Finite Group Databasebeta

← C3×#10588564 #13142192

Subgroups — C3×#10588564 #13142192

Order 1728.

348 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63459C211yesmaximal, characteristic
2C3×C4⋊C4×D9C211yesmaximal, characteristic
3C3×C9×#107C211yesmaximal, characteristic, complemented
4C3×(C2×D9)⋊C8C211yesmaximal, characteristic
5C3×#63376C211yesmaximal, characteristic, complemented
6C3×#63368C211yesmaximal, characteristic, complemented
7C3×#63366C211yesmaximal, characteristic, complemented
8C3×#4884—31nomaximal
9#10588564C311yesmaximal, characteristic, direct product factor
10C2×C3×C4×D9—21no—
11C2×C3×C4×D9C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16C3×(C2×D9)⋊C4—21no—
17C3×C4⋊Dic9—21no—
18C3×C4⋊Dic9C2211yescharacteristic
19C3×Dic9⋊C4—21no—
20C2×C3×C9⋊C8C2211yescharacteristic
21C3×#897—31no—
22C3×D3×C4⋊C4—31no—
23C32×#107D311yescharacteristic
24C3×(C2×D3)⋊C8—31no—
25C3×#816—31no—
26C3×#810—31no—
27C3×#807—31no—
28#63459C2×C311yescharacteristic
29C4⋊C4×D9C2×C311yescharacteristic
30C9×#107—21no—
31C9×#107C2×C311yescharacteristic, complemented
32(C2×D9)⋊C8C2×C311yescharacteristic
33#63376C2×C311yescharacteristic, complemented
34#63368C2×C311yescharacteristic, complemented
35#63366C2×C311yescharacteristic, complemented
36C22×C3×D9D411yescharacteristic
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3×Dic9—21no—
41C2×C3×Dic9—21no—
42C2×C3×Dic9D411yescharacteristic
43C3×C4×D9—21no—
44C3×C4×D9—21no—
45C3×C4×D9—42no—
46C3×Dic18—21no—
47C3×Dic18—21no—
48C3×C8×C9—21no—
49C3×C9⋊C8—21no—
50C3×#484—91no—
51#4884—31no—
52C2×C3×C4×D3—31no—
53C2×C3×C4×D3—61no—
54C2×C3×Dic6—31no—
55C2×C32×C8C2×D311yescharacteristic
56C32×C4⋊C4C2×D311yescharacteristic
57C32×C4⋊C4C2×D311yescharacteristic
58C3×(C2×D3)⋊C4—61no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—61no—
61C3×Dic3⋊C4—61no—
62C2×C3×C3⋊C8—31no—
63C2×C4×D9—21no—
64C2×C4×D9C22×C311yescharacteristic
65C2×Dic18C22×C311yescharacteristic
66C2×C8×C9—21no—
67C2×C8×C9C22×C311yescharacteristic
68C9×C4⋊C4—21no—
69C9×C4⋊C4—21no—
70C9×C4⋊C4C22×C311yescharacteristic
71C9×C4⋊C4C22×C311yescharacteristic
72(C2×D9)⋊C4—21no—
73C4⋊Dic9—21no—
74C4⋊Dic9C22×C311yescharacteristic
75Dic9⋊C4—21no—
76C2×C9⋊C8C22×C311yescharacteristic
77C22×C3×C9C2×D411yescharacteristic
78C2×C3×D9—21no—
79C2×C3×D9—21no—
80C2×C3×D9Q1612yes—
81C3×C4×C9—41no—
82C3×C4×C9—41no—
83C3×C4×C9#4111yescharacteristic
84C3×C4×C9#4111yescharacteristic
85C3×Dic9—21no—
86C3×Dic9—41no—
87C3×Dic9—41no—
88C3×C22⋊Q8—91no—
89C2×C3×C4⋊C4—91no—
90#897—31no—
91D3×C4⋊C4—31no—
92C3×#107—21no—
93C3×#107—91no—
94C3×#107D911yescharacteristic, complemented
95C3×#107C3×D311yescharacteristic
96C3×#103—91no—
97C3×#103—91no—
98C3×C22⋊C8—91no—
99(C2×D3)⋊C8—31no—
100#816—31no—
101#810—31no—
