Finite Group Databasebeta

← C3×#10590549 #13157532

Subgroups — C3×#10590549 #13157532

Order 1728.

708 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#887C211yesmaximal, characteristic, complemented
2C3×#63911C211yesmaximal, characteristic
3C2×C3×#3717C211yesmaximal, characteristic, complemented
4C2×C3×#3715C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5027—31nomaximal
9C3×#4762—31nomaximal
10#10590549C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×#626—21no—
15C2×C3×#626—21no—
16C33×C4⋊C4C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×Dic3⋊C4—21no—
19C32×C4×Dic3—21no—
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3717—22no—
23C3×#3715—22no—
24C32×C4×Q8—31no—
25C2×C3×#215—31no—
26C2×C3×#215—31no—
27C3×#940—31no—
28C2×C3×#214—31no—
29C2×C3×#204—31no—
30C3×#887—21no—
31C3×#887D311yescharacteristic, complemented
32C3×#887C2×C311yescharacteristic, complemented
33C3×#874—31no—
34#63911C2×C311yescharacteristic
35C2×#3717C2×C311yescharacteristic, complemented
36C2×#3715C2×C311yescharacteristic, complemented
37C32×#102—31no—
38#63636C2×C311yescharacteristic, complemented
39C3×Dic3⋊C8—31no—
40C3×#809—21no—
41C3×#809—31no—
42C3×#809—31no—
43C3×#809—61no—
44C3×#809D311yescharacteristic, complemented
45C3×#809C2×C311yescharacteristic, complemented
46C3×C4⋊(C3⋊C8)—31no—
47Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C32×D12—21no—
58C32×D12—21no—
59C32×Dic6—21no—
60C32×Dic6D412yescomplemented
61C3×#626—42no—
62C3×#626—42no—
63C3×C3⋊(C3⋊C8)—21no—
64C32×C3⋊C8—21no—
65C3×#460—91no—
66#5027—31no—
67#4762—31no—
68C2×C32×Q8—31no—
69C2×C32×D4—31no—
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
71C2×C3×#67—61no—
72C2×C3×#67—61no—
73C2×C3×D12—21no—
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×C3×D12—61no—
78C2×C3×D12—61no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×D12C22×C311yescharacteristic
81C2×C3×Dic6—21no—
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C3×Dic6C22×C311yescharacteristic
84C2×#626—21no—
85C2×#626—21no—
86C2×C32×C8—31no—
87C32×C4⋊C4—21no—
88C32×C4⋊C4—21no—
89C32×C4⋊C4—21no—
90C32×C4⋊C4—41no—
91C32×C4⋊C4—61no—
92C32×C4⋊C4C2×D311yescharacteristic
93C32×C4⋊C4C2×D311yescharacteristic
94C32×C4⋊C4C22×C311yescharacteristic
95C32×C42—31no—
96C32×C42—61no—
97C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
98C3×#215—62no—
99C3×#215—62no—
100C3×#214—62no—
101C3×Dic3⋊C4—22no—
102C3×Dic3⋊C4—21no—
103C3×Dic3⋊C4—21no—
104C3×C4×Dic3—22no—
105C3×C4×Dic3—21no—
106C3×C4×Dic3—21no—
107C3×C4×Dic3—21no—
108C3×C4×Dic3C2×D311yescharacteristic
109C3×C4×Dic3C22×C311yescharacteristic
110C2×C3×C3⋊C8—21no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—31no—
113C2×C3×C3⋊C8—61no—
114C2×C3×C3⋊C8C2×D311yescharacteristic
115C2×C3×C3⋊C8C22×C311yescharacteristic
116C3×#204—62no—
117#3717—22no—
118#3715—22no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C2×C32×D3—41no—
124C2×C32×D3—41no—
125C2×C32×D3—41no—
126C33×C4—41no—
127C33×C4C2×D411yescharacteristic
128C33×C4#4111yescharacteristic
129C32×Dic3—22no—
130C32×Dic3—21no—
131C32×Dic3SD1612yescomplemented
132C2×C3×SD16—91no—
133C2×C3×SD16—91no—
134C3×#127—91no—
135C3×C4×Q8—31no—
136C3×C4×Q8—31no—
137C3×C4×Q8—61no—
138C2×#215—31no—
139C2×#215—31no—
140#940—31no—
141C2×#214—31no—
142C2×#204—31no—
143#887C3×D311yescharacteristic, complemented
