Finite Group Databasebeta

← C3×#10590889 #13157852

Subgroups — C3×#10590889 #13157852

Order 1728.

530 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63918C212yesmaximal
2C3×#63644C211yesmaximal, characteristic
3C32×#817C211yesmaximal, characteristic
4C32×C8×Dic3C211yesmaximal, characteristic
5C3×#63573C212yesmaximal
6C3×#4855—31nomaximal
7C3×#4680—31nomaximal
8#10590889C311yesmaximal, characteristic, direct product factor
9C2×C3×#627—22no—
10C2×C33×C8C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C32×C4⋊Dic3C2212yes—
13C32×C4×Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2211yescharacteristic
15C3×Dic3⋊Dic3—22no—
16C3×#888—32no—
17C3×#869—32no—
18#63918C2×C312yes—
19C32×#106—31no—
20C32×C4×C8—31no—
21#63644C2×C311yescharacteristic
22C3×#817—21no—
23C3×#817—31no—
24C3×#817—31no—
25C3×#817—32no—
26C3×#817—61no—
27C3×#817D311yescharacteristic
28C3×#817C2×C311yescharacteristic
29C3×C8×Dic3—21no—
30C3×C8×Dic3D311yescharacteristic
31C3×C8×Dic3C2×C311yescharacteristic
32C3×#808—32no—
33#63573C2×C312yes—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—22no—
36C2×C32×Dic3—22no—
37C2×C32×Dic3D412yes—
38C3×#627—44no—
39C33×C8Q812yescomplemented
40C32×C3⋊C8Q812yescomplemented
41C3×#498—91no—
42#4855—31no—
43#4680—31no—
44C2×C3×Dic6—62no—
45C2×C3×Dic6—62no—
46C2×#627—22no—
47C2×C32×C8—21no—
48C2×C32×C8—21no—
49C2×C32×C8—21no—
50C2×C32×C8—31no—
51C2×C32×C8—41no—
52C2×C32×C8C2×D311yescharacteristic
53C2×C32×C8C2×D311yescharacteristic
54C2×C32×C8C22×C311yescharacteristic
55C32×C4⋊C4—32no—
56C32×C42—31no—
57C4⋊(C3⋊Dic3)C22×C312yes—
58C3×C4⋊Dic3—22no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—32no—
61C3×C4⋊Dic3—62no—
62C3×C4⋊Dic3—62no—
63C3×C4⋊Dic3C2×D312yes—
64C3×C4⋊Dic3C22×C312yes—
65C3×Dic3⋊C4—62no—
66C3×C4×Dic3—21no—
67C3×C4×Dic3C2×D311yescharacteristic
68C3×C4×Dic3C22×C311yescharacteristic
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8C2×D311yescharacteristic
71C2×C3×C3⋊C8C22×C311yescharacteristic
72Dic3⋊Dic3—22no—
73C22×C33C2×D411yescharacteristic
74C33×C4C2×Q811yescharacteristic
75C33×C4C2×Q811yescharacteristic
76C3×C3⋊Dic3—42no—
77C32×Dic3—42no—
78C32×Dic3SD1614yes—
79C3×C4⋊Q8—92no—
80#888—32no—
81#869—32no—
82C3×#106—31no—
83C3×#106—31no—
84C3×#106—61no—
85C3×#106—92no—
86C3×#106—91no—
87C3×C4×C8—31no—
88C3×C4×C8—31no—
89C3×C4×C8—61no—
90#817—31no—
91#817—31no—
92#817—32no—
93#817—61no—
94#817C3×D311yescharacteristic
95C8×Dic3C3×D311yescharacteristic
96#808—32no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—32no—
101C2×C32×C4—41no—
102C2×C32×C4—62no—
103C2×C32×C4C22×D311yescharacteristic
104C2×C32×C4C22×D311yescharacteristic
105C2×C32×C4C23×C311yescharacteristic
106C2×C3⋊Dic3—22no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—42no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—122no—
