Finite Group Databasebeta

← C3×#10588680 #13142669

Subgroups — C3×#10588680 #13142669

Order 1728.

364 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#117C211yesmaximal, characteristic
2C3×#63486C211yesmaximal, characteristic
3C3×#63389C211yesmaximal, characteristic
4C3×#63368C211yesmaximal, characteristic, complemented
5C3×#63368C211yesmaximal, characteristic, complemented
6C3×#63367C211yesmaximal, characteristic, complemented
7C3×#63366C211yesmaximal, characteristic, complemented
8C3×#4996—31nomaximal
9#10588680C311yesmaximal, characteristic, direct product factor
10C22×C3×C4×C9C2211yescharacteristic
11C2×C3×Dic18C2211yescharacteristic
12C3×C9×C4⋊C4C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C3×C9×C22⋊C4—21no—
15C3×C42×C9—21no—
16C3×#3676—21no—
17C3×C4⋊Dic9C2211yescharacteristic
18C3×Dic9⋊C4—21no—
19C2×C3×C9⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21C32×#117D311yescharacteristic
22C3×#924—31no—
23C3×#830—31no—
24C3×#810—31no—
25C3×#810—31no—
26C3×#808—31no—
27C3×#807—31no—
28C9×#117—21no—
29C9×#117C2×C311yescharacteristic
30#63486C2×C311yescharacteristic
31#63389C2×C311yescharacteristic
32#63368C2×C311yescharacteristic, complemented
33#63368C2×C311yescharacteristic, complemented
34#63367C2×C311yescharacteristic, complemented
35#63366C2×C311yescharacteristic, complemented
36C23×C3×C9D411yescharacteristic
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9D411yescharacteristic
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3×Dic9—21no—
44C2×C3×Dic9—21no—
45C3×Dic18—21no—
46C3×Dic18—21no—
47C3×C9⋊C8—21no—
48C3×C9⋊C8—21no—
49C3×#485—91no—
50#4996—31no—
51C22×C32×C4C2×D311yescharacteristic
52C2×C3×Dic6—31no—
53C32×C4⋊C4C2×D311yescharacteristic
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C22⋊C4—21no—
56C32×C42—21no—
57C3×#217—61no—
58C3×C4⋊Dic3—31no—
59C3×Dic3⋊C4—61no—
60C2×C3×C3⋊C8—31no—
61C2×C3×C3⋊C8—31no—
62C22×C4×C9—21no—
63C22×C4×C9C22×C311yescharacteristic
64C2×Dic18C22×C311yescharacteristic
65C9×C4⋊C4—21no—
66C9×C4⋊C4—21no—
67C9×C4⋊C4C22×C311yescharacteristic
68C9×C4⋊C4C22×C311yescharacteristic
69C9×C22⋊C4—22no—
70C9×C22⋊C4—21no—
71C42×C9—22no—
72C42×C9—21no—
73#3676—21no—
74C4⋊Dic9C22×C311yescharacteristic
75Dic9⋊C4—21no—
76C2×C9⋊C8C22×C311yescharacteristic
77C2×C9⋊C8C22×C311yescharacteristic
78C22×C3×C9—21no—
79C22×C3×C9—21no—
80C22×C3×C9—21no—
81C22×C3×C9C2×D411yescharacteristic
82C3×C4×C9—21no—
83C3×C4×C9—41no—
84C3×C4×C9—41no—
85C3×C4×C9#4111yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C3×Dic9—41no—
88C3×Dic9—41no—
89C3×C22⋊Q8—91no—
90C3×#117—21no—
91C3×#117D911yescharacteristic
92C3×#117C3×D311yescharacteristic
93#924—31no—
94C3×#107—91no—
95C3×#106—91no—
96C3×#103—91no—
97C3×#103—91no—
98C3×C22⋊C8—91no—
99#830—31no—
100#810—31no—
101#810—31no—
102#808—31no—
103#807—31no—
104C23×C32#6711yescharacteristic
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4#6711yescharacteristic
110C2×C32×C4C22×D311yescharacteristic
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C3×Dic6—61no—
114C3×Dic6—61no—
115C23×C9—21no—
116C23×C9C3×D411yescharacteristic
117C3×C3⋊C8—61no—
118C3×C3⋊C8—61no—
119C2×C4×C9—22no—
120C2×C4×C9—22no—
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9—22no—
124C2×C4×C9—22no—
125C2×C4×C9—21no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—21no—
129C2×C4×C9C3×D411yescharacteristic
130C2×C4×C9C23×C311yescharacteristic
131C2×Dic9—21no—
132C2×Dic9—21no—
133Dic18—21no—
134Dic18—21no—
135C9⋊C8—21no—
136C9⋊C8—21no—
137#485—91noSylow
138C2×C3×C9—41no—
139C2×C3×C9#12311yescharacteristic
140C2×C3×C9#13511yescharacteristic
141C2×C3×C9#13711yescharacteristic
142C2×C3×Q8—91no—
143C22×C3×C4—21no—
144C22×C3×C4C2×D911yescharacteristic
145C22×C3×C4C2×C3×D311yescharacteristic
146C2×Dic6—31no—
147C2×C3×C8—91no—
148C2×C3×C8—91no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—181no—
153C3×C4⋊C4C2×D911yescharacteristic
154C3×C4⋊C4C2×D911yescharacteristic
155C3×C4⋊C4C2×C3×D311yescharacteristic
156C3×C4⋊C4C2×C3×D311yescharacteristic
157C3×C22⋊C4—22no—
158C3×C22⋊C4—21no—
159C3×C22⋊C4—21no—
