Finite Group Databasebeta

← #29708965

Subgroups — #29708965

Order 15616.

888 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#430×D61C211yesmaximal, characteristic, complemented
2#21454337C211yesmaximal, characteristic
3#21454337C211yesmaximal, characteristic
4C2×#16839745C211yesmaximal, characteristic
5#21447287C211yesmaximal, characteristic, complemented
6#21447286C211yesmaximal, characteristic, complemented
7#21447283C211yesmaximal, characteristic, complemented
8#21447282C211yesmaximal, characteristic, complemented
9#21447281C211yesmaximal, characteristic, complemented
10#21447278C211yesmaximal, characteristic, complemented
11#21447277C211yesmaximal, characteristic, complemented
12#344×D61C211yesmaximal, characteristic, complemented
13#344×D61C211yesmaximal, characteristic, complemented
14H(3,ℤ/4)×D61C211yesmaximal, characteristic, complemented
15C61×#2082C211yesmaximal, characteristic, complemented
16C22×#11758226C2211yescharacteristic
17C2×M16×D61C2211yescharacteristic
18C2×M16×D61C2211yescharacteristic
19C2×M16×D61C2211yescharacteristic
20C2×M16×D61C2211yescharacteristic
21#117×D61C2211yescharacteristic
22#117×D61C2211yescharacteristic
23#16839811C2211yescharacteristic
24#16839810C2211yescharacteristic
25#16839810C2211yescharacteristic
26#16839809—22no—
27#108×D61—22no—
28#16839771C2211yescharacteristic
29#16839771C2211yescharacteristic
30#16839768C2211yescharacteristic
31#16839768C2211yescharacteristic
32#16839766C2211yescharacteristic
33#16839766C2211yescharacteristic
34#16839762—22no—
35#16839762—22no—
36#16839757—22no—
37#16839757—22no—
38#16839745—22no—
39#16839745—22no—
40C2×#14519696C2211yescharacteristic
41C2×#14519693C2211yescharacteristic
42C2×#14519693C2211yescharacteristic
43#16839534C2211yescharacteristic
44#16839534C2211yescharacteristic
45#16839533C2211yescharacteristic
46#16839533C2211yescharacteristic
47#16839532C2211yescharacteristic
48#16839532C2211yescharacteristic
49#16839519C2211yescharacteristic
50#16839518C2211yescharacteristic
51#16839518C2211yescharacteristic
52C61×#430C2211yescharacteristic
53C61×#419C2211yescharacteristic
54C61×#419C2211yescharacteristic
55C2×#101×C61C2211yescharacteristic
56C61×#344C2211yescharacteristic
57C61×#344C2211yescharacteristic
58C61×H(3,ℤ/4)C2211yescharacteristic
59C2×#11758226—21no—
60C2×#11758226—21no—
61C2×#11758226—21no—
62C2×#11758226—21no—
63C2×#11758226—21no—
64C2×#11758226—21no—
65C2×#11758226—21no—
66C2×#11758226—21no—
67C2×#11758226—21no—
68C2×#11758226—21no—
69C2×#11758226D412yes—
70C2×#11758226D412yes—
71C22×D244—21no—
72C22×D244C2311yescharacteristic
73C22×C4×D61—21no—
74C22×C4×D61C2311yescharacteristic
75C2×#11758194C2311yescharacteristic
76C2×#11758194C2311yescharacteristic
77C2×#11758194C2311yescharacteristic
