Finite Group Databasebeta

← #29647571

Subgroups — #29647571

Order 15616.

1046 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459846C211yesmaximal, characteristic
2#21454853C211yesmaximal, characteristic, complemented
3#21448457C211yesmaximal, characteristic, complemented
4#21444385C212yesmaximal, complemented
5#21444372C212yesmaximal, complemented
6#21444359C212yesmaximal, complemented
7#21444346C212yesmaximal, complemented
8#21444342C211yesmaximal, characteristic, complemented
9#21444327C211yesmaximal, characteristic, complemented
10#21444306C211yesmaximal, characteristic, complemented
11C61×#1502C211yesmaximal, characteristic, complemented
12#16840478C411yescharacteristic, complemented
13#16840473C411yescharacteristic, complemented
14#16840464—21no—
15#16840462C412yescomplemented
16#16840432C411yescharacteristic, complemented
17#16840407C412yescomplemented
18#16840400—22no—
19#16840397C411yescharacteristic, complemented
20#16840396—21no—
21#16840363—21no—
22#16840346—22no—
23#16840296C2211yescharacteristic
24#16840272C2212yes—
25#121×D61C2212yes—
26#16840234—21no—
27C2×C4×D244C2211yescharacteristic
28#16839841—21no—
29#16839832—22no—
30C8×D244—21no—
31#16839819C2211yescharacteristic, complemented
32#16839724C2212yescomplemented
33C22⋊C8×D61C2212yescomplemented
34#16839663—21no—
35#16839658—21no—
36#16839653—22no—
37#16839624C2212yescomplemented
38#16839621C2212yescomplemented
39#16839608C2211yescharacteristic, complemented
40#16839500C2211yescharacteristic, complemented
41#16839500C2211yescharacteristic, complemented
42#16839498C2212yescomplemented
43#16839497C2211yescharacteristic
44#16839497C2211yescharacteristic
45#16839496C2212yescomplemented
46#16839496C2212yescomplemented
47#16839495C2212yes—
48C61×#532C2211yescharacteristic
49C61×#431C2211yescharacteristic, complemented
50C4×C4⋊C4×C61C2211yescharacteristic, complemented
51C61×#327C2212yescomplemented
52C61×#327C2212yescomplemented
53C2×#11758224C2×C412yes—
54C22×D244D411yescharacteristic
55C2×D4×D61—22no—
56C2×D4×D61—21no—
57C2×D4×D61C2×C412yes—
58C22×C4×D61C2312yes—
59C2×#11758189—22no—
60C2×C8×D61—22no—
61#14519773—21no—
62#14519771—21no—
63#14519770—22no—
64#14519769C2×C411yescharacteristic
65#14519768—21no—
66#14519766—22no—
67#14519765C2×C411yescharacteristic
68#14519761—22no—
69#14519760—21no—
70#14519760—22no—
71#14519759—22no—
72#14519757C2×C412yes—
73#14519756C2311yescharacteristic
74#14519753—22no—
75#14519752—21no—
76#14519752C2×C412yes—
77#14519751—22no—
78#14519751C2311yescharacteristic
79#14519750—22no—
80#14519750—22no—
81#14519749—22no—
82#14519749—22no—
83#14519748—21no—
84#14519748C2×C412yes—
85#14519748C2×C412yes—
86#14519747—22no—
87#14519745—22no—
88#14519745—22no—
89C4×Dic122C2×C411yescharacteristic
90C4×D244C2×C412yes—
91C4×D244C2×C411yescharacteristic
92C4×D244C2×C411yescharacteristic
93C4×D244C2×C411yescharacteristic
94C4×D244D411yescharacteristic
95C4×D244D411yescharacteristic
96C4×D244D412yes—
97D4×Dic61—22no—
98D4×Dic61—21no—
99#14519739—21no—
100C4×C22⋊D61—21no—
101C4×C22⋊D61—22no—
102#14519737—22no—
103#14519735—21no—
104#14519735C2×C412yes—
105#14519734—21no—
106C2×C4⋊Dic61D411yescharacteristic
107C2×#11758196—22no—
108C4⋊C4×D61C2×C412yes—
109C2×#11758198—22no—
110C2×#11758198—22no—
111C2×#11758198—22no—
112C2×#11758198D412yes—
113C22⋊C4×D61—22no—
114C2×C4×Dic61—21no—
115C4⋊#92418—21no—
116C22⋊#92418C2312yes—
117#14519689—22no—
118#14519689—22no—
119#14519688—21no—
120#14519684—21no—
121#14519684—22no—
122#14519684C2311yescharacteristic
123#14519684C2311yescharacteristic
124C2×#41×C61—21no—
125C4⋊Q8×C61C2×C411yescharacteristic
126#127×C61C2×C411yescharacteristic
127#124×C61—21no—
128#121×C61—21no—
