Finite Group Databasebeta

← C3×#10590547 #13157530

Subgroups — C3×#10590547 #13157530

Order 1728.

780 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#893C211yesmaximal, characteristic, complemented
2C3×#63911C211yesmaximal, characteristic, complemented
3C2×C3×#3714C211yesmaximal, characteristic, complemented
4C2×C3×#3713C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5015—31nomaximal
9C3×#4786—31nomaximal
10#10590547C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×C4×D3—21no—
15C2×C3×#626—21no—
16C2×C3×#626—21no—
17C33×C4⋊C4C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×(C2×D3)⋊C4—21no—
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3714—22no—
23C3×#3713—22no—
24C32×C4×D4—31no—
25C3×#940—31no—
26C2×C3×#213—31no—
27C2×C3×#213—31no—
28C2×C3×#213—31no—
29C2×C3×D24—31no—
30C3×#893—21no—
31C3×#893D311yescharacteristic, complemented
32C3×#893C2×C311yescharacteristic, complemented
33C3×#874—31no—
34#63911C2×C311yescharacteristic, complemented
35C2×#3714C2×C311yescharacteristic, complemented
36C2×#3713C2×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—
50C22×C32×D3—21no—
51C2×C33×C4—21no—
52C2×C33×C4C2311yescharacteristic
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C32×D12—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×D12D412yescomplemented
61C32×C4×D3—42no—
62C3×#626—42no—
63C3×#626—42no—
64C3×C3⋊(C3⋊C8)—21no—
65C32×C3⋊C8—21no—
66C3×#459—91no—
67#5015—31no—
68#4786—31no—
69C2×C32×D4—31no—
70C2×C32×D4—31no—
71C22×C32×C4—61no—
72C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
73C2×C3×#67—61no—
74C2×C3×#67—61no—
75C2×C3×D12—21no—
76C2×C3×D12—21no—
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—61no—
80C2×C3×D12—61no—
81C2×C3×D12—61no—
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×D12C2×D311yescharacteristic
84C2×C3×D12C22×C311yescharacteristic
85C2×C3×D12C22×C311yescharacteristic
86C2×C3×C4×D3—22no—
87C2×C3×C4×D3—21no—
88C2×C3×C4×D3—21no—
89C2×#626—21no—
90C2×#626—21no—
91C2×C32×C8—31no—
92C32×C4⋊C4—21no—
93C32×C4⋊C4—21no—
94C32×C4⋊C4—21no—
95C32×C4⋊C4—41no—
96C32×C4⋊C4C2×D311yescharacteristic
97C32×C4⋊C4C2×D311yescharacteristic
98C32×C4⋊C4C22×C311yescharacteristic
99C32×C22⋊C4—61no—
100C32×C42—31no—
101C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
102C3×#213—62no—
103C3×#213—62no—
104C3×#213—62no—
105C3×(C2×D3)⋊C4—22no—
106C3×(C2×D3)⋊C4—21no—
107C3×(C2×D3)⋊C4—21no—
108C3×C4×Dic3—21no—
109C3×C4×Dic3C2×D311yescharacteristic
110C3×C4×Dic3C22×C311yescharacteristic
111C2×C3×C3⋊C8—21no—
112C2×C3×C3⋊C8—31no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—61no—
115C2×C3×C3⋊C8C2×D311yescharacteristic
116C2×C3×C3⋊C8C22×C311yescharacteristic
117C3×D24—62no—
118#3714—22no—
119#3713—22no—
120C22×C33C2×D411yescharacteristic
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—41no—
124C2×C32×D3—22no—
125C2×C32×D3—41no—
126C2×C32×D3—41no—
127C2×C32×D3—41no—
128C2×C32×D3—41no—
129C2×C32×D3—41no—
130C33×C4—41no—
131C33×C4C2×D411yescharacteristic
132C33×C4#4111yescharacteristic
133C32×Dic3—21no—
134C32×Dic3D812yescomplemented
135C2×C3×D8—91no—
136C2×C3×D8—91no—
137C3×#127—91no—
138C3×C4×D4—31no—
139C3×C4×D4—31no—
140C3×C4×D4—61no—
141#940—31no—
142C2×#213—31no—
143C2×#213—31no—
144C2×#213—31no—
145C2×D24—31no—
146#893C3×D311yescharacteristic, complemented
147#874—31no—
148C3×C4⋊C8—91no—
149C3×#102—31no—
150C3×#102—31no—
151C3×#102—61no—
152C3×#102—91no—
