Finite Group Databasebeta

← C3×#10592368 #13162709

Subgroups — C3×#10592368 #13162709

Order 1728.

1444 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3752C211yesmaximal, characteristic
2C2×C3×#3752C211yesmaximal, characteristic
3C2×C3×#3752C211yesmaximal, characteristic
4C2×C3×#3750C211yesmaximal, characteristic, complemented
5C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic, complemented
6C33×#95C211yesmaximal, characteristic, complemented
7C3×#63658C211yesmaximal, characteristic, complemented
8#10592368C311yesmaximal, characteristic, direct product factor
9C3×#4669—34nomaximal
10C23×C3×C3⋊D3C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C3×#3752—22no—
18C3×#3752—22no—
19C3×#3752C414yescomplemented
20C3×#3750—22no—
21C3×C4×C3⋊Dic3—22no—
22C2×#3752C2×C311yescharacteristic
23C2×#3752C2×C311yescharacteristic
24C2×#3752C2×C311yescharacteristic
25C2×#3750C2×C311yescharacteristic, complemented
26C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
27C2×C3×(C2×D3)⋊C4—34no—
28C2×C3×(C2×D3)⋊C4—34no—
29C2×C3×(C2×D3)⋊C4—34no—
30C2×C3×Dic3⋊C4—34no—
31C2×C3×C4×Dic3—34no—
32C32×#95—24no—
33C32×#95D314yescomplemented
34C32×#95C2×C311yescharacteristic, complemented
35#63658C2×C311yescharacteristic, complemented
36C3×C3⋊#95—34no—
37C23×C33C2311yescharacteristic
38C22×C3×C3⋊D3—21no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3—21no—
44C22×C3×C3⋊D3C2×C412yes—
45C2×C33×C4—21no—
46C2×C33×C4—21no—
47C2×C33×C4—21no—
48C2×C33×C4—21no—
49C2×C33×C4—21no—
50C2×C33×C4—21no—
51C2×C33×C4—21no—
52C2×C33×C4—21no—
53C2×C33×C4C2×C412yes—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3—21no—
57C2×C3×C3⋊Dic3—21no—
58C2×C3×C3⋊Dic3—21no—
59C2×C3×C3⋊Dic3—21no—
60C2×C3×C3⋊Dic3C2×C412yes—
61C2×C3×C3⋊Dic3D412yes—
62C2×C3×C3⋊Dic3D412yes—
63C3×#388—91no—
64#4669—34no—
65C23×C3⋊D3C22×C311yescharacteristic
66C23×C3×D3—34no—
67C22×C32×C4—24no—
68C22×C32×C4—24no—
69C22×C32×C4—24no—
70C22×C32×C4C2×D314yes—
71C22×C32×C4C2×D314yes—
72C22×C32×C4C2×D314yes—
73C22×C32×C4C22×C311yescharacteristic
74C22×C32×C4C22×C311yescharacteristic
75C22×C32×C4C22×C311yescharacteristic
76C22×C3⋊Dic3C22×C311yescharacteristic
77C22×C3⋊Dic3C22×C311yescharacteristic
78C22×C3⋊Dic3C22×C311yescharacteristic
79C22×C3×Dic3—34no—
80C22×C3×Dic3—34no—
81C22×C3×Dic3—34no—
82#3752—22no—
83#3752—22no—
84#3752C3×C414yescomplemented
85#3750—22no—
86C4×C3⋊Dic3—22no—
87C3×(C2×D3)⋊C4—316no—
88C3×(C2×D3)⋊C4—68no—
89C3×(C2×D3)⋊C4—68no—
90C3×Dic3⋊C4—68no—
91C3×C4×Dic3—68no—
92C22×C33C22×C411yescharacteristic
93C22×C33C2×D411yescharacteristic
94C22×C33C2×D411yescharacteristic
95C22×C33#4111yescharacteristic
96C22×C33#4111yescharacteristic
97C22×C33#4111yescharacteristic
98C22×C33#4111yescharacteristic
99C2×C3×C3⋊D3—22no—
100C2×C3×C3⋊D3—22no—
101C2×C3×C3⋊D3—22no—
102C2×C3×C3⋊D3—41no—
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3—41no—
105C2×C3×C3⋊D3—41no—
106C33×C4—22no—
107C33×C4—41no—
108C33×C4—41no—
109C3×C3⋊Dic3—22no—
110C3×C3⋊Dic3—22no—
111C3×C3⋊Dic3—22no—
112C2×C3×C4⋊C4—91no—
