Finite Group Databasebeta

← #29647828

Subgroups — #29647828

Order 15616.

1002 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459658C211yesmaximal, characteristic
2#21448792C211yesmaximal, characteristic, complemented
3#21444376C212yesmaximal, complemented
4#21444358C212yesmaximal, complemented
5#21444353C211yesmaximal, characteristic, complemented
6#21444349C211yesmaximal, characteristic, complemented
7#21444345C212yesmaximal, complemented
8#21444340C211yesmaximal, characteristic, complemented
9#21444336C211yesmaximal, characteristic, complemented
10#21444311C212yesmaximal, complemented
11C61×#2973C211yesmaximal, characteristic, complemented
12#16840461—22no—
13#16840458C412yescomplemented
14#16840452C412yescomplemented
15#16840441—22no—
16#16840421C412yescomplemented
17#16840408C412yescomplemented
18#16840359—21no—
19#16840350—21no—
20#16840321—21no—
21#16840296C2212yes—
22#16840273C2212yes—
23C2×#14519756C2211yescharacteristic
24C2×#14519751C2211yescharacteristic
25#16840235—21no—
26#16839665—21no—
27#16839655—21no—
28#16839655—21no—
29#16839628C2211yescharacteristic, complemented
30#16839625C2211yescharacteristic
31#16839625C2211yescharacteristic, complemented
32C4⋊C4×Dic61—21no—
33C4×C4⋊Dic61C2211yescharacteristic, complemented
34C4×C22⋊Dic61C2211yescharacteristic
35C4×C22⋊Dic61C2211yescharacteristic, complemented
36#16839500C2212yescomplemented
37#16839500C2212yescomplemented
38#16839500C2212yescomplemented
39#16839500C2212yescomplemented
40#16839497C2212yes—
41#16839497C2212yes—
42#16839497C2212yes—
43#16839497C2212yes—
44C2×C4×D4×C61C2211yescharacteristic
45C61×#435—22no—
46C61×#435—22no—
47C61×#432C2212yescomplemented
48C61×#407C2211yescharacteristic
49C61×#407C2211yescharacteristic
50C2×C22⋊C8×C61C2212yescomplemented
51C2×#11758222C2×C412yes—
52C22×Dic122C2×C411yescharacteristic
53C22×D244C2×C411yescharacteristic
54C22×C22⋊D61—21no—
55#14519772—22no—
56#14519771—22no—
57#14519770—22no—
58#14519769C2×C412yes—
59#14519767—22no—
60#14519765C2×C412yes—
61#14519763—22no—
62#14519762—22no—
63#14519760—22no—
64#14519760—22no—
65#14519759—22no—
66#14519758—22no—
67#14519756—21no—
68#14519756—21no—
69#14519756C2×C412yes—
70#14519756C2×C412yes—
71#14519756C2312yes—
72#14519755—22no—
73#14519751—21no—
74#14519751—21no—
75#14519751C2×C412yes—
76#14519751C2×C412yes—
77#14519751C2312yes—
78#14519750—22no—
79#14519749—22no—
80#14519749—22no—
81#14519746—22no—
82#14519745—22no—
83C4×Dic122C2×C412yes—
84C4×D244C2×C412yes—
85C4×C22⋊D61—22no—
86C4×C22⋊D61—22no—
87C4×C22⋊D61—22no—
88#14519737—22no—
89#14519737—22no—
90#14519736C2×C412yes—
91#14519734—22no—
92#14519733—22no—
93C2×C4⋊Dic61—21no—
94C2×C4⋊Dic61—21no—
95C2×C4⋊Dic61C2×C411yescharacteristic
96C2×C4⋊Dic61C2×C411yescharacteristic
97C2×#11758196—21no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×C22⋊Dic61—21no—
101C2×C22⋊Dic61—21no—
102C2×C22⋊Dic61—21no—
103C2×#11758198—21no—
104C22⋊C4×D61—22no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684C2311yescharacteristic
112#14519684C2311yescharacteristic
113#14519684C2311yescharacteristic
114#14519684C2311yescharacteristic
115C22×D4×C61D411yescharacteristic
116C23×C4×C61C2311yescharacteristic
117C23×C4×C61C2311yescharacteristic
118C2×M16×C61—21no—
119C2×M16×C61—21no—
120C22×C8×C61—22no—
