Finite Group Databasebeta

← C2×C3×#63574 #13158156

Subgroups — C2×C3×#63574 #13158156

Order 1728.

834 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C2×C32×C4⋊Dic3C212yesmaximal
3C3×#63574C214yesmaximal, direct product factor
4C2×C3×#808—32nomaximal
5C2×#63574C311yesmaximal, characteristic, direct product factor
6C22×C33×C4C2211yescharacteristic
7C22×C32×Dic3—22no—
8C2×C3×C3⋊(C3⋊C8)C414yescomplemented
9C2×C3×C3⋊(C3⋊C8)C2212yes—
10C32×C4⋊Dic3—22no—
11C32×C4⋊Dic3C2214yes—
12C2×C32×C4⋊C4—32no—
13C22×C3⋊(C3⋊C8)C2×C311yescharacteristic
14C2×C3×C4⋊Dic3—22no—
15C2×C3×C4⋊Dic3D312yes—
16C2×C3×C4⋊Dic3C2×C312yes—
17C22×C3×C3⋊C8—32no—
18C22×C3×C3⋊C8—61no—
19C3×#808—38no—
20#63574C2×C314yesdirect product factor
21C23×C33D411yescharacteristic
22C2×C33×C4C2×C412yes—
23C2×C33×C4D411yescharacteristic
24C2×C33×C4Q812yes—
25C2×C33×C4C2311yescharacteristic
26C2×C32×Dic3—24no—
27C2×C32×Dic3—44no—
28C3×C3⋊(C3⋊C8)C2×C414yescomplemented
29C2×C3×#106—91no—
30C2×#808—32no—
31C22×C32×C4—21no—
32C22×C32×C4—22no—
33C22×C32×C4—41no—
34C22×C32×C4—62no—
35C22×C32×C4C2×D312yes—
36C22×C32×C4C22×C311yescharacteristic
37C22×C3×Dic3—24no—
38C22×C3×Dic3—22no—
39C22×C3×Dic3—22no—
40C32×C4⋊C4—34no—
41C32×C4⋊C4—62no—
42C2×C3⋊(C3⋊C8)C3×C414yescomplemented
43C2×C3⋊(C3⋊C8)C22×C312yes—
44C3×C4⋊Dic3—24no—
45C3×C4⋊Dic3—24no—
46C3×C4⋊Dic3—22no—
47C3×C4⋊Dic3—22no—
48C3×C4⋊Dic3C2×D314yes—
49C3×C4⋊Dic3C22×C314yes—
50C2×C3×C3⋊C8—38no—
51C2×C3×C3⋊C8—34no—
52C2×C3×C3⋊C8—64no—
53C2×C3×C3⋊C8—62no—
54C22×C33C4⋊C412yes—
55C22×C33SD1614yes—
56C22×C33C2×D411yescharacteristic
57C33×C4C4⋊C412yes—
58C33×C4C22×C411yescharacteristic
59C33×C4C2×Q811yescharacteristic
60C32×Dic3—44no—
61C22×C3×C8—91no—
62C2×C3×C4⋊C4—32no—
63C2×C3×C4⋊C4—32no—
64C2×C3×C4⋊C4—62no—
65C2×C4⋊Dic3C3×D312yes—
66C22×C3⋊C8—32no—
67C22×C3⋊C8—61no—
68C3×#106—94no—
69#808—38no—
70C23×C32—21no—
71C23×C32—22no—
72C23×C32—41no—
73C23×C32#6712yes—
74C23×C32C3×D411yescharacteristic
75C2×C32×C4—22no—
76C2×C32×C4—21no—
77C2×C32×C4—22no—
78C2×C32×C4—24no—
79C2×C32×C4—24no—
80C2×C32×C4—22no—
81C2×C32×C4—21no—
82C2×C32×C4—22no—
83C2×C32×C4—42no—
84C2×C32×C4—41no—
85C2×C32×C4—42no—
86C2×C32×C4—41no—
87C2×C32×C4—64no—
88C2×C32×C4—124no—
89C2×C32×C4Dic614yes—
90C2×C32×C4C4×D314yes—
91C2×C32×C4#6712yes—
92C2×C32×C4C2×C3×C412yes—
93C2×C32×C4C3×D411yescharacteristic
94C2×C32×C4C3×Q812yes—
95C2×C32×C4C22×D312yes—
96C2×C32×C4C23×C311yescharacteristic
97C2×C3×Dic3—28no—
98C2×C3×Dic3—24no—
99C2×C3×Dic3—24no—
100C2×C3×Dic3—48no—
101C2×C3×Dic3—44no—
102C2×C3×Dic3—44no—
103C3⋊(C3⋊C8)C2×C3×C414yescomplemented
104C3×C3⋊C8—38no—
105C3×C3⋊C8—64no—
106C2×#106—91noSylow
