Finite Group Databasebeta

← C3×#10591090 #13158118

Subgroups — C3×#10591090 #13158118

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
1C32×#930C211yesmaximal, characteristic
2C32×#926C211yesmaximal, characteristic, complemented
3C3×#63657C211yesmaximal, characteristic, complemented
4C3×#63576C211yesmaximal, characteristic, complemented
5C3×#63574C211yesmaximal, characteristic, complemented
6C3×#63562C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8C3×#5035—31nomaximal
9C3×#5003—31nomaximal
10#10591090C311yesmaximal, characteristic, direct product factor
11C22×C33×C4C2211yescharacteristic
12C2×C32×#67—21no—
13C2×C32×D12C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×#217—21no—
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C32×C4⋊Dic3C2211yescharacteristic
21C32×C4×Dic3—21no—
22C32×#121—31no—
23C32×#117—31no—
24C3×#930—21no—
25C3×#930D311yescharacteristic
26C3×#930C2×C311yescharacteristic
27C3×#926—21no—
28C3×#926D311yescharacteristic, complemented
29C3×#926C2×C311yescharacteristic, complemented
30#63657C2×C311yescharacteristic, complemented
31C3×#832—31no—
32C3×#832—31no—
33C3×#830—31no—
34C3×#830—31no—
35C3×#830—61no—
36C3×#809—31no—
37C3×#809—31no—
38C3×#808—31no—
39C3×#808—31no—
40C3×#807—31no—
41C3×#807—31no—
42#63576C2×C311yescharacteristic, complemented
43#63574C2×C311yescharacteristic, complemented
44#63562C2×C311yescharacteristic, complemented
45#63562C2×C311yescharacteristic, complemented
46C23×C33D411yescharacteristic
47C22×C32×D3—21no—
48C2×C33×C4—21no—
49C2×C33×C4—21no—
50C2×C33×C4D411yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C32×#67—42no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C3×#482—91no—
61#5035—31no—
62#5003—31no—
63C2×C32×D4—31no—
64C2×C32×D4—61no—
65C22×C32×C4—21no—
66C22×C32×C4—21no—
67C22×C32×C4—21no—
68C22×C32×C4—41no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C2×D311yescharacteristic
71C22×C32×C4C22×C311yescharacteristic
72C2×C3×#67—22no—
73C2×C3×#67—21no—
74C2×C3×#67—21no—
75C2×C3×D12—21no—
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×D12C22×C311yescharacteristic
78C32×C4⋊C4—31no—
79C32×C4⋊C4—31no—
80C32×C4⋊C4—31no—
81C32×C22⋊C4—61no—
82C32×C22⋊C4—61no—
83C32×C42—61no—
84C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
85C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
86C3×#217—22no—
87C3×#217—21no—
88C3×#217—21no—
89C3×(C2×D3)⋊C4—22no—
90C3×(C2×D3)⋊C4—21no—
91C3×(C2×D3)⋊C4—21no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×C4⋊Dic3C22×C311yescharacteristic
99C3×C4⋊Dic3C22×C311yescharacteristic
100C3×C4⋊Dic3C22×C311yescharacteristic
101C3×C4×Dic3—22no—
102C3×C4×Dic3—21no—
103C3×C4×Dic3—21no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—61no—
109C2×C3×C3⋊C8—61no—
110C22×C33—21no—
111C22×C33—21no—
112C22×C33—21no—
113C22×C33C2×D411yescharacteristic
114C2×C32×D3—41no—
115C2×C32×D3—41no—
116C2×C32×D3—41no—
117C33×C4—21no—
118C33×C4#4111yescharacteristic
119C33×C4#4111yescharacteristic
120C32×Dic3—41no—
121C32×Dic3—41no—
122C32×Dic3—41no—
123C3×#121—31no—
124C3×#121—31no—
125C3×#121—61no—
126C3×#117—31no—
127C3×#117—31no—
128C3×#117—61no—
129#930C3×D311yescharacteristic
130#926C3×D311yescharacteristic, complemented
131C3×#107—91no—
132C3×#106—91no—
133C3×#102—91no—
134C3×#102—91no—
135C3×C22⋊C8—91no—
136#832—31no—
137#832—31no—
138#830—31no—
139#830—31no—
140#830—61no—
141#809—31no—
142#809—31no—
143#808—31no—
144#808—31no—
145#807—31no—
146#807—31no—
147C23×C32—21no—
148C23×C32—21no—
149C23×C32—21no—
150C23×C32—41no—
151C23×C32—61no—
152C23×C32#6711yescharacteristic
153C23×C32#6711yescharacteristic
154C23×C32C3×D411yescharacteristic
155C22×C3×D3—22no—
156C22×C3×D3—21no—
157C22×C3×D3—21no—
158C32×D4—61no—
159C32×D4—61no—
160C32×D4—122no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—22no—
164C2×C32×C4—22no—
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—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—41no—