114C2×C3×Dic3D1212yes—
115C2×C3×Dic3C3×D412yes—
116C3×Dic6—124no—
117C3×Dic6—124no—
118#627—44no—
119C32×C8—22no—
120C32×C8—22no—
121C32×C8—22no—
122C32×C8—32no—
123C32×C8—42no—
124C32×C8Dic612yescomplemented
125C32×C8Dic612yescomplemented
126C32×C8C3×Q812yescomplemented
127C3×C3⋊C8—22no—
128C3×C3⋊C8Dic612yescomplemented
129C3×C3⋊C8C3×Q812yescomplemented
130#498—91noSylow
131C2×C33C4⋊Q811yescharacteristic
132C2×C33C2×SD1612yes—
133C2×C3×Q8—182no—
134C2×Dic6—62no—
135C2×Dic6—62no—
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C2×C3×C8—21no—
139C2×C3×C8—31no—
140C2×C3×C8—31no—
141C2×C3×C8—41no—
142C2×C3×C8—61no—
143C2×C3×C8D3211yescharacteristic
144C2×C3×C8C2×C3×D311yescharacteristic
145C2×C3×C8C2×C3×D311yescharacteristic
146C3×C4⋊C4—32no—
147C3×C4⋊C4—32no—
148C3×C4⋊C4—62no—
149C3×C4⋊C4—92no—
150C3×C4⋊C4—182no—
151C3×C42—31no—
152C3×C42—31no—
153C3×C42—61no—
154C4⋊Dic3—32no—
155C4⋊Dic3—32no—
156C4⋊Dic3—62no—
157C4⋊Dic3—62no—
158C4⋊Dic3C2×C3×D312yes—
159Dic3⋊C4—62no—
160C4×Dic3C2×C3×D311yescharacteristic
161C2×C3⋊C8C2×C3×D311yescharacteristic
162C22×C32—21no—
163C22×C32—21no—
164C22×C32—21no—
165C22×C32—41no—
166C22×C32C2×D1211yescharacteristic
167C22×C32D3×D411yescharacteristic
168C22×C32C2×C3×D411yescharacteristic
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—21no—
172C32×C4—21no—
173C32×C4—21no—
174C32×C4—21no—
175C32×C4—34no—
176C32×C4—41no—
177C32×C4—41no—
178C32×C4—122no—
179C32×C4C2×Dic611yescharacteristic
180C32×C4C2×Dic611yescharacteristic
181C32×C4C2×Dic611yescharacteristic
182C32×C4D3×Q811yescharacteristic
183C32×C4C2×C3×Q811yescharacteristic
184C32×C4C2×C3×Q811yescharacteristic
185C3⋊Dic3—42no—
186C3×Dic3—24no—
187C3×Dic3—42no—
188C3×Dic3—42no—
189C3×Dic3—82no—
190C3×Dic3—122no—
191C3×Dic3—122no—
192C3×Dic3—242no—
193C3×Dic3#20414yes—
194C3×Dic3C3×SD1614yes—
195C4⋊Q8—92no—
196#106—31no—
197#106—92no—
198#106—91no—
199C4×C8—31no—
200C33#49811yescharacteristic, complemented, Sylow
201C3×Q8—364no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—32no—
206C2×C3×C4—32no—
207C2×C3×C4—41no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—122no—
212C2×C3×C4—182no—
213C2×C3×C4C2×D3211yescharacteristic
214C2×C3×C4C22×C3×D311yescharacteristic
215C2×C3×C4C22×C3×D311yescharacteristic
216C2×Dic3—22no—
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—122no—
220C2×Dic3C3×D1212yes—
221Dic6—124no—
222Dic6—124no—
223C3×C8—22no—
224C3×C8—22no—
225C3×C8—22no—
226C3×C8—32no—
227C3×C8—32no—
228C3×C8—42no—
229C3×C8—62no—
230C3×C8#62712yescomplemented
231C3×C8C3×Dic612yescomplemented
232C3×C8C3×Dic612yescomplemented
233C3⋊C8C3×Dic612yescomplemented
234C2×C32—21no—
235C2×C32—22no—
236C2×C32—22no—
237C2×C32—22no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32—41no—
241C2×C32—42no—
242C2×C32#86911yescharacteristic
