Finite Group Databasebeta

← #30615927

Subgroups — #30615927

Order 15616.

1026 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840777C211yesmaximal, characteristic, complemented
2C2×#16840759C211yesmaximal, characteristic
3D61⋊#408C211yesmaximal, characteristic
4C2×#16840727C211yesmaximal, characteristic, complemented
5#21461903C212yesmaximal, complemented
6#21461902C212yesmaximal, complemented
7#21461901C212yesmaximal, complemented
8#21461900C212yesmaximal, complemented
9#21461498C211yesmaximal, characteristic, complemented
10#21461487C211yesmaximal, characteristic, complemented
11#21454335C211yesmaximal, characteristic
12C22×#11758213C2211yescharacteristic
13C2×D61⋊M16C2211yescharacteristic
14C22×D61⋊C8C2211yescharacteristic
15C2×#14519856C2211yescharacteristic
16#16840777—24no—
17C2×#14519848C2212yescomplemented
18C2×#14519847C2212yescomplemented
19#16840759—24no—
20C2×#14519842C2212yes—
21C2×#14519841C2212yes—
22#16840737C2211yescharacteristic
23#16840737C2211yescharacteristic
24#16840736C2211yescharacteristic
25#16840736C2211yescharacteristic
26#16840735—21no—
27#16840735—21no—
28D61⋊(C22⋊C8)—21no—
29D61⋊(C22⋊C8)—21no—
30#16840727—21no—
31#16840727—21no—
32#16840727C2212yescomplemented
33#16840727C2212yescomplemented
34C2×#14519832—21no—
35C2×C4×#92420—21no—
36#16840632C2212yescomplemented
37#16840632C2212yescomplemented
38(C2×#6923)⋊C8C2212yescomplemented
39(C2×#6923)⋊C8C2212yescomplemented
40C2×#41×D61C2211yescharacteristic
41C2×M16×D61C2211yescharacteristic
42C2×#14519735C2211yescharacteristic
43#16839769C412yescomplemented
44#16839765C412yescomplemented
45#16839764C411yescharacteristic, complemented
46#16839763C411yescharacteristic, complemented
47#16839761C411yescharacteristic, complemented
48#16839757C2212yes—
49#16839755C2212yes—
50C61×#417C411yescharacteristic, complemented
51C2×#11758213—21no—
52C2×#11758213—22no—
53C2×#11758213D414yes—
54C2×#11758213C2311yescharacteristic
55C2×#11758213C2311yescharacteristic
56C2×#11758213C2312yes—
57D61⋊M16—22no—
58D61⋊M16—22no—
59C2×#11758217—21no—
60C2×D61⋊C8—22no—
61C2×D61⋊C8—22no—
62C2×D61⋊C8—21no—
63C2×D61⋊C8D411yescharacteristic
64C2×D61⋊C8D411yescharacteristic
65C2×D61⋊C8C2311yescharacteristic
66C2×D61⋊C8C2311yescharacteristic
67C22×#92420—21no—
68C22×#92420D411yescharacteristic
69C22×#92420D411yescharacteristic
70#14519856—22no—
71#14519856—22no—
72C2×C4⋊#6923—21no—
73C2×C22⋊#6923—21no—
74C2×C4×#6923C2312yes—
75#14519848—24no—
76#14519847—24no—
77#14519842—24no—
78#14519841—24no—
79#14519834—21no—
80#14519834—21no—
81#14519834—21no—
82#14519834—21no—
83C22⋊#92420—21no—
84C22⋊#92420—21no—
85C22⋊#92420—21no—
86C22⋊#92420—21no—
87#14519832—22no—
88#14519832—22no—
89C4×#92420—22no—
90C4×#92420—22no—
91C2×#11758226C2311yescharacteristic
92C2×#11758222—21no—
93C2×#11758222C2×C411yescharacteristic
94C2×#11758224—21no—
95C2×#11758224C2311yescharacteristic
96#41×D61—22no—
97#41×D61—22no—
98#41×D61—22no—
99#41×D61—22no—
100C2×Q8×D61C2×C411yescharacteristic
