Finite Group Databasebeta

← #29656503

Subgroups — #29656503

Order 15616.

1324 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457197C211yesmaximal, characteristic, complemented
2#21456269C211yesmaximal, characteristic, complemented
3#21455879C211yesmaximal, characteristic, complemented
4#21444946C212yesmaximal, complemented
5#21444935C212yesmaximal, complemented
6#21444884C212yesmaximal, complemented
7#21444873C212yesmaximal, complemented
8#21444487C211yesmaximal, characteristic, complemented
9#21444258C211yesmaximal, characteristic, complemented
10#21444236C211yesmaximal, characteristic, complemented
11C61×#3498C211yesmaximal, characteristic, complemented
12C2×#14519803—22no—
13#16840312C2211yescharacteristic, complemented
14#16840149—22no—
15#16840138C2212yescomplemented
16#16840134C2212yescomplemented
17#16840086—21no—
18#16840057C2211yescharacteristic, complemented
19#16840052C2211yescharacteristic, complemented
20#16840052C2212yescomplemented
21#16840049—21no—
22#16840013—22no—
23#16839962C2211yescharacteristic, complemented
24#16839957C2211yescharacteristic, complemented
25#16839957C2212yescomplemented
26#16839954—21no—
27#16839907—21no—
28C4×#11758201—21no—
29C4×#11758199—21no—
30#16839815C2211yescharacteristic, complemented
31#16839786C2211yescharacteristic, complemented
32C4×#11758194C2211yescharacteristic, complemented
33#16839506C2212yescomplemented
34#16839506C2212yescomplemented
35Q8⋊#92418C2212yescomplemented
36D4⋊#92418C2212yescomplemented
37D4⋊#92418C2212yescomplemented
38D4⋊#92418C2212yescomplemented
39C2×H(5,𝔽2)×C61—21no—
40C61×#550—21no—
41C61×#550C2212yes—
42D42×C61—22no—
43D42×C61—22no—
44D42×C61—21no—
45D42×C61C2212yes—
46C61×#535—22no—
47C61×#535—21no—
48C61×#532—21no—
49C61×#532C2211yescharacteristic
50C2×#127×C61—21no—
51C2×#127×C61C2211yescharacteristic
52C4×#41×C61C2211yescharacteristic
53C2×#11758222—21no—
54#14519803—48no—
55C2×#11758200—24no—
56C2×#11758201—21no—
57C2×#11758199—21no—
58C2×#11758199—24no—
59C2×#11758199—22no—
60C2×#11758194—21no—
61C2×#11758194—21no—
62#14519777C2311yescharacteristic
63#14519774C2311yescharacteristic
64#14519765—21no—
65#14519751—21no—
66#14519751—21no—
67C4×D244—21no—
68C4×D244—21no—
69#14519710—21no—
70#14519708—22no—
71#14519708—21no—
72C4⋊#92418—21no—
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2312yes—
75C4⋊#92418C2311yescharacteristic
76#14519701—21no—
77#14519698—22no—
78#14519698—21no—
79#14519697—22no—
80#14519697—21no—
81#14519697—21no—
82C22⋊#92418—21no—
83#14519688—21no—
84C4×#92418—21no—
85C4×#92418C2312yes—
86H(5,𝔽2)×C61—44no—
87H(5,𝔽2)×C61—44no—
88C2×#41×C61—22no—
89C2×#41×C61—22no—
90C2×#41×C61—21no—
91C2×#41×C61D411yescharacteristic
92C22×D4×C61—22no—
93C22×D4×C61—24no—
94C22×D4×C61—22no—
95C22×D4×C61—22no—
96C22×D4×C61—22no—
97C22×D4×C61—21no—
98C22×D4×C61—21no—
99C22×D4×C61D412yes—
100C4⋊Q8×C61D411yescharacteristic
101#127×C61—22no—
102#127×C61—22no—
103#127×C61—22no—
104#127×C61—21no—
105#127×C61—21no—
106#127×C61D411yescharacteristic
107#127×C61D412yes—
108#127×C61C2312yes—
109#124×C61—22no—
110#124×C61—21no—
111#121×C61—22no—
112#121×C61—24no—
113#121×C61—24no—
114#121×C61—21no—
115#121×C61—22no—
116#121×C61—22no—
117#121×C61—22no—
118#121×C61D412yes—
119#120×C61—24no—
120#120×C61—24no—
121#120×C61—24no—
122C4×Q8×C61D411yescharacteristic
123C4×Q8×C61C2311yescharacteristic
124C4×D4×C61—22no—