144#874—31no—
145C3×C4⋊C8—91no—
146C3×#102—31no—
147C3×#102—31no—
148C3×#102—61no—
149C3×#102—91no—
150Dic3⋊C8—31no—
151#809—31no—
152#809—31no—
153#809—61no—
154#809C3×D311yescharacteristic, complemented
155C4⋊(C3⋊C8)—31no—
156C23×C32—61no—
157C22×C3⋊D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—41no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C22×C3×D3—121no—
164C32×Q8—32no—
165C32×Q8—61no—
166C32×D4—61no—
167C32×D4—61no—
168C2×C32×C4—21no—
169C2×C32×C4—22no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—31no—
176C2×C32×C4—31no—
177C2×C32×C4—41no—
178C2×C32×C4—41no—
179C2×C32×C4—41no—
180C2×C32×C4—42no—
181C2×C32×C4—61no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C23×C311yescharacteristic
185C4⋊(C3⋊D3)—21no—
186C4⋊(C3⋊D3)—21no—
187C3×#67—122no—
188C3×#67—122no—
189C2×C3×Dic3—22no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3#6711yescharacteristic
195C2×C3×Dic3#6711yescharacteristic
196C2×C3×Dic3C3×D411yescharacteristic
197C2×C3×Dic3C3×D411yescharacteristic
198C3×D12—22no—
199C3×D12—21no—
200C3×D12—21no—
201C3×D12—21no—
202C3×D12—21no—
203C3×D12—41no—
204C3×D12—62no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—121no—
210C3×D12—122no—
211C3×D12—122no—
212C3×Dic6—22no—
213C3×Dic6—21no—
214C3×Dic6—21no—
215C3×Dic6—22no—
216C3×Dic6D1212yescomplemented
217C3×Dic6C3×D412yescomplemented
218#626—42no—
219#626—42no—
220C32×C8—61no—
221C3⋊(C3⋊C8)—21no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—41no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—121no—
228#460—91noSylow
229C2×C33#12111yescharacteristic
230C2×C33C2×SD1611yescharacteristic
231C2×C33AGL(1,ℤ/8)11yescharacteristic
232C3×C3⋊D3—81no—
233C32×D3—81no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—61no—
237C2×C3×D4—31no—
238C2×C3×D4—31no—
239C2×C3×D4—61no—
240C2×C3×D4—91no—
241C2×C3×D4—181no—
242C2×C3×D4—181no—
243C2×#67—61no—
244C2×#67—61no—
245C2×D12—31no—
246C2×D12—31no—
247C2×D12—61no—
248C2×D12—61no—
249C2×D12—61no—
250C2×D12C2×C3×D311yescharacteristic
251C2×Dic6C2×C3×D311yescharacteristic
252C3×SD16—182no—
253C3×SD16—182no—
254C2×C3×C8—31no—
255C2×C3×C8—31no—
256C2×C3×C8—61no—
257C2×C3×C8—91no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—41no—
262C3×C4⋊C4—61no—
263C3×C4⋊C4—61no—
264C3×C4⋊C4—62no—
265C3×C4⋊C4D3211yescharacteristic
266C3×C4⋊C4C2×C3×D311yescharacteristic
267C3×C4⋊C4C2×C3×D311yescharacteristic
268C3×C42—31no—
269C3×C42—31no—
270C3×C42—61no—
271C3×C42—61no—
272C3×C42—61no—
273C3×C42—62no—
274#215—62no—
275#215—62no—
276#214—62no—
277Dic3⋊C4—21no—
278C4×Dic3—21no—
279C4×Dic3C2×C3×D311yescharacteristic
280C2×C3⋊C8—31no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8C2×C3×D311yescharacteristic
284#204—62no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—41no—
289C22×C32—121no—
290C22×C32—121no—
291C22×C32—121no—
292C22×C32D3×D411yescharacteristic
293C22×C32C2×#6711yescharacteristic
294C22×C32C2×C3×D411yescharacteristic
295C2×C3⋊D3—41no—
296C2×C3⋊D3—41no—
297C2×C3⋊D3—41no—
298C2×C3×D3—42no—
299C2×C3×D3—41no—
300C2×C3×D3—41no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C2×C3×D3—81no—
306C2×C3×D3—81no—
307C2×C3×D3—122no—
308C2×C3×D3—121no—
309C2×C3×D3—121no—
310C2×C3×D3—121no—
311C2×C3×D3—121no—
312C2×C3×D3—121no—
313C2×C3×D3—121no—
314C2×C3×D3—241no—
315C2×C3×D3—241no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—32no—
