Finite Group Databasebeta

← #30615930

Subgroups — #30615930

Order 15616.

1050 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840777C211yesmaximal, characteristic, complemented
2C2×D61⋊#104C211yesmaximal, characteristic
3#21462404C211yesmaximal, characteristic
4#21462385C211yesmaximal, characteristic, complemented
5#21461903C212yesmaximal, complemented
6#21461902C212yesmaximal, complemented
7#21461901C212yesmaximal, complemented
8#21461900C212yesmaximal, complemented
9C2×#16840632C211yesmaximal, characteristic, complemented
10#21461481C211yesmaximal, characteristic, complemented
11#21454335C211yesmaximal, characteristic
12C22×#11758213C2211yescharacteristic
13C2×D61⋊M16C2211yescharacteristic
14C2×D61⋊M16C2211yescharacteristic
15C22×C4×#6923C2211yescharacteristic
16#16840777—24no—
17C2×#14519848C2212yescomplemented
18C2×#14519847C2212yescomplemented
19C2×#14519842C2211yescharacteristic
20C2×#14519842C2211yescharacteristic
21D61⋊#104—24no—
22C2×#14519841C2211yescharacteristic
23C2×#14519841C2211yescharacteristic
24#16840737C2212yes—
25#16840736C2212yes—
26#16840735—22no—
27#16840733—21no—
28#16840733—21no—
29#16840727—21no—
30#16840727—21no—
31#16840727C2212yescomplemented
32#16840727C2212yescomplemented
33C4×#11758217—22no—
34#16840632C2212yescomplemented
35#16840632C2212yescomplemented
36(C2×#6923)⋊C8C2212yescomplemented
37(C2×#6923)⋊C8C2212yescomplemented
38C2×#41×D61C2211yescharacteristic
39C2×M16×D61C2211yescharacteristic
40C2×#14519735C2211yescharacteristic
41#16839769C412yescomplemented
42#16839765C412yescomplemented
43#16839764C411yescharacteristic, complemented
44#16839763C411yescharacteristic, complemented
45#16839761C411yescharacteristic, complemented
46#16839757C2212yes—
47#16839755C2212yes—
48C61×#417C411yescharacteristic, complemented
49C23×#6923—21no—
50C2×#11758213—22no—
51C2×#11758213—22no—
52C2×#11758213—21no—
53C2×#11758213D411yescharacteristic
54C2×#11758213D411yescharacteristic
55C2×#11758213C2312yes—
56C2×#11758213C2311yescharacteristic
57C2×#11758213C2311yescharacteristic
58D61⋊M16—22no—
59D61⋊M16—22no—
60D61⋊M16—22no—
61D61⋊M16D414yes—
62C2×#11758217—21no—
63C2×#11758217D412yes—
64C2×D61⋊C8—21no—
65C2×D61⋊C8C2312yes—
66C22×#92420—21no—
67C2×C4×#6923—22no—
68C2×C4×#6923—22no—
69C2×C4×#6923—21no—
70C2×C4×#6923C2311yescharacteristic
71C2×C4×#6923C2311yescharacteristic
72#14519848—24no—
73#14519847—24no—
74#14519842—22no—
75#14519842—22no—
76#14519841—22no—
77#14519841—22no—
78#14519834—22no—
79#14519834—22no—
80C22⋊#92420—22no—
81C22⋊#92420—22no—
82#14519832—22no—
83#14519832—22no—
84C4×#92420—22no—
85C4×#92420—22no—
86C2×#11758226C2311yescharacteristic
87C2×#11758222—21no—
88C2×#11758222C2×C411yescharacteristic
89C2×#11758224—21no—
90C2×#11758224C2311yescharacteristic
91#41×D61—22no—
92#41×D61—22no—
93#41×D61—22no—
94#41×D61—22no—
95C2×Q8×D61C2×C411yescharacteristic
96C2×D4×D61—21no—
97C2×D4×D61C2×C411yescharacteristic
98C22×C4×D61—21no—
