D311yescharacteristic
97C3×C4×C8—31no—
98#832C2×C911yescharacteristic
99#816C2×C911yescharacteristic, complemented
100C8×Dic3C2×C911yescharacteristic, complemented
101#810C2×C911yescharacteristic, complemented
102C23×C32—21no—
103C32×D4—21no—
104C32×D4—21no—
105C2×C32×C4—21no—
106C2×C32×C4C23×C311yescharacteristic
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3C3×D411yescharacteristic
110C2×C3×Dic3C3×D411yescharacteristic
111C3×Dic6—21no—
112C3×Dic6—21no—
113C23×C9—21no—
114C23×C9—41no—
115C32×C8—21no—
116C3×C3⋊C8—21no—
117Q8×C9—61no—
118Q8×C9—61no—
119D4×C9—22no—
120D4×C9—21no—
121D4×C9—21no—
122D4×C9—41no—
123C2×C4×C9—21no—
124C2×C4×C9—21no—
125C2×C4×C9—31no—
126C2×C4×C9—31no—
127C2×C4×C9—41no—
128C2×C4×C9—61no—
129C2×C4×C9—61no—
130C2×C4×C9C22×D311yescharacteristic
131C8×C9—21no—
132C8×C9—41no—
133C8×C9—61no—
134#488—31noSylow
135C2×C3×C9—81no—
136C2×C3×C9#12411yescharacteristic
137C2×C3×C9#13511yescharacteristic
138C2×C3×C9#13511yescharacteristic
139C2×C3×Q8—31no—
140C2×C3×D4—21no—
141C2×C3×D4C22×C911yescharacteristic
142C2×C3×D4C2×C3×D311yescharacteristic
143C2×Dic6C22×C911yescharacteristic
144C2×C3×C8—21no—
145C2×C3×C8—31no—
146C2×C3×C8C22×C911yescharacteristic
147C2×C3×C8C2×C3×D311yescharacteristic
148C3×C4⋊C4—21no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4—61no—
151C3×C4⋊C4—61no—
152C3×C4⋊C4C22×C911yescharacteristic
153C3×C4⋊C4C2×C3×D311yescharacteristic
154C3×C22⋊C4—61no—
155C3×C22⋊C4—61no—
156C3×C42—31no—
157#217—21no—
158#217—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3C22×C911yescharacteristic
161Dic3⋊C4—21no—
162C4×Dic3C22×C911yescharacteristic
163C2×C3⋊C8C22×C911yescharacteristic
164C22×C32—41no—
165C22×C32—41no—
166C22×C32—41no—
167C22×C32C2×C3×D411yescharacteristic
168C32×C4—41no—
169C32×C4C3×#4111yescharacteristic
170C32×C4C3×#4111yescharacteristic
171C3×Dic3—21no—
172C3×Dic3—21no—
173C3×Dic3—41no—
174C3×Dic3—41no—
175C22×C9—21no—
176C22×C9—42no—
177C22×C9—41no—
178C22×C9—41no—
179C22×C9—41no—
180C22×C9—81no—
181C22×C9—81no—
182C22×C9D3×D411yescharacteristic
183C4×C9—21no—
184C4×C9—21no—
185C4×C9—41no—
186C4×C9—61no—
187C4×C9—61no—
188C4×C9—81no—
189C4×C9—121no—
190C4×C9—121no—
191C4×C9#23511yescharacteristic
192C4×C9#23711yescharacteristic
193#125—31no—
194#124—31no—
195#103—31no—
196#103—31no—
197#102—31no—
198#102D3×C911yescharacteristic
199C4×C8—31no—
200C3×C9#48811yescharacteristic, complemented, Sylow
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—41no—
204C3×Q8—61no—
205C3×Q8—61no—
206C3×D4—22no—
207C3×D4—21no—
208C3×D4—21no—
209C3×D4—21no—
210C3×D4—21no—
211C3×D4—41no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—41no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4C23×C911yescharacteristic
221C2×C3×C4C22×C3×D311yescharacteristic
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3D4×C911yescharacteristic
225C2×Dic3D4×C911yescharacteristic
226Dic6—21no—
227Dic6—21no—
228C3×C8—21no—
