Finite Group Databasebeta

← C3×#10591454 #13158516

Subgroups — C3×#10591454 #13158516

Order 1728.

794 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64147C211yesmaximal, characteristic, complemented
2C2×C32×#213C211yesmaximal, characteristic, complemented
3C3×#63916C211yesmaximal, characteristic, complemented
4C2×C3×#3712C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8C3×#5159—31nomaximal
9C3×#4782—31nomaximal
10#10591454C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×C3×#626—21no—
16C3×#3757—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C32×#213—22no—
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×(C2×D3)⋊Dic3—21no—
23C3×#3712—22no—
24C2×C32×D8—31no—
25#64147C2×C311yescharacteristic, complemented
26C3×#938—31no—
27C3×#938—31no—
28C3×#938—31no—
29C3×#938—61no—
30C2×C3×#213—21no—
31C2×C3×#213D311yescharacteristic, complemented
32C2×C3×#213C2×C311yescharacteristic, complemented
33C2×C3×ΓL(1,𝔽25)—31no—
34C2×C3×ΓL(1,𝔽25)—31no—
35C3×#895—31no—
36#63916C2×C311yescharacteristic, complemented
37C2×#3712C2×C311yescharacteristic, complemented
38C32×#102—31no—
39C3×#832—21no—
40C3×#832—31no—
41C3×#832D311yescharacteristic, complemented
42C3×#832C2×C311yescharacteristic, complemented
43C3×#821—31no—
44C3×#817—31no—
45C3×#808—31no—
46#63573C2×C311yescharacteristic, complemented
47#63563C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C3×C3⋊D3D411yescharacteristic
50C22×C32×D3—21no—
51C33×D4—21no—
52C33×D4—21no—
53C2×C33×C4C2311yescharacteristic
54C3×#638—42no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3D411yescharacteristic
57C3×C4×C3⋊D3—21no—
58C3×C4×C3⋊D3—21no—
59C2×C32×Dic3—21no—
60C32×D12—21no—
61C32×D12—21no—
62C3×#626—42no—
63C32×C3⋊C8—21no—
64C32×C3⋊C8D412yescomplemented
65C3×#469—91no—
66#5159—31no—
67#4782—31no—
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—31no—
72C2×C32×D4—41no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C22×C311yescharacteristic
76C2×#638—21no—
77C2×C4×C3⋊D3C22×C311yescharacteristic
78C2×C3×#67—61no—
79C2×C3×#67—61no—
80C2×C3×#67—61no—
81C2×C3×#67—62no—
82C2×C3×D12—21no—
83C2×C3×D12—61no—
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×D12C22×C311yescharacteristic
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—61no—
89C2×#626—21no—
90C32×D8—62no—
91C2×C32×C8—31no—
92C2×C32×C8—31no—
93C32×C4⋊C4—31no—
94#3757—21no—
95C4⋊(C3⋊Dic3)C22×C311yescharacteristic
96C3×#217—61no—
97C3×#217—61no—
98C3×#217—61no—
99C3×#217—62no—
100C3×#213—24no—
101C3×#213—22no—
102C3×#213—22no—
103C3×(C2×D3)⋊C4—61no—
104C3×C4⋊Dic3—21no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—61no—
108C3×C4⋊Dic3C2×D311yescharacteristic
109C3×C4⋊Dic3C22×C311yescharacteristic
110C2×C3×C3⋊C8—21no—
111C2×C3×C3⋊C8—21no—
112C2×C3×C3⋊C8C2×D311yescharacteristic
113C2×C3×C3⋊C8C2×D311yescharacteristic
114C2×C3×C3⋊C8C22×C311yescharacteristic
115C2×C3×C3⋊C8C22×C311yescharacteristic