78C2×#11758194C2311yescharacteristic
79C2×#11758189—21no—
80C2×#11758189—21no—
81C2×#11758189—21no—
82C2×#11758189D412yes—
83M16×D61—21no—
84M16×D61—21no—
85M16×D61—21no—
86M16×D61—21no—
87M16×D61—21no—
88M16×D61—21no—
89M16×D61—21no—
90M16×D61—21no—
91M16×D61—21no—
92M16×D61—21no—
93M16×D61—21no—
94M16×D61—21no—
95M16×D61—21no—
96M16×D61—21no—
97M16×D61—21no—
98M16×D61—21no—
99C2×C8×D61—21no—
100C2×C8×D61—21no—
101C2×C8×D61—21no—
102C2×C8×D61D412yes—
103#14519736C2311yescharacteristic
104#14519736C2311yescharacteristic
105#14519735—21no—
106#14519735—21no—
107#14519734—21no—
108#14519734—21no—
109#14519733—21no—
110#14519733—21no—
111C4⋊C4×D61—21no—
112C4⋊C4×D61—21no—
113C22⋊C4×D61—21no—
114C22⋊C4×D61—21no—
115C42×D61—21no—
116C42×D61—21no—
117#14519713—22no—
118#14519712—22no—
119#14519712—22no—
120#14519701—22no—
121#14519701—22no—
122#14519699—22no—
123#14519699—22no—
124#14519697—22no—
125#14519697—22no—
126#14519696—22no—
127#14519693—22no—
128#14519693—22no—
129C22×Q8×C61C2311yescharacteristic
130C2×M16×C61C2311yescharacteristic
131C2×M16×C61C2311yescharacteristic
132C2×M16×C61C2311yescharacteristic
133C2×M16×C61C2311yescharacteristic
134#117×C61C2311yescharacteristic
135#117×C61C2311yescharacteristic
136#108×C61—22no—
137#103×C61—22no—
138#103×C61—22no—
139#101×C61—22no—
140C23×D61—21no—
141C23×D61#4111yescharacteristic
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D412yes—
148C22×C4×C61—21no—
149C22×C4×C61C2411yescharacteristic
150C22×Dic61#4111yescharacteristic
151#11758226—22no—
152#11758226—22no—
153#11758226—22no—
154#11758226—22no—
155#11758226—22no—
156#11758226—22no—
157#11758226—22no—
158#11758226—22no—
159#11758226—41no—
160#11758226—41no—
161#11758226—41no—
162#11758226—41no—
163#11758226—41no—
164#11758226—41no—
165#11758226—41no—
166#11758226—41no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—22no—
172C2×D244—22no—
173C2×D244—21no—
174C2×D244—41no—
175C2×D244—41no—
176C2×D244—41no—
177C2×D244—41no—
178C2×D244C2×D412yes—
179C2×C4×D61—22no—
180C2×C4×D61—22no—
181C2×C4×D61—41no—
182C2×C4×D61—41no—
183C2×C4×D61—41no—
184C2×C4×D61—41no—
185C2×C4×D61—41no—
186C2×C4×D61—41no—
187C2×C4×D61C2×D411yescharacteristic
188C2×C4×D61C2×D411yescharacteristic
189C2×C4×D61C2×D411yescharacteristic
190C2×C4×D61C2×D411yescharacteristic
191C2×C4×D61#4111yescharacteristic
192C2×C4×D61#4111yescharacteristic
193C2×C4×D61#4111yescharacteristic
194C2×C4×D61#4111yescharacteristic
195C2×C4×D61#4111yescharacteristic
196C2×C4×D61#4111yescharacteristic
197C2×C4×D61#4111yescharacteristic
198C2×C4×D61#4111yescharacteristic
199M16×C61—21no—
200M16×C61—21no—
201M16×C61—21no—
202M16×C61—21no—
203M16×C61—21no—
204M16×C61—21no—
205M16×C61—21no—
206M16×C61—21no—
207C2×C8×C61—21no—
208C2×C8×C61—21no—
209C2×C8×C61—21no—
210C2×C8×C61C2×D412yes—