102#807—31no—
103C22×C3×D3—31no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3×Dic3—31no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C3×C4×D3—61no—
111C3×C4×D3—61no—
112C3×C4×D3—122no—
113C3×Dic6—61no—
114C3×Dic6—61no—
115C22×D9C3×D411yescharacteristic
116C32×C8—21no—
117C3×C3⋊C8—61no—
118C2×C4×C9—21no—
119C2×C4×C9—21no—
120C2×C4×C9—21no—
121C2×C4×C9—41no—
122C2×C4×C9—41no—
123C2×C4×C9C23×C311yescharacteristic
124C2×Dic9—21no—
125C2×Dic9—21no—
126C2×Dic9C3×D411yescharacteristic
127C4×D9—21no—
128C4×D9—21no—
129C4×D9—42no—
130Dic18—21no—
131Dic18—21no—
132C8×C9—22no—
133C8×C9—21no—
134C9⋊C8—21no—
135#484—91noSylow
136C2×C3×C9#12311yescharacteristic
137C2×C3×C9C2×Q1611yescharacteristic
138C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
139C3×D9—22no—
140C2×C3×Q8—91no—
141C22×C3×C4—91no—
142C22×C3×C4—181no—
143C2×C4×D3—31no—
144C2×C4×D3—61no—
145C2×Dic6—31no—
146C2×C3×C8—21no—
147C2×C3×C8—91no—
148C2×C3×C8C2×D911yescharacteristic
149C2×C3×C8C2×C3×D311yescharacteristic
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—91no—
153C3×C4⋊C4—181no—
154C3×C4⋊C4—181no—
155C3×C4⋊C4C2×D911yescharacteristic
156C3×C4⋊C4C2×D911yescharacteristic
157C3×C4⋊C4C2×C3×D311yescharacteristic
158C3×C4⋊C4C2×C3×D311yescharacteristic
159C3×C22⋊C4—181no—
160(C2×D3)⋊C4—61no—
161C4⋊Dic3—31no—
162C4⋊Dic3—61no—
163Dic3⋊C4—61no—
164C2×C3⋊C8—31no—
165C22×C32D3×D411yescharacteristic
166C2×C3×D3—32no—
167C2×C3×D3—61no—
168C2×C3×D3—61no—
169C32×C4—41no—
170C32×C4—41no—
171C32×C4#23511yescharacteristic
172C32×C4#23911yescharacteristic
173C3×Dic3—61no—
174C3×Dic3—121no—
175C3×Dic3—121no—
176C22×C9—21no—
177C22×C9C2×C3×D411yescharacteristic
178C2×D9—21no—
179C2×D9—21no—
180C2×D9C3×Q1612yes—
181C4×C9—21no—
182C4×C9—21no—
183C4×C9—41no—
184C4×C9—41no—
185C4×C9—81no—
186C4×C9—81no—
187C4×C9C3×#4111yescharacteristic
188C4×C9C3×#4111yescharacteristic
189Dic9—21no—
190Dic9—41no—
191Dic9—41no—
192C22⋊Q8—91no—
193C2×C4⋊C4—91no—
194#107—91no—
195#107C3×D911yescharacteristic, complemented
196#103—91no—
197#103—91no—
198C22⋊C8—91no—
199C3×C9#48411yescharacteristic, complemented, Sylow
200C23×C3—91no—
201C22×D3—31no—
202C3×Q8—181no—
203C3×Q8—181no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—91no—
212C2×C3×C4—181no—
213C2×C3×C4—181no—
214C2×C3×C4—181no—
215C2×C3×C4—181no—
216C2×C3×C4—362no—
217C2×C3×C4C22×D911yescharacteristic
218C2×C3×C4C22×C3×D311yescharacteristic
219C2×Dic3—31no—
220C2×Dic3—61no—
221C2×Dic3—61no—
222C4×D3—61no—
223C4×D3—61no—
224C4×D3—122no—
225Dic6—61no—
226Dic6—61no—
227C3×C8—22no—
228C3×C8—21no—
229C3×C8—21no—
230C3×C8—181no—
231C3⋊C8—61no—
232C2×C32#89411yescharacteristic
233C2×C32#91111yescharacteristic
234C2×C32D3×Q1611yescharacteristic
235C3×D3—62no—
236C2×C9—21no—
237C2×C9—21no—