116C3×ΓL(1,𝔽25)—62no—
117C3×ΓL(1,𝔽25)—62no—
118(C2×D3)⋊Dic3—21no—
119#3712—22no—
120C22×C33—41no—
121C22×C33—41no—
122C22×C33—41no—
123C22×C33C2×D411yescharacteristic
124C2×C3×C3⋊D3—21no—
125C2×C3×C3⋊D3—21no—
126C2×C3×C3⋊D3—21no—
127C2×C32×D3—41no—
128C2×C32×D3—41no—
129C2×C32×D3—41no—
130C33×C4C2×D411yescharacteristic
131C33×C4#4111yescharacteristic
132C3×C3⋊Dic3—21no—
133C3×C3⋊Dic3—41no—
134C32×Dic3—41no—
135C2×C3×D8—31no—
136C2×C3×D8—31no—
137C2×C3×D8—61no—
138C2×C3×M16—91no—
139C3×#121—91no—
140C3×#121—91no—
141#938—31no—
142#938—31no—
143#938—31no—
144#938—61no—
145C2×#213C3×D311yescharacteristic, complemented
146C2×ΓL(1,𝔽25)—31no—
147C2×ΓL(1,𝔽25)—31no—
148#895—31no—
149C3×#106—91no—
150C3×#102—31no—
151C3×#102—31no—
152C3×#102—61no—
153C3×#102—91no—
154#832—31no—
155#832C3×D311yescharacteristic, complemented
156#821—31no—
157#817—31no—
158#808—31no—
159C23×C32—21no—
160C23×C32—22no—
161C23×C32—21no—
162C23×C32—21no—
163C23×C32—41no—
164C23×C32—41no—
165C23×C32—42no—
166C23×C32—61no—
167C22×C3⋊D3C3×D411yescharacteristic
168C22×C3×D3—21no—
169C22×C3×D3—21no—
170C22×C3×D3—31no—
171C22×C3×D3—31no—
172C22×C3×D3—41no—
173C22×C3×D3—61no—
174C32×D4—21no—
175C32×D4—21no—
176C32×D4—22no—
177C32×D4—22no—
178C32×D4—21no—
179C32×D4—21no—
180C32×D4—21no—
181C32×D4—21no—
182C32×D4—41no—
183C32×D4—41no—
184C32×D4—41no—
185C32×D4—41no—
186C32×D4—42no—
187C32×D4—42no—
188C32×D4—61no—
189C32×D4—61no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4—41no—
194C2×C32×C4—61no—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C23×C311yescharacteristic
198#638—42no—
199C2×C3⋊Dic3—21no—
200C2×C3⋊Dic3C3×D411yescharacteristic
201C4×C3⋊D3—21no—
202C4×C3⋊D3—21no—
203C3×#67—122no—
204C3×#67—124no—
205C3×#67—122no—
206C3×#67—122no—
207C2×C3×Dic3—21no—
208C2×C3×Dic3—21no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—41no—
212C2×C3×Dic3—61no—
213C2×C3×Dic3—62no—
214C2×C3×Dic3—61no—
215C2×C3×Dic3—61no—
216C3×D12—22no—
217C3×D12—21no—
218C3×D12—21no—
219C3×D12—21no—
220C3×D12—21no—
221C3×D12—41no—
222C3×D12—122no—
223C3×C4×D3—61no—
224C3×C4×D3—61no—
225C3×C4×D3—61no—
226C3×C4×D3—61no—
227C3×C4×D3—121no—
228C3×C4×D3—121no—
229#626—42no—
230C32×C8—32no—
231C32×C8—61no—
232C3×C3⋊C8—21no—
233C3×C3⋊C8—21no—
234C3×C3⋊C8—22no—
235C3×C3⋊C8—41no—
236C3×C3⋊C8#6712yescomplemented
237C3×C3⋊C8C3×D412yescomplemented
238#469—91noSylow
239C2×C33—81no—
240C2×C33#12111yescharacteristic
241C2×C33AGL(1,ℤ/8)11yescharacteristic
242C2×C33AGL(1,ℤ/8)11yescharacteristic
243C3×C3⋊D3—41no—
244C32×D3—81no—
245C2×C3×D4—21no—
246C2×C3×D4—21no—
247C2×C3×D4—21no—
248C2×C3×D4—31no—
249C2×C3×D4—31no—
250C2×C3×D4—41no—
251C2×C3×D4—61no—
252C2×C3×D4—181no—
253C2×C3×D4—181no—
254C2×C3×D4D3211yescharacteristic
255C2×C3×D4C2×C3×D311yescharacteristic
256C2×C3×D4C2×C3×D311yescharacteristic
257C22×C3×C4—91no—
258C2×#67—61no—
259C2×#67—61no—
260C2×#67—61no—
261C2×#67—62no—
262C2×D12—61no—
263C2×D12C2×C3×D311yescharacteristic