211C4⋊C4×C61—21no—
212C4⋊C4×C61—21no—
213C22⋊C4×C61—21no—
214C22⋊C4×C61—21no—
215C42×C61—21no—
216C42×C61—21no—
217C22⋊Dic61—21no—
218C22⋊Dic61—21no—
219#11758198—41no—
220#11758198—41no—
221C4⋊Dic61—21no—
222C4⋊Dic61—21no—
223#11758196—41no—
224#11758196—41no—
225C4×Dic61—21no—
226C4×Dic61—21no—
227C4×Dic61—41no—
228C4×Dic61—41no—
229#11758194—21no—
230#11758194—21no—
231#11758194—21no—
232#11758194—21no—
233#11758194—21no—
234#11758194—21no—
235#11758194—21no—
236#11758194—21no—
237C2×#92418—21no—
238C2×#92418—21no—
239C2×#92418—21no—
240C2×#92418C2×D412yes—
241#11758189—22no—
242#11758189—22no—
243#11758189—41no—
244#11758189—41no—
245#11758189—41no—
246#11758189—41no—
247#11758189—41no—
248#11758189—41no—
249C8×D61—22no—
250C8×D61—22no—
251C8×D61—41no—
252C8×D61—41no—
253C8×D61—41no—
254C8×D61—41no—
255C8×D61—41no—
256C8×D61—41no—
257C23×C61C2×#4111yescharacteristic
258C22×D61—22no—
259C22×D61—41no—
260C22×D61—41no—
261C22×D61—81no—
262C22×D61—82no—
263C22×D61#12311yescharacteristic
264C22×D61#12311yescharacteristic
265C22×D61#12411yescharacteristic
266C22×D61#12411yescharacteristic
267C22×D61#12611yescharacteristic
268C22×D61#12611yescharacteristic
269Q8×C61—22no—
270Q8×C61—22no—
271Q8×C61—41no—
272Q8×C61—41no—
273C2×C4×C61—22no—
274C2×C4×C61—41no—
275C2×C4×C61—41no—
276C2×C4×C61—41no—
277C2×C4×C61—41no—
278C2×C4×C61C22×D411yescharacteristic
279C2×C4×C61C22×D411yescharacteristic
280C2×C4×C61C2×#4111yescharacteristic
281C2×C4×C61C2×#4111yescharacteristic
282C2×C4×C61C2×#4111yescharacteristic
283C2×C4×C61C2×#4111yescharacteristic
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61#12311yescharacteristic
287C2×Dic61#12311yescharacteristic
288C2×Dic61#12411yescharacteristic
289C2×Dic61#12411yescharacteristic
290C2×Dic61#12611yescharacteristic
291C2×Dic61#12611yescharacteristic
292D244—22no—
293D244—22no—
294D244—42no—
295D244—42no—
296D244—42no—
297D244—42no—
298D244—41no—
299D244—41no—
300C4×D61—42no—
301C4×D61—42no—
302C4×D61—42no—
303C4×D61—42no—
304C4×D61—82no—
305C4×D61—82no—
306C4×D61#12111yescharacteristic
307C4×D61#12111yescharacteristic
308C4×D61#12111yescharacteristic
309C4×D61#12111yescharacteristic
310C4×D61#12311yescharacteristic
311C4×D61#12311yescharacteristic
312C4×D61#12311yescharacteristic
313C4×D61#12311yescharacteristic
314C4×D61#12311yescharacteristic
315C4×D61#12311yescharacteristic
316C4×D61#12311yescharacteristic
317C4×D61#12311yescharacteristic
318C4×D61#12611yescharacteristic
319C4×D61#12611yescharacteristic
320C4×D61#12611yescharacteristic
321C4×D61#12611yescharacteristic
322C8×C61—22no—
323C8×C61—41no—
324C8×C61—41no—
325C8×C61—41no—
326#92418—22no—
327#92418—41no—
328#92418—41no—
329#92418—41no—
330#23652—611nomaximal, Sylow
331C22×C61—41no—
332C22×C61C2×#12311yescharacteristic