176C2×C32×C4—41no—
177C2×C32×C4—41no—
178C2×C32×C4—41no—
179C2×C32×C4—41no—
180C2×C32×C4—41no—
181C2×C32×C4—42no—
182C2×C32×C4—42no—
183C2×C32×C4—61no—
184C2×C32×C4—61no—
185C2×C32×C4—61no—
186C2×C32×C4#6711yescharacteristic
187C2×C32×C4#6711yescharacteristic
188C2×C32×C4C3×D411yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C23×C311yescharacteristic
192C3×#67—44no—
193C3×#67—42no—
194C3×#67—42no—
195C2×C3×Dic3—22no—
196C2×C3×Dic3—22no—
197C2×C3×Dic3—22no—
198C2×C3×Dic3—21no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—21no—
202C2×C3×Dic3—21no—
203C2×C3×Dic3—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—41no—
209C3×D12—41no—
210C3⋊(C3⋊C8)—21no—
211C3⋊(C3⋊C8)—21no—
212C3×C3⋊C8—61no—
213C3×C3⋊C8—61no—
214C3×C3⋊C8—61no—
215C3×C3⋊C8—61no—
216C3×C3⋊C8—121no—
217C3×C3⋊C8—121no—
218#482—91noSylow
219C2×C33—41no—
220C2×C33#12311yescharacteristic
221C2×C33#13511yescharacteristic
222C2×C33AGL(1,ℤ/8)11yescharacteristic
223C32×D3—81no—
224C2×C3×D4—31no—
225C2×C3×D4—31no—
226C2×C3×D4—61no—
227C2×C3×D4—62no—
228C2×C3×D4—61no—
229C2×C3×D4—61no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—41no—
234C22×C3×C4D3211yescharacteristic
235C22×C3×C4C2×C3×D311yescharacteristic
236C22×C3×C4C2×C3×D311yescharacteristic
237C2×#67—21no—
238C2×D12C2×C3×D311yescharacteristic
239C2×C3×C8—91no—
240C2×C3×C8—91no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—31no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—61no—
248C3×C4⋊C4—61no—
249C3×C4⋊C4—61no—
250C3×C22⋊C4—62no—
251C3×C22⋊C4—62no—
252C3×C22⋊C4—61no—
253C3×C22⋊C4—61no—
254C3×C22⋊C4—61no—
255C3×C22⋊C4—61no—
256C3×C42—62no—
257C3×C42—61no—
258C3×C42—61no—
259#217—21no—
260(C2×D3)⋊C4—21no—
261C4⋊Dic3C2×C3×D311yescharacteristic
262C4⋊Dic3C2×C3×D311yescharacteristic
263C4⋊Dic3C2×C3×D311yescharacteristic
264C4×Dic3—21no—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—31no—
268C2×C3⋊C8—31no—
269C2×C3⋊C8—61no—
270C2×C3⋊C8—61no—
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×C32—121no—
297C22×C32—121no—
298C22×C32—121no—
299C22×C32C2×#6711yescharacteristic
300C22×C32C2×#6711yescharacteristic
301C22×C32C2×C3×D411yescharacteristic
302C2×C3×D3—42no—
303C2×C3×D3—42no—
304C2×C3×D3—42no—
305C2×C3×D3—41no—
306C2×C3×D3—41no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C2×C3×D3—41no—
310C2×C3×D3—41no—
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—121no—
329C32×C4#23511yescharacteristic
330C32×C4#23511yescharacteristic
331C32×C4#23711yescharacteristic
332C32×C4#23711yescharacteristic
333C32×C4C3×#4111yescharacteristic
334C32×C4C3×#4111yescharacteristic
335C3×Dic3—42no—
336C3×Dic3—42no—
337C3×Dic3—42no—
338C3×Dic3—41no—
339C3×Dic3—41no—
340C3×Dic3—41no—
341C3×Dic3—41no—
342C3×Dic3—41no—
343C3×Dic3—41no—
344#121—31no—
345#117—31no—
346#107—91no—
347#106—91no—
348#102—91no—
349#102—91no—
350C22⋊C8—91no—
351C33#48211yescharacteristic, complemented, Sylow
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—41no—
356C23×C3—62no—
357C23×C3—61no—
358C23×C3—61no—
359C23×C3#62511yescharacteristic
360C23×C3C3×#6711yescharacteristic
361C23×C3C3×#6711yescharacteristic
362C22×D3—21no—
363C3×D4—61no—
364C3×D4—61no—
365C3×D4—61no—
366C3×D4—61no—
367C3×D4—121no—
368C3×D4—121no—
369C3×D4—124no—
370C3×D4—122no—
371C3×D4—122no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
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—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—42no—
391C2×C3×C4—42no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—62no—
395C2×C3×C4—62no—
396C2×C3×C4—62no—
397C2×C3×C4—61no—
398C2×C3×C4—61no—
399C2×C3×C4—61no—
400C2×C3×C4—61no—
401C2×C3×C4—61no—
402C2×C3×C4—61no—
403C2×C3×C4#62511yescharacteristic
404C2×C3×C4C3×#6711yescharacteristic
405C2×C3×C4C3×#6711yescharacteristic
406C2×C3×C4C2×D3211yescharacteristic
407C2×C3×C4C22×C3×D311yescharacteristic
