Finite Group Databasebeta

← C3×#10588561 #13142189

Subgroups — C3×#10588561 #13142189

Order 1728.

390 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3663C211yesmaximal, characteristic
2C3×#63454C211yesmaximal, characteristic
3C3×C4⋊C4×D9C211yesmaximal, characteristic
4C3×C9×#107C211yesmaximal, characteristic, complemented
5C3×#63377C211yesmaximal, characteristic, complemented
6C3×#63367C211yesmaximal, characteristic, complemented
7C3×#63366C211yesmaximal, characteristic, complemented
8C3×#4881—31nomaximal
9#10588561C311yesmaximal, characteristic, direct product factor
10C2×C3×C4×D9—21no—
11C2×C3×C4×D9C2211yescharacteristic
12C2×C3×C8×C9C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×(C2×D9)⋊C4—21no—
16C3×C4⋊Dic9C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C3×Dic9⋊C4—21no—
19C3×C4×Dic9—21no—
20C2×C3×C9⋊C8C2211yescharacteristic
21C3×#3663C414yescomplemented
22C2×C3×ΓL(1,𝔽25)—31no—
23C3×#892—31no—
24C3×D3×C4⋊C4—31no—
25C32×#107D311yescharacteristic
26C3×#817—31no—
27C3×#808—31no—
28C3×#807—31no—
29C2×#3663C2×C311yescharacteristic
30#63454C2×C311yescharacteristic
31C4⋊C4×D9C2×C311yescharacteristic
32C9×#107—21no—
33C9×#107C2×C311yescharacteristic, complemented
34#63377C2×C311yescharacteristic, complemented
35#63367C2×C311yescharacteristic, complemented
36#63366C2×C311yescharacteristic, complemented
37C22×C3×D9D411yescharacteristic
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3×Dic9—21no—
42C2×C3×Dic9—21no—
43C2×C3×Dic9D411yescharacteristic
44C3×C4×D9—42no—
45C3×C4×D9C2×C412yes—
46C3×C4×D9Q812yes—
47C3×C8×C9C2×C412yescomplemented
48C3×C9⋊C8C2×C412yescomplemented
49C3×#428—91no—
50#4881—31no—
51C2×C3×C4×D3—31no—
52C2×C3×C4×D3—61no—
53C2×C32×C8C2×D311yescharacteristic
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C3×(C2×D3)⋊C4—61no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×Dic3⋊C4—61no—
60C3×C4×Dic3—61no—
61C2×C3×C3⋊C8—31no—
62C3×ΓL(1,𝔽25)—34no—
63C2×C4×D9—21no—
64C2×C4×D9C22×C311yescharacteristic
65C2×C8×C9—21no—
66C2×C8×C9C22×C311yescharacteristic
67C9×C4⋊C4—21no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4C22×C311yescharacteristic
70C9×C4⋊C4C22×C311yescharacteristic
71(C2×D9)⋊C4—21no—
72C4⋊Dic9C22×C311yescharacteristic
73C4⋊Dic9C22×C311yescharacteristic
74Dic9⋊C4—21no—
75C4×Dic9—21no—
76C2×C9⋊C8C22×C311yescharacteristic
77#3663C3×C414yescomplemented
78C22×C3×C9C2×D411yescharacteristic
79C2×C3×D9—21no—
80C2×C3×D9—21no—
81C2×C3×D9C4⋊C412yes—
82C3×C4×C9—41no—
83C3×C4×C9—41no—
84C3×C4×C9C22×C411yescharacteristic
85C3×C4×C9C2×Q811yescharacteristic
86C3×Dic9—41no—
87C3×Dic9—41no—
88C3×Dic9C4⋊C412yes—
89C2×C3×M16—91no—
90C3×#117—91no—
91C2×C3×C4⋊C4—91no—
92C2×ΓL(1,𝔽25)—31no—
93#892—31no—
94D3×C4⋊C4—31no—
95C3×#107—21no—
96C3×#107—91no—
97C3×#107D911yescharacteristic, complemented
98C3×#107C3×D311yescharacteristic
99C3×#106—91no—
