Finite Group Databasebeta

← C3×#10591257 #13158268

Subgroups — C3×#10591257 #13158268

Order 1728.

1090 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×#3724C212yesmaximal
3C2×C3×Dic32C212yesmaximal
4C32×C3⋊#95C212yesmaximal
5C3×#4649—32nomaximal
6#10591257C311yesmaximal, characteristic, direct product factor
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C32×Dic3C2214yes—
11C3×C4×C3⋊Dic3C414yescomplemented
12C3×#3724C418yescomplemented
13C3×Dic32—24no—
14C2×C4×C3⋊Dic3C2×C311yescharacteristic
15C2×C3×Dic3⋊C4—34no—
16C2×C3×C4×Dic3—32no—
17C2×C3×C4×Dic3—34no—
18C2×C3×C4×Dic3—61no—
19C2×#3724C2×C312yes—
20C2×Dic32C2×C312yes—
21C32×#95—32no—
22C3×C3⋊#95—22no—
23C3×C3⋊#95D312yes—
24C3×C3⋊#95C2×C312yes—
25C23×C33C2311yescharacteristic
26C2×C33×C4—22no—
27C2×C33×C4C2×C412yes—
28C2×C3×C3⋊Dic3C2×C412yes—
29C2×C3×C3⋊Dic3C2×C414yes—
30C2×C3×C3⋊Dic3C2×C412yes—
31C2×C3×C3⋊Dic3D412yes—
32C2×C3×C3⋊Dic3Q812yes—
33C2×C32×Dic3—24no—
34C2×C32×Dic3—24no—
35C2×C32×Dic3C2×C418yes—
36C3×C4×C4⋊C4—91no—
37#4649—32no—
38C22×C32×C4—21no—
39C22×C32×C4—22no—
40C22×C32×C4—34no—
41C22×C32×C4—41no—
42C22×C32×C4C2×D312yes—
43C22×C32×C4C22×C311yescharacteristic
44C22×C3⋊Dic3C22×C311yescharacteristic
45C22×C3⋊Dic3C22×C311yescharacteristic
46C22×C3×Dic3—24no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—61no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3C2×D314yes—
52C22×C3×Dic3C22×C314yes—
53C4×C3⋊Dic3C3×C414yescomplemented
54C3×Dic3⋊C4—316no—
55C3×C4×Dic3—38no—
56C3×C4×Dic3—68no—
57C3×C4×Dic3—64no—
58#3724C3×C418yescomplemented
59Dic32—24no—
60C22×C33C22×C411yescharacteristic
61C22×C33C22×C412yes—
62C22×C33C2×D411yescharacteristic
63C22×C33C2×Q811yescharacteristic
64C22×C33#4112yes—
65C33×C4—22no—
66C3×C3⋊Dic3C4214yescomplemented
67C3×C3⋊Dic3C4⋊C414yescomplemented
68C32×Dic3—28no—
69C2×C3×C4⋊C4—92no—
70C2×C3×C42—91no—
71C2×C3×C42—92no—
72C2×Dic3⋊C4—34no—
73C2×C4×Dic3—32no—
74C2×C4×Dic3—34no—
75C2×C4×Dic3—61no—
76C3×#95—32no—
77C3×#95—32no—
78C3×#95—62no—
79C3⋊#95C3×D312yes—
80C23×C32—21no—
81C23×C32—22no—
82C23×C32—41no—
83C23×C32C22×D312yes—
84C23×C32C23×C311yescharacteristic
85C2×C32×C4—22no—
86C2×C32×C4—24no—
87C2×C32×C4—24no—
88C2×C32×C4—22no—
89C2×C32×C4—24no—
90C2×C32×C4—38no—
91C2×C32×C4—42no—
92C2×C32×C4—44no—
93C2×C32×C4—44no—
94C2×C32×C4—64no—
95C2×C32×C4—64no—
96C2×C32×C4C4×D314yes—
97C2×C32×C4C2×C3×C412yes—
98C2×C3⋊Dic3C2×C3×C414yes—
99C2×C3⋊Dic3C2×C3×C412yes—
100C2×C3⋊Dic3C2×C3×C412yes—
101C2×C3⋊Dic3C3×D412yes—
102C2×C3⋊Dic3C3×Q812yes—
103C2×C3×Dic3—24no—
104C2×C3×Dic3—24no—
105C2×C3×Dic3—24no—
106C2×C3×Dic3—24no—
107C2×C3×Dic3—28no—
108C2×C3×Dic3—34no—
109C2×C3×Dic3—34no—
110C2×C3×Dic3—38no—
111C2×C3×Dic3—34no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—44no—
114C2×C3×Dic3—44no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—64no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3C4×D318yes—
