Finite Group Databasebeta

← #29729554

Subgroups — #29729554

Order 15616.

848 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
1C2×#16839886C211yesmaximal, characteristic, complemented
2#21451308C211yesmaximal, characteristic, complemented
3#21451173C211yesmaximal, characteristic, complemented
4#21447212C211yesmaximal, characteristic, complemented
5#21447191C211yesmaximal, characteristic, complemented
6#21447132C211yesmaximal, characteristic, complemented
7#21447127C211yesmaximal, characteristic
8#21446686C211yesmaximal, characteristic
9#21446675C211yesmaximal, characteristic, complemented
10#21446631C211yesmaximal, characteristic, complemented
11#21446589C211yesmaximal, characteristic, complemented
12#21446499C211yesmaximal, characteristic, complemented
13#21446492C211yesmaximal, characteristic, complemented
14C4×#14519698C211yesmaximal, characteristic, complemented
15C2×C61×#463C211yesmaximal, characteristic
16C2×C4×Dic122C2211yescharacteristic
17#16839886—24no—
18C2×#14519711C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839698—21no—
32#16839687—21no—
33#16839674—21no—
34#16839674—21no—
35#16839669C2211yescharacteristic
36#16839669C2211yescharacteristic
37#16839665C2211yescharacteristic
38#16839665C2211yescharacteristic
39#16839665C2211yescharacteristic
40#16839665C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839658C2211yescharacteristic
43#16839656—21no—
44#16839656—21no—
45#16839635—21no—
46#16839635—21no—
47#16839628C2211yescharacteristic
48C4×C4⋊Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C22×D8×C61C2211yescharacteristic
55C2×#124×C61C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C61×#463—24no—
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C22×Dic122D411yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C4×Dic122—22no—
65C4×Dic122—22no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196D411yescharacteristic
74C2×#11758196D411yescharacteristic
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519711—41no—
87#14519709—41no—
88#14519706—22no—
89#14519706—22no—
90#14519701—22no—
91#14519701—41no—
92#14519698—22no—
93#14519698—22no—
94#14519698—22no—
95#14519698—41no—
96C4×#92418—22no—
97C8×Dic61—22no—
98C8×Dic61—22no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108C22×Q8×C61C2311yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C23×C4×C61—21no—
112C2×SD16×C61—22no—
113C2×SD16×C61—22no—
114C2×SD16×C61—41no—
115C2×D8×C61—22no—
116C2×D8×C61—22no—
117C2×D8×C61—41no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120#124×C61—22no—
121#124×C61—22no—
122C4×D4×C61—22no—
123C4×D4×C61—22no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61D414yes—
130#103×C61—22no—
131#102×C61—22no—
132C24×C61—21no—
133C24×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×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61#4112yes—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—42no—
155C22×C4×C61—81no—
156C22×C4×C61—81no—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2411yescharacteristic
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C2×Dic122—22no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171SD16×C61—44no—
172D8×C61—44no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D412yes—
178C2×C8×C61C2×D412yes—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C22⋊C4×C61—42no—
182C22⋊C4×C61—42no—
183C22⋊C4×C61—42no—
184C42×C61C2×D412yes—
185C42×C61#4112yes—
186C22⋊Dic61—42no—
187C22⋊Dic61—42no—
188C22⋊Dic61—42no—
189C22⋊Dic61—42no—
190C22⋊Dic61—81no—
191C22⋊Dic61—81no—
192C22⋊Dic61—81no—
193C22⋊Dic61—81no—
194C4⋊Dic61—22no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203#11758196—22no—
204#11758196—22no—
205#11758196—81no—
206#11758196—81no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—81no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418#4112yes—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61C22×D411yescharacteristic
233Q8×C61—41no—
234Q8×C61—41no—
235Q8×C61—41no—
236Q8×C61—41no—
237D4×C61—22no—
238D4×C61—22no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—41no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—84no—
256C2×C4×C61#12112yes—
257C2×C4×C61#12112yes—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61#13512yes—
287C2×Dic61#13512yes—
288Dic122—28no—
289C8×C61—22no—
290C8×C61—22no—
291#92418—22no—
292#92418—41no—
293#25296—611nomaximal, Sylow
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61C2×#12111yescharacteristic
309C22×C61#54811yescharacteristic
310C22×C61#55011yescharacteristic
311C22×C61C2×#13511yescharacteristic
312C22×C61C2×#13511yescharacteristic
313C22×C61#57611yescharacteristic
314C22×C61#57711yescharacteristic
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—81no—
318C4×C61—81no—
319C4×C61#52911yescharacteristic
320C4×C61#54211yescharacteristic
321C4×C61#54611yescharacteristic
322C4×C61#54811yescharacteristic
323Dic61—22no—
324Dic61—22no—
325Dic61—41no—
326Dic61—41no—
327Dic61—81no—
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331C2×#463D6111yescharacteristic
332C2×C8×Q8—611no—
333#2592—611no—
334#2580—611no—
335#2068—611no—
336#2065—611no—
337#2055—611no—
338#2054—611no—
339#1970—611no—
340#1968—611no—
341#1958—611no—
342#1949—611no—
343#1934—611no—
344#1933—611no—
345C4×#102—611no—
346C2×C61—82no—