129#121×C61C2312yes—
130C4×D4×C61—21no—
131C4×D4×C61C2×C412yes—
132C2×C4⋊C4×C61—21no—
133C2×C42×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C22⋊C8×C61C2311yescharacteristic
137C22⋊C8×C61C2311yescharacteristic
138C4×C8×C61—21no—
139#95×C61C2312yes—
140C23×D61—22no—
141C23×D61C2×D412yes—
142#41×C61—81no—
143#41×C61—81no—
144C2×Q8×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61C22×C412yes—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—21no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C22×Dic61—22no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61C2×D412yes—
163#11758224—82no—
164D4×D61—44no—
165D4×D61—82no—
166D4×D61—82no—
167D4×D61—82no—
168C2×D244—22no—
169C2×D244—42no—
170C2×D244C22⋊C411yescharacteristic
171C2×D244C22⋊C411yescharacteristic
172C2×D244C22⋊C412yes—
173C2×D244C22×C411yescharacteristic
174C2×C4×D61—21no—
175C2×C4×D61—22no—
176C2×C4×D61—82no—
177C2×C4×D61C22×C412yes—
178C2×C4×D61C22×C412yes—
179C2×C4×D61C2×D412yes—
180C2×C4×D61C2×D412yes—
181C2×Dic122C22×C411yescharacteristic
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—42no—
186C4⋊C4×C61C22×C412yes—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C42×C61—81no—
190C42×C61C22×C411yescharacteristic
191C42×C61C22×C411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C22⋊Dic61—21no—
195C22⋊Dic61—22no—
196C22⋊Dic61—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—21no—
200#11758198—21no—
201#11758198—21no—
202#11758198—21no—
203#11758198—22no—
204#11758198—82no—
205#11758198—82no—
206#11758198C22⋊C412yes—
207#11758198C22⋊C412yes—
208#11758198C22⋊C412yes—
209#11758198C22⋊C412yes—
210C4⋊Dic61—22no—
211C4⋊Dic61C22⋊C411yescharacteristic
212C4⋊Dic61C22⋊C411yescharacteristic
213C4⋊Dic61C22⋊C412yes—
214C4⋊Dic61C22×C411yescharacteristic
215C4⋊Dic61C22×C411yescharacteristic
216#11758196—21no—
217#11758196—21no—
218#11758196—21no—
219#11758196—21no—
220#11758196—22no—
221#11758196—82no—
222C4×Dic61—21no—
223C4×Dic61—22no—
224C4×Dic61—42no—
225C2×#92418—22no—
226#11758189—82no—
227C8×D61—82no—
228C23×C61—22no—
229C23×C61C22×D411yescharacteristic
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—21no—
233C22×D61—42no—
234C22×D61—42no—
235C22×D61—44no—
236C22×D61—82no—
237C22×D61—82no—
238C22×D61C2×C22⋊C412yes—
239C22×D61C2×C22⋊C412yes—
240Q8×C61—81no—
241D4×C61—42no—
242D4×C61—42no—
243D4×C61—42no—
244D4×C61—42no—
245D4×C61—81no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—82no—
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C23×C411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C22⋊D61—44no—
262C22⋊D61—41no—
263C22⋊D61—41no—
264C22⋊D61—82no—
265C22⋊D61—82no—
266C22⋊D61—82no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—42no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—82no—
278C2×Dic61C2×C22⋊C412yes—
279C2×Dic61C2×C22⋊C412yes—
280D244—22no—
281D244—22no—
282D244—42no—
283D244—41no—
284D244—82no—
285C4×D61—42no—
286C4×D61—42no—
287C4×D61—42no—
288C4×D61—42no—
289C4×D61—42no—
290C4×D61—82no—
291Dic122—41no—
292C8×C61—41no—
293C8×C61—41no—
294#92418—42no—
295#17599—611nomaximal, Sylow
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61C22×C22⋊C411yescharacteristic
302C22×C61#57611yescharacteristic
303C22×C61#57711yescharacteristic
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—41no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—82no—
314C2×D61—82no—
315C2×D61—164no—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—21no—
319C4×C61—21no—
320C4×C61—21no—
321C4×C61—42no—
322C4×C61—82no—
323C4×C61#42112yes—
324Dic61—22no—
325Dic61—22no—
326Dic61—41no—
327Dic61—41no—
328Dic61—41no—
329Dic61—81no—
330Dic61—81no—
331#3519—611no—
332#2974—611no—
333#2319—611no—
