Finite Group Databasebeta

← #29730113

Subgroups — #29730113

Order 15616.

824 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
1C2×#16840121C211yesmaximal, characteristic
2#21451031C211yesmaximal, characteristic, complemented
3#21448894C211yesmaximal, characteristic
4#21447240C211yesmaximal, characteristic, complemented
5#21447235C211yesmaximal, characteristic
6#21447139C211yesmaximal, characteristic, complemented
7#21447132C211yesmaximal, characteristic
8#21446766C211yesmaximal, characteristic, complemented
9#21446540C211yesmaximal, characteristic, complemented
10#21446162C211yesmaximal, characteristic, complemented
11#21446155C211yesmaximal, characteristic
12#21445832C211yesmaximal, characteristic, complemented
13#21445827C211yesmaximal, characteristic
14#21445649C211yesmaximal, characteristic, complemented
15C2×C61×#464C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19#16840121—24no—
20C2×#14519706C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29#16839701—21no—
30#16839701—21no—
31#16839669C2211yescharacteristic
32#16839669C2211yescharacteristic
33#16839669C2211yescharacteristic
34#16839668—21no—
35#16839668—21no—
36#16839665C2211yescharacteristic
37#16839655—21no—
38#16839655—21no—
39#16839637—21no—
40#16839627C2211yescharacteristic
41#16839627C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C22⋊C4×Dic61—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×Q16×C61C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×#124×C61C2211yescharacteristic
56C2×C4×Q8×C61C2211yescharacteristic
57C61×#464—24no—
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C22×Dic122—21no—
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519777—22no—
66#14519777—22no—
67Q8×Dic61—22no—
68Q8×Dic61—41no—
69D4×Dic61—22no—
70D4×Dic61—41no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61D411yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519706—41no—
87#14519706—41no—
88#14519701—22no—
89#14519701—22no—
90#14519701—22no—
91#14519701—22no—
92#14519698—22no—
93#14519698—22no—
94#14519688—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108C22×Q8×C61C2311yescharacteristic
109C22×Q8×C61C2311yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C2×Q16×C61—22no—
112C2×Q16×C61—22no—
113C2×Q16×C61—41no—
114C2×SD16×C61—22no—
115C2×SD16×C61—22no—
116C2×SD16×C61—41no—
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119#124×C61—22no—
120#124×C61D414yes—
121C4×Q8×C61—22no—
122C4×Q8×C61—22no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127C2×C42×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—22no—
130#103×C61—22no—
131#102×C61—22no—
132C24×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C2×D412yes—
144C2×Q8×C61#4112yes—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×D412yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—81no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C2×Dic122—42no—
169C2×Dic122—42no—
170C2×Dic122—42no—
171Q16×C61—44no—
172SD16×C61—44no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—42no—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—24no—
184C42×C61—42no—
185C42×C61C2×D412yes—
186C42×C61#4112yes—
187C22⋊Dic61—42no—
188C22⋊Dic61—42no—
189C22⋊Dic61—81no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—41no—
198C4⋊Dic61—41no—
199C4⋊Dic61—42no—
200C4⋊Dic61—42no—
201C4⋊Dic61#4112yes—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—42no—
207C4×Dic61—81no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C23×C61—22no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61C22×D411yescharacteristic
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—41no—
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61—41no—
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—41no—
233D4×C61—41no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61#12112yes—
248C2×C4×C61#12112yes—
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C2×C4×C612−1+411yescharacteristic
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—22no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—41no—
269C2×Dic61—41no—
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—84no—
280Dic122—84no—
281C8×C61—41no—
282C8×C61—41no—
283#92418—41no—
284#92418—41no—
285#25337—611nomaximal, Sylow
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61C2×#12111yescharacteristic
294C22×C61#54611yescharacteristic
295C22×C61#54811yescharacteristic
296C22×C61#57511yescharacteristic
297C22×C61#57511yescharacteristic
298C22×C61#57711yescharacteristic
299C22×C61#57811yescharacteristic
300C4×C61—22no—
301C4×C61—22no—
302C4×C61—42no—
303C4×C61—42no—
304C4×C61—81no—
305C4×C61#52911yescharacteristic
306C4×C61#54211yescharacteristic
307C4×C61#54611yescharacteristic
308C4×C61#54711yescharacteristic
309Dic61—41no—
310Dic61—41no—
311Dic61—42no—
312Dic61—41no—
313Dic61—41no—
314Dic61—81no—
315Dic61—81no—
316Dic61—81no—
317C2×#489—611no—
318C2×#464D6111yescharacteristic
319#2568—611no—
320#2367—611no—
321#2074—611no—
322#2073—611no—
323#2056—611no—
324#2055—611no—
325#1987—611no—
326#1940—611no—
327#1878—611no—
328#1877—611no—
329#1805—611no—
330#1804—611no—
331#1763—611no—
332C2×C61—82no—
