Finite Group Databasebeta

← C3×#10590504 #13157477

Subgroups — C3×#10590504 #13157477

Order 1728.

726 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#895C211yesmaximal, characteristic, complemented
2C3×#63908C211yesmaximal, characteristic, complemented
3C2×C3×#3713C211yesmaximal, characteristic, complemented
4C2×C3×#3711C211yesmaximal, characteristic, complemented
5C3×#63635C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8C3×#5039—31nomaximal
9C3×#4782—31nomaximal
10#10590504C311yesmaximal, characteristic, direct product factor
11C2×C32×D12—21no—
12C2×C32×D12C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×C4×D3C2211yescharacteristic
15C2×C3×#625—21no—
16C33×C4⋊C4C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×(C2×D3)⋊C4—21no—
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×(C2×D3)⋊Dic3—21no—
22C3×#3713—22no—
23C3×#3711—22no—
24C32×#121—31no—
25C3×#938—31no—
26C2×C3×#213—31no—
27C2×C3×#213—31no—
28C3×#930—31no—
29C2×C3×#208—31no—
30C2×C3×ΓL(1,𝔽25)—31no—
31C3×#895—21no—
32C3×#895D311yescharacteristic, complemented
33C3×#895C2×C311yescharacteristic, complemented
34#63908C2×C311yescharacteristic, complemented
35C2×#3713C2×C311yescharacteristic, complemented
36C2×#3711C2×C311yescharacteristic, complemented
37C32×#102—31no—
38#63635C2×C311yescharacteristic, complemented
39C3×#832—31no—
40C3×#821—31no—
41C3×#809—21no—
42C3×#809D311yescharacteristic, complemented
43C3×#809C2×C311yescharacteristic, complemented
44C3×#808—31no—
45C3×#808—31no—
46C3×#808—61no—
47#63563C2×C311yescharacteristic, complemented
48C22×C32×D3—21no—
49C22×C32×D3—21no—
50C22×C32×D3D411yescharacteristic
51C2×C33×C4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C32×D12—21no—
56C32×D12—21no—
57C32×D12—21no—
58C32×D12—21no—
59C32×D12—42no—
60C32×C4×D3—21no—
61C32×C4×D3—21no—
62C3×#625—42no—
63C3×C3⋊(C3⋊C8)D412yescomplemented
64C32×C3⋊C8—21no—
65C3×#469—91no—
66#5039—31no—
67#4782—31no—
68C2×C32×D4—31no—
69C2×C32×D4—31no—
70C2×C32×D4—61no—
71C22×C32×C4—31no—
72C2×C3×#67—61no—
73C2×C3×#67—61no—
74C2×C3×D12—21no—
75C2×C3×D12—22no—
76C2×C3×D12—21no—
77C2×C3×D12—21no—
78C2×C3×D12—21no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×D12C22×C311yescharacteristic
82C2×C3×D12C22×C311yescharacteristic
83C2×C3×C4×D3—21no—
84C2×C3×C4×D3C2×D311yescharacteristic
85C2×C3×C4×D3C22×C311yescharacteristic
86C2×#625—21no—
87C2×C32×C8—31no—
88C32×C4⋊C4—21no—
89C32×C4⋊C4—21no—
90C32×C4⋊C4—21no—
91C32×C4⋊C4—41no—
92C32×C4⋊C4C2×D311yescharacteristic
93C32×C4⋊C4C2×D311yescharacteristic
94C32×C4⋊C4C22×C311yescharacteristic
95C32×C22⋊C4—61no—
96C4⋊(C3⋊Dic3)C22×C311yescharacteristic
97C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
98C3×#217—61no—
99C3×#213—62no—
100C3×#213—62no—
101C3×(C2×D3)⋊C4—22no—
102C3×(C2×D3)⋊C4—21no—
