Finite Group Databasebeta

← C3×#10591120 #13158152

Subgroups — C3×#10591120 #13158152

Order 1728.

802 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C32×#208C212yesmaximal
3C3×#63572C214yesmaximal, complemented
4C3×#5118—32nomaximal
5#10591120C311yesmaximal, characteristic, direct product factor
6C22×C33×C4C2211yescharacteristic
7C22×C3×C3⋊Dic3C412yes—
8C3×C4×C3⋊Dic3C411yescharacteristic
9C3×C4×C3⋊Dic3C411yescharacteristic
10C3×C4×C3⋊Dic3C2212yes—
11C32×#208—24no—
12C2×C32×C3⋊C8C2214yes—
13C2×C32×M16—32no—
14C2×C4×C3⋊Dic3C2×C311yescharacteristic
15C2×C3×C4×Dic3—32no—
16C2×C3×C4×Dic3—61no—
17C2×C3×#208—22no—
18C2×C3×#208D312yes—
19C2×C3×#208C2×C312yes—
20C3×Dic3⋊C8—38no—
21#63572C2×C314yescomplemented
22C23×C33C2×C411yescharacteristic
23C2×C33×C4C2×C411yescharacteristic
24C2×C33×C4D412yes—
25C2×C33×C4Q812yes—
26C2×C33×C4C2311yescharacteristic
27C2×C3×C3⋊Dic3—24no—
28C2×C3×C3⋊Dic3C2×C412yes—
29C2×C3×C3⋊Dic3C2×C412yes—
30C32×C3⋊C8—24no—
31C3×#423—91no—
32#5118—32no—
33C22×C32×C4—21no—
34C22×C32×C4—22no—
35C22×C32×C4—41no—
36C22×C32×C4C2×D312yes—
37C22×C32×C4C22×C311yescharacteristic
38C22×C3⋊Dic3C3×C412yes—
39C22×C3×Dic3—34no—
40C22×C3×Dic3—62no—
41C32×M16—64no—
42C2×C32×C8—34no—
43C4×C3⋊Dic3C3×C411yescharacteristic
44C4×C3⋊Dic3C3×C411yescharacteristic
45C4×C3⋊Dic3C22×C312yes—
46C3×C4×Dic3—32no—
47C3×C4×Dic3—32no—
48C3×C4×Dic3—34no—
49C3×C4×Dic3—61no—
50C3×C4×Dic3—61no—
51C3×C4×Dic3—62no—
52C3×#208—28no—
53C3×#208—24no—
54C3×#208—24no—
55C2×C3×C3⋊C8—24no—
56C2×C3×C3⋊C8C2×D314yes—
57C2×C3×C3⋊C8C22×C314yes—
58C22×C33—22no—
59C22×C33C4⋊C412yes—
60C22×C33C22×C411yescharacteristic
61C33×C4C4⋊C412yes—
62C33×C4C2×D411yescharacteristic
63C33×C4C2×Q811yescharacteristic
64C3×C3⋊Dic3—22no—
65C3×C3⋊Dic3M1614yescomplemented
66C2×C3×M16—32no—
67C2×C3×M16—32no—
68C2×C3×M16—62no—
69C2×C3×C42—91no—
70C2×C4×Dic3—32no—
71C2×C4×Dic3—61no—
72C2×#208C3×D312yes—
73C3×C4⋊C8—94no—
74Dic3⋊C8—38no—
75C23×C32—21no—
76C23×C32—22no—
77C23×C32—41no—
78C23×C32C4×D312yes—
79C23×C32C2×C3×C411yescharacteristic
80C2×C32×C4—21no—
81C2×C32×C4—22no—
82C2×C32×C4—22no—
83C2×C32×C4—24no—
84C2×C32×C4—22no—
85C2×C32×C4—24no—
86C2×C32×C4—21no—
87C2×C32×C4—22no—
88C2×C32×C4—41no—
89C2×C32×C4—42no—
90C2×C32×C4—42no—
91C2×C32×C4—41no—
92C2×C32×C4Dic614yes—
93C2×C32×C4C4×D312yes—
94C2×C32×C4#6714yes—
95C2×C32×C4C2×C3×C411yescharacteristic
96C2×C32×C4C3×D412yes—
97C2×C32×C4C3×Q812yes—
98C2×C32×C4C22×D312yes—
99C2×C32×C4C23×C311yescharacteristic
100C2×C3⋊Dic3—24no—
