Finite Group Databasebeta

← C3×#10591360 #13158311

Subgroups — C3×#10591360 #13158311

Order 1728.

930 conjugacy classes of subgroups.

The subgroup lattice has 364 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
3C32×C3⋊#95C212yesmaximal
4C3×#63579C212yesmaximal
5C3×#4650—32nomaximal
6#10591360C311yesmaximal, characteristic, direct product factor
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C32×Dic3C2214yes—
11C3×C4×C3⋊Dic3C414yescomplemented
12C3×#3724—24no—
13C2×C4×C3⋊Dic3C2×C311yescharacteristic
14C2×C3×Dic3⋊C4—34no—
15C2×C3×C4×Dic3—32no—
16C2×C3×C4×Dic3—61no—
17C2×#3724C2×C312yes—
18C32×#95—32no—
19C3×C3⋊#95—22no—
20C3×C3⋊#95—34no—
21C3×C3⋊#95D312yes—
22C3×C3⋊#95C2×C312yes—
23#63579C2×C312yes—
24C23×C33C2311yescharacteristic
25C2×C33×C4—22no—
26C2×C33×C4C2×C412yes—
27C2×C3×C3⋊Dic3—22no—
28C2×C3×C3⋊Dic3C2×C412yes—
29C2×C3×C3⋊Dic3C2×C412yes—
30C2×C3×C3⋊Dic3D412yes—
31C2×C3×C3⋊Dic3Q812yes—
32C2×C32×Dic3—24no—
33C2×C32×Dic3—24no—
34C2×C32×Dic3—24no—
35C3×#382—91no—
36#4650—32no—
37C22×C32×C4—21no—
38C22×C32×C4—22no—
39C22×C32×C4—34no—
40C22×C32×C4—41no—
41C22×C32×C4C2×D312yes—
42C22×C32×C4C22×C311yescharacteristic
43C22×C3⋊Dic3C22×C311yescharacteristic
44C22×C3⋊Dic3C22×C311yescharacteristic
45C22×C3×Dic3—24no—
46C22×C3×Dic3—32no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—61no—
49C22×C3×Dic3—61no—
50C22×C3×Dic3C2×D314yes—
51C22×C3×Dic3C22×C314yes—
52C4×C3⋊Dic3C3×C414yescomplemented
53C3×Dic3⋊C4—68no—
54C3×C4×Dic3—38no—
55C3×C4×Dic3—64no—
56#3724—24no—
57C22×C33C22×C411yescharacteristic
58C22×C33C2×D411yescharacteristic
59C22×C33C2×Q811yescharacteristic
60C22×C33#4112yes—
61C22×C33#4112yes—
62C33×C4—22no—
63C3×C3⋊Dic3—22no—
64C3×C3⋊Dic3C4⋊C414yes—
65C32×Dic3—44no—
66C2×C3×C4⋊C4—92no—
67C2×C3×C42—91no—
68C2×Dic3⋊C4—34no—
69C2×C4×Dic3—32no—
70C2×C4×Dic3—61no—
71C3×#95—32no—
72C3×#95—32no—
73C3×#95—62no—
74C3×#95—92no—
75C3⋊#95—34no—
76C3⋊#95C3×D312yes—
77C23×C32—21no—
78C23×C32—22no—
79C23×C32—41no—
80C23×C32C22×D312yes—
81C23×C32C23×C311yescharacteristic
82C2×C32×C4—22no—
83C2×C32×C4—24no—
84C2×C32×C4—24no—
85C2×C32×C4—22no—
86C2×C32×C4—24no—
87C2×C32×C4—42no—
88C2×C32×C4—44no—
89C2×C32×C4—44no—
90C2×C32×C4—64no—
91C2×C32×C4—64no—
92C2×C32×C4—64no—
93C2×C32×C4C4×D314yes—
94C2×C32×C4C2×C3×C412yes—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3C2×C3×C412yes—
97C2×C3⋊Dic3C2×C3×C412yes—
98C2×C3⋊Dic3C3×D412yes—
99C2×C3⋊Dic3C3×Q812yes—
100C2×C3×Dic3—28no—
101C2×C3×Dic3—24no—
102C2×C3×Dic3—24no—
103C2×C3×Dic3—24no—
104C2×C3×Dic3—24no—
105C2×C3×Dic3—24no—
106C2×C3×Dic3—24no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—34no—
109C2×C3×Dic3—34no—
110C2×C3×Dic3—34no—
111C2×C3×Dic3—44no—
