Finite Group Databasebeta

← #30571026

Subgroups — #30571026

Order 15616.

2038 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
1#21459624C211yesmaximal, characteristic, complemented
2C2×#16840432C211yesmaximal, characteristic
3#21458799C214yesmaximal, complemented
4#21458797C214yesmaximal, complemented
5#21458487C214yesmaximal, complemented
6#21458485C214yesmaximal, complemented
7#21456760C211yesmaximal, characteristic, complemented
8#21456758C214yesmaximal, complemented
9#21455828C211yesmaximal, characteristic, complemented
10#21455823C211yesmaximal, characteristic, complemented
11#21455822C214yesmaximal, complemented
12#21454899C211yesmaximal, characteristic, complemented
13C61×#3208C211yesmaximal, characteristic, complemented
14C22×#11758224—21no—
15Q8×Dic122C2212yescomplemented
16#16840492C2212yescomplemented
17#16840490—22no—
18#16840490C2212yescomplemented
19#16840489—22no—
20#16840487C2212yescomplemented
21#16840432—22no—
22#16840432—22no—
23#16840432C2214yes—
24#16840432C2214yes—
25#16840391—22no—
26#16840390—21no—
27#16840319C2211yescharacteristic
28#16840314C2214yes—
29C2×#14519773C2212yes—
30C2×#14519749—22no—
31C2×#14519748—21no—
32#16840239—21no—
33C2×C4×Dic122C2211yescharacteristic
34C2×C4×D244C2211yescharacteristic
35C2×D4×Dic61C2212yes—
36#16840134C2212yescomplemented
37#16840133C2218yescomplemented
38#16840108C2212yescomplemented
39#16840107C2218yescomplemented
40#16840093C2212yes—
41#16840090C2212yes—
42#16840053C2212yescomplemented
43#16840050C2218yescomplemented
44#16840049C2212yescomplemented
45#16839968C2212yescomplemented
46#16839965C2218yescomplemented
47#16839964C2212yescomplemented
48#16839961C2218yescomplemented
49#16839960C2218yescomplemented
50#16839959C2214yescomplemented
51#16839915C2212yes—
52#16839913C2214yes—
53#16839912C2212yes—
54C8×Dic122C2214yescomplemented
55#16839886C2214yescomplemented
56#16839853C2218yescomplemented
57C4×#11758190C2214yescomplemented
58SD16×Dic61C2218yescomplemented
59C4×#11758201C2214yescomplemented
60#16839815C2211yescharacteristic
61#16839812C2214yes—
62C61×#533C2211yescharacteristic
63C2×#127×C61C2211yescharacteristic
64C61×#492C2212yescomplemented
65C61×#486C2212yescomplemented
66C61×#460C2214yescomplemented
67C61×#450C2214yes—
68C61×#431C2211yescharacteristic
69C2×#11758224—24no—
70C2×#11758224—24no—
71C2×#11758224—48no—
72C22×Dic122D411yescharacteristic
73C22×D244C2311yescharacteristic
74C22×C22⋊D61—22no—
75C22×C22⋊D61—22no—
76C23×Dic61—22no—
77C22×C4×D61—21no—
78C2×#11758190C2314yes—
79C2×#11758200C2318yes—
80C2×#11758201C2314yes—
81#14519777—22no—
82#14519777C2311yescharacteristic
83#14519777C2311yescharacteristic
84#14519776—22no—
85#14519776—24no—
86#14519776C2314yes—
87#14519775—24no—
88#14519775C2314yes—
89#14519773—24no—
90#14519773C2318yes—
91#14519769—21no—
92#14519769—22no—
93#14519768—22no—
94#14519767—24no—
95#14519767—24no—
96#14519767—24no—
97#14519756—21no—
98#14519756C2312yes—
99#14519754—24no—
100#14519749—28no—
101#14519749—28no—
102#14519748—24no—
103#14519748—24no—
