Finite Group Databasebeta

← C3×#10591415 #13158422

Subgroups — C3×#10591415 #13158422

Order 1728.

704 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#938C211yesmaximal, characteristic
2C3×D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic, complemented
4C32×#832C211yesmaximal, characteristic, complemented
5C3×#63574C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5034—31nomaximal
9C3×#4777—31nomaximal
10#10591415C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C32×#67—21no—
13C2×C32×C4×D3C2211yescharacteristic
14C2×C3×D3×Dic3—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#217—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C32×#121—31no—
23C3×#938—21no—
24C3×#938D311yescharacteristic
25C3×#938C2×C311yescharacteristic
26C2×C3×C4⋊Dic3—31no—
27C3×D3×C4⋊C4—31no—
28D3×C4⋊Dic3C2×C311yescharacteristic, complemented
29C32×#102—31no—
30#63659C2×C311yescharacteristic, complemented
31C3×#832—21no—
32C3×#832—31no—
33C3×#832—31no—
34C3×#832—61no—
35C3×#832D311yescharacteristic, complemented
36C3×#832C2×C311yescharacteristic, complemented
37C3×#830—31no—
38C3×(C2×D3)⋊C8—31no—
39C3×#817—31no—
40C3×#808—31no—
41C3×#808—31no—
42C3×#808—31no—
43#63574C2×C311yescharacteristic, complemented
44#63573C2×C311yescharacteristic, complemented
45(C2×D3)⋊(C3⋊C8)C2×C311yescharacteristic, complemented
46C23×C33—21no—
47C22×C32×D3D411yescharacteristic
48C33×D4—21no—
49C33×D4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C32×#67—42no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C32×C4×D3—21no—
57C32×C4×D3—21no—
58C3×D3×Dic3—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61C3×#481—91no—
62#5034—31no—
63#4777—31no—
64C2×C32×D4—21no—
65C2×C32×D4—21no—
66C2×C32×D4—21no—
67C2×C32×D4—41no—
68C2×C32×D4—61no—
69C2×C32×D4C2×D311yescharacteristic
70C2×C32×D4C2×D311yescharacteristic
71C2×C32×D4C22×C311yescharacteristic
72C22×C32×C4—31no—
73C2×C3×#67—22no—
74C2×C3×#67—21no—
75C2×C3×#67—21no—
76C22×C3×Dic3—61no—
77C2×C3×C4×D3—21no—
78C2×C3×C4×D3—61no—
79C2×C3×C4×D3C2×D311yescharacteristic
80C2×C3×C4×D3C22×C311yescharacteristic
81C2×D3×Dic3—21no—
82C2×C32×C8—31no—
83C32×C4⋊C4—31no—
84C32×C4⋊C4—31no—
85C32×C22⋊C4—61no—
86C4⋊(C3⋊Dic3)C22×C311yescharacteristic
87C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
88C3×#217—22no—
89C3×#217—21no—
90C3×#217—21no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—61no—
96C3×C4⋊Dic3—61no—
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×C4⋊Dic3C22×C311yescharacteristic
100C3×C4⋊Dic3C22×C311yescharacteristic
101C3×Dic3⋊C4—61no—
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
108Dic3⋊Dic3—21no—
109C22×C33—41no—
110C22×C33—41no—
111C22×C33—41no—
112C22×C33C2×D411yescharacteristic
113C2×C32×D3—21no—
114C2×C32×D3—21no—
115C2×C32×D3SD1612yes—
116C33×C4#4111yescharacteristic
117C33×C4#4111yescharacteristic
118C3×C3⋊Dic3—41no—
119C32×Dic3—21no—
120C32×Dic3—41no—
121C32×Dic3—41no—
122C3×#121—31no—
123C3×#121—31no—
124C3×#121—61no—
125C2×C3×C4⋊C4—91no—
126#938C3×D311yescharacteristic
127C2×C4⋊Dic3—31no—
128D3×C4⋊C4—31no—
129C3×#106—91no—
130C3×#106—91no—
131C3×#102—31no—
132C3×#102—31no—
133C3×#102—61no—
134C3×#102—91no—
135C3×C22⋊C8—91no—
136#832—31no—
137#832—31no—