99C22×C4×D61C2311yescharacteristic
100C2×#11758194C2×C411yescharacteristic
101C2×#11758189C2×C412yes—
102M16×D61—22no—
103M16×D61—22no—
104C2×C8×D61C2312yes—
105#14519735—21no—
106#14519735—21no—
107#14519735C2×C412yes—
108#14519735C2312yes—
109C2×C4⋊Dic61C2×C411yescharacteristic
110C2×#11758198—21no—
111C2×C4×Dic61—21no—
112#14519702C2×C412yescomplemented
113#14519701C2×C412yescomplemented
114#14519700C2×C412yescomplemented
115#14519699C2×C412yescomplemented
116#14519698C2×C412yescomplemented
117#14519697C2×C412yescomplemented
118C2×#41×C61C2×C411yescharacteristic
119C2×M16×C61C2×C411yescharacteristic
120C2×C4⋊C4×C61C2×C411yescharacteristic
121#103×C61C2×C412yescomplemented
122#102×C61C2×C412yescomplemented
123C23×D61—21no—
124C23×D61#4111yescharacteristic
125C22×#6923—21no—
126C22×#6923—21no—
127C22×#6923—42no—
128C22×#6923—42no—
129C22×#6923—42no—
130#41×C61—22no—
131#41×C61—22no—
132C2×Q8×C61C22×C411yescharacteristic
133C2×D4×C61—21no—
134C2×D4×C61C22×C411yescharacteristic
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C22×C411yescharacteristic
138C2×C22⋊D61—21no—
139C2×C22⋊D61—21no—
140C2×C22⋊D61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61C2×D411yescharacteristic
144#11758226—21no—
145#11758226—21no—
146#11758226—21no—
147#11758226—21no—
148Q8×D61—22no—
149Q8×D61—22no—
150#11758224—21no—
151#11758224—21no—
152#11758224—21no—
153#11758224—21no—
154#11758224—42no—
155#11758224—42no—
156D4×D61—22no—
157D4×D61—22no—
158D4×D61—42no—
159D4×D61—42no—
160#11758222—22no—
161#11758222—22no—
162#11758222—42no—
163#11758222—42no—
164C2×D244—21no—
165C2×D244C22×C411yescharacteristic
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61C22×C411yescharacteristic
172C2×C4×D61C2×D412yes—
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61#4112yes—
175C2×C4×D61#4112yes—
176C2×C4×D61#4112yes—
177C2×C4×D61#4111yescharacteristic
178C2×C4×D61C2411yescharacteristic
179C2×Dic122—21no—
180C2×Dic122C22×C411yescharacteristic
181#11758217—24no—
182#11758217—24no—
183C2×#92420—21no—
184C2×#92420—21no—
185C2×#92420—22no—
186C2×#92420—42no—
187C2×#92420C2×D412yes—
188C2×#92420C2×D412yes—
189C4×#6923—21no—
190C4×#6923—21no—
191C4×#6923—21no—
192C4×#6923—21no—
193C4×#6923—42no—
194#11758213—22no—
195#11758213—22no—
196#11758213—21no—
197#11758213—21no—
198#11758213—22no—
199#11758213—22no—
200#11758213—42no—
201#11758213C2×D412yes—
202#11758213C2×D412yes—
203D61⋊C8—24no—
204D61⋊C8—22no—
205D61⋊C8—22no—
206M16×C61—22no—
207C2×C8×C61C22×C412yes—
208C4⋊C4×C61—21no—
209C4⋊C4×C61C22×C412yes—
210#11758198—22no—
211#11758198—22no—
212C4⋊Dic61—21no—
213C4⋊Dic61C22×C412yes—
214C4×Dic61—22no—
215C4×Dic61—22no—
216#11758194—22no—
217C2×#92418C22×C412yes—
218#11758189—24no—
219C8×D61—22no—
220C8×D61—22no—
221C23×C61—21no—
222C23×C61C4×D411yescharacteristic
223C22×D61—21no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—42no—
227C22×D61—81no—
228C22×D61—81no—
229C22×D61C4×D411yescharacteristic
230C22×D61#12412yes—
