Finite Group Databasebeta

← #29718981

Subgroups — #29718981

Order 15616.

858 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455768C211yesmaximal, characteristic, complemented
2C2×#16839924C211yesmaximal, characteristic
3#21454479C211yesmaximal, characteristic, complemented
4#21454338C211yesmaximal, characteristic
5#21447182C211yesmaximal, characteristic, complemented
6#21447175C211yesmaximal, characteristic, complemented
7#21447085C211yesmaximal, characteristic, complemented
8#21447080C211yesmaximal, characteristic, complemented
9#21446731C211yesmaximal, characteristic, complemented
10#21446719C211yesmaximal, characteristic, complemented
11#21446483C211yesmaximal, characteristic, complemented
12#21446476C211yesmaximal, characteristic, complemented
13#21446285C211yesmaximal, characteristic, complemented
14#21446023C211yesmaximal, characteristic, complemented
15C61×#1785C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C22×#11758194C2211yescharacteristic
19C2×#14519756C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16840104—21no—
22#16840104—21no—
23#16840098—21no—
24#16840093—21no—
25#16839924—21no—
26#16839924—21no—
27#16839924—21no—
28#16839924—21no—
29#16839820—21no—
30#16839820—21no—
31#16839816—21no—
32#16839812—21no—
33C2×#14519712C2211yescharacteristic
34C2×#14519712C2211yescharacteristic
35#16839804C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839790C2211yescharacteristic
38#16839790C2211yescharacteristic
39#16839772C2211yescharacteristic
40#16839772C2211yescharacteristic
41#16839771C2211yescharacteristic
42#16839771C2211yescharacteristic
43#16839767C2211yescharacteristic
44#16839767C2211yescharacteristic
45#16839763—21no—
46#16839763—21no—
47#16839758—21no—
48#16839758—21no—
49C2×#14519701C2211yescharacteristic
50C2×#14519698C2211yescharacteristic
51C2×#14519695C2211yescharacteristic
52C2×#14519695C2211yescharacteristic
53C2×C22⋊#92418C2211yescharacteristic
54#16839720C2211yescharacteristic
55#16839720C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839526C2211yescharacteristic
59#16839526C2211yescharacteristic
60C22×#41×C61C2211yescharacteristic
61C2×#101×C61C2211yescharacteristic
62C2×#101×C61C2211yescharacteristic
63C2×#100×C61C2211yescharacteristic
64C2×#100×C61C2211yescharacteristic
65C61×#407C2211yescharacteristic
66C61×#407C2211yescharacteristic
67C22×Dic122C2311yescharacteristic
68C2×#11758202—22no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758202—41no—
72C2×#11758200—22no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758200—41no—
76C2×#11758194—21no—
77C2×#11758194—41no—
78C2×#11758194—41no—
79C2×#11758194C2311yescharacteristic
80C2×#11758194C2311yescharacteristic
81C22×#92418—21no—
82C22×#92418C2311yescharacteristic