138#832—61no—
139#832C3×D311yescharacteristic, complemented
140#830—31no—
141(C2×D3)⋊C8—31no—
142#817—31no—
143#808—31no—
144#808—31no—
145#808—31no—
146C23×C32—21no—
147C23×C32—22no—
148C23×C32—21no—
149C23×C32—21no—
150C23×C32—31no—
151C23×C32—41no—
152C23×C32—41no—
153C23×C32—42no—
154C22×C3×D3—21no—
155C22×C3×D3#6711yescharacteristic
156C22×C3×D3C3×D411yescharacteristic
157C32×D4—21no—
158C32×D4—21no—
159C32×D4—22no—
160C32×D4—22no—
161C32×D4—21no—
162C32×D4—21no—
163C32×D4—21no—
164C32×D4—21no—
165C32×D4—41no—
166C32×D4—41no—
167C32×D4—41no—
168C32×D4—41no—
169C32×D4—42no—
170C32×D4—42no—
171C32×D4—122no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—31no—
176C2×C32×C4—41no—
177C2×C32×C4—61no—
178C2×C32×C4—61no—
179C2×C32×C4—61no—
180C2×C32×C4—61no—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C23×C311yescharacteristic
184C2×C3⋊Dic3—21no—
185C3×#67—44no—
186C3×#67—42no—
187C3×#67—42no—
188C2×C3×Dic3—22no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—41no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—121no—
198C2×C3×Dic3—122no—
199C2×C3×Dic3#6711yescharacteristic
200C2×C3×Dic3C3×D411yescharacteristic
201C3×C4×D3—21no—
202C3×C4×D3—21no—
203C3×C4×D3—21no—
204C3×C4×D3—21no—
205C3×C4×D3—41no—
206C3×C4×D3—41no—
207C3×C4×D3—122no—
208D3×Dic3—42no—
209C32×C8—61no—
210C3⋊(C3⋊C8)—21no—
211C3×C3⋊C8—21no—
212C3×C3⋊C8—21no—
213C3×C3⋊C8—41no—
214C3×C3⋊C8—61no—
215C3×C3⋊C8—61no—
216C3×C3⋊C8—121no—
217#481—91noSylow
218C2×C33—81no—
219C2×C33#12311yescharacteristic
220C2×C33C2×SD1611yescharacteristic
221C2×C33AGL(1,ℤ/8)11yescharacteristic
222C32×D3—22no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—41no—
227C2×C3×D4—61no—
228C2×C3×D4—61no—
229C2×C3×D4—62no—
230C2×C3×D4D3211yescharacteristic
231C2×C3×D4C2×C3×D311yescharacteristic
232C2×C3×D4C2×C3×D311yescharacteristic
233C22×C3×C4—31no—
234C22×C3×C4—31no—
235C22×C3×C4—61no—
236C22×C3×C4—181no—
237C2×#67—21no—
238C22×Dic3—61no—
239C2×C4×D3—61no—
240C2×C4×D3C2×C3×D311yescharacteristic
241C2×C3×C8—31no—
242C2×C3×C8—31no—
243C2×C3×C8—61no—
244C2×C3×C8—91no—
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×C4⋊C4—181no—
253C3×C22⋊C4—61no—
254C3×C22⋊C4—61no—
255C3×C22⋊C4—62no—
256#217—21no—
257C4⋊Dic3—31no—
258C4⋊Dic3—31no—
259C4⋊Dic3—61no—
260C4⋊Dic3—61no—
261C4⋊Dic3C2×C3×D311yescharacteristic
262C4⋊Dic3C2×C3×D311yescharacteristic
263Dic3⋊C4—61no—
264C2×C3⋊C8—31no—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—61no—
267C2×C3⋊C8C2×C3×D311yescharacteristic
268C22×C32—21no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—32no—
272C22×C32—41no—
273C22×C32—42no—
274C22×C32—42no—
275C22×C32—42no—
276C22×C32—44no—
277C22×C32—41no—
278C22×C32—41no—
279C22×C32—41no—
280C22×C32—41no—
281C22×C32—41no—
282C22×C32—42no—
283C22×C32—42no—
284C22×C32—41no—
285C22×C32—41no—
286C22×C32—41no—
287C22×C32—41no—
288C22×C32—61no—
289C22×C32—61no—
290C22×C32—81no—
291C22×C32—81no—
292C22×C32—81no—
293C22×C32—81no—
294C22×C32—82no—
295C22×C32—82no—
296C22×C32D3×D411yescharacteristic
297C22×C32C2×#6711yescharacteristic
298C22×C32C2×C3×D411yescharacteristic
299C2×C3×D3—21no—
300C2×C3×D3—21no—
301C2×C3×D3—21no—
302C2×C3×D3—21no—
303C2×C3×D3—22no—
304C2×C3×D3—41no—