107C2×C33#10614yescomplemented
108C2×C33C2×C4⋊C411yescharacteristic
109C2×C33C2×SD1612yes—
110C22×C3×C4—21no—
111C22×C3×C4—22no—
112C22×C3×C4—41no—
113C22×C3×C4—64no—
114C22×C3×C4—62no—
115C22×C3×C4—62no—
116C22×C3×C4D3211yescharacteristic
117C22×C3×C4C2×C3×D312yes—
118C22×Dic3—22no—
119C2×C3×C8—94no—
120C2×C3×C8—92no—
121C3×C4⋊C4—34no—
122C3×C4⋊C4—34no—
123C3×C4⋊C4—64no—
124C3×C4⋊C4—64no—
125C3×C4⋊C4—62no—
126C3×C4⋊C4—62no—
127C4⋊Dic3—22no—
128C4⋊Dic3C2×C3×D314yes—
129C2×C3⋊C8—38no—
130C2×C3⋊C8—34no—
131C2×C3⋊C8—64no—
132C2×C3⋊C8—62no—
133C22×C32—24no—
134C22×C32—22no—
135C22×C32—24no—
136C22×C32—24no—
137C22×C32—24no—
138C22×C32—21no—
139C22×C32—22no—
140C22×C32—42no—
141C22×C32—44no—
142C22×C32—41no—
143C22×C32Dic3⋊C414yes—
144C22×C32#21414yes—
145C22×C32#21514yes—
146C22×C32C3×C4⋊C412yes—
147C22×C32C3×SD1614yes—
148C22×C32C2×#6712yes—
149C22×C32C2×C3×D411yescharacteristic
150C32×C4—22no—
151C32×C4—24no—
152C32×C4—21no—
153C32×C4—21no—
154C32×C4—22no—
155C32×C4—22no—
156C32×C4—42no—
157C32×C4—41no—
158C32×C4—41no—
159C32×C4—124no—
160C32×C4Dic3⋊C414yes—
161C32×C4C3×C4⋊C412yes—
162C32×C4C2×Dic612yes—
163C32×C4C2×C4×D312yes—
164C32×C4C22×C3×C411yescharacteristic
165C32×C4C2×C3×Q811yescharacteristic
166C3×Dic3—48no—
167C3×Dic3—44no—
168C3×Dic3—44no—
169C22×C8—91no—
170C2×C4⋊C4—32no—
171#106—94no—
172C33C2×#10611yescharacteristic, complemented, Sylow
173C23×C3—22no—
174C23×C3—21no—
175C23×C3—41no—
176C23×C3#62511yescharacteristic
177C23×C3C3×#6712yes—
178C2×C3×C4—24no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—21no—
182C2×C3×C4—24no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—21no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—42no—
189C2×C3×C4—41no—
190C2×C3×C4—68no—
191C2×C3×C4—64no—
192C2×C3×C4—64no—
193C2×C3×C4—128no—
194C2×C3×C4—124no—
195C2×C3×C4—124no—
196C2×C3×C4#62412yes—
197C2×C3×C4#62511yescharacteristic
198C2×C3×C4#62712yes—
199C2×C3×C4C3×Dic614yes—
200C2×C3×C4C3×C4×D314yes—
201C2×C3×C4C3×#6712yes—
202C2×C3×C4C2×D3211yescharacteristic
203C2×C3×C4C22×C3×D312yes—
204C2×Dic3—24no—
205C2×Dic3—44no—
206C3×C8—94no—
207C3⋊C8—38no—
208C3⋊C8—64no—
209C2×C32—22no—
210C2×C32—22no—
211C2×C32—22no—
212C2×C32—22no—
213C2×C32—24no—
214C2×C32—28no—
215C2×C32—21no—
216C2×C32—44no—
217C2×C32—41no—
218C2×C32—42no—
219C2×C32#80818yescomplemented
220C2×C32C3×#10614yescomplemented
221C2×C32C2×Dic3⋊C412yes—
222C2×C32C2×#21412yes—
223C2×C32C2×#21512yes—
224C2×C32C2×C3×C4⋊C411yescharacteristic
225C2×C32C2×C3×SD1612yes—
226C22×C4—62no—
227C22×C4C3×D3211yescharacteristic
228C2×C8—94no—
229C2×C8—92no—
230C4⋊C4—34no—
231C4⋊C4—62no—
232C22×C3—22no—