333C22×C61C2×#12411yescharacteristic
334C22×C61C2×#12611yescharacteristic
335C2×D61—41no—
336C2×D61—41no—
337C2×D61—41no—
338C2×D61—41no—
339C2×D61—41no—
340C2×D61—41no—
341C2×D61—42no—
342C2×D61—82no—
343C2×D61—82no—
344C2×D61—82no—
345C2×D61#39711yescharacteristic
346C2×D61#39711yescharacteristic
347C2×D61#39711yescharacteristic
348C2×D61#39711yescharacteristic
349C4×C61—42no—
350C4×C61—81no—
351C4×C61—81no—
352C4×C61C2×#12111yescharacteristic
353C4×C61C2×#12311yescharacteristic
354C4×C61C2×#12311yescharacteristic
355C4×C61C2×#12611yescharacteristic
356Dic61—81no—
357Dic61—81no—
358Dic61#39711yescharacteristic
359Dic61#39711yescharacteristic
360Dic61#39711yescharacteristic
361Dic61#39711yescharacteristic
362C2×#430—611no—
363#2896—611no—
364#2896—611no—
365C2×#413—611no—
366#2082—611no—
367#2082—611no—
368#2082—611no—
369#2082D6111yescharacteristic, complemented
370#2081—611no—
371#2081—611no—
372#2081—611no—
373#2081—611no—
374C2×#344—611no—
375C2×#344—611no—
376C2×H(3,ℤ/4)—611no—
377C2×C61—21no—
378C2×C61—21no—
379C2×C61—21no—
380C2×C61C2×#39711yescharacteristic
381D61—21no—
382D61—21no—
383D61—21no—
384D61—21no—
385D61—82no—
386C22×#41—611no—
387C22×M16—611no—
388C22×M16—611no—
389C22×M16—611no—
390C22×M16—611no—
391C2×#117—611no—
392C2×#117—611no—
393#430—611no—
394#430—611no—
395#430—611no—
396#430—1222no—
397#430C2×D6111yescharacteristic
398C2×#108—1222no—
399#419—611no—
400#419—611no—
401#419C2×D6111yescharacteristic
402#419C2×D6111yescharacteristic
403#418—611no—
404#418—611no—
405#418—611no—
406#418—611no—
407#417—1222no—
408#417—1222no—
409#416—1222no—
410#416—1222no—
411#413—1222no—
412#413—1222no—
413C2×#101—611no—
414C2×#101C2×D6111yescharacteristic
415C2×#100—611no—
416C2×#100—611no—
417#344—611no—
418#344—611no—
419#344—611no—
420#344—611no—
421#344—611no—
422#344—611no—
423#344C2×D6111yescharacteristic
424#344C2×D6111yescharacteristic
425H(3,ℤ/4)—611no—
426H(3,ℤ/4)—611no—
427H(3,ℤ/4)—611no—
428H(3,ℤ/4)C2×D6111yescharacteristic
429C61#2365211yescharacteristic, complemented, Sylow
430C2×#41—612no—
431C2×#41—612no—
432C2×#41—1221no—
433C2×#41—1221no—
434C2×#41—1221no—
435C2×#41—1221no—
436C2×#41—1221no—
437C2×#41—1221no—
438C2×#41—1221no—
439C2×#41—1221no—
440C2×#41—1221no—
441C2×#41—1221no—
442C22×Q8C22×D6111yescharacteristic
443C22×D4—611no—
444C22×D4—1221no—
445C23×C4—611no—
446C23×C4—1221no—
447C2×M16—611no—
448C2×M16—612no—
449C2×M16—611no—
450C2×M16—611no—
451C2×M16—611no—
452C2×M16—1221no—
453C2×M16—1221no—
454C2×M16—1221no—
455C2×M16—1221no—
456C2×M16—1221no—
457C2×M16—1221no—
458C2×M16—1221no—
459C2×M16—1221no—
460C2×M16—1221no—
461C2×M16—1221no—
462C2×M16—1221no—
463C2×M16—1221no—
464C2×M16—1221no—
465C2×M16—1221no—
466C2×M16—1221no—
467C2×M16—1221no—
468C2×M16—1221no—
469C2×M16—1221no—