103C3×(C2×D3)⋊C4—21no—
104C3×(C2×D3)⋊C4—61no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—61no—
108C3×#208—62no—
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—61no—
113C2×C3×C3⋊C8C2×D311yescharacteristic
114C2×C3×C3⋊C8C22×C311yescharacteristic
115C3×ΓL(1,𝔽25)—62no—
116(C2×D3)⋊Dic3—21no—
117#3713—22no—
118#3711—22no—
119C22×C33C2×D411yescharacteristic
120C2×C32×D3—21no—
121C2×C32×D3—21no—
122C2×C32×D3—21no—
123C2×C32×D3—41no—
124C2×C32×D3—41no—
125C2×C32×D3—41no—
126C2×C32×D3—41no—
127C2×C32×D3—41no—
128C2×C32×D3—41no—
129C33×C4—41no—
130C33×C4C2×D411yescharacteristic
131C33×C4#4111yescharacteristic
132C3×C3⋊Dic3—41no—
133C32×Dic3—21no—
134C2×C3×D8—91no—
135C2×C3×M16—91no—
136C3×#121—31no—
137C3×#121—31no—
138C3×#121—61no—
139C3×#121—91no—
140#938—31no—
141C2×#213—31no—
142C2×#213—31no—
143#930—31no—
144C2×#208—31no—
145C2×ΓL(1,𝔽25)—31no—
146#895C3×D311yescharacteristic, complemented
147C3×#106—91no—
148C3×#102—31no—
149C3×#102—31no—
150C3×#102—61no—
151C3×#102—91no—
152#832—31no—
153#821—31no—
154#809C3×D311yescharacteristic, complemented
155#808—31no—
156#808—31no—
157#808—61no—
158C23×C32—31no—
159C23×C32—61no—
160C23×C32—61no—
161C22×C3×D3—22no—
162C22×C3×D3—21no—
163C22×C3×D3—21no—
164C22×C3×D3—21no—
165C22×C3×D3—21no—
166C22×C3×D3—21no—
167C22×C3×D3—41no—
168C22×C3×D3#6711yescharacteristic
169C22×C3×D3C3×D411yescharacteristic
170C32×D4—61no—
171C32×D4—61no—
172C32×D4—61no—
173C32×D4—61no—
174C32×D4—122no—
175C2×C32×C4—21no—
176C2×C32×C4—22no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—31no—
183C2×C32×C4—41no—
184C2×C32×C4—41no—
185C2×C32×C4—41no—
186C2×C32×C4—42no—
187C2×C32×C4—61no—
188C2×C32×C4—61no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C23×C311yescharacteristic
192C2×C3⋊Dic3—21no—
193C3×#67—122no—
194C3×#67—122no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—121no—
199C2×C3×Dic3#6711yescharacteristic
200C2×C3×Dic3C3×D411yescharacteristic
201C3×D12—22no—
202C3×D12—21no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—21no—
209C3×D12—21no—
210C3×D12—41no—
211C3×D12—41no—
212C3×D12—41no—
213C3×D12—44no—
214C3×D12—42no—
215C3×D12—42no—
216C3×C4×D3—21no—
217C3×C4×D3—21no—
218C3×C4×D3—21no—
219C3×C4×D3—21no—
220C3×C4×D3—41no—
221C3×C4×D3—41no—
222#625—42no—
223C32×C8—61no—
224C3⋊(C3⋊C8)C3×D412yescomplemented
225C3×C3⋊C8—21no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—32no—
228C3×C3⋊C8—32no—
229C3×C3⋊C8—41no—
230C3×C3⋊C8—62no—
231#469—91noSylow
232C2×C33#12111yescharacteristic
233C2×C33AGL(1,ℤ/8)11yescharacteristic
234C2×C33AGL(1,ℤ/8)11yescharacteristic
235C32×D3—41no—
236C32×D3—81no—
237C32×D3—81no—
238C2×C3×D4—31no—
239C2×C3×D4—31no—
240C2×C3×D4—31no—
241C2×C3×D4—31no—
242C2×C3×D4—61no—
243C2×C3×D4—61no—
244C2×C3×D4—61no—
245C2×C3×D4—61no—
246C2×C3×D4—62no—