153Dic3⋊C8—31no—
154#809—31no—
155#809—31no—
156#809—61no—
157#809C3×D311yescharacteristic, complemented
158C4⋊(C3⋊C8)—31no—
159C23×C32—61no—
160C23×C32—61no—
161C22×C3⋊D3—21no—
162C22×C3×D3—22no—
163C22×C3×D3—21no—
164C22×C3×D3—21no—
165C22×C3×D3—21no—
166C22×C3×D3—21no—
167C22×C3×D3—41no—
168C22×C3×D3—61no—
169C22×C3×D3—61no—
170C22×C3×D3—121no—
171C32×D4—32no—
172C32×D4—61no—
173C32×D4—61no—
174C32×D4—61no—
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—31no—
184C2×C32×C4—41no—
185C2×C32×C4—41no—
186C2×C32×C4—41no—
187C2×C32×C4—42no—
188C2×C32×C4—122no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C23×C311yescharacteristic
192C4⋊(C3⋊D3)—21no—
193C4⋊(C3⋊D3)—21no—
194C3×#67—122no—
195C3×#67—122no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3#6711yescharacteristic
199C2×C3×Dic3#6711yescharacteristic
200C2×C3×Dic3C3×D411yescharacteristic
201C2×C3×Dic3C3×D411yescharacteristic
202C3×D12—22no—
203C3×D12—22no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—21no—
209C3×D12—21no—
210C3×D12—22no—
211C3×D12—41no—
212C3×D12—62no—
213C3×D12—61no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—61no—
217C3×D12—121no—
218C3×D12—122no—
219C3×D12—122no—
220C3×D12D1212yescomplemented
221C3×D12C3×D412yescomplemented
222C3×C4×D3—44no—
223C3×C4×D3—42no—
224C3×C4×D3—42no—
225#626—42no—
226#626—42no—
227C32×C8—61no—
228C3⋊(C3⋊C8)—21no—
229C3×C3⋊C8—21no—
230C3×C3⋊C8—21no—
231C3×C3⋊C8—41no—
232C3×C3⋊C8—61no—
233C3×C3⋊C8—61no—
234C3×C3⋊C8—121no—
235#459—91noSylow
236C2×C33#12111yescharacteristic
237C2×C33C2×D811yescharacteristic
238C2×C33AGL(1,ℤ/8)11yescharacteristic
239C3×C3⋊D3—81no—
240C32×D3—42no—
241C32×D3—81no—
242C2×C3×D4—31no—
243C2×C3×D4—31no—
244C2×C3×D4—31no—
245C2×C3×D4—31no—
246C2×C3×D4—61no—
247C2×C3×D4—61no—
248C2×C3×D4—91no—
249C2×C3×D4—181no—
250C2×C3×D4—181no—
251C22×C3×C4—61no—
252C22×C3×C4—61no—
253C22×C3×C4—62no—
254C2×#67—61no—
255C2×#67—61no—
256C2×D12—31no—
257C2×D12—31no—
258C2×D12—61no—
259C2×D12—61no—
260C2×D12—61no—
261C2×D12C2×C3×D311yescharacteristic
262C2×D12C2×C3×D311yescharacteristic
263C2×C4×D3—21no—
264C3×D8—182no—
265C3×D8—182no—
266C2×C3×C8—31no—
267C2×C3×C8—31no—
268C2×C3×C8—61no—
269C2×C3×C8—91no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—21no—
272C3×C4⋊C4—21no—
273C3×C4⋊C4—41no—
274C3×C4⋊C4D3211yescharacteristic
275C3×C4⋊C4C2×C3×D311yescharacteristic
276C3×C4⋊C4C2×C3×D311yescharacteristic
277C3×C22⋊C4—61no—
278C3×C22⋊C4—61no—
279C3×C22⋊C4—62no—
280C3×C42—31no—
281C3×C42—31no—
282C3×C42—61no—
283#213—62no—
284#213—62no—
285#213—62no—
286(C2×D3)⋊C4—21no—
287C4×Dic3C2×C3×D311yescharacteristic
288C2×C3⋊C8—31no—
289C2×C3⋊C8—31no—
290C2×C3⋊C8—61no—
291C2×C3⋊C8C2×C3×D311yescharacteristic
292D24—62no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—41no—
297C22×C32—62no—
298C22×C32—121no—
299C22×C32—121no—
300C22×C32—121no—
301C22×C32—121no—
302C22×C32—121no—
303C22×C32D3×D411yescharacteristic
304C22×C32C2×#6711yescharacteristic
305C22×C32C2×C3×D411yescharacteristic
306C2×C3⋊D3—41no—
307C2×C3⋊D3—41no—
308C2×C3⋊D3—41no—
309C2×C3×D3—24no—
310C2×C3×D3—22no—
311C2×C3×D3—22no—
312C2×C3×D3—42no—
313C2×C3×D3—42no—
314C2×C3×D3—41no—
315C2×C3×D3—41no—
316C2×C3×D3—41no—
317C2×C3×D3—41no—
318C2×C3×D3—42no—
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—81no—
326C2×C3×D3—81no—
327C2×C3×D3—122no—
328C2×C3×D3—121no—
329C2×C3×D3—121no—
330C2×C3×D3—121no—
331C2×C3×D3—121no—