333C2×C61#242711yescharacteristic
334C2×C61#304011yescharacteristic
335C2×C61#304611yescharacteristic
336C2×C61#331111yescharacteristic
337C2×C61#331311yescharacteristic
338C2×C61#337111yescharacteristic
339C2×C61#337711yescharacteristic
340C22×Q16C2×D6111yescharacteristic
341C22×SD16C2×D6111yescharacteristic
342C2×C4⋊Q8—611no—
343C2×#124C2×D6111yescharacteristic
344C2×C4×Q8—611no—
345C2×C4×Q8C2×D6111yescharacteristic
346C2×C4×D4—611no—
347#489—1224no—
348#464—24no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8—611no—
351C2×C4⋊C8C2×D6111yescharacteristic
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103C2×D6111yescharacteristic
357C2×#102—611no—
358C2×#102—611no—
359C2×#102C2×D6111yescharacteristic
360C2×#97—611no—
361#400—1221no—
362#400—1221no—
363#391—611no—
364#391—611no—
365#391—611no—
366#391—1221no—
367#391—1221no—
368#390—611no—
369#388—1221no—
370#388—1221no—
371#385—1221no—
372#382—611no—
373#382—611no—
374C4×C4⋊C4—611no—
375C4×C4⋊C4—611no—
376C4×C22⋊C4—1221no—
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#336—611no—
384#336—611no—
385C61#2533711yescharacteristic, complemented, Sylow
386C22×Q8—1221no—
387C22×Q8C22×D6111yescharacteristic
388C22×Q8C22×D6111yescharacteristic
389C22×D4C22×D6111yescharacteristic
390C23×C4—1221no—
391C2×Q16—22no—
392C2×Q16—22no—
393C2×Q16—41no—
394C2×SD16—22no—
395C2×SD16—22no—
396C2×SD16—41no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8C22×D6111yescharacteristic
400C22×C8C22×D6111yescharacteristic
401C4⋊Q8—1222no—
402C4⋊Q8—1222no—
403#124—22no—
404#124C22⋊D6114yes—
405C4×Q8—22no—
406C4×Q8—22no—
407C4×Q8—1222no—
408C4×Q8—2441no—
409C4×D4—1222no—
410C4×D4—2441no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C42—21no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—1221no—
430C2×C42—1221no—
431C2×C42C22×D6111yescharacteristic
432C4⋊C8—22no—
433C4⋊C8—2441no—
434C4⋊C8—2441no—
435#103—22no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#102—22no—
441#102—1222no—
442#102—1222no—
443#97—1222no—
444#95—611no—
445#95—611no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457C24—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—2442no—
469C2×Q8—2442no—
470C2×Q8—2442no—
471C2×Q8#1175822412yes—
472C2×Q8C2×C22⋊D6112yes—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4C2×C22⋊D6112yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—4881no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4D4×D6111yescharacteristic
494C22×C4D4×D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496Q16—44no—
497SD16—44no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—42no—
513C4⋊C4—612no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—2441no—
522C4⋊C4—2441no—
523C4⋊C4—2442no—
524C4⋊C4—2442no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—4881no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C42—42no—
531C42—1221no—
532C42—1221no—
533C42—1221no—
534C42—1221no—
535C42—4881no—
536C42—2442no—
537C42#1175822412yes—
538C42C2×C22⋊D6112yes—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C2×D4×D6111yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—22no—
550Q8—22no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—4884no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1222no—
586C2×C4—4884no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4#1451974912yes—
599C2×C4#1451974912yes—
600C2×C4C22×C22⋊D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4C2×#1175822411yescharacteristic
603C2×C4C2×#1175822411yescharacteristic
604C2×C4#1451982211yescharacteristic
605C2×C4#1451982411yescharacteristic
606C8—41no—
607C8—41no—
608C8—2441no—
609C8—2441no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#1451974911yescharacteristic
618C22#1684040911yescharacteristic
619C22#1684041811yescharacteristic
620C22#1684056111yescharacteristic
621C22#1684056111yescharacteristic
622C22#1684056911yescharacteristic
623C22#1684057611yescharacteristic
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—81no—
629C4—4881no—
630C4—4881no—
631C4—4881no—
632C4—2441no—
633C4—2441no—
634C4—2442no—
635C4—2441no—
636C4—2441no—
637C4#1684029311yescharacteristic
638C4#1684037911yescharacteristic
639C4#1684040611yescharacteristic
640C4#1684041111yescharacteristic
641C2—82no—
642C2#2144939811yescharacteristic
643C2#2145531511yescharacteristic
644C2#2145536711yescharacteristic
645C2#2145786211yescharacteristic
646C2#2145789211yescharacteristic
647C2#2145858411yescharacteristic
648C2#2145865811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#464C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#464D61
hypocenterC61#25337
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
2. #21451031C2semidirect
4. #21447240C2semidirect
6. #21447139C2semidirect
8. #21446766C2semidirect
9. #21446540C2semidirect
10. #21446162C2semidirect
12. #21445832C2semidirect
14. #21445649C2semidirect
385. C61#25337semidirect

Maximal subgroups

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

subgroupindex
C2×#168401212
#214510312
#214488942
#214472402
#214472352
#214471392
#214471322
#214467662
#214465402
#214461622
#214461552
#214458322
#214458272
#214456492
C2×C61×#4642
#2533761