Finite Group Databasebeta

← C3×#10596000 #13179283

Subgroups — C3×#10596000 #13179283

Order 1728.

1588 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×#117C211yesmaximal, characteristic, complemented
2C3×C4×#638C211yesmaximal, characteristic
3C3×#64133C211yesmaximal, characteristic, complemented
4C3×#64106C211yesmaximal, characteristic, complemented
5C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
6C3×#64099C211yesmaximal, characteristic, complemented
7C3×#64096C211yesmaximal, characteristic, complemented
8C3×#64092C212yesmaximal
9C3×#64088C212yesmaximal, complemented
10C3×#64081C212yesmaximal, complemented
11C3×#64079C212yesmaximal, complemented
12#10596000C311yesmaximal, characteristic, direct product factor
13C3×#5529—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×C3×#638C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20C33×C4⋊C4C2211yescharacteristic
21C33×C22⋊C4C2212yes—
22C33×C42C2212yes—
23C3×#3757C2211yescharacteristic
24C3×#3757C2212yes—
25C3×#3752C2211yescharacteristic
26C3×#3752C2212yes—
27C3×C4⋊(C3⋊Dic3)C2212yes—
28C3×C4⋊(C3⋊Dic3)C2212yes—
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
30C3×#3750C2211yescharacteristic
31C3×#3750C2212yes—
32C3×#3750C2212yes—
33C3×#3750C2212yes—
34C3×#3750C2212yes—
35C3×C4×C3⋊Dic3C2212yes—
36C3×C4×C3⋊Dic3C2211yescharacteristic
37C32×#117—24no—
38C32×#117D314yescomplemented
39C32×#117C2×C311yescharacteristic, complemented
40C4×#638C2×C311yescharacteristic
41#64133C2×C311yescharacteristic, complemented
42#64106C2×C311yescharacteristic, complemented
43C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
44#64099C2×C311yescharacteristic, complemented
45#64096C2×C311yescharacteristic, complemented
46#64092C2×C312yes—
47#64088C2×C312yescomplemented
48#64081C2×C312yescomplemented
49#64079C2×C312yescomplemented
50C3×C4×#67—34no—
51C3×#924—34no—
52C3×#897—34no—
53C3×D3×C4⋊C4—34no—
54C3×#890—34no—
55C3×#887—34no—
56C3×#883—38no—
57C3×#879—38no—
58C3×#872—38no—
59C3×#870—38no—
60C23×C33C2311yescharacteristic
61C22×C3×C3⋊D3C2311yescharacteristic
62C2×C33×C4—22no—
63C2×C33×C4C2312yes—
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2312yes—
66C2×C33×C4C2311yescharacteristic
67C3×#638—22no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2312yes—
73C3×C4×C3⋊D3—24no—
74C3×C4×C3⋊D3—22no—
75C3×#634—22no—
76C3×#552—91no—
77#5529—34no—
78C22×C32×C4—24no—
79C22×C32×C4C2×D314yes—
80C22×C32×C4C22×C311yescharacteristic
81C2×#638C22×C311yescharacteristic
82C2×C4×C3⋊D3C22×C312yes—
83C2×C4×C3⋊D3C22×C311yescharacteristic
84C2×#634C22×C311yescharacteristic
85C2×C3×#67—34no—
86C2×C3×C4×D3—38no—
87C2×C3×C4×D3—34no—
88C2×C3×Dic6—34no—
89C32×C4⋊C4—24no—
90C32×C4⋊C4—24no—
91C32×C4⋊C4C2×D314yes—
92C32×C4⋊C4C2×D314yes—
93C32×C4⋊C4C22×C311yescharacteristic
94C32×C4⋊C4C22×C311yescharacteristic
95C32×C22⋊C4—28no—
96C32×C22⋊C4C2×D318yes—
97C32×C22⋊C4C22×C312yes—
98C32×C42—28no—
99C32×C42C2×D318yes—
100C32×C42C22×C312yes—
101#3757C22×C311yescharacteristic
102#3757C22×C312yes—