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61—22no—
128C4×D4×C61—21no—
129C4×D4×C61D411yescharacteristic
130C4×D4×C61C2312yes—
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#117×C61D411yescharacteristic
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C24×C61—24no—
137C24×C61—22no—
138#41×C61—22no—
139#41×C61—22no—
140#41×C61—48no—
141#41×C61—42no—
142#41×C61—42no—
143#41×C61—44no—
144#41×C61—44no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—24no—
149C2×D4×C61—24no—
150C2×D4×C61—22no—
151C2×D4×C61—24no—
152C2×D4×C61—24no—
153C2×D4×C61—22no—
154C2×D4×C61—21no—
155C2×D4×C61—22no—
156C2×D4×C61—21no—
157C2×D4×C61—22no—
158C2×D4×C61—22no—
159C2×D4×C61—22no—
160C2×D4×C61—48no—
161C2×D4×C61—42no—
162C2×D4×C61—42no—
163C2×D4×C61—44no—
164C2×D4×C61—416no—
165C2×D4×C61—44no—
166C2×D4×C61—44no—
167C2×D4×C61—81no—
168C2×D4×C61—81no—
169C2×D4×C61—82no—
170C2×D4×C61—82no—
171C2×D4×C61—84no—
172C2×D4×C61C2×D412yes—
173C2×D4×C61C2×D411yescharacteristic
174C2×D4×C61C2×D414yes—
175C22×C4×C61—21no—
176C22×C4×C61—24no—
177C22×C4×C61—21no—
178C22×C4×C61—21no—
179C22×C4×C61—22no—
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2×D411yescharacteristic
183C2×C22⋊D61—41no—
184#11758222—44no—
185C2×D244—21no—
186C2×D244—21no—
187C2×D244—21no—
188C2×C4×D61—41no—
189C2×Dic122—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61C2×D411yescharacteristic
194C4⋊C4×C61C2×D411yescharacteristic
195C4⋊C4×C61C2×D412yes—
196C22⋊C4×C61—24no—
197C22⋊C4×C61—24no—
198C22⋊C4×C61—21no—
199C22⋊C4×C61—21no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—21no—
202C22⋊C4×C61—22no—
203C42×C61—21no—
204C42×C61—21no—
205C42×C61—21no—
206C42×C61C2×D411yescharacteristic
207C42×C61C2×D411yescharacteristic
208C42×C61C2×D411yescharacteristic
209C42×C61C2411yescharacteristic
210#11758201—42no—
211#11758200—48no—
212#11758199—48no—
213#11758199—42no—
214#11758199—82no—
215#11758198—41no—
216#11758198—41no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219#11758194—24no—
220#11758194—81no—
221C2×#92418—22no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—22no—
225C23×C61—21no—
226C23×C61—22no—
227C23×C61—24no—
228C23×C61—44no—
229C23×C61—48no—
230C23×C61—82no—
231C23×C61—82no—
232C23×C61—84no—
233C23×C61—84no—
234C23×C61#12011yescharacteristic
235C22×D61—41no—
236Q8×C61—21no—
237Q8×C61—42no—
238Q8×C61AGL(1,ℤ/8)12yescomplemented
239D4×C61—24no—
240D4×C61—21no—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61—44no—
244D4×C61—44no—
245D4×C61—42no—
246D4×C61—48no—
247D4×C61—44no—
248D4×C61—48no—
249D4×C61—42no—
250D4×C61—42no—
251D4×C61—42no—
252D4×C61—42no—
253D4×C61—44no—
254D4×C61—42no—
255D4×C61—82no—
256D4×C61—84no—
257D4×C61—84no—
258D4×C61AGL(1,ℤ/8)12yescomplemented
259C2×C4×C61—24no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—22no—
266C2×C4×C61—21no—
267C2×C4×C61—21no—
268C2×C4×C61—22no—
269C2×C4×C61—48no—
270C2×C4×C61—42no—
271C2×C4×C61—42no—
272C2×C4×C61—41no—
273C2×C4×C61—41no—
274C2×C4×C61—42no—
275C2×C4×C61#12011yescharacteristic
276C2×C4×C61#12011yescharacteristic
277C2×C4×C61#12011yescharacteristic
278C2×C4×C61C22×D411yescharacteristic
279C2×C4×C61C22×D411yescharacteristic
280C2×C4×C61C22×D411yescharacteristic
281C22⋊D61—82no—
282C2×Dic61—41no—
283D244—42no—
284D244—41no—
285D244—41no—
286D244—41no—
287D244—41no—
288C4×D61—82no—
289Dic122—42no—
290#92418—24no—
291#92418—41no—
