Finite Group Databasebeta

← C3×#10596012 #13179294

Subgroups — C3×#10596012 #13179294

Order 1728.

1660 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×D4C211yesmaximal, characteristic
2C3×#64144C212yesmaximal
3C3×#64135C211yesmaximal, characteristic, complemented
4C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5C3×#64133C211yesmaximal, characteristic, complemented
6C3×#64133C211yesmaximal, characteristic, complemented
7C3×#64098C212yesmaximal, complemented
8C3×#64088C212yesmaximal, complemented
9C3×#64086C212yesmaximal, complemented
10C3×#64079C211yesmaximal, characteristic, complemented
11C3×C4×#634C211yesmaximal, characteristic, complemented
12#10596012C311yesmaximal, characteristic, direct product factor
13C3×#5541—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C33×C4C2211yescharacteristic
16C22×C33×C4C2211yescharacteristic
17C22×C3×C3⋊Dic3C2212yes—
18C2×C3×#634C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20C33×C22⋊C4C2211yescharacteristic
21C33×C22⋊C4C2211yescharacteristic
22C33×C42C2211yescharacteristic
23C3×#3757C2212yes—
24C3×#3757C2212yes—
25C3×#3757C2212yes—
26C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
27C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
29C3×C4⋊(C3⋊Dic3)C2212yes—
30C3×#3750C2212yes—
31C3×#3750C2212yes—
32C3×#3750C2212yes—
33C3×#3750C2212yes—
34C3×#3750C2212yes—
35C3×C4×C3⋊Dic3C2212yes—
36C3×C4×C3⋊Dic3C2212yes—
37C32×C4×D4—24no—
38C32×C4×D4D314yes—
39C32×C4×D4C2×C311yescharacteristic
40#64144C2×C312yes—
41#64135C2×C311yescharacteristic, complemented
42C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
43#64133C2×C311yescharacteristic, complemented
44#64133C2×C311yescharacteristic, complemented
45#64098C2×C312yescomplemented
46#64088C2×C312yescomplemented
47#64086C2×C312yescomplemented
48#64079C2×C311yescharacteristic, complemented
49C4×#634C2×C311yescharacteristic, complemented
50C3×#935—38no—
51C3×#926—34no—
52C2×C3×C4⋊Dic3—34no—
53C3×#924—34no—
54C3×#924—34no—
55C3×#889—38no—
56C3×#879—38no—
57C3×#877—38no—
58C3×#870—34no—
59C3×C4×Dic6—34no—
60C23×C33C2311yescharacteristic
61C23×C33C2311yescharacteristic
62C33×D4—22no—
63C2×C33×C4—22no—
64C2×C33×C4—22no—
65C2×C33×C4C2311yescharacteristic
66C2×C33×C4C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C2×C33×C4C2311yescharacteristic
70C2×C3×C3⋊Dic3—24no—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2312yes—
75C3×#634—22no—
76C3×#552—91no—
77#5541—34no—
78C2×C32×D4—24no—
79C2×C32×D4C2×D314yes—
80C2×C32×D4C22×C311yescharacteristic
81C22×C32×C4—24no—
82C22×C32×C4—24no—
83C22×C32×C4C2×D314yes—
84C22×C32×C4C2×D314yes—
85C22×C32×C4C22×C311yescharacteristic
86C22×C32×C4C22×C311yescharacteristic
87C22×C3⋊Dic3C22×C312yes—
88C2×#634C22×C311yescharacteristic
89C22×C3×Dic3—38no—
90C2×C3×Dic6—34no—
91C32×C4⋊C4—24no—
92C32×C4⋊C4C2×D314yes—
93C32×C4⋊C4C22×C311yescharacteristic
94C32×C22⋊C4—24no—
95C32×C22⋊C4—24no—
96C32×C22⋊C4C2×D314yes—
97C32×C22⋊C4C2×D314yes—
98C32×C22⋊C4C22×C311yescharacteristic
99C32×C22⋊C4C22×C311yescharacteristic
100C32×C42—24no—
101C32×C42C2×D314yes—
