Finite Group Databasebeta

← C3×#10591060 #13158088

Subgroups — C3×#10591060 #13158088

Order 1728.

866 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
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C32×#208C212yesmaximal
3C3×#63555C212yesmaximal, complemented
4C3×#63553C212yesmaximal, complemented
5C3×Dic3×M16—32nomaximal
6#10591060C311yesmaximal, characteristic, direct product factor
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C412yes—
9C3×C4×C3⋊Dic3C412yes—
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×#208C418yescomplemented
13C2×C32×C3⋊C8C2214yes—
14C2×C32×M16—32no—
15C2×C4×C3⋊Dic3C2×C311yescharacteristic
16C2×C3×C4×Dic3—32no—
17C2×C3×C4×Dic3—61no—
18C2×C3×#208—22no—
19C2×C3×#208D312yes—
20C2×C3×#208C2×C312yes—
21C3×#815—34no—
22C3×C8×Dic3—34no—
23#63555C2×C312yescomplemented
24#63553C2×C312yescomplemented
25C23×C33C2×C411yescharacteristic
26C2×C33×C4C2×C411yescharacteristic
27C2×C33×C4C2×C414yes—
28C2×C33×C4C2311yescharacteristic
29C2×C3×C3⋊Dic3—24no—
30C2×C3×C3⋊Dic3C2×C412yes—
31C2×C3×C3⋊Dic3C2×C412yes—
32C32×C3⋊C8C2×C418yescomplemented
33C3×C4×M16—91no—
34Dic3×M16—32no—
35C22×C32×C4—21no—
36C22×C32×C4—22no—
37C22×C32×C4—41no—
38C22×C32×C4C2×D312yes—
39C22×C32×C4C22×C311yescharacteristic
40C22×C3⋊Dic3C3×C412yes—
41C22×C3×Dic3—34no—
42C22×C3×Dic3—62no—
43C32×M16—38no—
44C2×C32×C8—34no—
45C4×C3⋊Dic3C3×C412yes—
46C4×C3⋊Dic3C22×C311yescharacteristic
47C4×C3⋊Dic3C22×C311yescharacteristic
48C3×C4×Dic3—34no—
49C3×C4×Dic3—32no—
50C3×C4×Dic3—32no—
51C3×C4×Dic3—62no—
52C3×C4×Dic3—61no—
53C3×C4×Dic3—61no—
54C3×#208—28no—
55C3×#208Dic318yescomplemented
56C3×#208C3×C418yescomplemented
57C2×C3×C3⋊C8—24no—
58C2×C3×C3⋊C8C2×D314yes—
59C2×C3×C3⋊C8C22×C314yes—
60C22×C33—22no—
61C22×C33C4212yes—
62C22×C33C22×C411yescharacteristic
63C33×C4C4212yes—
64C33×C4C22×C412yes—
65C3×C3⋊Dic3—22no—
66C3×C3⋊Dic3M1614yescomplemented
67C2×C3×M16—32no—
68C2×C3×M16—32no—
69C2×C3×M16—62no—
70C2×C3×C42—91no—
71C2×C4×Dic3—32no—
72C2×C4×Dic3—61no—
73C2×#208C3×D312yes—
74C3×#97—92no—
75C3×C4×C8—92no—
76#815—34no—
77C8×Dic3—34no—
78C23×C32—21no—
79C23×C32—22no—
80C23×C32—41no—
81C23×C32C4×D312yes—
82C23×C32C2×C3×C411yescharacteristic
83C2×C32×C4—24no—
84C2×C32×C4—21no—
85C2×C32×C4—24no—
86C2×C32×C4—22no—
87C2×C32×C4—24no—
88C2×C32×C4—21no—
89C2×C32×C4—22no—
90C2×C32×C4—44no—
91C2×C32×C4—41no—
92C2×C32×C4—41no—
93C2×C32×C4C4×D314yes—
94C2×C32×C4C4×D312yes—
95C2×C32×C4C2×Dic314yes—
96C2×C32×C4C2×C3×C411yescharacteristic
97C2×C32×C4C2×C3×C414yes—
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—38no—
110C3×C3⋊C8—28no—
111C3×C3⋊C8C2×Dic318yescomplemented
112C3×C3⋊C8C2×C3×C418yescomplemented
113C4×M16—91noSylow
114C2×C33—22no—
115C2×C33C2×C4211yescharacteristic
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—38no—
126C3×M16—38no—
127C3×M16—68no—
128C2×C3×C8—34no—
129C2×C3×C8—34no—
130C2×C3×C8—64no—
131C3×C42—92no—
132C3×C42—91no—
133C3×C42—91no—
134C4×Dic3—34no—
135C4×Dic3—32no—
136C4×Dic3—32no—
137C4×Dic3—62no—
138C4×Dic3—61no—
139C4×Dic3—61no—
140#208C3×Dic318yescomplemented
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×C32C4×Dic314yes—
154C22×C32C3×C4212yes—
155C22×C32C2×C4×D312yes—
156C22×C32C22×C3×C411yescharacteristic
157C32×C4—22no—
158C32×C4—24no—
159C32×C4—22no—
160C32×C4—22no—
161C32×C4—22no—
162C32×C4—42no—
163C32×C4—42no—
164C32×C4C4×Dic314yes—
165C32×C4C3×C4212yes—
166C32×C4C2×C4×D312yes—
167C32×C4C22×Dic312yes—
168C32×C4C22×C3×C412yes—
169C3⋊Dic3—22no—
170C3⋊Dic3C3×M1614yescomplemented
171C3×Dic3—38no—
172C3×Dic3—64no—
173C3×Dic3—64no—
174C3×Dic3—64no—
175C2×M16—32no—
176C2×C42—91no—
177#97—92no—
178C4×C8—92no—