112C2×C3×Dic3—44no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—64no—
118C2×C3×Dic3—64no—
119#382—91noSylow
120C2×C33C2×C4⋊C411yescharacteristic
121C2×C33#11711yescharacteristic
122C2×C33#11711yescharacteristic
123C2×C33#12412yes—
124C2×C33#12512yes—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—34no—
128C22×C3×C4—34no—
129C22×C3×C4—41no—
130C22×C3×C4—64no—
131C22×C3×C4—91no—
132C22×C3×C4—91no—
133C22×C3×C4D3211yescharacteristic
134C22×C3×C4C2×C3×D312yes—
135C22×Dic3—32no—
136C22×Dic3—32no—
137C22×Dic3—61no—
138C22×Dic3—61no—
139C22×Dic3C2×C3×D314yes—
140C3×C4⋊C4—184no—
141C3×C42—94no—
142Dic3⋊C4—68no—
143C4×Dic3—38no—
144C4×Dic3—64no—
145C22×C32—21no—
146C22×C32—22no—
147C22×C32—21no—
148C22×C32—22no—
149C22×C32—21no—
150C22×C32—22no—
151C22×C32—24no—
152C22×C32—22no—
153C22×C32—24no—
154C22×C32—22no—
155C22×C32—41no—
156C22×C32—41no—
157C22×C32—41no—
158C22×C32—42no—
159C22×C32—42no—
160C22×C32C2×C4×D312yes—
161C22×C32#23514yes—
162C22×C32D3×D412yes—
163C22×C32#23714yes—
164C22×C32D3×Q812yes—
165C22×C32C22×C3×C411yescharacteristic
166C22×C32C2×C3×D411yescharacteristic
167C22×C32C2×C3×Q811yescharacteristic
168C22×C32C3×#4112yes—
169C22×C32C3×#4112yes—
170C32×C4—22no—
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—44no—
174C32×C4—44no—
175C32×C4—124no—
176C3⋊Dic3—22no—
177C3⋊Dic3C3×C4⋊C414yes—
178C3×Dic3—38no—
179C3×Dic3—48no—
180C3×Dic3—44no—
181C3×Dic3—44no—
182C3×Dic3—64no—
183C3×Dic3—64no—
184C3×Dic3—64no—
185C2×C4⋊C4—92no—
186C2×C42—91no—
187#95—32no—
188#95—92no—
189C33#38211yescharacteristic, complemented, Sylow
190C23×C3—22no—
191C23×C3—21no—
192C23×C3—41no—
193C23×C3C2×D3211yescharacteristic
194C23×C3C22×C3×D312yes—
195C2×C3×C4—24no—
196C2×C3×C4—22no—
197C2×C3×C4—24no—
198C2×C3×C4—22no—
199C2×C3×C4—24no—
200C2×C3×C4—44no—
201C2×C3×C4—44no—
202C2×C3×C4—42no—
203C2×C3×C4—68no—
204C2×C3×C4—64no—
205C2×C3×C4—64no—
206C2×C3×C4—64no—
207C2×C3×C4—64no—
208C2×C3×C4—64no—
209C2×C3×C4—64no—
210C2×C3×C4—92no—
211C2×C3×C4—92no—
212C2×C3×C4—92no—
213C2×C3×C4—92no—
214C2×C3×C4—124no—
215C2×C3×C4—124no—
216C2×C3×C4—182no—
217C2×C3×C4#62412yes—
218C2×C3×C4C3×C4×D314yes—
219C2×Dic3—24no—
220C2×Dic3—24no—
221C2×Dic3—24no—
222C2×Dic3—34no—
223C2×Dic3—34no—
224C2×Dic3—34no—
225C2×Dic3—34no—
226C2×Dic3—62no—
227C2×Dic3—62no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230C2×Dic3—64no—
231C2×Dic3—64no—
232C2×C32—22no—
233C2×C32—24no—
234C2×C32—24no—
235C2×C32—22no—
236C2×C32—22no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32—22no—
240C2×C32—21no—
241C2×C32—22no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32—42no—
246C2×C32—42no—
247C2×C32#87212yes—
248C2×C32#87712yes—
249C2×C32#88514yes—
250C2×C32#88914yes—
251C2×C32D3×C4⋊C412yes—
