Finite Group Databasebeta

← C3×#10592751 #13163085

Subgroups — C3×#10592751 #13163085

Order 1728.

956 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
1C33×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3C3×#64133C211yesmaximal, characteristic
4C2×C3×#3748C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63637C211yesmaximal, characteristic, complemented
7C3×#63635C211yesmaximal, characteristic, complemented
8#10592751C311yesmaximal, characteristic, direct product factor
9C3×#5052—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C33×C4⋊C4—21no—
14C33×C4⋊C4C2211yescharacteristic
15C33×C22⋊C4—21no—
16C3×#3757—21no—
17C3×#3756—22no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×#3750—21no—
20C3×#3748—22no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C32×C22⋊Q8—24no—
24C32×C22⋊Q8D314yes—
25C32×C22⋊Q8C2×C311yescharacteristic
26C2×#3756C2×C311yescharacteristic, complemented
27#64133C2×C311yescharacteristic
28C2×#3748C2×C311yescharacteristic
29C2×C3×#216—34no—
30C3×#924—34no—
31C2×C3×#208—34no—
32#63662C2×C311yescharacteristic, complemented
33#63637C2×C311yescharacteristic, complemented
34#63635C2×C311yescharacteristic, complemented
35C3×#835—34no—
36C3×#810—34no—
37C3×#808—34no—
38C23×C33D411yescharacteristic
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4—21no—
43C2×C33×C4—21no—
44C2×C33×C4—21no—
45C2×C33×C4D411yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C3×#634—21no—
50C3×#634—21no—
51C3×C3⋊(C3⋊C8)—21no—
52C3×C3⋊(C3⋊C8)D412yes—
53C3×#470—91no—
54#5052—34no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C22×C311yescharacteristic
58C22×C32×C4—24no—
59C22×C32×C4C2×D314yes—
60C22×C32×C4C22×C311yescharacteristic
61C2×#634C22×C311yescharacteristic
62C2×C3×Dic6—34no—
63C32×C4⋊C4—28no—
64C32×C4⋊C4—24no—
65C32×C4⋊C4—24no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2×D314yes—
68C32×C4⋊C4C22×C311yescharacteristic
69C32×C22⋊C4—28no—
70C32×C22⋊C4—24no—
71C32×C22⋊C4—21no—
72#3757—21no—
73#3756—22no—
74C4⋊(C3⋊Dic3)C22×C311yescharacteristic
75#3750—21no—
76#3748—22no—
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C3×#217—64no—
80C3×#216—68no—
81C3×C4⋊Dic3—34no—
82C3×Dic3⋊C4—64no—
83C3×#208—68no—
84C2×C3×C3⋊C8—34no—
85C2×C3×C3⋊C8—34no—
86C22×C33—21no—
87C22×C33—21no—
88C22×C33—21no—
89C22×C33C2×D411yescharacteristic
90C33×C4—21no—
91C33×C4—41no—
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3×C3⋊Dic3—41no—
96C3×C3⋊Dic3—41no—
97C2×C3×Q16—91no—
98C2×C3×M16—91no—
99C3×C22⋊Q8—24no—
100C3×C22⋊Q8—91no—
101C3×C22⋊Q8C3×D314yes—
102C3×C22⋊Q8C3⋊D311yescharacteristic
103C2×#216—34no—
104#924—34no—
105C2×#208—34no—
106C3×#106—91no—
107C3×#103—91no—
108C3×#103—91no—
109#835—34no—
110#810—34no—
111#808—34no—
112C23×C32—24no—
113C23×C32#6714yes—
114C23×C32C3×D411yescharacteristic
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—24no—
118C32×Q8—24no—
119C32×Q8—44no—
120C32×Q8—44no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—28no—
124C2×C32×C4—28no—
125C2×C32×C4—24no—
126C2×C32×C4—24no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4—44no—
134C2×C32×C4—44no—
135C2×C32×C4#6714yes—
136C2×C32×C4C3×D411yescharacteristic
137C2×C32×C4C22×D314yes—
138C2×C32×C4C23×C311yescharacteristic
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—21no—
141#634—21no—
142#634—21no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—64no—
145C3×Dic6—64no—
146C3×Dic6—64no—
147C3⋊(C3⋊C8)—21no—
148C3⋊(C3⋊C8)C3×D412yes—
149C3×C3⋊C8—38no—
150C3×C3⋊C8—64no—
151#470—91noSylow
152C2×C33—41no—
153C2×C33#12111yescharacteristic
154C2×C33#13711yescharacteristic
155C2×C33#13711yescharacteristic
156C2×C3×Q8—24no—
157C2×C3×Q8—91no—
158C2×C3×Q8C2×C3×D314yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160C22×C3×C4—24no—
161C22×C3×C4C2×C3×D314yes—
162C22×C3×C4C2×C3⋊D311yescharacteristic
163C2×Dic6—34no—
164C3×Q16—182no—
165C3×M16—182no—
166C2×C3×C8—91no—
167C2×C3×C8—91no—
168C3×C4⋊C4—28no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—91no—
173C3×C4⋊C4—181no—
174C3×C4⋊C4C2×C3×D314yes—
175C3×C4⋊C4C2×C3⋊D311yescharacteristic
176C3×C22⋊C4—28no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—21no—
179C3×C22⋊C4—181no—
180#217—64no—
181#216—68no—
182C4⋊Dic3—34no—
183Dic3⋊C4—64no—
184#208—68no—
185C2×C3⋊C8—34no—
186C2×C3⋊C8—34no—