121C4×D4×C61C2×C412yes—
122C4×D4×C61C2×C412yes—
123C4×D4×C61D412yes—
124C4×D4×C61D412yes—
125C2×C4⋊C4×C61D411yescharacteristic
126C2×C22⋊C4×C61D411yescharacteristic
127C2×C22⋊C4×C61D411yescharacteristic
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—22no—
130C22⋊C8×C61—22no—
131C22⋊C8×C61—21no—
132C22⋊C8×C61—21no—
133C22⋊C8×C61C2312yes—
134C22⋊C8×C61C2312yes—
135#97×C61—22no—
136C24×C61C2×D411yescharacteristic
137C24×C61C2×D411yescharacteristic
138C23×D61—21no—
139C2×D4×C61—41no—
140C2×D4×C61—41no—
141C2×D4×C61C22⋊C412yes—
142C2×D4×C61C22⋊C412yes—
143C22×C4×C61—81no—
144C22×C4×C61—81no—
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C2×C22⋊D61—42no—
161C2×C22⋊D61—42no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169#11758222—82no—
170C2×D244—21no—
171C2×D244—81no—
172C2×D244C22×C412yes—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×Dic122—21no—
176C2×Dic122—81no—
177C2×Dic122C22×C412yes—
178M16×C61—81no—
179M16×C61—81no—
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C2×C8×C61—82no—
183C4⋊C4×C61C22⋊C412yes—
184C4⋊C4×C61C22⋊C412yes—
185C22⋊C4×C61C22⋊C412yes—
186C22⋊C4×C61C22⋊C412yes—
187C22⋊C4×C61C22⋊C412yes—
188C22⋊C4×C61C22⋊C412yes—
189C42×C61C22×C412yes—
190C42×C61C2×D412yes—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61—42no—
196C22⋊Dic61—42no—
197C22⋊Dic61—81no—
198C22⋊Dic61—81no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—42no—
208C4⋊Dic61—42no—
209C4⋊Dic61C22×C412yes—
210C4⋊Dic61C22×C412yes—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216#11758196—22no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—42no—
220C4×Dic61—81no—
221C4×Dic61—81no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61C2×C22⋊C411yescharacteristic
231C23×C61C2×C22⋊C412yes—
232C23×C61C2×C22⋊C411yescharacteristic
233C23×C61C22×D411yescharacteristic
234C22×D61—22no—
235C22×D61—42no—
236C22×D61—81no—
237C22×D61—81no—
238D4×C61—42no—
239D4×C61—42no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—41no—
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C22⋊D61—44no—
260C22⋊D61—42no—
261C22⋊D61—42no—
262C22⋊D61—82no—
263C22⋊D61—82no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—41no—
269C2×Dic61—42no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—81no—
281C2×Dic61—81no—
282D244—42no—
283D244—81no—
284C4×D61—82no—
285C4×D61—82no—
286Dic122—42no—
287Dic122—81no—
288C8×C61—42no—
289C8×C61—42no—
290#17625—611nomaximal, Sylow
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—22no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—22no—
299C22×C61—21no—
300C22×C61—22no—
301C22×C61—21no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61C22×C22⋊C411yescharacteristic
307C22×C61#57511yescharacteristic
308C22×C61#57511yescharacteristic
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—162no—
313C4×C61—21no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—22no—
317C4×C61—21no—
318C4×C61—22no—
319C4×C61—22no—
320Dic61—42no—
321Dic61—42no—
322Dic61—42no—
323Dic61—42no—
324Dic61—81no—
325Dic61—81no—
326Dic61—81no—
327#3505—611no—
328#2973D6111yescharacteristic, complemented
329#2357—611no—
330#1516—612no—
331#1512—612no—
332#1511—611no—
333#1510—611no—
334#1509—612no—
335#1508—611no—
336#1507—611no—
337#1501—612no—
338C2×C61—21no—
339C2×C61—21no—
340C2×C61—41no—