102C32×C42C22×C311yescharacteristic
103#3757C22×C312yes—
104#3757C22×C312yes—
105#3757C22×C312yes—
106C4⋊(C3⋊Dic3)C22×C311yescharacteristic
107C4⋊(C3⋊Dic3)C22×C311yescharacteristic
108C4⋊(C3⋊Dic3)C22×C311yescharacteristic
109C4⋊(C3⋊Dic3)C22×C312yes—
110#3750C22×C312yes—
111#3750C22×C312yes—
112#3750C22×C312yes—
113#3750C22×C312yes—
114#3750C22×C312yes—
115C4×C3⋊Dic3C22×C312yes—
116C4×C3⋊Dic3C22×C312yes—
117C3×#217—38no—
118C3×#217—38no—
119C3×#217—38no—
120C3×C4⋊Dic3—38no—
121C3×C4⋊Dic3—34no—
122C3×C4⋊Dic3—34no—
123C3×C4⋊Dic3—34no—
124C3×Dic3⋊C4—38no—
125C3×Dic3⋊C4—38no—
126C3×Dic3⋊C4—38no—
127C3×Dic3⋊C4—38no—
128C3×Dic3⋊C4—38no—
129C3×C4×Dic3—38no—
130C3×C4×Dic3—38no—
131C22×C33—21no—
132C22×C33—21no—
133C22×C33—21no—
134C22×C33—21no—
135C22×C33—21no—
136C22×C33#4112yes—
137C22×C33C2411yescharacteristic
138C33×C4—21no—
139C33×C4—21no—
140C33×C4—21no—
141C33×C4—21no—
142C33×C4#4112yes—
143C3×C3⋊Dic3—22no—
144C3×C3⋊Dic3—22no—
145C3×C3⋊Dic3—22no—
146C3×C3⋊Dic3—22no—
147C3×#126—92no—
148C3×#125—92no—
149C3×#125—91no—
150C3×#123—92no—
151C3×C22⋊Q8—91no—
152C3×C22⋊Q8—91no—
153C3×C4×Q8—91no—
154C3×C4×D4—24no—
155C3×C4×D4C3×D314yes—
156C3×C4×D4C3⋊D311yescharacteristic
157C3×#117—91no—
158C3×#117—92no—
159C2×C3×C4⋊C4—91no—
160#935—38no—
161#926—34no—
162C2×C4⋊Dic3—34no—
163#924—34no—
164#924—34no—
165#889—38no—
166#879—38no—
167#877—38no—
168#870—34no—
169C4×Dic6—34no—
170C23×C32—24no—
171C23×C32—24no—
172C23×C32C22×D314yes—
173C23×C32C22×D314yes—
174C23×C32C23×C311yescharacteristic
175C23×C32C23×C311yescharacteristic
176C32×D4—216no—
177C32×D4—28no—
178C32×D4—22no—
179C2×C32×C4—22no—
180C2×C32×C4—22no—
181C2×C32×C4—24no—
182C2×C32×C4—24no—
183C2×C32×C4—24no—
184C2×C32×C4—24no—
185C2×C32×C4—24no—
186C2×C32×C4—28no—
187C2×C32×C4—28no—
188C2×C32×C4—48no—
189C2×C32×C4—48no—
190C2×C32×C4C22×D314yes—
191C2×C32×C4C22×D314yes—
192C2×C32×C4C22×D314yes—
193C2×C32×C4C22×D314yes—
194C2×C32×C4C22×D314yes—
195C2×C32×C4C23×C311yescharacteristic
196C2×C32×C4C23×C311yescharacteristic
197C2×C32×C4C23×C311yescharacteristic
198C2×C32×C4C23×C311yescharacteristic
199C2×C32×C4C23×C311yescharacteristic
200C2×C3⋊Dic3—24no—
201C2×C3⋊Dic3C23×C312yes—
202C2×C3⋊Dic3C23×C312yes—
203C2×C3⋊Dic3C23×C312yes—
204C2×C3⋊Dic3C23×C312yes—
205#634—22no—
206C2×C3×Dic3—38no—
207C2×C3×Dic3—38no—
208C2×C3×Dic3—38no—
209C2×C3×Dic3—38no—
210C2×C3×Dic3—616no—
211C3×Dic6—68no—
212#552—91noSylow
213C2×C33—22no—
214C2×C33—41no—
215C2×C33C2×#4111yescharacteristic
216C2×C33C2×#4111yescharacteristic
217C2×C332−1+411yescharacteristic
218C2×C3×Q8—91no—
219C2×C3×D4—24no—
220C2×C3×D4C2×C3×D314yes—
221C2×C3×D4C2×C3⋊D311yescharacteristic
222C22×C3×C4—24no—
223C22×C3×C4—24no—
224C22×C3×C4—92no—
225C22×C3×C4C2×C3×D314yes—
226C22×C3×C4C2×C3×D314yes—
227C22×C3×C4C2×C3⋊D311yescharacteristic
228C22×C3×C4C2×C3⋊D311yescharacteristic
229C22×Dic3—38no—
230C2×Dic6—34no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—92no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—91no—
239C3×C4⋊C4—92no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4C2×C3×D314yes—
242C3×C4⋊C4C2×C3⋊D311yescharacteristic