83C22×#92418C2311yescharacteristic
84#14519756—21no—
85#14519756—21no—
86#14519756—21no—
87#14519756—21no—
88#14519736—21no—
89#14519736—21no—
90#14519736D411yescharacteristic
91#14519736D411yescharacteristic
92#14519736C2311yescharacteristic
93#14519736C2311yescharacteristic
94C2×C4⋊Dic61D411yescharacteristic
95C2×C4⋊Dic61D411yescharacteristic
96C2×C4⋊Dic61C2311yescharacteristic
97C2×#11758196—21no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×C4×Dic61—21no—
101#14519712—41no—
102#14519712—41no—
103#14519711—21no—
104#14519711—21no—
105#14519709—21no—
106#14519709—21no—
107#14519706—21no—
108#14519706—21no—
109#14519706—21no—
110#14519706—21no—
111#14519702—21no—
112#14519702—21no—
113#14519702—21no—
114#14519702—21no—
115#14519701—21no—
116#14519701—21no—
117#14519701—21no—
118#14519701—21no—
119#14519701—21no—
120#14519701—21no—
121#14519698—21no—
122#14519698—21no—
123#14519698—21no—
124#14519698—21no—
125#14519698—21no—
126#14519698—21no—
127#14519695—41no—
128#14519695—41no—
129C22⋊#92418D411yescharacteristic
130C22⋊#92418D411yescharacteristic
131C22⋊#92418D411yescharacteristic
132C22⋊#92418D411yescharacteristic
133#14519687—21no—
134#14519687—21no—
135C8×Dic61—21no—
136C8×Dic61—21no—
137C2×#41×C61—22no—
138C2×#41×C61—22no—
139C2×#41×C61—21no—
140C2×#41×C61—21no—
141C2×#41×C61—21no—
142C2×#41×C61—21no—
143C2×#41×C61—41no—
144C2×#41×C61—41no—
145C22×Q8×C61C2311yescharacteristic
146C22×D4×C61—21no—
147C22×D4×C61C2311yescharacteristic
148C23×C4×C61—21no—
149C23×C4×C61C2311yescharacteristic
150C2×M16×C61C2311yescharacteristic
151C2×M16×C61C2311yescharacteristic
152C2×M16×C61C2311yescharacteristic
153C2×M16×C61C2311yescharacteristic
154#101×C61—22no—
155#101×C61—41no—
156#100×C61—22no—
157#100×C61—41no—
158C22⋊C8×C61—21no—
159C22⋊C8×C61—21no—
160C22⋊C8×C61—21no—
161C22⋊C8×C61—21no—
162C24×C61—21no—
163C24×C61#4111yescharacteristic
164#41×C61—24no—
165#41×C61—24no—
166#41×C61—42no—
167#41×C61—42no—
168#41×C61—42no—
169#41×C61—42no—
170#41×C61—42no—
171#41×C61—42no—
172C2×Q8×C61—21no—
173C2×Q8×C61—21no—
174C2×Q8×C61—21no—
175C2×Q8×C61—21no—
176C2×Q8×C61—21no—
177C2×Q8×C61#4112yes—
178C2×D4×C61—21no—
179C2×D4×C61—21no—
180C2×D4×C61—21no—
181C2×D4×C61—21no—
182C2×D4×C61—21no—
183C2×D4×C61—42no—
184C2×D4×C61—42no—
185C2×D4×C61—41no—
186C2×D4×C61—41no—
187C2×D4×C61#4112yes—
188C22×C4×C61—21no—
189C22×C4×C61—21no—
190C22×C4×C61—21no—
191C22×C4×C61—21no—
192C22×C4×C61—21no—
193C22×C4×C61—42no—
194C22×C4×C61—42no—
195C22×C4×C61—41no—
196C22×C4×C61—41no—
197C22×C4×C61C2×D411yescharacteristic
198C22×C4×C61C2×D411yescharacteristic
199C22×C4×C61C2×D411yescharacteristic
200C22×C4×C61C2×D411yescharacteristic
201C22×C4×C61#4111yescharacteristic
202C22×C4×C61C2411yescharacteristic
203C22×Dic61—21no—
