Finite Group Databasebeta

← #29737585

Subgroups — #29737585

Order 15616.

842 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455685C211yesmaximal, characteristic, complemented
2#21455674C211yesmaximal, characteristic, complemented
3#21454352C211yesmaximal, characteristic
4#21454352C211yesmaximal, characteristic
5#21447367C211yesmaximal, characteristic, complemented
6#21447182C211yesmaximal, characteristic, complemented
7#21447182C211yesmaximal, characteristic, complemented
8#21446980C211yesmaximal, characteristic, complemented
9#21446092C211yesmaximal, characteristic, complemented
10#21446092C211yesmaximal, characteristic, complemented
11#21446054C211yesmaximal, characteristic, complemented
12#21446054C211yesmaximal, characteristic, complemented
13#21445881C211yesmaximal, characteristic, complemented
14#21445876C211yesmaximal, characteristic, complemented
15C61×#1793C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18#16840393C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16840245C2211yescharacteristic
21#16840071—21no—
22#16840067—21no—
23#16840064—21no—
24#16840060—21no—
25#16840055—21no—
26#16840050—21no—
27#16840045—21no—
28#16840040—21no—
29C2×#14519712C2211yescharacteristic
30C2×#14519712C2211yescharacteristic
31#16839804C2211yescharacteristic
32#16839804C2211yescharacteristic
33C2×#14519710C2211yescharacteristic
34C2×#14519708C2211yescharacteristic
35#16839790C2211yescharacteristic
36#16839790C2211yescharacteristic
37#16839771—21no—
38#16839771—21no—
39#16839771C2211yescharacteristic
40#16839771C2211yescharacteristic
41#16839763—21no—
42#16839763—21no—
43#16839763C2211yescharacteristic
44#16839763C2211yescharacteristic
45#16839758—21no—
46#16839758—21no—
47#16839758C2211yescharacteristic
48#16839758C2211yescharacteristic
49C2×#14519701—21no—
50C2×#14519701—21no—
51C2×#14519701C2211yescharacteristic
52C2×#14519701C2211yescharacteristic
53#16839628C2211yescharacteristic
54#16839531C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839522C2211yescharacteristic
58#16839522C2211yescharacteristic
59#16839517C2211yescharacteristic
60C2×2−1+4×C61C2211yescharacteristic
61C2×#104×C61C2211yescharacteristic
62C2×#104×C61C2211yescharacteristic
63C61×#417C2211yescharacteristic
64C61×#417C2211yescharacteristic
65C2×#101×C61C2211yescharacteristic
66C61×#391C2211yescharacteristic
67#14519784—22no—
68#14519784—22no—
69#14519784—41no—
70#14519784—41no—
71C2×#11758194—21no—
72C2×#11758194—21no—
73C2×#11758194C2311yescharacteristic
74C2×#11758194C2311yescharacteristic
75C22×#92418—21no—
76C22×#92418—21no—
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79#14519776—21no—
80#14519776—21no—
81#14519768—21no—
82#14519768—21no—
83#14519756—41no—
84Q8×Dic61—21no—
85Q8×Dic61—21no—
86D4×Dic61—21no—
87D4×Dic61—21no—
88D4×Dic61—41no—
89D4×Dic61—41no—
90#14519736—21no—
91#14519736—21no—
92#14519736C2311yescharacteristic
93#14519736C2311yescharacteristic
94C2×C4⋊Dic61—21no—
95C2×C4⋊Dic61—21no—
96C2×C4⋊Dic61C2311yescharacteristic
97C2×C4⋊Dic61C2311yescharacteristic
98#14519712—41no—