101C2×D4×D61—21no—
102C2×D4×D61C2×C411yescharacteristic
103C22×C4×D61—21no—
104C22×C4×D61C2311yescharacteristic
105C2×#11758194C2×C411yescharacteristic
106C2×#11758189C2×C412yes—
107M16×D61—22no—
108M16×D61—22no—
109C2×C8×D61C2312yes—
110#14519735—21no—
111#14519735—21no—
112#14519735C2×C412yes—
113#14519735C2312yes—
114C2×C4⋊Dic61C2×C411yescharacteristic
115C2×#11758198—21no—
116C2×C4×Dic61—21no—
117#14519702C2×C412yes—
118#14519701C2×C412yes—
119#14519700C2×C412yes—
120#14519699C2×C412yes—
121#14519698C2×C412yes—
122#14519697C2×C412yes—
123C2×#41×C61C2×C411yescharacteristic
124C2×M16×C61C2×C411yescharacteristic
125C2×C4⋊C4×C61C2×C411yescharacteristic
126#103×C61C2×C412yes—
127#102×C61C2×C412yes—
128C23×D61—21no—
129C23×D61#4111yescharacteristic
130C22×#6923—22no—
131#41×C61—22no—
132#41×C61—22no—
133C2×Q8×C61C22×C411yescharacteristic
134C2×D4×C61—21no—
135C2×D4×C61C22×C411yescharacteristic
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C22×C411yescharacteristic
139C2×C22⋊D61—21no—
140C2×C22⋊D61—21no—
141C2×C22⋊D61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D411yescharacteristic
145#11758226—21no—
146#11758226—21no—
147#11758226—21no—
148#11758226—21no—
149Q8×D61—22no—
150Q8×D61—22no—
151#11758224—21no—
152#11758224—21no—
153#11758224—21no—
154#11758224—21no—
155#11758224—42no—
156#11758224—42no—
157D4×D61—22no—
158D4×D61—22no—
159D4×D61—42no—
160D4×D61—42no—
161#11758222—22no—
162#11758222—22no—
163#11758222—42no—
164#11758222—42no—
165C2×D244—21no—
166C2×D244C22×C411yescharacteristic
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61C22×C411yescharacteristic
173C2×C4×D61C2×D412yes—
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61#4112yes—
176C2×C4×D61#4112yes—
177C2×C4×D61#4112yes—
178C2×C4×D61#4111yescharacteristic
179C2×C4×D61C2411yescharacteristic
180C2×Dic122—21no—
181C2×Dic122C22×C411yescharacteristic
182C22⋊#6923—42no—
183#11758217—42no—
184C2×#92420—24no—
185C2×#92420—22no—
186C2×#92420—22no—
187C2×#92420—22no—
188C2×#92420—21no—
189C2×#92420—21no—
190C2×#92420C2×D411yescharacteristic
191C2×#92420C2×D411yescharacteristic
192C2×#92420C2×D411yescharacteristic
193C2×#92420C2×D411yescharacteristic
194C4⋊#6923—42no—
195C4×#6923—22no—
196C4×#6923—22no—
197#11758213—24no—
198#11758213—22no—
199#11758213—21no—
200#11758213—21no—
201#11758213—21no—
202#11758213—21no—
203#11758213C2×D414yes—
204D61⋊C8—22no—
205D61⋊C8—22no—
206D61⋊C8—21no—
207D61⋊C8—21no—
208D61⋊C8—21no—
209D61⋊C8—21no—
210D61⋊C8—42no—
211M16×C61—22no—
212C2×C8×C61C22×C412yes—
213C4⋊C4×C61—21no—
214C4⋊C4×C61C22×C412yes—
215#11758198—22no—
216#11758198—22no—
217C4⋊Dic61—21no—
218C4⋊Dic61C22×C412yes—
219C4×Dic61—22no—
220C4×Dic61—22no—
221#11758194—22no—
222C2×#92418C22×C412yes—
223#11758189—24no—
224C8×D61—22no—
225C8×D61—22no—
226C23×C61—21no—
227C23×C61C4×D411yescharacteristic
228C22×D61—21no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—42no—
232C22×D61—42no—
233C22×D61—42no—