103#3752C22×C311yescharacteristic
104#3752C22×C312yes—
105C4⋊(C3⋊Dic3)C22×C312yes—
106C4⋊(C3⋊Dic3)C22×C312yes—
107C4⋊(C3⋊Dic3)C22×C311yescharacteristic
108#3750C22×C311yescharacteristic
109#3750C22×C312yes—
110#3750C22×C312yes—
111#3750C22×C312yes—
112#3750C22×C312yes—
113C4×C3⋊Dic3C22×C312yes—
114C4×C3⋊Dic3C22×C311yescharacteristic
115C3×#217—34no—
116C3×#217—38no—
117C3×(C2×D3)⋊C4—38no—
118C3×(C2×D3)⋊C4—34no—
119C3×C4⋊Dic3—38no—
120C3×C4⋊Dic3—38no—
121C3×C4⋊Dic3—34no—
122C3×Dic3⋊C4—38no—
123C3×Dic3⋊C4—38no—
124C3×Dic3⋊C4—38no—
125C3×Dic3⋊C4—38no—
126C3×Dic3⋊C4—34no—
127C3×C4×Dic3—38no—
128C3×C4×Dic3—34no—
129C22×C33—21no—
130C22×C33—21no—
131C22×C33—21no—
132C22×C33C2411yescharacteristic
133C2×C3×C3⋊D3—21no—
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3#4112yes—
136C33×C4—21no—
137C33×C4—22no—
138C33×C4—22no—
139C33×C4#4112yes—
140C3×C3⋊Dic3—22no—
141C3×C3⋊Dic3—21no—
142C3×C3⋊Dic3—22no—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3—21no—
145C3×#126—92no—
146C3×#125—91no—
147C3×#125—92no—
148C3×#123—92no—
149C3×C22⋊Q8—91no—
150C3×C22⋊Q8—91no—
151C3×C4×Q8—91no—
152C3×C4×D4—91no—
153C3×#117—24no—
154C3×#117—92no—
155C3×#117C3×D314yescomplemented
156C3×#117C3⋊D311yescharacteristic, complemented
157C2×C3×C4⋊C4—91no—
158C4×#67—34no—
159#924—34no—
160#897—34no—
161D3×C4⋊C4—34no—
162#890—34no—
163#887—34no—
164#883—38no—
165#879—38no—
166#872—38no—
167#870—38no—
168C23×C32—24no—
169C23×C32C22×D314yes—
170C23×C32C23×C311yescharacteristic
171C22×C3⋊D3C23×C311yescharacteristic
172C22×C3×D3—34no—
173C2×C32×C4—216no—
174C2×C32×C4—28no—
175C2×C32×C4—22no—
176C2×C32×C4—24no—
177C2×C32×C4—24no—
178C2×C32×C4—28no—
179C2×C32×C4—28no—
180C2×C32×C4C22×D318yes—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D318yes—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C23×C312yes—
185C2×C32×C4C23×C311yescharacteristic
186C2×C32×C4C23×C312yes—
187C2×C32×C4C23×C311yescharacteristic
188#638—22no—
189C2×C3⋊Dic3C23×C311yescharacteristic
190C2×C3⋊Dic3C23×C311yescharacteristic
191C2×C3⋊Dic3C23×C312yes—
192C2×C3⋊Dic3C23×C311yescharacteristic
193C2×C3⋊Dic3C23×C312yes—
194C4×C3⋊D3—24no—
195C4×C3⋊D3—22no—
196#634—22no—
197C3×#67—68no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—34no—
200C2×C3×Dic3—38no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—38no—
203C3×C4×D3—616no—
204C3×C4×D3—68no—
205C3×Dic6—68no—
206#552—91noSylow
207C2×C33—41no—
208C2×C33C2×#4111yescharacteristic
209C2×C33C2×#4111yescharacteristic
210C2×C332−1+411yescharacteristic
211C3×C3⋊D3—22no—
212C2×C3×Q8—91no—
213C2×C3×D4—91no—
214C22×C3×C4—24no—
215C22×C3×C4—92no—
216C22×C3×C4—91no—
217C22×C3×C4C2×C3×D314yes—
218C22×C3×C4C2×C3⋊D311yescharacteristic
219C2×#67—34no—
220C2×C4×D3—38no—
221C2×C4×D3—34no—
222C2×Dic6—34no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4—92no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—92no—
230C3×C4⋊C4—91no—
231C3×C4⋊C4—92no—
232C3×C4⋊C4—92no—
233C3×C4⋊C4C2×C3×D314yes—
234C3×C4⋊C4C2×C3×D314yes—
235C3×C4⋊C4C2×C3⋊D311yescharacteristic
