Finite Group Databasebeta

← C3×#10588725 #13142862

Subgroups — C3×#10588725 #13142862

Order 1728.

362 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C22⋊Q8C211yesmaximal, characteristic
2C3×#63488C211yesmaximal, characteristic
3C3×#63395C211yesmaximal, characteristic, complemented
4C3×#63395C211yesmaximal, characteristic, complemented
5C3×#63389C211yesmaximal, characteristic
6C3×#63367C211yesmaximal, characteristic, complemented
7C3×#63366C211yesmaximal, characteristic, complemented
8C3×#5045—31nomaximal
9#10588725C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C3×C9×C4⋊C4—21no—
13C3×C9×C4⋊C4C2211yescharacteristic
14C3×C9×C22⋊C4—21no—
15C3×#3676—21no—
16C3×C4⋊Dic9C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C3×C4×Dic9—21no—
19C2×C3×C9⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21C32×C22⋊Q8D311yescharacteristic
22C3×#926—31no—
23C3×#835—31no—
24C3×#835—31no—
25C3×#830—31no—
26C3×#808—31no—
27C3×#807—31no—
28C9×C22⋊Q8—21no—
29C9×C22⋊Q8C2×C311yescharacteristic
30#63488C2×C311yescharacteristic
31#63395C2×C311yescharacteristic, complemented
32#63395C2×C311yescharacteristic, complemented
33#63389C2×C311yescharacteristic
34#63367C2×C311yescharacteristic, complemented
35#63366C2×C311yescharacteristic, complemented
36C23×C3×C9D411yescharacteristic
37C3×Q8×C9—21no—
38C3×Q8×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9D411yescharacteristic
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3×Dic9—21no—
46C2×C3×Dic9—21no—
47C3×C9⋊C8—21no—
48C3×C9⋊C8—21no—
49C3×#485—91no—
50#5045—31no—
51C2×C32×Q8C2×D311yescharacteristic
52C22×C32×C4C2×D311yescharacteristic
53C32×C4⋊C4—21no—
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C22⋊C4—21no—
56C3×#217—61no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×C4×Dic3—61no—
60C2×C3×C3⋊C8—31no—
61C2×C3×C3⋊C8—31no—
62C2×Q8×C9—21no—
63C2×Q8×C9C22×C311yescharacteristic
64C22×C4×C9—21no—
65C22×C4×C9C22×C311yescharacteristic
66C9×C4⋊C4—22no—
67C9×C4⋊C4—21no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4C22×C311yescharacteristic
70C9×C22⋊C4—22no—
71C9×C22⋊C4—21no—
72#3676—21no—
73C4⋊Dic9C22×C311yescharacteristic
74C4⋊Dic9C22×C311yescharacteristic
75C4×Dic9—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—21no—
90C3×C22⋊Q8D911yescharacteristic
91C3×C22⋊Q8C3×D311yescharacteristic
92C3×#117—91no—
93#926—31no—
94C3×#107—91no—
95C3×#106—91no—
96C3×#103—91no—
97C3×#103—91no—
98C3×C22⋊C8—91no—
99#835—31no—
100#835—31no—
101#830—31no—
102#808—31no—
103#807—31no—
104C23×C32#6711yescharacteristic
105C32×Q8—21no—
106C32×Q8—21no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4#6711yescharacteristic
112C2×C32×C4C22×D311yescharacteristic
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C23×C9—21no—
116C23×C9C3×D411yescharacteristic
117C3×C3⋊C8—61no—
118C3×C3⋊C8—61no—
119Q8×C9—21no—
120Q8×C9—21no—
121Q8×C9—41no—
122Q8×C9—41no—
123C2×C4×C9—21no—
124C2×C4×C9—21no—
125C2×C4×C9—22no—
126C2×C4×C9—22no—
127C2×C4×C9—21no—
128C2×C4×C9—21no—
129C2×C4×C9—21no—
130C2×C4×C9—21no—
131C2×C4×C9—41no—
132C2×C4×C9—41no—
133C2×C4×C9C3×D411yescharacteristic
134C2×C4×C9C23×C311yescharacteristic
135C2×Dic9—21no—
136C2×Dic9—21no—
137C9⋊C8—21no—
138C9⋊C8—21no—
139#485—91noSylow
140C2×C3×C9—41no—
141C2×C3×C9#12311yescharacteristic
142C2×C3×C9#13511yescharacteristic
143C2×C3×C9#13711yescharacteristic
144C2×C3×Q8—21no—
145C2×C3×Q8C2×D911yescharacteristic
146C2×C3×Q8C2×C3×D311yescharacteristic
147C22×C3×C4—21no—
148C22×C3×C4C2×D911yescharacteristic
149C22×C3×C4C2×C3×D311yescharacteristic
150C2×C3×C8—91no—
151C2×C3×C8—91no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4C2×D911yescharacteristic
159C3×C4⋊C4C2×C3×D311yescharacteristic
160C3×C22⋊C4—22no—
161C3×C22⋊C4—21no—
162C3×C22⋊C4—21no—
163C3×C22⋊C4—181no—
164C3×C42—181no—
165#217—61no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4×Dic3—61no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8—31no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32C2×#6711yescharacteristic
