Finite Group Databasebeta

← C3×#10590423 #13157471

Subgroups — C3×#10590423 #13157471

Order 1728.

660 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D3×C4⋊C4C211yesmaximal, characteristic, complemented
2C3×#63909C211yesmaximal, characteristic
3C3×#63636C211yesmaximal, characteristic, complemented
4C32×#808C211yesmaximal, characteristic, complemented
5C3×#63574C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#4968—31nomaximal
9C3×#4863—31nomaximal
10#10590423C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×C4×D3—21no—
13C2×C32×C4×D3C2211yescharacteristic
14C2×C3×#626—21no—
15C33×C4⋊C4C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×#3722—21no—
22C2×C32×C4⋊C4—31no—
23C3×#930—31no—
24C3×#895—31no—
25C3×D3×C4⋊C4—21no—
26C3×D3×C4⋊C4D311yescharacteristic, complemented
27C3×D3×C4⋊C4C2×C311yescharacteristic, complemented
28#63909C2×C311yescharacteristic
29C32×#106—31no—
30#63636C2×C311yescharacteristic, complemented
31C3×#830—31no—
32C3×#821—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×#809—31no—
35C3×#809—31no—
36C3×#809—31no—
37C3×#809—61no—
38C3×#808—21no—
39C3×#808—31no—
40C3×#808—31no—
41C3×#808D311yescharacteristic, complemented
42C3×#808C2×C311yescharacteristic, complemented
43#63574C2×C311yescharacteristic, complemented
44#63564C2×C311yescharacteristic, complemented
45(C2×D3)⋊(C3⋊C8)C2×C311yescharacteristic, complemented
46C22×C3×C3⋊D3—21no—
47C22×C32×D3D411yescharacteristic
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C32×C4×D3—21no—
56C32×C4×D3—21no—
57C32×C4×D3—42no—
58C3×#626—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61C3×#481—91no—
62#4968—31no—
63#4863—31no—
64C22×C32×C4—31no—
65C22×C32×C4—61no—
66C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
67C2×C3×#67—61no—
68C2×C3×D12—31no—
69C2×C3×D12—31no—
70C2×C3×D12—61no—
71C2×C3×D12—61no—
72C2×C3×C4×D3—21no—
73C2×C3×C4×D3—22no—
74C2×C3×C4×D3—21no—
75C2×C3×C4×D3—21no—
76C2×C3×C4×D3C2×D311yescharacteristic
77C2×C3×C4×D3C22×C311yescharacteristic
78C2×#626—21no—
79C2×C32×C8—31no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4—21no—
83C32×C4⋊C4—31no—
84C32×C4⋊C4—31no—
85C32×C4⋊C4—41no—
86C32×C4⋊C4—61no—
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C2×D311yescharacteristic
89C32×C4⋊C4C22×C311yescharacteristic
90C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
91C3×(C2×D3)⋊C4—61no—
92C3×(C2×D3)⋊C4—61no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×C4⋊Dic3C22×C311yescharacteristic
98C3×C4⋊Dic3C22×C311yescharacteristic
99C3×Dic3⋊C4—22no—
100C3×Dic3⋊C4—21no—
101C3×Dic3⋊C4—21no—
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C22×C311yescharacteristic
108#3722—21no—
109C22×C33C2×D411yescharacteristic
110C2×C3×C3⋊D3—41no—
111C2×C3×C3⋊D3—41no—
112C2×C3×C3⋊D3—41no—
113C2×C32×D3—21no—
114C2×C32×D3—21no—
115C2×C32×D3SD1612yes—
116C33×C4—41no—
117C33×C4#4111yescharacteristic
118C33×C4#4111yescharacteristic
119C32×Dic3—21no—
120C32×Dic3—41no—
121C32×Dic3—41no—
122C3×#121—91no—
123C2×C3×C4⋊C4—31no—
124C2×C3×C4⋊C4—31no—
125C2×C3×C4⋊C4—61no—
126#930—31no—
127#895—31no—
