Finite Group Databasebeta

← #29681113

Subgroups — #29681113

Order 15616.

1458 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459020C211yesmaximal, characteristic
2C2×#16839735C211yesmaximal, characteristic
3#21448442C211yesmaximal, characteristic, complemented
4#21448439C211yesmaximal, characteristic, complemented
5#21448407C211yesmaximal, characteristic
6#21446101C218yesmaximal, complemented
7#21445551C211yesmaximal, characteristic, complemented
8C61×#1740C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10#16840243—21no—
11#16840243—21no—
12#16840242—21no—
13#16840242—21no—
14#16840239—21no—
15#16840236—21no—
16#16840235C2214yes—
17C4×#11758222—22no—
18#16840230—22no—
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#117×D61—22no—
22C22×C4⋊Dic61C2211yescharacteristic
23#16839735C418yescomplemented
24C2×#14519691C2214yes—
25#16839668—21no—
26#16839664—21no—
27#16839661—22no—
28#16839660—22no—
29#16839639C2214yes—
30#16839637C2214yes—
31#16839634C2214yes—
32#16839632—21no—
33#16839631—21no—
34#16839628—21no—
35#16839627—21no—
36#16839619—22no—
37#16839528C2214yes—
38C2×#117×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C61×#342C2214yes—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222C2×C414yes—
49C2×#11758222Q814yes—
50C22×Dic122D411yescharacteristic
51C22×D244D411yescharacteristic
52C22×C22⋊D61C2312yes—
53C23×Dic61C2312yes—
54C22×C4×D61D412yes—
55C4×Dic122—22no—
56C4×Dic122—22no—
57#14519742—22no—
58#14519742—22no—
59C4×D244—22no—
60C4×D244—22no—
61#14519739—22no—
62#14519739—22no—
63C4×C22⋊D61—24no—
64C4×C22⋊D61—24no—
65#14519737—24no—
66#14519737—24no—
67#14519736—21no—
68#14519736—21no—
69#14519736—21no—
70#14519736—21no—
71#14519736—22no—
72#14519736C2×C414yescomplemented
73#14519735—22no—
74#14519735—22no—
75#14519734—24no—
76#14519733—24no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—24no—
85C2×C4⋊Dic61D411yescharacteristic
86C2×C4⋊Dic61D411yescharacteristic
87C2×#11758196—24no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61—22no—
90C2×C22⋊Dic61C2312yes—
91C2×C22⋊Dic61C2312yes—
92C2×#11758198—24no—
93C22⋊C4×D61—24no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61D412yes—
96C42×D61—24no—
97#14519691C2×C4116yescomplemented
98#14519684—24no—
99#14519684—24no—
100#14519684—24no—
101C23×C4×C61C2311yescharacteristic
102#117×C61—21no—
103#117×C61—21no—
104#117×C61—21no—
105#117×C61—21no—
106#117×C61—22no—
107#117×C61—22no—
108#117×C61—22no—
109#117×C61C2×C414yescomplemented
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61D411yescharacteristic
113C2×C4⋊C4×C61D411yescharacteristic
114C2×C22⋊C4×C61C2312yes—
115C2×C22⋊C4×C61C2312yes—
116C2×C22⋊C4×C61C2312yes—
117C2×C42×C61—21no—
118C2×C42×C61—21no—
119C2×C42×C61D412yes—
120C24×C61C2411yescharacteristic
121C23×D61C2×D412yes—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×Q812yes—
134C22×C4×C61#4112yes—
135C22×C4×C61#4112yes—
136C2×C22⋊D61—22no—
137C2×C22⋊D61—22no—
138C2×C22⋊D61—22no—
139C2×C22⋊D61—22no—
140C2×C22⋊D61C22×C414yes—
141C2×C22⋊D61C2×Q814yes—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—82no—
147C22×Dic61C2×D412yes—
148C22×Dic61C2×D412yes—
149C22×Dic61C2×D412yes—
150#11758222—24no—
151#11758222—24no—