332C2×C3×D3—121no—
333C2×C3×D3—121no—
334C2×C3×D3—241no—
335C2×C3×D3—241no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—21no—
342C32×C4—32no—
343C32×C4—41no—
344C32×C4—41no—
345C32×C4—41no—
346C32×C4—42no—
347C32×C4—41no—
348C32×C4—41no—
349C32×C4—61no—
350C32×C4—81no—
351C32×C4—81no—
352C32×C4—82no—
353C32×C4C2×D1211yescharacteristic
354C32×C4#23511yescharacteristic
355C32×C4#23511yescharacteristic
356C32×C4D3×D411yescharacteristic
357C32×C4C2×C3×D411yescharacteristic
358C32×C4C3×#4111yescharacteristic
359C3×Dic3—21no—
360C3×Dic3—21no—
361C3×Dic3—22no—
362C3×Dic3—41no—
363C3×Dic3#21312yescomplemented
364C3×Dic3C3×D812yescomplemented
365C2×D8—91no—
366C2×D8—91no—
367#127—91no—
368C4×D4—31no—
369C4⋊C8—91no—
370#102—31no—
371#102—91no—
372C33#45911yescharacteristic, complemented, Sylow
373C23×C3—62no—
374C23×C3—61no—
375C23×C3—61no—
376C23×C3—61no—
377C23×C3—61no—
378C23×C3—121no—
379C23×C3—181no—
380C22×D3—21no—
381C22×D3—21no—
382C22×D3—61no—
383C22×D3—61no—
384C22×D3—121no—
385C3×D4—32no—
386C3×D4—32no—
387C3×D4—62no—
388C3×D4—62no—
389C3×D4—62no—
390C3×D4—61no—
391C3×D4—61no—
392C3×D4—61no—
393C3×D4—61no—
394C3×D4—61no—
395C3×D4—61no—
396C3×D4—121no—
397C3×D4—181no—
398C3×D4—181no—
399C3×D4—362no—
400C3×D4—362no—
401C2×C3×C4—22no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—31no—
409C2×C3×C4—31no—
410C2×C3×C4—31no—
411C2×C3×C4—31no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—42no—
415C2×C3×C4—41no—
416C2×C3×C4—61no—
417C2×C3×C4—61no—
418C2×C3×C4—122no—
419C2×C3×C4—122no—
420C2×C3×C4—124no—
421C2×C3×C4C2×D3211yescharacteristic
422C2×C3×C4C22×C3×D311yescharacteristic
423C2×C3×C4C22×C3×D311yescharacteristic
424#67—122no—
425#67—122no—
426C2×Dic3C3×#6711yescharacteristic
427C2×Dic3C3×#6711yescharacteristic
428D12—21no—
429D12—21no—
430D12—21no—
431D12—62no—
432D12—61no—
433D12—61no—
434D12—61no—
435D12—61no—
436D12—121no—
437D12—122no—
438D12—122no—
439D12C3×D1212yescomplemented
440C4×D3—42no—
441C3×C8—61no—
442C3×C8—61no—
443C3×C8—121no—
444C3×C8—181no—
445C3⋊C8—21no—
446C3⋊C8—61no—
447C3⋊C8—61no—
448C3⋊C8—121no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32—41no—
459C2×C32—41no—
460C2×C32—41no—
461C2×C32—122no—
462C2×C32—241no—
463C2×C32#88411yescharacteristic
464C2×C32D3×D811yescharacteristic
465C2×C32#91411yescharacteristic
466C2×C32#93011yescharacteristic
467C2×C32C2×#21311yescharacteristic
468C2×C32#94911yescharacteristic
469C2×C32C3×#12111yescharacteristic
470C2×C32C2×C3×D811yescharacteristic
471C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
472C3⋊D3—81no—
473C3×D3—44no—
474C3×D3—42no—
475C3×D3—42no—
476C3×D3—82no—
477C3×D3—81no—
478C3×D3—81no—
479C3×D3—242no—
480C3×D3—241no—
481C3×D3—241no—
482C2×D4—31no—
483C2×D4—31no—
484C2×D4—91no—
485C2×D4—181no—
486C2×D4—181no—
487C22×C4—61no—
488D8—182no—
489D8—182no—
490C2×C8—31no—
491C2×C8—91no—
492C4⋊C4C3×D3211yescharacteristic
493C22⋊C4—61no—
494C42—31no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—41no—
499C22×C3—64no—
500C22×C3—62no—
501C22×C3—62no—
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—241no—
516C22×C3—241no—
517C22×C3—361no—
518C22×C3—361no—
519C22×C3—361no—
520C22×C3D3×#6711yescharacteristic
521C22×C3C3×D3×D411yescharacteristic
522C22×C3C2×C3×#6711yescharacteristic
523C2×D3—22no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—122no—
530C2×D3—121no—
531C2×D3—121no—
532C2×D3—121no—
