Finite Group Databasebeta

← C3×#10591136 #13158164

Subgroups — C3×#10591136 #13158164

Order 1728.

698 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64139C211yesmaximal, characteristic
2C32×#926C211yesmaximal, characteristic, complemented
3C32×#830C211yesmaximal, characteristic, complemented
4C3×#63575C211yesmaximal, characteristic, complemented
5C3×#63573C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8C3×#5003—31nomaximal
9C3×#4737—31nomaximal
10#10591136C311yesmaximal, characteristic, direct product factor
11C22×C33×C4C2211yescharacteristic
12C2×C3×#638—21no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C3×#3752—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×#217—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3—21no—
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C32×#117—31no—
23#64139C2×C311yescharacteristic
24C3×#930—31no—
25C3×#930—31no—
26C3×#930—61no—
27C3×#926—21no—
28C3×#926D311yescharacteristic, complemented
29C3×#926C2×C311yescharacteristic, complemented
30C32×C22⋊C8—31no—
31C3×#830—21no—
32C3×#830D311yescharacteristic, complemented
33C3×#830C2×C311yescharacteristic, complemented
34C3×#821—31no—
35C3×#821—31no—
36C3×#818—31no—
37C3×#817—31no—
38C3×#809—31no—
39C3×#809—31no—
40C3×#808—31no—
41C3×#807—31no—
42#63575C2×C311yescharacteristic, complemented
43#63573C2×C311yescharacteristic, complemented
44#63564C2×C311yescharacteristic, complemented
45#63564C2×C311yescharacteristic, complemented
46C23×C33D411yescharacteristic
47C22×C3×C3⋊D3—21no—
48C2×C33×C4—21no—
49C2×C33×C4—21no—
50C2×C33×C4D411yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C3×#638—42no—
53C2×C3×C3⋊Dic3—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3—21no—
58C32×C3⋊C8—21no—
59C32×C3⋊C8—21no—
60C3×#482—91no—
61#5003—31no—
62#4737—31no—
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4—21no—
66C22×C32×C4—41no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C22×C311yescharacteristic
70C2×#638—21no—
71C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
72C2×C3×#67—62no—
73C2×C3×#67—61no—
74C2×C3×#67—61no—
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C32×C8—31no—
79C2×C32×C8—31no—
80C32×C4⋊C4—31no—
81C32×C4⋊C4—31no—
82C32×C22⋊C4—61no—
83C32×C42—61no—
84#3752—21no—
85C4⋊(C3⋊Dic3)C22×C311yescharacteristic
86C3×#217—22no—
87C3×#217—21no—
88C3×#217—21no—
89C3×(C2×D3)⋊C4—62no—
90C3×(C2×D3)⋊C4—61no—
91C3×(C2×D3)⋊C4—61no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—61no—
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×C4⋊Dic3C22×C311yescharacteristic
100C3×C4⋊Dic3C22×C311yescharacteristic
101C3×C4×Dic3—22no—
102C3×C4×Dic3—21no—
103C3×C4×Dic3—21no—
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8—21no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C22×C311yescharacteristic
109C2×C3×C3⋊C8C22×C311yescharacteristic
110C22×C33—21no—
111C22×C33—21no—
112C22×C33—21no—
113C22×C33C2×D411yescharacteristic
114C2×C3×C3⋊D3—41no—
115C2×C3×C3⋊D3—41no—
116C2×C3×C3⋊D3—41no—
117C33×C4—21no—
118C33×C4#4111yescharacteristic
119C33×C4#4111yescharacteristic
120C3×C3⋊Dic3—41no—
121C32×Dic3—41no—
122C32×Dic3—41no—
123C3×#121—91no—