204C22×Dic61—21no—
205C22×Dic61—21no—
206C22×Dic61—21no—
207C2×Dic122—21no—
208C2×Dic122—21no—
209C2×Dic122—21no—
210C2×Dic122—21no—
211C2×Dic122—21no—
212C2×Dic122—21no—
213M16×C61—21no—
214M16×C61—21no—
215M16×C61—21no—
216M16×C61—21no—
217M16×C61—21no—
218M16×C61—21no—
219M16×C61—21no—
220M16×C61—21no—
221C2×C8×C61—21no—
222C2×C8×C61—21no—
223C2×C8×C61—21no—
224C2×C8×C61—21no—
225C22⋊Dic61—21no—
226C22⋊Dic61—21no—
227C22⋊Dic61—21no—
228C22⋊Dic61—21no—
229C22⋊Dic61—41no—
230C22⋊Dic61—41no—
231#11758202—42no—
232#11758202—42no—
233#11758200—42no—
234#11758200—42no—
235C4⋊Dic61—21no—
236C4⋊Dic61—21no—
237C4⋊Dic61—21no—
238C4⋊Dic61—21no—
239C4⋊Dic61C2×D411yescharacteristic
240C4⋊Dic61C2×D411yescharacteristic
241C4⋊Dic61C2×D411yescharacteristic
242C4⋊Dic61C2×D411yescharacteristic
243#11758196—41no—
244#11758196—41no—
245C4×Dic61—21no—
246C4×Dic61—21no—
247C4×Dic61—21no—
248C4×Dic61—21no—
249#11758194—21no—
250#11758194—21no—
251#11758194—21no—
252#11758194—21no—
253#11758194—42no—
254C2×#92418—21no—
255C2×#92418—21no—
256C2×#92418—21no—
257C2×#92418—21no—
258C2×#92418—21no—
259C2×#92418—21no—
260C2×#92418—42no—
261C2×#92418C2×D411yescharacteristic
262C2×#92418C2×D411yescharacteristic
263C2×#92418C2×D411yescharacteristic
264C2×#92418C2×D411yescharacteristic
265C23×C61—21no—
266C23×C61—21no—
267C23×C61—22no—
268C23×C61—41no—
269C23×C61—41no—
270C23×C61—41no—
271C23×C61—41no—
272C23×C61—81no—
273C23×C61—81no—
274C23×C61#12111yescharacteristic
275C23×C61#12111yescharacteristic
276C23×C61C2×#4111yescharacteristic
277Q8×C61—22no—
278Q8×C61—22no—
279Q8×C61—41no—
280Q8×C61—41no—
281D4×C61—22no—
282D4×C61—22no—
283D4×C61—44no—
284D4×C61—44no—
285D4×C61—41no—
286D4×C61—41no—
287C2×C4×C61—21no—
288C2×C4×C61—21no—
289C2×C4×C61—21no—
290C2×C4×C61—21no—
291C2×C4×C61—21no—
292C2×C4×C61—21no—
293C2×C4×C61—21no—
294C2×C4×C61—21no—
295C2×C4×C61—21no—
296C2×C4×C61—21no—
297C2×C4×C61—22no—
298C2×C4×C61—44no—
299C2×C4×C61—44no—
300C2×C4×C61—41no—
301C2×C4×C61—41no—
302C2×C4×C61#12111yescharacteristic
303C2×C4×C61#12111yescharacteristic
304C2×C4×C61C22×D411yescharacteristic
305C2×C4×C61C22×D411yescharacteristic
306C2×C4×C61C2×#4111yescharacteristic
307C2×C4×C61C2×#4111yescharacteristic
308C2×C4×C61H(5,𝔽2)11yescharacteristic
309C2×C4×C61H(5,𝔽2)11yescharacteristic
310C2×Dic61—21no—
311C2×Dic61—21no—
312C2×Dic61—21no—
313C2×Dic61—21no—
314C2×Dic61—42no—
315C2×Dic61—41no—
316C2×Dic61—41no—
317C2×Dic61—41no—
318C2×Dic61—41no—
319C2×Dic61—81no—
320C2×Dic61—81no—
321Dic122—41no—
322Dic122—41no—
323Dic122—41no—
324Dic122—41no—
325C8×C61—41no—
326C8×C61—41no—
327C8×C61—41no—
328C8×C61—41no—
329#92418—21no—
330#92418—21no—
331#92418—21no—
332#92418—21no—
333#92418—41no—
334#92418—41no—
335#24525—611nomaximal, Sylow
336C22×C61—21no—