341C2×C61—41no—
342C2×C61—42no—
343C2×C61#288311yescharacteristic
344C2×C61#289611yescharacteristic
345C2×C61#290111yescharacteristic
346D61—82no—
347#553—612no—
348#553—1222no—
349#552—612no—
350#551—1222no—
351#548—612no—
352#546—612no—
353#539—1221no—
354#538—1221no—
355#534—1221no—
356#529—612no—
357#525—612no—
358C2×C22⋊Q8—611no—
359C2×#121—611no—
360#518—1221no—
361C2×C4×D4C2×D6111yescharacteristic
362#435—22no—
363#435—22no—
364#432C2×D6112yescomplemented
365#407C2×D6111yescharacteristic
366#407C2×D6111yescharacteristic
367C2×C22⋊C8C2×D6112yescomplemented
368#390—1221no—
369#388—1221no—
370#388—1221no—
371#382—611no—
372#381—611no—
373#381—611no—
374C4×C4⋊C4—611no—
375C4×C4⋊C4—1221no—
376C4×C22⋊C4—611no—
377C4×C22⋊C4—611no—
378#328—612no—
379#328—612no—
380#328—612no—
381#328—612no—
382#327—612no—
383#327—612no—
384#327—612no—
385#327—612no—
386C61#1762511yescharacteristic, complemented, Sylow
387C2×#41—612no—
388C22×Q8—611no—
389C22×D4—611no—
390C22×D4—1221no—
391C22×D4D24411yescharacteristic
392C23×C4C22×D6111yescharacteristic
393C23×C4C22×D6111yescharacteristic
394C2×M16—21no—
395C2×M16—21no—
396C22×C8—22no—
397#126—1222no—
398#126—1222no—
399#126—1222no—
400#125—612no—
401#125—1222no—
402#124—612no—
403#124—1222no—
404#123—1222no—
405#123—1222no—
406#123—1222no—
407#123—1222no—
408#123—1222no—
409C22⋊Q8—612no—
410C22⋊Q8—612no—
411C22⋊Q8—612no—
412C22⋊Q8—1221no—
413C22⋊Q8—1221no—
414C22⋊Q8—1222no—
415#121—612no—
416#121—612no—
417#121—612no—
418#121—1222no—
419#121—1222no—
420#121—1221no—
421#121—1221no—
422#121—1222no—
423#120—1222no—
424#120—1222no—
425C4×Q8—612no—
426C4×D4—612no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1222no—
432C4×D4C4×D6112yes—
433C4×D4C4×D6112yes—
434C4×D4C22⋊D6112yes—
435C4×D4C22⋊D6112yes—
436#117—612no—
437#117—1222no—
438#117—1222no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4D24411yescharacteristic
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1222no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4D24411yescharacteristic
453C2×C22⋊C4D24411yescharacteristic
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42—1221no—
457C2×C42C22×D6111yescharacteristic
458C4⋊C8—22no—
459C22⋊C8—22no—
460C22⋊C8—21no—
461C22⋊C8—21no—
462C22⋊C8C22×D6112yes—
463C22⋊C8C22×D6112yes—
464#97—22no—
465#95—611no—
466#95—611no—
467#95—611no—
468#95—611no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24—1221no—
473C24C2×D24411yescharacteristic
474C24C2×D24411yescharacteristic
475#41—4882no—
476C2×Q8—612no—
477C2×Q8—1221no—
478C2×Q8—4881no—
479C2×D4—41no—
480C2×D4—41no—
481C2×D4—612no—
482C2×D4—1222no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—1222no—
486C2×D4—1221no—
487C2×D4—4881no—
488C2×D4—2442no—
489C2×D4—2442no—
490C2×D4#1175819812yes—
491C2×D4#1175819812yes—
492C22×C4—81no—
493C22×C4—81no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4C2×C4×D6111yescharacteristic
504C22×C4C2×C4×D6111yescharacteristic
505C22×C4C2×C4×D6112yes—
506C22×C4C2×D24411yescharacteristic
507C22×C4C2×D24411yescharacteristic
508C22×C4C2×D24411yescharacteristic
509C22×C4C2×D24411yescharacteristic
510C22×C4C2×C22⋊D6111yescharacteristic
511C22×C4C2×C22⋊D6111yescharacteristic
512C22×C4C2×C22⋊D6112yes—
513C22×C4C23×D6111yescharacteristic
514M16—81no—