234C22×D61C4×D411yescharacteristic
235C22×D61#12412yes—
236C22×D61#12612yes—
237C22×D61C2×#4111yescharacteristic
238C2×#6923—42no—
239C2×#6923—42no—
240C2×#6923—42no—
241Q8×C61—21no—
242Q8×C61—21no—
243D4×C61—21no—
244D4×C61—21no—
245D4×C61—42no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61#11712yes—
251C2×C4×C61C4×D412yes—
252C2×C4×C61C4×D411yescharacteristic
253C2×C4×C61C23×C411yescharacteristic
254C22⋊D61—42no—
255C22⋊D61—41no—
256C22⋊D61—41no—
257C22⋊D61—42no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—42no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61C4×D411yescharacteristic
264C2×Dic61#12112yes—
265C2×Dic61#12312yes—
266C2×Dic61C22×D411yescharacteristic
267D244—21no—
268D244—21no—
269D244—41no—
270D244—41no—
271C4×D61—42no—
272C4×D61—42no—
273C4×D61—41no—
274C4×D61—41no—
275C4×D61#11712yes—
276C4×D61C4×D412yes—
277C4×D61#12112yes—
278C4×D61#12312yes—
279C4×D61#12412yes—
280C4×D61#12612yes—
281C4×D61C22×D411yescharacteristic
282C4×D61C2×#4111yescharacteristic
283C4×D61C2×#4111yescharacteristic
284C4×D61C2×#4111yescharacteristic
285Dic122—21no—
286Dic122—21no—
287Dic122—42no—
288#92420—24no—
289#92420—21no—
290#92420—21no—
291#92420—21no—
292#92420—21no—
293#92420—41no—
294#92420—41no—
295C8×C61—22no—
296#92418—22no—
297#21070—611nomaximal, Sylow
298C22×C61—21no—
299C22×C61—21no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61#38812yes—
304C22×C61C2×C4×D411yescharacteristic
305C2×D61—21no—
306C2×D61—21no—
307C2×D61—21no—
308C2×D61—21no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—84no—
315C2×D61#38812yes—
316C2×D61C2×#12411yescharacteristic
317C2×D61C2×#12611yescharacteristic
318#6923—82no—
319C4×C61—42no—
320C4×C61—41no—
321C4×C61#38812yes—
322C4×C61C2×#11711yescharacteristic
323C4×C61C2×C4×D411yescharacteristic
324Dic61—42no—
325Dic61—41no—
326Dic61#38812yes—
327Dic61C2×#12111yescharacteristic
328Dic61C2×#12311yescharacteristic
329C2×#430—611no—
330C2×#421—611no—
331#2896—611no—
332C2×#408—611no—
333C2×#405—611no—
334#1929—612no—
335#1929—612no—
336#1929—612no—
337#1929—612no—
338#1733—611no—
339#1729—611no—
340C2×C61—22no—
341C2×C61—81no—
342C2×C61C2×#38811yescharacteristic
343C2×C61#295511yescharacteristic
344C2×C61#295511yescharacteristic
345D61—21no—
346D61—22no—
347D61—21no—
348D61—81no—
349C22×#41—611no—
350C22×M16—611no—
351C22×M16—611no—
352C22×M16—611no—
353C23×C8—611no—
354C2×#117—611no—
355C2×#117—611no—
356#430—1224no—
357C2×#108—612no—
358C2×#108—612no—
359#421—1224no—
360C2×#104—612no—
361C2×#104—612no—
362#419—612no—
363#418—612no—
364#417—611no—
365#417—611no—
366#417—611no—
367#417#692311yescharacteristic, complemented
368#416—612no—
369#416—612no—
370#408—611no—
371#408—611no—
372#408—611no—
373#408—611no—
374#408—1221no—
375#408—1221no—
376C2×C22⋊C8—1221no—
377C2×C22⋊C8—1221no—
378#405—612no—
379#405—612no—
380#405—1221no—
381#405—1221no—
382C2×#97—1221no—
383C2×C4×C8—1221no—
384#339—612no—
385#339—612no—
386#336—612no—
387#336—612no—