470C2×M16—1221no—
471C2×M16C22×D6111yescharacteristic
472C2×M16C22×D6111yescharacteristic
473C2×M16C22×D6111yescharacteristic
474C2×M16C22×D6111yescharacteristic
475C22×C8—612no—
476C22×C8—1221no—
477C22×C8—1221no—
478C22×C8—1221no—
479#117—611no—
480#117—611no—
481#117—1221no—
482#117—1221no—
483#117—1221no—
484#117—1221no—
485#117—1221no—
486#117—1221no—
487#117C22×D6111yescharacteristic
488#117C22×D6111yescharacteristic
489C2×C4⋊C4—1221no—
490C2×C4⋊C4—1221no—
491C2×C22⋊C4—1221no—
492C2×C22⋊C4—1221no—
493C2×C42—1221no—
494C2×C42—1221no—
495#108—22no—
496#108—1222no—
497#108—1222no—
498#108—1222no—
499#103—22no—
500#103—22no—
501#103—1222no—
502#103—1222no—
503#102—1222no—
504#102—1222no—
505#102—1222no—
506#102—1222no—
507#101—22no—
508#101—1222no—
509#100—1222no—
510#100—1222no—
511C24—611no—
512C24—1221no—
513#41—1222no—
514#41—1222no—
515#41—1222no—
516#41—1222no—
517#41—1222no—
518#41—1222no—
519#41—1222no—
520#41—1222no—
521#41—2441no—
522#41—2441no—
523#41—2441no—
524#41—2441no—
525#41—2441no—
526#41—2441no—
527#41—2441no—
528#41—2441no—
529C2×Q8—21no—
530C2×Q8—21no—
531C2×Q8—21no—
532C2×Q8—21no—
533C2×Q8—21no—
534C2×Q8D4×D6112yes—
535C2×D4—612no—
536C2×D4—1222no—
537C2×D4—1222no—
538C2×D4—1221no—
539C2×D4—1221no—
540C2×D4—1221no—
541C2×D4—1221no—
542C2×D4—1221no—
543C2×D4—2441no—
544C2×D4—2441no—
545C2×D4—2441no—
546C2×D4—2441no—
547C22×C4—21no—
548C22×C4—611no—
549C22×C4—611no—
550C22×C4—611no—
551C22×C4—611no—
552C22×C4—611no—
553C22×C4—611no—
554C22×C4—611no—
555C22×C4—611no—
556C22×C4—611no—
557C22×C4—611no—
558C22×C4—611no—
559C22×C4—611no—
560C22×C4—611no—
561C22×C4—1222no—
562C22×C4—1222no—
563C22×C4—2441no—
564C22×C4—2441no—
565C22×C4—2441no—
566C22×C4—2441no—
567C22×C4—2441no—
568C22×C4—2441no—
569C22×C4C23×D6111yescharacteristic
570M16—21no—
571M16—21no—
572M16—21no—
573M16—21no—
574M16—21no—
575M16—21no—
576M16—21no—
577M16—21no—
578M16—1222no—
579M16—1222no—
580M16—1221no—
581M16—1221no—
582M16—1221no—
583M16—1221no—
584M16—1221no—
585M16—1221no—
586M16—1221no—
587M16—1221no—
588M16—2441no—
589M16—2441no—
590M16—2441no—
591M16—2441no—
592M16—2441no—
593M16—2441no—
594C2×C8—21no—
595C2×C8—21no—
596C2×C8—21no—
597C2×C8—612no—
598C2×C8—1222no—
599C2×C8—1222no—
600C2×C8—1221no—
601C2×C8—1221no—
602C2×C8—1221no—
603C2×C8—2441no—
604C2×C8—2441no—
605C2×C8—2441no—
606C2×C8—2441no—
607C2×C8—2441no—
608C2×C8—2441no—
609C2×C8D4×D6112yes—
610C4⋊C4—21no—
611C4⋊C4—21no—
612C4⋊C4—1221no—
613C4⋊C4—1221no—
614C4⋊C4—2441no—
615C4⋊C4—2441no—
616C22⋊C4—21no—
617C22⋊C4—21no—
618C22⋊C4—1221no—
619C22⋊C4—1221no—
620C22⋊C4—2441no—
621C22⋊C4—2441no—
622C42—21no—
623C42—21no—
624C42—1221no—
625C42—1221no—
626C42—2441no—
627C42—2441no—
628C23—611no—
629C23—611no—
630C23—611no—
631C23—611no—
632C23—611no—