305C2×C3×D3—41no—
306C2×C3×D3#21412yes—
307C2×C3×D3C3×SD1612yes—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—41no—
315C32×C4—41no—
316C32×C4—61no—
317C32×C4—121no—
318C32×C4—121no—
319C32×C4#23511yescharacteristic
320C32×C4#23711yescharacteristic
321C32×C4#23711yescharacteristic
322C32×C4#23711yescharacteristic
323C32×C4C3×#4111yescharacteristic
324C32×C4C3×#4111yescharacteristic
325C3⋊Dic3—41no—
326C3×Dic3—21no—
327C3×Dic3—21no—
328C3×Dic3—41no—
329C3×Dic3—42no—
330C3×Dic3—41no—
331C3×Dic3—41no—
332C3×Dic3—41no—
333C3×Dic3—41no—
334C3×Dic3—81no—
335C3×Dic3—121no—
336C3×Dic3—121no—
337C3×Dic3—241no—
338#121—31no—
339C2×C4⋊C4—91no—
340#106—91no—
341#106—91no—
342#102—31no—
343#102—91no—
344C22⋊C8—91no—
345C33#48111yescharacteristic, complemented, Sylow
346C23×C3—22no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—31no—
351C23×C3—31no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—42no—
355C23×C3—61no—
356C22×D3C3×#6711yescharacteristic
357C3×D4—22no—
358C3×D4—22no—
359C3×D4—21no—
360C3×D4—21no—
361C3×D4—21no—
362C3×D4—21no—
363C3×D4—21no—
364C3×D4—21no—
365C3×D4—41no—
366C3×D4—41no—
367C3×D4—41no—
368C3×D4—41no—
369C3×D4—42no—
370C3×D4—42no—
371C3×D4—122no—
372C3×D4—122no—
373C3×D4—124no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—31no—
378C2×C3×C4—31no—
379C2×C3×C4—41no—
380C2×C3×C4—61no—
381C2×C3×C4—61no—
382C2×C3×C4—61no—
383C2×C3×C4—61no—
384C2×C3×C4—61no—
385C2×C3×C4—62no—
386C2×C3×C4—61no—
387C2×C3×C4—61no—
388C2×C3×C4—61no—
389C2×C3×C4—61no—
390C2×C3×C4—121no—
391C2×C3×C4—121no—
392C2×C3×C4—121no—
393C2×C3×C4—181no—
394C2×C3×C4—362no—
395C2×C3×C4C2×D3211yescharacteristic
396C2×C3×C4C22×C3×D311yescharacteristic
397C2×C3×C4C22×C3×D311yescharacteristic
398#67—42no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—61no—
402C2×Dic3—61no—
403C2×Dic3—121no—
404C2×Dic3—122no—
405C2×Dic3C3×#6711yescharacteristic
406C4×D3—21no—
407C4×D3—21no—
408C4×D3—122no—
409C3×C8—61no—
410C3×C8—61no—
411C3×C8—121no—
412C3×C8—181no—
413C3⋊C8—21no—
414C3⋊C8—61no—
415C3⋊C8—61no—
416C3⋊C8—121no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—41no—
427C2×C32—41no—
428C2×C32—41no—
429C2×C32—62no—
430C2×C32—82no—
431C2×C32—82no—
432C2×C32—82no—
433C2×C32—84no—
434C2×C32—81no—
435C2×C32—81no—
436C2×C32—81no—
437C2×C32#88311yescharacteristic
438C2×C32#91111yescharacteristic
439C2×C32D3×SD1611yescharacteristic
440C2×C32C2×#21411yescharacteristic
441C2×C32#93511yescharacteristic
442C2×C32#94911yescharacteristic
443C2×C32C3×#12311yescharacteristic
444C2×C32C2×C3×SD1611yescharacteristic
445C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
446C3×D3—22no—
447C3×D3—22no—
448C3×D3—42no—
449C2×D4—61no—
450C2×D4C3×D3211yescharacteristic
451C22×C4—31no—
452C22×C4—181no—
453C2×C8—31no—
454C2×C8—91no—
455C4⋊C4—31no—
456C4⋊C4—31no—
457C4⋊C4—91no—
458C4⋊C4—181no—
459C22⋊C4—61no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—32no—
464C22×C3—32no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—44no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—62no—
482C22×C3—61no—
483C22×C3—61no—
484C22×C3—61no—
485C22×C3—61no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—82no—
491C22×C3—82no—
492C22×C3—121no—
493C22×C3—121no—
