Finite Group Databasebeta

← #29689347

Subgroups — #29689347

Order 15616.

1474 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#124×D61C211yesmaximal, characteristic
2C2×#16839817C211yesmaximal, characteristic, complemented
3#21448224C211yesmaximal, characteristic, complemented
4#21446750C212yesmaximal, complemented
5#21446748C212yesmaximal, complemented
6#21446636C211yesmaximal, characteristic, complemented
7#21446635C211yesmaximal, characteristic, complemented
8#21446633C211yesmaximal, characteristic, complemented
9#21445752C212yesmaximal, complemented
10#21445748C212yesmaximal
11C61×#1959C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C22×D4×D61C2211yescharacteristic
14C2×#14519766C2211yescharacteristic
15C2×#14519765C2211yescharacteristic
16C2×#14519764C2211yescharacteristic
17C2×#14519763—21no—
18C2×#14519763—21no—
19#124×D61—24no—
20#124×D61C418yescomplemented
21C2×#14519735—21no—
22C2×C22⋊C4×D61—21no—
23C2×C22⋊C4×D61—21no—
24C2×C42×D61C2211yescharacteristic
25#16839817—24no—
26C2×#14519706C2212yescomplemented
27C2×#14519702C2212yescomplemented
28C2×#14519701C2212yescomplemented
29C2×#14519700C2212yescomplemented
30C2×#14519699C2212yes—
31C2×#14519698C2212yes—
32C2×#14519697C2212yes—
33C2×#14519689C2212yes—
34C2×#14519688C2211yescharacteristic, complemented
35#16839632C2211yescharacteristic, complemented
36#16839627—21no—
37#16839619—21no—
38#16839619C2212yes—
39C4⋊C4×Dic61—21no—
40C4⋊C4×Dic61C2212yescomplemented
41C2×#124×C61C2211yescharacteristic
42C2×#103×C61C2212yescomplemented
43C2×#102×C61C2212yes—
44C2×#97×C61C2211yescharacteristic, complemented
45C61×#384C2211yescharacteristic, complemented
46C24×D61—21no—
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C2×Q8×D61—22no—
50C2×Q8×D61—42no—
51C2×Q8×D61—42no—
52C2×Q8×D61C2×C414yes—
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C2×D4×D61—22no—
56C2×D4×D61—42no—
57C2×D4×D61—42no—
58C2×D4×D61C2×C414yes—
59C22×C4×D61—21no—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×#92418C2312yes—
64#14519766—22no—
65#14519766C2×C414yes—
66#14519765—22no—
67#14519765C2×C414yes—
68#14519764—22no—
69#14519764C2×C414yes—
70#14519763—28no—
71#14519763—28no—
72#14519735—44no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2312yes—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—22no—
83C22⋊C4×D61—28no—
84C22⋊C4×D61—28no—
85C2×C4×Dic61—22no—
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C42×D61C2×C414yes—
90C42×D61D414yes—
91#14519706—24no—
92#14519702—42no—
93#14519701—42no—
94#14519700—42no—
95#14519699—42no—
96#14519698—42no—
97#14519697—42no—
98#14519689—24no—
99#14519688—22no—
100#14519684—22no—
101#14519684—22no—
102C22×Q8×C61C2311yescharacteristic
103C22×D4×C61C2311yescharacteristic
104C22×C8×C61C2312yes—
105#124×C61—22no—
106#124×C61C2×C414yes—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61C2312yes—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C42×C61C2311yescharacteristic
113#103×C61—42no—
114#102×C61—42no—
115#97×C61—22no—
116C24×C61—21no—
117C23×D61—21no—
118C23×D61—24no—
119C23×D61—42no—
120C23×D61—42no—
121C23×D61—42no—
122C23×D61—42no—
123C23×D61—42no—
124C23×D61—42no—
125C23×D61C2×D411yescharacteristic
126C2×Q8×C61—21no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61C22×C412yes—