388C61#2107011yescharacteristic, complemented, Sylow
389C2×#41—611no—
390C2×#41—611no—
391C2×#41—611no—
392C2×#41—1221no—
393C2×#41—1221no—
394C2×#41—1222no—
395C2×#41—1222no—
396C2×#41—1222no—
397C2×#41—1222no—
398C2×#41C2×#692311yescharacteristic
399C22×Q8—611no—
400C22×D4—611no—
401C22×D4—1221no—
402C23×C4—611no—
403C23×C4—1221no—
404C2×M16—611no—
405C2×M16—612no—
406C2×M16—614no—
407C2×M16—612no—
408C2×M16—611no—
409C2×M16—611no—
410C2×M16—1222no—
411C2×M16—1222no—
412C2×M16—1222no—
413C2×M16—1221no—
414C2×M16—1221no—
415C2×M16—1222no—
416C2×M16—1222no—
417C2×M16C2×#692311yescharacteristic
418C22×C8—611no—
419C22×C8—611no—
420C22×C8—611no—
421C22×C8—611no—
422C22×C8—611no—
423C22×C8—611no—
424C22×C8—612no—
425C22×C8—1221no—
426C22×C8—1222no—
427C22×C8—1222no—
428C22×C8—1221no—
429#117—612no—
430#117—612no—
431#117—1221no—
432#117—1221no—
433#117—1222no—
434#117—1222no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4C2×#692311yescharacteristic
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C42—612no—
441C2×C42—1221no—
442#108—1224no—
443#108—1224no—
444#104—1224no—
445#104—1224no—
446#103—612no—
447#103—612no—
448#103—612no—
449#103C2×#692312yes—
450#102—612no—
451#102—612no—
452#102—612no—
453#102C2×#692312yes—
454C22⋊C8—1221no—
455C22⋊C8—1221no—
456C22⋊C8—1221no—
457C22⋊C8—1221no—
458C22⋊C8—1221no—
459C22⋊C8—1221no—
460C22⋊C8—1221no—
461C22⋊C8—1221no—
462#97—1222no—
463#97—1222no—
464C4×C8—1222no—
465C4×C8—1222no—
466C24—611no—
467C24—1221no—
468#41—22no—
469#41—22no—
470#41—1221no—
471#41—1221no—
472#41—1221no—
473#41—1221no—
474#41—1221no—
475#41—1221no—
476#41—1221no—
477#41—1221no—
478#41—1222no—
479#41—1222no—
480#41—2442no—
481#41—2442no—
482#41—2442no—
483#41—2442no—
484C2×Q8—611no—
485C2×Q8—1221no—
486C2×Q8—1222no—
487C2×Q8—1222no—
488C2×Q8C22×#692311yescharacteristic
489C2×D4—21no—
490C2×D4—611no—
491C2×D4—1221no—
492C2×D4—1222no—
493C2×D4—1222no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—2442no—
498C2×D4—2442no—
499C2×D4C22×#692311yescharacteristic
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—611no—
503C22×C4—612no—
504C22×C4—612no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—612no—
508C22×C4—612no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1222no—
517C22×C4—2442no—
518C22×C4—2442no—
519C22×C4C22×#692311yescharacteristic
520M16—22no—
521M16—614no—
522M16—1224no—
523M16—1224no—
524M16—1222no—
525M16—1222no—
526M16—1221no—
527M16—1221no—
528M16—1221no—
529M16—1221no—
530M16—2442no—
531C2×C8—611no—
532C2×C8—611no—
533C2×C8—611no—
534C2×C8—611no—
535C2×C8—612no—
536C2×C8—1222no—
537C2×C8—1222no—
538C2×C8—1222no—
539C2×C8—1222no—
540C2×C8—1222no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1222no—
546C2×C8—1222no—
547C2×C8—1224no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—2442no—
551C2×C8C22×#692312yes—
552C4⋊C4—21no—
553C4⋊C4—612no—
554C4⋊C4—1221no—
555C4⋊C4—2442no—
556C4⋊C4C22×#692312yes—
557C22⋊C4—1222no—
558C22⋊C4—1222no—
