Finite Group Databasebeta

← #29647412

Subgroups — #29647412

Order 15616.

1024 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840293C211yesmaximal, characteristic
2#21454448C211yesmaximal, characteristic, complemented
3#21448213C211yesmaximal, characteristic, complemented
4#21444393C214yesmaximal, complemented
5#21444354C214yesmaximal, complemented
6#21444340C211yesmaximal, characteristic, complemented
7#21444338C211yesmaximal, characteristic, complemented
8#21444335C211yesmaximal, characteristic
9C61×#1507C211yesmaximal, characteristic
10#16840293—21no—
11#16840293—21no—
12#16840293—21no—
13#16840293—21no—
14#16840293C412yescomplemented
15#16840293C412yescomplemented
16#16840293C2212yes—
17#16840293C2212yes—
18C2×#14519770—21no—
19C2×#14519768C411yescharacteristic, complemented
20C2×#14519764C411yescharacteristic, complemented
21C2×#14519758—21no—
22C2×#14519756C2211yescharacteristic
23C2×#14519742C411yescharacteristic, complemented
24C2×D4×Dic61—21no—
25C2×D4×Dic61C411yescharacteristic, complemented
26C2×#14519733—21no—
27C22×C4×Dic61C2211yescharacteristic
28#16839816—21no—
29#16839816C2212yescomplemented
30#16839812—21no—
31#16839812C2212yescomplemented
32#16839785—21no—
33#16839726C2211yescharacteristic
34M16×Dic61—21no—
35#16839629C2212yescomplemented
36#16839626C2212yescomplemented
37#16839623—21no—
38#16839620—21no—
39C4×#11758196—21no—
40C4×C22⋊Dic61C2211yescharacteristic
41#16839608—21no—
42#16839500C2214yescomplemented
43#16839498C2214yescomplemented
44#16839496C2214yes—
45C2×#121×C61C2211yescharacteristic
46C61×#407C2211yescharacteristic
47C4×C22⋊C4×C61C2211yescharacteristic
48C61×#327C2214yes—
49C22×Dic122C2×C411yescharacteristic
50C23×Dic61—21no—
51C23×Dic61D411yescharacteristic
52C23×Dic61D411yescharacteristic
53C2×#11758194—21no—
54#14519770—22no—
55#14519770—22no—
56#14519770—22no—
57#14519770—22no—
58#14519768—21no—
59#14519768—21no—
60#14519768C2×C412yes—
61#14519768C2×C412yes—
62#14519764—21no—
63#14519764—21no—
64#14519764C2×C412yes—
65#14519764C2×C412yes—
66#14519758—22no—
67#14519758—22no—
68#14519758—22no—
69#14519758—22no—
70#14519756—21no—
71#14519756—21no—
72#14519756C2×C412yes—
73#14519756C2312yes—
74#14519742—21no—
75#14519742—21no—
76#14519742C2×C412yes—
77#14519742C2×C412yes—
78D4×Dic61—22no—
79D4×Dic61—22no—
80D4×Dic61—22no—
81D4×Dic61—22no—
82D4×Dic61—21no—
83D4×Dic61—21no—
84D4×Dic61C2×C412yes—
85D4×Dic61C2×C412yes—
86#14519733—22no—
87#14519733—22no—
88#14519733—22no—
89#14519733—22no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61C2×C411yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—22no—
96C2×C22⋊Dic61—21no—
97C2×C22⋊Dic61—21no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×C22⋊Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61C2×C411yescharacteristic
104C2×C4×Dic61C2×C411yescharacteristic
105C2×C4×Dic61C2×C411yescharacteristic
106C2×C4×Dic61C2×C411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115#14519706—22no—
116#14519706—22no—
117C22⋊#92418—21no—
118C22⋊#92418C2312yes—
119#14519687—22no—
120C8×Dic61—22no—
121#14519684—22no—
122#14519684—22no—
123#14519684—22no—
124#14519684C2312yes—
125C22×D4×C61—21no—
126C22×D4×C61C2×C411yescharacteristic
127C23×C4×C61C2311yescharacteristic
128C2×M16×C61—21no—
129#121×C61—21no—
130#121×C61—21no—
131#121×C61C2×C412yes—
132#121×C61C2312yes—
133C2×C4⋊C4×C61C2×C411yescharacteristic
134C2×C22⋊C4×C61—21no—
135C2×C22⋊C4×C61—21no—
136C2×C42×C61—21no—
137C22⋊C8×C61—21no—
138C22⋊C8×C61C2312yes—
139#95×C61C2312yes—
140C24×C61—21no—
141C24×C61C2×D411yescharacteristic
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—21no—
145C2×D4×C61—42no—
146C2×D4×C61—44no—
147C2×D4×C61C22×C412yes—
148C22×C4×C61—21no—
149C22×C4×C61—22no—
150C22×C4×C61—42no—
151C22×C4×C61C22×C411yescharacteristic
152C22×C4×C61C22×C411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61—44no—
165C22×Dic61C22⋊C411yescharacteristic
166C22×Dic61C22⋊C411yescharacteristic
167C22×Dic61C22⋊C411yescharacteristic