494C22×C3D3×#6711yescharacteristic
495C22×C3C3×D3×D411yescharacteristic
496C22×C3C2×C3×#6711yescharacteristic
497C2×D3—21no—
498C2×D3—21no—
499C2×D3C3×#21412yes—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—61no—
509C3×C4—61no—
510C3×C4—121no—
511C3×C4—122no—
512C3×C4—121no—
513C3×C4—121no—
514C3×C4—121no—
515C3×C4—121no—
516C3×C4—241no—
517C3×C4—361no—
518C3×C4#380411yescharacteristic
519C3×C4#380511yescharacteristic
520C3×C4C3×#23511yescharacteristic
521C3×C4C3×#23711yescharacteristic
522C3×C4C3×#23711yescharacteristic
523C3×C4C3×#23711yescharacteristic
524Dic3—21no—
525Dic3—41no—
526Dic3—41no—
527Dic3—121no—
528Dic3—121no—
529Dic3—241no—
530C32—21no—
531C32—21no—
532C32—21no—
533C32—41no—
534C32#477711yescharacteristic, complemented
535C32#503411yescharacteristic, complemented
536C32C3×#48111yescharacteristic, complemented
537C23—21no—
538C23—31no—
539D4—21no—
540D4—21no—
541D4—122no—
542C2×C4—31no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—181no—
548C2×C4—362no—
549C2×C4C2×C3×D3211yescharacteristic
550C8—61no—
551C8—181no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—41no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—62no—
565C2×C3—62no—
566C2×C3—82no—
567C2×C3—82no—
568C2×C3—82no—
569C2×C3—84no—
570C2×C3—81no—
571C2×C3—81no—
572C2×C3—81no—
573C2×C3—122no—
574C2×C3#6392511yescharacteristic
575C2×C3D3×#21411yescharacteristic
576C2×C3#6396911yescharacteristic
577C2×C3C3×#88311yescharacteristic
578C2×C3C3×#91111yescharacteristic
579C2×C3C3×D3×SD1611yescharacteristic
580C2×C3C2×C3×#21411yescharacteristic
581C2×C3C3×#93511yescharacteristic
582C2×C3C3×#94911yescharacteristic
583D3—22no—
584C22—32no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—61no—
589C22—61no—
590C22C3×D3×#6711yescharacteristic
591C4—61no—
592C4—121no—
593C4—121no—
594C4—361no—
595C4C3×#380411yescharacteristic
596C4C3×#380511yescharacteristic
597C3—21no—
598C3—21no—
599C3—21no—
600C3—41no—
601C3#1059141511yescharacteristic, direct product factor
602C3C3×#477711yescharacteristic, complemented
603C3C3×#503411yescharacteristic, complemented
604C2—62no—
605C2—81no—
606C2C3×#6392511yescharacteristic
607C2C3×D3×#21411yescharacteristic
608C2C3×#6396911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC32×#938C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#481
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
2. C3×D3×C4⋊Dic3C2semidirect
3. C3×#63659C2semidirect
4. C32×#832C2semidirect
5. C3×#63574C2semidirect
6. C3×#63573C2semidirect
7. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
10. #10591415C3direct (N × H)
28. D3×C4⋊Dic3C2×C3semidirect
30. #63659C2×C3semidirect
35. C3×#832D3semidirect
36. C3×#832C2×C3semidirect
43. #63574C2×C3semidirect
44. #63573C2×C3semidirect
45. (C2×D3)⋊(C3⋊C8)C2×C3semidirect
139. #832C3×D3semidirect
345. C33#481semidirect
534. C32#4777semidirect
535. C32#5034semidirect
536. C32C3×#481semidirect
601. C3#10591415direct (N × H)
602. C3C3×#4777semidirect
603. C3C3×#5034semidirect

Maximal subgroups

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

subgroupindex
C32×#9382
C3×D3×C4⋊Dic32
C3×#636592
C32×#8322
C3×#635742
C3×#635732
C3×(C2×D3)⋊(C3⋊C8)2
C3×#50343
C3×#47773
#105914153