559C22⋊C4—2442no—
560C42—1222no—
561C42—1222no—
562C42—1222no—
563C42—1222no—
564C23—21no—
565C23—611no—
566C23—611no—
567C23—612no—
568C23—612no—
569C23—1221no—
570C23—2441no—
571C23—2441no—
572C23—2442no—
573C23—2442no—
574C23—2442no—
575C23D4×#692311yescharacteristic
576Q8—21no—
577Q8—21no—
578Q8—1221no—
579Q8—1221no—
580Q8—2442no—
581D4—21no—
582D4—21no—
583D4—42no—
584D4—1221no—
585D4—1221no—
586D4—2441no—
587D4—2441no—
588D4—2442no—
589D4—2441no—
590D4—2441no—
591D4—2442no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—612no—
597C2×C4—612no—
598C2×C4—611no—
599C2×C4—611no—
600C2×C4—612no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—611no—
606C2×C4—611no—
607C2×C4—612no—
608C2×C4—611no—
609C2×C4—611no—
610C2×C4—1222no—
611C2×C4—1221no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2442no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2442no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4#1451985612yes—
623C2×C4D4×#692311yescharacteristic
624C2×C4D4×#692312yes—
625C2×C4C23×#692311yescharacteristic
626C8—22no—
627C8—1221no—
628C8—1221no—
629C8—1221no—
630C8—1221no—
631C8—1224no—
632C8—1222no—
633C8—2441no—
634C8—2441no—
635C22—21no—
636C22—21no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—611no—
641C22—612no—
642C22—611no—
643C22—1221no—
644C22—1221no—
645C22—1221no—
646C22—1221no—
647C22—4884no—
648C22—2442no—
649C22—2442no—
650C22—2441no—
651C22—2441no—
652C22—2441no—
653C22#1684071212yes—
654C22C2×D4×#692311yescharacteristic
655C4—41no—
656C4—42no—
657C4—611no—
658C4—611no—
659C4—612no—
660C4—4882no—
661C4—2441no—
662C4—2442no—
663C4#1684071212yes—
664C4C2×#1451985611yescharacteristic
665C4C2×D4×#692311yescharacteristic
666C2—22no—
667C2—81no—
668C2—1221no—
669C2—1222no—
670C2—1221no—
671C2—4881no—
672C2C2×#1684071211yescharacteristic
673C2#2146259111yescharacteristic
674C2#2146259211yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#417C4
socleC22×C61C2×C4×D4
prime radicalC2×#41×D61C22
cosocleC2×C4×D61C24
hypercenter#417#6923
hypocenterC61#21070
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16840777C2semidirect
4. C2×#16840727C2semidirect
5. #21461903C2semidirect
6. #21461902C2semidirect
7. #21461901C2semidirect
8. #21461900C2semidirect
9. #21461498C2semidirect
10. #21461487C2semidirect
17. C2×#14519848C22semidirect
18. C2×#14519847C22semidirect
32. #16840727C22semidirect
33. #16840727C22semidirect
36. #16840632C22semidirect
37. #16840632C22semidirect
38. (C2×#6923)⋊C8C22semidirect
39. (C2×#6923)⋊C8C22semidirect
43. #16839769C4semidirect
44. #16839765C4semidirect
45. #16839764C4semidirect
46. #16839763C4semidirect
47. #16839761C4semidirect
50. C61×#417C4semidirect
367. #417#6923semidirect
388. C61#21070semidirect

Maximal subgroups

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

subgroupindex
C2×#168407772
C2×#168407592
D61⋊#4082
C2×#168407272
#214619032
#214619022
#214619012
#214619002
#214614982
#214614872
#214543352
#2107061