243C3×C22⋊C4—24no—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4C2×C3×D314yes—
249C3×C22⋊C4C2×C3×D314yes—
250C3×C22⋊C4C2×C3⋊D311yescharacteristic
251C3×C22⋊C4C2×C3⋊D311yescharacteristic
252C3×C42—24no—
253C3×C42—92no—
254C3×C42—92no—
255C3×C42C2×C3×D314yes—
256C3×C42C2×C3⋊D311yescharacteristic
257#217—38no—
258#217—38no—
259#217—38no—
260C4⋊Dic3—34no—
261C4⋊Dic3—34no—
262C4⋊Dic3—34no—
263C4⋊Dic3—38no—
264Dic3⋊C4—38no—
265Dic3⋊C4—38no—
266Dic3⋊C4—38no—
267Dic3⋊C4—38no—
268Dic3⋊C4—38no—
269C4×Dic3—38no—
270C4×Dic3—38no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—28no—
274C22×C32—24no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—24no—
279C22×C32—28no—
280C22×C32—24no—
281C22×C32—24no—
282C22×C32—24no—
283C22×C32—24no—
284C22×C32—44no—
285C22×C32—44no—
286C22×C32—44no—
287C22×C32—44no—
288C22×C32#23518yes—
289C22×C32C3×#4112yes—
290C22×C32C23×D314yes—
291C22×C32C24×C311yescharacteristic
292C32×C4—28no—
293C32×C4—24no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—28no—
299C32×C4—24no—
300C32×C4—24no—
301C32×C4—24no—
302C32×C4—44no—
303C32×C4—44no—
304C32×C4—44no—
305C32×C4#23718yes—
306C32×C4C3×#4112yes—
307C3⋊Dic3—22no—
308C3⋊Dic3—22no—
309C3⋊Dic3—22no—
310C3⋊Dic3—22no—
311C3×Dic3—68no—
312C3×Dic3—68no—
313C3×Dic3—68no—
314C3×Dic3—68no—
315#126—92no—
316#125—92no—
317#125—91no—
318#123—92no—
319C22⋊Q8—91no—
320C22⋊Q8—91no—
321C4×Q8—91no—
322C4×D4C3×C3⋊D311yescharacteristic
323#117—91no—
324#117—92no—
325C2×C4⋊C4—91no—
326C33#55211yescharacteristic, complemented, Sylow
327C23×C3—24no—
328C23×C3—24no—
329C23×C3C22×C3×D314yes—
330C23×C3C22×C3×D314yes—
331C23×C3C22×C3⋊D311yescharacteristic
332C23×C3C22×C3⋊D311yescharacteristic
333C3×Q8—182no—
334C3×D4—216no—
335C3×D4—28no—
336C3×D4—22no—
337C2×C3×C4—28no—
338C2×C3×C4—28no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—48no—
347C2×C3×C4—48no—
348C2×C3×C4—92no—
349C2×C3×C4—92no—
350C2×C3×C4—92no—
351C2×C3×C4—92no—
352C2×C3×C4—184no—
353C2×C3×C4C22×C3×D314yes—
354C2×C3×C4C22×C3×D314yes—
355C2×C3×C4C22×C3×D314yes—
356C2×C3×C4C22×C3×D314yes—
357C2×C3×C4C22×C3×D314yes—
358C2×C3×C4C22×C3⋊D311yescharacteristic
359C2×C3×C4C22×C3⋊D311yescharacteristic
360C2×C3×C4C22×C3⋊D311yescharacteristic
361C2×C3×C4C22×C3⋊D311yescharacteristic
362C2×C3×C4C22×C3⋊D311yescharacteristic
363C2×Dic3—38no—
364C2×Dic3—38no—
365C2×Dic3—38no—
366C2×Dic3—38no—
367C2×Dic3—616no—
368Dic6—68no—
369C2×C32—22no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32—24no—
373C2×C32—28no—
374C2×C32—48no—
375C2×C32—48no—
376C2×C32—44no—
377C2×C32—41no—
378C2×C32C2×#23514yes—
379C2×C32C2×#23714yes—
380C2×C32#101014yes—
381C2×C32C2×C3×#4111yescharacteristic
382C2×C32C2×C3×#4111yescharacteristic
383C2×C32C3×2−1+411yescharacteristic
384C2×Q8—91no—
385C2×D4C2×C3×C3⋊D311yescharacteristic
386C22×C4—92no—
387C22×C4C2×C3×C3⋊D311yescharacteristic
388C22×C4C2×C3×C3⋊D311yescharacteristic
389C4⋊C4—91no—
390C4⋊C4—91no—
391C4⋊C4—92no—
392C4⋊C4—92no—
393C4⋊C4—91no—
394C4⋊C4—92no—
395C4⋊C4—92no—
396C4⋊C4—92no—
397C4⋊C4—92no—
398C4⋊C4C2×C3×C3⋊D311yescharacteristic
399C22⋊C4—92no—