152#11758222—24no—
153#11758222—24no—
154#11758222—24no—
155#11758222—24no—
156#11758222—24no—
157#11758222—24no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—22no—
161C2×D244C4⋊C414yescomplemented
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—24no—
165C2×C4×D61—24no—
166C2×C4×D61—24no—
167C2×C4×D61C4⋊C418yescomplemented
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—22no—
171C2×Dic122C4⋊C414yescomplemented
172C4⋊C4×C61—24no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—42no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61C22×C414yes—
184C42×C61—24no—
185C42×C61—22no—
186C42×C61—22no—
187C42×C61—42no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61C22×C414yes—
192#11758198—24no—
193#11758198—24no—
194#11758198—84no—
195C4⋊Dic61—24no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61—41no—
202C4⋊Dic61—41no—
203C4⋊Dic61—41no—
204C4⋊Dic61—84no—
205#11758196—24no—
206#11758196—24no—
207#11758196—84no—
208C4×Dic61—24no—
209C4×Dic61—24no—
210C4×Dic61—24no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61C23×C411yescharacteristic
216C23×C61C22×D411yescharacteristic
217C23×C61C22×D411yescharacteristic
218C23×C61C22×D411yescharacteristic
219C23×C61C22×Q811yescharacteristic
220C23×C61C2×#4111yescharacteristic
221C23×C61C2×#4111yescharacteristic
222C22×D61—22no—
223C22×D61—22no—
224C22×D61—42no—
225C22×D61—42no—
226C22×D61C2×C4⋊C414yes—
227C2×C4×C61—24no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61C2×C4⋊C412yes—
238C2×C4×C61C2×C4⋊C412yes—
239C2×C4×C61C4×D412yes—
240C2×C4×C61C4×D412yes—
241C2×C4×C61C4×D412yes—
242C2×C4×C61C4×D412yes—
243C2×C4×C61#12011yescharacteristic
244C2×C4×C61#12011yescharacteristic
245C2×C4×C61#12011yescharacteristic
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61C22⋊Q812yes—
248C2×C4×C61C22⋊Q812yes—
249C2×C4×C61C22⋊Q812yes—
250C2×C4×C61C22⋊Q812yes—
251C2×C4×C61#12312yes—
252C2×C4×C61#12312yes—
253C22⋊D61—24no—
254C22⋊D61—24no—
255C22⋊D61—24no—
256C22⋊D61—24no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—24no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—44no—
267C2×Dic61—44no—
268C2×Dic61—44no—
269C2×Dic61—42no—
270C2×Dic61C2×C4⋊C414yes—
271D244—24no—
272D244—24no—
273C4×D61—28no—
274C4×D61—28no—
275C4×D61—84no—
276C4×D61—84no—
277Dic122—24no—
278Dic122—24no—
279#20890—611nomaximal, Sylow
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61C22×C4⋊C411yescharacteristic
291C22×C61C2×C4×D411yescharacteristic
292C22×C61C2×C4×D411yescharacteristic
293C22×C61C2×#12011yescharacteristic
294C22×C61C2×C22⋊Q811yescharacteristic
295C22×C61C2×C22⋊Q811yescharacteristic
296C22×C61C2×#12311yescharacteristic
297C2×D61—24no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—44no—
301C2×D61—82no—
302C4×C61—44no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—82no—
306C4×C61#38512yes—
307C4×C61#38512yes—
308C4×C61#38512yes—
309C4×C61#38512yes—
310Dic61—24no—
311Dic61—44no—
312Dic61—42no—
313Dic61—42no—
314Dic61—82no—
315#3428—611no—
316C2×#410—611no—
317#2317—611no—
318#2317—611no—
319#2311—611no—
320#1867—618no—
321#1740—611no—
322#1740D6111yescharacteristic, complemented
323C2×C61—21no—
324C2×C61—21no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61—21no—