236C3×C4⋊C4C2×C3⋊D311yescharacteristic
237C3×C22⋊C4—28no—
238C3×C22⋊C4—91no—
239C3×C22⋊C4—92no—
240C3×C22⋊C4—91no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4C2×C3×D318yes—
243C3×C22⋊C4C2×C3⋊D312yes—
244C3×C42—28no—
245C3×C42—92no—
246C3×C42—91no—
247C3×C42C2×C3×D318yes—
248C3×C42C2×C3⋊D312yes—
249#217—34no—
250#217—38no—
251(C2×D3)⋊C4—34no—
252(C2×D3)⋊C4—38no—
253C4⋊Dic3—38no—
254C4⋊Dic3—38no—
255C4⋊Dic3—34no—
256Dic3⋊C4—34no—
257Dic3⋊C4—38no—
258Dic3⋊C4—38no—
259Dic3⋊C4—38no—
260Dic3⋊C4—38no—
261C4×Dic3—38no—
262C4×Dic3—34no—
263C22×C32—28no—
264C22×C32—24no—
265C22×C32—21no—
266C22×C32—21no—
267C22×C32—21no—
268C22×C32—24no—
269C22×C32—24no—
270C22×C32—24no—
271C22×C32—44no—
272C22×C32—44no—
273C22×C32C23×D314yes—
274C22×C32C24×C311yescharacteristic
275C2×C3⋊D3—21no—
276C2×C3⋊D3—21no—
277C2×C3⋊D3C3×#4112yes—
278C2×C3×D3—38no—
279C2×C3×D3—64no—
280C2×C3×D3—64no—
281C32×C4—28no—
282C32×C4—24no—
283C32×C4—22no—
284C32×C4—21no—
285C32×C4—22no—
286C32×C4—28no—
287C32×C4—28no—
288C32×C4—28no—
289C32×C4—48no—
290C32×C4—48no—
291C32×C4#23518yes—
292C32×C4C3×#4112yes—
293C3⋊Dic3—22no—
294C3⋊Dic3—21no—
295C3⋊Dic3—22no—
296C3⋊Dic3—21no—
297C3⋊Dic3—21no—
298C3×Dic3—68no—
299C3×Dic3—64no—
300C3×Dic3—68no—
301C3×Dic3—64no—
302C3×Dic3—64no—
303#126—92no—
304#125—91no—
305#125—92no—
306#123—92no—
307C22⋊Q8—91no—
308C22⋊Q8—91no—
309C4×Q8—91no—
310C4×D4—91no—
311#117—92no—
312#117C3×C3⋊D311yescharacteristic, complemented
313C2×C4⋊C4—91no—
314C33#55211yescharacteristic, complemented, Sylow
315C23×C3—24no—
316C23×C3—91no—
317C23×C3C22×C3×D314yes—
318C23×C3C22×C3⋊D311yescharacteristic
319C22×D3—34no—
320C3×Q8—182no—
321C3×D4—182no—
322C2×C3×C4—216no—
323C2×C3×C4—28no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—28no—
328C2×C3×C4—28no—
329C2×C3×C4—91no—
330C2×C3×C4—92no—
331C2×C3×C4—91no—
332C2×C3×C4—91no—
333C2×C3×C4—92no—
334C2×C3×C4—184no—
335C2×C3×C4—182no—
336C2×C3×C4C22×C3×D318yes—
337C2×C3×C4C22×C3×D314yes—
338C2×C3×C4C22×C3×D318yes—
339C2×C3×C4C22×C3×D314yes—
340C2×C3×C4C22×C3⋊D312yes—
341C2×C3×C4C22×C3⋊D311yescharacteristic
342C2×C3×C4C22×C3⋊D312yes—
343C2×C3×C4C22×C3⋊D311yescharacteristic
344#67—68no—
345C2×Dic3—34no—
346C2×Dic3—34no—
347C2×Dic3—38no—
348C2×Dic3—34no—
349C2×Dic3—38no—
350C4×D3—616no—
351C4×D3—68no—
352Dic6—68no—
353C2×C32—24no—
354C2×C32—24no—
355C2×C32—24no—
356C2×C32—48no—
357C2×C32—44no—
358C2×C32—41no—
359C2×C32C2×#23514yes—
360C2×C32D3×#4114yes—
361C2×C32#101014yes—
362C2×C32C2×C3×#4111yescharacteristic
363C2×C32C2×C3×#4111yescharacteristic
364C2×C32C3×2−1+411yescharacteristic
365C3⋊D3—22no—
366C3×D3—68no—
367C2×Q8—91no—
368C2×D4—91no—
369C22×C4—92no—
370C22×C4—91no—
371C22×C4C2×C3×C3⋊D311yescharacteristic
372C4⋊C4—91no—
373C4⋊C4—92no—
374C4⋊C4—92no—
375C4⋊C4—92no—
376C4⋊C4—91no—
377C4⋊C4—92no—
378C4⋊C4—92no—
379C4⋊C4—92no—
380C4⋊C4C2×C3×C3⋊D311yescharacteristic
381C4⋊C4C2×C3×C3⋊D311yescharacteristic
382C22⋊C4—91no—
383C22⋊C4—92no—
384C22⋊C4—91no—
385C22⋊C4—92no—
386C22⋊C4C2×C3×C3⋊D312yes—
387C42—92no—
388C42—91no—