238C2×C9—21no—
239C2×C9C3×#12311yescharacteristic
240C2×C9C2×C3×Q1611yescharacteristic
241C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
242D9—22no—
243C2×Q8—91no—
244C22×C4—91no—
245C22×C4—181no—
246C2×C8—91no—
247C2×C8C2×C3×D911yescharacteristic
248C4⋊C4—91no—
249C4⋊C4—181no—
250C4⋊C4—181no—
251C4⋊C4C2×C3×D911yescharacteristic
252C4⋊C4C2×C3×D911yescharacteristic
253C22⋊C4—181no—
254C22×C3—21no—
255C22×C3—92no—
256C22×C3—181no—
257C22×C3—181no—
258C22×C3D4×D911yescharacteristic
259C22×C3C3×D3×D411yescharacteristic
260C2×D3—32no—
261C2×D3—61no—
262C2×D3—61no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—81no—
270C3×C4—81no—
271C3×C4—181no—
272C3×C4—361no—
273C3×C4—361no—
274C3×C4#369711yescharacteristic
275C3×C4#370111yescharacteristic
276C3×C4C3×#23511yescharacteristic
277C3×C4C3×#23911yescharacteristic
278Dic3—61no—
279Dic3—121no—
280Dic3—121no—
281C32#488411yescharacteristic
282C9—21no—
283C9C3×#48411yescharacteristic, complemented
284C23—91no—
285Q8—181no—
286Q8—181no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—91no—
290C2×C4—181no—
291C2×C4—181no—
292C2×C4—181no—
293C2×C4—181no—
294C2×C4—362no—
295C2×C4C22×C3×D911yescharacteristic
296C8—21no—
297C8—181no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—182no—
302C2×C3#6345611yescharacteristic
303C2×C3#6347311yescharacteristic
304C2×C3Q16×D911yescharacteristic
305C2×C3C3×#89411yescharacteristic
306C2×C3C3×#91111yescharacteristic
307C2×C3C3×D3×Q1611yescharacteristic
308D3—62no—
309C22—92no—
310C22—181no—
311C22—181no—
312C22C3×D4×D911yescharacteristic
313C4—41no—
314C4—41no—
315C4—181no—
316C4—361no—
317C4—361no—
318C4C3×#369711yescharacteristic
319C4C3×#370111yescharacteristic
320C3—21no—
321C3#1058856411yescharacteristic, direct product factor
322C3C3×#488411yescharacteristic
323C2—182no—
324C2C3×#6345611yescharacteristic
325C2C3×#6347311yescharacteristic
326C2C3×Q16×D911yescharacteristic
subgroupG/H
centerC22×C3D4×D9
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#107C2
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
hypercenterC3×#107D9
hypocenterC9C3×#484
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#107

Complements

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

NH ≅ G/Ntype
3. C3×C9×#107C2semidirect
5. C3×#63376C2semidirect
6. C3×#63368C2semidirect
7. C3×#63366C2semidirect
9. #10588564C3direct (N × H)
31. C9×#107C2×C3semidirect
33. #63376C2×C3semidirect
34. #63368C2×C3semidirect
35. #63366C2×C3semidirect
94. C3×#107D9semidirect
195. #107C3×D9semidirect
199. C3×C9#484semidirect
283. C9C3×#484semidirect
321. C3#10588564direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#634592
C3×C4⋊C4×D92
C3×C9×#1072
C3×(C2×D9)⋊C82
C3×#633762
C3×#633682
C3×#633662
C3×#48843
#105885643