Finite Group Databasebeta

← #29753216

Subgroups — #29753216

Order 15616.

762 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450406C211yesmaximal, characteristic
2#21448876C211yesmaximal, characteristic, complemented
3#21447359C211yesmaximal, characteristic, complemented
4#21447135C211yesmaximal, characteristic
5#21447064C211yesmaximal, characteristic, complemented
6#21446900C211yesmaximal, characteristic
7#21446450C211yesmaximal, characteristic, complemented
8#21446449C211yesmaximal, characteristic
9#21446399C211yesmaximal, characteristic, complemented
10#21446376C211yesmaximal, characteristic, complemented
11#21446177C211yesmaximal, characteristic
12#21446176C211yesmaximal, characteristic
13#21445822C211yesmaximal, characteristic, complemented
14C4×#14519702C211yesmaximal, characteristic, complemented
15C61×#1968C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×#14519770—21no—
19C2×Q8×Dic61C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29#16839690—21no—
30#16839685—21no—
31#16839667C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839659—21no—
36#16839650—21no—
37#16839649—21no—
38#16839644—21no—
39#16839631C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×#11758196C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43#16839613—21no—
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C2×#11758200—22no—
61C2×#11758200—41no—
62C2×#11758200—41no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519775—22no—
66#14519775—22no—
67#14519770—44no—
68Q8×Dic61—22no—
69Q8×Dic61—41no—
70C4×Dic122—22no—
71C4×Dic122—41no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196D411yescharacteristic
83C2×#11758196D411yescharacteristic
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519708—41no—
95#14519706—41no—
96#14519702—22no—
97#14519702—22no—
98#14519701—22no—
99#14519701—41no—
100#14519700—41no—
101#14519698—22no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×Q8×C61C2311yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C2×SD16×C61—22no—
112C2×SD16×C61—22no—
113C2×SD16×C61—41no—
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#106×C61—22no—
122#103×C61—41no—
123#102×C61—41no—
124C4×C8×C61—41no—
125#95×C61—21no—
126#95×C61—21no—
127C24×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61#4112yes—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122#4112yes—
160SD16×C61—44no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61#4112yes—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C22⋊C4×C61—42no—
172C22⋊C4×C61—81no—
173C42×C61—21no—
174C42×C61—21no—
175C22⋊Dic61—42no—
176C22⋊Dic61—42no—
177C22⋊Dic61—42no—
178C22⋊Dic61—81no—
179#11758200—44no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—42no—
186C4⋊Dic61—42no—
187C4⋊Dic61#4112yes—
188#11758196—24no—
189#11758196—24no—
190#11758196—41no—
191#11758196—42no—
192#11758196—42no—
193#11758196—81no—
194#11758196SD1614yescomplemented
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—42no—
200C4×Dic61—42no—
201C4×Dic61—42no—
202C4×Dic61—42no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61C22×D411yescharacteristic
217Q8×C61—22no—
218Q8×C61—22no—
219Q8×C61—41no—
220Q8×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61C2×SD1612yes—
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—22no—
252C2×Dic61—21no—
253C2×Dic61—22no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61C2×SD1612yes—
269C2×Dic61C2×SD1612yes—
270Dic122—22no—
271Dic122—22no—
272Dic122—41no—
273Dic122—41no—
274C8×C61—22no—
275C8×C61—41no—
276#92418—41no—
277#92418—41no—
278#27066—611nomaximal, Sylow
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61#52511yescharacteristic
287C22×C61#54611yescharacteristic
288C22×C61#55011yescharacteristic
289C22×C61C22×SD1611yescharacteristic
290C22×C61#57511yescharacteristic
291C22×C61#57711yescharacteristic
292C22×C61#57811yescharacteristic
293C4×C61—22no—
294C4×C61—42no—
295C4×C61—41no—
296C4×C61—81no—
297C4×C61—81no—
298C4×C61#52911yescharacteristic
299C4×C61#53811yescharacteristic
300C4×C61#55111yescharacteristic
301C4×C61#55311yescharacteristic
302Dic61—22no—
303Dic61—22no—
304Dic61—41no—
305Dic61—42no—
306Dic61—41no—
307Dic61—42no—
308Dic61—81no—
309Dic61—81no—
310#2512—611no—
311#2365—611no—
312#2096—611no—
313#2055—611no—
314#2043—611no—
315#2013—611no—
316#1968D6111yescharacteristic
317#1927—611no—
318#1926—611no—
319#1919—611no—
320#1916—611no—
321#1880—611no—
322#1880—611no—
323#1803—611no—
324C4×#103—611no—
325C2×C61—161no—
326C2×C61#246711yescharacteristic
327C2×C61#310311yescharacteristic
328C2×C61#312111yescharacteristic
329C2×C61#329811yescharacteristic
330C2×C61#330711yescharacteristic
331C2×C61#339211yescharacteristic
332C2×C61#340211yescharacteristic
333C22×SD16—611no—
334C22×SD16C2×D6111yescharacteristic
335C2×C4⋊Q8—611no—
336C2×#126—1221no—
337C2×C4×Q8—611no—