337C22×C61—21no—
338C22×C61—21no—
339C22×C61—21no—
340C22×C61—21no—
341C22×C61—21no—
342C22×C61—21no—
343C22×C61—21no—
344C22×C61—42no—
345C22×C61—81no—
346C22×C61—81no—
347C22×C61—82no—
348C22×C61—82no—
349C22×C61—81no—
350C22×C61—81no—
351C22×C61C2×#12111yescharacteristic
352C22×C61#55111yescharacteristic
353C22×C61#55111yescharacteristic
354C4×C61—21no—
355C4×C61—21no—
356C4×C61—42no—
357C4×C61#52511yescharacteristic
358C4×C61#53511yescharacteristic
359C4×C61#54611yescharacteristic
360C4×C61#54611yescharacteristic
361Dic61—41no—
362Dic61—41no—
363Dic61—41no—
364Dic61—41no—
365Dic61—81no—
366Dic61—81no—
367#3095—611no—
368C2×#452—611no—
369#2924—611no—
370#2896—611no—
371#2064—611no—
372#2063—611no—
373#2047—611no—
374#2046—611no—
375#1979—611no—
376#1977—611no—
377#1931—611no—
378#1930—611no—
379#1901—611no—
380#1855—611no—
381#1785D6111yescharacteristic
382C2×C61—21no—
383C2×C61—21no—
384C2×C61—41no—
385C2×C61—41no—
386C2×C61—82no—
387C2×C61#241111yescharacteristic
388C2×C61#306711yescharacteristic
389C2×C61#306911yescharacteristic
390C22×#41C2×D6111yescharacteristic
391C22×Q16—611no—
392C22×SD16—611no—
393C22×M16—611no—
394C2×C22⋊Q8—611no—
395C2×#117—611no—
396#485—1221no—
397#485—1221no—
398#484—1221no—
399#483—1221no—
400#452—1221no—
401#452—1221no—
402#452—1221no—
403#452—1221no—
404#433—1221no—
405#433—1221no—
406#432—1221no—
407#431—1221no—
408C2×#108—611no—
409C2×#108—611no—
410#428—611no—
411#428—611no—
412#424—611no—
413#424—611no—
414#419—611no—
415#419—611no—
416#419—611no—
417#419—611no—
418#418—611no—
419#418—611no—
420#417—1221no—
421#417—1221no—
422#416—1221no—
423#416—1221no—
424C2×#103—611no—
425C2×#102—611no—
426C2×#101—611no—
427C2×#101—611no—
428C2×#101C2×D6111yescharacteristic
429C2×#101C2×D6111yescharacteristic
430C2×#100C2×D6111yescharacteristic
431C2×#100C2×D6111yescharacteristic
432#407C2×D6111yescharacteristic
433#407C2×D6111yescharacteristic
434C2×C22⋊C8—611no—
435#405—611no—
436#405—611no—
437#343—611no—
438#343—611no—
439#341—611no—
440#341—611no—
441C61#2452511yescharacteristic, complemented, Sylow
442C2×#41—22no—
443C2×#41—22no—
444C2×#41—21no—
445C2×#41—21no—
446C2×#41—21no—
447C2×#41—21no—
448C2×#41—41no—
449C2×#41—41no—
450C22×Q8—611no—
451C22×Q8C22×D6111yescharacteristic
452C22×D4—21no—
453C22×D4C22×D6111yescharacteristic
454C23×C4—21no—
455C23×C4C22×D6111yescharacteristic
456C2×Q16—1221no—
457C2×Q16—1221no—
458C2×Q16—1222no—
459C2×Q16—2441no—
460C2×SD16—1221no—
461C2×SD16—1221no—
462C2×SD16—1222no—
463C2×SD16—2441no—
464C2×M16—611no—
465C2×M16—611no—
466C2×M16—1221no—
467C2×M16—2441no—
468C2×M16—2441no—
469C2×M16C22×D6111yescharacteristic
470C2×M16C22×D6111yescharacteristic
471C2×M16C22×D6111yescharacteristic
472C2×M16C22×D6111yescharacteristic
