Finite Group Databasebeta

← C3×#10588496 #13141875

Subgroups — C3×#10588496 #13141875

Order 1728.

372 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63508C211yesmaximal, characteristic
2C2×C3×#3674C211yesmaximal, characteristic, complemented
3C2×C3×#3663C211yesmaximal, characteristic
4C3×#63459C211yesmaximal, characteristic
5C3×C9×#103C211yesmaximal, characteristic, complemented
6C3×#63376C211yesmaximal, characteristic, complemented
7C3×#63367C211yesmaximal, characteristic, complemented
8C3×#4816—31nomaximal
9#10588496C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×C4×D9C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×#3674—22no—
16C3×(C2×D9)⋊C4—21no—
17C3×(C2×D9)⋊C4—21no—
18C3×C4⋊Dic9—21no—
19C3×C4⋊Dic9C2211yescharacteristic
20C3×Dic9⋊C4—21no—
21C2×C3×C9⋊C8C2211yescharacteristic
22C3×#3663—22no—
23C3×#946—31no—
24C2×C3×#216—31no—
25C2×C3×ΓL(1,𝔽25)—31no—
26C3×#897—31no—
27C32×#103D311yescharacteristic
28C3×#816—31no—
29C3×#808—31no—
30#63508C2×C311yescharacteristic
31C2×#3674C2×C311yescharacteristic, complemented
32C2×#3663C2×C311yescharacteristic
33#63459C2×C311yescharacteristic
34C9×#103—21no—
35C9×#103C2×C311yescharacteristic, complemented
36#63376C2×C311yescharacteristic, complemented
37#63367C2×C311yescharacteristic, complemented
38C22×C3×D9D411yescharacteristic
39C3×Q8×C9—21no—
40C3×Q8×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3×Dic9—21no—
45C2×C3×Dic9—21no—
46C2×C3×Dic9D411yescharacteristic
47C3×C4×D9—21no—
48C3×C4×D9—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51C3×C8×C9—21no—
52C3×C9⋊C8D412yes—
53C3×#470—91no—
54#4816—31no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C3×C4×D3—31no—
57C2×C3×Dic6—31no—
58C2×C32×C8C2×D311yescharacteristic
59C32×C4⋊C4C2×D311yescharacteristic
60C3×#216—62no—
61C3×(C2×D3)⋊C4—61no—
62C3×(C2×D3)⋊C4—61no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—61no—
65C3×Dic3⋊C4—61no—
66C2×C3×C3⋊C8—31no—
67C3×ΓL(1,𝔽25)—62no—
68C2×Q8×C9—21no—
69C2×Q8×C9C22×C311yescharacteristic
70C2×C4×D9C22×C311yescharacteristic
71C2×Dic18C22×C311yescharacteristic
72C2×C8×C9—21no—
73C2×C8×C9C22×C311yescharacteristic
74C9×C4⋊C4—21no—
75C9×C4⋊C4C22×C311yescharacteristic
76#3674—22no—
77(C2×D9)⋊C4—21no—
78(C2×D9)⋊C4—21no—
79C4⋊Dic9—21no—
80C4⋊Dic9C22×C311yescharacteristic
81Dic9⋊C4—21no—
82C2×C9⋊C8C22×C311yescharacteristic
83#3663—22no—
84C22×C3×C9C2×D411yescharacteristic
85C2×C3×D9—21no—
86C2×C3×D9—21no—
87C2×C3×D9—21no—
88C3×C4×C9—41no—
89C3×C4×C9—41no—
90C3×C4×C9C2×D411yescharacteristic
91C3×C4×C9#4111yescharacteristic
92C3×Dic9—21no—
93C3×Dic9—41no—
94C3×Dic9—41no—
95C2×C3×Q16—91no—
96C2×C3×M16—91no—
97C3×C22⋊Q8—91no—
98C3×C22⋊Q8—91no—
99#946—31no—
100C2×#216—31no—
101C2×ΓL(1,𝔽25)—31no—
102#897—31no—
103C3×#106—91no—
104C3×#103—21no—
105C3×#103—91no—
106C3×#103D911yescharacteristic, complemented