187C22×C32—28no—
188C22×C32—24no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—24no—
193C22×C32—24no—
194C22×C32—24no—
195C22×C32—44no—
196C22×C32—44no—
197C22×C32C2×#6714yes—
198C22×C32C2×C3×D411yescharacteristic
199C32×C4—21no—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—44no—
204C32×C4—48no—
205C32×C4—48no—
206C32×C4—44no—
207C32×C4—44no—
208C32×C4—41no—
209C32×C4—41no—
210C32×C4D3×D414yes—
211C32×C4#23714yes—
212C32×C4C2×C3×D411yescharacteristic
213C32×C4C3×#4111yescharacteristic
214C3⋊Dic3—41no—
215C3⋊Dic3—41no—
216C3×Dic3—124no—
217C3×Dic3—124no—
218C2×Q16—91no—
219C2×M16—91no—
220C22⋊Q8—91no—
221C22⋊Q8C3×C3⋊D311yescharacteristic
222#106—91no—
223#103—91no—
224#103—91no—
225C33#47011yescharacteristic, complemented, Sylow
226C23×C3—24no—
227C23×C3C3×#6714yes—
228C23×C3#63811yescharacteristic
229C3×Q8—24no—
230C3×Q8—24no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—44no—
234C3×Q8—44no—
235C3×Q8—181no—
236C3×Q8—181no—
237C2×C3×C4—24no—
238C2×C3×C4—24no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—28no—
242C2×C3×C4—28no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—44no—
250C2×C3×C4—44no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4C3×#6714yes—
254C2×C3×C4#63811yescharacteristic
255C2×C3×C4C22×C3×D314yes—
256C2×C3×C4C22×C3⋊D311yescharacteristic
257C2×Dic3—64no—
258C2×Dic3—64no—
259Dic6—64no—
260Dic6—64no—
261C3×C8—92no—
262C3×C8—181no—
263C3⋊C8—38no—
264C3⋊C8—64no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—48no—
269C2×C32—44no—
270C2×C32—41no—
271C2×C32#93914yes—
272C2×C32#94214yes—
273C2×C32#95114yes—
274C2×C32C3×#12111yescharacteristic
275C2×C32C3×#13711yescharacteristic
276C2×C32C3×#13711yescharacteristic
277C2×Q8—91no—
278C2×Q8C2×C3×C3⋊D311yescharacteristic
279C22×C4C2×C3×C3⋊D311yescharacteristic
280Q16—182no—
281M16—182no—
282C2×C8—91no—
283C2×C8—91no—
284C4⋊C4—21no—
285C4⋊C4—91no—
286C4⋊C4—181no—
287C4⋊C4C2×C3×C3⋊D311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—181no—
290C22×C3—28no—
291C22×C3—24no—
292C22×C3—24no—
293C22×C3—24no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—24no—
298C22×C3—44no—
299C22×C3—44no—
300C22×C3C2×C3×#6714yes—
301C22×C3C2×#63811yescharacteristic
302C3×C4—24no—
303C3×C4—21no—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—44no—
307C3×C4—48no—
308C3×C4—48no—
309C3×C4—44no—
310C3×C4—44no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—361no—
314C3×C4—361no—
315C3×C4C3×D3×D414yes—
316C3×C4C3×#23714yes—
317C3×C4D4×C3⋊D311yescharacteristic
318C3×C4#383011yescharacteristic
319Dic3—124no—
320Dic3—124no—
321C32—24no—
322C32#505214yescomplemented
323C32C3×#47011yescharacteristic, complemented
324C23C3×#63811yescharacteristic
325Q8—21no—
326Q8—21no—
327Q8—181no—
328Q8—181no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4C3×#63811yescharacteristic
336C2×C4C22×C3×C3⋊D311yescharacteristic
337C8—92no—
338C8—181no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3—48no—
343C2×C3—44no—
344C2×C3—41no—
345C2×C3C3×#93914yes—
346C2×C3C3×#94214yes—
347C2×C3C3×#95114yes—
348C2×C3#6414811yescharacteristic
349C2×C3#6415111yescharacteristic
350C2×C3#6416011yescharacteristic
351C22—21no—
352C22—21no—
353C22—21no—
354C22C2×C3×#63811yescharacteristic
355C4—21no—
356C4—41no—
357C4—41no—
358C4—361no—
359C4—361no—
360C4C3×D4×C3⋊D311yescharacteristic
361C4C3×#383011yescharacteristic
362C3—24no—
363C3C3×#505214yescomplemented
364C3#1059275111yescharacteristic, direct product factor
365C2—41no—
366C2C3×#6414811yescharacteristic
367C2C3×#6415111yescharacteristic
368C2C3×#6416011yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C22⋊Q8C2
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC3×C22⋊Q8C3⋊D3
hypocenterC32C3×#470
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3756C2semidirect
5. C3×#63662C2semidirect
6. C3×#63637C2semidirect
7. C3×#63635C2semidirect
8. #10592751C3direct (N × H)
26. C2×#3756C2×C3semidirect
32. #63662C2×C3semidirect
33. #63637C2×C3semidirect
34. #63635C2×C3semidirect
225. C33#470semidirect
322. C32#5052semidirect
323. C32C3×#470semidirect
363. C3C3×#5052semidirect
364. C3#10592751direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C22⋊Q82
C2×C3×#37562
C3×#641332
C2×C3×#37482
C3×#636622
C3×#636372
C3×#636352
#105927513
C3×#50523