400C22⋊C4—92no—
401C22⋊C4—92no—
402C22⋊C4C2×C3×C3⋊D311yescharacteristic
403C22⋊C4C2×C3×C3⋊D311yescharacteristic
404C42—92no—
405C42—92no—
406C42C2×C3×C3⋊D311yescharacteristic
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—28no—
412C22×C3—24no—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—24no—
419C22×C3—28no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3C3×#23518yes—
425C22×C3#382812yes—
426C22×C3C23×C3×D314yes—
427C22×C3C23×C3⋊D311yescharacteristic
428C3×C4—28no—
429C3×C4—24no—
430C3×C4—24no—
431C3×C4—24no—
432C3×C4—24no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—28no—
438C3×C4—44no—
439C3×C4—44no—
440C3×C4—44no—
441C3×C4—182no—
442C3×C4—182no—
443C3×C4—182no—
444C3×C4—182no—
445C3×C4C3×#23718yes—
446C3×C4#383012yes—
447Dic3—68no—
448Dic3—68no—
449Dic3—68no—
450Dic3—68no—
451C32—24no—
452C32#554114yescomplemented
453C32C3×#55211yescharacteristic, complemented
454C23C22×C3×C3⋊D311yescharacteristic
455C23C22×C3×C3⋊D311yescharacteristic
456Q8—182no—
457D4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—92no—
461C2×C4—92no—
462C2×C4—92no—
463C2×C4—92no—
464C2×C4—184no—
465C2×C4C22×C3×C3⋊D311yescharacteristic
466C2×C4C22×C3×C3⋊D311yescharacteristic
467C2×C4C22×C3×C3⋊D311yescharacteristic
468C2×C4C22×C3×C3⋊D311yescharacteristic
469C2×C4C22×C3×C3⋊D311yescharacteristic
470C2×C3—28no—
471C2×C3—22no—
472C2×C3—24no—
473C2×C3—24no—
474C2×C3—24no—
475C2×C3—48no—
476C2×C3—48no—
477C2×C3—44no—
478C2×C3—41no—
479C2×C3C2×C3×#23514yes—
480C2×C3C2×C3×#23714yes—
481C2×C3C3×#101014yes—
482C2×C3C2×#382811yescharacteristic
483C2×C3C2×#383011yescharacteristic
484C2×C3#6436711yescharacteristic
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22C3×#382812yes—
491C22C23×C3×C3⋊D311yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—182no—
497C4—182no—
498C4—182no—
499C4—182no—
500C4C3×#383012yes—
501C3—24no—
502C3C3×#554114yescomplemented
503C3#1059601211yescharacteristic, direct product factor
504C2—22no—
505C2—41no—
506C2C2×C3×#382811yescharacteristic
507C2C2×C3×#383011yescharacteristic
508C2C3×#6436711yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C24
prime radicalC2×C33×D4C22
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
3. C3×#64135C2semidirect
4. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
5. C3×#64133C2semidirect
6. C3×#64133C2semidirect
7. C3×#64098C2semidirect
8. C3×#64088C2semidirect
9. C3×#64086C2semidirect
10. C3×#64079C2semidirect
11. C3×C4×#634C2semidirect
12. #10596012C3direct (N × H)
41. #64135C2×C3semidirect
42. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
43. #64133C2×C3semidirect
44. #64133C2×C3semidirect
45. #64098C2×C3semidirect
46. #64088C2×C3semidirect
47. #64086C2×C3semidirect
48. #64079C2×C3semidirect
49. C4×#634C2×C3semidirect
326. C33#552semidirect
452. C32#5541semidirect
453. C32C3×#552semidirect
502. C3C3×#5541semidirect
503. C3#10596012direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4×D42
C3×#641442
C3×#641352
C2×C3×C4⋊(C3⋊Dic3)2
C3×#641332
C3×#640982
C3×#640882
C3×#640862
C3×#640792
C3×C4×#6342
#105960123
C3×#55413