Finite Group Databasebeta

← #29736573

Subgroups — #29736573

Order 15616.

1442 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840121C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3#21450992C211yesmaximal, characteristic, complemented
4#21447019C212yesmaximal, complemented
5#21446998C212yesmaximal, complemented
6#21446752C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21445759C212yesmaximal, complemented
9#21445750C212yesmaximal, complemented
10#21445644C211yesmaximal, characteristic, complemented
11C2×C61×#546C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic, complemented
13#16840121—24no—
14#16840042—24no—
15C2×#14519710C2212yescomplemented
16C2×C4⋊#92418C2211yescharacteristic, complemented
17C2×#14519701C2212yescomplemented
18C2×#14519701C2212yescomplemented
19C2×#14519698C2212yes—
20C2×#14519698C2212yes—
21C2×#14519698C2212yes—
22C2×#14519698C2212yes—
23C2×C22⋊#92418C2212yes—
24C2×C22⋊#92418C2212yescomplemented
25C2×#14519688C2211yescharacteristic, complemented
26#16839674—21no—
27#16839674—21no—
28#16839665—21no—
29#16839627—21no—
30#16839622C2212yes—
31#16839622C2212yescomplemented
32C4×C4⋊Dic61C2211yescharacteristic, complemented
33#16839524C2212yescomplemented
34C22×#41×C61C2211yescharacteristic
35C23×D4×C61C2211yescharacteristic
36C61×#546—24no—
37C61×#546—24no—
38C2×#124×C61C2211yescharacteristic
39C2×#123×C61—21no—
40C2×C22⋊Q8×C61C2211yescharacteristic
41C2×#121×C61—21no—
42C2×#121×C61C2211yescharacteristic
43C2×#120×C61—21no—
44C2×C4×D4×C61C2211yescharacteristic
45C2×C4×D4×C61C2211yescharacteristic
46C22×C22⋊C4×C61—21no—
47C22×Dic122—21no—
48C22×#92418C2312yes—
49C22×#92418C2312yes—
50#14519777—22no—
51#14519777—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2312yes—
55C2×C22⋊Dic61—22no—
56C2×C22⋊Dic61—22no—
57C2×C4×Dic61—21no—
58#14519710—24no—
59C4⋊#92418—22no—
60#14519701—24no—
61#14519701—42no—
62#14519698—24no—
63#14519698—24no—
64#14519698—24no—
65#14519698—42no—
66C22⋊#92418—42no—
67C22⋊#92418—42no—
68#14519688—22no—
69#14519684—21no—
70#14519684—22no—
71#14519684—22no—
72C25×C61—21no—
73C2×#41×C61—22no—
74C2×#41×C61—22no—
75C2×#41×C61—24no—
76C2×#41×C61—42no—
77C22×Q8×C61C2311yescharacteristic
78C22×D4×C61—21no—
79C22×D4×C61—21no—
80C22×D4×C61—22no—
81C22×D4×C61—22no—
82C22×D4×C61—24no—
83C22×D4×C61—21no—
84C22×D4×C61—42no—
85C22×D4×C61D411yescharacteristic
86C22×D4×C61C2311yescharacteristic
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61—21no—
89C23×C4×C61—21no—
90C23×C4×C61C2311yescharacteristic
91C23×C4×C61C2311yescharacteristic
92#124×C61—22no—
93#124×C61D414yes—
94#123×C61—28no—
95C22⋊Q8×C61—22no—
96C22⋊Q8×C61—22no—
97#121×C61—28no—
98#121×C61—22no—
99#121×C61—22no—
100#120×C61—28no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61D414yes—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—28no—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C22⋊C4×C61D411yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C24×C61—21no—
118C24×C61—24no—
119C24×C61—21no—
120C24×C61—44no—
121C24×C61—42no—
122C24×C61—42no—
123C24×C61—82no—
124C24×C61C2×D411yescharacteristic
125C24×C61C2×D411yescharacteristic
126#41×C61—48no—
127#41×C61—44no—
128#41×C61—44no—
129C2×Q8×C61—21no—
130C2×Q8×C61—22no—
131C2×Q8×C61—22no—
132C2×Q8×C61C2×D412yes—
133C2×D4×C61—24no—
134C2×D4×C61—24no—
135C2×D4×C61—22no—
136C2×D4×C61—24no—
137C2×D4×C61—22no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—22no—
141C2×D4×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61—41no—
144C2×D4×C61—44no—
145C2×D4×C61—48no—
146C2×D4×C61—44no—
147C2×D4×C61—44no—
148C2×D4×C61—44no—
149C2×D4×C61—82no—
150C2×D4×C61C2×D412yes—
151C2×D4×C61C2×D412yes—
152C2×D4×C61#4114yes—
153C22×C4×C61—24no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—22no—
161C22×C4×C61—22no—
162C22×C4×C61—22no—
163C22×C4×C61—22no—
164C22×C4×C61—24no—
165C22×C4×C61—44no—
166C22×C4×C61—82no—
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2411yescharacteristic
172C22×Dic61—22no—
173C22×Dic61—22no—
174C2×Dic122—42no—
175C2×Dic122—82no—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C2×D412yes—
180C22⋊C4×C61—24no—
181C22⋊C4×C61—24no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—24no—
