Finite Group Databasebeta

← C9×#4765 #13141728

Subgroups — C9×#4765 #13141728

Order 1728.

350 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#936C211yesmaximal, characteristic
2C9×#888C211yesmaximal, characteristic, complemented
3C3×C9×#102C211yesmaximal, characteristic
4C9×#832C211yesmaximal, characteristic
5C9×#816C211yesmaximal, characteristic, complemented
6C9×#815C211yesmaximal, characteristic, complemented
7C9×#810C211yesmaximal, characteristic, complemented
8C3×#4765C311yesmaximal, characteristic
9C9×#489—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×C9×Dic6—21no—
12C2×C9×Dic6C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C9×#217—21no—
16C9×#217—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C9×Dic3⋊C4—21no—
19C4×C9×Dic3C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C3×#936C2×C311yescharacteristic
22C3×#888C2×C311yescharacteristic
23C32×#102C2×C311yescharacteristic
24C3×#832C2×C311yescharacteristic
25C3×#816C2×C311yescharacteristic
26C3×#815C2×C311yescharacteristic
27C3×#810C2×C311yescharacteristic
28C9×C4⋊Q8—31no—
29C9×#124—31no—
30C9×#103—31no—
31C9×#103—31no—
32C9×#102—21no—
33C9×#102—31no—
34C9×#102D311yescharacteristic
35C9×#97—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—
47C9×Dic6—42no—
48C3×C8×C9—21no—
49C9×C3⋊C8—21no—
50C3×#489—31no—
51#4765C911yescharacteristic, direct product factor
52C2×C32×D4C22×C311yescharacteristic
53C2×C3×Dic6—21no—
54C2×C3×Dic6C22×C311yescharacteristic
55C2×C32×C8C22×C311yescharacteristic
56C32×C4⋊C4C22×C311yescharacteristic
57C3×#217—21no—
58C3×#217—21no—
59C3×C4⋊Dic3C22×C311yescharacteristic
60C3×Dic3⋊C4—21no—
61C3×C4×Dic3C22×C311yescharacteristic
62C2×C3×C3⋊C8C22×C311yescharacteristic
63C2×Q8×C9—31no—
64C2×Q8×C9—61no—
65C2×D4×C9—21no—
66C2×D4×C9C2×D311yescharacteristic
67C2×C8×C9—21no—
68C2×C8×C9—31no—
69C2×C8×C9C2×D311yescharacteristic
70C9×C4⋊C4—21no—
71C9×C4⋊C4—31no—
72C9×C4⋊C4—61no—
73C9×C4⋊C4C2×D311yescharacteristic
74C9×C22⋊C4—61no—
75C9×C22⋊C4—61no—
76C42×C9—31no—
77C22×C3×C9—41no—
78C22×C3×C9—41no—
79C22×C3×C9—41no—
80C22×C3×C9C2×D411yescharacteristic
81C3×C4×C9—41no—
82C3×C4×C9#4111yescharacteristic
83C3×C4×C9#4111yescharacteristic
84C9×Dic3—21no—
85C9×Dic3—21no—
86C9×Dic3—41no—
87C9×Dic3—41no—
88C3×C4⋊Q8—31no—
89C3×#124—31no—
90#936C2×C911yescharacteristic
91#888C2×C911yescharacteristic, complemented
92C3×#103—31no—
93C3×#103—31no—
94C3×#102—21no—
95C3×#102—31no—
96C3×#102C2×C911yescharacteristic
97C3×#102C3×D311yescharacteristic
98C3×#97—31no—
99#832C2×C911yescharacteristic
100#816C2×C911yescharacteristic, complemented
101#815C2×C911yescharacteristic, complemented
102#810C2×C911yescharacteristic, complemented
103C23×C32—21no—
104C32×D4—21no—
105C32×D4—21no—
106C2×C32×C4—21no—
107C2×C32×C4C23×C311yescharacteristic
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3C3×D411yescharacteristic
111C2×C3×Dic3C3×D411yescharacteristic
112C3×Dic6—21no—
113C3×Dic6—21no—
114C3×Dic6—42no—
115C23×C9—21no—
116C23×C9—41no—
117C32×C8—21no—
118C3×C3⋊C8—21no—
119Q8×C9—61no—
120Q8×C9—61no—
121Q8×C9—122no—
122D4×C9—22no—
123D4×C9—21no—
124D4×C9—21no—
125D4×C9—41no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—31no—
129C2×C4×C9—31no—
130C2×C4×C9—41no—
131C2×C4×C9—61no—
132C2×C4×C9—61no—
133C2×C4×C9C22×D311yescharacteristic
134C8×C9—21no—
135C8×C9—41no—
136C8×C9—61no—
137#489—31noSylow
138C2×C3×C9—81no—
139C2×C3×C9#12411yescharacteristic
140C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
141C2×C3×C9#13711yescharacteristic
142C2×C3×Q8—31no—
143C2×C3×Q8—61no—
144C2×C3×D4—21no—
145C2×C3×D4C22×C911yescharacteristic
146C2×C3×D4C2×C3×D311yescharacteristic
147C2×Dic6—21no—
148C2×Dic6C22×C911yescharacteristic
149C2×C3×C8—21no—
150C2×C3×C8—31no—
151C2×C3×C8C22×C911yescharacteristic
152C2×C3×C8C2×C3×D311yescharacteristic
153C3×C4⋊C4—21no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4C22×C911yescharacteristic
157C3×C4⋊C4C2×C3×D311yescharacteristic
158C3×C22⋊C4—61no—
159C3×C22⋊C4—61no—
160C3×C42—31no—
161#217—21no—
162#217—21no—
163C4⋊Dic3C22×C911yescharacteristic
164Dic3⋊C4—21no—
