Finite Group Databasebeta

← #29698484

Subgroups — #29698484

Order 15616.

862 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839876C211yesmaximal, characteristic
2#21454700C211yesmaximal, characteristic
3#21454698C211yesmaximal, characteristic
4#21454413C211yesmaximal, characteristic, complemented
5#21454412C211yesmaximal, characteristic, complemented
6#21446598C211yesmaximal, characteristic, complemented
7#21446596C211yesmaximal, characteristic, complemented
8#21446047C211yesmaximal, characteristic, complemented
9#21446046C211yesmaximal, characteristic, complemented
10#21446044C211yesmaximal, characteristic, complemented
11#21445975C211yesmaximal, characteristic, complemented
12#21445877C211yesmaximal, characteristic, complemented
13#21445876C211yesmaximal, characteristic, complemented
14#21445697C211yesmaximal, characteristic, complemented
15C61×#1859C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C2×#14519785C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16840243C2211yescharacteristic
21#16840242C2211yescharacteristic
22#16839876C418yescomplemented
23#16839872—21no—
24#16839871—21no—
25#16839867—21no—
26#16839866—21no—
27#16839853—21no—
28C4×#11758201—21no—
29C4×#11758200—21no—
30#16839847—21no—
31C2×#14519712C2211yescharacteristic
32C2×#14519712C2211yescharacteristic
33#16839805—21no—
34#16839805C2211yescharacteristic
35#16839804C2211yescharacteristic
36#16839803—21no—
37#16839802—21no—
38#16839802C2211yescharacteristic
39#16839801—21no—
40#16839799—21no—
41#16839798—21no—
42#16839798C2211yescharacteristic
43C2×#14519708—21no—
44C2×#14519708C2211yescharacteristic
45C2×#14519708C2211yescharacteristic
46C2×#14519708C2211yescharacteristic
47C2×#14519708C2211yescharacteristic
48#106×D61—21no—
49C2×#14519705C2211yescharacteristic
50#16839764C2211yescharacteristic
51#16839763C2211yescharacteristic
52#16839761C2211yescharacteristic
53#16839720C2211yescharacteristic
54#16839718C2211yescharacteristic
55C2×C4×#92418C2211yescharacteristic
56#16839632C2211yescharacteristic
57#16839531C2211yescharacteristic
58#16839530C2211yescharacteristic
59#16839530C2211yescharacteristic
60#16839522C2211yescharacteristic
61C61×#520C2211yescharacteristic
62C61×#428C2211yescharacteristic
63C2×#104×C61C2211yescharacteristic
64C61×#417C2211yescharacteristic
65C61×#384C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#339C2211yescharacteristic
68C2×#11758222C2311yescharacteristic
69#14519797C2×C414yes—
70#14519797Q814yes—
71C2×#11758202D411yescharacteristic
72C2×#11758200D411yescharacteristic
73C2×#11758201D411yescharacteristic
74C2×#11758199D411yescharacteristic
75#14519785—22no—
76#14519785C2×C414yescomplemented
77#14519784—22no—
78#14519784C2×C414yescomplemented
79C2×#11758194—21no—
80C2×#11758194C2311yescharacteristic
81C2×#11758194C2311yescharacteristic
82C2×#11758189—21no—
83C22×#92418—21no—
84C22×#92418C2311yescharacteristic
85C22×#92418C2311yescharacteristic
86C2×C8×D61—21no—
87Q8×Dic61—21no—
88C4×Dic122—21no—
89#14519742—21no—
