Finite Group Databasebeta

← #29689153

Subgroups — #29689153

Order 15616.

1362 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459148C211yesmaximal, characteristic
2#21448310C211yesmaximal, characteristic, complemented
3#21446265C212yesmaximal, complemented
4#21446260C212yesmaximal, complemented
5#21446121C214yesmaximal, complemented
6#21445916C212yesmaximal, complemented
7#21445762C212yesmaximal, complemented
8C61×#2879C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10#16840340C414yescomplemented
11#16840336—21no—
12#16840333C412yescomplemented
13#16840329—21no—
14#16840326C412yescomplemented
15#16840321—21no—
16#16840321—21no—
17#16840308—22no—
18#16840273C2212yes—
19#16840273C2212yes—
20#16840269—22no—
21C2×#14519736C2211yescharacteristic
22C2×#14519736C2211yescharacteristic, complemented
23C22×C4×Dic61C2211yescharacteristic, complemented
24C2×#14519703C2214yes—
25C2×#14519703C2214yes—
26#16839669—21no—
27#16839665—21no—
28#16839642—21no—
29#16839642—21no—
30C4×C22⋊Dic61C2212yes—
31C4×C22⋊Dic61C2212yescomplemented
32#16839523C2214yescomplemented
33#16839523C2214yescomplemented
34C22×#41×C61C2211yescharacteristic
35C22×M16×C61C2212yescomplemented
36C61×#408—22no—
37C61×#408—22no—
38C61×#407C2212yes—
39C61×#407C2212yescomplemented
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—42no—
43C2×#11758222—42no—
44C2×#11758222C2×C412yescomplemented
45C2×#11758222C2312yes—
46C22×Dic122C2×C411yescharacteristic, complemented
47C22×D244C2×C411yescharacteristic, complemented
48C22×C22⋊D61—21no—
49C23×Dic61—21no—
50C22×C4×D61—21no—
51#14519775—22no—
52#14519773—22no—
53#14519766—22no—
54#14519764—22no—
55#14519760—22no—
56#14519760—22no—
57#14519758—22no—
58#14519758—22no—
59#14519756C2×C414yes—
60#14519756C2×C414yescomplemented
61#14519753—24no—
62#14519751C2×C414yes—
63#14519751C2×C414yescomplemented
64#14519749—24no—
65#14519749—24no—
66#14519748—24no—
67#14519746—22no—
68#14519746—22no—
69#14519745—22no—
70#14519745—22no—
71C4×C22⋊D61—24no—
72C4×C22⋊D61—24no—
73#14519736—21no—
74#14519736—21no—
75#14519736—42no—
76#14519736C2×C412yescomplemented
77#14519736C2312yes—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2×C412yescomplemented
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—22no—
83C2×C22⋊Dic61—22no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—42no—
87C2×C4×Dic61C2312yes—
88C2×C4×Dic61C2312yes—
89#14519703—44no—
90#14519703—44no—
91#14519684—22no—
92#14519684—22no—
93C2×#41×C61—42no—
94C2×#41×C61—42no—
95C2×#41×C61C2×C412yes—
96C2×#41×C61C2×C412yes—
97C2×#41×C61D412yes—
98C2×#41×C61D412yes—
99C22×Q8×C61D411yescharacteristic
100C22×D4×C61D411yescharacteristic
101C22×D4×C61D411yescharacteristic
102C22×D4×C61D411yescharacteristic
103C23×C4×C61C2311yescharacteristic
104C23×C4×C61C2311yescharacteristic
105C23×C4×C61C2311yescharacteristic
106C2×M16×C61—22no—
107C2×M16×C61—42no—
108C2×M16×C61—42no—
109C2×M16×C61C2314yes—
110C22×C8×C61—22no—
111C22⋊C8×C61—22no—
112C22⋊C8×C61—22no—
113C22⋊C8×C61—24no—
114C24×C61C2×D411yescharacteristic
115C24×C61C2×D411yescharacteristic
116C24×C61C2×D411yescharacteristic
117C23×D61—21no—
118#41×C61—44no—
119#41×C61—44no—
120#41×C61—42no—
121#41×C61—42no—
122#41×C61—42no—
123#41×C61—42no—
124C2×Q8×C61—41no—
125C2×Q8×C61—41no—