633C23—611no—
634C23—1222no—
635C23—4882no—
636C23—4881no—
637C23—2441no—
638C23—2441no—
639C23#41×D6111yescharacteristic
640Q8—22no—
641Q8—22no—
642Q8—41no—
643Q8—41no—
644D4—1222no—
645D4—1222no—
646D4—2442no—
647D4—2442no—
648D4—2442no—
649D4—2442no—
650D4—2441no—
651D4—2441no—
652C2×C4—22no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—611no—
658C2×C4—611no—
659C2×C4—611no—
660C2×C4—611no—
661C2×C4—611no—
662C2×C4—611no—
663C2×C4—611no—
664C2×C4—611no—
665C2×C4—611no—
666C2×C4—611no—
667C2×C4—611no—
668C2×C4—611no—
669C2×C4—611no—
670C2×C4—611no—
671C2×C4—611no—
672C2×C4—611no—
673C2×C4—611no—
674C2×C4—611no—
675C2×C4—611no—
676C2×C4—611no—
677C2×C4—611no—
678C2×C4—611no—
679C2×C4—4882no—
680C2×C4—4882no—
681C2×C4—2442no—
682C2×C4—2442no—
683C2×C4—2442no—
684C2×C4—2442no—
685C2×C4—2441no—
686C2×C4—2441no—
687C2×C4C2×D4×D6111yescharacteristic
688C2×C4C2×D4×D6111yescharacteristic
689C2×C4#41×D6111yescharacteristic
690C2×C4#41×D6111yescharacteristic
691C2×C4#41×D6111yescharacteristic
692C2×C4#41×D6111yescharacteristic
693C8—22no—
694C8—41no—
695C8—41no—
696C8—41no—
697C8—1222no—
698C8—2441no—
699C8—2441no—
700C8—2441no—
701C22—41no—
702C22—611no—
703C22—611no—
704C22—611no—
705C22—611no—
706C22—4882no—
707C22—4882no—
708C22—4882no—
709C22—2441no—
710C22—2441no—
711C22—2441no—
712C22—2441no—
713C22—2441no—
714C22—2441no—
715C22—2442no—
716C22#123×D6111yescharacteristic
717C22#124×D6111yescharacteristic
718C22#126×D6111yescharacteristic
719C4—42no—
720C4—81no—
721C4—81no—
722C4—611no—
723C4—611no—
724C4—611no—
725C4—611no—
726C4—4881no—
727C4—4881no—
728C4#121×D6111yescharacteristic
729C4#123×D6111yescharacteristic
730C4#123×D6111yescharacteristic
731C4#126×D6111yescharacteristic
732C2—21no—
733C2—21no—
734C2—21no—
735C2—1221no—
736C2—1221no—
737C2—1221no—
738C2—1221no—
739C2—4882no—
740C2#397×D6111yescharacteristic
subgroupG/H
centerC2#397×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2082C2
socleC2×C61C2×#397
prime radicalC22×#11758226C22
hypercenter#2082D61
hypocenterC61#23652
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2082

Complements

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

NH ≅ G/Ntype
1. #430×D61C2semidirect
5. #21447287C2semidirect
6. #21447286C2semidirect
7. #21447283C2semidirect
8. #21447282C2semidirect
9. #21447281C2semidirect
10. #21447278C2semidirect
11. #21447277C2semidirect
12. #344×D61C2semidirect
13. #344×D61C2semidirect
14. H(3,ℤ/4)×D61C2semidirect
15. C61×#2082C2semidirect
369. #2082D61semidirect
429. C61#23652semidirect

Maximal subgroups

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

subgroupindex
#430×D612
#214543372
C2×#168397452
#214472872
#214472862
#214472832
#214472822
#214472812
#214472782
#214472772
#344×D612
H(3,ℤ/4)×D612
C61×#20822
#2365261