124C3×#117—31no—
125C3×#117—31no—
126C3×#117—61no—
127#930—31no—
128#930—31no—
129#930—61no—
130#926C3×D311yescharacteristic, complemented
131C3×#107—91no—
132C3×#106—91no—
133C3×#102—91no—
134C3×#102—91no—
135C3×C22⋊C8—31no—
136C3×C22⋊C8—31no—
137C3×C22⋊C8—61no—
138#830C3×D311yescharacteristic, complemented
139#821—31no—
140#821—31no—
141#818—31no—
142#817—31no—
143#809—31no—
144#809—31no—
145#808—31no—
146#807—31no—
147C23×C32—21no—
148C23×C32—21no—
149C23×C32—21no—
150C23×C32—41no—
151C23×C32D1211yescharacteristic
152C23×C32#6711yescharacteristic
153C23×C32C3×D411yescharacteristic
154C22×C3⋊D3—21no—
155C22×C3×D3—62no—
156C22×C3×D3—61no—
157C22×C3×D3—61no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—22no—
161C2×C32×C4—22no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—41no—
173C2×C32×C4—41no—
174C2×C32×C4—41no—
175C2×C32×C4—41no—
176C2×C32×C4—41no—
177C2×C32×C4—41no—
178C2×C32×C4—42no—
179C2×C32×C4—42no—
180C2×C32×C4—61no—
181C2×C32×C4—61no—
182C2×C32×C4D1211yescharacteristic
183C2×C32×C4#6711yescharacteristic
184C2×C32×C4C3×D411yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C23×C311yescharacteristic
188#638—42no—
189C2×C3⋊Dic3—21no—
190C4⋊(C3⋊D3)—21no—
191C4⋊(C3⋊D3)—21no—
192C3×#67—124no—
193C3×#67—122no—
194C3×#67—122no—
195C2×C3×Dic3—22no—
196C2×C3×Dic3—22no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—21no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—62no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—61no—
204C3×D12—61no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—121no—
209C3×D12—121no—
210C32×C8—61no—
211C32×C8—61no—
212C3×C3⋊C8—21no—
213C3×C3⋊C8—21no—
214C3×C3⋊C8—21no—
215C3×C3⋊C8—21no—
216C3×C3⋊C8—41no—
217C3×C3⋊C8—41no—
218#482—91noSylow
219C2×C33—41no—
220C2×C33#12311yescharacteristic
221C2×C33#13511yescharacteristic
222C2×C33AGL(1,ℤ/8)11yescharacteristic
223C3×C3⋊D3—81no—
224C2×C3×D4—91no—
225C2×C3×D4—181no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—41no—
230C22×C3×C4D3211yescharacteristic
231C22×C3×C4C2×C3×D311yescharacteristic
232C22×C3×C4C2×C3×D311yescharacteristic
233C2×#67—62no—
234C2×#67—61no—
235C2×#67—61no—
236C2×D12—31no—
237C2×D12—31no—
238C2×D12—61no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—31no—
242C2×C3×C8—31no—
243C2×C3×C8—61no—
244C2×C3×C8—61no—
245C3×C4⋊C4—31no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—61no—
250C3×C4⋊C4—61no—
251C3×C4⋊C4—91no—
252C3×C22⋊C4—62no—
253C3×C22⋊C4—61no—
254C3×C22⋊C4—61no—
255C3×C22⋊C4—181no—
256C3×C42—62no—
257C3×C42—61no—
258C3×C42—61no—
259#217—21no—
260(C2×D3)⋊C4—62no—
261(C2×D3)⋊C4—61no—
262(C2×D3)⋊C4—61no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—61no—
266C4⋊Dic3C2×C3×D311yescharacteristic
267C4⋊Dic3C2×C3×D311yescharacteristic
268C4×Dic3—21no—
269C2×C3⋊C8C2×C3×D311yescharacteristic
270C2×C3⋊C8C2×C3×D311yescharacteristic
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—22no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—41no—
287C22×C32—41no—
288C22×C32—41no—
289C22×C32—41no—
290C22×C32—41no—
291C22×C32—41no—
292C22×C32—41no—
293C22×C32—42no—
294C22×C32—42no—
295C22×C32—42no—
296C22×C32C2×D1211yescharacteristic