126C2×Q8×C61C22⋊C412yes—
127C2×Q8×C61C22⋊C412yes—
128C2×D4×C61—41no—
129C2×D4×C61—41no—
130C2×D4×C61—41no—
131C2×D4×C61—41no—
132C2×D4×C61—41no—
133C2×D4×C61—41no—
134C2×D4×C61C22⋊C412yes—
135C2×D4×C61C22⋊C412yes—
136C2×D4×C61C22⋊C412yes—
137C2×D4×C61C22⋊C412yes—
138C2×D4×C61C22⋊C412yes—
139C2×D4×C61C22⋊C412yes—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C22×C412yes—
149C22×C4×C61C22×C412yes—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D412yes—
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C2×C22⋊D61—24no—
161C2×C22⋊D61—24no—
162C2×C22⋊D61—42no—
163C2×C22⋊D61—42no—
164C22×Dic61—21no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—44no—
169C22×Dic61—42no—
170#11758222—24no—
171#11758222—24no—
172#11758222—44no—
173#11758222—42no—
174#11758222—42no—
175#11758222—44no—
176C2×D244—21no—
177C2×D244—42no—
178C2×D244C22×C412yes—
179C2×C4×D61—22no—
180C2×C4×D61—22no—
181C2×C4×D61—44no—
182C2×Dic122—21no—
183C2×Dic122—42no—
184C2×Dic122C22×C412yes—
185M16×C61—24no—
186M16×C61—24no—
187M16×C61—44no—
188C2×C8×C61—24no—
189C2×C8×C61—44no—
190C22⋊Dic61—22no—
191C22⋊Dic61—24no—
192C22⋊Dic61—24no—
193C22⋊Dic61—22no—
194C22⋊Dic61—44no—
195C22⋊Dic61—44no—
196C22⋊Dic61—42no—
197#11758198—24no—
198#11758198—24no—
199C4⋊Dic61—22no—
200C4⋊Dic61—42no—
201C4⋊Dic61C22×C414yes—
202#11758196—24no—
203#11758196—24no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—24no—
207C4×Dic61—24no—
208C4×Dic61—42no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—42no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61C2×C22⋊C411yescharacteristic
219C23×C61C2×C22⋊C412yes—
220C23×C61C2×C22⋊C412yes—
221C23×C61C2×C22⋊C411yescharacteristic
222C23×C61C22×D411yescharacteristic
223C22×D61—22no—
224C22×D61—42no—
225C22×D61—82no—
226Q8×C61—42no—
227Q8×C61—42no—
228D4×C61—42no—
229D4×C61—42no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61—42no—
233D4×C61—42no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—24no—
243C2×C4×C61—24no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C412yes—
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61H(5,𝔽2)12yes—
257C2×C4×C61H(5,𝔽2)12yes—
258C22⋊D61—44no—
259C22⋊D61—42no—
260C22⋊D61—42no—
261C22⋊D61—84no—
262C22⋊D61—84no—
263C2×Dic61—24no—
264C2×Dic61—24no—
265C2×Dic61—22no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—48no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—82no—
274D244—24no—
275D244—42no—
276C4×D61—44no—
277C4×D61—44no—
278Dic122—24no—
279Dic122—42no—
280C8×C61—44no—
281#21775—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—24no—
289C22×C61—24no—
290C22×C61—22no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61C22×C22⋊C411yescharacteristic
298C22×C61#53411yescharacteristic
299C22×C61#53411yescharacteristic
300C2×D61—42no—
301C2×D61—44no—
302C2×D61—84no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—22no—
306C4×C61—22no—
307C4×C61#51812yes—
308C4×C61#53412yes—
309Dic61—44no—
310Dic61—42no—
311Dic61—44no—
312Dic61—82no—
313#3449—611no—
314#2879D6111yescharacteristic, complemented
315#2297—611no—
316#1897—612no—
317#1896—612no—
318#1871—614no—
319#1826—612no—
320#1789—612no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—41no—
324C2×C61—42no—
325C2×C61—41no—
326C2×C61—42no—