328C2×C61—21no—
329C2×C61C2×#38511yescharacteristic
330C2×C61#302711yescharacteristic
331C2×C61#302811yescharacteristic
332D61—44no—
333C22×#41—611no—
334#520—1221no—
335#520—1221no—
336#520—1221no—
337#520—1221no—
338#519—1221no—
339#518—614no—
340#518—1221no—
341C4×#41—1222no—
342C4×#41—1222no—
343C2×#117—611no—
344C2×#117—611no—
345C2×#117—1222no—
346C2×#117C2×D6111yescharacteristic
347C2×#117C2×D6111yescharacteristic
348C2×#117C2×D6111yescharacteristic
349C22×C4⋊C4—611no—
350#410—618no—
351C2×#99—614no—
352#391—1221no—
353#390—1221no—
354#389—1222no—
355#389—1222no—
356#386—614no—
357#385—614no—
358#385—614no—
359#384—1221no—
360#384—1221no—
361#382—1221no—
362#382—1221no—
363#380—1222no—
364#342—614no—
365#342C2×D6114yes—
366C61#2089011yescharacteristic, complemented, Sylow
367C2×#41—614no—
368C2×#41—614no—
369C2×#41—1221no—
370C2×#41—1221no—
371C2×#41—1222no—
372C2×#41—1221no—
373C2×#41—1221no—
374C2×#41—1222no—
375C22×Q8—611no—
376C22×D4—612no—
377C22×D4—611no—
378C23×C4—612no—
379C23×C4—612no—
380C23×C4C22×D6111yescharacteristic
381C4×Q8—1222no—
382C4×Q8—1222no—
383C4×Q8—1222no—
384C4×Q8—1222no—
385C4×D4—1222no—
386C4×D4—1222no—
387C4×D4—1224no—
388C4×D4—1224no—
389C4×D4—1222no—
390C4×D4—1222no—
391C4×D4—1224no—
392C4×D4—1224no—
393#117—21no—
394#117—21no—
395#117—21no—
396#117—21no—
397#117—22no—
398#117—22no—
399#117—22no—
400#117—614no—
401#117—1224no—
402#117—1224no—
403#117—1222no—
404#117—1222no—
405#117—1221no—
406#117—1221no—
407#117—1222no—
408#117—1221no—
409#117—1221no—
410#117C4×D6114yescomplemented
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1224no—
419C2×C4⋊C4—1224no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4D24411yescharacteristic
427C2×C4⋊C4D24411yescharacteristic
428C2×C22⋊C4—612no—
429C2×C22⋊C4—612no—
430C2×C22⋊C4—1224no—
431C2×C22⋊C4—1224no—
432C2×C22⋊C4C22×D6112yes—
433C2×C22⋊C4C22×D6112yes—
434C2×C22⋊C4C22×D6112yes—
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—612no—
438C2×C42—1224no—
439C2×C42—1221no—
440C2×C42D24412yes—
441#99—6116no—
442#95—1224no—
443#95—1224no—
444#95—1224no—
445C24—612no—
446C24C23×D6111yescharacteristic
447#41—1224no—
448#41—1224no—
449#41—1224no—
450#41—1224no—
451#41—1224no—
452#41—1224no—
453#41—1224no—
454#41—1224no—
455C2×Q8—614no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1222no—
459C2×D4—614no—
460C2×D4—614no—
461C2×D4—614no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—1222no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1222no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—612no—
472C22×C4—618no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—1224no—
476C22×C4—1224no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1224no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—4882no—
485C22×C4C2×C4×D6112yes—
486C22×C4C2×D24412yes—
487C22×C4C2×D24412yes—
488C22×C4C2×D24411yescharacteristic
489C22×C4C2×D24411yescharacteristic
490C22×C4#1175822212yes—
491C22×C4D4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493C22×C4D4×D6111yescharacteristic
494C22×C4D4×D6111yescharacteristic
495C22×C4#1175822412yes—
496C22×C4Q8×D6112yes—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—1224no—
504C4⋊C4—1224no—