160C3×C22⋊C4—181no—
161C3×C42—22no—
162C3×C42—21no—
163C3×C42—21no—
164#217—61no—
165C4⋊Dic3—31no—
166Dic3⋊C4—61no—
167C2×C3⋊C8—31no—
168C2×C3⋊C8—31no—
169C22×C32—21no—
170C22×C32—21no—
171C22×C32—21no—
172C22×C32C2×#6711yescharacteristic
173C32×C4—21no—
174C32×C4—41no—
175C32×C4—41no—
176C32×C4#23511yescharacteristic
177C32×C4#23511yescharacteristic
178C3×Dic3—121no—
179C3×Dic3—121no—
180C22×C9—22no—
181C22×C9—21no—
182C22×C9—21no—
183C22×C9—21no—
184C22×C9—21no—
185C22×C9—41no—
186C22×C9—41no—
187C22×C9C2×C3×D411yescharacteristic
188C4×C9—22no—
189C4×C9—21no—
190C4×C9—21no—
191C4×C9—21no—
192C4×C9—42no—
193C4×C9—42no—
194C4×C9—41no—
195C4×C9—41no—
196C4×C9C3×#4111yescharacteristic
197C4×C9C3×#4111yescharacteristic
198Dic9—41no—
199Dic9—41no—
200C22⋊Q8—91no—
201#117C3×D911yescharacteristic
202#107—91no—
203#106—91no—
204#103—91no—
205#103—91no—
206C22⋊C8—91no—
207C3×C9#48511yescharacteristic, complemented, Sylow
208C23×C3—21no—
209C23×C3#61111yescharacteristic
210C23×C3C3×#6711yescharacteristic
211C3×Q8—181no—
212C3×Q8—181no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4#61111yescharacteristic
230C2×C3×C4C22×D911yescharacteristic
231C2×C3×C4C3×#6711yescharacteristic
232C2×C3×C4C22×C3×D311yescharacteristic
233C2×Dic3—61no—
234C2×Dic3—61no—
235Dic6—61no—
236Dic6—61no—
237C3×C8—181no—
238C3×C8—181no—
239C3⋊C8—61no—
240C3⋊C8—61no—
241C2×C32—41no—
242C2×C32#92911yescharacteristic
243C2×C32#95011yescharacteristic
244C2×C32#95111yescharacteristic
245C2×C9—21no—
246C2×C9—21no—
247C2×C9—21no—
248C2×C9—42no—
249C2×C9—41no—
250C2×C9C3×#12311yescharacteristic
251C2×C9C3×#13511yescharacteristic
252C2×C9C3×#13711yescharacteristic
253C2×Q8—91no—
254C22×C4C2×C3×D911yescharacteristic
255C2×C8—91no—
256C2×C8—91no—
257C4⋊C4—91no—
258C4⋊C4—181no—
259C4⋊C4C2×C3×D911yescharacteristic
260C4⋊C4C2×C3×D911yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—181no—
263C42—21no—
264C22×C3—22no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3C2×#61111yescharacteristic
275C22×C3C2×C3×#6711yescharacteristic
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—42no—
282C3×C4—42no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4—361no—
288C3×C4—361no—
289C3×C4#369711yescharacteristic
290C3×C4#369711yescharacteristic
291C3×C4C3×#23511yescharacteristic
292C3×C4C3×#23511yescharacteristic
293Dic3—121no—
294Dic3—121no—
295C32#499611yescharacteristic
296C9—21no—
297C9C3×#48511yescharacteristic, complemented
298C23C3×#61111yescharacteristic
299Q8—181no—
300Q8—181no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—181no—
306C2×C4—181no—
307C2×C4C3×#61111yescharacteristic
308C2×C4C22×C3×D911yescharacteristic
309C8—181no—
310C8—181no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—42no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3#6349111yescharacteristic
318C2×C3#6351111yescharacteristic
319C2×C3#6351311yescharacteristic
320C2×C3C3×#92911yescharacteristic
321C2×C3C3×#95011yescharacteristic
322C2×C3C3×#95111yescharacteristic
323C22—21no—
324C22—21no—
325C22—21no—
326C22C2×C3×#61111yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—361no—
331C4—361no—
332C4C3×#369711yescharacteristic
333C4C3×#369711yescharacteristic
334C3—21no—
335C3#1058868011yescharacteristic, direct product factor
336C3C3×#499611yescharacteristic
337C2—41no—
338C2C3×#6349111yescharacteristic
339C2C3×#6351111yescharacteristic
340C2C3×#6351311yescharacteristic
subgroupG/H
centerC22×C3C2×#611
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#117C2
socleC22×C32C2×#67
prime radicalC23×C32#67
hypercenterC3×#117D9
hypocenterC9C3×#485
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
4. C3×#63368C2semidirect
5. C3×#63368C2semidirect
6. C3×#63367C2semidirect
7. C3×#63366C2semidirect
9. #10588680C3direct (N × H)
32. #63368C2×C3semidirect
33. #63368C2×C3semidirect
34. #63367C2×C3semidirect
35. #63366C2×C3semidirect
207. C3×C9#485semidirect
297. C9C3×#485semidirect
335. C3#10588680direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1172
C3×#634862
C3×#633892
C3×#633682
C3×#633672
C3×#633662
C3×#49963
#105886803