165C4×Dic3C22×C911yescharacteristic
166C2×C3⋊C8C22×C911yescharacteristic
167C22×C32—41no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32C2×C3×D411yescharacteristic
171C32×C4—41no—
172C32×C4C3×#4111yescharacteristic
173C32×C4C3×#4111yescharacteristic
174C3×Dic3—21no—
175C3×Dic3—21no—
176C3×Dic3—41no—
177C3×Dic3—41no—
178C22×C9—21no—
179C22×C9—42no—
180C22×C9—41no—
181C22×C9—41no—
182C22×C9—41no—
183C22×C9—81no—
184C22×C9—81no—
185C22×C9D3×D411yescharacteristic
186C4×C9—21no—
187C4×C9—21no—
188C4×C9—41no—
189C4×C9—61no—
190C4×C9—61no—
191C4×C9—81no—
192C4×C9—121no—
193C4×C9—121no—
194C4×C9#23511yescharacteristic
195C4×C9#23711yescharacteristic
196C4⋊Q8—31no—
197#124—31no—
198#103—31no—
199#103—31no—
200#102—31no—
201#102D3×C911yescharacteristic
202#97—31no—
203C3×C9#48911yescharacteristic, complemented, Sylow
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—41no—
207C3×Q8—61no—
208C3×Q8—61no—
209C3×Q8—122no—
210C3×D4—22no—
211C3×D4—21no—
212C3×D4—21no—
213C3×D4—21no—
214C3×D4—21no—
215C3×D4—41no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—41no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4C23×C911yescharacteristic
225C2×C3×C4C22×C3×D311yescharacteristic
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3D4×C911yescharacteristic
229C2×Dic3D4×C911yescharacteristic
230Dic6—21no—
231Dic6—21no—
232Dic6—42no—
233C3×C8—21no—
234C3×C8—21no—
235C3×C8—41no—
236C3×C8—61no—
237C3⋊C8—21no—
238C2×C32—81no—
239C2×C32C3×#12411yescharacteristic
240C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
241C2×C32C3×#13711yescharacteristic
242C2×C9—21no—
243C2×C9—21no—
244C2×C9—21no—
245C2×C9—82no—
246C2×C9—81no—
247C2×C9#88511yescharacteristic
248C2×C9#91111yescharacteristic
249C2×C9#91511yescharacteristic
250C2×Q8—31no—
251C2×Q8—61no—
252C2×D4C2×D3×C911yescharacteristic
253C2×C8—31no—
254C2×C8C2×D3×C911yescharacteristic
255C4⋊C4—31no—
256C4⋊C4—61no—
257C4⋊C4C2×D3×C911yescharacteristic
258C22⋊C4—61no—
259C22⋊C4—61no—
260C42—31no—
261C22×C3—21no—
262C22×C3—42no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—81no—
270C22×C3—81no—
271C22×C3C2×D4×C911yescharacteristic
272C22×C3C3×D3×D411yescharacteristic
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—61no—
278C3×C4—61no—
279C3×C4—81no—
280C3×C4—121no—
281C3×C4—121no—
282C3×C4C9×#4111yescharacteristic
283C3×C4C9×#4111yescharacteristic
284C3×C4C3×#23511yescharacteristic
285C3×C4C3×#23711yescharacteristic
286Dic3—21no—
287Dic3—21no—
288Dic3—41no—
289Dic3—41no—
290C32C3×#48911yescharacteristic
291C9—21no—
292C9#476511yescharacteristic, direct product factor
293C23—21no—
294Q8—61no—
295Q8—61no—
296Q8—122no—
297D4—21no—
298D4—21no—
299C2×C4—21no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4C22×D3×C911yescharacteristic
305C8—21no—
306C8—61no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—82no—
311C2×C3—81no—
312C2×C3—81no—
313C2×C3C9×#12411yescharacteristic
314C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
315C2×C3C9×#13711yescharacteristic
316C2×C3C3×#88511yescharacteristic
317C2×C3C3×#91111yescharacteristic
318C2×C3C3×#91511yescharacteristic
319C22—41no—
320C22—41no—
321C22—41no—
322C22D3×D4×C911yescharacteristic
323C4—41no—
324C4—61no—
325C4—61no—
326C4—121no—
327C4—121no—
328C4C9×#23511yescharacteristic
329C4C9×#23711yescharacteristic
330C3—21no—
331C3C9×#48911yescharacteristic, complemented
332C3C3×#476511yescharacteristic
333C2—81no—
334C2C9×#88511yescharacteristic
335C2C9×#91111yescharacteristic
336C2C9×#91511yescharacteristic
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×#489
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×#888C2semidirect
5. C9×#816C2semidirect
6. C9×#815C2semidirect
7. C9×#810C2semidirect
51. #4765C9direct (N × H)
91. #888C2×C9semidirect
100. #816C2×C9semidirect
101. #815C2×C9semidirect
102. #810C2×C9semidirect
203. C3×C9#489semidirect
292. C9#4765direct (N × H)
331. C3C9×#489semidirect

Maximal subgroups

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

subgroupindex
C9×#9362
C9×#8882
C3×C9×#1022
C9×#8322
C9×#8162
C9×#8152
C9×#8102
C3×#47653
C9×#4893