292#92418—41no—
293#92418—42no—
294#18566—611nomaximal, Sylow
295C22×C61—21no—
296C22×C61—22no—
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—44no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—44no—
305C22×C61—44no—
306C22×C61—42no—
307C22×C61—164no—
308C22×C61—168no—
309C22×C61C2×#12011yescharacteristic
310C2×D61—81no—
311C2×D61—81no—
312C2×D61—81no—
313C4×C61—21no—
314C4×C61—22no—
315C4×C61—21no—
316C4×C61—21no—
317C4×C61—42no—
318C4×C61—41no—
319C4×C61—41no—
320C4×C61#45312yes—
321C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
322C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
323C4×C61#57611yescharacteristic
324C4×C61#57711yescharacteristic
325Dic61—81no—
326#3498D6111yescharacteristic, complemented
327#3253—611no—
328#3151—611no—
329#3106—611no—
330#1631—612no—
331#1629—612no—
332#1621—612no—
333#1619—612no—
334#1545—611no—
335#1489—611no—
336#1484—611no—
337C2×C61—41no—
338C2×C61—42no—
339C2×C61—82no—
340C2×C61—84no—
341C2×C61#300311yescharacteristic
342C2×C61#301211yescharacteristic
343C2×C61C2×#45311yescharacteristic
344D61—161no—
345C2×H(5,𝔽2)—21no—
346C2×AGL(1,ℤ/8)—1222no—
347#550—21no—
348#550C2×D6112yes—
349D42—22no—
350D42—22no—
351D42—21no—
352D42C2×D6112yes—
353#535—22no—
354#535—21no—
355#532—21no—
356#532—611no—
357#532C2×D6111yescharacteristic
358C2×#127—21no—
359C2×#127C2×D6111yescharacteristic
360C4×#41C2×D6111yescharacteristic
361#496—1222no—
362#493—612no—
363#492—612no—
364#482—1221no—
365#477—611no—
366#476—611no—
367#476—612no—
368#476—1221no—
369#469—1222no—
370#460—611no—
371#459—611no—
372#459—612no—
373#459—1221no—
374#449—1221no—
375C4×SD16—1221no—
376C4×D8—1221no—
377#431—611no—
378#423—611no—
379C4×M16—611no—
380#331—612no—
381#331—612no—
382Q8⋊C8—612no—
383D4⋊C8—612no—
384D4⋊C8—612no—
385D4⋊C8—612no—
386C61#1856611yescharacteristic, complemented, Sylow
387H(5,𝔽2)—44no—
388H(5,𝔽2)—44no—
389C2×#41—22no—
390C2×#41—22no—
391C2×#41—21no—
392C2×#41—1221no—
393C2×#41C22⋊D6111yescharacteristic
394C22×D4—22no—
395C22×D4—24no—
396C22×D4—22no—
397C22×D4—22no—
398C22×D4—22no—
399C22×D4—21no—
400C22×D4—21no—
401C22×D4C22⋊D6112yes—
402AGL(1,ℤ/8)—2448no—
403C2×SD16—1221no—
404C2×SD16—1224no—
405C2×D8—1222no—
406C2×D8—1224no—
407C2×D8—1221no—
408C2×M16—1221no—
409C2×M16—1221no—
410C4⋊Q8—611no—
411C4⋊Q8C22⋊D6111yescharacteristic
412#127—22no—
413#127—22no—
414#127—22no—
415#127—21no—
416#127—21no—
417#127—611no—
418#127C22⋊D6111yescharacteristic
419#127C22⋊D6112yes—
420#127C22×D6112yes—
421#124—22no—
422#124—21no—
423#124—1221no—
424#121—22no—
425#121—24no—
426#121—24no—
427#121—21no—
428#121—22no—
429#121—22no—
430#121—22no—
431#121—1221no—
432#121—1221no—
433#121C22⋊D6112yes—
434#120—24no—
435#120—24no—
436#120—24no—
437C4×Q8C22⋊D6111yescharacteristic
438C4×Q8C22×D6111yescharacteristic
439C4×D4—22no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4—22no—
443C4×D4—21no—
444C4×D4—1221no—
445C4×D4—1221no—
446C4×D4C22⋊D6111yescharacteristic
447C4×D4C22×D6111yescharacteristic
448C4×D4C22×D6112yes—
449#117—21no—
450#117C22⋊D6111yescharacteristic
451C2×C42—21no—
452C2×C42C22×D6111yescharacteristic
453#107—1221no—
454#106—1221no—
455#106—1222no—
456C4⋊C8—611no—
457C4⋊C8—611no—
458C4⋊C8—612no—
459C4⋊C8—1221no—
460#103—1221no—
461#102—1221no—
462#102—1222no—
463#102—1221no—
464#102—1221no—
465#102—1222no—
466C22⋊C8—1221no—
467#97—1221no—
468C4×C8—612no—
469C4×C8—1221no—
