Finite Group Databasebeta

← #29712644

Subgroups — #29712644

Order 15616.

1844 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519748C211yesmaximal, characteristic
2C2×#16840074C212yesmaximal, complemented
3#380×D61C211yesmaximal, characteristic, complemented
4#21447000C212yesmaximal, complemented
5#21446999C212yesmaximal, complemented
6#21446133C212yesmaximal, complemented
7#21446009C211yesmaximal, characteristic, complemented
8#21446006C211yesmaximal, characteristic, complemented
9#21445750C211yesmaximal, characteristic, complemented
10#21445728C211yesmaximal, characteristic, complemented
11C61×#1786C211yesmaximal, characteristic, complemented
12C23×C22⋊D61—21no—
13C23×C4×D61C2211yescharacteristic
14C2×#14519748—28no—
15C2×#14519748—21no—
16C2×#14519748—21no—
17C2×#14519748C2212yes—
18C2×#14519748C2212yes—
19C22×C4⋊Dic61C2211yescharacteristic
20C2×C4⋊C4×D61C2211yescharacteristic, complemented
21C2×C4⋊C4×D61C2211yescharacteristic, complemented
22C22×C22⋊Dic61—21no—
23C2×C22⋊C4×D61—21no—
24#16840074—28no—
25C2×#14519711C2212yescomplemented
26C2×#14519710C2212yescomplemented
27C2×#14519698C2212yescomplemented
28C2×#14519698C2212yescomplemented
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519689C2212yescomplemented
31C2×#14519689C2212yescomplemented
32#16839622C2211yescharacteristic
33#16839621—21no—
34#16839620—21no—
35#16839619C2211yescharacteristic, complemented
36#16839619C2211yescharacteristic, complemented
37#95×D61—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40C23×D4×C61C2211yescharacteristic
41C2×#102×C61C2212yescomplemented
42C2×#102×C61C2212yescomplemented
43C2×C22⋊C8×C61C2211yescharacteristic
44C61×#380C2211yescharacteristic
45C24×D61D411yescharacteristic
46C22×C22⋊D61—28no—
47C22×C22⋊D61—22no—
48C22×C22⋊D61—22no—
49C22×C22⋊D61—44no—
50C22×C22⋊D61—44no—
51C23×Dic61—21no—
52C23×Dic61D411yescharacteristic
53C22×C4×D61—22no—
54C22×C4×D61—22no—
55C22×C4×D61—24no—
56C22×C4×D61—21no—
57C22×C4×D61—21no—
58C22×C4×D61—42no—
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×#92418C2312yes—
64#14519748—28no—
65#14519748—28no—
66#14519748—24no—
67#14519748—24no—
68#14519748—24no—
69#14519748D418yes—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—24no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C4⋊C4×D61—24no—
78C4⋊C4×D61—42no—
79C2×C22⋊Dic61—28no—
80C2×C22⋊Dic61—22no—
81C2×C22⋊Dic61—22no—
82C2×C22⋊Dic61—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C22⋊C4×D61—44no—
86#14519711—24no—
87#14519710—24no—
88#14519698—24no—
89#14519698—42no—
90C22⋊#92418—22no—
91#14519689—24no—
92#14519689—42no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96C25×C61—21no—
97C22×D4×C61—24no—
98C22×D4×C61—21no—
99C22×D4×C61—22no—
100C22×D4×C61—22no—
101C22×D4×C61—42no—
102C22×D4×C61C2312yes—
103C23×C4×C61C2311yescharacteristic
104C22×C8×C61C2312yes—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C22⋊C4×C61—21no—
108#102×C61—24no—
109#102×C61—42no—
110C22⋊C8×C61—22no—
111#95×C61—21no—
112C24×C61—24no—
113C24×C61—22no—
114C24×C61—42no—
115C24×C61—42no—
116C24×C61—42no—
117C24×C61—42no—
118C24×C61—42no—
119C24×C61—42no—
120C24×C61C2×D411yescharacteristic
121C23×D61—22no—
122C23×D61—22no—
123C23×D61—22no—
124C23×D61—22no—
125C23×D61—41no—
126C23×D61—41no—
127C23×D61D814yes—
128C23×D61C2×D411yescharacteristic
129C23×D61C2×D411yescharacteristic
130C2×D4×C61—24no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—48no—
137C2×D4×C61—48no—
138C2×D4×C61—42no—
139C2×D4×C61—42no—
140C2×D4×C61C2×D414yes—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61#4112yes—
147C22×C4×C61#4112yes—
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—28no—
150C2×C22⋊D61—28no—
151C2×C22⋊D61—432no—
152C2×C22⋊D61—44no—
153C2×C22⋊D61—44no—
154C2×C22⋊D61—44no—
155C2×C22⋊D61—44no—
156C22×Dic61—24no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—41no—
163C22×Dic61—42no—
164C22×Dic61—42no—
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—22no—
177C2×C4×D61—22no—
178C2×C4×D61—48no—
179C2×C4×D61—48no—
180C2×C4×D61—41no—
181C2×C4×D61—41no—
182C2×C4×D61—42no—
183C2×C4×D61—42no—
184C2×C4×D61—42no—
185C2×C4×D61—42no—
186C2×C4×D61—42no—
187C2×C4×D61—42no—