90C4×D244—21no—
91D4×Dic61—21no—
92D4×Dic61—41no—
93#14519739—21no—
94C4×C22⋊D61—41no—
95#14519737—41no—
96#14519736—21no—
97#14519736C2311yescharacteristic
98#14519735—21no—
99#14519735—21no—
100C2×C4⋊Dic61—21no—
101C2×C4⋊Dic61—41no—
102C2×C4⋊Dic61C2311yescharacteristic
103C2×C4⋊Dic61C2311yescharacteristic
104C4⋊C4×D61—21no—
105C4⋊C4×D61—21no—
106#14519712—22no—
107#14519712C2×C414yescomplemented
108#14519711—21no—
109#14519710—21no—
110#14519710—21no—
111#14519710D411yescharacteristic
112#14519710D411yescharacteristic
113#14519709—21no—
114#14519708—21no—
115#14519708—21no—
116#14519708—21no—
117#14519708—21no—
118#14519708—21no—
119#14519708—21no—
120#14519708—21no—
121#14519708—21no—
122#14519708—21no—
123#14519708—21no—
124#14519708D411yescharacteristic
125#14519708D411yescharacteristic
126#14519705C2×C414yescomplemented
127#14519702—21no—
128#14519701—21no—
129#14519700—21no—
130#14519699—21no—
131#14519698—21no—
132#14519697—21no—
133#14519688—21no—
134#14519687—21no—
135C4×#92418—21no—
136C4×#92418C2×C414yescomplemented
137C8×Dic61—21no—
138C2×#41×C61C2311yescharacteristic
139C2×M16×C61C2311yescharacteristic
140C2×M16×C61C2311yescharacteristic
141C4×Q8×C61—21no—
142C4×D4×C61—21no—
143C4×D4×C61—41no—
144#117×C61—21no—
145#117×C61C2311yescharacteristic
146C2×C4⋊C4×C61—21no—
147C2×C4⋊C4×C61—41no—
148C2×C4⋊C4×C61C2311yescharacteristic
149C2×C4⋊C4×C61C2311yescharacteristic
150C2×C42×C61C2311yescharacteristic
151#107×C61—21no—
152#106×C61—21no—
153#104×C61C2×C414yescomplemented
154#103×C61—21no—
155#102×C61—21no—
156#41×C61—21no—
157#41×C61—21no—
158#41×C61C22×C412yes—
159#41×C61C2×Q812yes—
160C2×Q8×C61C2×D411yescharacteristic
161C2×D4×C61—21no—
162C2×D4×C61C2×D411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—41no—
168C22×C4×C61C2411yescharacteristic
169C2×C22⋊D61—21no—
170C22×Dic61—21no—
171C22×Dic61—21no—
172C22×Dic61—41no—
173#11758222—21no—
174#11758222—21no—
175#11758222C22×C412yes—
176#11758222C2×Q812yes—
177C2×D244C2×D411yescharacteristic
178C2×C4×D61—21no—
179C2×C4×D61—41no—
180C2×C4×D61—41no—
181C2×Dic122C2×D411yescharacteristic
182M16×C61—21no—
183M16×C61—21no—
184M16×C61—21no—
185M16×C61C22×C412yes—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—41no—
193C4⋊C4×C61—82no—
194C4⋊C4×C61C2×D411yescharacteristic
195C4⋊C4×C61C2×D411yescharacteristic
196C22⋊C4×C61—21no—
197C22⋊C4×C61—41no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—41no—
201C42×C61C22×C412yes—
202C22⋊Dic61—21no—
203C22⋊Dic61—41no—
204#11758202C4⋊C414yescomplemented
205#11758201C4⋊C414yescomplemented
206#11758200C4⋊C414yescomplemented
207#11758199C4⋊C414yescomplemented
208#11758198—41no—
209#11758198—41no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—41no—
215C4⋊Dic61—82no—
216C4⋊Dic61C2×D411yescharacteristic
217C4⋊Dic61C2×D411yescharacteristic
218#11758196—41no—
219#11758196—41no—
220C4×Dic61—21no—
221C4×Dic61—41no—
222C4×Dic61—41no—
223C4×Dic61—41no—
224#11758194—21no—
225#11758194—24no—
226#11758194—21no—
227#11758194—21no—