334#1519—612no—
335#1516—612no—
336#1513—612no—
337#1510—612no—
338#1509—611no—
339#1505—611no—
340#1502D6111yescharacteristic, complemented
341#1500—611no—
342C2×C61—21no—
343C2×C61—21no—
344C2×C61—82no—
345C2×C61#288311yescharacteristic
346C2×C61#289711yescharacteristic
347C2×C61C2×#42111yescharacteristic
348D61—42no—
349D61—42no—
350D61—81no—
351#555—611no—
352#554—611no—
353#553—612no—
354#553—1221no—
355#550—611no—
356#546—612no—
357#546—1222no—
358D42—611no—
359D42—1221no—
360#540—1221no—
361#538—1222no—
362#532C2×D6111yescharacteristic
363#529—611no—
364#525—612no—
365C2×#121—612no—
366#518—1221no—
367C2×C4×D4—611no—
368#436—1221no—
369#435—1222no—
370C8×D4—1221no—
371#432—611no—
372#431C2×D6111yescharacteristic, complemented
373#407—612no—
374C2×C22⋊C8—612no—
375#390—1221no—
376#389—1221no—
377#388—1222no—
378#381—612no—
379#380—612no—
380C4×C4⋊C4C2×D6111yescharacteristic, complemented
381#376—611no—
382#328—611no—
383#328—611no—
384#328—612no—
385#327—612no—
386#327—612no—
387#327—611no—
388#327—611no—
389#327—612no—
390#327C2×D6112yescomplemented
391#327C2×D6112yescomplemented
392C61#1759911yescharacteristic, complemented, Sylow
393C2×#41—21no—
394C2×#41—612no—
395C22×D4—612no—
396C22×D4—611no—
397C22×D4—1221no—
398C22×D4—1222no—
399C23×C4—612no—
400C2×M16—1222no—
401C22×C8—1222no—
402C4⋊Q8C4×D6111yescharacteristic
403#127—1221no—
404#127C4×D6111yescharacteristic
405#126—1222no—
406#126—1221no—
407#125—611no—
408#125—1221no—
409#124—21no—
410#124—611no—
411#124—1222no—
412#123—1222no—
413#123—1222no—
414#123—1222no—
415#123—1221no—
416C22⋊Q8—611no—
417C22⋊Q8—612no—
418C22⋊Q8—1222no—
419#121—21no—
420#121—612no—
421#121—612no—
422#121—611no—
423#121—612no—
424#121—1222no—
425#121—1222no—
426#121—1222no—
427#121—1222no—
428#121—1221no—
429#121—1221no—
430#121—1222no—
431#121C22×D6112yes—
432#120—1222no—
433#120—1222no—
434#120—1222no—
435C4×Q8—611no—
436C4×D4—21no—
437C4×D4—611no—
438C4×D4—612no—
439C4×D4—611no—
440C4×D4—611no—
441C4×D4—611no—
442C4×D4—612no—
443C4×D4—611no—
444C4×D4—1222no—
445C4×D4—1222no—
446C4×D4—1221no—
447C4×D4—1221no—
448C4×D4—1222no—
449C4×D4—1221no—
450C4×D4C4×D6112yes—
451#117—612no—
452#117—1221no—
453#117—1221no—
454C2×C4⋊C4—21no—
455C2×C4⋊C4—611no—
456C2×C4⋊C4—612no—
457C2×C4⋊C4—1222no—
458C2×C22⋊C4—612no—
459C2×C22⋊C4—1222no—
460C2×C22⋊C4—1222no—
461C2×C22⋊C4—1222no—
462C2×C22⋊C4—1222no—
463C2×C42—21no—
464C2×C42—1221no—
465C2×C42C22×D6111yescharacteristic
466C4⋊C8—21no—
467C4⋊C8—1221no—
468C22⋊C8—612no—
469C22⋊C8—1222no—
470C22⋊C8—1222no—
471C22⋊C8C22×D6111yescharacteristic
472C22⋊C8C22×D6111yescharacteristic
473#97—1221no—
474C4×C8—21no—
475#95—611no—
476#95—611no—
477#95—1222no—
478#95—1221no—
479#95C22×D6112yes—
480C24—612no—
481C24—1222no—
482#41—81no—
483#41—81no—
484#41—4882no—
485C2×Q8—21no—
486C2×Q8—611no—
487C2×D4—22no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—611no—
491C2×D4—611no—
492C2×D4—611no—
493C2×D4—612no—
494C2×D4—1222no—
495C2×D4—1222no—
496C2×D4—1222no—
497C2×D4—1222no—
498C2×D4—1222no—
499C2×D4—1221no—
500C2×D4—4882no—
501C2×D4—4882no—
502C2×D4—4882no—
503C2×D4—2442no—
504C2×D4—2444no—
505C2×D4C2×C4×D6112yes—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—612no—
509C22×C4—612no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—612no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—4882no—
519C22×C4D4×D6111yescharacteristic
520C22×C4D4×D6111yescharacteristic
521C22×C4C23×D6111yescharacteristic
522M16—4882no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—1222no—
526C2×C8—4882no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—611no—
530C4⋊C4—611no—
531C4⋊C4—612no—
532C4⋊C4—611no—
533C4⋊C4—611no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—4882no—