327C2×C61#230811yescharacteristic
328C2×C61#290112yes—
329D61—82no—
330C22×#41—611no—
331C22×#41C2×D6111yescharacteristic
332C22×M16C2×D6112yescomplemented
333#537—614no—
334#536—612no—
335#536—1221no—
336#535—612no—
337#535—1221no—
338#534—1221no—
339#534—1221no—
340#532—1222no—
341#525—612no—
342#525—612no—
343#525—1222no—
344C2×#117—611no—
345C2×#117—611no—
346C22×C42—611no—
347C2×#104—614no—
348C2×#104—614no—
349#408—22no—
350#408—22no—
351#407C2×D6112yes—
352#407C2×D6112yescomplemented
353#391—1221no—
354#390—1221no—
355#386—1221no—
356#386—1221no—
357C4×C22⋊C4—612no—
358C4×C22⋊C4—612no—
359#339—614no—
360#339—614no—
361C61#2177511yescharacteristic, complemented, Sylow
362C2×#41—42no—
363C2×#41—42no—
364C2×#41—612no—
365C2×#41—612no—
366C2×#41—1221no—
367C2×#41—1221no—
368C2×#41—2442no—
369C2×#41—2442no—
370C2×#41C4×D6112yes—
371C2×#41C4×D6112yes—
372C2×#41C22⋊D6112yes—
373C2×#41C22⋊D6112yes—
374C22×Q8—611no—
375C22×Q8D24411yescharacteristic
376C22×D4—611no—
377C22×D4—1221no—
378C22×D4D24411yescharacteristic
379C22×D4D24411yescharacteristic
380C22×D4D24411yescharacteristic
381C23×C4—1221no—
382C23×C4—1221no—
383C23×C4C22×D6111yescharacteristic
384C23×C4C22×D6111yescharacteristic
385C23×C4C22×D6111yescharacteristic
386C2×M16—22no—
387C2×M16—42no—
388C2×M16—42no—
389C2×M16C22×D6114yes—
390C22×C8—22no—
391C4⋊Q8—1222no—
392#127—1222no—
393#124—1222no—
394#124—1222no—
395#123—1222no—
396#123—1222no—
397#123—1222no—
398#123—1222no—
399C22⋊Q8—614no—
400C22⋊Q8—614no—
401C22⋊Q8—1224no—
402#121—614no—
403#121—614no—
404#121—1224no—
405#121—1224no—
406#121—1224no—
407#120—1222no—
408#120—1222no—
409#120—1222no—
410#120—1222no—
411C4×D4—1224no—
412C4×D4—1224no—
413#117—612no—
414#117—612no—
415#117—1221no—
416#117—1221no—
417#117—2442no—
418C2×C4⋊C4—612no—
419C2×C4⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1222no—
424C2×C42—612no—
425C2×C42—612no—
426C2×C42—1221no—
427C2×C42—1221no—
428C2×C42—2442no—
429#104—2444no—
430#104—2444no—
431C22⋊C8—22no—
432C22⋊C8—22no—
433C22⋊C8—24no—
434#95—1222no—
435#95—1222no—
436C24—1221no—
437C24C2×D24411yescharacteristic
438C24C2×D24411yescharacteristic
439C24C2×D24411yescharacteristic
440#41—44no—
441#41—44no—
442#41—42no—
443#41—42no—
444#41—42no—
445#41—42no—
446#41—1224no—
447#41—1224no—
448#41—2444no—
449#41—2442no—
450#41—2442no—
451#41—2444no—
452C2×Q8—41no—
453C2×Q8—41no—
454C2×Q8—612no—
455C2×Q8—1221no—
456C2×Q8—2442no—
457C2×Q8#1175819812yes—
458C2×Q8#1175819812yes—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—612no—
466C2×D4—1224no—
467C2×D4—1224no—
468C2×D4—1222no—
469C2×D4—1222no—
470C2×D4—1221no—
471C2×D4—2442no—
472C2×D4—2442no—
473C2×D4—2442no—
474C2×D4#1175819812yes—
475C2×D4#1175819812yes—
476C2×D4#1175819812yes—
477C2×D4#1175819812yes—
478C2×D4#1175819812yes—
479C2×D4#1175819812yes—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1221no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—2444no—
493C22×C4—2444no—
494C22×C4—2442no—
495C22×C4C2×C4×D6111yescharacteristic
496C22×C4C2×C4×D6111yescharacteristic
497C22×C4C2×C4×D6112yes—
498C22×C4C2×C4×D6112yes—
499C22×C4C2×D24411yescharacteristic
500C22×C4C2×D24411yescharacteristic
501C22×C4C2×D24411yescharacteristic
502C22×C4C2×C22⋊D6111yescharacteristic
503C22×C4C2×C22⋊D6111yescharacteristic