247C2×C3×D4—181no—
248C22×C3×C4—31no—
249C22×C3×C4—31no—
250C22×C3×C4—61no—
251C2×#67—61no—
252C2×#67—61no—
253C2×D12—21no—
254C2×D12C2×C3×D311yescharacteristic
255C2×D12C2×C3×D311yescharacteristic
256C2×C4×D3C2×C3×D311yescharacteristic
257C3×D8—182no—
258C3×M16—182no—
259C2×C3×C8—31no—
260C2×C3×C8—31no—
261C2×C3×C8—61no—
262C2×C3×C8—91no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—41no—
267C3×C4⋊C4—91no—
268C3×C4⋊C4D3211yescharacteristic
269C3×C4⋊C4C2×C3×D311yescharacteristic
270C3×C4⋊C4C2×C3×D311yescharacteristic
271C3×C22⋊C4—61no—
272C3×C22⋊C4—61no—
273C3×C22⋊C4—62no—
274C3×C22⋊C4—181no—
275#217—61no—
276#213—62no—
277#213—62no—
278(C2×D3)⋊C4—21no—
279(C2×D3)⋊C4—61no—
280C4⋊Dic3—31no—
281C4⋊Dic3—31no—
282C4⋊Dic3—61no—
283#208—62no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—61no—
287C2×C3⋊C8C2×C3×D311yescharacteristic
288ΓL(1,𝔽25)—62no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—41no—
293C22×C32—61no—
294C22×C32—61no—
295C22×C32—61no—
296C22×C32—121no—
297C22×C32—121no—
298C22×C32—121no—
299C22×C32—121no—
300C22×C32—121no—
301C22×C32—121no—
302C22×C32D3×D411yescharacteristic
303C22×C32C2×#6711yescharacteristic
304C22×C32C2×C3×D411yescharacteristic
305C2×C3×D3—22no—
306C2×C3×D3—21no—
307C2×C3×D3—21no—
308C2×C3×D3—21no—
309C2×C3×D3—21no—
310C2×C3×D3—21no—
311C2×C3×D3—21no—
312C2×C3×D3—41no—
313C2×C3×D3—41no—
314C2×C3×D3—42no—
315C2×C3×D3—42no—
316C2×C3×D3—42no—
317C2×C3×D3—42no—
318C2×C3×D3—41no—
319C2×C3×D3—41no—
320C2×C3×D3—41no—
321C2×C3×D3—41no—
322C2×C3×D3—41no—
323C2×C3×D3—41no—
324C2×C3×D3—41no—
325C2×C3×D3—41no—
326C2×C3×D3—41no—
327C2×C3×D3—41no—
328C2×C3×D3—41no—
329C2×C3×D3—41no—
330C2×C3×D3—81no—
331C2×C3×D3—81no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—41no—
339C32×C4—41no—
340C32×C4—41no—
341C32×C4—42no—
342C32×C4—41no—
343C32×C4—41no—
344C32×C4—61no—
345C32×C4—81no—
346C32×C4—81no—
347C32×C4—82no—
348C32×C4#23511yescharacteristic
349C32×C4D3×D411yescharacteristic
350C32×C4D3×D411yescharacteristic
351C32×C4#23711yescharacteristic
352C32×C4C2×C3×D411yescharacteristic
353C32×C4C3×#4111yescharacteristic
354C3⋊Dic3—41no—
355C3×Dic3—21no—
356C3×Dic3—21no—
357C3×Dic3—41no—
358C3×Dic3—121no—
359C3×Dic3—121no—
360C3×Dic3—241no—
361C2×D8—91no—
362C2×M16—91no—
363#121—31no—
364#121—91no—
365#106—91no—
366#102—31no—
367#102—91no—
368C33#46911yescharacteristic, complemented, Sylow
369C23×C3—31no—
370C23×C3—31no—
371C23×C3—61no—
372C23×C3—62no—
373C23×C3—61no—
374C23×C3—61no—
375C23×C3—61no—
376C23×C3—61no—
377C23×C3—121no—
378C22×D3—21no—
379C22×D3—21no—
380C22×D3C3×#6711yescharacteristic
381C3×D4—62no—
382C3×D4—61no—
383C3×D4—61no—
384C3×D4—61no—
385C3×D4—61no—
386C3×D4—61no—
387C3×D4—61no—
388C3×D4—61no—
389C3×D4—61no—
390C3×D4—121no—
391C3×D4—121no—
392C3×D4—121no—