228#11758194C22×C412yes—
229C2×#92418—21no—
230C2×#92418—24no—
231C2×#92418—21no—
232C2×#92418C22×C412yes—
233C2×#92418C22×C412yes—
234C2×#92418C2×D411yescharacteristic
235C2×#92418C2×D411yescharacteristic
236C2×#92418C2×D411yescharacteristic
237C2×#92418C2×D411yescharacteristic
238C2×#92418C2×Q812yes—
239C2×#92418#4112yes—
240#11758189—24no—
241C8×D61—24no—
242C23×C61—21no—
243C23×C61H(5,𝔽2)11yescharacteristic
244C22×D61—21no—
245Q8×C61—21no—
246Q8×C61C2×C4⋊C412yes—
247D4×C61—22no—
248D4×C61—22no—
249D4×C61—21no—
250D4×C61C2×C4⋊C412yes—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—21no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—81no—
262C2×C4×C61—81no—
263C2×C4×C61—161no—
264C2×C4×C61C23×C411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×Q811yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C612−1+411yescharacteristic
270C22⋊D61—22no—
271C22⋊D61—22no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—81no—
278C2×Dic61—81no—
279C2×Dic61—161no—
280D244—21no—
281D244C2×C4⋊C412yes—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—161no—
285C4×D61—161no—
286Dic122—21no—
287Dic122C2×C4⋊C412yes—
288C8×C61—22no—
289C8×C61—41no—
290#92418—22no—
291#92418—41no—
292#92418C2×C4⋊C412yes—
293#92418C2×C4⋊C412yes—
294#92418C4×D412yes—
295#92418C4×D412yes—
296#22689—611nomaximal, Sylow
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—22no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61#52011yescharacteristic
303C22×C61D4211yescharacteristic
304C22×C61#55411yescharacteristic
305C2×D61—22no—
306C2×D61—41no—
307C2×D61—41no—
308C4×C61—24no—
309C4×C61—22no—
310C4×C61—41no—
311C4×C61—41no—
312C4×C61—81no—
313C4×C61—81no—
314C4×C61C22×C4⋊C411yescharacteristic
315C4×C61C2×C4×D411yescharacteristic
316C4×C61#54711yescharacteristic
317C4×C61D4×Q811yescharacteristic
318Dic61—22no—
319Dic61—41no—
320Dic61—41no—
321Dic61—81no—
322Dic61—81no—
323C2×#443—611no—
324#2952—611no—
325#2952—611no—
326#2912—611no—
327#2912—611no—
328#1951—611no—
329#1951—611no—
330#1859—611no—
331#1859—611no—
332#1859—611no—
333#1859D6111yescharacteristic, complemented
334#1844—611no—
335#1815—611no—
336#1815—611no—
337#1772—611no—
338C2×C61—21no—
339C2×C61—21no—
340C2×C61—42no—
341C2×C61D4×C4⋊C411yescharacteristic
342C2×C61#329111yescharacteristic
343C2×C61#329111yescharacteristic
344D61—42no—
345C2×#135—611no—
346C2×#131—611no—
347C2×#131—611no—
348#520—611no—
349#520—611no—
350#520—611no—
351#520C2×D6111yescharacteristic
352#443—618no—
353#442—1221no—
354#442—1221no—
355#441—1221no—
356#441—1221no—
357C4×SD16—1221no—
358C4×SD16—1221no—
359C4×SD16—1221no—
360C4×SD16—1221no—
361C2×#108—611no—
362C2×#108—611no—
363#428—611no—
364#428—611no—
365#428—611no—
366#428—1221no—
367#428—1221no—
368#428—1221no—
369#428—1221no—
370#428C2×D6111yescharacteristic
371#427—611no—
372#427—1221no—
373#427—1221no—
374C2×#106—611no—
375C2×#106—611no—
376C2×#106—611no—
377C2×#106—611no—
378C2×#106—1221no—
379C2×#106—1221no—
380C2×#104—611no—
381C2×#104C2×D6111yescharacteristic
