Finite Group Databasebeta

← C2×C3×#63576 #13158165

Subgroups — C2×C3×#63576 #13158165

Order 1728.

834 conjugacy classes of subgroups.

The subgroup lattice has 330 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×#63576C214yesmaximal, direct product factor
4C2×C3×#807—32nomaximal
5C2×#63576C311yesmaximal, 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×#807—38no—
20#63576C2×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×#107—91no—
30C2×#807—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×C33D812yes—
56C22×C33Q1612yes—
57C22×C33C2×D411yescharacteristic
58C33×C4C4⋊C412yes—
59C33×C4C22×C411yescharacteristic
60C33×C4C2×Q811yescharacteristic
61C32×Dic3—44no—
62C22×C3×C8—91no—
63C2×C3×C4⋊C4—32no—
64C2×C3×C4⋊C4—32no—
65C2×C3×C4⋊C4—62no—
66C2×C4⋊Dic3C3×D312yes—
67C22×C3⋊C8—32no—
68C22×C3⋊C8—61no—
69C3×#107—94no—
70#807—38no—
71C23×C32—21no—
72C23×C32—22no—
73C23×C32—41no—
74C23×C32#6712yes—
75C23×C32C3×D411yescharacteristic
76C2×C32×C4—22no—
77C2×C32×C4—21no—
78C2×C32×C4—22no—
79C2×C32×C4—24no—
80C2×C32×C4—24no—
81C2×C32×C4—22no—
82C2×C32×C4—21no—
83C2×C32×C4—22no—
84C2×C32×C4—42no—
85C2×C32×C4—41no—
86C2×C32×C4—42no—
87C2×C32×C4—41no—
88C2×C32×C4—64no—
89C2×C32×C4—124no—
90C2×C32×C4Dic614yes—
91C2×C32×C4C4×D314yes—
92C2×C32×C4#6712yes—
93C2×C32×C4C2×C3×C412yes—
94C2×C32×C4C3×D411yescharacteristic
95C2×C32×C4C3×Q812yes—
96C2×C32×C4C22×D312yes—
97C2×C32×C4C23×C311yescharacteristic
98C2×C3×Dic3—28no—
99C2×C3×Dic3—24no—
100C2×C3×Dic3—24no—
101C2×C3×Dic3—48no—
102C2×C3×Dic3—44no—
103C2×C3×Dic3—44no—
104C3⋊(C3⋊C8)C2×C3×C414yescomplemented
105C3×C3⋊C8—38no—
106C3×C3⋊C8—64no—
107C2×#107—91noSylow
108C2×C33#10714yescomplemented
109C2×C33C2×C4⋊C411yescharacteristic
110C2×C33C2×D811yescharacteristic
111C2×C33C2×Q1611yescharacteristic
112C22×C3×C4—21no—
113C22×C3×C4—22no—
114C22×C3×C4—41no—
115C22×C3×C4—64no—
116C22×C3×C4—62no—
117C22×C3×C4—62no—
118C22×C3×C4D3211yescharacteristic
119C22×C3×C4C2×C3×D312yes—
120C22×Dic3—22no—
121C2×C3×C8—94no—
122C2×C3×C8—92no—
123C3×C4⋊C4—34no—
124C3×C4⋊C4—34no—
125C3×C4⋊C4—64no—
126C3×C4⋊C4—64no—
127C3×C4⋊C4—62no—
128C3×C4⋊C4—62no—
129C4⋊Dic3—22no—
130C4⋊Dic3C2×C3×D314yes—
131C2×C3⋊C8—38no—
132C2×C3⋊C8—34no—
133C2×C3⋊C8—64no—
134C2×C3⋊C8—62no—
135C22×C32—22no—
136C22×C32—22no—
137C22×C32—22no—
138C22×C32—24no—
139C22×C32—24no—
140C22×C32—24no—
141C22×C32—21no—
142C22×C32—22no—
143C22×C32—42no—
144C22×C32—42no—
145C22×C32—42no—
146C22×C32—41no—
147C22×C32Dic3⋊C414yes—
148C22×C32#21314yes—
149C22×C32#21614yes—
150C22×C32C3×C4⋊C412yes—
151C22×C32C3×D812yes—
152C22×C32C3×Q1612yes—
153C22×C32C2×#6712yes—
154C22×C32C2×C3×D411yescharacteristic
155C32×C4—22no—
156C32×C4—24no—
157C32×C4—21no—
158C32×C4—21no—
159C32×C4—22no—
160C32×C4—22no—
161C32×C4—42no—
162C32×C4—41no—
163C32×C4—41no—
164C32×C4—124no—
165C32×C4Dic3⋊C414yes—
166C32×C4C3×C4⋊C412yes—
167C32×C4C2×Dic612yes—
168C32×C4C2×C4×D312yes—
169C32×C4C22×C3×C411yescharacteristic
170C32×C4C2×C3×Q811yescharacteristic
171C3×Dic3—48no—
172C3×Dic3—44no—
173C3×Dic3—44no—
174C22×C8—91no—
175C2×C4⋊C4—32no—
176#107—94no—
177C33C2×#10711yescharacteristic, complemented, Sylow
178C23×C3—22no—
179C23×C3—21no—
180C23×C3—41no—
181C23×C3#62511yescharacteristic
182C23×C3C3×#6712yes—
183C2×C3×C4—24no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—21no—
187C2×C3×C4—24no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—21no—
191C2×C3×C4—42no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4—41no—
195C2×C3×C4—68no—
196C2×C3×C4—64no—
197C2×C3×C4—64no—
198C2×C3×C4—128no—
199C2×C3×C4—124no—
200C2×C3×C4—124no—
