Finite Group Databasebeta

← C3×#10592439 #13162778

Subgroups — C3×#10592439 #13162778

Order 1728.

1156 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic
2C3×#64086C212yesmaximal, complemented
3C33×#99C211yesmaximal, characteristic, complemented
4C3×#63661C211yesmaximal, characteristic, complemented
5C3×#63633C212yesmaximal, complemented
6#10592439C311yesmaximal, characteristic, direct product factor
7C3×#4740—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638C411yescharacteristic, complemented
10C2×C3×#638C2211yescharacteristic
11C22×C3×C3⋊Dic3C411yescharacteristic, complemented
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C3×#3830—22no—
14C3×#3830—22no—
15C2×C3×C4×C3⋊D3C411yescharacteristic, complemented
16C2×C3×#634C411yescharacteristic, complemented
17C33×C22⋊C4C2212yes—
18C3×#3757C2212yes—
19C3×#3750—22no—
20C3×C4×C3⋊Dic3—22no—
21C2×#3830C2×C311yescharacteristic
22C2×C3×#237—34no—
23#64086C2×C312yescomplemented
24C3×#877—38no—
25C32×#99—24no—
26C32×#99D314yescomplemented
27C32×#99C2×C311yescharacteristic, complemented
28#63661C2×C311yescharacteristic, complemented
29#63633C2×C312yescomplemented
30C3×#834—34no—
31C3×#806—38no—
32C23×C33C2×C411yescharacteristic
33C23×C33C2311yescharacteristic
34C22×C3×C3⋊D3C2×C411yescharacteristic
35C33×D4—22no—
36C2×C33×C4—22no—
37C2×C33×C4C2×C411yescharacteristic
38C3×#638—22no—
39C3×#638—22no—
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3C2×C411yescharacteristic
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C2×C3×C3⋊Dic3C2×C411yescharacteristic
46C2×C3×C3⋊Dic3D412yes—
47C2×C3×C3⋊Dic3D412yes—
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3—21no—
50C3×#634—22no—
51C3×#410—91no—
52#4740—34no—
53C2×C32×D4—24no—
54C2×C32×D4C2×D314yes—
55C2×C32×D4C22×C311yescharacteristic
56C2×#638C3×C411yescharacteristic, complemented
57C2×#638C22×C311yescharacteristic
58C22×C3⋊Dic3C3×C411yescharacteristic, complemented
59C22×C3⋊Dic3C22×C311yescharacteristic
60#3830—22no—
61#3830—22no—
62C2×C4×C3⋊D3C3×C411yescharacteristic, complemented
63C2×#634C3×C411yescharacteristic, complemented
64C2×C3×#67—34no—
65C2×C3×#67—34no—
66C22×C3×Dic3—34no—
67C22×C3×Dic3—34no—
68C3×#237—68no—
69C3×#237—68no—
70C2×C3×C4×D3—34no—
71C2×C3×Dic6—34no—
72C32×C22⋊C4—28no—
73C32×C22⋊C4C2×D318yes—
74C32×C22⋊C4C22×C312yes—
75#3757C22×C312yes—
76#3750—22no—
77C4×C3⋊Dic3—22no—
78C3×#217—38no—
79C3×Dic3⋊C4—68no—
80C3×C4×Dic3—68no—
81C22×C33—21no—
82C22×C33—41no—
83C22×C33—41no—
84C22×C33C22×C411yescharacteristic
85C22×C33C2×D412yes—
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—41no—
88C33×C4—21no—
89C33×C4—42no—
90C3×C3⋊Dic3—21no—
91C3×C3⋊Dic3—42no—
92C3×C3⋊Dic3C22⋊C412yes—
93C3×C3⋊Dic3C22⋊C411yescharacteristic
94C3×C3⋊Dic3C22⋊C411yescharacteristic
95C2×C3×#41—91no—
96C2×#237—34no—
97C3×#117—92no—
98#877—38no—
99C3×#99—24no—
100C3×#99—91no—
101C3×#99—92no—
102C3×#99C3×D314yescomplemented
103C3×#99C3⋊D311yescharacteristic, complemented
104#834—34no—
105#806—38no—
106C23×C32—24no—
107C23×C32—24no—
108C23×C32C4×D314yes—
109C23×C32C2×C3×C411yescharacteristic
110C23×C32C22×D314yes—
111C23×C32C23×C311yescharacteristic
112C22×C3⋊D3C2×C3×C411yescharacteristic