104Q8×Dic61C2314yes—
105C4×Dic122—22no—
106C4×Dic122—22no—
107C4×Dic122—21no—
108C4×Dic122D412yes—
109C4×Dic122D412yes—
110C4×Dic122C2312yes—
111C4×Dic122C2312yes—
112C4×Dic122C2312yes—
113#14519742—22no—
114#14519742—22no—
115#14519742—24no—
116#14519742C2314yes—
117C4×D244—21no—
118C4×D244—21no—
119C4×D244C2312yes—
120C4×D244C2312yes—
121D4×Dic61—24no—
122D4×Dic61C2318yes—
123#14519736—21no—
124#14519733—24no—
125C2×C4⋊Dic61D411yescharacteristic
126C2×C4⋊Dic61C2311yescharacteristic
127C2×#11758196D412yes—
128C2×C22⋊Dic61—22no—
129C2×#11758198—21no—
130C2×C4×Dic61C2312yes—
131#14519709C2314yes—
132#14519708C2314yes—
133C4⋊#92418C2312yes—
134#14519706C2318yes—
135#14519702C2314yes—
136#14519701C2314yes—
137#14519699C2314yes—
138#14519698C2318yes—
139#14519697C2314yes—
140C22⋊#92418C2312yes—
141C4×#92418C2312yes—
142C8×Dic61C2318yes—
143C22×D4×C61—21no—
144C22×D4×C61—21no—
145C22×D4×C61C2312yes—
146C2×SD16×C61C2318yes—
147C4⋊Q8×C61C2311yescharacteristic
148C4⋊Q8×C61C2311yescharacteristic
149#127×C61—21no—
150#127×C61—21no—
151#127×C61C2312yes—
152#127×C61C2312yes—
153#125×C61—21no—
154C22⋊Q8×C61—21no—
155C22⋊Q8×C61C2312yes—
156C4×Q8×C61—21no—
157C4×Q8×C61C2312yes—
158#117×C61—21no—
159C2×C42×C61C2311yescharacteristic
160C4⋊C8×C61C2312yes—
161#102×C61C2318yes—
162C22⋊C8×C61C2312yes—
163C4×C8×C61C2312yes—
164C24×C61—22no—
165C23×D61—21no—
166C2×Q8×C61—21no—
167C2×Q8×C61C2412yes—
168C2×D4×C61—22no—
169C2×D4×C61—22no—
170C2×D4×C61—22no—
171C2×D4×C61—22no—
172C2×D4×C61—22no—
173C2×D4×C61—44no—
174C2×D4×C61—44no—
175C2×D4×C61—82no—
176C2×D4×C61C2414yes—
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2411yescharacteristic
180C2×C22⋊D61—28no—
181C2×C22⋊D61—28no—
182C2×C22⋊D61—48no—
183C2×C22⋊D61—84no—
184C22×Dic61—21no—
185C22×Dic61—28no—
186C22×Dic61—48no—
187C22×Dic61C2×D412yes—
188C22×Dic61C2×D412yes—
189#11758224—416no—
190#11758224—416no—
191C2×D244—21no—
192C2×D244—42no—
193C2×D244C2412yes—
194C2×C4×D61—22no—
195C2×C4×D61—22no—
196C2×C4×D61—44no—
197C2×Dic122—21no—
198C2×Dic122—24no—
199C2×Dic122SD1618yescomplemented
200C2×Dic122C2×D412yes—
201C2×Dic122C2×D412yes—
202C2×Dic122C2412yes—
203SD16×C61—48no—
204C2×C8×C61C2414yes—
205C4⋊C4×C61—21no—
206C4⋊C4×C61—21no—
207C4⋊C4×C61—21no—
208C4⋊C4×C61—22no—
209C4⋊C4×C61—22no—
210C4⋊C4×C61C2412yes—
211C22⋊C4×C61—22no—
212C42×C61—22no—
213C42×C61C2×D411yescharacteristic
214C42×C61C2×D411yescharacteristic
215C42×C61C2411yescharacteristic
216C42×C61C2411yescharacteristic
217C22⋊Dic61—22no—
218C22⋊Dic61—28no—
219#11758201—44no—
220#11758200—48no—
221#11758198—22no—
222#11758198—22no—
223C4⋊Dic61—21no—
224C4⋊Dic61—21no—
225C4⋊Dic61—21no—
226C4⋊Dic61—21no—
227C4⋊Dic61—24no—
228C4⋊Dic61C2×D412yes—
229C4⋊Dic61C2×D412yes—
230C4⋊Dic61C2412yes—
231C4⋊Dic61C2412yes—
232#11758196—22no—
233#11758196—22no—
234#11758196—24no—
235#11758196—24no—
236#11758196—24no—
237#11758196—24no—
238#11758196C2×D414yes—
239#11758196C2×D414yes—
240C4×Dic61—22no—
241C4×Dic61—24no—
242C4×Dic61—24no—
243C4×Dic61C2×D414yes—
244C4×Dic61C2414yes—