101C2×C3⋊Dic3C2×C3×C412yes—
102C2×C3⋊Dic3C2×C3×C412yes—
103C2×C3×Dic3—34no—
104C2×C3×Dic3—34no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—68no—
108C2×C3×Dic3—68no—
109C32×C8—64no—
110C3×C3⋊C8—28no—
111C3×C3⋊C8—24no—
112C3×C3⋊C8—24no—
113#423—91noSylow
114C2×C33—22no—
115C2×C33C2×C4⋊C411yescharacteristic
116C2×C33C2×M1612yes—
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—41no—
120C22×C3×C4—92no—
121C22×C3×C4D3211yescharacteristic
122C22×C3×C4C2×C3×D312yes—
123C22×Dic3—34no—
124C22×Dic3—62no—
125C3×M16—68no—
126C3×M16—64no—
127C3×M16—64no—
128C2×C3×C8—34no—
129C2×C3×C8—34no—
130C2×C3×C8—64no—
131C3×C42—91no—
132C3×C42—91no—
133C3×C42—92no—
134C4×Dic3—32no—
135C4×Dic3—32no—
136C4×Dic3—34no—
137C4×Dic3—61no—
138C4×Dic3—61no—
139C4×Dic3—62no—
140#208—24no—
141C2×C3⋊C8C2×C3×D314yes—
142C22×C32—22no—
143C22×C32—24no—
144C22×C32—24no—
145C22×C32—22no—
146C22×C32—24no—
147C22×C32—21no—
148C22×C32—22no—
149C22×C32—42no—
150C22×C32—41no—
151C22×C32—44no—
152C22×C32—44no—
153C22×C32Dic3⋊C414yes—
154C22×C32C3×C4⋊C412yes—
155C22×C32C2×C4×D312yes—
156C22×C32C22×C3×C411yescharacteristic
157C32×C4—22no—
158C32×C4—24no—
159C32×C4—21no—
160C32×C4—21no—
161C32×C4—22no—
162C32×C4—22no—
163C32×C4—42no—
164C32×C4—41no—
165C32×C4—41no—
166C32×C4Dic3⋊C414yes—
167C32×C4C3×C4⋊C412yes—
168C32×C4C2×Dic612yes—
169C32×C4C2×#6712yes—
170C32×C4C2×C3×D411yescharacteristic
171C32×C4C2×C3×Q811yescharacteristic
172C3⋊Dic3—22no—
173C3⋊Dic3C3×M1614yescomplemented
174C3×Dic3—38no—
175C3×Dic3—64no—
176C3×Dic3—64no—
177C3×Dic3—64no—
178C2×M16—32no—
179C2×C42—91no—
180C4⋊C8—94no—
181C33#42311yescharacteristic, complemented, Sylow
182C23×C3—21no—
183C23×C3—22no—
184C23×C3—41no—
185C23×C3#62411yescharacteristic
186C23×C3C3×C4×D312yes—
187C2×C3×C4—22no—
188C2×C3×C4—21no—
189C2×C3×C4—24no—
190C2×C3×C4—22no—
191C2×C3×C4—24no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—21no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—41no—
199C2×C3×C4—92no—
200C2×C3×C4—92no—
201C2×C3×C4—184no—
202C2×C3×C4#62411yescharacteristic
203C2×C3×C4#62512yes—
204C2×C3×C4#62712yes—
205C2×C3×C4C3×Dic614yes—
206C2×C3×C4C3×C4×D312yes—
207C2×C3×C4C3×#6714yes—
208C2×C3×C4C2×D3211yescharacteristic
209C2×C3×C4C22×C3×D312yes—
210C2×Dic3—34no—
211C2×Dic3—34no—
212C2×Dic3—62no—
213C2×Dic3—62no—
214C2×Dic3—68no—
215C2×Dic3—68no—
216C3×C8—68no—
217C3×C8—64no—
218C3×C8—64no—
219C3⋊C8—24no—
220C2×C32—22no—
221C2×C32—24no—
222C2×C32—24no—
223C2×C32—24no—
224C2×C32—22no—
225C2×C32—22no—
226C2×C32—21no—