175C32×C4—21no—
176C32×C4—41no—
177C32×C4—41no—
178C32×C4#23711yescharacteristic
179C32×C4#23711yescharacteristic
180C3×Dic3—121no—
181C3×Dic3—121no—
182C22×C9—22no—
183C22×C9—21no—
184C22×C9—21no—
185C22×C9—21no—
186C22×C9—21no—
187C22×C9—41no—
188C22×C9—41no—
189C22×C9C2×C3×D411yescharacteristic
190C4×C9—21no—
191C4×C9—21no—
192C4×C9—21no—
193C4×C9—41no—
194C4×C9—42no—
195C4×C9—42no—
196C4×C9—41no—
197C4×C9—41no—
198C4×C9C3×#4111yescharacteristic
199C4×C9C3×#4111yescharacteristic
200Dic9—41no—
201Dic9—41no—
202C22⋊Q8C3×D911yescharacteristic
203#117—91no—
204#107—91no—
205#106—91no—
206#103—91no—
207#103—91no—
208C22⋊C8—91no—
209C3×C9#48511yescharacteristic, complemented, Sylow
210C23×C3—21no—
211C23×C3#61111yescharacteristic
212C23×C3C3×#6711yescharacteristic
213C3×Q8—21no—
214C3×Q8—21no—
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8—41no—
218C3×Q8—41no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4#61111yescharacteristic
236C2×C3×C4C22×D911yescharacteristic
237C2×C3×C4C3×#6711yescharacteristic
238C2×C3×C4C22×C3×D311yescharacteristic
239C2×Dic3—61no—
240C2×Dic3—61no—
241C3×C8—181no—
242C3×C8—181no—
243C3⋊C8—61no—
244C3⋊C8—61no—
245C2×C32—41no—
246C2×C32#93511yescharacteristic
247C2×C32#94211yescharacteristic
248C2×C32#95011yescharacteristic
249C2×C9—21no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9—42no—
253C2×C9—41no—
254C2×C9C3×#12311yescharacteristic
255C2×C9C3×#13511yescharacteristic
256C2×C9C3×#13711yescharacteristic
257C2×Q8C2×C3×D911yescharacteristic
258C22×C4C2×C3×D911yescharacteristic
259C2×C8—91no—
260C2×C8—91no—
261C4⋊C4—21no—
262C4⋊C4—91no—
263C4⋊C4—91no—
264C4⋊C4C2×C3×D911yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—181no—
267C42—181no—
268C22×C3—22no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3C2×#61111yescharacteristic
279C22×C3C2×C3×#6711yescharacteristic
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—41no—
285C3×C4—42no—
286C3×C4—42no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—361no—
292C3×C4—361no—
293C3×C4#369911yescharacteristic
294C3×C4#369911yescharacteristic
295C3×C4C3×#23711yescharacteristic
296C3×C4C3×#23711yescharacteristic
297Dic3—121no—
298Dic3—121no—
299C32#504511yescharacteristic
300C9—21no—
301C9C3×#48511yescharacteristic, complemented
302C23C3×#61111yescharacteristic
303Q8—21no—
304Q8—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—181no—
310C2×C4—181no—
311C2×C4C3×#61111yescharacteristic
312C2×C4C22×C3×D911yescharacteristic
313C8—181no—
314C8—181no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—42no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3#6349711yescharacteristic
322C2×C3#6350511yescharacteristic
323C2×C3#6351311yescharacteristic
324C2×C3C3×#93511yescharacteristic
325C2×C3C3×#94211yescharacteristic
326C2×C3C3×#95011yescharacteristic
327C22—21no—
328C22—21no—
329C22—21no—
330C22C2×C3×#61111yescharacteristic
331C4—21no—
332C4—41no—
333C4—41no—
334C4—361no—
335C4—361no—
336C4C3×#369911yescharacteristic
337C4C3×#369911yescharacteristic
338C3—21no—
339C3#1058872511yescharacteristic, direct product factor
340C3C3×#504511yescharacteristic
341C2—41no—
342C2C3×#6349711yescharacteristic
343C2C3×#6350511yescharacteristic
344C2C3×#6351311yescharacteristic
subgroupG/H
centerC22×C3C2×#611
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C2×#67
prime radicalC23×C32#67
hypercenterC3×C22⋊Q8D9
hypocenterC9C3×#485
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C3×#63395C2semidirect
4. C3×#63395C2semidirect
6. C3×#63367C2semidirect
7. C3×#63366C2semidirect
9. #10588725C3direct (N × H)
31. #63395C2×C3semidirect
32. #63395C2×C3semidirect
34. #63367C2×C3semidirect
35. #63366C2×C3semidirect
209. C3×C9#485semidirect
301. C9C3×#485semidirect
339. C3#10588725direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×C22⋊Q82
C3×#634882
C3×#633952
C3×#633892
C3×#633672
C3×#633662
C3×#50453
#105887253