323C32×C4—41no—
324C32×C4—41no—
325C32×C4—41no—
326C32×C4—42no—
327C32×C4—41no—
328C32×C4—41no—
329C32×C4—62no—
330C32×C4—61no—
331C32×C4—81no—
332C32×C4—81no—
333C32×C4—82no—
334C32×C4C2×D1211yescharacteristic
335C32×C4#23511yescharacteristic
336C32×C4#23511yescharacteristic
337C32×C4D3×D411yescharacteristic
338C32×C4C2×C3×D411yescharacteristic
339C32×C4C3×#4111yescharacteristic
340C3×Dic3—24no—
341C3×Dic3—22no—
342C3×Dic3—21no—
343C3×Dic3—22no—
344C3×Dic3—21no—
345C3×Dic3—22no—
346C3×Dic3—41no—
347C3×Dic3#21512yescomplemented
348C3×Dic3C3×SD1612yescomplemented
349C2×SD16—91no—
350C2×SD16—91no—
351#127—91no—
352C4×Q8—31no—
353C4⋊C8—91no—
354#102—31no—
355#102—91no—
356C33#46011yescharacteristic, complemented, Sylow
357C23×C3—61no—
358C23×C3—61no—
359C23×C3—121no—
360C23×C3—181no—
361C22×D3—21no—
362C22×D3—61no—
363C22×D3—61no—
364C22×D3—121no—
365C3×Q8—32no—
366C3×Q8—32no—
367C3×Q8—62no—
368C3×Q8—62no—
369C3×Q8—61no—
370C3×Q8—61no—
371C3×D4—62no—
372C3×D4—61no—
373C3×D4—61no—
374C3×D4—61no—
375C3×D4—61no—
376C3×D4—121no—
377C3×D4—181no—
378C3×D4—181no—
379C3×D4—362no—
380C3×D4—362no—
381C2×C3×C4—22no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—31no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—31no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—42no—
395C2×C3×C4—41no—
396C2×C3×C4—61no—
397C2×C3×C4—61no—
398C2×C3×C4—62no—
399C2×C3×C4—61no—
400C2×C3×C4—61no—
401C2×C3×C4C2×D3211yescharacteristic
402C2×C3×C4C22×C3×D311yescharacteristic
403C2×C3×C4C22×C3×D311yescharacteristic
404#67—122no—
405#67—122no—
406C2×Dic3—21no—
407C2×Dic3C3×#6711yescharacteristic
408C2×Dic3C3×#6711yescharacteristic
409D12—21no—
410D12—21no—
411D12—62no—
412D12—61no—
413D12—61no—
414D12—61no—
415D12—61no—
416D12—121no—
417D12—122no—
418D12—122no—
419Dic6—21no—
420Dic6C3×D1212yescomplemented
421C3×C8—61no—
422C3×C8—61no—
423C3×C8—121no—
424C3×C8—181no—
425C3⋊C8—21no—
426C3⋊C8—61no—
427C3⋊C8—61no—
428C3⋊C8—121no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—41no—
439C2×C32—41no—
440C2×C32—41no—
441C2×C32—241no—
442C2×C32#88411yescharacteristic
443C2×C32#91111yescharacteristic
444C2×C32D3×SD1611yescharacteristic
445C2×C32#93011yescharacteristic
446C2×C32C2×#21511yescharacteristic
447C2×C32#94911yescharacteristic
448C2×C32C3×#12111yescharacteristic
449C2×C32C2×C3×SD1611yescharacteristic
450C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
451C3⋊D3—81no—
452C3×D3—82no—
453C3×D3—81no—
454C3×D3—81no—
455C3×D3—242no—
456C3×D3—241no—
457C3×D3—241no—
458C2×Q8—31no—
459C2×D4—31no—
460C2×D4—91no—
461C2×D4—181no—
462C2×D4—181no—
463SD16—182no—
464SD16—182no—
465C2×C8—31no—
466C2×C8—91no—
467C4⋊C4—61no—
468C4⋊C4C3×D3211yescharacteristic
469C42—31no—
470C42—61no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—41no—
475C22×C3—122no—
476C22×C3—121no—
477C22×C3—121no—
478C22×C3—121no—
479C22×C3—121no—
480C22×C3—121no—
481C22×C3—121no—
482C22×C3—241no—
483C22×C3—241no—
484C22×C3—361no—
485C22×C3—361no—
486C22×C3—361no—
487C22×C3D3×#6711yescharacteristic
488C22×C3C3×D3×D411yescharacteristic
489C22×C3C2×C3×#6711yescharacteristic
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—122no—
494C2×D3—121no—
495C2×D3—121no—
496C2×D3—121no—
497C2×D3—121no—
498C2×D3—121no—
499C2×D3—121no—
500C2×D3—241no—