99#14519712—41no—
100#14519711—21no—
101#14519711—21no—
102#14519710—21no—
103#14519710—21no—
104#14519709—21no—
105#14519709—21no—
106#14519708—21no—
107#14519708—21no—
108#14519706—21no—
109#14519706—21no—
110#14519706—21no—
111#14519706—21no—
112#14519701—21no—
113#14519701—21no—
114#14519701—21no—
115#14519701—21no—
116#14519701—21no—
117#14519701—21no—
118#14519701—21no—
119#14519701—21no—
120#14519701—21no—
121#14519701—21no—
122#14519701D411yescharacteristic
123#14519701D411yescharacteristic
124#14519701D411yescharacteristic
125#14519701D411yescharacteristic
126#14519698—21no—
127#14519698—21no—
128#14519698D411yescharacteristic
129#14519698D411yescharacteristic
130#14519698D411yescharacteristic
131#14519698D411yescharacteristic
132#14519684—41no—
1332−1+4×C61—22no—
1342−1+4×C61—22no—
1352−1+4×C61—41no—
1362−1+4×C61Q814yescomplemented
137C2×#41×C61—21no—
138C2×#41×C61—21no—
139C2×#41×C61—41no—
140C2×#41×C61C2311yescharacteristic
141C2×#41×C61C2311yescharacteristic
142C22×Q8×C61—21no—
143C22×Q8×C61—21no—
144C22×Q8×C61C2311yescharacteristic
145C2×M16×C61C2311yescharacteristic
146C2×M16×C61C2311yescharacteristic
147C2×C4⋊C4×C61C2311yescharacteristic
148C2×C42×C61C2311yescharacteristic
149#104×C61—22no—
150#104×C61—22no—
151#103×C61—21no—
152#103×C61—21no—
153#102×C61—21no—
154#102×C61—21no—
155#101×C61—41no—
156#95×C61—41no—
157#41×C61—21no—
158#41×C61—21no—
159#41×C61—21no—
160#41×C61—21no—
161#41×C61—24no—
162#41×C61—24no—
163#41×C61—44no—
164#41×C61—42no—
165#41×C61—42no—
166#41×C61—44no—
167#41×C61C2×Q812yes—
168#41×C61C2×Q812yes—
169#41×C61#4112yes—
170#41×C61#4112yes—
171C2×Q8×C61—21no—
172C2×Q8×C61—24no—
173C2×Q8×C61—24no—
174C2×Q8×C61—21no—
175C2×Q8×C61—21no—
176C2×Q8×C61—21no—
177C2×Q8×C61—21no—
178C2×Q8×C61—41no—
179C2×Q8×C61—42no—
180C2×Q8×C61—42no—
181C2×Q8×C61C2×D411yescharacteristic
182C2×Q8×C61C2×D411yescharacteristic
183C2×Q8×C61C2×Q812yes—
184C2×D4×C61—21no—
185C2×D4×C61—21no—
186C2×D4×C61—41no—
187C2×D4×C61C2×D411yescharacteristic
188C2×D4×C61C2×D411yescharacteristic
189C22×C4×C61—21no—
190C22×C4×C61—21no—
191C22×C4×C61—21no—
192C22×C4×C61—21no—
193C22×C4×C61—21no—
194C22×C4×C61—41no—
195C22×C4×C61—41no—
196C22×C4×C61C2411yescharacteristic
197C22×Dic61—21no—
198C22×Dic61—21no—
199C22×Dic61—41no—
200C22×Dic61—41no—
201C2×Dic122—41no—
202M16×C61—21no—
203M16×C61—21no—
204M16×C61—21no—
205M16×C61—21no—
206C2×C8×C61—21no—
207C2×C8×C61—21no—
208C4⋊C4×C61—21no—
209C4⋊C4×C61—21no—
210C42×C61—21no—
211C42×C61—21no—
212C22⋊Dic61—21no—
213C22⋊Dic61—21no—
214C22⋊Dic61—41no—
215C22⋊Dic61—41no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220C4⋊Dic61—41no—
221C4⋊Dic61—41no—
222C4⋊Dic61—41no—
223C4⋊Dic61C2×D411yescharacteristic
224C4⋊Dic61C2×D411yescharacteristic
225C4⋊Dic61C2×D411yescharacteristic
226C4⋊Dic61C2×D411yescharacteristic
227#11758196—41no—
228#11758196—41no—
229C4×Dic61—21no—
230C4×Dic61—21no—
231C4×Dic61—41no—
232C4×Dic61—41no—
233#11758194—21no—
234#11758194—21no—
235#11758194—21no—
236#11758194—21no—
237#11758194—42no—
238#11758194—42no—
239C2×#92418—21no—
240C2×#92418—21no—
