Finite Group Databasebeta

← C3×#10591559 #13158562

Subgroups — C3×#10591559 #13158562

Order 1728.

580 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#946C211yesmaximal, characteristic
2C3×#63841C211yesmaximal, characteristic
3C3×#63662C211yesmaximal, characteristic, complemented
4C32×#835C211yesmaximal, characteristic, complemented
5C3×#63576C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8C3×#5045—31nomaximal
9C3×#4815—31nomaximal
10#10591559C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×(C2×D3)⋊C4—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×Dic3⋊C4—21no—
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×(C2×D3)⋊Dic3—21no—
21C3×Dic32—21no—
22C32×C22⋊Q8—31no—
23C3×#946—21no—
24C3×#946D311yescharacteristic
25C3×#946C2×C311yescharacteristic
26C3×#926—31no—
27C3×#892—31no—
28#63841C2×C311yescharacteristic
29C32×#103—31no—
30#63662C2×C311yescharacteristic, complemented
31C3×#835—21no—
32C3×#835—31no—
33C3×#835—31no—
34C3×#835—61no—
35C3×#835D311yescharacteristic, complemented
36C3×#835C2×C311yescharacteristic, complemented
37C3×#830—31no—
38C3×(C2×D3)⋊C8—31no—
39C3×#817—31no—
40C3×#808—31no—
41C3×#807—31no—
42C3×#807—31no—
43#63576C2×C311yescharacteristic, complemented
44#63573C2×C311yescharacteristic, complemented
45(C2×D3)⋊(C3⋊C8)C2×C311yescharacteristic
46C22×C32×D3D411yescharacteristic
47C33×Q8—21no—
48C33×Q8—21no—
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C32×C4×D3—21no—
56C32×C4×D3—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59C3×#485—91no—
60#5045—31no—
61#4815—31no—
62C2×C32×Q8—21no—
63C2×C32×Q8—21no—
64C2×C32×Q8—21no—
65C2×C32×Q8—41no—
66C2×C32×Q8C2×D311yescharacteristic
67C2×C32×Q8C2×D311yescharacteristic
68C2×C32×Q8C22×C311yescharacteristic
69C22×C32×C4—31no—
70C2×C3×C4×D3—21no—
71C2×C3×C4×D3C2×D311yescharacteristic
72C2×C3×C4×D3C22×C311yescharacteristic
73C2×C32×C8—31no—
74C32×C4⋊C4—31no—
75C32×C4⋊C4—31no—
76C32×C4⋊C4—61no—
77C32×C22⋊C4—61no—
78C4⋊(C3⋊Dic3)C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C3×#217—61no—
81C3×(C2×D3)⋊C4—22no—
82C3×(C2×D3)⋊C4—21no—
83C3×(C2×D3)⋊C4—21no—
84C3×(C2×D3)⋊C4—61no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C22×C311yescharacteristic
93C3×C4⋊Dic3C22×C311yescharacteristic
94C3×Dic3⋊C4—22no—
95C3×Dic3⋊C4—21no—
96C3×Dic3⋊C4—21no—
97C3×C4×Dic3—61no—
98C3×C4×Dic3—61no—
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C22×C311yescharacteristic
105(C2×D3)⋊Dic3—21no—
106Dic32—21no—
107C22×C33C2×D411yescharacteristic
108C2×C32×D3—21no—
109C2×C32×D3—21no—
110C2×C32×D3—21no—
111C33×C4—41no—
112C33×C4#4111yescharacteristic
113C33×C4#4111yescharacteristic
114C3×C3⋊Dic3—41no—
115C32×Dic3—21no—
116C32×Dic3—41no—
117C32×Dic3—41no—
118C3×C22⋊Q8—31no—
119C3×C22⋊Q8—31no—
120C3×C22⋊Q8—61no—
121C3×#117—91no—
122#946C3×D311yescharacteristic
123#926—31no—
124#892—31no—
125C3×#107—91no—
126C3×#106—91no—
127C3×#103—31no—
128C3×#103—31no—
129C3×#103—61no—
130C3×#103—91no—
131C3×C22⋊C8—91no—
132#835—31no—
133#835—31no—
134#835—61no—
135#835C3×D311yescharacteristic, complemented
136#830—31no—
137(C2×D3)⋊C8—31no—
138#817—31no—
139#808—31no—
140#807—31no—
141#807—31no—
142C23×C32—31no—
143C22×C3×D3—21no—
144C22×C3×D3#6711yescharacteristic
145C22×C3×D3C3×D411yescharacteristic
146C32×Q8—21no—
147C32×Q8—21no—
148C32×Q8—22no—
149C32×Q8—22no—
150C32×Q8—21no—
151C32×Q8—21no—
152C32×Q8—21no—
153C32×Q8—21no—
154C32×Q8—41no—
155C32×Q8—41no—
156C32×Q8—41no—
157C32×Q8—41no—
158C32×Q8—42no—
159C32×Q8—42no—
160C2×C32×C4—21no—
161C2×C32×C4—22no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—31no—
168C2×C32×C4—41no—
169C2×C32×C4—41no—
170C2×C32×C4—41no—
171C2×C32×C4—42no—
172C2×C32×C4—61no—
173C2×C32×C4—61no—
174C2×C32×C4—61no—