179C33C4×M1611yescharacteristic, complemented, Sylow
180C23×C3—21no—
181C23×C3—22no—
182C23×C3—41no—
183C23×C3#62411yescharacteristic
184C23×C3C3×C4×D312yes—
185C2×C3×C4—24no—
186C2×C3×C4—24no—
187C2×C3×C4—24no—
188C2×C3×C4—22no—
189C2×C3×C4—21no—
190C2×C3×C4—22no—
191C2×C3×C4—21no—
192C2×C3×C4—44no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—92no—
196C2×C3×C4—92no—
197C2×C3×C4—184no—
198C2×C3×C4D3×Dic314yes—
199C2×C3×C4#62411yescharacteristic
200C2×C3×C4C3×C4×D314yes—
201C2×C3×C4C3×C4×D312yes—
202C2×C3×C4C2×C3×Dic314yes—
203C2×C3×C4C2×D3211yescharacteristic
204C2×C3×C4C22×C3×D312yes—
205C2×Dic3—34no—
206C2×Dic3—34no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—68no—
210C2×Dic3—68no—
211C3×C8—38no—
212C3×C8—38no—
213C3×C8—68no—
214C3⋊C8C2×C3×Dic318yescomplemented
215C2×C32—22no—
216C2×C32—24no—
217C2×C32—24no—
218C2×C32—24no—
219C2×C32—22no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—41no—
223C2×C32—42no—
224C2×C32—44no—
225C2×C32—44no—
226C2×C32D3×M1614yes—
227C2×C32C2×C4×Dic312yes—
228C2×C32C2×C3×C4211yescharacteristic
229C2×C32C2×C3×M1612yes—
230C22×C4—92no—
231C22×C4C3×D3211yescharacteristic
232M16—38no—
233C2×C8—34no—
234C42—92no—
235C42—91no—
236C42—91no—
237C22×C3—24no—
238C22×C3—24no—
239C22×C3—22no—
240C22×C3—21no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—24no—
244C22×C3—44no—
245C22×C3—44no—
246C22×C3—41no—
247C22×C3—42no—
248C22×C3Dic3212yes—
249C22×C3C3×C4×Dic314yes—
250C22×C3C2×#62411yescharacteristic
251C22×C3C2×C3×C4×D312yes—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—24no—
256C3×C4—22no—
257C3×C4—42no—
258C3×C4—42no—
259C3×C4—94no—
260C3×C4—182no—
261C3×C4Dic3212yes—
262C3×C4C3×C4×Dic314yes—
263C3×C4C2×D3×Dic312yes—
264C3×C4C2×C3×C4×D312yes—
265C3×C4C22×C3×Dic312yes—
266Dic3—38no—
267Dic3—64no—
268Dic3—64no—
269Dic3—64no—
270C32—21no—
271C32—22no—
272C32—41no—
273C32Dic3×M1612yescomplemented
274C32C3×C4×M1611yescharacteristic, complemented
275C23C3×#62411yescharacteristic
276C2×C4—92no—
277C2×C4—92no—
278C2×C4—184no—
279C2×C4C3×D3×Dic314yes—
280C2×C4C3×#62411yescharacteristic
281C2×C4C2×C3×D3211yescharacteristic
282C8—38no—
283C2×C3—24no—
284C2×C3—22no—
285C2×C3—24no—
286C2×C3—22no—
287C2×C3—21no—
288C2×C3—22no—
289C2×C3—24no—
290C2×C3—44no—
291C2×C3—44no—
292C2×C3—41no—
293C2×C3—42no—
294C2×C3#6381812yes—
295C2×C3C2×Dic3211yescharacteristic
296C2×C3C3×D3×M1614yes—
297C2×C3C2×C3×C4×Dic312yes—
298C22—22no—
299C22C3×Dic3212yes—
300C22C2×C3×#62411yescharacteristic
301C4—94no—
302C4—182no—
303C4C3×Dic3212yes—
304C4C2×C3×D3×Dic312yes—
305C3—21no—
306C3—22no—
307C3—41no—
308C3#1059106011yescharacteristic, direct product factor
309C3C3×Dic3×M1612yescomplemented
310C2—22no—
311C2C3×#6381812yes—
312C2C2×C3×Dic3211yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC2×C32C2×C3×C42
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC23×C33C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×C4×M16
LCSG ▷ C2×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×#63555C2semidirect
4. C3×#63553C2semidirect
6. #10591060C3direct (N × H)
12. C32×#208C4semidirect
23. #63555C2×C3semidirect
24. #63553C2×C3semidirect
32. C32×C3⋊C8C2×C4semidirect
55. C3×#208Dic3semidirect
56. C3×#208C3×C4semidirect
66. C3×C3⋊Dic3M16semidirect
111. C3×C3⋊C8C2×Dic3semidirect
112. C3×C3⋊C8C2×C3×C4semidirect
140. #208C3×Dic3semidirect
170. C3⋊Dic3C3×M16semidirect
179. C33C4×M16semidirect
214. C3⋊C8C2×C3×Dic3semidirect
273. C32Dic3×M16semidirect
274. C32C3×C4×M16semidirect
308. C3#10591060direct (N × H)
309. C3C3×Dic3×M16semidirect

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×#635552
C3×#635532
C3×Dic3×M163
#105910603