473C22×C8—611no—
474C22×C8—611no—
475C22×C8—1221no—
476C22⋊Q8—1221no—
477C22⋊Q8—1221no—
478C22⋊Q8—1221no—
479C22⋊Q8—1221no—
480#117—611no—
481#117—611no—
482#117—611no—
483#117—611no—
484#117—1221no—
485#117—1221no—
486C2×C4⋊C4—611no—
487C2×C4⋊C4—611no—
488C2×C4⋊C4—611no—
489C2×C4⋊C4—1221no—
490C2×C22⋊C4—1221no—
491C2×C22⋊C4—1221no—
492C2×C42—1221no—
493#108—2441no—
494#108—2441no—
495#107—1221no—
496#107—1221no—
497#106—1221no—
498#106—1221no—
499C4⋊C8—1221no—
500C4⋊C8—1221no—
501C4⋊C8—1221no—
502C4⋊C8—1221no—
503#103—1221no—
504#103—1221no—
505#103—1221no—
506#103—1221no—
507#103—1221no—
508#103—1221no—
509#103—1221no—
510#103—1221no—
511#103—1221no—
512#103—1221no—
513#102—1221no—
514#102—1221no—
515#102—1221no—
516#102—1221no—
517#102—1221no—
518#102—1221no—
519#101—22no—
520#101—41no—
521#101—2441no—
522#101—2441no—
523#100—22no—
524#100—41no—
525C22⋊C8—21no—
526C22⋊C8—21no—
527C22⋊C8—21no—
528C22⋊C8—21no—
529C22⋊C8—611no—
530C22⋊C8—611no—
531C22⋊C8—611no—
532C22⋊C8—611no—
533#97—1221no—
534#97—1221no—
535C4×C8—1221no—
536C4×C8—1221no—
537C24—21no—
538C24#1175822211yescharacteristic
539#41—24no—
540#41—24no—
541#41—42no—
542#41—42no—
543#41—42no—
544#41—42no—
545#41—42no—
546#41—42no—
547C2×Q8—21no—
548C2×Q8—21no—
549C2×Q8—21no—
550C2×Q8—21no—
551C2×Q8—21no—
552C2×Q8—1221no—
553C2×Q8—1221no—
554C2×Q8—1221no—
555C2×Q8—1221no—
556C2×Q8—1221no—
557C2×Q8—1221no—
558C2×Q8#1175822412yes—
559C2×D4—21no—
560C2×D4—21no—
561C2×D4—21no—
562C2×D4—21no—
563C2×D4—21no—
564C2×D4—42no—
565C2×D4—42no—
566C2×D4—41no—
567C2×D4—41no—
568C2×D4#1175822412yes—
569C22×C4—21no—
570C22×C4—21no—
571C22×C4—21no—
572C22×C4—21no—
573C22×C4—21no—
574C22×C4—42no—
575C22×C4—42no—
576C22×C4—41no—
577C22×C4—41no—
578C22×C4—1221no—
579C22×C4—1221no—
580C22×C4—1221no—
581C22×C4—1221no—
582C22×C4#1175822211yescharacteristic
583C22×C4D4×D6111yescharacteristic
584C22×C4D4×D6111yescharacteristic
585C22×C4D4×D6111yescharacteristic
586C22×C4D4×D6111yescharacteristic
587C22×C4C23×D6111yescharacteristic
588Q16—2442no—
589Q16—2442no—
590SD16—2442no—
591SD16—2442no—
592M16—21no—
593M16—21no—
594M16—21no—
595M16—21no—
596M16—21no—
597M16—21no—
598M16—21no—
599M16—21no—
600M16—1221no—
601M16—1221no—
602M16—1221no—
603M16—1221no—
604M16—2442no—
605C2×C8—21no—
606C2×C8—21no—
607C2×C8—21no—
608C2×C8—21no—
609C2×C8—611no—
610C2×C8—611no—
611C2×C8—611no—
612C2×C8—611no—
613C2×C8—1221no—
614C2×C8—1221no—
615C2×C8—1221no—
616C2×C8—1221no—
617C2×C8—1221no—
618C2×C8—1221no—
619C2×C8—2442no—
620C4⋊C4—611no—
621C4⋊C4—611no—
622C4⋊C4—611no—
623C4⋊C4—611no—
624C4⋊C4—1221no—
625C4⋊C4—1221no—
626C4⋊C4—1221no—
627C4⋊C4—1221no—
628C4⋊C4—2441no—
629C4⋊C4—2441no—
630C22⋊C4—1221no—
631C22⋊C4—1221no—
632C22⋊C4—1221no—
633C22⋊C4—1221no—
634C22⋊C4—2441no—