107C3×#103C3×D311yescharacteristic
108#816—31no—
109#808—31no—
110C22×C3×D3—31no—
111C32×Q8—21no—
112C32×Q8—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3×Dic3—31no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C3×C4×D3—61no—
120C3×C4×D3—61no—
121C3×Dic6—61no—
122C3×Dic6—61no—
123C22×D9C3×D411yescharacteristic
124C32×C8—21no—
125C3×C3⋊C8—32no—
126Q8×C9—22no—
127Q8×C9—21no—
128Q8×C9—21no—
129Q8×C9—41no—
130C2×C4×C9—21no—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—41no—
134C2×C4×C9—41no—
135C2×C4×C9C23×C311yescharacteristic
136C2×Dic9—21no—
137C2×Dic9—21no—
138C2×Dic9C3×D411yescharacteristic
139C4×D9—21no—
140C4×D9—21no—
141Dic18—21no—
142Dic18—21no—
143C8×C9—21no—
144C8×C9—41no—
145C9⋊C8C3×D412yes—
146#470—91noSylow
147C2×C3×C9#12111yescharacteristic
148C2×C3×C9#13711yescharacteristic
149C2×C3×C9#13711yescharacteristic
150C3×D9—41no—
151C2×C3×Q8—21no—
152C2×C3×Q8—91no—
153C2×C3×Q8C2×D911yescharacteristic
154C2×C3×Q8C2×C3×D311yescharacteristic
155C22×C3×C4—91no—
156C2×C4×D3—31no—
157C2×Dic6—31no—
158C3×Q16—182no—
159C3×M16—182no—
160C2×C3×C8—21no—
161C2×C3×C8—91no—
162C2×C3×C8C2×D911yescharacteristic
163C2×C3×C8C2×C3×D311yescharacteristic
164C3×C4⋊C4—21no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—181no—
167C3×C4⋊C4—181no—
168C3×C4⋊C4C2×D911yescharacteristic
169C3×C4⋊C4C2×C3×D311yescharacteristic
170C3×C22⋊C4—181no—
171C3×C22⋊C4—181no—
172#216—62no—
173(C2×D3)⋊C4—61no—
174(C2×D3)⋊C4—61no—
175C4⋊Dic3—31no—
176C4⋊Dic3—61no—
177Dic3⋊C4—61no—
178C2×C3⋊C8—31no—
179ΓL(1,𝔽25)—62no—
180C22×C32D3×D411yescharacteristic
181C2×C3×D3—61no—
182C2×C3×D3—61no—
183C2×C3×D3—61no—
184C32×C4—41no—
185C32×C4—41no—
186C32×C4#23511yescharacteristic
187C32×C4D3×D411yescharacteristic
188C3×Dic3—61no—
189C3×Dic3—121no—
190C3×Dic3—121no—
191C22×C9—21no—
192C22×C9C2×C3×D411yescharacteristic
193C2×D9—21no—
194C2×D9—21no—
195C2×D9—21no—
196C4×C9—21no—
197C4×C9—21no—
198C4×C9—42no—
199C4×C9—41no—
200C4×C9—41no—
201C4×C9—81no—
202C4×C9C2×C3×D411yescharacteristic
203C4×C9C3×#4111yescharacteristic
204Dic9—21no—
205Dic9—41no—
206Dic9—41no—
207C2×Q16—91no—
208C2×M16—91no—
209C22⋊Q8—91no—
210C22⋊Q8—91no—
211#106—91no—
212#103—91no—
213#103C3×D911yescharacteristic, complemented
214C3×C9#47011yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C22×D3—31no—
217C3×Q8—22no—
218C3×Q8—21no—
219C3×Q8—21no—
220C3×Q8—21no—
221C3×Q8—21no—
222C3×Q8—41no—
223C3×Q8—181no—
224C3×Q8—181no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—91no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4C22×D911yescharacteristic
238C2×C3×C4C22×C3×D311yescharacteristic
239C2×Dic3—31no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C4×D3—61no—
243C4×D3—61no—
244Dic6—61no—
245Dic6—61no—
246C3×C8—21no—
247C3×C8—21no—
248C3×C8—41no—
249C3×C8—92no—
250C3⋊C8—32no—
251C2×C32#88411yescharacteristic
