Finite Group Databasebeta

← C3×#10590304 #13157321

Subgroups — C3×#10590304 #13157321

Order 1728.

530 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64079C211yesmaximal, characteristic
2C32×#870C211yesmaximal, characteristic
3C32×C4×C3⋊C8C211yesmaximal, characteristic
4C3×#63575C212yesmaximal
5C3×#63573C212yesmaximal
6C3×#5065—31nomaximal
7C3×#4683—31nomaximal
8#10590304C311yesmaximal, characteristic, direct product factor
9C33×C42C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)C2212yes—
11C3×#3750—22no—
12C32×C4⋊Dic3C2212yes—
13C32×Dic3⋊C4—22no—
14C2×C32×C3⋊C8C2211yescharacteristic
15C2×C32×C3⋊C8C2211yescharacteristic
16C32×#125—31no—
17#64079C2×C311yescharacteristic
18C3×#870—21no—
19C3×#870—31no—
20C3×#870—31no—
21C3×#870—61no—
22C3×#870D311yescharacteristic
23C3×#870C2×C311yescharacteristic
24C32×C4×C8—31no—
25C3×#818—32no—
26C3×#817—32no—
27C3×#808—32no—
28C3×#807—32no—
29C3×C4×C3⋊C8—21no—
30C3×C4×C3⋊C8D311yescharacteristic
31C3×C4×C3⋊C8C2×C311yescharacteristic
32#63575C2×C312yes—
33#63573C2×C312yes—
34C2×C33×C4D412yes—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3—22no—
37C2×C32×Dic3—22no—
38C32×C3⋊C8Q812yes—
39C32×C3⋊C8Q812yes—
40C3×#499—91no—
41#5065—31no—
42#4683—31no—
43C2×C32×C8—31no—
44C2×C32×C8—31no—
45C32×C4⋊C4—32no—
46C32×C4⋊C4—62no—
47C32×C42—21no—
48C32×C42—21no—
49C32×C42—21no—
50C32×C42—41no—
51C32×C42C2×D311yescharacteristic
52C32×C42C2×D311yescharacteristic
53C32×C42C22×C311yescharacteristic
54C4⋊(C3⋊Dic3)C22×C312yes—
55#3750—22no—
56C3×C4⋊Dic3—22no—
57C3×C4⋊Dic3—32no—
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—62no—
60C3×C4⋊Dic3C2×D312yes—
61C3×C4⋊Dic3C22×C312yes—
62C3×Dic3⋊C4—24no—
63C3×Dic3⋊C4—22no—
64C3×Dic3⋊C4—22no—
65C3×Dic3⋊C4—64no—
66C3×Dic3⋊C4—62no—
67C3×Dic3⋊C4—62no—
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8C2×D311yescharacteristic
71C2×C3×C3⋊C8C2×D311yescharacteristic
72C2×C3×C3⋊C8C22×C311yescharacteristic
73C2×C3×C3⋊C8C22×C311yescharacteristic
74C22×C33C2×D411yescharacteristic
75C33×C4—22no—
76C33×C4C2×Q811yescharacteristic
77C33×C4C2×Q811yescharacteristic
78C3×C3⋊Dic3—42no—
79C32×Dic3—42no—
80C3×#125—31no—
81C3×#125—31no—
82C3×#125—61no—
83C3×#125—91no—
84#870—31no—
85#870—31no—
86#870—61no—
87#870C3×D311yescharacteristic
88C3×#107—92no—
89C3×#106—92no—
90C3×C4×C8—31no—
91C3×C4×C8—31no—
92C3×C4×C8—61no—
93#818—32no—
94#817—32no—
95#808—32no—
96#807—32no—
97C4×C3⋊C8C3×D311yescharacteristic
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—22no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—42no—
105C2×C32×C4—41no—
106C2×C32×C4—62no—
107C2×C32×C4D1212yes—
108C2×C32×C4#6712yes—
109C2×C32×C4C3×D412yes—
110C2×C32×C4C22×D311yescharacteristic
111C2×C32×C4C22×D311yescharacteristic
112C2×C32×C4C23×C311yescharacteristic
113C2×C3⋊Dic3—22no—
114C2×C3×Dic3—24no—
115C2×C3×Dic3—22no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—64no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C32×C8—32no—
121C32×C8—32no—
122C3×C3⋊C8—22no—
123C3×C3⋊C8—22no—
124C3×C3⋊C8Dic612yes—
125C3×C3⋊C8Dic612yes—
126C3×C3⋊C8C3×Q812yes—
127C3×C3⋊C8C3×Q812yes—
128#499—91noSylow
129C2×C33C4⋊Q811yescharacteristic
130C2×C33#13512yes—
131C2×C3×C8—31no—
132C2×C3×C8—31no—
133C2×C3×C8—31no—
134C2×C3×C8—31no—
135C2×C3×C8—61no—
136C2×C3×C8—61no—
137C3×C4⋊C4—32no—
138C3×C4⋊C4—32no—
139C3×C4⋊C4—62no—
140C3×C4⋊C4—64no—
141C3×C4⋊C4—62no—
142C3×C4⋊C4—62no—
143C3×C4⋊C4—92no—
144C3×C4⋊C4—182no—
145C3×C42—21no—
146C3×C42—21no—
147C3×C42—21no—
148C3×C42—41no—
149C3×C42D3211yescharacteristic
150C3×C42C2×C3×D311yescharacteristic
151C3×C42C2×C3×D311yescharacteristic
152C4⋊Dic3—32no—
153C4⋊Dic3—32no—
154C4⋊Dic3—62no—
155C4⋊Dic3C2×C3×D312yes—
156Dic3⋊C4—22no—
157Dic3⋊C4—64no—
158Dic3⋊C4—62no—
159Dic3⋊C4—62no—
160C2×C3⋊C8C2×C3×D311yescharacteristic
161C2×C3⋊C8C2×C3×D311yescharacteristic
162C22×C32—21no—
163C22×C32—21no—
164C22×C32—21no—
165C22×C32—41no—
166C22×C32C2×D1211yescharacteristic