233C22×C3—21no—
234C22×C3—24no—
235C22×C3—24no—
236C22×C3—24no—
237C22×C3—24no—
238C22×C3—22no—
239C22×C3—41no—
240C22×C3—44no—
241C22×C3—42no—
242C22×C3#371514yes—
243C22×C3#372412yes—
244C22×C3C3×Dic3⋊C414yes—
245C22×C3C3×#21414yes—
246C22×C3C3×#21514yes—
247C22×C3C2×#62511yescharacteristic
248C22×C3C2×C3×#6712yes—
249C3×C4—21no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—21no—
253C3×C4—24no—
254C3×C4—22no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—42no—
258C3×C4—128no—
259C3×C4—124no—
260C3×C4—124no—
261C3×C4#372412yes—
262C3×C4C3×Dic3⋊C414yes—
263C3×C4C2×#62411yescharacteristic
264C3×C4C2×#62711yescharacteristic
265C3×C4C2×C3×Dic612yes—
266C3×C4C2×C3×C4×D312yes—
267Dic3—44no—
268C32—22no—
269C32—21no—
270C32—41no—
271C32C2×#80812yescomplemented
272C32C2×C3×#10611yescharacteristic, complemented
273C23C3×#62511yescharacteristic
274C2×C4—64no—
275C2×C4—124no—
276C2×C4C3×#62412yes—
277C2×C4C3×#62511yescharacteristic
278C2×C4C3×#62712yes—
279C2×C4C2×C3×D3211yescharacteristic
280C8—94no—
281C2×C3—28no—
282C2×C3—24no—
283C2×C3—22no—
284C2×C3—21no—
285C2×C3—22no—
286C2×C3—22no—
287C2×C3—22no—
288C2×C3—44no—
289C2×C3—41no—
290C2×C3—42no—
291C2×C3#6357414yesdirect product factor
292C2×C3C3×#80818yescomplemented
293C2×C3C2×#371512yes—
294C2×C3C2×#372411yescharacteristic
295C2×C3C2×C3×Dic3⋊C412yes—
296C2×C3C2×C3×#21412yes—
297C2×C3C2×C3×#21512yes—
298C22C3×#371514yes—
299C22C3×#372412yes—
300C22C2×C3×#62511yescharacteristic
301C4—124no—
302C4C3×#372412yes—
303C4C2×C3×#62411yescharacteristic
304C4C2×C3×#62711yescharacteristic
305C3—22no—
306C3—21no—
307C3—41no—
308C3C2×#6357411yescharacteristic, direct product factor
309C3C2×C3×#80812yescomplemented
310C2C3×#6357414yesdirect product factor
311C2C2×C3×#371512yes—
312C2C2×C3×#372411yescharacteristic
subgroupG/H
centerC23×C3#625
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33D4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×#106
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63574C2direct (N × H)
5. C2×#63574C3direct (N × H)
8. C2×C3×C3⋊(C3⋊C8)C4semidirect
20. #63574C2×C3direct (N × H)
28. C3×C3⋊(C3⋊C8)C2×C4semidirect
42. C2×C3⋊(C3⋊C8)C3×C4semidirect
103. C3⋊(C3⋊C8)C2×C3×C4semidirect
107. C2×C33#106semidirect
172. C33C2×#106semidirect
219. C2×C32#808semidirect
220. C2×C32C3×#106semidirect
271. C32C2×#808semidirect
272. C32C2×C3×#106semidirect
291. C2×C3#63574direct (N × H)
292. C2×C3C3×#808semidirect
308. C3C2×#63574direct (N × H)
309. C3C2×C3×#808semidirect
310. C2C3×#63574direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊(C3⋊C8)2
C2×C32×C4⋊Dic32
C3×#635742
C2×C3×#8083
C2×#635743