229C3×C8—21no—
230C3×C8—41no—
231C3×C8—61no—
232C3⋊C8—21no—
233C2×C32—81no—
234C2×C32C3×#12411yescharacteristic
235C2×C32C3×#13511yescharacteristic
236C2×C32C3×#13511yescharacteristic
237C2×C9—21no—
238C2×C9—21no—
239C2×C9—21no—
240C2×C9—82no—
241C2×C9—81no—
242C2×C9#88511yescharacteristic
243C2×C9#91211yescharacteristic
244C2×C9#91611yescharacteristic
245C2×Q8—31no—
246C2×D4C2×D3×C911yescharacteristic
247C2×C8—31no—
248C2×C8C2×D3×C911yescharacteristic
249C4⋊C4—31no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4C2×D3×C911yescharacteristic
253C22⋊C4—61no—
254C22⋊C4—61no—
255C42—31no—
256C22×C3—21no—
257C22×C3—42no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—81no—
265C22×C3—81no—
266C22×C3C2×D4×C911yescharacteristic
267C22×C3C3×D3×D411yescharacteristic
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—61no—
273C3×C4—61no—
274C3×C4—81no—
275C3×C4—121no—
276C3×C4—121no—
277C3×C4C9×#4111yescharacteristic
278C3×C4C9×#4111yescharacteristic
279C3×C4C3×#23511yescharacteristic
280C3×C4C3×#23711yescharacteristic
281Dic3—21no—
282Dic3—21no—
283Dic3—41no—
284Dic3—41no—
285C32C3×#48811yescharacteristic
286C9—21no—
287C9#476811yescharacteristic, direct product factor
288C23—21no—
289Q8—61no—
290Q8—61no—
291D4—21no—
292D4—21no—
293C2×C4—21no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4C22×D3×C911yescharacteristic
299C8—21no—
300C8—61no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—82no—
305C2×C3—81no—
306C2×C3—81no—
307C2×C3C9×#12411yescharacteristic
308C2×C3C9×#13511yescharacteristic
309C2×C3C9×#13511yescharacteristic
310C2×C3C3×#88511yescharacteristic
311C2×C3C3×#91211yescharacteristic
312C2×C3C3×#91611yescharacteristic
313C22—41no—
314C22—41no—
315C22—41no—
316C22D3×D4×C911yescharacteristic
317C4—41no—
318C4—61no—
319C4—61no—
320C4—121no—
321C4—121no—
322C4C9×#23511yescharacteristic
323C4C9×#23711yescharacteristic
324C3—21no—
325C3C9×#48811yescharacteristic, complemented
326C3C3×#476811yescharacteristic
327C2—81no—
328C2C9×#88511yescharacteristic
329C2C9×#91211yescharacteristic
330C2C9×#91611yescharacteristic
subgroupG/H
centerC22×C9D3×D4
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#102C2
socleC22×C32C2×C3×D4
prime radicalC2×C32×D4C22×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#102D3
hypocenterC3C9×#488
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C9×#102

Complements

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

NH ≅ G/Ntype
2. C9×#890C2semidirect
5. C9×#816C2semidirect
6. C8×C9×Dic3C2semidirect
7. C9×#810C2semidirect
50. #4768C9direct (N × H)
90. #890C2×C9semidirect
99. #816C2×C9semidirect
100. C8×Dic3C2×C9semidirect
101. #810C2×C9semidirect
200. C3×C9#488semidirect
287. C9#4768direct (N × H)
325. C3C9×#488semidirect

Maximal subgroups

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

subgroupindex
C9×#9362
C9×#8902
C3×C9×#1022
C9×#8322
C9×#8162
C8×C9×Dic32
C9×#8102
C3×#47683
C9×#4883