393C3×D4—124no—
394C3×D4—122no—
395C3×D4—122no—
396C3×D4—362no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—31no—
405C2×C3×C4—31no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—42no—
409C2×C3×C4—41no—
410C2×C3×C4—61no—
411C2×C3×C4—61no—
412C2×C3×C4—61no—
413C2×C3×C4—61no—
414C2×C3×C4—61no—
415C2×C3×C4—121no—
416C2×C3×C4—121no—
417C2×C3×C4—181no—
418C2×C3×C4C2×D3211yescharacteristic
419C2×C3×C4C22×C3×D311yescharacteristic
420C2×C3×C4C22×C3×D311yescharacteristic
421#67—122no—
422#67—122no—
423C2×Dic3—61no—
424C2×Dic3—61no—
425C2×Dic3—121no—
426C2×Dic3C3×#6711yescharacteristic
427D12—21no—
428D12—21no—
429D12—21no—
430D12—21no—
431D12—42no—
432C4×D3—21no—
433C4×D3—21no—
434C3×C8—61no—
435C3×C8—61no—
436C3×C8—92no—
437C3×C8—121no—
438C3⋊C8—21no—
439C3⋊C8—32no—
440C3⋊C8—32no—
441C3⋊C8—62no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—41no—
452C2×C32—41no—
453C2×C32—41no—
454C2×C32—121no—
455C2×C32—241no—
456C2×C32—241no—
457C2×C32#88411yescharacteristic
458C2×C32#91111yescharacteristic
459C2×C32#91411yescharacteristic
460C2×C32#93211yescharacteristic
461C2×C32#93911yescharacteristic
462C2×C32#94911yescharacteristic
463C2×C32C3×#12111yescharacteristic
464C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
465C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
466C3×D3—42no—
467C3×D3—41no—
468C3×D3—41no—
469C3×D3—82no—
470C3×D3—82no—
471C3×D3—81no—
472C3×D3—81no—
473C3×D3—81no—
474C3×D3—81no—
475C2×D4—31no—
476C2×D4—31no—
477C2×D4—61no—
478C2×D4—181no—
479C22×C4—31no—
480D8—182no—
481M16—182no—
482C2×C8—31no—
483C2×C8—91no—
484C4⋊C4—91no—
485C4⋊C4C3×D3211yescharacteristic
486C22⋊C4—61no—
487C22⋊C4—181no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—41no—
492C22×C3—62no—
493C22×C3—61no—
494C22×C3—61no—
495C22×C3—61no—
496C22×C3—61no—
497C22×C3—61no—
498C22×C3—61no—
499C22×C3—121no—
500C22×C3—121no—
501C22×C3—122no—
502C22×C3—122no—
503C22×C3—122no—
504C22×C3—122no—
505C22×C3—121no—
506C22×C3—121no—
507C22×C3—121no—
508C22×C3—121no—
509C22×C3—121no—
510C22×C3—121no—
511C22×C3—121no—
512C22×C3—121no—
513C22×C3—121no—
514C22×C3—121no—
515C22×C3—121no—
516C22×C3—121no—
517C22×C3—241no—
518C22×C3—241no—
519C22×C3D3×#6711yescharacteristic
520C22×C3C3×D3×D411yescharacteristic
521C22×C3C2×C3×#6711yescharacteristic
522C2×D3—21no—
523C2×D3—21no—
524C2×D3—21no—
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—41no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—42no—
538C3×C4—41no—
539C3×C4—41no—
540C3×C4—41no—
541C3×C4—41no—
542C3×C4—41no—
543C3×C4—61no—
544C3×C4—61no—
545C3×C4—81no—
546C3×C4—81no—
547C3×C4—82no—
548C3×C4—121no—
549C3×C4—361no—
550C3×C4#379511yescharacteristic
551C3×C4#380211yescharacteristic
552C3×C4C3×#23511yescharacteristic
553C3×C4C3×D3×D411yescharacteristic