227C2×C32—41no—
228C2×C32—42no—
229C2×C32—44no—
230C2×C32—44no—
231C2×C32D3×M1614yes—
232C2×C32C2×Dic3⋊C412yes—
233C2×C32C2×C3×C4⋊C411yescharacteristic
234C2×C32C2×C3×M1612yes—
235C22×C4—92no—
236C22×C4C3×D3211yescharacteristic
237M16—64no—
238C2×C8—34no—
239C42—91no—
240C42—91no—
241C42—92no—
242C22×C3—24no—
243C22×C3—24no—
244C22×C3—22no—
245C22×C3—21no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—24no—
249C22×C3—44no—
250C22×C3—44no—
251C22×C3—41no—
252C22×C3—42no—
253C22×C3#372412yes—
254C22×C3C3×Dic3⋊C414yes—
255C22×C3C2×#62411yescharacteristic
256C22×C3C2×C3×C4×D312yes—
257C3×C4—22no—
258C3×C4—21no—
259C3×C4—22no—
260C3×C4—21no—
261C3×C4—24no—
262C3×C4—22no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—42no—
266C3×C4—94no—
267C3×C4—182no—
268C3×C4#372412yes—
269C3×C4C3×Dic3⋊C414yes—
270C3×C4C2×#62511yescharacteristic
271C3×C4C2×#62711yescharacteristic
272C3×C4C2×C3×Dic612yes—
273C3×C4C2×C3×#6712yes—
274Dic3—38no—
275Dic3—64no—
276Dic3—64no—
277Dic3—64no—
278C32—21no—
279C32—22no—
280C32—41no—
281C32#511812yescomplemented
282C32C3×#42311yescharacteristic, complemented
283C23C3×#62411yescharacteristic
284C2×C4—92no—
285C2×C4—92no—
286C2×C4—184no—
287C2×C4C3×#62411yescharacteristic
288C2×C4C3×#62512yes—
289C2×C4C3×#62712yes—
290C2×C4C2×C3×D3211yescharacteristic
291C8—64no—
292C2×C3—24no—
293C2×C3—22no—
294C2×C3—24no—
295C2×C3—22no—
296C2×C3—21no—
297C2×C3—22no—
298C2×C3—24no—
299C2×C3—44no—
300C2×C3—44no—
301C2×C3—41no—
302C2×C3—42no—
303C2×C3#6381812yes—
304C2×C3C2×#372411yescharacteristic
305C2×C3C3×D3×M1614yes—
306C2×C3C2×C3×Dic3⋊C412yes—
307C22—22no—
308C22C3×#372412yes—
309C22C2×C3×#62411yescharacteristic
310C4—94no—
311C4—182no—
312C4C3×#372412yes—
313C4C2×C3×#62511yescharacteristic
314C4C2×C3×#62711yescharacteristic
315C3—21no—
316C3—22no—
317C3—41no—
318C3#1059112011yescharacteristic, direct product factor
319C3C3×#511812yescomplemented
320C2—22no—
321C2C3×#6381812yes—
322C2C2×C3×#372411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC23×C33C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#423
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63572C2semidirect
5. #10591120C3direct (N × H)
21. #63572C2×C3semidirect
65. C3×C3⋊Dic3M16semidirect
173. C3⋊Dic3C3×M16semidirect
181. C33#423semidirect
281. C32#5118semidirect
282. C32C3×#423semidirect
318. C3#10591120direct (N × H)
319. C3C3×#5118semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊Dic32
C2×C32×#2082
C3×#635722
C3×#51183
#105911203