338C2×C4×Q8—611no—
339C2×#106—611no—
340C2×#106C2×D6111yescharacteristic
341C2×C4⋊C8—611no—
342C2×#103—611no—
343C2×#103—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#103C2×D6111yescharacteristic
348C2×#102—611no—
349C2×#102C2×D6111yescharacteristic
350C2×C4×C8C2×D6111yescharacteristic
351#397—1221no—
352#396—1221no—
353#391—611no—
354#391C2×D6111yescharacteristic
355#390—611no—
356#390—611no—
357#390C2×D6111yescharacteristic
358#389—611no—
359#389—1221no—
360#388—1221no—
361#387—1221no—
362#387—1221no—
363#384—611no—
364C4×C4⋊C4—611no—
365C4×C4⋊C4—611no—
366C4×C4⋊C4—611no—
367C4×C4⋊C4—1221no—
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#336—611no—
373#336—611no—
374#336—611no—
375#336—611no—
376C61#2706611yescharacteristic, complemented, Sylow
377C22×Q8—611no—
378C22×Q8C22×D6111yescharacteristic
379C22×D4C22×D6111yescharacteristic
380C2×SD16—22no—
381C2×SD16—22no—
382C2×SD16—41no—
383C2×SD16—1222no—
384C2×SD16—2441no—
385C2×SD16—2441no—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8C22×D6111yescharacteristic
389C22×C8C22×D6111yescharacteristic
390C4⋊Q8—1222no—
391C4⋊Q8—1222no—
392#126—2444no—
393C4×Q8—1222no—
394C4×Q8—1222no—
395C4×Q8—2441no—
396C4×Q8—2441no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—1221no—
420C2×C42—1221no—
421C2×C42—1221no—
422C2×C42—1221no—
423C2×C42C22×D6111yescharacteristic
424#106—22no—
425#106—2441no—
426C4⋊C8—2441no—
427#103—41no—
428#103—1222no—
429#103—1222no—
430#103—1222no—
431#103—2441no—
432#103—2441no—
433#102—41no—
434#102—1222no—
435C4×C8—41no—
436#95—21no—
437#95—21no—
438#95—611no—
439#95—611no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445C24—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—612no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8#1175822212yes—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4D4×D6111yescharacteristic
476C22×C4D4×D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478SD16—44no—
479SD16—2444no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8#1175822412yes—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—614no—
497C4⋊C4—612no—
498C4⋊C4—1224no—
499C4⋊C4—1224no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—4881no—
506C4⋊C4—2441no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4—2442no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C22⋊C4—4881no—
514C22⋊C4—2442no—
515C22⋊C4—2442no—
516C22⋊C4—2442no—
517C42—21no—
518C42—21no—
519C42—1221no—
520C42—1221no—
521C42—1221no—
522C42—1221no—
523C42—2442no—
524C42—2442no—
525C42—2442no—
526C42—2442no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23C2×D4×D6111yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—41no—
538Q8—41no—
539Q8—1222no—
540Q8—1222no—
541Q8—2441no—
542Q8—2441no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—612no—
562C2×C4—612no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1222no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1222no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4SD16×D6112yes—
586C2×C4#41×D6111yescharacteristic
587C2×C4#41×D6111yescharacteristic
588C2×C4C2×#1175822411yescharacteristic
589C2×C4C2×#1175822211yescharacteristic
590C2×C4#1451982211yescharacteristic
591C2×C4#1451982211yescharacteristic
592C8—22no—
593C8—41no—
594C8—2441no—
595C8—2441no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#1684027111yescharacteristic
604C22#1684040111yescharacteristic
605C22#1684043211yescharacteristic
606C22C2×SD16×D6111yescharacteristic
607C22#1684055911yescharacteristic
608C22#1684056811yescharacteristic
609C22#1684057611yescharacteristic
610C4—22no—
611C4—41no—
612C4—42no—
613C4—81no—
614C4—81no—
615C4—1222no—
616C4—1222no—
617C4—4881no—
618C4—4881no—
619C4—2441no—
620C4—2442no—
621C4—2442no—
622C4—2441no—
623C4#1684029111yescharacteristic
624C4#1684034411yescharacteristic
625C4#1684043811yescharacteristic
626C4#1684046111yescharacteristic
627C2—161no—
628C2#2144987611yescharacteristic
629C2#2145584811yescharacteristic
630C2#2145604411yescharacteristic
631C2#2145767911yescharacteristic
632C2#2145780111yescharacteristic
633C2#2145879711yescharacteristic
634C2#2145890111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1968C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1968D61
hypocenterC61#27066
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. #21448876C2semidirect
3. #21447359C2semidirect
5. #21447064C2semidirect
7. #21446450C2semidirect
9. #21446399C2semidirect
10. #21446376C2semidirect
13. #21445822C2semidirect
14. C4×#14519702C2semidirect
194. #11758196SD16semidirect
376. C61#27066semidirect

Maximal subgroups

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

subgroupindex
#214504062
#214488762
#214473592
#214471352
#214470642
#214469002
#214464502
#214464492
#214463992
#214463762
#214461772
#214461762
#214458222
C4×#145197022
C61×#19682
#2706661