554C3×C4C3×D3×D411yescharacteristic
555C3×C4C3×#23711yescharacteristic
556Dic3—21no—
557Dic3—121no—
558Dic3—121no—
559Dic3—241no—
560C32—21no—
561C32—21no—
562C32—21no—
563C32—41no—
564C32#478211yescharacteristic, complemented
565C32#503911yescharacteristic, complemented
566C32C3×#46911yescharacteristic, complemented
567C23—31no—
568C23—61no—
569C23—61no—
570D4—61no—
571D4—61no—
572D4—61no—
573D4—61no—
574D4—122no—
575D4—362no—
576C2×C4—21no—
577C2×C4—31no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—181no—
581C2×C4C2×C3×D3211yescharacteristic
582C8—61no—
583C8—92no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—41no—
594C2×C3—41no—
595C2×C3—41no—
596C2×C3—122no—
597C2×C3—121no—
598C2×C3—121no—
599C2×C3—242no—
600C2×C3—242no—
601C2×C3—241no—
602C2×C3—241no—
603C2×C3—241no—
604C2×C3—241no—
605C2×C3#6388511yescharacteristic
606C2×C3#6393211yescharacteristic
607C2×C3#6393911yescharacteristic
608C2×C3C3×#88411yescharacteristic
609C2×C3C3×#91111yescharacteristic
610C2×C3C3×#91411yescharacteristic
611C2×C3C3×#93211yescharacteristic
612C2×C3C3×#93911yescharacteristic
613C2×C3C3×#94911yescharacteristic
614D3—41no—
615D3—81no—
616D3—81no—
617C22—61no—
618C22—61no—
619C22—61no—
620C22—121no—
621C22—121no—
622C22—121no—
623C22—121no—
624C22—121no—
625C22—121no—
626C22C3×D3×#6711yescharacteristic
627C4—41no—
628C4—61no—
629C4—361no—
630C4C3×#379511yescharacteristic
631C4C3×#380211yescharacteristic
632C3—21no—
633C3—21no—
634C3—21no—
635C3—41no—
636C3#1059050411yescharacteristic, direct product factor
637C3C3×#478211yescharacteristic, complemented
638C3C3×#503911yescharacteristic, complemented
639C2—121no—
640C2—241no—
641C2—241no—
642C2C3×#6388511yescharacteristic
643C2C3×#6393211yescharacteristic
644C2C3×#6393911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#469
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×#895C2semidirect
2. C3×#63908C2semidirect
3. C2×C3×#3713C2semidirect
4. C2×C3×#3711C2semidirect
5. C3×#63635C2semidirect
6. C32×#809C2semidirect
7. C3×#63563C2semidirect
10. #10590504C3direct (N × H)
32. C3×#895D3semidirect
33. C3×#895C2×C3semidirect
34. #63908C2×C3semidirect
35. C2×#3713C2×C3semidirect
36. C2×#3711C2×C3semidirect
38. #63635C2×C3semidirect
42. C3×#809D3semidirect
43. C3×#809C2×C3semidirect
47. #63563C2×C3semidirect
63. C3×C3⋊(C3⋊C8)D4semidirect
146. #895C3×D3semidirect
154. #809C3×D3semidirect
224. C3⋊(C3⋊C8)C3×D4semidirect
368. C33#469semidirect
564. C32#4782semidirect
565. C32#5039semidirect
566. C32C3×#469semidirect
636. C3#10590504direct (N × H)
637. C3C3×#4782semidirect
638. C3C3×#5039semidirect

Maximal subgroups

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

subgroupindex
C32×#8952
C3×#639082
C2×C3×#37132
C2×C3×#37112
C3×#636352
C32×#8092
C3×#635632
C3×#50393
C3×#47823
#105905043