100C3×#106—91no—
101#817—31no—
102#808—31no—
103#807—31no—
104C22×C3×D3—31no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3×Dic3—31no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C3×C4×D3—32no—
112C3×C4×D3—32no—
113C3×C4×D3—122no—
114C22×D9C3×D411yescharacteristic
115C32×C8C4×D312yes—
116C3×C3⋊C8—32no—
117C2×C4×C9—21no—
118C2×C4×C9—21no—
119C2×C4×C9—21no—
120C2×C4×C9—41no—
121C2×C4×C9—41no—
122C2×C4×C9C23×C311yescharacteristic
123C2×Dic9—21no—
124C2×Dic9—21no—
125C2×Dic9C3×D411yescharacteristic
126C4×D9—42no—
127C4×D9C2×C3×C412yes—
128C4×D9C3×Q812yes—
129C8×C9—22no—
130C8×C9C2×C3×C412yescomplemented
131C9⋊C8C2×C3×C412yescomplemented
132#428—91noSylow
133C2×C3×C9C2×C4⋊C411yescharacteristic
134C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
135C2×C3×C9#13711yescharacteristic
136C3×D9—22no—
137C22×C3×C4—91no—
138C22×C3×C4—181no—
139C2×C4×D3—31no—
140C2×C4×D3—61no—
141C3×M16—94no—
142C2×C3×C8—21no—
143C2×C3×C8—91no—
144C2×C3×C8C2×D911yescharacteristic
145C2×C3×C8C2×C3×D311yescharacteristic
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—181no—
151C3×C4⋊C4C2×D911yescharacteristic
152C3×C4⋊C4C2×D911yescharacteristic
153C3×C4⋊C4C2×C3×D311yescharacteristic
154C3×C4⋊C4C2×C3×D311yescharacteristic
155C3×C22⋊C4—181no—
156C3×C42—181no—
157(C2×D3)⋊C4—61no—
158C4⋊Dic3—31no—
159C4⋊Dic3—31no—
160Dic3⋊C4—61no—
161C4×Dic3—61no—
162C2×C3⋊C8—31no—
163ΓL(1,𝔽25)—34no—
164C22×C32D3×D411yescharacteristic
165C2×C3×D3—32no—
166C2×C3×D3—61no—
167C2×C3×D3—61no—
168C32×C4—41no—
169C32×C4—41no—
170C32×C4C2×C4×D311yescharacteristic
171C32×C4D3×Q811yescharacteristic
172C3×Dic3—32no—
173C3×Dic3—121no—
174C3×Dic3—121no—
175C22×C9—21no—
176C22×C9C2×C3×D411yescharacteristic
177C2×D9—21no—
178C2×D9—21no—
179C2×D9C3×C4⋊C412yes—
180C4×C9—21no—
181C4×C9—21no—
182C4×C9—41no—
183C4×C9—41no—
184C4×C9—81no—
185C4×C9—81no—
186C4×C9C22×C3×C411yescharacteristic
187C4×C9C2×C3×Q811yescharacteristic
188Dic9—41no—
189Dic9—41no—
190Dic9C3×C4⋊C412yes—
191C2×M16—91no—
192#117—91no—
193C2×C4⋊C4—91no—
194#107—91no—
195#107C3×D911yescharacteristic, complemented
196#106—91no—
197#106—91no—
198C3×C9#42811yescharacteristic, complemented, Sylow
199C23×C3—91no—
200C22×D3—31no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—91no—
209C2×C3×C4—92no—
210C2×C3×C4—92no—
211C2×C3×C4—181no—
212C2×C3×C4—181no—
213C2×C3×C4—362no—
214C2×C3×C4C22×D911yescharacteristic
215C2×C3×C4C22×C3×D311yescharacteristic
216C2×Dic3—31no—
217C2×Dic3—61no—
218C2×Dic3—61no—
219C4×D3—32no—
220C4×D3—32no—
221C4×D3—122no—
222C3×C8—22no—
223C3×C8—92no—
224C3×C8C4×D912yescomplemented
225C3×C8C3×C4×D312yes—
226C3⋊C8—32no—
227C2×C32D3×C4⋊C411yescharacteristic
228C2×C32#91111yescharacteristic
229C2×C32#91811yescharacteristic
230C3×D3—62no—
231C2×C9—21no—
232C2×C9—21no—
233C2×C9—21no—
234C2×C9C2×C3×C4⋊C411yescharacteristic
235C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