241C2×#92418—21no—
242C2×#92418—21no—
243C2×#92418—21no—
244C2×#92418—21no—
245C2×#92418—42no—
246C2×#92418—42no—
247C2×#92418C2×D411yescharacteristic
248C2×#92418C2×D411yescharacteristic
249C2×#92418C2×D411yescharacteristic
250C2×#92418C2×D411yescharacteristic
251C23×C61—21no—
252C23×C61—21no—
253C23×C61H(5,𝔽2)11yescharacteristic
254Q8×C61—21no—
255Q8×C61—21no—
256Q8×C61—22no—
257Q8×C61—22no—
258Q8×C61—44no—
259Q8×C61—41no—
260Q8×C61—41no—
261Q8×C61C22⋊Q812yes—
262Q8×C61C22⋊Q812yes—
263D4×C61—22no—
264D4×C61—22no—
265D4×C61—22no—
266D4×C61—22no—
267D4×C61—21no—
268D4×C61—21no—
269D4×C61—44no—
270D4×C61C22⋊Q812yes—
271D4×C61C22⋊Q812yes—
272C2×C4×C61—21no—
273C2×C4×C61—21no—
274C2×C4×C61—22no—
275C2×C4×C61—22no—
276C2×C4×C61—22no—
277C2×C4×C61—22no—
278C2×C4×C61—22no—
279C2×C4×C61—44no—
280C2×C4×C61—44no—
281C2×C4×C61—41no—
282C2×C4×C61—41no—
283C2×C4×C61—41no—
284C2×C4×C61—41no—
285C2×C4×C61—41no—
286C2×C4×C61—41no—
287C2×C4×C61—81no—
288C2×C4×C61C22×D411yescharacteristic
289C2×C4×C61C22×D411yescharacteristic
290C2×C4×C61C22×Q811yescharacteristic
291C2×C4×C61C2×#4111yescharacteristic
292C2×C4×C61C2×#4111yescharacteristic
293C2×C4×C61H(5,𝔽2)11yescharacteristic
294C2×Dic61—21no—
295C2×Dic61—21no—
296C2×Dic61—21no—
297C2×Dic61—21no—
298C2×Dic61—41no—
299C2×Dic61—41no—
300C2×Dic61—81no—
301C2×Dic61—81no—
302C2×Dic61—161no—
303C2×Dic61—161no—
304Dic122—161no—
305C8×C61—41no—
306C8×C61—41no—
307#92418—21no—
308#92418—21no—
309#92418—21no—
310#92418—21no—
311#92418—41no—
312#92418—41no—
313#25985—611nomaximal, Sylow
314C22×C61—21no—
315C22×C61—21no—
316C22×C61—22no—
317C22×C61—22no—
318C22×C61—41no—
319C22×C61—41no—
320C22×C61—41no—
321C22×C61—41no—
322C22×C61#54311yescharacteristic
323C22×C61D4211yescharacteristic
324C22×C61#55111yescharacteristic
325C4×C61—22no—
326C4×C61—22no—
327C4×C61—42no—
328C4×C61—81no—
329C4×C61—81no—
330C4×C61C2×C22⋊Q811yescharacteristic
331C4×C61C2×C22⋊Q811yescharacteristic
332C4×C61#54611yescharacteristic
333C4×C61#54611yescharacteristic
334Dic61—41no—
335Dic61—41no—
336Dic61—41no—
337Dic61—41no—
338Dic61—81no—
339Dic61—81no—
340#3084—611no—
341#3083—611no—
342#2898—611no—
343#2898—611no—
344#2097—611no—
345#2064—611no—
346#2064—611no—
347#2028—611no—
348#1866—611no—
349#1866—611no—
350#1860—611no—
351#1860—611no—
352#1816—611no—
353#1815—611no—
354#1793D6111yescharacteristic
355C2×C61—21no—
356C2×C61—21no—
357C2×C61—42no—
358C2×C61—42no—
359C2×C61#243711yescharacteristic
360C2×C61#328911yescharacteristic
361C2×C61#329011yescharacteristic
362C2×2−1+4C2×D6111yescharacteristic
363C2×#131—611no—
364C2×#131—611no—
365#544—611no—
366#520—611no—
367#520—611no—
368#479—1221no—
369#479—1221no—
370#478—1221no—
371#478—1221no—
372#477—1221no—
373#476—1221no—
374#475—1221no—
375#474—1221no—
376C2×#108—611no—
377C2×#108—611no—
378#428—611no—
379#428—611no—
380C2×#107—611no—
381C2×#106—611no—
382#424—611no—
383#424—611no—
384C2×#104C2×D6111yescharacteristic
385C2×#104C2×D6111yescharacteristic
386#419—611no—
387#419—611no—
388#419—1221no—
389#419—1221no—
390#417—611no—