113C2×C3×C22⋊C4—91no—
114C2×C3×C22⋊C4—91no—
115C2×C3×C22⋊C4—91no—
116C2×C3×C42—91no—
117C2×(C2×D3)⋊C4—34no—
118C2×(C2×D3)⋊C4—34no—
119C2×(C2×D3)⋊C4—34no—
120C2×Dic3⋊C4—34no—
121C2×C4×Dic3—34no—
122C3×#95—24no—
123C3×#95—91no—
124C3×#95C3×D314yescomplemented
125C3×#95C3⋊D311yescharacteristic, complemented
126C3⋊#95—34no—
127C23×C32—24no—
128C23×C32C22×D314yes—
129C23×C32C23×C311yescharacteristic
130C22×C3⋊D3—21no—
131C22×C3⋊D3—21no—
132C22×C3⋊D3—21no—
133C22×C3⋊D3—21no—
134C22×C3⋊D3—21no—
135C22×C3⋊D3—21no—
136C22×C3⋊D3C2×C3×C412yes—
137C22×C3×D3—38no—
138C22×C3×D3—64no—
139C22×C3×D3—64no—
140C22×C3×D3—64no—
141C22×C3×D3—64no—
142C22×C3×D3—64no—
143C22×C3×D3—64no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—28no—
147C2×C32×C4—28no—
148C2×C32×C4—24no—
149C2×C32×C4—24no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—28no—
157C2×C32×C4—24no—
158C2×C32×C4—24no—
159C2×C32×C4—24no—
160C2×C32×C4—24no—
161C2×C32×C4—24no—
162C2×C32×C4—24no—
163C2×C32×C4—44no—
164C2×C32×C4—44no—
165C2×C32×C4—44no—
166C2×C32×C4—44no—
167C2×C32×C4—44no—
168C2×C32×C4—44no—
169C2×C32×C4C4×D318yes—
170C2×C32×C4C2×C3×C412yes—
171C2×C3⋊Dic3—21no—
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3—21no—
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3C2×C3×C412yes—
178C2×C3⋊Dic3C3×D412yes—
179C2×C3⋊Dic3C3×D412yes—
180C2×C3×Dic3—38no—
181C2×C3×Dic3—38no—
182C2×C3×Dic3—38no—
183C2×C3×Dic3—64no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3—64no—
186C2×C3×Dic3—64no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3—64no—
189#388—91noSylow
190C2×C33#11711yescharacteristic
191C2×C33C4×D411yescharacteristic
192C2×C33C4×D411yescharacteristic
193C2×C33#12111yescharacteristic
194C2×C33#12311yescharacteristic
195C2×C33#12411yescharacteristic
196C2×C33#12611yescharacteristic
197C3×C3⋊D3—42no—
198C24×C3—91no—
199C23×D3—34no—
200C22×C3×C4—24no—
201C22×C3×C4—24no—
202C22×C3×C4—24no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C22×C3×C4—91no—
206C22×C3×C4C2×C3×D314yes—
207C22×C3×C4C2×C3×D314yes—
208C22×C3×C4C2×C3×D314yes—
209C22×C3×C4C2×C3⋊D311yescharacteristic
210C22×C3×C4C2×C3⋊D311yescharacteristic
211C22×C3×C4C2×C3⋊D311yescharacteristic
212C22×Dic3—34no—
213C22×Dic3—34no—
214C22×Dic3—34no—
215C3×C4⋊C4—182no—
216C3×C22⋊C4—94no—
217C3×C22⋊C4—182no—
218C3×C22⋊C4—182no—
219C3×C42—182no—
220(C2×D3)⋊C4—316no—
221(C2×D3)⋊C4—68no—
222(C2×D3)⋊C4—68no—
223Dic3⋊C4—68no—
224C4×Dic3—68no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32—24no—
228C22×C32—24no—
229C22×C32—24no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32C2×C4×D314yes—
233C22×C32#23514yes—
234C22×C32#23514yes—
235C22×C32D3×D414yes—
236C22×C32D3×D414yes—
237C22×C32#23714yes—
238C22×C32#23914yes—
239C22×C32C22×C3×C411yescharacteristic
240C22×C32C2×C3×D411yescharacteristic
241C22×C32C2×C3×D411yescharacteristic
242C22×C32C3×#4111yescharacteristic
243C22×C32C3×#4111yescharacteristic
244C22×C32C3×#4111yescharacteristic
245C22×C32C3×#4111yescharacteristic
246C2×C3⋊D3—22no—
247C2×C3⋊D3—22no—
248C2×C3⋊D3—22no—