504C22×C4C2×C22⋊D6112yes—
505C22×C4C2×C22⋊D6112yes—
506C22×C4C23×D6111yescharacteristic
507M16—24no—
508M16—24no—
509M16—44no—
510C2×C8—24no—
511C2×C8—44no—
512C4⋊C4—614no—
513C4⋊C4—1222no—
514C4⋊C4—1224no—
515C4⋊C4—1224no—
516C4⋊C4—2442no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1224no—
520C22⋊C4—1224no—
521C22⋊C4—1224no—
522C22⋊C4—1224no—
523C22⋊C4—2444no—
524C22⋊C4—2444no—
525C22⋊C4—2442no—
526C42—1222no—
527C42—1222no—
528C42—1224no—
529C42—1224no—
530C42—2442no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—1222no—
541C23—4882no—
542C23—2442no—
543C23C2×#1175819811yescharacteristic
544C23C2×#1175819812yes—
545C23C2×#1175819812yes—
546C23C2×#1175819811yescharacteristic
547C23C22×D24411yescharacteristic
548Q8—42no—
549Q8—42no—
550Q8—1224no—
551Q8—2442no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—1224no—
559D4—4884no—
560D4—4884no—
561D4—2442no—
562D4—2444no—
563D4—2442no—
564D4—2442no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—1224no—
582C2×C4—1224no—
583C2×C4—1222no—
584C2×C4—4882no—
585C2×C4—2444no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2448no—
592C2×C4—2442no—
593C2×C4—2444no—
594C2×C4C2×#1175819811yescharacteristic
595C2×C4C2×#1175819812yes—
596C2×C4C2×#1175819812yes—
597C2×C4C2×#1175819811yescharacteristic
598C2×C4C22×C4×D6111yescharacteristic
599C2×C4C22×C22⋊D6111yescharacteristic
600C2×C4#1451982212yes—
601C2×C4#1451982212yes—
602C8—44no—
603C22—22no—
604C22—22no—
605C22—24no—
606C22—24no—
607C22—22no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—4884no—
619C22—2444no—
620C22—2442no—
621C22C22×#1175819811yescharacteristic
622C22#1684032411yescharacteristic
623C22#1684032411yescharacteristic
624C4—21no—
625C4—21no—
626C4—22no—
627C4—22no—
628C4—4882no—
629C4—2444no—
630C4—2444no—
631C4—2442no—
632C4#1684023512yes—
633C4#1684032112yes—
634C2—21no—
635C2—21no—
636C2—42no—
637C2—41no—
638C2—41no—
639C2—42no—
640C2—4882no—
641C2#2144839411yescharacteristic
642C2#2145436412yes—
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2879C2
socleC22×C61C22×C22⋊C4
prime radical#21459148C2
cosocleC22×C4×C61C24
hypercenter#2879D61
hypocenterC61#21775
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×#41 ◁ #2879

Complements

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

NH ≅ G/Ntype
2. #21448310C2semidirect
3. #21446265C2semidirect
4. #21446260C2semidirect
5. #21446121C2semidirect
6. #21445916C2semidirect
7. #21445762C2semidirect
8. C61×#2879C2semidirect
10. #16840340C4semidirect
12. #16840333C4semidirect
14. #16840326C4semidirect
22. C2×#14519736C22semidirect
23. C22×C4×Dic61C22semidirect
31. C4×C22⋊Dic61C22semidirect
32. #16839523C22semidirect
33. #16839523C22semidirect
35. C22×M16×C61C22semidirect
39. C61×#407C22semidirect
44. C2×#11758222C2×C4semidirect
46. C22×Dic122C2×C4semidirect
47. C22×D244C2×C4semidirect
60. #14519756C2×C4semidirect
63. #14519751C2×C4semidirect
76. #14519736C2×C4semidirect
79. C2×C4⋊Dic61C2×C4semidirect
314. #2879D61semidirect
332. C22×M16C2×D61semidirect
352. #407C2×D61semidirect
361. C61#21775semidirect

Maximal subgroups

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

subgroupindex
#214591482
#214483102
#214462652
#214462602
#214461212
#214459162
#214457622
C61×#28792
#2177561