128D3×C4⋊C4C3×D311yescharacteristic, complemented
129C3×#106—31no—
130C3×#106—31no—
131C3×#106—61no—
132C3×#106—91no—
133C3×#102—91no—
134C3×#102—91no—
135C3×C22⋊C8—91no—
136#830—31no—
137#821—31no—
138(C2×D3)⋊C8—31no—
139#809—31no—
140#809—31no—
141#809—31no—
142#809—61no—
143#808—31no—
144#808—31no—
145#808C3×D311yescharacteristic, complemented
146C23×C32—31no—
147C22×C3⋊D3—21no—
148C22×C3×D3—21no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—121no—
152C22×C3×D3#6711yescharacteristic
153C22×C3×D3C3×D411yescharacteristic
154C2×C32×C4—21no—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—31no—
162C2×C32×C4—41no—
163C2×C32×C4—41no—
164C2×C32×C4—41no—
165C2×C32×C4—42no—
166C2×C32×C4—61no—
167C2×C32×C4—61no—
168C2×C32×C4—61no—
169C2×C32×C4—61no—
170C2×C32×C4—122no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C23×C311yescharacteristic
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)—21no—
176C3×#67—122no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—41no—
184C2×C3×Dic3#6711yescharacteristic
185C2×C3×Dic3C3×D411yescharacteristic
186C3×D12—62no—
187C3×D12—61no—
188C3×D12—61no—
189C3×D12—61no—
190C3×D12—61no—
191C3×D12—121no—
192C3×D12—122no—
193C3×C4×D3—22no—
194C3×C4×D3—22no—
195C3×C4×D3—21no—
196C3×C4×D3—21no—
197C3×C4×D3—21no—
198C3×C4×D3—21no—
199C3×C4×D3—44no—
200C3×C4×D3—42no—
201C3×C4×D3—42no—
202#626—42no—
203C32×C8—61no—
204C3⋊(C3⋊C8)—21no—
205C3×C3⋊C8—22no—
206C3×C3⋊C8—21no—
207C3×C3⋊C8—21no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—61no—
210C3×C3⋊C8—121no—
211#481—91noSylow
212C2×C33#12311yescharacteristic
213C2×C33C2×SD1611yescharacteristic
214C2×C33AGL(1,ℤ/8)11yescharacteristic
215C3×C3⋊D3—81no—
216C32×D3—22no—
217C2×C3×D4—91no—
218C2×C3×D4—181no—
219C22×C3×C4—31no—
220C22×C3×C4—31no—
221C22×C3×C4—61no—
222C22×C3×C4—61no—
223C22×C3×C4—61no—
224C22×C3×C4—62no—
225C2×#67—61no—
226C2×D12—31no—
227C2×D12—31no—
228C2×D12—61no—
229C2×D12—61no—
230C2×C4×D3—21no—
231C2×C4×D3C2×C3×D311yescharacteristic
232C2×C3×C8—31no—
233C2×C3×C8—31no—
234C2×C3×C8—61no—
235C2×C3×C8—91no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—61no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—61no—
247C3×C4⋊C4—61no—
248C3×C4⋊C4—62no—
249C3×C4⋊C4D3211yescharacteristic
250C3×C4⋊C4C2×C3×D311yescharacteristic
251C3×C4⋊C4C2×C3×D311yescharacteristic
252C3×C22⋊C4—181no—
253(C2×D3)⋊C4—61no—
254(C2×D3)⋊C4—61no—
255C4⋊Dic3C2×C3×D311yescharacteristic
256C4⋊Dic3C2×C3×D311yescharacteristic
257Dic3⋊C4—21no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—31no—
260C2×C3⋊C8—61no—
261C2×C3⋊C8C2×C3×D311yescharacteristic
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—32no—
266C22×C32—41no—
267C22×C32—61no—
268C22×C32—61no—
269C22×C32D3×D411yescharacteristic
270C22×C32C2×#6711yescharacteristic
271C22×C32C2×C3×D411yescharacteristic
272C2×C3⋊D3—41no—
273C2×C3⋊D3—41no—
274C2×C3⋊D3—41no—
275C2×C3×D3—22no—
276C2×C3×D3—22no—
277C2×C3×D3—21no—
278C2×C3×D3—21no—
279C2×C3×D3—21no—
280C2×C3×D3—21no—
281C2×C3×D3—22no—
282C2×C3×D3—122no—
283C2×C3×D3—121no—
284C2×C3×D3—121no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C2×C3×D3—241no—