245C2×#92418C2414yes—
246#11758190—44no—
247C23×C61—24no—
248C23×C61—44no—
249C23×C61—44no—
250C23×C61—44no—
251C23×C61C22×D411yescharacteristic
252C22×D61—22no—
253C22×D61—42no—
254C22×D61—42no—
255C22×D61—42no—
256Q8×C61—24no—
257Q8×C61—81no—
258D4×C61—28no—
259D4×C61—44no—
260D4×C61—44no—
261D4×C61—44no—
262D4×C61—82no—
263D4×C61—82no—
264C2×C4×C61—22no—
265C2×C4×C61—22no—
266C2×C4×C61—22no—
267C2×C4×C61—41no—
268C2×C4×C61C2×SD1614yes—
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C2511yescharacteristic
275C22⋊D61—416no—
276C22⋊D61—88no—
277C2×Dic61—22no—
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282C2×Dic61—416no—
283C2×Dic61C2×SD1618yes—
284C2×Dic61C22×D414yes—
285C2×Dic61C22×D414yes—
286D244—22no—
287D244—22no—
288D244—42no—
289C4×D61—44no—
290C4×D61—44no—
291Dic122—24no—
292Dic122—84no—
293Dic122—81no—
294Dic122C2×SD1618yes—
295Dic122C2×SD1618yes—
296C8×C61—24no—
297#92418—24no—
298#61881—611nomaximal, Sylow
299C22×C61—21no—
300C22×C61—21no—
301C22×C61—44no—
302C22×C61—44no—
303C22×C61—44no—
304C22×C61—88no—
305C22×C61C22×SD1612yes—
306C22×C61C23×D411yescharacteristic
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—84no—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—21no—
314C4×C61—42no—
315C4×C61—42no—
316C4×C61C22×SD1612yes—
317C4×C61C22×SD1612yes—
318C4×C61#58512yes—
319Dic61—24no—
320Dic61—42no—
321Dic61—42no—
322Dic61—42no—
323Dic61C22×SD1618yes—
324#3502—611no—
325C2×#550—611no—
326#3392—614no—
327#3392—614no—
328#3362—614no—
329#3362—614no—
330#3208—614no—
331#3208—611no—
332#3208D6111yescharacteristic, complemented
333#3101—614no—
334#3101—611no—
335#3101—611no—
336#2982—611no—
337C2×C61—22no—
338C2×C61—84no—
339C2×C61#348711yescharacteristic
340C2×C61C23×SD1611yescharacteristic
341C2×C61#358511yescharacteristic
342D61—82no—
343C22×#41—1221no—
344Q82—612no—
345Q82—612no—
346#557—612no—
347#557—612no—
348#557—1222no—
349#557—1222no—
350#550—614no—
351#550—614no—
352#550—1222no—
353#550—1222no—
354#544—1221no—
355#544—1222no—
356#533—614no—
357#533—611no—
358#533C2×D6111yescharacteristic
359C2×#127—612no—
360C2×#127C2×D6111yescharacteristic
361C2×#121—1221no—
362C2×#121—1222no—
363#519—1221no—
364C2×C4×Q8—611no—
365C2×C4×D4—612no—
366C2×C4×D4—611no—
367#492—618no—
368#492—612no—
369#492C2×D6112yescomplemented
370#486—612no—
371#486—618no—
372#486C2×D6112yescomplemented
373#483—612no—
374#483—612no—
375#476—612no—
376#476—612no—
377#476—618no—
378#461—618no—
379#461—612no—
380#461—612no—
381#460—618no—
382#460—614no—
383#460—618no—
384#460C2×D6114yescomplemented
385#450—614no—
386#450—612no—
387#450—612no—
388#450C2×D6114yes—
389C8×Q8—614no—
390C8×Q8—614no—
391C4×SD16—618no—
392C4×SD16—614no—
393C4×SD16—618no—
394C4×SD16—614no—
395#431—611no—
396#431—614no—
397#431C2×D6111yescharacteristic
398C61#6188111yescharacteristic, complemented, Sylow
399C2×#41—1224no—
400C2×#41—1224no—
401C2×#41—2448no—
402C22×Q8—611no—
403C22×D4—21no—
404C22×D4—21no—
405C22×D4—611no—
406C22×D4—1222no—
407C22×D4—1222no—