113C22×C3×D3—34no—
114C32×D4—22no—
115C32×D4—28no—
116C32×D4—48no—
117C2×C32×C4—22no—
118C2×C32×C4—24no—
119C2×C32×C4—28no—
120C2×C32×C4—48no—
121C2×C32×C4C4×D314yes—
122C2×C32×C4C2×C3×C411yescharacteristic
123#638—22no—
124#638—22no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—22no—
128C2×C3⋊Dic3C2×C3×C411yescharacteristic
129C2×C3⋊Dic3C2×C3×C411yescharacteristic
130C2×C3⋊Dic3C2×C3×C411yescharacteristic
131C2×C3⋊Dic3C3×D412yes—
132C2×C3⋊Dic3C3×D412yes—
133C4×C3⋊D3—21no—
134C4×C3⋊D3—21no—
135#634—22no—
136C3×#67—68no—
137C3×#67—68no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—38no—
141C2×C3×Dic3—38no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3—68no—
146C3×C4×D3—64no—
147C3×C4×D3—64no—
148C3×Dic6—68no—
149#410—91noSylow
150C2×C33—22no—
151C2×C33—21no—
152C2×C33—41no—
153C2×C33C2×C22⋊C411yescharacteristic
154C3×C3⋊D3—41no—
155C3×#41—182no—
156C3×#41—182no—
157C2×C3×Q8—91no—
158C2×C3×D4—24no—
159C2×C3×D4—91no—
160C2×C3×D4—91no—
161C2×C3×D4C2×C3×D314yes—
162C2×C3×D4C2×C3⋊D311yescharacteristic
163C22×C3×C4—91no—
164C22×C3×C4—91no—
165C22×C3×C4—91no—
166C2×#67—34no—
167C2×#67—34no—
168C22×Dic3—34no—
169C22×Dic3—34no—
170#237—68no—
171#237—68no—
172C2×C4×D3—34no—
173C2×Dic6—34no—
174C3×C4⋊C4—182no—
175C3×C22⋊C4—28no—
176C3×C22⋊C4—92no—
177C3×C22⋊C4C2×C3×D318yes—
178C3×C22⋊C4C2×C3⋊D312yes—
179C3×C42—182no—
180#217—38no—
181Dic3⋊C4—68no—
182C4×Dic3—68no—
183C22×C32—21no—
184C22×C32—24no—
185C22×C32—28no—
186C22×C32—24no—
187C22×C32—44no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—44no—
191C22×C32—44no—
192C22×C32—84no—
193C22×C32—84no—
194C22×C32C2×C4×D314yes—
195C22×C32D3×D418yes—
196C22×C32C22×C3×C411yescharacteristic
197C22×C32C2×C3×D412yes—
198C2×C3⋊D3—21no—
199C2×C3⋊D3—41no—
200C2×C3×D3—64no—
201C2×C3×D3—124no—
202C32×C4—21no—
203C32×C4—24no—
204C32×C4—44no—
205C32×C4—42no—
206C32×C4—48no—
207C32×C4—88no—
208C3⋊Dic3—21no—
209C3⋊Dic3—42no—
210C3⋊Dic3C3×C22⋊C412yes—
211C3⋊Dic3C3×C22⋊C411yescharacteristic
212C3⋊Dic3C3×C22⋊C411yescharacteristic
213C3×Dic3—38no—
214C3×Dic3—34no—
215C3×Dic3—34no—
216C3×Dic3—64no—
217C3×Dic3—128no—
218C2×#41—91no—
219#117—92no—
220#99—91no—
221#99—92no—
222#99C3×C3⋊D311yescharacteristic, complemented
223C33#41011yescharacteristic, complemented, Sylow
224C23×C3—24no—
225C23×C3—24no—
226C23×C3—91no—
227C23×C3C3×C4×D314yes—
228C23×C3C4×C3⋊D311yescharacteristic
229C23×C3C22×C3×D314yes—
230C23×C3C22×C3⋊D311yescharacteristic
231C22×D3—34no—
232C3×Q8—182no—
233C3×D4—28no—
234C3×D4—22no—
235C3×D4—48no—
236C3×D4—182no—
237C3×D4—182no—
238C2×C3×C4—28no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—48no—
242C2×C3×C4—91no—
243C2×C3×C4—91no—
244C2×C3×C4—92no—
245C2×C3×C4—92no—
246C2×C3×C4—91no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4—182no—
252C2×C3×C4C3×C4×D314yes—
253C2×C3×C4C4×C3⋊D311yescharacteristic
254#67—68no—
255#67—68no—
256C2×Dic3—34no—
257C2×Dic3—34no—
258C2×Dic3—38no—
259C2×Dic3—38no—
260C2×Dic3—34no—
261C2×Dic3—64no—
262C2×Dic3—64no—
263C2×Dic3—68no—
264C4×D3—64no—
265C4×D3—64no—
266Dic6—68no—
267C2×C32—22no—
268C2×C32—21no—
269C2×C32—24no—
270C2×C32—28no—
271C2×C32—24no—
272C2×C32—48no—
273C2×C32—44no—
274C2×C32—41no—
275C2×C32—44no—