236C2×C9C3×#13711yescharacteristic
237D9—22no—
238C22×C4—91no—
239C22×C4—181no—
240M16—94no—
241C2×C8—91no—
242C2×C8C2×C3×D911yescharacteristic
243C4⋊C4—91no—
244C4⋊C4—91no—
245C4⋊C4—181no—
246C4⋊C4C2×C3×D911yescharacteristic
247C4⋊C4C2×C3×D911yescharacteristic
248C22⋊C4—181no—
249C42—181no—
250C22×C3—21no—
251C22×C3—92no—
252C22×C3—181no—
253C22×C3—181no—
254C22×C3D4×D911yescharacteristic
255C22×C3C3×D3×D411yescharacteristic
256C2×D3—32no—
257C2×D3—61no—
258C2×D3—61no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—81no—
266C3×C4—81no—
267C3×C4—92no—
268C3×C4—361no—
269C3×C4—361no—
270C3×C4C2×C4×D911yescharacteristic
271C3×C4Q8×D911yescharacteristic
272C3×C4C2×C3×C4×D311yescharacteristic
273C3×C4C3×D3×Q811yescharacteristic
274Dic3—32no—
275Dic3—121no—
276Dic3—121no—
277C32#488111yescharacteristic
278C9—21no—
279C9C3×#42811yescharacteristic, complemented
280C23—91no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—91no—
284C2×C4—92no—
285C2×C4—92no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—362no—
289C2×C4C22×C3×D911yescharacteristic
290C8—92no—
291C8C3×C4×D912yescomplemented
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—182no—
296C2×C3C4⋊C4×D911yescharacteristic
297C2×C3#6347311yescharacteristic
298C2×C3#6348011yescharacteristic
299C2×C3C3×D3×C4⋊C411yescharacteristic
300C2×C3C3×#91111yescharacteristic
301C2×C3C3×#91811yescharacteristic
302D3—62no—
303C22—92no—
304C22—181no—
305C22—181no—
306C22C3×D4×D911yescharacteristic
307C4—41no—
308C4—41no—
309C4—92no—
310C4—361no—
311C4—361no—
312C4C2×C3×C4×D911yescharacteristic
313C4C3×Q8×D911yescharacteristic
314C3—21no—
315C3#1058856111yescharacteristic, direct product factor
316C3C3×#488111yescharacteristic
317C2—182no—
318C2C3×C4⋊C4×D911yescharacteristic
319C2C3×#6347311yescharacteristic
320C2C3×#6348011yescharacteristic
subgroupG/H
centerC22×C3D4×D9
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#107C2
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
cosocleC2×C4×C9C23×C3
hypercenterC3×#107D9
hypocenterC9C3×#428
LCSG ▷ 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
4. C3×C9×#107C2semidirect
5. C3×#63377C2semidirect
6. C3×#63367C2semidirect
7. C3×#63366C2semidirect
9. #10588561C3direct (N × H)
21. C3×#3663C4semidirect
33. C9×#107C2×C3semidirect
34. #63377C2×C3semidirect
35. #63367C2×C3semidirect
36. #63366C2×C3semidirect
47. C3×C8×C9C2×C4semidirect
48. C3×C9⋊C8C2×C4semidirect
77. #3663C3×C4semidirect
97. C3×#107D9semidirect
130. C8×C9C2×C3×C4semidirect
131. C9⋊C8C2×C3×C4semidirect
195. #107C3×D9semidirect
198. C3×C9#428semidirect
224. C3×C8C4×D9semidirect
279. C9C3×#428semidirect
291. C8C3×C4×D9semidirect
315. C3#10588561direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36632
C3×#634542
C3×C4⋊C4×D92
C3×C9×#1072
C3×#633772
C3×#633672
C3×#633662
C3×#48813
#105885613