168C22×Dic61C22⋊C411yescharacteristic
169C22×Dic61C22⋊C411yescharacteristic
170C22×Dic61C22⋊C411yescharacteristic
171C22×Dic61C22⋊C411yescharacteristic
172C22×Dic61C22⋊C411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174C22×Dic61C2×D411yescharacteristic
175C22×Dic61C2×D411yescharacteristic
176C22×Dic61C2×D411yescharacteristic
177C2×Dic122—21no—
178C2×Dic122—42no—
179C2×Dic122C22×C412yes—
180M16×C61—42no—
181C2×C8×C61—22no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C22×C412yes—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—42no—
187C42×C61—42no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C22⋊Dic61—42no—
197C4⋊Dic61—21no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61C22×C412yes—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—44no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—42no—
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22×C411yescharacteristic
220C4×Dic61C22×C411yescharacteristic
221C4×Dic61C22×C411yescharacteristic
222C4×Dic61C22×C411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227#11758194—42no—
228C2×#92418—22no—
229C23×C61—22no—
230C23×C61—22no—
231C23×C61—22no—
232C23×C61—41no—
233C23×C61—42no—
234C23×C61—42no—
235C23×C61—42no—
236C23×C61C2×C22⋊C411yescharacteristic
237C23×C61C2×C22⋊C411yescharacteristic
238C23×C61C22×D411yescharacteristic
239D4×C61—42no—
240D4×C61—44no—
241D4×C61—44no—
242D4×C61—42no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C23×C411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—44no—
273C2×Dic61—44no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C2×C22⋊C411yescharacteristic
281C2×Dic61C2×C22⋊C411yescharacteristic
282C2×Dic61C2×C22⋊C411yescharacteristic
283C2×Dic61C2×C22⋊C411yescharacteristic
284C2×Dic61C2×C22⋊C411yescharacteristic
285C2×Dic61C2×C22⋊C411yescharacteristic
286C2×Dic61C2×C22⋊C411yescharacteristic
287Dic122—42no—
288Dic122—42no—
289C8×C61—42no—
290#92418—42no—
291#17579—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—44no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—84no—
304C22×C61#42112yes—
305C22×C61C22×C22⋊C411yescharacteristic
306C22×C61#57512yes—
307C4×C61—21no—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—42no—
311C4×C61—82no—
312C4×C61#41012yes—
313Dic61—21no—
314Dic61—21no—
315Dic61—21no—
316Dic61—21no—
317Dic61—21no—
318Dic61—21no—
319Dic61—21no—
320Dic61—21no—
321Dic61—42no—
322Dic61—42no—
323Dic61—82no—
324C2×#529—611no—
325#2920—611no—
326#2283—611no—
327#1521—614no—
328#1512—614no—
329#1508—611no—
330#1508—611no—
331#1507—611no—
332#1507D6111yescharacteristic
333C2×C61—22no—
334C2×C61—22no—
335C2×C61—41no—
336C2×C61—82no—
337C2×C61C2×#41011yescharacteristic
338C2×C61#289611yescharacteristic
339C2×C61C2×#42111yescharacteristic
340#529—612no—
341#529—612no—
342#529—612no—
343#529—612no—
344#529—1221no—
345#529—1221no—
346#529—1221no—
347#529—1221no—
348C2×#126—1221no—
349C2×#125—611no—
350C2×#124—611no—
351C2×#123—1221no—
352C2×C22⋊Q8—611no—
353C2×#121C2×D6111yescharacteristic
354C2×C4×Q8—611no—
355C2×C4×D4—611no—
356C2×C4×D4—1221no—
357C2×#117—1221no—
358C22×C42—611no—
359#432—612no—
360#432—1221no—
361#431—612no—
362#431—1221no—
363#423—1221no—
364#407—611no—
365#407C2×D6111yescharacteristic
366C4×M16—1221no—
367#383—612no—
368#382—612no—
369#381—1221no—
370#380—1221no—
371C4×C4⋊C4—1221no—
372C4×C22⋊C4—611no—
373C4×C22⋊C4C2×D6111yescharacteristic
374#376—1221no—
375#328—614no—
376#328—614no—
377#327—614no—
378#327C2×D6114yes—
379C61#1757911yescharacteristic, complemented, Sylow
380C22×Q8—611no—
381C22×D4—21no—
382C22×D4C4×D6111yescharacteristic
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—1221no—
386C23×C4C22×D6111yescharacteristic
387C2×M16—21no—
388C2×M16—1221no—
389#126—1222no—
390#126—1222no—
391#126—1222no—
392#126—1222no—
393#125—612no—
394#125—612no—
395#125—1221no—
396#125—1221no—
397#124—612no—
398#124—612no—
399#124—1221no—
400#124—1221no—
401#123—1222no—
402#123—1222no—
403#123—1222no—
404#123—1222no—
405C22⋊Q8—612no—
406C22⋊Q8—612no—