470C24—24no—
471C24—22no—
472#41—22no—
473#41—22no—
474#41—48no—
475#41—42no—
476#41—42no—
477#41—44no—
478#41—44no—
479#41—2444no—
480C2×Q8—21no—
481C2×Q8—1221no—
482C2×Q8C2×C22⋊D6111yescharacteristic
483C2×D4—21no—
484C2×D4—24no—
485C2×D4—24no—
486C2×D4—22no—
487C2×D4—24no—
488C2×D4—24no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—22no—
492C2×D4—21no—
493C2×D4—22no—
494C2×D4—21no—
495C2×D4—22no—
496C2×D4—48no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4—44no—
500C2×D4—416no—
501C2×D4—44no—
502C2×D4—44no—
503C2×D4—81no—
504C2×D4—81no—
505C2×D4—82no—
506C2×D4—82no—
507C2×D4—84no—
508C2×D4—1221no—
509C2×D4—1221no—
510C2×D4—1221no—
511C2×D4—2441no—
512C2×D4C2×C22⋊D6112yes—
513C2×D4C2×C22⋊D6111yescharacteristic
514C2×D4C2×C22⋊D6114yes—
515C22×C4—21no—
516C22×C4—24no—
517C22×C4—21no—
518C22×C4—22no—
519C22×C4—21no—
520C22×C4—2441no—
521C22×C4C2×C22⋊D6111yescharacteristic
522C22×C4C2×C22⋊D6111yescharacteristic
523C22×C4C2×C22⋊D6111yescharacteristic
524SD16—2448no—
525SD16—2442no—
526D8—4882no—
527D8—2448no—
528D8—2442no—
529M16—1224no—
530M16—4881no—
531C2×C8—1222no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1222no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4C2×C22⋊D6111yescharacteristic
541C4⋊C4C2×C22⋊D6112yes—
542C4⋊C4C2×C22⋊D6111yescharacteristic
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—22no—
550C22⋊C4—2441no—
551C22⋊C4—2441no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42C2×C22⋊D6111yescharacteristic
556C42C2×C22⋊D6111yescharacteristic
557C42C2×C22⋊D6111yescharacteristic
558C42C23×D6111yescharacteristic
559C23—21no—
560C23—22no—
561C23—24no—
562C23—44no—
563C23—48no—
564C23—82no—
565C23—82no—
566C23—84no—
567C23—84no—
568C23—2441no—
569C23#1451974611yescharacteristic
570Q8—21no—
571Q8—42no—
572Q8—2442no—
573Q8#1451980312yescomplemented
574D4—24no—
575D4—21no—
576D4—22no—
577D4—22no—
578D4—44no—
579D4—44no—
580D4—42no—
581D4—48no—
582D4—44no—
583D4—48no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—44no—
589D4—42no—
590D4—82no—
591D4—84no—
592D4—84no—
593D4—4882no—
594D4—2442no—
595D4—2441no—
596D4—2441no—
597D4—2441no—
598D4—2441no—
599D4#1451980312yescomplemented
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—4882no—
617C2×C4—2441no—
618C2×C4#1451974611yescharacteristic
619C2×C4#1451974611yescharacteristic
620C2×C4#1451974611yescharacteristic
621C2×C4C22×C22⋊D6111yescharacteristic
622C2×C4C22×C22⋊D6111yescharacteristic
623C2×C4C22×C22⋊D6111yescharacteristic
624C8—1224no—
625C8—2441no—
626C8—2441no—
627C8—2442no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—42no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—44no—
639C22—42no—
640C22—164no—
641C22—168no—
642C22—4881no—
643C22—4881no—
644C22—4881no—
645C22C2×#1451974611yescharacteristic
646C4—21no—
647C4—21no—
648C4—21no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—42no—
653C4—4881no—
654C4#1683993012yes—
655C4C2×#1451980311yescharacteristic
656C4C2×#1451980311yescharacteristic
657C4#1684056511yescharacteristic
658C4#1684057311yescharacteristic
659C2—41no—
660C2—42no—
661C2—82no—
662C2—84no—
663C2—9761no—
664C2#2145504611yescharacteristic
665C2#2145512311yescharacteristic
666C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3498C2
socleC22×C61C2×#120
hypercenter#3498D61
hypocenterC61#18566
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3498