231C22×D61#12612yes—
232C22×D61C2×#4111yescharacteristic
233C2×#6923—22no—
234C2×#6923—22no—
235C2×#6923—41no—
236C2×#6923—41no—
237C2×#6923—48no—
238C2×#6923—41no—
239C2×#6923—41no—
240Q8×C61—21no—
241Q8×C61—21no—
242D4×C61—21no—
243D4×C61—21no—
244D4×C61—42no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—42no—
248C2×C4×C61—81no—
249C2×C4×C61#11712yes—
250C2×C4×C61C4×D412yes—
251C2×C4×C61C4×D411yescharacteristic
252C2×C4×C61C23×C411yescharacteristic
253C22⋊D61—42no—
254C22⋊D61—41no—
255C22⋊D61—41no—
256C22⋊D61—42no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—81no—
262C2×Dic61C4×D411yescharacteristic
263C2×Dic61#12112yes—
264C2×Dic61#12312yes—
265C2×Dic61C22×D411yescharacteristic
266D244—21no—
267D244—21no—
268D244—41no—
269D244—41no—
270C4×D61—42no—
271C4×D61—42no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61#11712yes—
275C4×D61C4×D412yes—
276C4×D61#12112yes—
277C4×D61#12312yes—
278C4×D61#12412yes—
279C4×D61#12612yes—
280C4×D61C22×D411yescharacteristic
281C4×D61C2×#4111yescharacteristic
282C4×D61C2×#4111yescharacteristic
283C4×D61C2×#4111yescharacteristic
284Dic122—21no—
285Dic122—21no—
286Dic122—42no—
287#92420—22no—
288#92420—22no—
289#92420—22no—
290#92420—22no—
291#92420—42no—
292C8×C61—22no—
293#92418—22no—
294#21071—611nomaximal, Sylow
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61#38812yes—
301C22×C61C2×C4×D411yescharacteristic
302C2×D61—21no—
303C2×D61—21no—
304C2×D61—21no—
305C2×D61—21no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—84no—
312C2×D61#38812yes—
313C2×D61C2×#12411yescharacteristic
314C2×D61C2×#12611yescharacteristic
315#6923—42no—
316#6923—42no—
317C4×C61—42no—
318C4×C61—41no—
319C4×C61#38812yes—
320C4×C61C2×#11711yescharacteristic
321C4×C61C2×C4×D411yescharacteristic
322Dic61—42no—
323Dic61—41no—
324Dic61#38812yes—
325Dic61C2×#12111yescharacteristic
326Dic61C2×#12311yescharacteristic
327C2×#430—611no—
328C22×#104—611no—
329#2896—611no—
330#2879—611no—
331#2873—611no—
332#1929—612no—
333#1929—612no—
334#1929—612no—
335#1929—612no—
336C2×#339—611no—
337#1728—611no—
338C2×C61—22no—
339C2×C61—81no—
340C2×C61C2×#38811yescharacteristic
341C2×C61#295611yescharacteristic
342C2×C61#295611yescharacteristic
343D61—21no—
344D61—22no—
345D61—21no—
346D61—81no—
347C22×#41—611no—
348C22×M16—611no—
349C22×M16—611no—
350C22×M16—611no—
351C22×M16—611no—
352C2×#117—611no—
353C22×C42—611no—
354#430—1224no—
355C2×#108—612no—
356C2×#108—612no—
357C2×#104—611no—
358C2×#104—611no—
359C2×#104—611no—
360C2×#104—611no—
361C2×#104—1224no—
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—612no—
371#408—612no—
372#408—1222no—
373#407—1221no—
374#407—1221no—
375#405—612no—
376#405—612no—
377#405—1221no—
378#405—1221no—
379C4×M16—1222no—
380#339—612no—
381#339—612no—
382#336—612no—
383#336—612no—
384C61#2107111yescharacteristic, complemented, Sylow
385C2×#41—611no—
386C2×#41—611no—
387C2×#41—611no—
388C2×#41—1221no—
389C2×#41—1221no—
390C2×#41—1222no—