408C22×D4C22×D6112yes—
409C23×C4—1222no—
410C23×C4—1221no—
411C2×SD16—614no—
412C2×SD16—618no—
413C2×SD16—614no—
414C2×SD16C22×D6118yes—
415C4⋊Q8—614no—
416C4⋊Q8—614no—
417C4⋊Q8—611no—
418C4⋊Q8—611no—
419C4⋊Q8—1222no—
420C4⋊Q8—1224no—
421C4⋊Q8—1222no—
422C4⋊Q8—1224no—
423C4⋊Q8C22×D6111yescharacteristic
424C4⋊Q8C22×D6111yescharacteristic
425#127—21no—
426#127—21no—
427#127—618no—
428#127—1224no—
429#127C22×D6112yes—
430#127C22×D6112yes—
431#125—21no—
432#125—1222no—
433#125—1224no—
434#125—1224no—
435#125—1224no—
436#125—1222no—
437#125—1221no—
438C22⋊Q8—21no—
439C22⋊Q8—612no—
440C22⋊Q8—1224no—
441C22⋊Q8—1221no—
442C22⋊Q8C22×D6112yes—
443#121—1224no—
444#121—1224no—
445#121—1228no—
446#121—1228no—
447C4×Q8—21no—
448C4×Q8—612no—
449C4×Q8—612no—
450C4×Q8—612no—
451C4×Q8—612no—
452C4×Q8—614no—
453C4×Q8—614no—
454C4×Q8—612no—
455C4×Q8—1222no—
456C4×Q8—1222no—
457C4×Q8—1224no—
458C4×Q8—1222no—
459C4×Q8—1222no—
460C4×Q8—1221no—
461C4×Q8C22×D6112yes—
462C4×D4—618no—
463C4×D4—612no—
464C4×D4—612no—
465C4×D4—1224no—
466C4×D4—1221no—
467C4×D4—1221no—
468#117—21no—
469#117—1224no—
470#117—1221no—
471C2×C4⋊C4—612no—
472C2×C4⋊C4—611no—
473C2×C4⋊C4—611no—
474C2×C22⋊C4—1222no—
475C2×C22⋊C4—1221no—
476C2×C42—612no—
477C2×C42C22×D6111yescharacteristic
478#106—614no—
479#106—614no—
480C4⋊C8—618no—
481C4⋊C8—612no—
482C4⋊C8C22×D6112yes—
483#103—614no—
484#103—614no—
485#102—618no—
486#102—614no—
487#102—614no—
488#102C22×D6118yes—
489C22⋊C8—612no—
490C22⋊C8C22×D6112yes—
491C4×C8—618no—
492C4×C8—612no—
493C4×C8C22×D6112yes—
494C24—22no—
495C24—1221no—
496#41—24416no—
497#41—24416no—
498C2×Q8—21no—
499C2×Q8—612no—
500C2×Q8—612no—
501C2×Q8—618no—
502C2×Q8—612no—
503C2×Q8—1224no—
504C2×Q8—1221no—
505C2×Q8C23×D6112yes—
506C2×D4—22no—
507C2×D4—22no—
508C2×D4—22no—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—44no—
512C2×D4—44no—
513C2×D4—82no—
514C2×D4—612no—
515C2×D4—1228no—
516C2×D4—1228no—
517C2×D4—1221no—
518C2×D4—4884no—
519C2×D4—2442no—
520C2×D4—2448no—
521C2×D4C23×D6114yes—
522C22×C4—612no—
523C22×C4—612no—
524C22×C4—1228no—
525C22×C4—1221no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4—2448no—
529C22×C4—2444no—
530C22×C4D4×D6111yescharacteristic
531C22×C4D4×D6111yescharacteristic
532C22×C4C23×D6111yescharacteristic
533SD16—48no—
534SD16—2444no—
535SD16—2448no—
536SD16—2444no—
537C2×C8—614no—
538C2×C8C23×D6114yes—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—614no—
545C4⋊C4—614no—
546C4⋊C4—612no—
547C4⋊C4—612no—
548C4⋊C4—612no—
549C4⋊C4—612no—
550C4⋊C4—1224no—
551C4⋊C4—1224no—
552C4⋊C4—1224no—
553C4⋊C4—1224no—
554C4⋊C4—1224no—
555C4⋊C4—1221no—
556C4⋊C4—1222no—
557C4⋊C4—1222no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4C23×D6112yes—
562C22⋊C4—22no—
563C22⋊C4—1228no—
564C22⋊C4—1222no—
565C22⋊C4—1222no—
566C22⋊C4—1222no—
567C42—22no—
568C42—614no—
569C42—614no—
570C42—1224no—
571C42—1224no—
572C42—1222no—
573C42D4×D6111yescharacteristic
574C42D4×D6111yescharacteristic
575C42C23×D6111yescharacteristic
576C42C23×D6111yescharacteristic