252C2×C32#91511yescharacteristic
253C2×C32#91811yescharacteristic
254C3×D3—121no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9C3×#12111yescharacteristic
259C2×C9C3×#13711yescharacteristic
260C2×C9C3×#13711yescharacteristic
261D9—41no—
262C2×Q8—91no—
263C2×Q8C2×C3×D911yescharacteristic
264C22×C4—91no—
265Q16—182no—
266M16—182no—
267C2×C8—91no—
268C2×C8C2×C3×D911yescharacteristic
269C4⋊C4—91no—
270C4⋊C4—181no—
271C4⋊C4—181no—
272C4⋊C4C2×C3×D911yescharacteristic
273C22⋊C4—181no—
274C22⋊C4—181no—
275C22×C3—21no—
276C22×C3—181no—
277C22×C3—181no—
278C22×C3—181no—
279C22×C3D4×D911yescharacteristic
280C22×C3C3×D3×D411yescharacteristic
281C2×D3—61no—
282C2×D3—61no—
283C2×D3—61no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—42no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—81no—
292C3×C4—181no—
293C3×C4—361no—
294C3×C4—361no—
295C3×C4#369711yescharacteristic
296C3×C4D4×D911yescharacteristic
297C3×C4C3×#23511yescharacteristic
298C3×C4C3×D3×D411yescharacteristic
299Dic3—61no—
300Dic3—121no—
301Dic3—121no—
302C32#481611yescharacteristic
303C9—21no—
304C9C3×#47011yescharacteristic, complemented
305C23—91no—
306Q8—21no—
307Q8—21no—
308Q8—181no—
309Q8—181no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—91no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4—181no—
317C2×C4C22×C3×D911yescharacteristic
318C8—21no—
319C8—92no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—361no—
324C2×C3#6344611yescharacteristic
325C2×C3#6347711yescharacteristic
326C2×C3#6348011yescharacteristic
327C2×C3C3×#88411yescharacteristic
328C2×C3C3×#91511yescharacteristic
329C2×C3C3×#91811yescharacteristic
330D3—121no—
331C22—181no—
332C22—181no—
333C22—181no—
334C22C3×D4×D911yescharacteristic
335C4—41no—
336C4—41no—
337C4—181no—
338C4—361no—
339C4—361no—
340C4C3×#369711yescharacteristic
341C4C3×D4×D911yescharacteristic
342C3—21no—
343C3#1058849611yescharacteristic, direct product factor
344C3C3×#481611yescharacteristic
345C2—361no—
346C2C3×#6344611yescharacteristic
347C2C3×#6347711yescharacteristic
348C2C3×#6348011yescharacteristic
subgroupG/H
centerC22×C3D4×D9
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#103C2
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
hypercenterC3×#103D9
hypocenterC9C3×#470
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3674C2semidirect
5. C3×C9×#103C2semidirect
6. C3×#63376C2semidirect
7. C3×#63367C2semidirect
9. #10588496C3direct (N × H)
31. C2×#3674C2×C3semidirect
35. C9×#103C2×C3semidirect
36. #63376C2×C3semidirect
37. #63367C2×C3semidirect
106. C3×#103D9semidirect
213. #103C3×D9semidirect
214. C3×C9#470semidirect
304. C9C3×#470semidirect
343. C3#10588496direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#635082
C2×C3×#36742
C2×C3×#36632
C3×#634592
C3×C9×#1032
C3×#633762
C3×#633672
C3×#48163
#105884963