391C2×#41—1222no—
392C2×#41—1222no—
393C2×#41—1222no—
394C2×#41C2×#692311yescharacteristic
395C22×Q8—611no—
396C22×D4—611no—
397C22×D4—1221no—
398C23×C4—611no—
399C23×C4—1221no—
400C23×C4—1221no—
401C2×M16—612no—
402C2×M16—611no—
403C2×M16—611no—
404C2×M16—611no—
405C2×M16—612no—
406C2×M16—614no—
407C2×M16—611no—
408C2×M16—611no—
409C2×M16—612no—
410C2×M16—1222no—
411C2×M16—1222no—
412C2×M16—1221no—
413C2×M16—1222no—
414C2×M16—1222no—
415C2×M16—1222no—
416C2×M16—1221no—
417C2×M16—1222no—
418C2×M16—1222no—
419C2×M16C2×#692311yescharacteristic
420C22×C8—612no—
421C22×C8—612no—
422C22×C8—1221no—
423C22×C8—1221no—
424#117—612no—
425#117—612no—
426#117—1221no—
427#117—1221no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4C2×#692311yescharacteristic
430C2×C22⋊C4—1221no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—1221no—
434C2×C42—1222no—
435C2×C42—1222no—
436C2×C42—1221no—
437#108—1224no—
438#108—1224no—
439#104—1222no—
440#104—1222no—
441#104—1222no—
442#104—1222no—
443#103—612no—
444#103—612no—
445#103—612no—
446#103C2×#692312yescomplemented
447#102—612no—
448#102—612no—
449#102—612no—
450#102C2×#692312yescomplemented
451C22⋊C8—1222no—
452C22⋊C8—1222no—
453C22⋊C8—1222no—
454C22⋊C8—1222no—
455#97—1222no—
456#97—1222no—
457C4×C8—1222no—
458C4×C8—1222no—
459C24—611no—
460C24—1221no—
461#41—22no—
462#41—22no—
463#41—1221no—
464#41—1221no—
465#41—1221no—
466#41—1221no—
467#41—1221no—
468#41—1221no—
469#41—1221no—
470#41—1221no—
471#41—1222no—
472#41—1222no—
473#41—2442no—
474#41—2442no—
475#41—2442no—
476#41—2442no—
477C2×Q8—611no—
478C2×Q8—1221no—
479C2×Q8—1222no—
480C2×Q8—1222no—
481C2×Q8C22×#692311yescharacteristic
482C2×D4—21no—
483C2×D4—611no—
484C2×D4—1221no—
485C2×D4—1222no—
486C2×D4—1222no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—2442no—
491C2×D4—2442no—
492C2×D4C22×#692311yescharacteristic
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—611no—
496C22×C4—612no—
497C22×C4—612no—
498C22×C4—611no—
499C22×C4—612no—
500C22×C4—611no—
501C22×C4—612no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—2442no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4C22×#692311yescharacteristic
517M16—22no—
518M16—612no—
519M16—612no—
520M16—1224no—
521M16—1222no—
522M16—1222no—
523M16—1221no—
524M16—1221no—
525M16—1224no—
526M16—1224no—
527M16—1222no—
528M16—1222no—
529M16—1222no—
530M16—2442no—
531C2×C8—612no—
532C2×C8—612no—
533C2×C8—612no—
534C2×C8—1222no—
535C2×C8—1222no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1222no—
539C2×C8—1222no—
540C2×C8—1224no—
541C2×C8—1222no—
542C2×C8—2442no—
543C2×C8C22×#692312yes—
544C4⋊C4—21no—
545C4⋊C4—612no—
546C4⋊C4—1221no—
547C4⋊C4C22×#692312yes—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C42—1222no—
551C42—1222no—
552C42—1221no—
553C42—1221no—
554C42—1221no—
555C42—1221no—
556C42—2442no—
557C23—21no—
558C23—611no—
559C23—611no—
560C23—612no—
561C23—612no—
562C23—1221no—
563C23—4881no—
564C23—4881no—