201C2×C3×C4#62412yes—
202C2×C3×C4#62511yescharacteristic
203C2×C3×C4#62712yes—
204C2×C3×C4C3×Dic614yes—
205C2×C3×C4C3×C4×D314yes—
206C2×C3×C4C3×#6712yes—
207C2×C3×C4C2×D3211yescharacteristic
208C2×C3×C4C22×C3×D312yes—
209C2×Dic3—24no—
210C2×Dic3—44no—
211C3×C8—94no—
212C3⋊C8—38no—
213C3⋊C8—64no—
214C2×C32—22no—
215C2×C32—22no—
216C2×C32—22no—
217C2×C32—21no—
218C2×C32—24no—
219C2×C32—21no—
220C2×C32—28no—
221C2×C32—21no—
222C2×C32—44no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32#80718yescomplemented
227C2×C32C3×#10714yescomplemented
228C2×C32C2×Dic3⋊C412yes—
229C2×C32C2×#21312yes—
230C2×C32C2×#21612yes—
231C2×C32C2×C3×C4⋊C411yescharacteristic
232C2×C32C2×C3×D811yescharacteristic
233C2×C32C2×C3×Q1611yescharacteristic
234C22×C4—62no—
235C22×C4C3×D3211yescharacteristic
236C2×C8—94no—
237C2×C8—92no—
238C4⋊C4—34no—
239C4⋊C4—62no—
240C22×C3—22no—
241C22×C3—21no—
242C22×C3—24no—
243C22×C3—22no—
244C22×C3—24no—
245C22×C3—22no—
246C22×C3—24no—
247C22×C3—22no—
248C22×C3—41no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3—42no—
252C22×C3#371312yes—
253C22×C3#371812yes—
254C22×C3#372412yes—
255C22×C3C3×Dic3⋊C414yes—
256C22×C3C3×#21314yes—
257C22×C3C3×#21614yes—
258C22×C3C2×#62511yescharacteristic
259C22×C3C2×C3×#6712yes—
260C3×C4—21no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—21no—
264C3×C4—24no—
265C3×C4—22no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—42no—
269C3×C4—128no—
270C3×C4—124no—
271C3×C4—124no—
272C3×C4#372412yes—
273C3×C4C3×Dic3⋊C414yes—
274C3×C4C2×#62411yescharacteristic
275C3×C4C2×#62711yescharacteristic
276C3×C4C2×C3×Dic612yes—
277C3×C4C2×C3×C4×D312yes—
278Dic3—44no—
279C32—22no—
280C32—21no—
281C32—41no—
282C32C2×#80712yescomplemented
283C32C2×C3×#10711yescharacteristic, complemented
284C23C3×#62511yescharacteristic
285C2×C4—64no—
286C2×C4—124no—
287C2×C4C3×#62412yes—
288C2×C4C3×#62511yescharacteristic
289C2×C4C3×#62712yes—
290C2×C4C2×C3×D3211yescharacteristic
291C8—94no—
292C2×C3—28no—
293C2×C3—24no—
294C2×C3—22no—
295C2×C3—21no—
296C2×C3—22no—
297C2×C3—21no—
298C2×C3—22no—
299C2×C3—21no—
300C2×C3—44no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3—41no—
304C2×C3#6357614yesdirect product factor
305C2×C3C3×#80718yescomplemented
306C2×C3C2×#371311yescharacteristic
307C2×C3C2×#371811yescharacteristic
308C2×C3C2×#372411yescharacteristic
309C2×C3C2×C3×Dic3⋊C412yes—
310C2×C3C2×C3×#21312yes—
311C2×C3C2×C3×#21612yes—
312C22C3×#371312yes—
313C22C3×#371812yes—
314C22C3×#372412yes—
315C22C2×C3×#62511yescharacteristic
316C4—124no—
317C4C3×#372412yes—
318C4C2×C3×#62411yescharacteristic
319C4C2×C3×#62711yescharacteristic
320C3—22no—
321C3—21no—
322C3—41no—
323C3C2×#6357611yescharacteristic, direct product factor
324C3C2×C3×#80712yescomplemented
325C2C3×#6357614yesdirect product factor
326C2C2×C3×#371311yescharacteristic
327C2C2×C3×#371811yescharacteristic
328C2C2×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×#107
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×#63576C2direct (N × H)
5. C2×#63576C3direct (N × H)
8. C2×C3×C3⋊(C3⋊C8)C4semidirect
20. #63576C2×C3direct (N × H)
28. C3×C3⋊(C3⋊C8)C2×C4semidirect
42. C2×C3⋊(C3⋊C8)C3×C4semidirect
104. C3⋊(C3⋊C8)C2×C3×C4semidirect
108. C2×C33#107semidirect
177. C33C2×#107semidirect
226. C2×C32#807semidirect
227. C2×C32C3×#107semidirect
282. C32C2×#807semidirect
283. C32C2×C3×#107semidirect
304. C2×C3#63576direct (N × H)
305. C2×C3C3×#807semidirect
323. C3C2×#63576direct (N × H)
324. C3C2×C3×#807semidirect
325. C2C3×#63576direct (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×#635762
C2×C3×#8073
C2×#635763