186C22⋊C4×C61—22no—
187C42×C61C2×D412yes—
188C42×C61C2×D412yes—
189C22⋊Dic61—82no—
190C22⋊Dic61—82no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—42no—
196C4×Dic61—42no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—24no—
207C23×C61—21no—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—21no—
211C23×C61—22no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—22no—
215C23×C61—48no—
216C23×C61—48no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—42no—
220C23×C61—42no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—42no—
224C23×C61—88no—
225C23×C61—84no—
226C23×C61—88no—
227C23×C61#13712yes—
228C23×C61C22×D411yescharacteristic
229Q8×C61—24no—
230Q8×C61—41no—
231Q8×C61—41no—
232D4×C61—22no—
233D4×C61—24no—
234D4×C61—24no—
235D4×C61—44no—
236D4×C61—48no—
237D4×C61—48no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—84no—
241D4×C61#12414yes—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—48no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61—84no—
261C2×C4×C61#12012yes—
262C2×C4×C61#12012yes—
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C2×#4112yes—
266C2×C4×C61H(5,𝔽2)12yes—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273Dic122—84no—
274#92418—42no—
275#92418—42no—
276#25897—611nomaximal, Sylow
277C22×C61—24no—
278C22×C61—22no—
279C22×C61—24no—
280C22×C61—41no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—41no—
284C22×C61—44no—
285C22×C61—42no—
286C22×C61—44no—
287C22×C61—41no—
288C22×C61—42no—
289C22×C61—41no—
290C22×C61—42no—
291C22×C61—41no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—816no—
296C22×C61—82no—
297C22×C61—82no—
298C22×C61C2×#12011yescharacteristic
299C22×C61#54612yes—
300C22×C61C2×#13711yescharacteristic
301C22×C61#57511yescharacteristic
302C22×C61#57612yes—
303C4×C61—22no—
304C4×C61—22no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—42no—
308C4×C61—42no—
309C4×C61C2×#12411yescharacteristic
310C4×C61#53511yescharacteristic
311C4×C61#54612yes—
312Dic61—82no—
313Dic61—82no—
314C2×#546D6111yescharacteristic
315C2×#489—611no—
316C2×#474—611no—
317#2563—611no—
318#2036—612no—
319#2033—612no—
320#1983—611no—
321#1955—611no—
322#1788—612no—
323#1786—612no—
324#1762—611no—
325C2×C61—42no—
326C2×C61—44no—
327C2×C61—81no—
328C2×C61—82no—
329C2×C61#243511yescharacteristic
330C2×C61#300711yescharacteristic
331C2×C61#300811yescharacteristic
332C2×C61#331412yes—
333C2×C61#331912yes—
334C22×#41C2×D6111yescharacteristic
335C23×D4C2×D6111yescharacteristic
336#546—24no—
337#546—24no—
338C2×C4⋊Q8—611no—
339C2×#124C2×D6111yescharacteristic
340C2×#123—21no—
341C2×C22⋊Q8C2×D6111yescharacteristic
342C2×#121—21no—
343C2×#121C2×D6111yescharacteristic
344C2×#120—21no—
345C2×C4×D4C2×D6111yescharacteristic
346C2×C4×D4C2×D6111yescharacteristic
347C22×C22⋊C4—21no—
348#489—1224no—
349#474—1224no—
350C2×#107—612no—
351C2×C4⋊C8—611no—
352C2×#103—612no—
353C2×#103—612no—
354C2×#102—612no—
355C2×#102—612no—
356C2×#102—612no—
357C2×#102—612no—
358C2×C22⋊C8—612no—
359C2×C22⋊C8—612no—
360C2×#97—611no—
361#393—1221no—
362#393—1221no—
363#390—1221no—
364#382—1221no—
365#380—612no—
366#380—612no—
367C4×C4⋊C4—611no—
368#340—612no—
369C61#2589711yescharacteristic, complemented, Sylow
370C25—21no—
371C2×#41—22no—
372C2×#41—22no—
373C2×#41—24no—
374C2×#41—42no—
375C22×Q8—1221no—
376C22×Q8C22×D6111yescharacteristic
377C22×D4—21no—
378C22×D4—21no—
379C22×D4—22no—
380C22×D4—22no—
381C22×D4—24no—
382C22×D4—21no—
383C22×D4—42no—
384C22×D4C22⋊D6111yescharacteristic
385C22×D4C22×D6111yescharacteristic
386C22×D4C22×D6111yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C23×C4C22×D6111yescharacteristic
390C23×C4C22×D6111yescharacteristic
391C22×C8—612no—
392C22×C8—612no—
393C4⋊Q8—1222no—
394C4⋊Q8—1222no—
395#124—22no—
396#124C22⋊D6114yes—
397#123—28no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400#121—28no—
401#121—22no—
402#121—22no—
403#120—28no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4C22⋊D6114yes—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—612no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4C22⋊D6111yescharacteristic