408C2×C3×C4C22×C3×D311yescharacteristic
409#67—42no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413D12—21no—
414D12—21no—
415C3×C8—181no—
416C3×C8—181no—
417C3⋊C8—61no—
418C3⋊C8—61no—
419C3⋊C8—61no—
420C3⋊C8—61no—
421C3⋊C8—121no—
422C3⋊C8—121no—
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—241no—
443C2×C32#92911yescharacteristic
444C2×C32#93211yescharacteristic
445C2×C32#93511yescharacteristic
446C2×C32#94911yescharacteristic
447C2×C32#95011yescharacteristic
448C2×C32#95011yescharacteristic
449C2×C32C3×#12311yescharacteristic
450C2×C32C3×#13511yescharacteristic
451C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
452C3×D3—82no—
453C3×D3—81no—
454C3×D3—81no—
455C2×D4—31no—
456C2×D4—61no—
457C22×C4C3×D3211yescharacteristic
458C2×C8—91no—
459C2×C8—91no—
460C4⋊C4—31no—
461C4⋊C4—31no—
462C4⋊C4—31no—
463C22⋊C4—61no—
464C22⋊C4—61no—
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—122no—
492C22×C3—122no—
493C22×C3—122no—
494C22×C3—121no—
495C22×C3—121no—
496C22×C3—121no—
497C22×C3—121no—
498C22×C3—121no—
499C22×C3—121no—
500C22×C3C2×#62511yescharacteristic
501C22×C3C2×C3×#6711yescharacteristic
502C22×C3C2×C3×#6711yescharacteristic
503C2×D3—41no—
504C2×D3—41no—
505C2×D3—41no—
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—122no—
524C3×C4—121no—
525C3×C4—121no—
526C3×C4—121no—
527C3×C4—121no—
528C3×C4—121no—
529C3×C4—121no—
530C3×C4#380411yescharacteristic
531C3×C4#380411yescharacteristic
532C3×C4C3×#23511yescharacteristic
533C3×C4C3×#23511yescharacteristic
534C3×C4C3×#23711yescharacteristic
535C3×C4C3×#23711yescharacteristic
536Dic3—41no—
537Dic3—41no—
538Dic3—41no—
539C32—21no—
540C32—21no—
541C32—21no—
542C32—41no—
543C32#500311yescharacteristic, complemented
544C32#503511yescharacteristic, complemented
545C32C3×#48211yescharacteristic, complemented
546C23—61no—
547C23C3×#62511yescharacteristic
548D4—61no—
549D4—61no—
550D4—122no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4C3×#62511yescharacteristic
557C2×C4C2×C3×D3211yescharacteristic
558C8—181no—
559C8—181no—
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—242no—
580C2×C3—241no—
581C2×C3—241no—
582C2×C3#6382211yescharacteristic
583C2×C3#6382311yescharacteristic
584C2×C3#6396111yescharacteristic
585C2×C3C3×#92911yescharacteristic
586C2×C3C3×#93211yescharacteristic
587C2×C3C3×#93511yescharacteristic
588C2×C3C3×#94911yescharacteristic
589C2×C3C3×#95011yescharacteristic
590C2×C3C3×#95011yescharacteristic
591D3—81no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—121no—
596C22—121no—
597C22—121no—
598C22C2×C3×#62511yescharacteristic
599C4—21no—
600C4—121no—
601C4—121no—
602C4—121no—
603C4C3×#380411yescharacteristic
604C4C3×#380411yescharacteristic
605C3—21no—
606C3—21no—
607C3—21no—
608C3—41no—
609C3#1059109011yescharacteristic, direct product factor
610C3C3×#500311yescharacteristic, complemented
611C3C3×#503511yescharacteristic, complemented
612C2—41no—
613C2—241no—
614C2C3×#6382211yescharacteristic
615C2C3×#6382311yescharacteristic
616C2C3×#6396111yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC32×#930C2
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. C3×#63657C2semidirect
4. C3×#63576C2semidirect
5. C3×#63574C2semidirect
6. C3×#63562C2semidirect
7. C3×#63562C2semidirect
10. #10591090C3direct (N × H)
28. C3×#926D3semidirect
29. C3×#926C2×C3semidirect
30. #63657C2×C3semidirect
42. #63576C2×C3semidirect
43. #63574C2×C3semidirect
44. #63562C2×C3semidirect
45. #63562C2×C3semidirect
130. #926C3×D3semidirect
351. C33#482semidirect
543. C32#5003semidirect
544. C32#5035semidirect
545. C32C3×#482semidirect
609. C3#10591090direct (N × H)
610. C3C3×#5003semidirect
611. C3C3×#5035semidirect

Maximal subgroups

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

subgroupindex
C32×#9302
C32×#9262
C3×#636572
C3×#635762
C3×#635742
C3×#635622
C3×#50353
C3×#50033
#105910903