264C2×C4×D3—31no—
265C2×C4×D3—31no—
266C2×C4×D3—61no—
267C3×D8—62no—
268C3×D8—62no—
269C3×D8—64no—
270C3×M16—182no—
271C2×C3×C8—31no—
272C2×C3×C8—31no—
273C2×C3×C8—31no—
274C2×C3×C8—31no—
275C2×C3×C8—61no—
276C2×C3×C8—61no—
277C3×C4⋊C4—31no—
278C3×C4⋊C4—31no—
279C3×C4⋊C4—61no—
280C3×C4⋊C4—91no—
281C3×C22⋊C4—181no—
282C3×C22⋊C4—181no—
283#217—61no—
284#217—61no—
285#217—61no—
286#217—62no—
287#213—22no—
288(C2×D3)⋊C4—61no—
289C4⋊Dic3—31no—
290C4⋊Dic3—31no—
291C4⋊Dic3—61no—
292C4⋊Dic3C2×C3×D311yescharacteristic
293C2×C3⋊C8C2×C3×D311yescharacteristic
294C2×C3⋊C8C2×C3×D311yescharacteristic
295ΓL(1,𝔽25)—62no—
296ΓL(1,𝔽25)—62no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—41no—
301C22×C32—42no—
302C22×C32—42no—
303C22×C32—42no—
304C22×C32—44no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32—41no—
309C22×C32—41no—
310C22×C32—42no—
311C22×C32—42no—
312C22×C32—41no—
313C22×C32—41no—
314C22×C32—41no—
315C22×C32—41no—
316C22×C32—81no—
317C22×C32—81no—
318C22×C32—81no—
319C22×C32—81no—
320C22×C32—82no—
321C22×C32—82no—
322C22×C32—121no—
323C22×C32—121no—
324C22×C32—121no—
325C22×C32D3×D411yescharacteristic
326C22×C32D3×D411yescharacteristic
327C22×C32C2×C3×D411yescharacteristic
328C2×C3⋊D3—21no—
329C2×C3⋊D3—21no—
330C2×C3⋊D3—21no—
331C2×C3×D3—42no—
332C2×C3×D3—41no—
333C2×C3×D3—41no—
334C2×C3×D3—41no—
335C2×C3×D3—41no—
336C2×C3×D3—41no—
337C2×C3×D3—41no—
338C2×C3×D3—62no—
339C2×C3×D3—61no—
340C2×C3×D3—61no—
341C2×C3×D3—61no—
342C2×C3×D3—61no—
343C2×C3×D3—61no—
344C2×C3×D3—61no—
345C2×C3×D3—81no—
346C2×C3×D3—81no—
347C2×C3×D3—121no—
348C2×C3×D3—121no—
349C32×C4—21no—
350C32×C4—21no—
351C32×C4—21no—
352C32×C4—21no—
353C32×C4—21no—
354C32×C4—21no—
355C32×C4—41no—
356C32×C4—41no—
357C32×C4—121no—
358C32×C4#23511yescharacteristic
359C32×C4D3×D411yescharacteristic
360C32×C4#23711yescharacteristic
361C32×C4C2×#6711yescharacteristic
362C32×C4C2×C3×D411yescharacteristic
363C32×C4C3×#4111yescharacteristic
364C3⋊Dic3—21no—
365C3⋊Dic3—41no—
366C3×Dic3—41no—
367C3×Dic3—41no—
368C3×Dic3—61no—
369C3×Dic3—61no—
370C3×Dic3—81no—
371C3×Dic3—121no—
372C3×Dic3—122no—
373C3×Dic3—121no—
374C3×Dic3—121no—
375C2×D8—31no—
376C2×M16—91no—
377#121—91no—
378#121—91no—
379#106—91no—
380#102—31no—
381#102—91no—
382C33#46911yescharacteristic, complemented, Sylow
383C23×C3—22no—
384C23×C3—21no—
385C23×C3—21no—
386C23×C3—21no—
387C23×C3—41no—
388C23×C3—41no—
389C23×C3—42no—
390C23×C3—61no—
391C23×C3—61no—
392C23×C3—91no—
393C23×C3—121no—
394C22×D3—21no—
395C22×D3—31no—
396C22×D3—31no—
397C22×D3—61no—
398C3×D4—22no—
399C3×D4—22no—
400C3×D4—21no—
401C3×D4—21no—
402C3×D4—21no—
403C3×D4—21no—
404C3×D4—21no—
405C3×D4—21no—
406C3×D4—41no—
407C3×D4—41no—
408C3×D4—41no—
409C3×D4—41no—
410C3×D4—42no—
411C3×D4—42no—
412C3×D4—62no—
413C3×D4—61no—
414C3×D4—61no—
415C3×D4—61no—
416C3×D4—61no—
417C3×D4—121no—
418C3×D4—362no—
419C3×D4—362no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—41no—