505C4⋊C4—1224no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—4884no—
511C4⋊C4—4884no—
512C4⋊C4—2441no—
513C4⋊C4—2441no—
514C4⋊C4—2441no—
515C4⋊C4—2441no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—614no—
522C22⋊C4—1224no—
523C22⋊C4—1224no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—4884no—
528C22⋊C4C2×C4×D6114yes—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—42no—
533C42—1224no—
534C42—1224no—
535C42—1224no—
536C42—1222no—
537C42—1222no—
538C23—41no—
539C23—41no—
540C23—614no—
541C23—1222no—
542C23—1222no—
543C23—2442no—
544C23—2442no—
545C23C22×C4×D6111yescharacteristic
546C23C2×D4×D6111yescharacteristic
547C23C2×D4×D6111yescharacteristic
548C23C22×D24411yescharacteristic
549C23C2×Q8×D6111yescharacteristic
550C23C2×#1175822411yescharacteristic
551C23C2×#1175822211yescharacteristic
552Q8—1224no—
553Q8—1224no—
554D4—1224no—
555D4—1224no—
556D4—1224no—
557D4—1224no—
558D4—1224no—
559D4—1224no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—614no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1228no—
574C2×C4—1228no—
575C2×C4—1224no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—4884no—
579C2×C4—4884no—
580C2×C4—2444no—
581C2×C4—2444no—
582C2×C4—2444no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4C4⋊C4×D6112yes—
589C2×C4C4⋊C4×D6112yes—
590C2×C4#1451973712yes—
591C2×C4#1451973712yes—
592C2×C4C4×D24412yes—
593C2×C4C4×D24412yes—
594C2×C4#1451974711yescharacteristic
595C2×C4#1451974711yescharacteristic
596C2×C4#1451974711yescharacteristic
597C2×C4#1451974711yescharacteristic
598C2×C4#1451975512yes—
599C2×C4#1451975512yes—
600C2×C4#1451975712yes—
601C2×C4#1451975712yes—
602C2×C4#1451975912yes—
603C2×C4#1451975912yes—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—1224no—
615C22—4882no—
616C22—2444no—
617C22—2442no—
618C22—2442no—
619C22C2×C4⋊C4×D6111yescharacteristic
620C22C2×#1451973711yescharacteristic
621C22C2×C4×D24411yescharacteristic
622C22C2×#1451974711yescharacteristic
623C22C2×#1451975511yescharacteristic
624C22C2×#1451975711yescharacteristic
625C22C2×#1451975911yescharacteristic
626C4—44no—
627C4—42no—
628C4—42no—
629C4—82no—
630C4—1224no—
631C4—4882no—
632C4—2444no—
633C4—2442no—
634C4—2442no—
635C4#1683963612yes—
636C4#1683963612yes—
637C4#1683963612yes—
638C4#1683963612yes—
639C2—21no—
640C2—21no—
641C2—21no—
642C2—21no—
643C2—21no—
644C2—21no—
645C2—2444no—
646C2C2×#1683963611yescharacteristic
647C2#2145522211yescharacteristic
648C2#2145522611yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1740C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
cosocleC24×C61C24
hypercenter#1740D61
hypocenterC61#20890
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
3. #21448442C2semidirect
4. #21448439C2semidirect
6. #21446101C2semidirect
7. #21445551C2semidirect
8. C61×#1740C2semidirect
23. #16839735C4semidirect
72. #14519736C2×C4semidirect
97. #14519691C2×C4semidirect
109. #117×C61C2×C4semidirect
161. C2×D244C4⋊C4semidirect
167. C2×C4×D61C4⋊C4semidirect
171. C2×Dic122C4⋊C4semidirect
322. #1740D61semidirect
366. C61#20890semidirect
410. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214590202
C2×#168397352
#214484422
#214484392
#214484072
#214461012
#214455512
C61×#17402
#2089061