249C2×C3⋊D3—41no—
250C2×C3⋊D3—41no—
251C2×C3⋊D3—41no—
252C2×C3⋊D3—41no—
253C2×C3×D3—68no—
254C2×C3×D3—68no—
255C2×C3×D3—68no—
256C2×C3×D3—124no—
257C2×C3×D3—124no—
258C2×C3×D3—124no—
259C2×C3×D3—124no—
260C32×C4—22no—
261C32×C4—28no—
262C32×C4—48no—
263C32×C4—48no—
264C32×C4—48no—
265C32×C4—44no—
266C32×C4—44no—
267C32×C4—41no—
268C32×C4—41no—
269C3⋊Dic3—22no—
270C3⋊Dic3—22no—
271C3⋊Dic3—22no—
272C3×Dic3—68no—
273C3×Dic3—68no—
274C3×Dic3—68no—
275C2×C4⋊C4—91no—
276C2×C22⋊C4—91no—
277C2×C22⋊C4—91no—
278C2×C22⋊C4—91no—
279C2×C42—91no—
280#95—91no—
281#95C3×C3⋊D311yescharacteristic, complemented
282C33#38811yescharacteristic, complemented, Sylow
283C23×C3—24no—
284C23×C3—92no—
285C23×C3—181no—
286C23×C3—181no—
287C23×C3—181no—
288C23×C3—181no—
289C23×C3—181no—
290C23×C3—181no—
291C23×C3C22×C3×D314yes—
292C23×C3C22×C3⋊D311yescharacteristic
293C22×D3—38no—
294C22×D3—64no—
295C22×D3—64no—
296C22×D3—64no—
297C22×D3—64no—
298C22×D3—64no—
299C22×D3—64no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—28no—
305C2×C3×C4—28no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—28no—
319C2×C3×C4—44no—
320C2×C3×C4—44no—
321C2×C3×C4—44no—
322C2×C3×C4—44no—
323C2×C3×C4—44no—
324C2×C3×C4—44no—
325C2×C3×C4—92no—
326C2×C3×C4—92no—
327C2×C3×C4—92no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4—181no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4C3×C4×D318yes—
335C2×C3×C4C4×C3⋊D312yes—
336C2×Dic3—38no—
337C2×Dic3—38no—
338C2×Dic3—38no—
339C2×Dic3—64no—
340C2×Dic3—64no—
341C2×Dic3—64no—
342C2×Dic3—64no—
343C2×Dic3—64no—
344C2×Dic3—64no—
345C2×C32—24no—
346C2×C32—24no—
347C2×C32—24no—
348C2×C32—24no—
349C2×C32—24no—
350C2×C32—24no—
351C2×C32—24no—
352C2×C32#87214yes—
353C2×C32#67⋊C414yes—
354C2×C32#88414yes—
355C2×C32#88514yes—
356C2×C32#89314yes—
357C2×C32#89414yes—
358C2×C32#89814yes—
359C2×C32C3×#11711yescharacteristic
360C2×C32C3×C4×D411yescharacteristic
361C2×C32C3×C4×D411yescharacteristic
362C2×C32C3×#12111yescharacteristic
363C2×C32C3×#12311yescharacteristic
364C2×C32C3×#12411yescharacteristic
365C2×C32C3×#12611yescharacteristic
366C3⋊D3—42no—
367C3×D3—128no—
368C24—91no—
369C22×C4—91no—
370C22×C4—91no—
371C22×C4—91no—
372C22×C4C2×C3×C3⋊D311yescharacteristic
373C22×C4C2×C3×C3⋊D311yescharacteristic
374C22×C4C2×C3×C3⋊D311yescharacteristic
375C4⋊C4—182no—
376C22⋊C4—94no—
377C22⋊C4—182no—
378C22⋊C4—182no—
379C42—182no—
380C22×C3—24no—
381C22×C3—24no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—24no—
385C22×C3—24no—
386C22×C3—24no—
387C22×C3—182no—
388C22×C3—182no—
389C22×C3—182no—
390C22×C3—361no—
391C22×C3—361no—
392C22×C3—361no—
393C22×C3—361no—
394C22×C3C2×C3×C4×D314yes—
395C22×C3C3×#23514yes—
396C22×C3C3×#23514yes—
397C22×C3C3×D3×D414yes—
398C22×C3C3×D3×D414yes—
399C22×C3C3×#23714yes—
400C22×C3C3×#23914yes—
401C22×C3C2×C4×C3⋊D311yescharacteristic
402C22×C3#382811yescharacteristic
403C22×C3#382811yescharacteristic
404C22×C3D4×C3⋊D311yescharacteristic
405C22×C3D4×C3⋊D311yescharacteristic
406C22×C3#383011yescharacteristic