243C2×C32#88811yescharacteristic
244C2×C32C2×#20412yes—
245C2×C32D3×SD1612yes—
246C2×C32C3×C4⋊Q811yescharacteristic
247C2×C32C2×C3×SD1612yes—
248C2×Q8—182no—
249C2×C8—31no—
250C2×C8C3×D3211yescharacteristic
251C4⋊C4—32no—
252C4⋊C4—92no—
253C4⋊C4—182no—
254C42—31no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—41no—
259C22×C3D3×D1211yescharacteristic
260C22×C3C2×C3×D1211yescharacteristic
261C22×C3C3×D3×D411yescharacteristic
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—34no—
269C3×C4—34no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—64no—
273C3×C4—122no—
274C3×C4—122no—
275C3×C4—242no—
276C3×C4—362no—
277C3×C4D3×Dic611yescharacteristic
278C3×C4C2×#62711yescharacteristic
279C3×C4C2×C3×Dic611yescharacteristic
280C3×C4C2×C3×Dic611yescharacteristic
281C3×C4C2×C3×Dic611yescharacteristic
282C3×C4C3×D3×Q811yescharacteristic
283Dic3—42no—
284Dic3—122no—
285Dic3—122no—
286Dic3—242no—
287Dic3C3×#20414yes—
288C32—21no—
289C32—21no—
290C32—21no—
291C32—41no—
292C32#468011yescharacteristic, complemented
293C32#485511yescharacteristic, complemented
294C32C3×#49811yescharacteristic, complemented
295Q8—364no—
296C2×C4—32no—
297C2×C4—62no—
298C2×C4—182no—
299C2×C4C2×C3×D3211yescharacteristic
300C8—32no—
301C8C3×#62712yescomplemented
302C2×C3—22no—
303C2×C3—22no—
304C2×C3—22no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—42no—
309C2×C3—41no—
310C2×C3D3×#20412yes—
311C2×C3#6391811yescharacteristic
312C2×C3C3×#86911yescharacteristic
313C2×C3C3×#88811yescharacteristic
314C2×C3C2×C3×#20412yes—
315C2×C3C3×D3×SD1612yes—
316C22C3×D3×D1211yescharacteristic
317C4—34no—
318C4—122no—
319C4—362no—
320C4C3×D3×Dic611yescharacteristic
321C4C2×C3×#62711yescharacteristic
322C3—21no—
323C3—21no—
324C3—21no—
325C3—41no—
326C3#1059088911yescharacteristic, direct product factor
327C3C3×#468011yescharacteristic, complemented
328C3C3×#485511yescharacteristic, complemented
329C2C3×D3×#20412yes—
330C2C3×#6391811yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×C8D32
hypocenterC32C3×#498
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
8. #10590889C3direct (N × H)
39. C33×C8Q8semidirect
40. C32×C3⋊C8Q8semidirect
124. C32×C8Dic6semidirect
125. C32×C8Dic6semidirect
126. C32×C8C3×Q8semidirect
128. C3×C3⋊C8Dic6semidirect
129. C3×C3⋊C8C3×Q8semidirect
200. C33#498semidirect
230. C3×C8#627semidirect
231. C3×C8C3×Dic6semidirect
232. C3×C8C3×Dic6semidirect
233. C3⋊C8C3×Dic6semidirect
292. C32#4680semidirect
293. C32#4855semidirect
294. C32C3×#498semidirect
301. C8C3×#627semidirect
326. C3#10590889direct (N × H)
327. C3C3×#4680semidirect
328. C3C3×#4855semidirect

Maximal subgroups

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

subgroupindex
C3×#639182
C3×#636442
C32×#8172
C32×C8×Dic32
C3×#635732
C3×#48553
C3×#46803
#105908893