424C2×C3×C4—61no—
425C2×C3×C4—61no—
426C2×C3×C4—91no—
427C2×C3×C4—121no—
428C2×C3×C4—181no—
429C2×C3×C4—181no—
430C2×C3×C4—181no—
431C2×C3×C4C2×D3211yescharacteristic
432C2×C3×C4C22×C3×D311yescharacteristic
433C2×C3×C4C22×C3×D311yescharacteristic
434#67—122no—
435#67—124no—
436#67—122no—
437#67—122no—
438C2×Dic3—21no—
439C2×Dic3—31no—
440C2×Dic3—31no—
441C2×Dic3—61no—
442C2×Dic3—62no—
443C2×Dic3—61no—
444C2×Dic3—61no—
445D12—21no—
446D12—21no—
447D12—122no—
448C4×D3—61no—
449C4×D3—61no—
450C4×D3—61no—
451C4×D3—61no—
452C4×D3—121no—
453C4×D3—121no—
454C3×C8—32no—
455C3×C8—32no—
456C3×C8—62no—
457C3×C8—61no—
458C3×C8—61no—
459C3×C8—121no—
460C3⋊C8—21no—
461C3⋊C8C3×#6712yescomplemented
462C2×C32—21no—
463C2×C32—21no—
464C2×C32—21no—
465C2×C32—21no—
466C2×C32—21no—
467C2×C32—21no—
468C2×C32—21no—
469C2×C32—21no—
470C2×C32—21no—
471C2×C32—41no—
472C2×C32—41no—
473C2×C32—41no—
474C2×C32—82no—
475C2×C32—82no—
476C2×C32—82no—
477C2×C32—84no—
478C2×C32—81no—
479C2×C32—81no—
480C2×C32—81no—
481C2×C32—241no—
482C2×C32#88411yescharacteristic
483C2×C32#91111yescharacteristic
484C2×C32#91111yescharacteristic
485C2×C32#91111yescharacteristic
486C2×C32#91411yescharacteristic
487C2×C32#93811yescharacteristic
488C2×C32C3×#12111yescharacteristic
489C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
490C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
491C3⋊D3—41no—
492C3×D3—82no—
493C3×D3—81no—
494C3×D3—81no—
495C3×D3—122no—
496C3×D3—121no—
497C3×D3—121no—
498C2×D4—31no—
499C2×D4—181no—
500C2×D4—181no—
501C2×D4C3×D3211yescharacteristic
502C22×C4—91no—
503D8—62no—
504M16—182no—
505C2×C8—31no—
506C2×C8—31no—
507C4⋊C4—31no—
508C4⋊C4—91no—
509C22⋊C4—181no—
510C22⋊C4—181no—
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—44no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3—41no—
526C22×C3—41no—
527C22×C3—41no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3—82no—
535C22×C3—82no—
536C22×C3—122no—
537C22×C3—121no—
538C22×C3—121no—
539C22×C3—121no—
540C22×C3—121no—
541C22×C3—121no—
542C22×C3—121no—
543C22×C3—181no—
544C22×C3—181no—
545C22×C3—181no—
546C22×C3—241no—
547C22×C3—241no—
548C22×C3#381111yescharacteristic
549C22×C3C3×D3×D411yescharacteristic
550C22×C3C3×D3×D411yescharacteristic
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—41no—
554C2×D3—62no—
555C2×D3—61no—
556C2×D3—61no—
557C2×D3—61no—
558C2×D3—61no—
559C2×D3—61no—
560C2×D3—61no—
561C2×D3—121no—
562C2×D3—121no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—21no—
566C3×C4—21no—
567C3×C4—21no—
568C3×C4—21no—
569C3×C4—41no—
570C3×C4—41no—
571C3×C4—121no—
572C3×C4—121no—
573C3×C4—181no—
574C3×C4—241no—
575C3×C4—361no—
576C3×C4#380411yescharacteristic
577C3×C4D3×#6711yescharacteristic
578C3×C4C3×#23511yescharacteristic
579C3×C4C3×D3×D411yescharacteristic
580C3×C4C3×#23711yescharacteristic
581C3×C4C2×C3×#6711yescharacteristic
582Dic3—41no—