389C42C2×C3×C3⋊D312yes—
390C22×C3—28no—
391C22×C3—24no—
392C22×C3—24no—
393C22×C3—24no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—24no—
398C22×C3—44no—
399C22×C3—44no—
400C22×C3—92no—
401C22×C3—181no—
402C22×C3—181no—
403C22×C3C23×C3×D314yes—
404C22×C3C23×C3⋊D311yescharacteristic
405C2×D3—38no—
406C2×D3—64no—
407C2×D3—64no—
408C3×C4—28no—
409C3×C4—24no—
410C3×C4—28no—
411C3×C4—28no—
412C3×C4—21no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—28no—
416C3×C4—48no—
417C3×C4—48no—
418C3×C4—181no—
419C3×C4—182no—
420C3×C4—182no—
421C3×C4—181no—
422C3×C4—181no—
423C3×C4C3×#23518yes—
424C3×C4#382812yes—
425Dic3—68no—
426Dic3—64no—
427Dic3—68no—
428Dic3—64no—
429Dic3—64no—
430C32—24no—
431C32#552914yescomplemented
432C32C3×#55211yescharacteristic, complemented
433C23—91no—
434C23C22×C3×C3⋊D311yescharacteristic
435Q8—182no—
436D4—182no—
437C2×C4—22no—
438C2×C4—91no—
439C2×C4—92no—
440C2×C4—91no—
441C2×C4—91no—
442C2×C4—92no—
443C2×C4—184no—
444C2×C4—182no—
445C2×C4C22×C3×C3⋊D312yes—
446C2×C4C22×C3×C3⋊D311yescharacteristic
447C2×C4C22×C3×C3⋊D312yes—
448C2×C4C22×C3×C3⋊D311yescharacteristic
449C2×C3—24no—
450C2×C3—24no—
451C2×C3—24no—
452C2×C3—48no—
453C2×C3—44no—
454C2×C3—41no—
455C2×C3—182no—
456C2×C3C2×C3×#23514yes—
457C2×C3C3×D3×#4114yes—
458C2×C3C3×#101014yes—
459C2×C3C2×#382811yescharacteristic
460C2×C3#41×C3⋊D311yescharacteristic
461C2×C3#6436711yescharacteristic
462D3—68no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—92no—
467C22—181no—
468C22—181no—
469C22C23×C3×C3⋊D311yescharacteristic
470C4—21no—
471C4—22no—
472C4—22no—
473C4—181no—
474C4—182no—
475C4—182no—
476C4—181no—
477C4—181no—
478C4C3×#382812yes—
479C3—24no—
480C3C3×#552914yescomplemented
481C3#1059600011yescharacteristic, direct product factor
482C2—41no—
483C2—182no—
484C2C2×C3×#382811yescharacteristic
485C2C3×#41×C3⋊D311yescharacteristic
486C2C3×#6436711yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×#117C2
socleC22×C33C24
prime radicalC2×C3×#638C22
hypercenterC3×#117C3⋊D3
hypocenterC32C3×#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
1. C33×#117C2semidirect
3. C3×#64133C2semidirect
4. C3×#64106C2semidirect
5. C3×C4⋊C4×C3⋊D3C2semidirect
6. C3×#64099C2semidirect
7. C3×#64096C2semidirect
9. C3×#64088C2semidirect
10. C3×#64081C2semidirect
11. C3×#64079C2semidirect
12. #10596000C3direct (N × H)
38. C32×#117D3semidirect
39. C32×#117C2×C3semidirect
41. #64133C2×C3semidirect
42. #64106C2×C3semidirect
43. C4⋊C4×C3⋊D3C2×C3semidirect
44. #64099C2×C3semidirect
45. #64096C2×C3semidirect
47. #64088C2×C3semidirect
48. #64081C2×C3semidirect
49. #64079C2×C3semidirect
155. C3×#117C3×D3semidirect
156. C3×#117C3⋊D3semidirect
312. #117C3×C3⋊D3semidirect
314. C33#552semidirect
431. C32#5529semidirect
432. C32C3×#552semidirect
480. C3C3×#5529semidirect
481. C3#10596000direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1172
C3×C4×#6382
C3×#641332
C3×#641062
C3×C4⋊C4×C3⋊D32
C3×#640992
C3×#640962
C3×#640922
C3×#640882
C3×#640812
C3×#640792
#105960003
C3×#55293