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C22⋊C4—28no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1222no—
421C2×C22⋊C4—1222no—
422C2×C22⋊C4C22⋊D6111yescharacteristic
423C2×C22⋊C4C22⋊D6111yescharacteristic
424C2×C42—1221no—
425C2×C42C22×D6111yescharacteristic
426#107—1224no—
427C4⋊C8—1222no—
428#103—1224no—
429#103—2442no—
430#102—1224no—
431#102—1224no—
432#102—1224no—
433#102—2442no—
434C22⋊C8—2442no—
435C22⋊C8—2442no—
436#97—1222no—
437#95—1221no—
438#95—1222no—
439#95—1222no—
440C24—21no—
441C24—24no—
442C24—21no—
443C24—44no—
444C24—42no—
445C24—42no—
446C24—82no—
447C24C2×C22⋊D6111yescharacteristic
448C24C2×C22⋊D6111yescharacteristic
449#41—48no—
450#41—44no—
451#41—44no—
452C2×Q8—21no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—4882no—
456C2×Q8—2442no—
457C2×Q8C2×C22⋊D6112yes—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—41no—
469C2×D4—44no—
470C2×D4—48no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—44no—
474C2×D4—82no—
475C2×D4#1175822414yes—
476C2×D4C2×C22⋊D6112yes—
477C2×D4C2×C22⋊D6112yes—
478C22×C4—24no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—44no—
491C22×C4—82no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C23×D6111yescharacteristic
499C2×C8—1222no—
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—1222no—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—2442no—
513C4⋊C4C2×C22⋊D6112yes—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—22no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—4882no—
522C22⋊C4—4882no—
523C42—2442no—
524C42C2×C22⋊D6112yes—
525C42C2×C22⋊D6112yes—
526C23—21no—
527C23—22no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—48no—
539C23—48no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—88no—
548C23—84no—
549C23—88no—
550C23#1451980712yes—
551C23C22×C22⋊D6111yescharacteristic
552Q8—24no—
553Q8—41no—
554Q8—41no—
555Q8—4884no—
556D4—22no—
557D4—24no—
558D4—24no—
559D4—44no—
560D4—48no—
561D4—48no—
562D4—41no—
563D4—41no—
564D4—84no—
565D4#1451976414yes—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—84no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4#1451974612yes—
592C2×C4#1451974612yes—
593C2×C4C22×C22⋊D6111yescharacteristic
594C2×C4C22×C22⋊D6111yescharacteristic
595C2×C4C2×#1175822412yes—
596C2×C4#1451982212yes—
597C8—2442no—
598C8—2442no—
599C22—22no—
600C22—24no—
601C22—24no—
602C22—42no—
603C22—44no—
604C22—42no—
605C22—44no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—42no—
610C22—41no—
611C22—42no—
612C22—42no—
613C22—41no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—816no—
618C22—82no—
619C22—82no—
620C22C2×#1451974611yescharacteristic
621C22#1684040612yes—
622C22C2×#1451980711yescharacteristic
623C22#1684056011yescharacteristic
624C22#1684056512yes—
625C4—22no—
626C4—22no—
627C4—42no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—4882no—
632C4—4882no—
633C4C2×#1451976411yescharacteristic
634C4#1684032911yescharacteristic
635C4#1684040612yes—
636C2—42no—
637C2—44no—
638C2—81no—
639C2—82no—
640C2#2144946611yescharacteristic
641C2#2145507911yescharacteristic
642C2#2145508711yescharacteristic
643C2#2145790012yes—
644C2#2145797512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#546C2
socleC23×C61C22×D4
hypercenterC2×#546D61
hypocenterC61#25897
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#546

Complements

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

NH ≅ G/Ntype
1. C2×#16840121C2semidirect
2. C2×#16840042C2semidirect
3. #21450992C2semidirect
4. #21447019C2semidirect
5. #21446998C2semidirect
6. #21446752C2semidirect
7. #21446618C2semidirect
8. #21445759C2semidirect
9. #21445750C2semidirect
10. #21445644C2semidirect
12. C2×#14519777C22semidirect
15. C2×#14519710C22semidirect
16. C2×C4⋊#92418C22semidirect
17. C2×#14519701C22semidirect
18. C2×#14519701C22semidirect
24. C2×C22⋊#92418C22semidirect
25. C2×#14519688C22semidirect
31. #16839622C22semidirect
32. C4×C4⋊Dic61C22semidirect
33. #16839524C22semidirect
369. C61#25897semidirect

Maximal subgroups

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

subgroupindex
C2×#168401212
C2×#168400422
#214509922
#214470192
#214469982
#214467522
#214466182
#214457592
#214457502
#214456442
C2×C61×#5462
#2589761