121C2×C3×Dic3C2×C3×C418yes—
122C4×C4⋊C4—91noSylow
123C2×C33C2×C4211yescharacteristic
124C2×C33C2×C4⋊C411yescharacteristic
125C2×C33#11711yescharacteristic
126C2×C33C4×D412yes—
127C2×C33C4×Q812yes—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—34no—
131C22×C3×C4—34no—
132C22×C3×C4—41no—
133C22×C3×C4—64no—
134C22×C3×C4—91no—
135C22×C3×C4—91no—
136C22×C3×C4D3211yescharacteristic
137C22×C3×C4C2×C3×D312yes—
138C22×Dic3—32no—
139C22×Dic3—32no—
140C22×Dic3—61no—
141C22×Dic3—61no—
142C22×Dic3C2×C3×D314yes—
143C3×C4⋊C4—98no—
144C3×C42—94no—
145C3×C42—184no—
146Dic3⋊C4—316no—
147C4×Dic3—38no—
148C4×Dic3—64no—
149C4×Dic3—68no—
150C22×C32—21no—
151C22×C32—22no—
152C22×C32—21no—
153C22×C32—22no—
154C22×C32—21no—
155C22×C32—24no—
156C22×C32—22no—
157C22×C32—22no—
158C22×C32—24no—
159C22×C32—22no—
160C22×C32—41no—
161C22×C32—42no—
162C22×C32—41no—
163C22×C32—41no—
164C22×C32—42no—
165C22×C32C2×C4×D314yes—
166C22×C32C2×C4×D312yes—
167C22×C32D3×D412yes—
168C22×C32#23714yes—
169C22×C32D3×Q812yes—
170C22×C32C22×C3×C411yescharacteristic
171C22×C32C22×C3×C412yes—
172C22×C32C2×C3×D411yescharacteristic
173C22×C32C2×C3×Q811yescharacteristic
174C22×C32C3×#4112yes—
175C32×C4—22no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—44no—
179C32×C4—44no—
180C32×C4—68no—
181C3⋊Dic3C3×C4214yescomplemented
182C3⋊Dic3C3×C4⋊C414yescomplemented
183C3×Dic3—28no—
184C3×Dic3—28no—
185C3×Dic3—38no—
186C3×Dic3—38no—
187C3×Dic3—48no—
188C3×Dic3—64no—
189C3×Dic3—64no—
190C2×C4⋊C4—92no—
191C2×C42—91no—
192C2×C42—92no—
193#95—32no—
194C33C4×C4⋊C411yescharacteristic, complemented, Sylow
195C23×C3—22no—
196C23×C3—21no—
197C23×C3—41no—
198C23×C3C2×D3211yescharacteristic
199C23×C3C22×C3×D312yes—
200C2×C3×C4—24no—
201C2×C3×C4—22no—
202C2×C3×C4—24no—
203C2×C3×C4—22no—
204C2×C3×C4—24no—
205C2×C3×C4—38no—
206C2×C3×C4—38no—
207C2×C3×C4—44no—
208C2×C3×C4—44no—
209C2×C3×C4—42no—
210C2×C3×C4—68no—
211C2×C3×C4—64no—
212C2×C3×C4—64no—
213C2×C3×C4—64no—
214C2×C3×C4—64no—
215C2×C3×C4—92no—
216C2×C3×C4—94no—
217C2×C3×C4—92no—
218C2×C3×C4—92no—
219C2×C3×C4—92no—
220C2×C3×C4—124no—
221C2×C3×C4—124no—
222C2×C3×C4#62412yes—
223C2×C3×C4C3×C4×D314yes—
224C2×Dic3—24no—
225C2×Dic3—24no—
226C2×Dic3—34no—
227C2×Dic3—34no—
228C2×Dic3—38no—
229C2×Dic3—34no—
230C2×Dic3—34no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—64no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3C3×C4×D318yes—
237C2×C32—22no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32—22no—
241C2×C32—22no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—22no—
245C2×C32—21no—
246C2×C32—22no—
247C2×C32—41no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—42no—
251C2×C32—42no—
252C2×C32C42×D312yes—
253C2×C32#87712yes—
254C2×C32#67⋊C414yes—
255C2×C32#88714yes—
256C2×C32D3×C4⋊C412yes—
257C2×C32C2×C3×C4211yescharacteristic
258C2×C32C2×C3×C4⋊C411yescharacteristic
259C2×C32C3×#11711yescharacteristic