382#417—611no—
383#417—611no—
384#417—611no—
385#417C2×D6111yescharacteristic
386#405—611no—
387#405—611no—
388C2×C4×C8—611no—
389#384—611no—
390#384C2×D6111yescharacteristic
391#343—611no—
392#343—611no—
393#343—611no—
394#343C2×D6111yescharacteristic
395#339—611no—
396#339C2×D6111yescharacteristic
397C61#2268911yescharacteristic, complemented, Sylow
398C2×#41—611no—
399C2×#41C22×D6111yescharacteristic
400#135—614no—
401#135—614no—
402C2×Q16—611no—
403C2×SD16—611no—
404C2×SD16—611no—
405C2×D8—611no—
406#131—614no—
407#131—614no—
408#131—1222no—
409#131—1222no—
410C2×M16—611no—
411C2×M16—611no—
412C2×M16—1221no—
413C2×M16—1221no—
414C2×M16C22×D6111yescharacteristic
415C2×M16C22×D6111yescharacteristic
416C22×C8—611no—
417C22×C8—611no—
418C22×C8—1221no—
419C22×C8—1221no—
420C4×Q8—21no—
421C4×Q8—1221no—
422C4×Q8—1221no—
423C4×Q8—1221no—
424C4×D4—21no—
425C4×D4—41no—
426C4×D4—1221no—
427C4×D4—1221no—
428C4×D4—1221no—
429C4×D4—2441no—
430C4×D4—2441no—
431C4×D4—2441no—
432#117—21no—
433#117—611no—
434#117—1221no—
435#117—1221no—
436#117—1221no—
437#117C22×D6111yescharacteristic
438C2×C4⋊C4—21no—
439C2×C4⋊C4—41no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—2441no—
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C42C22×D6111yescharacteristic
449#108—614no—
450#108—1222no—
451#107—21no—
452#107—611no—
453#107—611no—
454#107—1221no—
455#107—1221no—
456#107—1221no—
457#106—21no—
458#106—611no—
459#106—611no—
460#106—1221no—
461#106—1221no—
462#106—1221no—
463#106—1221no—
464#106—1221no—
465#106—1221no—
466#106—1221no—
467#106—1221no—
468#106—1221no—
469#106—1221no—
470#106—1221no—
471#104—614no—
472#104C4×D6114yescomplemented
473#103—21no—
474#103—1221no—
475#103—1221no—
476#103—1221no—
477#102—21no—
478#102—1221no—
479#102—1221no—
480#102—1221no—
481#97—1221no—
482#97—1221no—
483C4×C8—614no—
484C4×C8—1221no—
485C4×C8—1221no—
486#41—21no—
487#41—21no—
488#41—612no—
489#41—612no—
490#41—1221no—
491#41—1221no—
492#41C2×C4×D6112yes—
493#41Q8×D6112yes—
494C2×Q8—611no—
495C2×Q8D4×D6111yescharacteristic
496C2×D4—21no—
497C2×D4—611no—
498C2×D4—1221no—
499C2×D4D4×D6111yescharacteristic
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—41no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—2441no—
509C22×C4—2441no—
510C22×C4—2441no—
511C22×C4C23×D6111yescharacteristic
512Q16—614no—
513SD16—614no—
514SD16—614no—
515D8—614no—
516M16—21no—
517M16—21no—
518M16—21no—
519M16—612no—
520M16—1224no—
521M16—1224no—
522M16—1221no—
523M16—1221no—
524M16—1221no—
525M16C2×C4×D6112yes—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—612no—
529C2×C8—612no—
530C2×C8—611no—
531C2×C8—611no—
532C2×C8—611no—
533C2×C8—611no—
534C2×C8—612no—
535C2×C8—612no—
536C2×C8—1224no—
537C2×C8—1224no—
538C2×C8—1221no—
539C2×C8—1221no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—41no—
545C4⋊C4—82no—
546C4⋊C4—611no—
547C4⋊C4—611no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—4882no—
553C4⋊C4—2441no—
554C4⋊C4—2441no—
555C4⋊C4—2441no—