347C2×C61—161no—
348C2×C61#242611yescharacteristic
349C2×C61#305011yescharacteristic
350C2×C61#306211yescharacteristic
351C2×C61#336011yescharacteristic
352C2×C61#336111yescharacteristic
353C2×C61#339411yescharacteristic
354C2×C61#339611yescharacteristic
355C22×SD16C2×D6111yescharacteristic
356C22×D8C2×D6111yescharacteristic
357C2×#124C2×D6111yescharacteristic
358C2×C4×Q8—611no—
359C2×C4×D4C2×D6111yescharacteristic
360#463—24no—
361C8×Q8—1224no—
362C2×#107—611no—
363C2×#106—611no—
364C2×C4⋊C8—611no—
365C2×C4⋊C8—611no—
366C2×C4⋊C8C2×D6111yescharacteristic
367C2×#103—611no—
368C2×#103—611no—
369C2×#103C2×D6111yescharacteristic
370C2×#102—611no—
371C2×#102—611no—
372C2×#102—611no—
373C2×#102—611no—
374C2×#102C2×D6111yescharacteristic
375C2×C4×C8—611no—
376C2×C4×C8—611no—
377C2×C4×C8—611no—
378#399—1221no—
379#397—1221no—
380#393—1221no—
381#393—1221no—
382#391—611no—
383#391—611no—
384#390—611no—
385#390—611no—
386#390—611no—
387#390—611no—
388#389—611no—
389#389—611no—
390#388—1221no—
391#388—1221no—
392#385—1221no—
393#385—1221no—
394#382—611no—
395C4×C4⋊C4—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400C61#2529611yescharacteristic, complemented, Sylow
401C22×Q8—611no—
402C22×Q8C22×D6111yescharacteristic
403C22×D4C22×D6111yescharacteristic
404C22×D4C22×D6111yescharacteristic
405C23×C4—21no—
406C2×SD16—22no—
407C2×SD16—22no—
408C2×SD16—41no—
409C2×D8—22no—
410C2×D8—22no—
411C2×D8—41no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416#124—22no—
417#124—22no—
418C4×Q8—1222no—
419C4×Q8—1222no—
420C4×D4—22no—
421C4×D4—22no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—1221no—
446C2×C42C22×D6111yescharacteristic
447#107—2441no—
448#106—2441no—
449C4⋊C8—1222no—
450C4⋊C8—1222no—
451C4⋊C8C22⋊D6114yes—
452#103—22no—
453#103—1222no—
454#103—2441no—
455#102—22no—
456#102—1222no—
457#102—1222no—
458#102—1222no—
459#102—2441no—
460C4×C8—1222no—
461C4×C8—1222no—
462C4×C8—1222no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24—21no—
473C24—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—1222no—
481C2×Q8—1222no—
482C2×Q8—1222no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4#1175822412yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—42no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511SD16—44no—
512D8—44no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—612no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8C2×C22⋊D6112yes—
524C2×C8C2×C22⋊D6112yes—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—4881no—
539C4⋊C4—4881no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—4881no—
544C22⋊C4—4881no—
545C22⋊C4—4881no—
546C22⋊C4—4881no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C22⋊C4—2442no—
550C22⋊C4—2442no—
551C42—1221no—
552C42—1221no—
553C42—1221no—
554C42—1221no—
555C42—4881no—
556C42#1175822412yes—
557C42C2×C22⋊D6112yes—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23C2×D4×D6111yescharacteristic
573Q8—41no—
574Q8—41no—
575Q8—41no—
576Q8—41no—
577Q8—1228no—
578D4—22no—
579D4—22no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—41no—
585D4—41no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—84no—
597C2×C4—612no—
598C2×C4—612no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4#1451974812yes—
622C2×C4#1451974812yes—
623C2×C4C22×C22⋊D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4C2×#1175822411yescharacteristic
626C2×C4C2×#1175822411yescharacteristic
627C2×C4#1451982211yescharacteristic
628C2×C4#1451982411yescharacteristic
629C8—22no—
630C8—22no—
631C8—1222no—
632C8—2441no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22C2×#1451974811yescharacteristic
648C22#1684041811yescharacteristic
649C22#1684043011yescharacteristic
650C22#135×D6111yescharacteristic
651C22#135×D6111yescharacteristic
652C22#1684056611yescharacteristic
653C22#1684057211yescharacteristic
654C4—22no—
655C4—22no—
656C4—81no—
657C4—81no—
658C4—1222no—
659C4—1222no—
660C4—4881no—
661C4—4881no—
662C4—4881no—
663C4—4881no—
664C4—2441no—
665C4—2441no—
666C4#1684029311yescharacteristic
667C4#1684037911yescharacteristic
668C4#1684040611yescharacteristic
669C4#1684041911yescharacteristic
670C2—82no—
671C2—161no—
672C2#2144938111yescharacteristic
673C2#2145539611yescharacteristic
674C2#2145549311yescharacteristic
675C2#2145846311yescharacteristic
676C2#2145847411yescharacteristic
677C2#2145881811yescharacteristic
678C2#2145882611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#463C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenterC2×#463D61
hypocenterC61#25296
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#463

Complements

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

NH ≅ G/Ntype
1. C2×#16839886C2semidirect
2. #21451308C2semidirect
3. #21451173C2semidirect
4. #21447212C2semidirect
5. #21447191C2semidirect
6. #21447132C2semidirect
9. #21446675C2semidirect
10. #21446631C2semidirect
11. #21446589C2semidirect
12. #21446499C2semidirect
13. #21446492C2semidirect
14. C4×#14519698C2semidirect
400. C61#25296semidirect

Maximal subgroups

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

subgroupindex
C2×#168398862
#214513082
#214511732
#214472122
#214471912
#214471322
#214471272
#214466862
#214466752
#214466312
#214465892
#214464992
#214464922
C4×#145196982
C2×C61×#4632
#2529661