130C2×D4×C61—21no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61C22×C412yes—
134C22×C4×C61—21no—
135C22×C4×C61—22no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C2×C22⋊D61—24no—
140C2×C22⋊D61—42no—
141C2×C22⋊D61—42no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147Q8×D61—48no—
148Q8×D61—48no—
149D4×D61—48no—
150D4×D61—48no—
151C2×D244—22no—
152C2×D244—22no—
153C2×D244—21no—
154C2×D244C22×C412yes—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—24no—
160C2×C4×D61—41no—
161C2×C4×D61—41no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61C22⋊C414yes—
165C2×C4×D61C22⋊C414yes—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C2×D412yes—
168C2×Dic122—22no—
169C2×Dic122—22no—
170C2×Dic122—21no—
171C2×Dic122—24no—
172C2×Dic122—42no—
173C2×Dic122—42no—
174C2×Dic122C22×C412yes—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—42no—
181C4⋊C4×C61—42no—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—24no—
184C42×C61C22×C412yes—
185C42×C61C2×D412yes—
186C22⋊Dic61—24no—
187C22⋊Dic61—24no—
188#11758198—24no—
189#11758198—24no—
190#11758198—24no—
191#11758198—24no—
192#11758198—44no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—42no—
196C4⋊Dic61—42no—
197C4×Dic61—44no—
198C4×Dic61C22⋊C414yes—
199C4×Dic61C22⋊C414yes—
200C4×Dic61C22×C412yes—
201C4×Dic61C2×D412yes—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C23×C61—22no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61C22×D411yescharacteristic
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—22no—
218C22×D61—48no—
219C22×D61—48no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—48no—
227C22×D61—84no—
228C22×D61—84no—
229C22×D61—84no—
230C22×D61—84no—
231C22×D61C2×C22⋊C412yes—
232C22×D61AGL(1,ℤ/8)12yes—
233C22×D61#13712yes—
234Q8×C61—24no—
235Q8×C61—42no—
236D4×C61—24no—
237D4×C61—42no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61C2×C22⋊C412yes—
249C2×C4×C61C2×C22⋊C412yes—
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61H(5,𝔽2)12yes—
253C2×C4×C61H(5,𝔽2)12yes—
254C22⋊D61—48no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61C2×C22⋊C412yes—
268C2×Dic61C2×C22⋊C412yes—
269C2×Dic61C2×C22⋊C412yes—
270D244—24no—
271D244—42no—
272C4×D61—28no—
273C4×D61—24no—
274C4×D61—24no—
275C4×D61—44no—
276C4×D61—48no—
277Dic122—24no—
278Dic122—48no—
279Dic122—42no—
280C8×C61—42no—
281#92418—42no—
282#21808—611nomaximal, Sylow
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61C22×C22⋊C411yescharacteristic
291C22×C61#53511yescharacteristic
292C22×C61#53511yescharacteristic
293C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C61C2×#13711yescharacteristic
295C22×C61#57511yescharacteristic
296C22×C61#57511yescharacteristic
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—816no—
307C2×D61—81no—
308C2×D61—81no—
309C2×D61—81no—
310C2×D61—81no—
311C2×D61#41714yes—
312C4×C61—22no—
313C4×C61—22no—
314C4×C61—42no—
315C4×C61—82no—
316C4×C61#51812yes—
317C4×C61#53512yes—
318Dic61—22no—
319Dic61—22no—
320Dic61—22no—
321Dic61—42no—
322Dic61—82no—
323C22×#124—611no—
324C2×#432—611no—
325#2285—611no—
326#1983—612no—
327#1983—612no—
328#1959—611no—
329#1959—611no—
330#1959—611no—
331#1959D6111yescharacteristic, complemented
332#1787—612no—
333#1786—612no—
334C2×C61—82no—
335C2×C61#230911yescharacteristic
336C2×C61C2×#41711yescharacteristic