290C2×C3×D3—241no—
291C2×C3×D3#21512yes—
292C2×C3×D3C3×SD1612yes—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4—41no—
302C32×C4—42no—
303C32×C4—41no—
304C32×C4—41no—
305C32×C4—61no—
306C32×C4—81no—
307C32×C4—81no—
308C32×C4—82no—
309C32×C4—121no—
310C32×C4—121no—
311C32×C4#23511yescharacteristic
312C32×C4#23511yescharacteristic
313C32×C4#23511yescharacteristic
314C32×C4#23911yescharacteristic
315C32×C4C3×#4111yescharacteristic
316C32×C4C3×#4111yescharacteristic
317C3×Dic3—22no—
318C3×Dic3—21no—
319C3×Dic3—21no—
320C3×Dic3—42no—
321C3×Dic3—41no—
322C3×Dic3—41no—
323C3×Dic3—41no—
324C3×Dic3—41no—
325C3×Dic3—81no—
326#121—91no—
327C2×C4⋊C4—31no—
328#106—31no—
329#106—91no—
330#102—91no—
331#102—91no—
332C22⋊C8—91no—
333C33#48111yescharacteristic, complemented, Sylow
334C23×C3—31no—
335C23×C3—31no—
336C23×C3—61no—
337C23×C3—181no—
338C22×D3—61no—
339C22×D3—61no—
340C22×D3—121no—
341C22×D3C3×#6711yescharacteristic
342C3×D4—181no—
343C3×D4—181no—
344C3×D4—362no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—61no—
359C2×C3×C4—62no—
360C2×C3×C4—62no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—62no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—121no—
371C2×C3×C4—124no—
372C2×C3×C4—122no—
373C2×C3×C4—122no—
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C22×C3×D311yescharacteristic
376C2×C3×C4C22×C3×D311yescharacteristic
377#67—122no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3C3×#6711yescharacteristic
381D12—62no—
382D12—61no—
383D12—61no—
384D12—61no—
385D12—61no—
386D12—121no—
387D12—122no—
388C4×D3—21no—
389C4×D3—21no—
390C4×D3—42no—
391C3×C8—62no—
392C3×C8—61no—
393C3×C8—61no—
394C3×C8—181no—
395C3⋊C8—21no—
396C3⋊C8—61no—
397C3⋊C8—61no—
398C3⋊C8—121no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—41no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—62no—
412C2×C32#89411yescharacteristic
413C2×C32D3×SD1611yescharacteristic
414C2×C32#91411yescharacteristic
415C2×C32#92911yescharacteristic
416C2×C32#93211yescharacteristic
417C2×C32C2×#21511yescharacteristic
418C2×C32C3×#12311yescharacteristic
419C2×C32C2×C3×SD1611yescharacteristic
420C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
421C3⋊D3—81no—
422C3×D3—24no—
423C3×D3—22no—
424C3×D3—22no—
425C3×D3—242no—
426C3×D3—241no—
427C3×D3—241no—
428C2×D4—91no—
429C2×D4—181no—
430C22×C4—31no—
431C22×C4—61no—
432C2×C8—31no—
433C2×C8—91no—
434C4⋊C4—31no—
435C4⋊C4—31no—
436C4⋊C4—61no—
437C4⋊C4C3×D3211yescharacteristic
438C22⋊C4—181no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—32no—
443C22×C3—32no—
444C22×C3—41no—
445C22×C3—62no—
446C22×C3—62no—
447C22×C3—62no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—361no—
453C22×C3—361no—
454C22×C3—361no—
455C22×C3D3×#6711yescharacteristic
456C22×C3C3×D3×D411yescharacteristic
457C22×C3C2×C3×#6711yescharacteristic
458C2×D3—21no—
459C2×D3—21no—
460C2×D3—122no—
461C2×D3—121no—
462C2×D3—121no—
463C2×D3—121no—
464C2×D3—121no—
465C2×D3—121no—
466C2×D3—121no—
467C2×D3—241no—
468C2×D3—241no—