260C2×C32C3×C4×D412yes—
261C2×C32C3×C4×Q812yes—
262C22×C4—34no—
263C22×C4—91no—
264C22×C4—91no—
265C22×C4C3×D3211yescharacteristic
266C4⋊C4—98no—
267C42—94no—
268C42—184no—
269C22×C3—21no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—24no—
273C22×C3—21no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—24no—
277C22×C3—21no—
278C22×C3—22no—
279C22×C3—41no—
280C22×C3—42no—
281C22×C3—41no—
282C22×C3—42no—
283C22×C3—41no—
284C22×C3#379711yescharacteristic
285C22×C3C4×D3212yes—
286C22×C3#380211yescharacteristic
287C22×C3#380512yes—
288C22×C3C2×#62411yescharacteristic
289C22×C3C2×C3×C4×D314yes—
290C22×C3C2×C3×C4×D312yes—
291C22×C3C3×D3×D412yes—
292C22×C3C3×#23714yes—
293C22×C3C3×D3×Q812yes—
294C3×C4—24no—
295C3×C4—24no—
296C3×C4—22no—
297C3×C4—44no—
298C3×C4—44no—
299C3×C4—68no—
300C3×C4—68no—
301C3×C4—94no—
302C3×C4—94no—
303C3×C4—128no—
304Dic3—28no—
305Dic3—38no—
306Dic3—38no—
307Dic3—64no—
308Dic3—64no—
309C32—22no—
310C32—21no—
311C32—41no—
312C32#464912yescomplemented
313C32C3×C4×C4⋊C411yescharacteristic, complemented
314C23C2×C3×D3211yescharacteristic
315C2×C4—22no—
316C2×C4—38no—
317C2×C4—64no—
318C2×C4—64no—
319C2×C4—92no—
320C2×C4—94no—
321C2×C4—92no—
322C2×C4—92no—
323C2×C4—92no—
324C2×C4C3×#62412yes—
325C2×C3—21no—
326C2×C3—22no—
327C2×C3—24no—
328C2×C3—22no—
329C2×C3—21no—
330C2×C3—22no—
331C2×C3—21no—
332C2×C3—22no—
333C2×C3—24no—
334C2×C3—22no—
335C2×C3—41no—
336C2×C3—42no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—42no—
340C2×C3#6383812yes—
341C2×C3C4×#62411yescharacteristic
342C2×C3#6390011yescharacteristic
343C2×C3#6390212yes—
344C2×C3#6395711yescharacteristic
345C2×C3C3×C42×D312yes—
346C2×C3C3×#87712yes—
347C2×C3C3×#67⋊C414yes—
348C2×C3C3×#88714yes—
349C2×C3C3×D3×C4⋊C412yes—
350C22C3×#379711yescharacteristic
351C22C3×C4×D3212yes—
352C22C3×#380211yescharacteristic
353C22C3×#380512yes—
354C22C2×C3×#62411yescharacteristic
355C4—22no—
356C4—68no—
357C4—94no—
358C4—94no—
359C3—22no—
360C3—21no—
361C3—41no—
362C3#1059125711yescharacteristic, direct product factor
363C3C3×#464912yescomplemented
364C2C3×#6383812yes—
365C2C3×C4×#62411yescharacteristic
366C2C3×#6390011yescharacteristic
367C2C3×#6390212yes—
368C2C3×#6395711yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC2×C32C2×C3×C42
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×C4×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
6. #10591257C3direct (N × H)
11. C3×C4×C3⋊Dic3C4semidirect
12. C3×#3724C4semidirect
53. C4×C3⋊Dic3C3×C4semidirect
58. #3724C3×C4semidirect
66. C3×C3⋊Dic3C42semidirect
67. C3×C3⋊Dic3C4⋊C4semidirect
181. C3⋊Dic3C3×C42semidirect
182. C3⋊Dic3C3×C4⋊C4semidirect
194. C33C4×C4⋊C4semidirect
312. C32#4649semidirect
313. C32C3×C4×C4⋊C4semidirect
362. C3#10591257direct (N × H)
363. C3C3×#4649semidirect

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×C3×#37242
C2×C3×Dic322
C32×C3⋊#952
C3×#46493
#105912573