175C2×C32×C4—61no—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C23×C311yescharacteristic
179C2×C3⋊Dic3—21no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—41no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—121no—
190C2×C3×Dic3#6711yescharacteristic
191C2×C3×Dic3C3×D411yescharacteristic
192C3×C4×D3—21no—
193C3×C4×D3—21no—
194C3×C4×D3—21no—
195C3×C4×D3—21no—
196C3×C4×D3—41no—
197C3×C4×D3—41no—
198C32×C8—61no—
199C3⋊(C3⋊C8)—21no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—41no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—121no—
206#485—91noSylow
207C2×C33#12311yescharacteristic
208C2×C33#13511yescharacteristic
209C2×C33#13711yescharacteristic
210C32×D3—41no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—41no—
215C2×C3×Q8D3211yescharacteristic
216C2×C3×Q8C2×C3×D311yescharacteristic
217C2×C3×Q8C2×C3×D311yescharacteristic
218C22×C3×C4—31no—
219C22×C3×C4—31no—
220C22×C3×C4—61no—
221C2×C4×D3C2×C3×D311yescharacteristic
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8—61no—
225C2×C3×C8—91no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—61no—
231C3×C4⋊C4—61no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4—62no—
235C3×C4⋊C4—91no—
236C3×C22⋊C4—61no—
237C3×C22⋊C4—61no—
238C3×C22⋊C4—62no—
239C3×C22⋊C4—181no—
240C3×C42—181no—
241#217—61no—
242(C2×D3)⋊C4—21no—
243(C2×D3)⋊C4—61no—
244C4⋊Dic3—31no—
245C4⋊Dic3—31no—
246C4⋊Dic3—61no—
247C4⋊Dic3C2×C3×D311yescharacteristic
248C4⋊Dic3C2×C3×D311yescharacteristic
249Dic3⋊C4—21no—
250C4×Dic3—61no—
251C4×Dic3—61no—
252C2×C3⋊C8—31no—
253C2×C3⋊C8—31no—
254C2×C3⋊C8—61no—
255C2×C3⋊C8C2×C3×D311yescharacteristic
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—41no—
260C22×C32—61no—
261C22×C32—61no—
262C22×C32—61no—
263C22×C32D3×D411yescharacteristic
264C22×C32C2×#6711yescharacteristic
265C22×C32C2×C3×D411yescharacteristic
266C2×C3×D3—22no—
267C2×C3×D3—21no—
268C2×C3×D3—21no—
269C2×C3×D3—21no—
270C2×C3×D3—21no—
271C2×C3×D3—21no—
272C2×C3×D3—21no—
273C2×C3×D3—41no—
274C2×C3×D3—41no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—41no—
282C32×C4—41no—
283C32×C4—42no—
284C32×C4—42no—
285C32×C4—42no—
286C32×C4—44no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4—41no—
290C32×C4—61no—
291C32×C4—121no—
292C32×C4—121no—
293C32×C4#23511yescharacteristic
294C32×C4#23711yescharacteristic
295C32×C4#23711yescharacteristic
296C32×C4#23711yescharacteristic
297C32×C4C3×#4111yescharacteristic
298C32×C4C3×#4111yescharacteristic
299C3⋊Dic3—41no—
300C3×Dic3—21no—
301C3×Dic3—21no—
302C3×Dic3—41no—
303C3×Dic3—42no—
304C3×Dic3—41no—
305C3×Dic3—41no—
306C3×Dic3—41no—
307C3×Dic3—41no—
308C3×Dic3—81no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311C3×Dic3—241no—
312C22⋊Q8—31no—
313#117—91no—
314#107—91no—
315#106—91no—
316#103—31no—
317#103—91no—
318C22⋊C8—91no—
319C33#48511yescharacteristic, complemented, Sylow
320C23×C3—31no—
321C23×C3—31no—
322C23×C3—61no—
323C22×D3C3×#6711yescharacteristic
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—21no—
327C3×Q8—21no—
328C3×Q8—21no—
329C3×Q8—21no—
330C3×Q8—21no—
331C3×Q8—21no—
332C3×Q8—41no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×Q8—41no—
336C3×Q8—42no—
337C3×Q8—42no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—62no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—121no—
362C2×C3×C4—121no—
363C2×C3×C4—121no—
364C2×C3×C4—181no—
365C2×C3×C4C2×D3211yescharacteristic
366C2×C3×C4C22×C3×D311yescharacteristic
367C2×C3×C4C22×C3×D311yescharacteristic
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—61no—
371C2×Dic3—61no—
372C2×Dic3—121no—
373C2×Dic3C3×#6711yescharacteristic
374C4×D3—21no—
375C4×D3—21no—
376C3×C8—61no—
377C3×C8—61no—
378C3×C8—121no—
379C3×C8—181no—
380C3⋊C8—21no—
381C3⋊C8—61no—
382C3⋊C8—61no—