252C2×C32C2×C3×C4⋊C411yescharacteristic
253C2×C32C3×#11711yescharacteristic
254C2×C32C3×#11711yescharacteristic
255C2×C32C3×#12412yes—
256C2×C32C3×#12512yes—
257C22×C4—34no—
258C22×C4—91no—
259C22×C4—91no—
260C22×C4C3×D3211yescharacteristic
261C4⋊C4—184no—
262C42—94no—
263C22×C3—21no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—24no—
267C22×C3—21no—
268C22×C3—22no—
269C22×C3—22no—
270C22×C3—24no—
271C22×C3—21no—
272C22×C3—22no—
273C22×C3—41no—
274C22×C3—42no—
275C22×C3—41no—
276C22×C3—42no—
277C22×C3—41no—
278C22×C3#379711yescharacteristic
279C22×C3#379812yes—
280C22×C3#380211yescharacteristic
281C22×C3#380512yes—
282C22×C3C2×#62411yescharacteristic
283C22×C3C2×C3×C4×D312yes—
284C22×C3C3×#23514yes—
285C22×C3C3×D3×D412yes—
286C22×C3C3×#23714yes—
287C22×C3C3×D3×Q812yes—
288C3×C4—24no—
289C3×C4—24no—
290C3×C4—22no—
291C3×C4—44no—
292C3×C4—44no—
293C3×C4—94no—
294C3×C4—128no—
295C3×C4—124no—
296C3×C4—124no—
297C3×C4—182no—
298Dic3—38no—
299Dic3—44no—
300Dic3—64no—
301Dic3—64no—
302Dic3—64no—
303C32—22no—
304C32—21no—
305C32—41no—
306C32#465012yescomplemented
307C32C3×#38211yescharacteristic, complemented
308C23C2×C3×D3211yescharacteristic
309C2×C4—22no—
310C2×C4—64no—
311C2×C4—64no—
312C2×C4—64no—
313C2×C4—92no—
314C2×C4—92no—
315C2×C4—92no—
316C2×C4—92no—
317C2×C4—182no—
318C2×C4C3×#62412yes—
319C2×C3—21no—
320C2×C3—22no—
321C2×C3—24no—
322C2×C3—22no—
323C2×C3—21no—
324C2×C3—22no—
325C2×C3—21no—
326C2×C3—22no—
327C2×C3—24no—
328C2×C3—22no—
329C2×C3—41no—
330C2×C3—42no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—42no—
334C2×C3#6387012yes—
335C2×C3#6387411yescharacteristic
336C2×C3#6390011yescharacteristic
337C2×C3#6391512yes—
338C2×C3#6395711yescharacteristic
339C2×C3C3×#87212yes—
340C2×C3C3×#87712yes—
341C2×C3C3×#88514yes—
342C2×C3C3×#88914yes—
343C2×C3C3×D3×C4⋊C412yes—
344C22C3×#379711yescharacteristic
345C22C3×#379812yes—
346C22C3×#380211yescharacteristic
347C22C3×#380512yes—
348C22C2×C3×#62411yescharacteristic
349C4—22no—
350C4—94no—
351C4—124no—
352C4—182no—
353C3—22no—
354C3—21no—
355C3—41no—
356C3#1059136011yescharacteristic, direct product factor
357C3C3×#465012yescomplemented
358C2C3×#6387012yes—
359C2C3×#6387411yescharacteristic
360C2C3×#6390011yescharacteristic
361C2C3×#6391512yes—
362C2C3×#6395711yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#382
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
6. #10591360C3direct (N × H)
11. C3×C4×C3⋊Dic3C4semidirect
52. C4×C3⋊Dic3C3×C4semidirect
189. C33#382semidirect
306. C32#4650semidirect
307. C32C3×#382semidirect
356. C3#10591360direct (N × H)
357. C3C3×#4650semidirect

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
C32×C3⋊#952
C3×#635792
C3×#46503
#105913603