515M16—81no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—82no—
519C4⋊C4—612no—
520C4⋊C4—612no—
521C4⋊C4—1222no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—2442no—
530C4⋊C4—2442no—
531C4⋊C4#1175819812yes—
532C4⋊C4#1175819812yes—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C22⋊C4—4881no—
544C22⋊C4—4881no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C22⋊C4#1175819812yes—
548C22⋊C4#1175819812yes—
549C22⋊C4#1175819812yes—
550C22⋊C4#1175819812yes—
551C42—1222no—
552C42—1222no—
553C42—4881no—
554C42—4881no—
555C42—2442no—
556C42C2×C4×D6112yes—
557C42C2×C22⋊D6112yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—1222no—
567C23—4881no—
568C23—4881no—
569C23—2442no—
570C23C2×#1175819811yescharacteristic
571C23C2×#1175819812yes—
572C23C2×#1175819811yescharacteristic
573C23C22×D24411yescharacteristic
574Q8—4881no—
575Q8—2442no—
576D4—42no—
577D4—42no—
578D4—4881no—
579D4—4882no—
580D4—4882no—
581D4—2442no—
582D4—2444no—
583D4—2442no—
584D4—2442no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—4882no—
602C2×C4—4882no—
603C2×C4—4881no—
604C2×C4—4881no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2442no—
616C2×C4—2441no—
617C2×C4C2×#1175819812yes—
618C2×C4C2×#1175819812yes—
619C2×C4C2×#1175819812yes—
620C2×C4C2×#1175819811yescharacteristic
621C2×C4C2×#1175819811yescharacteristic
622C2×C4C22×C4×D6111yescharacteristic
623C2×C4C22×C22⋊D6111yescharacteristic
624C8—42no—
625C8—42no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—22no—
631C22—21no—
632C22—22no—
633C22—21no—
634C22—21no—
635C22—22no—
636C22—21no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—9762no—
642C22—2442no—
643C22—2442no—
644C22—2442no—
645C22C22×#1175819811yescharacteristic
646C22#1684056211yescharacteristic
647C22#1684056211yescharacteristic
648C4—22no—
649C4—22no—
650C4—21no—
651C4—21no—
652C4—22no—
653C4—21no—
654C4—21no—
655C4—4881no—
656C4—4881no—
657C4—4881no—
658C4—2442no—
659C4—2442no—
660C4—2442no—
661C4—2442no—
662C2—21no—
663C2—21no—
664C2—41no—
665C2—41no—
666C2—42no—
667C2—4882no—
668C2#2145427911yescharacteristic
669C2#2145433911yescharacteristic
670C2#2145436411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2973C2
socleC22×C61C22×C22⋊C4
prime radical#21459658C2
cosocleC22×C4×C61C24
hypercenter#2973D61
hypocenterC61#17625
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #21448792C2semidirect
3. #21444376C2semidirect
4. #21444358C2semidirect
5. #21444353C2semidirect
6. #21444349C2semidirect
7. #21444345C2semidirect
8. #21444340C2semidirect
9. #21444336C2semidirect
10. #21444311C2semidirect
11. C61×#2973C2semidirect
13. #16840458C4semidirect
14. #16840452C4semidirect
16. #16840421C4semidirect
17. #16840408C4semidirect
29. #16839628C22semidirect
31. #16839625C22semidirect
33. C4×C4⋊Dic61C22semidirect
35. C4×C22⋊Dic61C22semidirect
36. #16839500C22semidirect
37. #16839500C22semidirect
38. #16839500C22semidirect
39. #16839500C22semidirect
47. C61×#432C22semidirect
50. C2×C22⋊C8×C61C22semidirect
328. #2973D61semidirect
364. #432C2×D61semidirect
367. C2×C22⋊C8C2×D61semidirect
386. C61#17625semidirect

Maximal subgroups

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

subgroupindex
#214596582
#214487922
#214443762
#214443582
#214443532
#214443492
#214443452
#214443402
#214443362
#214443112
C61×#29732
#1762561