501C2×D3—241no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—32no—
509C3×C4—32no—
510C3×C4—42no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4—41no—
516C3×C4—62no—
517C3×C4—64no—
518C3×C4—62no—
519C3×C4—61no—
520C3×C4—62no—
521C3×C4—61no—
522C3×C4—81no—
523C3×C4—81no—
524C3×C4—82no—
525C3×C4—121no—
526C3×C4#379811yescharacteristic
527C3×C4D3×D1211yescharacteristic
528C3×C4C2×C3×D1211yescharacteristic
529C3×C4C3×#23511yescharacteristic
530C3×C4C3×#23511yescharacteristic
531C3×C4C3×D3×D411yescharacteristic
532Dic3—22no—
533Dic3—21no—
534Dic3C3×#21512yescomplemented
535C32—21no—
536C32—21no—
537C32—21no—
538C32—41no—
539C32#476211yescharacteristic, complemented
540C32#502711yescharacteristic, complemented
541C32C3×#46011yescharacteristic, complemented
542C23—61no—
543C23—181no—
544Q8—32no—
545Q8—61no—
546D4—61no—
547D4—61no—
548D4—181no—
549D4—181no—
550D4—362no—
551D4—362no—
552C2×C4—21no—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—61no—
556C2×C4C2×C3×D3211yescharacteristic
557C8—61no—
558C8—181no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—41no—
569C2×C3—41no—
570C2×C3—41no—
571C2×C3—242no—
572C2×C3—241no—
573C2×C3—241no—
574C2×C3—721no—
575C2×C3#6390611yescharacteristic
576C2×C3#6392511yescharacteristic
577C2×C3D3×#21511yescharacteristic
578C2×C3C3×#88411yescharacteristic
579C2×C3C3×#91111yescharacteristic
580C2×C3C3×D3×SD1611yescharacteristic
581C2×C3C3×#93011yescharacteristic
582C2×C3C2×C3×#21511yescharacteristic
583C2×C3C3×#94911yescharacteristic
584D3—81no—
585D3—242no—
586D3—241no—
587D3—241no—
588C22—121no—
589C22—121no—
590C22—121no—
591C22—361no—
592C22—361no—
593C22—361no—
594C22C3×D3×#6711yescharacteristic
595C4—32no—
596C4—41no—
597C4—62no—
598C4—61no—
599C4C3×#379811yescharacteristic
600C4C3×D3×D1211yescharacteristic
601C3—21no—
602C3—21no—
603C3—21no—
604C3—41no—
605C3#1059054911yescharacteristic, direct product factor
606C3C3×#476211yescharacteristic, complemented
607C3C3×#502711yescharacteristic, complemented
608C2—241no—
609C2—721no—
610C2C3×#6390611yescharacteristic
611C2C3×#6392511yescharacteristic
612C2C3×D3×#21511yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#63911C2
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#460
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C32×#887C2semidirect
3. C2×C3×#3717C2semidirect
4. C2×C3×#3715C2semidirect
5. C3×#63636C2semidirect
6. C32×#809C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
10. #10590549C3direct (N × H)
31. C3×#887D3semidirect
32. C3×#887C2×C3semidirect
35. C2×#3717C2×C3semidirect
36. C2×#3715C2×C3semidirect
38. #63636C2×C3semidirect
44. C3×#809D3semidirect
45. C3×#809C2×C3semidirect
47. Dic3⋊(C3⋊C8)C2×C3semidirect
60. C32×Dic6D4semidirect
131. C32×Dic3SD16semidirect
143. #887C3×D3semidirect
154. #809C3×D3semidirect
216. C3×Dic6D12semidirect
217. C3×Dic6C3×D4semidirect
347. C3×Dic3#215semidirect
348. C3×Dic3C3×SD16semidirect
356. C33#460semidirect
420. Dic6C3×D12semidirect
534. Dic3C3×#215semidirect
539. C32#4762semidirect
540. C32#5027semidirect
541. C32C3×#460semidirect
605. C3#10590549direct (N × H)
606. C3C3×#4762semidirect
607. C3C3×#5027semidirect

Maximal subgroups

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

subgroupindex
C32×#8872
C3×#639112
C2×C3×#37172
C2×C3×#37152
C3×#636362
C32×#8092
C3×Dic3⋊(C3⋊C8)2
C3×#50273
C3×#47623
#105905493