276C2×C32—84no—
277C2×C32D3×C22⋊C414yes—
278C2×C32C2×C3×C22⋊C411yescharacteristic
279C3⋊D3—41no—
280C3×D3—124no—
281#41—182no—
282#41—182no—
283C2×Q8—91no—
284C2×D4—91no—
285C2×D4—91no—
286C2×D4C2×C3×C3⋊D311yescharacteristic
287C22×C4—91no—
288C22×C4—91no—
289C22×C4—91no—
290C4⋊C4—182no—
291C22⋊C4—92no—
292C22⋊C4C2×C3×C3⋊D312yes—
293C42—182no—
294C22×C3—24no—
295C22×C3—21no—
296C22×C3—24no—
297C22×C3—28no—
298C22×C3—44no—
299C22×C3—44no—
300C22×C3—41no—
301C22×C3—44no—
302C22×C3—41no—
303C22×C3—84no—
304C22×C3—84no—
305C22×C3—181no—
306C22×C3—361no—
307C22×C3C2×C3×C4×D314yes—
308C22×C3C3×D3×D418yes—
309C22×C3C2×C4×C3⋊D311yescharacteristic
310C22×C3D4×C3⋊D312yes—
311C2×D3—64no—
312C2×D3—124no—
313C3×C4—24no—
314C3×C4—21no—
315C3×C4—44no—
316C3×C4—42no—
317C3×C4—48no—
318C3×C4—88no—
319C3×C4—91no—
320C3×C4—91no—
321C3×C4—92no—
322C3×C4—181no—
323C3×C4—362no—
324Dic3—38no—
325Dic3—34no—
326Dic3—34no—
327Dic3—64no—
328Dic3—128no—
329C32—24no—
330C32#474014yescomplemented
331C32C3×#41011yescharacteristic, complemented
332C23—91no—
333C23C3×C4×C3⋊D311yescharacteristic
334C23C22×C3×C3⋊D311yescharacteristic
335Q8—182no—
336D4—22no—
337D4—182no—
338D4—182no—
339C2×C4—22no—
340C2×C4—91no—
341C2×C4—91no—
342C2×C4—92no—
343C2×C4—92no—
344C2×C4—91no—
345C2×C4—181no—
346C2×C4—181no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4—182no—
350C2×C4C3×C4×C3⋊D311yescharacteristic
351C2×C3—28no—
352C2×C3—24no—
353C2×C3—22no—
354C2×C3—21no—
355C2×C3—24no—
356C2×C3—48no—
357C2×C3—44no—
358C2×C3—44no—
359C2×C3—41no—
360C2×C3—84no—
361C2×C3—361no—
362C2×C3C3×D3×C22⋊C414yes—
363C2×C3C22⋊C4×C3⋊D311yescharacteristic
364D3—124no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—181no—
369C22—361no—
370C22C2×C3×C4×C3⋊D311yescharacteristic
371C22C3×D4×C3⋊D312yes—
372C4—21no—
373C4—42no—
374C4—91no—
375C4—91no—
376C4—92no—
377C4—181no—
378C4—362no—
379C3—24no—
380C3C3×#474014yescomplemented
381C3#1059243911yescharacteristic, direct product factor
382C2—22no—
383C2—21no—
384C2—41no—
385C2—361no—
386C2C3×C22⋊C4×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C3C22⋊C4×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC23C22×C3×C3⋊D3
FittingC33×#99C2
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3830C2
cosocleC23×C32C23×C3
hypercenterC3×#99C3⋊D3
hypocenterC32C3×#410
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
2. C3×#64086C2semidirect
3. C33×#99C2semidirect
4. C3×#63661C2semidirect
5. C3×#63633C2semidirect
6. #10592439C3direct (N × H)
9. C2×C3×#638C4semidirect
11. C22×C3×C3⋊Dic3C4semidirect
15. C2×C3×C4×C3⋊D3C4semidirect
16. C2×C3×#634C4semidirect
23. #64086C2×C3semidirect
26. C32×#99D3semidirect
27. C32×#99C2×C3semidirect
28. #63661C2×C3semidirect
29. #63633C2×C3semidirect
56. C2×#638C3×C4semidirect
58. C22×C3⋊Dic3C3×C4semidirect
62. C2×C4×C3⋊D3C3×C4semidirect
63. C2×#634C3×C4semidirect
102. C3×#99C3×D3semidirect
103. C3×#99C3⋊D3semidirect
222. #99C3×C3⋊D3semidirect
223. C33#410semidirect
330. C32#4740semidirect
331. C32C3×#410semidirect
380. C3C3×#4740semidirect
381. C3#10592439direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C3×#640862
C33×#992
C3×#636612
C3×#636332
#105924393
C3×#47403