337C2×C61#289611yescharacteristic
338C2×C61#321311yescharacteristic
339C2×C61#321311yescharacteristic
340C2×C61#321511yescharacteristic
341C2×C61#321511yescharacteristic
342D61—24no—
343D61—82no—
344C23×Q8—611no—
345C23×D4—611no—
346C2×#124—611no—
347C2×#124—611no—
348C2×#124—618no—
349C2×#124—611no—
350C2×#124—1221no—
351C2×#124—1221no—
352C2×#124—1224no—
353C2×#124C2×D6111yescharacteristic
354C2×#117—1221no—
355C22×C22⋊C4—1221no—
356C22×C22⋊C4—1221no—
357C22×C42—611no—
358#432—1224no—
359C2×C4⋊C8—612no—
360C2×#103—612no—
361C2×#103—612no—
362C2×#103—612no—
363C2×#103C2×D6112yescomplemented
364C2×#102—612no—
365C2×#102—612no—
366C2×#102—612no—
367C2×#102C2×D6112yes—
368C2×C22⋊C8—612no—
369C2×#97—611no—
370C2×#97C2×D6111yescharacteristic, complemented
371#384—611no—
372#384C2×D6111yescharacteristic, complemented
373#382—1221no—
374#380—612no—
375#380—1221no—
376C4×C4⋊C4—612no—
377C4×C4⋊C4—1221no—
378C61#2180811yescharacteristic, complemented, Sylow
379C25—1221no—
380C22×Q8—614no—
381C22×Q8—611no—
382C22×Q8—1221no—
383C22×Q8—1222no—
384C22×Q8—2442no—
385C22×Q8—2442no—
386C22×Q8C22×D6111yescharacteristic
387C22×D4—614no—
388C22×D4—611no—
389C22×D4—1222no—
390C22×D4—1221no—
391C22×D4—2442no—
392C22×D4—2442no—
393C22×D4C22×D6111yescharacteristic
394C23×C4—611no—
395C23×C4—611no—
396C23×C4—611no—
397C23×C4—1221no—
398C22×C8—612no—
399C22×C8C22×D6112yes—
400#124—22no—
401#124—614no—
402#124—614no—
403#124—614no—
404#124—1228no—
405#124—1228no—
406#124—1222no—
407#124—1222no—
408#124—1222no—
409#124C4×D6114yes—
410#117—2444no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22×D6112yes—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—1228no—
420C2×C22⋊C4—1228no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1222no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—614no—
432C2×C42—614no—
433C2×C42—1222no—
434C2×C42C22×D6111yescharacteristic
435C4⋊C8—1224no—
436#103—42no—
437#103—2442no—
438#103—2442no—
439#103—2442no—
440#102—42no—
441#102—2442no—
442#102—2442no—
443#102—2442no—
444C22⋊C8—1224no—
445#97—22no—
446#97—1222no—
447#95—1222no—
448#95—1222no—
449C24—21no—
450C24—611no—
451C24—1224no—
452C24—1221no—
453C24—2442no—
454C24—2442no—
455C24—2442no—
456C24—2442no—
457C24—2442no—
458C24—2442no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—612no—
463C2×Q8—1224no—
464C2×Q8—1221no—
465C2×Q8—1222no—
466C2×Q8—1222no—
467C2×Q8—2442no—
468C2×Q8—2442no—
469C2×Q8—2448no—
470C2×Q8—2448no—
471C2×Q8C2×C4×D6112yes—
472C2×D4—21no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—612no—
476C2×D4—1224no—
477C2×D4—1221no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—2448no—
481C2×D4—2448no—
482C2×D4—2442no—
483C2×D4—2442no—
484C2×D4C2×C4×D6112yes—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—612no—
491C22×C4—612no—
492C22×C4—614no—
493C22×C4—614no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1224no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1221no—
500C22×C4—1222no—
501C22×C4—2441no—
502C22×C4—2441no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4D4×D6111yescharacteristic
506C22×C4D4×D6111yescharacteristic
507C22×C4C23×D6111yescharacteristic
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—1222no—
512C2×C8—1222no—
513C2×C8—1222no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—1224no—