383C3⋊C8—121no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—41no—
394C2×C32—41no—
395C2×C32—41no—
396C2×C32—121no—
397C2×C32#88311yescharacteristic
398C2×C32#91611yescharacteristic
399C2×C32#91811yescharacteristic
400C2×C32#93511yescharacteristic
401C2×C32#94211yescharacteristic
402C2×C32#95011yescharacteristic
403C2×C32C3×#12311yescharacteristic
404C2×C32C3×#13511yescharacteristic
405C2×C32C3×#13711yescharacteristic
406C3×D3—42no—
407C3×D3—41no—
408C3×D3—41no—
409C2×Q8C3×D3211yescharacteristic
410C22×C4—31no—
411C2×C8—31no—
412C2×C8—91no—
413C4⋊C4—31no—
414C4⋊C4—31no—
415C4⋊C4—61no—
416C4⋊C4—91no—
417C22⋊C4—61no—
418C22⋊C4—181no—
419C42—181no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—41no—
424C22×C3—62no—
425C22×C3—61no—
426C22×C3—61no—
427C22×C3—61no—
428C22×C3—61no—
429C22×C3—61no—
430C22×C3—61no—
431C22×C3—121no—
432C22×C3—121no—
433C22×C3D3×#6711yescharacteristic
434C22×C3C3×D3×D411yescharacteristic
435C22×C3C2×C3×#6711yescharacteristic
436C2×D3—21no—
437C2×D3—21no—
438C2×D3—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—44no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—61no—
455C3×C4—61no—
456C3×C4—121no—
457C3×C4—122no—
458C3×C4—121no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—121no—
462C3×C4—241no—
463C3×C4—361no—
464C3×C4#380411yescharacteristic
465C3×C4#380511yescharacteristic
466C3×C4C3×#23511yescharacteristic
467C3×C4C3×#23711yescharacteristic
468C3×C4C3×#23711yescharacteristic
469C3×C4C3×#23711yescharacteristic
470Dic3—21no—
471Dic3—41no—
472Dic3—41no—
473Dic3—121no—
474Dic3—121no—
475Dic3—241no—
476C32—21no—
477C32—21no—
478C32—21no—
479C32—41no—
480C32#481511yescharacteristic, complemented
481C32#504511yescharacteristic, complemented
482C32C3×#48511yescharacteristic, complemented
483C23—31no—
484Q8—21no—
485Q8—21no—
486C2×C4—21no—
487C2×C4—31no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—181no—
493C2×C4C2×C3×D3211yescharacteristic
494C8—61no—
495C8—181no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—41no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—122no—
509C2×C3—121no—
510C2×C3—121no—
511C2×C3#6394711yescharacteristic
512C2×C3#6394811yescharacteristic
513C2×C3#6396911yescharacteristic
514C2×C3C3×#88311yescharacteristic
515C2×C3C3×#91611yescharacteristic
516C2×C3C3×#91811yescharacteristic
517C2×C3C3×#93511yescharacteristic
518C2×C3C3×#94211yescharacteristic
519C2×C3C3×#95011yescharacteristic
520D3—41no—
521C22—61no—
522C22—61no—
523C22—61no—
524C22C3×D3×#6711yescharacteristic
525C4—41no—
526C4—61no—
527C4—121no—
528C4—121no—
529C4—361no—
530C4C3×#380411yescharacteristic
531C4C3×#380511yescharacteristic
532C3—21no—
533C3—21no—
534C3—21no—
535C3—41no—
536C3#1059155911yescharacteristic, direct product factor
537C3C3×#481511yescharacteristic, complemented
538C3C3×#504511yescharacteristic, complemented
539C2—121no—
540C2C3×#6394711yescharacteristic
541C2C3×#6394811yescharacteristic
542C2C3×#6396911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC2×C3×Q8D32
hypocenterC32C3×#485
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#63662C2semidirect
4. C32×#835C2semidirect
5. C3×#63576C2semidirect
6. C3×#63573C2semidirect
10. #10591559C3direct (N × H)
30. #63662C2×C3semidirect
35. C3×#835D3semidirect
36. C3×#835C2×C3semidirect
43. #63576C2×C3semidirect
44. #63573C2×C3semidirect
135. #835C3×D3semidirect
319. C33#485semidirect
480. C32#4815semidirect
481. C32#5045semidirect
482. C32C3×#485semidirect
536. C3#10591559direct (N × H)
537. C3C3×#4815semidirect
538. C3C3×#5045semidirect

Maximal subgroups

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

subgroupindex
C32×#9462
C3×#638412
C3×#636622
C32×#8352
C3×#635762
C3×#635732
C3×(C2×D3)⋊(C3⋊C8)2
C3×#50453
C3×#48153
#105915593