533C2×D3—121no—
534C2×D3—121no—
535C2×D3—121no—
536C2×D3—241no—
537C2×D3—241no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—32no—
545C3×C4—32no—
546C3×C4—42no—
547C3×C4—41no—
548C3×C4—41no—
549C3×C4—41no—
550C3×C4—41no—
551C3×C4—41no—
552C3×C4—62no—
553C3×C4—61no—
554C3×C4—61no—
555C3×C4—81no—
556C3×C4—81no—
557C3×C4—82no—
558C3×C4—121no—
559C3×C4#379811yescharacteristic
560C3×C4D3×D1211yescharacteristic
561C3×C4C2×C3×D1211yescharacteristic
562C3×C4C3×#23511yescharacteristic
563C3×C4C3×#23511yescharacteristic
564C3×C4C3×D3×D411yescharacteristic
565Dic3—21no—
566Dic3C3×#21312yescomplemented
567C32—21no—
568C32—21no—
569C32—21no—
570C32—41no—
571C32#478611yescharacteristic, complemented
572C32#501511yescharacteristic, complemented
573C32C3×#45911yescharacteristic, complemented
574C23—61no—
575C23—61no—
576C23—181no—
577D4—32no—
578D4—61no—
579D4—61no—
580D4—61no—
581D4—181no—
582D4—181no—
583D4—362no—
584D4—362no—
585C2×C4—21no—
586C2×C4—31no—
587C2×C4—31no—
588C2×C4—122no—
589C2×C4C2×C3×D3211yescharacteristic
590C8—61no—
591C8—181no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—41no—
602C2×C3—41no—
603C2×C3—41no—
604C2×C3—124no—
605C2×C3—122no—
606C2×C3—122no—
607C2×C3—242no—
608C2×C3—241no—
609C2×C3—241no—
610C2×C3—721no—
611C2×C3#6390611yescharacteristic
612C2×C3D3×#21311yescharacteristic
613C2×C3#6393911yescharacteristic
614C2×C3C3×#88411yescharacteristic
615C2×C3C3×D3×D811yescharacteristic
616C2×C3C3×#91411yescharacteristic
617C2×C3C3×#93011yescharacteristic
618C2×C3C2×C3×#21311yescharacteristic
619C2×C3C3×#94911yescharacteristic
620D3—42no—
621D3—81no—
622D3—242no—
623D3—241no—
624D3—241no—
625C22—62no—
626C22—121no—
627C22—121no—
628C22—121no—
629C22—121no—
630C22—121no—
631C22—361no—
632C22—361no—
633C22—361no—
634C22C3×D3×#6711yescharacteristic
635C4—32no—
636C4—41no—
637C4—61no—
638C4C3×#379811yescharacteristic
639C4C3×D3×D1211yescharacteristic
640C3—21no—
641C3—21no—
642C3—21no—
643C3—41no—
644C3#1059054711yescharacteristic, direct product factor
645C3C3×#478611yescharacteristic, complemented
646C3C3×#501511yescharacteristic, complemented
647C2—122no—
648C2—241no—
649C2—721no—
650C2C3×#6390611yescharacteristic
651C2C3×D3×#21311yescharacteristic
652C2C3×#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×#459
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×#893C2semidirect
2. C3×#63911C2semidirect
3. C2×C3×#3714C2semidirect
4. C2×C3×#3713C2semidirect
5. C3×#63636C2semidirect
6. C32×#809C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
10. #10590547C3direct (N × H)
31. C3×#893D3semidirect
32. C3×#893C2×C3semidirect
34. #63911C2×C3semidirect
35. C2×#3714C2×C3semidirect
36. C2×#3713C2×C3semidirect
38. #63636C2×C3semidirect
44. C3×#809D3semidirect
45. C3×#809C2×C3semidirect
47. Dic3⋊(C3⋊C8)C2×C3semidirect
60. C32×D12D4semidirect
134. C32×Dic3D8semidirect
146. #893C3×D3semidirect
157. #809C3×D3semidirect
220. C3×D12D12semidirect
221. C3×D12C3×D4semidirect
363. C3×Dic3#213semidirect
364. C3×Dic3C3×D8semidirect
372. C33#459semidirect
439. D12C3×D12semidirect
566. Dic3C3×#213semidirect
571. C32#4786semidirect
572. C32#5015semidirect
573. C32C3×#459semidirect
644. C3#10590547direct (N × H)
645. C3C3×#4786semidirect
646. C3C3×#5015semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
C3×#639112
C2×C3×#37142
C2×C3×#37132
C3×#636362
C32×#8092
C3×Dic3⋊(C3⋊C8)2
C3×#50153
C3×#47863
#105905473