167C22×C32C2×#6711yescharacteristic
168C22×C32C2×C3×D411yescharacteristic
169C32×C4—22no—
170C32×C4—24no—
171C32×C4—22no—
172C32×C4—22no—
173C32×C4—21no—
174C32×C4—21no—
175C32×C4—21no—
176C32×C4—21no—
177C32×C4—21no—
178C32×C4—21no—
179C32×C4—41no—
180C32×C4—41no—
181C32×C4—42no—
182C32×C4—42no—
183C32×C4—44no—
184C32×C4—122no—
185C32×C4C2×Dic611yescharacteristic
186C32×C4C2×Dic611yescharacteristic
187C32×C4D3×Q811yescharacteristic
188C32×C4D3×Q811yescharacteristic
189C32×C4C2×C3×Q811yescharacteristic
190C32×C4C2×C3×Q811yescharacteristic
191C3⋊Dic3—42no—
192C3×Dic3—44no—
193C3×Dic3—42no—
194C3×Dic3—42no—
195C3×Dic3—124no—
196C3×Dic3—122no—
197C3×Dic3—122no—
198#125—31no—
199#125—91no—
200#107—92no—
201#106—92no—
202C4×C8—31no—
203C33#49911yescharacteristic, complemented, Sylow
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—64no—
213C2×C3×C4—62no—
214C2×C3×C4—62no—
215C2×C3×C4—182no—
216C2×C3×C4#62612yes—
217C2×C3×C4C3×D1212yes—
218C2×C3×C4C3×#6712yes—
219C2×C3×C4C2×D3211yescharacteristic
220C2×C3×C4C22×C3×D311yescharacteristic
221C2×C3×C4C22×C3×D311yescharacteristic
222C2×Dic3—22no—
223C2×Dic3—64no—
224C2×Dic3—62no—
225C2×Dic3—62no—
226C3×C8—32no—
227C3×C8—32no—
228C3×C8—32no—
229C3×C8—32no—
230C3×C8—62no—
231C3×C8—62no—
232C3⋊C8C3×Dic612yes—
233C3⋊C8C3×Dic612yes—
234C2×C32—22no—
235C2×C32—21no—
236C2×C32—22no—
237C2×C32—22no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32—41no—
241C2×C32—42no—
242C2×C32#86911yescharacteristic
243C2×C32#90412yes—
244C2×C32#94411yescharacteristic
245C2×C32#95012yes—
246C2×C32C3×C4⋊Q811yescharacteristic
247C2×C32C3×#13512yes—
248C2×C8—31no—
249C2×C8—31no—
250C4⋊C4—32no—
251C4⋊C4—62no—
252C4⋊C4—92no—
253C4⋊C4—182no—
254C42C3×D3211yescharacteristic
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—41no—
259C22×C3C2×#62611yescharacteristic
260C22×C3C2×C3×D1211yescharacteristic
261C22×C3C2×C3×#6711yescharacteristic
262C3×C4—24no—
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—22no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—42no—
273C3×C4—42no—
274C3×C4—44no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—124no—
278C3×C4—122no—
279C3×C4—122no—
280C3×C4—362no—
281C3×C4D3×Dic611yescharacteristic
282C3×C4D3×Dic611yescharacteristic
283C3×C4C2×C3×Dic611yescharacteristic
284C3×C4C2×C3×Dic611yescharacteristic
285C3×C4C3×D3×Q811yescharacteristic
286C3×C4C3×D3×Q811yescharacteristic
287Dic3—42no—
288Dic3—124no—
289Dic3—122no—
290Dic3—122no—
291C32—21no—
292C32—21no—
293C32—21no—
294C32—41no—
295C32#468311yescharacteristic, complemented
296C32#506511yescharacteristic, complemented
297C32C3×#49911yescharacteristic, complemented
298C2×C4—62no—
299C2×C4—182no—
300C2×C4C3×#62612yes—
301C2×C4C2×C3×D3211yescharacteristic
302C8—32no—
303C8—32no—
304C2×C3—22no—
305C2×C3—22no—
306C2×C3—22no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—42no—
311C2×C3—41no—
312C2×C3#6382912yes—
313C2×C3#6386611yescharacteristic
314C2×C3C3×#86911yescharacteristic
315C2×C3C3×#90412yes—
316C2×C3C3×#94411yescharacteristic
317C2×C3C3×#95012yes—
318C22C2×C3×#62611yescharacteristic
319C4—22no—
320C4—122no—
321C4—362no—
322C4C3×D3×Dic611yescharacteristic
323C4C3×D3×Dic611yescharacteristic
324C3—21no—
325C3—21no—
326C3—21no—
327C3—41no—
328C3#1059030411yescharacteristic, direct product factor
329C3C3×#468311yescharacteristic, complemented
330C3C3×#506511yescharacteristic, complemented
331C2C3×#6382912yes—
332C2C3×#6386611yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C42D32
hypocenterC32C3×#499
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
8. #10590304C3direct (N × H)
203. C33#499semidirect
295. C32#4683semidirect
296. C32#5065semidirect
297. C32C3×#499semidirect
328. C3#10590304direct (N × H)
329. C3C3×#4683semidirect
330. C3C3×#5065semidirect

Maximal subgroups

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

subgroupindex
C3×#640792
C32×#8702
C32×C4×C3⋊C82
C3×#635752
C3×#635732
C3×#50653
C3×#46833
#105903043