635C22⋊C4—2441no—
636C42—1221no—
637C42—1221no—
638C42—1221no—
639C42—1221no—
640C23—21no—
641C23—21no—
642C23—22no—
643C23—41no—
644C23—41no—
645C23—41no—
646C23—41no—
647C23—81no—
648C23—81no—
649C23#1451975011yescharacteristic
650C23#1451975011yescharacteristic
651C23C2×#1175822211yescharacteristic
652Q8—22no—
653Q8—22no—
654Q8—41no—
655Q8—41no—
656Q8—2441no—
657Q8—2441no—
658Q8—2441no—
659Q8—2441no—
660D4—22no—
661D4—22no—
662D4—44no—
663D4—44no—
664D4—41no—
665D4—41no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—21no—
674C2×C4—21no—
675C2×C4—21no—
676C2×C4—22no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—1221no—
682C2×C4—1221no—
683C2×C4—1221no—
684C2×C4—1221no—
685C2×C4—4881no—
686C2×C4—4881no—
687C2×C4—2441no—
688C2×C4—2441no—
689C2×C4—2441no—
690C2×C4—2441no—
691C2×C4—2442no—
692C2×C4#1451975011yescharacteristic
693C2×C4#1451975011yescharacteristic
694C2×C4C2×D4×D6111yescharacteristic
695C2×C4C2×D4×D6111yescharacteristic
696C2×C4C2×#1175822411yescharacteristic
697C2×C4C2×#1175822411yescharacteristic
698C2×C4#1451982211yescharacteristic
699C2×C4#1451982211yescharacteristic
700C8—41no—
701C8—41no—
702C8—41no—
703C8—41no—
704C8—1221no—
705C8—1221no—
706C8—1221no—
707C8—1221no—
708C8—2441no—
709C8—2441no—
710C22—21no—
711C22—21no—
712C22—21no—
713C22—21no—
714C22—21no—
715C22—21no—
716C22—21no—
717C22—21no—
718C22—42no—
719C22—81no—
720C22—81no—
721C22—82no—
722C22—82no—
723C22—81no—
724C22—81no—
725C22C2×#1451975011yescharacteristic
726C22#1684043811yescharacteristic
727C22#1684043811yescharacteristic
728C4—21no—
729C4—21no—
730C4—42no—
731C4—4881no—
732C4—4881no—
733C4—2441no—
734C4—2441no—
735C4—2441no—
736C4—2441no—
737C4#1684026811yescharacteristic
738C4#1684032711yescharacteristic
739C4#1684039911yescharacteristic
740C4#1684039911yescharacteristic
741C2—21no—
742C2—21no—
743C2—41no—
744C2—41no—
745C2—82no—
746C2#2144920611yescharacteristic
747C2#2145553811yescharacteristic
748C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1785C2
socleC22×C61C2×#121
prime radicalC22×#41×C61C22
hypercenter#1785D61
hypocenterC61#24525
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
1. #21455768C2semidirect
3. #21454479C2semidirect
5. #21447182C2semidirect
6. #21447175C2semidirect
7. #21447085C2semidirect
8. #21447080C2semidirect
9. #21446731C2semidirect
10. #21446719C2semidirect
11. #21446483C2semidirect
12. #21446476C2semidirect
13. #21446285C2semidirect
14. #21446023C2semidirect
441. C61#24525semidirect

Maximal subgroups

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

subgroupindex
#214557682
C2×#168399242
#214544792
#214543382
#214471822
#214471752
#214470852
#214470802
#214467312
#214467192
#214464832
#214464762
#214462852
#214460232
C61×#17852
#2452561