297C22×C32C2×#6711yescharacteristic
298C22×C32C2×C3×D411yescharacteristic
299C2×C3⋊D3—41no—
300C2×C3⋊D3—41no—
301C2×C3⋊D3—41no—
302C2×C3×D3—122no—
303C2×C3×D3—122no—
304C2×C3×D3—122no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—121no—
308C2×C3×D3—121no—
309C2×C3×D3—121no—
310C2×C3×D3—121no—
311C32×C4—21no—
312C32×C4—22no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—41no—
322C32×C4—41no—
323C32×C4—41no—
324C32×C4—41no—
325C32×C4—42no—
326C32×C4—121no—
327C32×C4—121no—
328C32×C4#23511yescharacteristic
329C32×C4#23511yescharacteristic
330C32×C4#23711yescharacteristic
331C32×C4#23711yescharacteristic
332C32×C4C3×#4111yescharacteristic
333C32×C4C3×#4111yescharacteristic
334C3⋊Dic3—41no—
335C3×Dic3—42no—
336C3×Dic3—42no—
337C3×Dic3—41no—
338C3×Dic3—41no—
339C3×Dic3—41no—
340C3×Dic3—41no—
341C3×Dic3—122no—
342C3×Dic3—121no—
343C3×Dic3—121no—
344#121—91no—
345#117—31no—
346#107—91no—
347#106—91no—
348#102—91no—
349#102—91no—
350C22⋊C8—31no—
351C33#48211yescharacteristic, complemented, Sylow
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—41no—
356C23×C3—181no—
357C23×C3#62611yescharacteristic
358C23×C3C3×D1211yescharacteristic
359C23×C3C3×#6711yescharacteristic
360C22×D3—62no—
361C22×D3—61no—
362C22×D3—61no—
363C3×D4—181no—
364C3×D4—181no—
365C3×D4—362no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—42no—
385C2×C3×C4—42no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—62no—
389C2×C3×C4—62no—
390C2×C3×C4—61no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—61no—
394C2×C3×C4—181no—
395C2×C3×C4#62611yescharacteristic
396C2×C3×C4C3×D1211yescharacteristic
397C2×C3×C4C3×#6711yescharacteristic
398C2×C3×C4C2×D3211yescharacteristic
399C2×C3×C4C22×C3×D311yescharacteristic
400C2×C3×C4C22×C3×D311yescharacteristic
401#67—124no—
402#67—122no—
403#67—122no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—62no—
407C2×Dic3—61no—
408C2×Dic3—61no—
409D12—61no—
410D12—61no—
411D12—61no—
412D12—61no—
413D12—121no—
414D12—121no—
415C3×C8—61no—
416C3×C8—61no—
417C3×C8—61no—
418C3×C8—61no—
419C3×C8—121no—
420C3×C8—121no—
421C3⋊C8—21no—
422C3⋊C8—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—41no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—42no—
436C2×C32—42no—
437C2×C32—42no—
438C2×C32—44no—
439C2×C32—41no—
440C2×C32—41no—
441C2×C32—41no—
442C2×C32#88611yescharacteristic
443C2×C32#90411yescharacteristic
444C2×C32#90811yescharacteristic
445C2×C32#92911yescharacteristic
446C2×C32#94911yescharacteristic
447C2×C32#95011yescharacteristic
448C2×C32C3×#12311yescharacteristic
449C2×C32C3×#13511yescharacteristic
450C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
451C3⋊D3—81no—
452C3×D3—242no—
453C3×D3—241no—
454C3×D3—241no—
455C2×D4—91no—
456C2×D4—181no—
457C22×C4C3×D3211yescharacteristic
458C2×C8—31no—
459C2×C8—31no—
460C4⋊C4—31no—
461C4⋊C4—31no—
462C4⋊C4—91no—
463C22⋊C4—61no—
464C22⋊C4—181no—
465C42—61no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—41no—
491C22×C3—361no—
492C22×C3—361no—
493C22×C3—361no—
494C22×C3C2×#62611yescharacteristic
495C22×C3C2×C3×D1211yescharacteristic
496C22×C3C2×C3×#6711yescharacteristic
497C2×D3—122no—
498C2×D3—122no—