525C22⋊C4—1224no—
526C22⋊C4—1224no—
527C22⋊C4—1224no—
528C22⋊C4—1224no—
529C22⋊C4—1224no—
530C22⋊C4—2444no—
531C42—612no—
532C42—612no—
533C42—614no—
534C42—614no—
535C42—2444no—
536C42C2×C4×D6112yes—
537C42D4×D6112yes—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—612no—
546C23—612no—
547C23—612no—
548C23—1221no—
549C23—1221no—
550C23—1221no—
551C23—1221no—
552C23—1222no—
553C23—4884no—
554C23—4884no—
555C23—4884no—
556C23—4884no—
557C23—2448no—
558C23—2448no—
559C23—2448no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23—2441no—
564C23—2441no—
565C23—2441no—
566C23C2×D4×D6111yescharacteristic
567Q8—24no—
568Q8—42no—
569Q8—1224no—
570Q8—2448no—
571Q8—2442no—
572D4—24no—
573D4—42no—
574D4—1224no—
575D4—2448no—
576D4—2442no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—1228no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1224no—
598C2×C4—1224no—
599C2×C4—1222no—
600C2×C4—2444no—
601C2×C4—2448no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4C22⋊C4×D6112yes—
608C2×C4C22⋊C4×D6112yes—
609C2×C4C22×C4×D6111yescharacteristic
610C2×C4C2×D4×D6111yescharacteristic
611C2×C4#1451982212yes—
612C2×C4#1451982312yes—
613C8—42no—
614C8—2442no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—614no—
623C22—1222no—
624C22—1222no—
625C22—1222no—
626C22—1222no—
627C22—1222no—
628C22—1222no—
629C22—48816no—
630C22—4881no—
631C22—4881no—
632C22—4881no—
633C22—4881no—
634C22—2442no—
635C22—2442no—
636C22—2442no—
637C22C2×C22⋊C4×D6111yescharacteristic
638C22#1684032511yescharacteristic
639C22#1684033011yescharacteristic
640C22AGL(1,ℤ/8)×D6111yescharacteristic
641C22#137×D6111yescharacteristic
642C22#1684055811yescharacteristic
643C22#1684056111yescharacteristic
644C4—22no—
645C4—22no—
646C4—42no—
647C4—82no—
648C4—1222no—
649C4—1222no—
650C4—1222no—
651C4—4882no—
652C4—2442no—
653C4#1684023412yes—
654C4#1684032812yes—
655C2—82no—
656C2—1224no—
657C2—4882no—
658C2#2144839811yescharacteristic
659C2#417×D6111yescharacteristic
660C2#2145433511yescharacteristic
661C2#2145680111yescharacteristic
662C2#2145680711yescharacteristic
663C2#2145681711yescharacteristic
664C2#2145682311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1959C2
socleC23×C61C22×D4
prime radicalC22×D4×D61C22
cosocleC22×C4×C61C24
hypercenter#1959D61
hypocenterC61#21808
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
2. C2×#16839817C2semidirect
3. #21448224C2semidirect
4. #21446750C2semidirect
5. #21446748C2semidirect
6. #21446636C2semidirect
7. #21446635C2semidirect
8. #21446633C2semidirect
9. #21445752C2semidirect
11. C61×#1959C2semidirect
20. #124×D61C4semidirect
26. C2×#14519706C22semidirect
27. C2×#14519702C22semidirect
28. C2×#14519701C22semidirect
29. C2×#14519700C22semidirect
34. C2×#14519688C22semidirect
35. #16839632C22semidirect
40. C4⋊C4×Dic61C22semidirect
42. C2×#103×C61C22semidirect
44. C2×#97×C61C22semidirect
45. C61×#384C22semidirect
331. #1959D61semidirect
363. C2×#103C2×D61semidirect
370. C2×#97C2×D61semidirect
372. #384C2×D61semidirect
378. C61#21808semidirect

Maximal subgroups

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

subgroupindex
C2×#124×D612
C2×#168398172
#214482242
#214467502
#214467482
#214466362
#214466352
#214466332
#214457522
#214457482
C61×#19592
#2180861