391#417—611no—
392#417—1221no—
393#417—1221no—
394#417C2×D6111yescharacteristic
395#417C2×D6111yescharacteristic
396#416—611no—
397#416—611no—
398#416—1221no—
399#416—1221no—
400C2×#103—611no—
401C2×#103—611no—
402C2×#103—1221no—
403C2×#103—1221no—
404C2×#101C2×D6111yescharacteristic
405#391C2×D6111yescharacteristic
406#382—611no—
407#343—611no—
408#343—611no—
409#343—611no—
410#339—611no—
411#339—611no—
412#336—611no—
413C61#2598511yescharacteristic, complemented, Sylow
4142−1+4—22no—
4152−1+4—22no—
4162−1+4—41no—
4172−1+4Dic12214yescomplemented
418C2×#41—21no—
419C2×#41—21no—
420C2×#41—41no—
421C2×#41C22×D6111yescharacteristic
422C2×#41C22×D6111yescharacteristic
423C22×Q8—21no—
424C22×Q8—21no—
425C22×Q8C22×D6111yescharacteristic
426#131—1222no—
427#131—1222no—
428#131—2441no—
429#131—2441no—
430C2×M16—611no—
431C2×M16—611no—
432C2×M16—1221no—
433C2×M16—1221no—
434C2×M16C22×D6111yescharacteristic
435C2×M16C22×D6111yescharacteristic
436C22×C8—611no—
437C22×C8—611no—
438C22×C8—1221no—
439C22×C8—1221no—
440C4⋊Q8—1221no—
441C4⋊Q8—1221no—
442#125—1221no—
443#125—1221no—
444C22⋊Q8—2441no—
445C4×Q8—1221no—
446C4×Q8—1221no—
447C4×D4—1221no—
448C4×D4—1221no—
449C4×D4—2441no—
450C4×D4—2441no—
451#117—611no—
452#117—611no—
453#117—1221no—
454#117—1221no—
455C2×C4⋊C4—611no—
456C2×C4⋊C4—611no—
457C2×C4⋊C4—1221no—
458C2×C4⋊C4—1221no—
459C2×C4⋊C4C22×D6111yescharacteristic
460C2×C42C22×D6111yescharacteristic
461#108—2441no—
462#108—2441no—
463#107—1221no—
464#107—1221no—
465#107—1221no—
466#107—1221no—
467#106—1221no—
468#106—1221no—
469#106—1221no—
470#106—1221no—
471C4⋊C8—1221no—
472C4⋊C8—1221no—
473C4⋊C8—1221no—
474C4⋊C8—1221no—
475#104—22no—
476#104—22no—
477#103—21no—
478#103—21no—
479#103—611no—
480#103—611no—
481#103—611no—
482#103—611no—
483#103—1221no—
484#103—1221no—
485#103—1221no—
486#103—1221no—
487#103—1221no—
488#103—1221no—
489#103—1221no—
490#103—1221no—
491#103—1221no—
492#103—1221no—
493#102—21no—
494#102—21no—
495#102—611no—
496#102—611no—
497#102—611no—
498#102—611no—
499#102—1221no—
500#102—1221no—
501#101—41no—
502#95—41no—
503#95—2441no—
504#41—21no—
505#41—21no—
506#41—21no—
507#41—21no—
508#41—24no—
509#41—24no—
510#41—44no—
511#41—42no—
512#41—42no—
513#41—44no—
514#41C2×Dic12212yes—
515#41C2×Dic12212yes—
516#41#1175822412yes—
517#41#1175822412yes—
518C2×Q8—21no—
519C2×Q8—24no—
520C2×Q8—24no—
521C2×Q8—21no—
522C2×Q8—21no—
523C2×Q8—21no—
524C2×Q8—21no—
525C2×Q8—41no—
526C2×Q8—42no—
527C2×Q8—42no—
528C2×Q8—2441no—
529C2×Q8C2×Dic12212yes—
530C2×Q8D4×D6111yescharacteristic
531C2×Q8D4×D6111yescharacteristic
532C2×D4—21no—
533C2×D4—21no—
534C2×D4—41no—
535C2×D4D4×D6111yescharacteristic
536C2×D4D4×D6111yescharacteristic
537C22×C4—21no—
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—21no—
541C22×C4—21no—
542C22×C4—41no—
543C22×C4—41no—
544C22×C4—1221no—
545C22×C4—1221no—
546C22×C4—2441no—
547C22×C4—2441no—
548C22×C4C23×D6111yescharacteristic
549M16—21no—
550M16—21no—
551M16—21no—
552M16—21no—
553M16—1221no—
554M16—1221no—
555M16—1221no—
556M16—1221no—
557M16—2442no—
558M16—2442no—