407C22⋊Q8—1221no—
408C22⋊Q8—1221no—
409#121—21no—
410#121—21no—
411#121C4×D6112yes—
412#121C22×D6112yes—
413C4×Q8—612no—
414C4×Q8—612no—
415C4×Q8—1221no—
416C4×Q8—1221no—
417C4×D4—612no—
418C4×D4—612no—
419C4×D4—1221no—
420C4×D4—1221no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425#117—1222no—
426#117—1222no—
427#117—1222no—
428#117—1222no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1222no—
435C2×C4⋊C4C4×D6111yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C42—21no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—611no—
456C2×C42—1221no—
457C2×C42—1221no—
458C4⋊C8—1222no—
459C4⋊C8—1222no—
460C22⋊C8—21no—
461C22⋊C8—612no—
462C22⋊C8—1221no—
463C22⋊C8C22×D6112yes—
464#97—1222no—
465C4×C8—1222no—
466#95—612no—
467#95—1222no—
468#95—1222no—
469#95—1222no—
470#95C22×D6112yes—
471C24—21no—
472C24D4×D6111yescharacteristic
473C2×Q8—612no—
474C2×Q8—1221no—
475C2×Q8—2442no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—21no—
479C2×D4—42no—
480C2×D4—44no—
481C2×D4C2×C4×D6112yes—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—42no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1222no—
502C22×C4—2442no—
503C22×C4—2442no—
504C22×C4—2444no—
505C22×C4C2×C4×D6111yescharacteristic
506C22×C4C2×C4×D6111yescharacteristic
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511M16—42no—
512M16—2442no—
513C2×C8—22no—
514C2×C8—1222no—
515C4⋊C4—21no—
516C4⋊C4—612no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1221no—
525C4⋊C4—1222no—
526C4⋊C4—2444no—
527C4⋊C4C2×C4×D6112yes—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—42no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—2442no—
540C42—42no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—611no—
550C42—611no—
551C42—611no—
552C42—611no—
553C42—611no—
554C42—611no—
555C42—611no—
556C42—611no—
557C42—1222no—
558C42—1222no—
559C42—2442no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23C22⋊C4×D6111yescharacteristic
568C23C22⋊C4×D6111yescharacteristic
569C23C2×D4×D6111yescharacteristic
570Q8—2442no—
571Q8—2442no—
572D4—42no—
573D4—44no—
574D4—44no—
575D4—42no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—611no—
592C2×C4—611no—
593C2×C4—611no—
594C2×C4—611no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2444no—
610C2×C4—2444no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4—2442no—
616C2×C4C22⋊C4×D6111yescharacteristic
617C2×C4C22⋊C4×D6111yescharacteristic
618C2×C4C22×C4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C8—42no—
621C8—2442no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—84no—
634C22#1683977712yes—
635C22C2×C22⋊C4×D6111yescharacteristic
636C22#1684055812yes—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—42no—
641C4—82no—
642C4—1221no—
643C4—1221no—
644C4—1221no—
645C4—1221no—
646C4—1221no—
647C4—1221no—
648C4—1221no—
649C4—1221no—
650C4—4882no—
651C4—2442no—
652C4—2442no—
653C4#1683973412yes—
654C2—22no—
655C2—22no—
656C2—41no—
657C2—82no—
658C2C2×#1683973411yescharacteristic
659C2#2145433311yescharacteristic
660C2C2×#1683977711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1507C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#121×C61C22
cosocleC22×C4×C61C24
hypercenter#1507D61
hypocenterC61#17579
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. #21454448C2semidirect
3. #21448213C2semidirect
4. #21444393C2semidirect
5. #21444354C2semidirect
6. #21444340C2semidirect
7. #21444338C2semidirect
14. #16840293C4semidirect
15. #16840293C4semidirect
19. C2×#14519768C4semidirect
20. C2×#14519764C4semidirect
23. C2×#14519742C4semidirect
25. C2×D4×Dic61C4semidirect
29. #16839816C22semidirect
31. #16839812C22semidirect
35. #16839629C22semidirect
36. #16839626C22semidirect
42. #16839500C22semidirect
43. #16839498C22semidirect
379. C61#17579semidirect

Maximal subgroups

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

subgroupindex
C2×#168402932
#214544482
#214482132
#214443932
#214443542
#214443402
#214443382
#214443352
C61×#15072
#1757961