469C2×D3C3×#21512yes—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—42no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—62no—
483C3×C4—61no—
484C3×C4—61no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4—82no—
488C3×C4—122no—
489C3×C4—121no—
490C3×C4—121no—
491C3×C4—121no—
492C3×C4—121no—
493C3×C4—241no—
494C3×C4#379811yescharacteristic
495C3×C4#379911yescharacteristic
496C3×C4C3×#23511yescharacteristic
497C3×C4C3×#23511yescharacteristic
498C3×C4C3×#23511yescharacteristic
499C3×C4C3×#23911yescharacteristic
500Dic3—21no—
501Dic3—41no—
502Dic3—41no—
503C32—21no—
504C32—21no—
505C32—21no—
506C32—41no—
507C32#486311yescharacteristic, complemented
508C32#496811yescharacteristic, complemented
509C32C3×#48111yescharacteristic, complemented
510C23—31no—
511C23—181no—
512D4—181no—
513D4—181no—
514D4—362no—
515C2×C4—21no—
516C2×C4—31no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—122no—
522C2×C4C2×C3×D3211yescharacteristic
523C8—61no—
524C8—181no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—41no—
535C2×C3—41no—
536C2×C3—41no—
537C2×C3—64no—
538C2×C3—62no—
539C2×C3—62no—
540C2×C3—721no—
541C2×C3#6385011yescharacteristic
542C2×C3#6393411yescharacteristic
543C2×C3D3×#21511yescharacteristic
544C2×C3C3×#89411yescharacteristic
545C2×C3C3×D3×SD1611yescharacteristic
546C2×C3C3×#91411yescharacteristic
547C2×C3C3×#92911yescharacteristic
548C2×C3C3×#93211yescharacteristic
549C2×C3C2×C3×#21511yescharacteristic
550D3—22no—
551D3—242no—
552D3—241no—
553D3—241no—
554C22—32no—
555C22—61no—
556C22—61no—
557C22—361no—
558C22—361no—
559C22—361no—
560C22C3×D3×#6711yescharacteristic
561C4—41no—
562C4—61no—
563C4—121no—
564C4—121no—
565C4C3×#379811yescharacteristic
566C4C3×#379911yescharacteristic
567C3—21no—
568C3—21no—
569C3—21no—
570C3—41no—
571C3#1059042311yescharacteristic, direct product factor
572C3C3×#486311yescharacteristic, complemented
573C3C3×#496811yescharacteristic, complemented
574C2—62no—
575C2—721no—
576C2C3×#6385011yescharacteristic
577C2C3×#6393411yescharacteristic
578C2C3×D3×#21511yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#63909C2
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#481
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×D3×C4⋊C4C2semidirect
3. C3×#63636C2semidirect
4. C32×#808C2semidirect
5. C3×#63574C2semidirect
6. C3×#63564C2semidirect
7. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
10. #10590423C3direct (N × H)
26. C3×D3×C4⋊C4D3semidirect
27. C3×D3×C4⋊C4C2×C3semidirect
30. #63636C2×C3semidirect
41. C3×#808D3semidirect
42. C3×#808C2×C3semidirect
43. #63574C2×C3semidirect
44. #63564C2×C3semidirect
45. (C2×D3)⋊(C3⋊C8)C2×C3semidirect
128. D3×C4⋊C4C3×D3semidirect
145. #808C3×D3semidirect
333. C33#481semidirect
507. C32#4863semidirect
508. C32#4968semidirect
509. C32C3×#481semidirect
571. C3#10590423direct (N × H)
572. C3C3×#4863semidirect
573. C3C3×#4968semidirect

Maximal subgroups

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

subgroupindex
C32×D3×C4⋊C42
C3×#639092
C3×#636362
C32×#8082
C3×#635742
C3×#635642
C3×(C2×D3)⋊(C3⋊C8)2
C3×#49683
C3×#48633
#105904233