559C2×C8—21no—
560C2×C8—21no—
561C2×C8—611no—
562C2×C8—611no—
563C2×C8—611no—
564C2×C8—611no—
565C2×C8—1221no—
566C2×C8—1221no—
567C2×C8—1221no—
568C2×C8—1221no—
569C2×C8—1221no—
570C2×C8—1221no—
571C2×C8—2442no—
572C2×C8—2442no—
573C4⋊C4—21no—
574C4⋊C4—21no—
575C4⋊C4—611no—
576C4⋊C4—611no—
577C4⋊C4—611no—
578C4⋊C4—611no—
579C4⋊C4—1221no—
580C4⋊C4—1221no—
581C4⋊C4—1221no—
582C4⋊C4—1221no—
583C4⋊C4—2441no—
584C4⋊C4—2441no—
585C4⋊C4—2441no—
586C4⋊C4—2441no—
587C4⋊C4—2441no—
588C22⋊C4—1221no—
589C22⋊C4—1221no—
590C22⋊C4—2441no—
591C22⋊C4—2441no—
592C42—21no—
593C42—21no—
594C42—1221no—
595C42—1221no—
596C42—2441no—
597C42—2441no—
598C23—21no—
599C23—21no—
600C23#1451982211yescharacteristic
601Q8—21no—
602Q8—21no—
603Q8—22no—
604Q8—22no—
605Q8—44no—
606Q8—41no—
607Q8—41no—
608Q8—9761no—
609Q8#1451975412yes—
610Q8#1451975412yes—
611D4—22no—
612D4—22no—
613D4—22no—
614D4—22no—
615D4—21no—
616D4—21no—
617D4—44no—
618D4#1451975412yes—
619D4#1451975412yes—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—81no—
636C2×C4—1221no—
637C2×C4—1221no—
638C2×C4—1221no—
639C2×C4—1221no—
640C2×C4—9761no—
641C2×C4—9761no—
642C2×C4—4881no—
643C2×C4—4881no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4C2×D4×D6111yescharacteristic
647C2×C4C2×D4×D6111yescharacteristic
648C2×C4C22×Dic12211yescharacteristic
649C2×C4C2×#1175822411yescharacteristic
650C2×C4C2×#1175822411yescharacteristic
651C2×C4#1451982211yescharacteristic
652C8—41no—
653C8—41no—
654C8—1221no—
655C8—1221no—
656C8—1221no—
657C8—1221no—
658C8—2441no—
659C8—2441no—
660C22—21no—
661C22—21no—
662C22—22no—
663C22—22no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22#1684038811yescharacteristic
669C22#1684039411yescharacteristic
670C22#1684044311yescharacteristic
671C4—22no—
672C4—22no—
673C4—42no—
674C4—81no—
675C4—81no—
676C4—4881no—
677C4—4881no—
678C4—2441no—
679C4—2441no—
680C4—2441no—
681C4—2441no—
682C4C2×#1451975411yescharacteristic
683C4C2×#1451975411yescharacteristic
684C4#1684039911yescharacteristic
685C4#1684039911yescharacteristic
686C2—21no—
687C2—21no—
688C2—42no—
689C2—42no—
690C2#2144948511yescharacteristic
691C2#2145758911yescharacteristic
692C2#2145759811yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1793C2
socleC22×C61D42
prime radicalC2×2−1+4×C61C22
hypercenter#1793D61
hypocenterC61#25985
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1793

Complements

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

NH ≅ G/Ntype
1. #21455685C2semidirect
2. #21455674C2semidirect
5. #21447367C2semidirect
6. #21447182C2semidirect
7. #21447182C2semidirect
8. #21446980C2semidirect
9. #21446092C2semidirect
10. #21446092C2semidirect
11. #21446054C2semidirect
12. #21446054C2semidirect
13. #21445881C2semidirect
14. #21445876C2semidirect
136. 2−1+4×C61Q8semidirect
413. C61#25985semidirect
417. 2−1+4Dic122semidirect

Maximal subgroups

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

subgroupindex
#214556852
#214556742
#214543522
#214473672
#214471822
#214469802
#214460922
#214460542
#214458812
#214458762
C61×#17932
#2598561