499C2×D3—122no—
500C2×D3—121no—
501C2×D3—121no—
502C2×D3—121no—
503C2×D3—121no—
504C2×D3—121no—
505C2×D3—121no—
506C3×C4—22no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—42no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—122no—
522C3×C4—122no—
523C3×C4—121no—
524C3×C4—121no—
525C3×C4—121no—
526C3×C4—121no—
527C3×C4—361no—
528C3×C4#380411yescharacteristic
529C3×C4#380411yescharacteristic
530C3×C4C3×#23511yescharacteristic
531C3×C4C3×#23511yescharacteristic
532C3×C4C3×#23711yescharacteristic
533C3×C4C3×#23711yescharacteristic
534Dic3—41no—
535Dic3—41no—
536Dic3—122no—
537Dic3—121no—
538Dic3—121no—
539C32—21no—
540C32—21no—
541C32—21no—
542C32—41no—
543C32#473711yescharacteristic, complemented
544C32#500311yescharacteristic, complemented
545C32C3×#48211yescharacteristic, complemented
546C23—181no—
547C23C3×#62611yescharacteristic
548D4—181no—
549D4—181no—
550D4—362no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—181no—
556C2×C4C3×#62611yescharacteristic
557C2×C4C2×C3×D3211yescharacteristic
558C8—61no—
559C8—61no—
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—21no—
569C2×C3—42no—
570C2×C3—42no—
571C2×C3—42no—
572C2×C3—44no—
573C2×C3—41no—
574C2×C3—41no—
575C2×C3—41no—
576C2×C3—41no—
577C2×C3—41no—
578C2×C3—41no—
579C2×C3—721no—
580C2×C3#6382511yescharacteristic
581C2×C3#6382911yescharacteristic
582C2×C3#6396611yescharacteristic
583C2×C3C3×#88611yescharacteristic
584C2×C3C3×#90411yescharacteristic
585C2×C3C3×#90811yescharacteristic
586C2×C3C3×#92911yescharacteristic
587C2×C3C3×#94911yescharacteristic
588C2×C3C3×#95011yescharacteristic
589D3—242no—
590D3—241no—
591D3—241no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—361no—
596C22—361no—
597C22—361no—
598C22C2×C3×#62611yescharacteristic
599C4—21no—
600C4—121no—
601C4—121no—
602C4—361no—
603C4C3×#380411yescharacteristic
604C4C3×#380411yescharacteristic
605C3—21no—
606C3—21no—
607C3—21no—
608C3—41no—
609C3#1059113611yescharacteristic, direct product factor
610C3C3×#473711yescharacteristic, complemented
611C3C3×#500311yescharacteristic, complemented
612C2—41no—
613C2—721no—
614C2C3×#6382511yescharacteristic
615C2C3×#6382911yescharacteristic
616C2C3×#6396611yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC3×#64139C2
hypercenterC22×C3×C4D32
hypocenterC32C3×#482
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C32×#926C2semidirect
3. C32×#830C2semidirect
4. C3×#63575C2semidirect
5. C3×#63573C2semidirect
6. C3×#63564C2semidirect
7. C3×#63564C2semidirect
10. #10591136C3direct (N × H)
28. C3×#926D3semidirect
29. C3×#926C2×C3semidirect
32. C3×#830D3semidirect
33. C3×#830C2×C3semidirect
42. #63575C2×C3semidirect
43. #63573C2×C3semidirect
44. #63564C2×C3semidirect
45. #63564C2×C3semidirect
130. #926C3×D3semidirect
138. #830C3×D3semidirect
351. C33#482semidirect
543. C32#4737semidirect
544. C32#5003semidirect
545. C32C3×#482semidirect
609. C3#10591136direct (N × H)
610. C3C3×#4737semidirect
611. C3C3×#5003semidirect

Maximal subgroups

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

subgroupindex
C3×#641392
C32×#9262
C32×#8302
C3×#635752
C3×#635732
C3×#635642
C3×#50033
C3×#47373
#105911363