577C23—24no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—1222no—
582C23—2442no—
583C23—2442no—
584C23—2442no—
585C23C2×D4×D6111yescharacteristic
586Q8—24no—
587Q8—81no—
588Q8—618no—
589Q8—618no—
590Q8—1224no—
591Q8—4881no—
592Q8—4884no—
593D4—28no—
594D4—44no—
595D4—44no—
596D4—44no—
597D4—82no—
598D4—82no—
599D4—1222no—
600D4—1222no—
601D4—4888no—
602D4—2442no—
603D4—24416no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—41no—
608C2×C4—618no—
609C2×C4—614no—
610C2×C4—614no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—24416no—
615C2×C4—2442no—
616C2×C4—2442no—
617C2×C4—2444no—
618C2×C4—2444no—
619C2×C4SD16×D6114yes—
620C2×C4C2×D4×D6111yescharacteristic
621C2×C4C2×D4×D6111yescharacteristic
622C2×C4C2×D4×D6111yescharacteristic
623C2×C4C2×D4×D6111yescharacteristic
624C2×C4C2×D4×D6111yescharacteristic
625C2×C4C24×D6111yescharacteristic
626C8—24no—
627C8—1224no—
628C22—21no—
629C22—21no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—88no—
634C22—4884no—
635C22—2442no—
636C22—2442no—
637C22—2442no—
638C22C2×SD16×D6112yes—
639C22C22×D4×D6111yescharacteristic
640C4—21no—
641C4—21no—
642C4—21no—
643C4—42no—
644C4—42no—
645C4—618no—
646C4—1224no—
647C4—2442no—
648C4—2442no—
649C4—2442no—
650C4C2×SD16×D6112yes—
651C4C2×SD16×D6112yes—
652C4#1684059712yes—
653C2—22no—
654C2—84no—
655C2—4882no—
656C2#2145947511yescharacteristic
657C2C22×SD16×D6111yescharacteristic
658C2#2146059511yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3208C2
socleC22×C61C23×D4
prime radicalC2×#16840432C2
hypercenter#3208D61
hypocenterC61#61881
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #21459624C2semidirect
3. #21458799C2semidirect
4. #21458797C2semidirect
5. #21458487C2semidirect
6. #21458485C2semidirect
7. #21456760C2semidirect
8. #21456758C2semidirect
9. #21455828C2semidirect
10. #21455823C2semidirect
11. #21455822C2semidirect
12. #21454899C2semidirect
13. C61×#3208C2semidirect
15. Q8×Dic122C22semidirect
16. #16840492C22semidirect
18. #16840490C22semidirect
20. #16840487C22semidirect
36. #16840134C22semidirect
37. #16840133C22semidirect
38. #16840108C22semidirect
39. #16840107C22semidirect
42. #16840053C22semidirect
43. #16840050C22semidirect
44. #16840049C22semidirect
45. #16839968C22semidirect
46. #16839965C22semidirect
47. #16839964C22semidirect
48. #16839961C22semidirect
49. #16839960C22semidirect
50. #16839959C22semidirect
54. C8×Dic122C22semidirect
55. #16839886C22semidirect
56. #16839853C22semidirect
57. C4×#11758190C22semidirect
58. SD16×Dic61C22semidirect
59. C4×#11758201C22semidirect
64. C61×#492C22semidirect
65. C61×#486C22semidirect
66. C61×#460C22semidirect
199. C2×Dic122SD16semidirect
332. #3208D61semidirect
369. #492C2×D61semidirect
372. #486C2×D61semidirect
384. #460C2×D61semidirect
398. C61#61881semidirect

Maximal subgroups

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

subgroupindex
#214596242
C2×#168404322
#214587992
#214587972
#214584872
#214584852
#214567602
#214567582
#214558282
#214558232
#214558222
#214548992
C61×#32082
#6188161