556C4⋊C4D4×D6111yescharacteristic
557C4⋊C4D4×D6111yescharacteristic
558C22⋊C4—21no—
559C22⋊C4—41no—
560C22⋊C4—1221no—
561C22⋊C4—2441no—
562C22⋊C4—2441no—
563C22⋊C4—2441no—
564C42—21no—
565C42—21no—
566C42—41no—
567C42—1221no—
568C42—2441no—
569C42—2441no—
570C42—2441no—
571C42C2×C4×D6112yes—
572C23—21no—
573C23—1221no—
574C23#1451982211yescharacteristic
575Q8—21no—
576Q8—612no—
577Q8—1221no—
578Q8C4⋊C4×D6112yes—
579D4—22no—
580D4—22no—
581D4—21no—
582D4—612no—
583D4—1222no—
584D4—1222no—
585D4—1221no—
586D4C4⋊C4×D6112yes—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—161no—
600C2×C4—1221no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—9761no—
606C2×C4—9761no—
607C2×C4—9761no—
608C2×C4—4881no—
609C2×C4—4881no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4C22×C4×D6111yescharacteristic
613C2×C4C2×D4×D6111yescharacteristic
614C2×C4C2×D4×D6111yescharacteristic
615C2×C4C2×Q8×D6111yescharacteristic
616C2×C4#41×D6111yescharacteristic
617C2×C4#1451982411yescharacteristic
618C8—22no—
619C8—41no—
620C8—612no—
621C8—612no—
622C8—612no—
623C8—612no—
624C8—1222no—
625C8—2441no—
626C22—21no—
627C22—21no—
628C22—22no—
629C22—41no—
630C22—41no—
631C22—1222no—
632C22—2441no—
633C22—2441no—
634C22#1684024111yescharacteristic
635C22#1684039411yescharacteristic
636C22#1684046611yescharacteristic
637C4—24no—
638C4—22no—
639C4—41no—
640C4—41no—
641C4—81no—
642C4—81no—
643C4—1222no—
644C4—4881no—
645C4—4881no—
646C4—2441no—
647C4—2441no—
648C4C2×C4⋊C4×D6111yescharacteristic
649C4C4×D4×D6111yescharacteristic
650C4#1684041011yescharacteristic
651C4#1684042511yescharacteristic
652C2—21no—
653C2—21no—
654C2—42no—
655C2—2442no—
656C2#2144871711yescharacteristic
657C2#2145760911yescharacteristic
658C2#2145761411yescharacteristic
subgroupG/H
centerC22#16840394
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1859C2
socleC22×C61D42
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1859D61
hypocenterC61#22689
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1859

Complements

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

NH ≅ G/Ntype
4. #21454413C2semidirect
5. #21454412C2semidirect
6. #21446598C2semidirect
7. #21446596C2semidirect
8. #21446047C2semidirect
9. #21446046C2semidirect
10. #21446044C2semidirect
11. #21445975C2semidirect
12. #21445877C2semidirect
13. #21445876C2semidirect
14. #21445697C2semidirect
15. C61×#1859C2semidirect
22. #16839876C4semidirect
76. #14519785C2×C4semidirect
78. #14519784C2×C4semidirect
107. #14519712C2×C4semidirect
126. #14519705C2×C4semidirect
136. C4×#92418C2×C4semidirect
153. #104×C61C2×C4semidirect
204. #11758202C4⋊C4semidirect
205. #11758201C4⋊C4semidirect
206. #11758200C4⋊C4semidirect
207. #11758199C4⋊C4semidirect
333. #1859D61semidirect
397. C61#22689semidirect
472. #104C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398762
#214547002
#214546982
#214544132
#214544122
#214465982
#214465962
#214460472
#214460462
#214460442
#214459752
#214458772
#214458762
#214456972
C61×#18592
#2268961