Complements

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

NH ≅ G/Ntype
1. #21457197C2semidirect
2. #21456269C2semidirect
3. #21455879C2semidirect
4. #21444946C2semidirect
5. #21444935C2semidirect
6. #21444884C2semidirect
7. #21444873C2semidirect
8. #21444487C2semidirect
9. #21444258C2semidirect
10. #21444236C2semidirect
11. C61×#3498C2semidirect
13. #16840312C22semidirect
15. #16840138C22semidirect
16. #16840134C22semidirect
18. #16840057C22semidirect
19. #16840052C22semidirect
20. #16840052C22semidirect
23. #16839962C22semidirect
24. #16839957C22semidirect
25. #16839957C22semidirect
30. #16839815C22semidirect
31. #16839786C22semidirect
32. C4×#11758194C22semidirect
33. #16839506C22semidirect
34. #16839506C22semidirect
35. Q8⋊#92418C22semidirect
36. D4⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
38. D4⋊#92418C22semidirect
238. Q8×C61AGL(1,ℤ/8)semidirect
258. D4×C61AGL(1,ℤ/8)semidirect
326. #3498D61semidirect
386. C61#18566semidirect
573. Q8#14519803semidirect
599. D4#14519803semidirect

Maximal subgroups

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

subgroupindex
#214571972
#214562692
#214558792
#214449462
#214449352
#214448842
#214448732
#214444872
#214442582
#214442362
C61×#34982
#1856661