583Dic3—61no—
584Dic3—61no—
585Dic3—121no—
586Dic3—122no—
587Dic3—121no—
588Dic3—121no—
589C32—21no—
590C32—21no—
591C32—21no—
592C32—41no—
593C32#478211yescharacteristic, complemented
594C32#515911yescharacteristic, complemented
595C32C3×#46911yescharacteristic, complemented
596C23—21no—
597C23—61no—
598C23—91no—
599D4—21no—
600D4—21no—
601D4—61no—
602D4—61no—
603D4—362no—
604D4—362no—
605C2×C4—61no—
606C2×C4—91no—
607C2×C4—181no—
608C2×C4—181no—
609C2×C4—181no—
610C2×C4C2×C3×D3211yescharacteristic
611C8—32no—
612C8—61no—
613C2×C3—21no—
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—21no—
620C2×C3—21no—
621C2×C3—21no—
622C2×C3—41no—
623C2×C3—41no—
624C2×C3—41no—
625C2×C3—82no—
626C2×C3—82no—
627C2×C3—82no—
628C2×C3—84no—
629C2×C3—81no—
630C2×C3—81no—
631C2×C3—81no—
632C2×C3—242no—
633C2×C3—241no—
634C2×C3—241no—
635C2×C3—361no—
636C2×C3#6392711yescharacteristic
637C2×C3#6393711yescharacteristic
638C2×C3#6395811yescharacteristic
639C2×C3C3×#88411yescharacteristic
640C2×C3C3×#91111yescharacteristic
641C2×C3C3×#91111yescharacteristic
642C2×C3C3×#91111yescharacteristic
643C2×C3C3×#91411yescharacteristic
644C2×C3C3×#93811yescharacteristic
645D3—81no—
646D3—122no—
647D3—121no—
648D3—121no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—121no—
653C22—121no—
654C22—121no—
655C22—181no—
656C22—181no—
657C22—181no—
658C22C3×#381111yescharacteristic
659C4—121no—
660C4—181no—
661C4—361no—
662C4C3×#380411yescharacteristic
663C4C3×D3×#6711yescharacteristic
664C3—21no—
665C3—21no—
666C3—21no—
667C3—41no—
668C3#1059145411yescharacteristic, direct product factor
669C3C3×#478211yescharacteristic, complemented
670C3C3×#515911yescharacteristic, complemented
671C2—81no—
672C2—241no—
673C2—361no—
674C2C3×#6392711yescharacteristic
675C2C3×#6393711yescharacteristic
676C2C3×#6395811yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#469
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64147C2semidirect
2. C2×C32×#213C2semidirect
3. C3×#63916C2semidirect
4. C2×C3×#3712C2semidirect
5. C32×#832C2semidirect
6. C3×#63573C2semidirect
7. C3×#63563C2semidirect
10. #10591454C3direct (N × H)
25. #64147C2×C3semidirect
31. C2×C3×#213D3semidirect
32. C2×C3×#213C2×C3semidirect
36. #63916C2×C3semidirect
37. C2×#3712C2×C3semidirect
41. C3×#832D3semidirect
42. C3×#832C2×C3semidirect
46. #63573C2×C3semidirect
47. #63563C2×C3semidirect
64. C32×C3⋊C8D4semidirect
145. C2×#213C3×D3semidirect
155. #832C3×D3semidirect
236. C3×C3⋊C8#67semidirect
237. C3×C3⋊C8C3×D4semidirect
382. C33#469semidirect
461. C3⋊C8C3×#67semidirect
593. C32#4782semidirect
594. C32#5159semidirect
595. C32C3×#469semidirect
668. C3#10591454direct (N × H)
669. C3C3×#4782semidirect
670. C3C3×#5159semidirect

Maximal subgroups

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

subgroupindex
C3×#641472
C2×C32×#2132
C3×#639162
C2×C3×#37122
C32×#8322
C3×#635732
C3×#635632
C3×#51593
C3×#47823
#105914543