407C22×C3#383211yescharacteristic
408C2×D3—68no—
409C2×D3—68no—
410C2×D3—68no—
411C2×D3—124no—
412C2×D3—124no—
413C2×D3—124no—
414C2×D3—124no—
415C3×C4—28no—
416C3×C4—22no—
417C3×C4—48no—
418C3×C4—48no—
419C3×C4—48no—
420C3×C4—44no—
421C3×C4—44no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—182no—
425C3×C4—182no—
426C3×C4—182no—
427Dic3—68no—
428Dic3—68no—
429Dic3—68no—
430C32—24no—
431C32#466914yescomplemented
432C32C3×#38811yescharacteristic, complemented
433C23—92no—
434C23—181no—
435C23—181no—
436C23—181no—
437C23—181no—
438C23—181no—
439C23—181no—
440C23C22×C3×C3⋊D311yescharacteristic
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—92no—
450C2×C4—92no—
451C2×C4—92no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—181no—
455C2×C4—181no—
456C2×C4—181no—
457C2×C4—181no—
458C2×C4C3×C4×C3⋊D312yes—
459C2×C3—24no—
460C2×C3—24no—
461C2×C3—24no—
462C2×C3—24no—
463C2×C3—24no—
464C2×C3—24no—
465C2×C3—24no—
466C2×C3—362no—
467C2×C3C3×#87214yes—
468C2×C3C3×#67⋊C414yes—
469C2×C3C3×#88414yes—
470C2×C3C3×#88514yes—
471C2×C3C3×#89314yes—
472C2×C3C3×#89414yes—
473C2×C3C3×#89814yes—
474C2×C3#6408111yescharacteristic
475C2×C3#6409011yescharacteristic
476C2×C3#6409311yescharacteristic
477C2×C3#6409411yescharacteristic
478C2×C3#6410211yescharacteristic
479C2×C3#6410311yescharacteristic
480C2×C3#6410711yescharacteristic
481D3—128no—
482C22—182no—
483C22—182no—
484C22—182no—
485C22—361no—
486C22—361no—
487C22—361no—
488C22—361no—
489C22C2×C3×C4×C3⋊D311yescharacteristic
490C22C3×#382811yescharacteristic
491C22C3×#382811yescharacteristic
492C22C3×D4×C3⋊D311yescharacteristic
493C22C3×D4×C3⋊D311yescharacteristic
494C22C3×#383011yescharacteristic
495C22C3×#383211yescharacteristic
496C4—22no—
497C4—41no—
498C4—41no—
499C4—182no—
500C4—182no—
501C4—182no—
502C3—24no—
503C3C3×#466914yescomplemented
504C3#1059236811yescharacteristic, direct product factor
505C2—362no—
506C2C3×#6408111yescharacteristic
507C2C3×#6409011yescharacteristic
508C2C3×#6409311yescharacteristic
509C2C3×#6409411yescharacteristic
510C2C3×#6410211yescharacteristic
511C2C3×#6410311yescharacteristic
512C2C3×#6410711yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC23C22×C3×C3⋊D3
FittingC33×#95C2
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC23×C32C23×C3
hypercenterC3×#95C3⋊D3
hypocenterC32C3×#388
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
4. C2×C3×#3750C2semidirect
5. C2×C3×C4×C3⋊Dic3C2semidirect
6. C33×#95C2semidirect
7. C3×#63658C2semidirect
8. #10592368C3direct (N × H)
19. C3×#3752C4semidirect
25. C2×#3750C2×C3semidirect
26. C2×C4×C3⋊Dic3C2×C3semidirect
33. C32×#95D3semidirect
34. C32×#95C2×C3semidirect
35. #63658C2×C3semidirect
84. #3752C3×C4semidirect
124. C3×#95C3×D3semidirect
125. C3×#95C3⋊D3semidirect
281. #95C3×C3⋊D3semidirect
282. C33#388semidirect
431. C32#4669semidirect
432. C32C3×#388semidirect
503. C3C3×#4669semidirect
504. C3#10592368direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37522
C2×C3×#37502
C2×C3×C4×C3⋊Dic32
C33×#952
C3×#636582
#105923683
C3×#46693