541C4⋊C4C2×C4×D6112yes—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—612no—
545C22⋊C4—612no—
546C22⋊C4—612no—
547C22⋊C4—612no—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C22⋊C4—1221no—
554C22⋊C4—1221no—
555C22⋊C4—1221no—
556C22⋊C4—1221no—
557C22⋊C4—1221no—
558C22⋊C4—4882no—
559C22⋊C4—4882no—
560C42—81no—
561C42—1222no—
562C42—1221no—
563C42—2442no—
564C42C2×C4×D6111yescharacteristic
565C42C2×C4×D6111yescharacteristic
566C42D4×D6111yescharacteristic
567C42D4×D6111yescharacteristic
568C23—22no—
569C23—612no—
570C23—612no—
571C23—1222no—
572C23—1222no—
573C23—1221no—
574C23—4882no—
575C23—4882no—
576C23—2442no—
577C23—2442no—
578C23—2444no—
579C23C2×D4×D6111yescharacteristic
580Q8—81no—
581Q8—2441no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—81no—
587D4—1222no—
588D4—1222no—
589D4—4882no—
590D4—4882no—
591D4—4882no—
592D4—4882no—
593D4—2442no—
594D4—2441no—
595D4—2444no—
596D4—2441no—
597D4—2441no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—82no—
607C2×C4—612no—
608C2×C4—612no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—4882no—
613C2×C4—4882no—
614C2×C4—2442no—
615C2×C4—2442no—
616C2×C4—2442no—
617C2×C4—2442no—
618C2×C4—2442no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2442no—
626C2×C4C22⋊C4×D6111yescharacteristic
627C2×C4C22⋊C4×D6111yescharacteristic
628C2×C4C22⋊C4×D6111yescharacteristic
629C2×C4C22⋊C4×D6111yescharacteristic
630C2×C4C22×C4×D6111yescharacteristic
631C2×C4C2×D4×D6111yescharacteristic
632C8—41no—
633C8—41no—
634C8—2442no—
635C22—21no—
636C22—21no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—1222no—
641C22—1222no—
642C22—1222no—
643C22—1222no—
644C22—1222no—
645C22—1222no—
646C22—9764no—
647C22—4882no—
648C22—4882no—
649C22—2441no—
650C22—2441no—
651C22—2441no—
652C22C2×C22⋊C4×D6111yescharacteristic
653C22#1684056311yescharacteristic
654C22#1684056811yescharacteristic
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—42no—
661C4—82no—
662C4—1222no—
663C4—1222no—
664C4—4881no—
665C4—4881no—
666C4—2441no—
667C4—2441no—
668C4—2441no—
669C4#1683977712yes—
670C2—21no—
671C2—21no—
672C2—82no—
673C2—4881no—
674C2—2442no—
675C2—2442no—
676C2#2145427811yescharacteristic
677C2#2145434011yescharacteristic
678C2C2×#1683977711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1502C2
socleC22×C61C22×C22⋊C4
prime radical#21459846C2
cosocleC22×C4×C61C24
hypercenter#1502D61
hypocenterC61#17599
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #21454853C2semidirect
3. #21448457C2semidirect
4. #21444385C2semidirect
5. #21444372C2semidirect
6. #21444359C2semidirect
7. #21444346C2semidirect
8. #21444342C2semidirect
9. #21444327C2semidirect
10. #21444306C2semidirect
11. C61×#1502C2semidirect
12. #16840478C4semidirect
13. #16840473C4semidirect
15. #16840462C4semidirect
16. #16840432C4semidirect
17. #16840407C4semidirect
19. #16840397C4semidirect
31. #16839819C22semidirect
32. #16839724C22semidirect
33. C22⋊C8×D61C22semidirect
37. #16839624C22semidirect
38. #16839621C22semidirect
39. #16839608C22semidirect
40. #16839500C22semidirect
41. #16839500C22semidirect
42. #16839498C22semidirect
45. #16839496C22semidirect
46. #16839496C22semidirect
49. C61×#431C22semidirect
50. C4×C4⋊C4×C61C22semidirect
51. C61×#327C22semidirect
52. C61×#327C22semidirect
340. #1502D61semidirect
372. #431C2×D61semidirect
380. C4×C4⋊C4C2×D61semidirect
390. #327C2×D61semidirect
391. #327C2×D61semidirect
392. C61#17599semidirect

Maximal subgroups

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

subgroupindex
#214598462
#214548532
#214484572
#214443852
#214443722
#214443592
#214443462
#214443422
#214443272
#214443062
C61×#15022
#1759961