565C23—2441no—
566C23—2441no—
567C23—2442no—
568C23D4×#692311yescharacteristic
569Q8—21no—
570Q8—21no—
571Q8—1221no—
572Q8—1221no—
573Q8—2442no—
574D4—21no—
575D4—21no—
576D4—42no—
577D4—1221no—
578D4—1221no—
579D4—2441no—
580D4—2441no—
581D4—2442no—
582D4—2441no—
583D4—2441no—
584D4—2442no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—42no—
588C2×C4—81no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—611no—
592C2×C4—611no—
593C2×C4—612no—
594C2×C4—612no—
595C2×C4—612no—
596C2×C4—612no—
597C2×C4—612no—
598C2×C4—611no—
599C2×C4—611no—
600C2×C4—612no—
601C2×C4—611no—
602C2×C4—611no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1221no—
607C2×C4—4881no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2442no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2448no—
618C2×C4—2442no—
619C2×C4#1451985612yes—
620C2×C4D4×#692311yescharacteristic
621C2×C4D4×#692312yes—
622C2×C4C23×#692311yescharacteristic
623C8—22no—
624C8—1222no—
625C8—1222no—
626C8—1222no—
627C8—1222no—
628C8—1222no—
629C8—2442no—
630C22—21no—
631C22—21no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—611no—
636C22—612no—
637C22—611no—
638C22—1221no—
639C22—1221no—
640C22—1221no—
641C22—1221no—
642C22—4884no—
643C22—2442no—
644C22—2442no—
645C22—2441no—
646C22—2441no—
647C22—2441no—
648C22#1684071212yes—
649C22C2×D4×#692311yescharacteristic
650C4—41no—
651C4—42no—
652C4—611no—
653C4—611no—
654C4—612no—
655C4—2442no—
656C4—2442no—
657C4—2441no—
658C4—2442no—
659C4#1684071212yes—
660C4C2×#1451985611yescharacteristic
661C4C2×D4×#692311yescharacteristic
662C2—22no—
663C2—81no—
664C2—1221no—
665C2—1222no—
666C2—1221no—
667C2—4881no—
668C2C2×#1684071211yescharacteristic
669C2#2146259311yescharacteristic
670C2#2146259411yescharacteristic
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#21071
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. #21462385C2semidirect
5. #21461903C2semidirect
6. #21461902C2semidirect
7. #21461901C2semidirect
8. #21461900C2semidirect
9. C2×#16840632C2semidirect
10. #21461481C2semidirect
17. C2×#14519848C22semidirect
18. C2×#14519847C22semidirect
31. #16840727C22semidirect
32. #16840727C22semidirect
34. #16840632C22semidirect
35. #16840632C22semidirect
36. (C2×#6923)⋊C8C22semidirect
37. (C2×#6923)⋊C8C22semidirect
41. #16839769C4semidirect
42. #16839765C4semidirect
43. #16839764C4semidirect
44. #16839763C4semidirect
45. #16839761C4semidirect
48. C61×#417C4semidirect
112. #14519702C2×C4semidirect
113. #14519701C2×C4semidirect
114. #14519700C2×C4semidirect
115. #14519699C2×C4semidirect
116. #14519698C2×C4semidirect
117. #14519697C2×C4semidirect
121. #103×C61C2×C4semidirect
122. #102×C61C2×C4semidirect
367. #417#6923semidirect
384. C61#21071semidirect
446. #103C2×#6923semidirect
450. #102C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407772
C2×D61⋊#1042
#214624042
#214623852
#214619032
#214619022
#214619012
#214619002
C2×#168406322
#214614812
#214543352
#2107161