188C2×C4×D61C2×D414yes—
189C2×C8×C61—22no—
190C2×C8×C61—22no—
191C2×C8×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—22no—
194C22⋊C4×C61—42no—
195C22⋊Dic61—28no—
196C22⋊Dic61—28no—
197C22⋊Dic61—42no—
198#11758198—42no—
199#11758198—42no—
200C4⋊Dic61—24no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61C2×D414yes—
205#11758196—42no—
206#11758196—42no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418—22no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—24no—
215C23×C61—48no—
216C23×C61—48no—
217C23×C61—48no—
218C23×C61—42no—
219C23×C61—42no—
220C23×C61—42no—
221C23×C61—42no—
222C23×C61—42no—
223C23×C61—42no—
224C23×C61—84no—
225C23×C61—84no—
226C23×C61—84no—
227C23×C61—84no—
228C23×C61#12312yes—
229C23×C61C2×D812yes—
230C23×C61#13712yes—
231C23×C61C22×D411yescharacteristic
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—21no—
238C22×D61—21no—
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—21no—
242C22×D61—28no—
243C22×D61—28no—
244C22×D61—44no—
245C22×D61—44no—
246C22×D61—44no—
247C22×D61—44no—
248C22×D61—44no—
249C22×D61#12012yes—
250C22×D61C2×D812yes—
251C22×D61C2×D812yes—
252C22×D61AGL(1,ℤ/8)12yes—
253D4×C61—24no—
254D4×C61—24no—
255D4×C61—48no—
256D4×C61—42no—
257D4×C61—42no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—24no—
261C2×C4×C61—24no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61#12312yes—
269C2×C4×C61C22×D412yes—
270C2×C4×C61C2×#4111yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61H(5,𝔽2)12yes—
273C22⋊D61—416no—
274C22⋊D61—416no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—24no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—44no—
282C2×Dic61—48no—
283C2×Dic61—42no—
284C2×Dic61—42no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61#12012yes—
292C4×D61—24no—
293C4×D61—24no—
294C4×D61—44no—
295C4×D61—44no—
296C4×D61—42no—
297C4×D61—42no—
298C4×D61—84no—
299C4×D61—84no—
300C8×C61—42no—
301#92418—42no—
302#24002—611nomaximal, Sylow
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—22no—
306C22×C61—22no—
307C22×C61—22no—
308C22×C61—22no—
309C22×C61—44no—
310C22×C61—44no—
311C22×C61—44no—
312C22×C61—816no—
313C22×C61—82no—
314C22×C61—82no—
315C22×C61—82no—
316C22×C61—82no—
317C22×C61#48014yescomplemented
318C22×C61C2×#12011yescharacteristic
319C22×C61C2×#12311yescharacteristic
320C22×C61#53411yescharacteristic
321C22×C61C22×D811yescharacteristic
322C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
323C22×C61C2×#13711yescharacteristic
324C22×C61#57511yescharacteristic
325C2×D61—22no—
326C2×D61—22no—
327C2×D61—22no—
328C2×D61—22no—
329C2×D61—22no—
330C2×D61—22no—
331C2×D61—22no—
332C2×D61—22no—
333C2×D61—22no—
334C2×D61—416no—
335C2×D61—41no—
336C2×D61—41no—
337C2×D61—41no—
338C2×D61—41no—
339C2×D61#44714yes—
340C4×C61—22no—
341C4×C61—81no—
342C4×C61—81no—
343C4×C61#54612yes—
344C4×C61#54612yes—
345Dic61—22no—
346Dic61—41no—
347Dic61—44no—
348Dic61—81no—
349Dic61—81no—
350C22×#121—611no—
351C2×#480—612no—
352C2×#380—611no—
353#2033—612no—
354#2033—612no—
355#1874—612no—
356#1852—611no—
357#1852—611no—
358#1786—611no—
359#1786D6111yescharacteristic, complemented
360#1779—611no—
361C2×C61—24no—
362C2×C61—84no—
363C2×C61#240511yescharacteristic
364C2×C61C2×#44711yescharacteristic
365C2×C61#300711yescharacteristic
366C2×C61C2×#48011yescharacteristic
367C2×C61#309411yescharacteristic
368C2×C61#318611yescharacteristic
369C2×C61#319311yescharacteristic
370D61—24no—
371D61—42no—
372C23×D4—1221no—
373C23×D4C2×D6111yescharacteristic
374C24×C4—611no—
375C2×#121—612no—
376C2×#121—612no—
377C2×#121—1228no—
378C2×#121—1221no—
379C2×#121—1221no—
380C22×C4⋊C4—611no—
381C22×C4⋊C4—611no—
382C22×C4⋊C4—611no—
383C22×C22⋊C4—1221no—
384C22×C22⋊C4—1221no—
385#480—1228no—
386C2×#107—612no—
387C2×#107—612no—
388C2×#102—612no—
389C2×#102—612no—
390C2×#102C2×D6112yescomplemented
391C2×#102C2×D6112yescomplemented
392C2×C22⋊C8—611no—
393C2×C22⋊C8—612no—
394C2×C22⋊C8—612no—
395C2×C22⋊C8C2×D6111yescharacteristic
396#380—611no—
397#380—611no—
398#380—611no—
399#380—1221no—
400#380—1221no—
401#380C2×D6111yescharacteristic
402C2×#95—1221no—
403#340—612no—
404#340—612no—
405C61#2400211yescharacteristic, complemented, Sylow
406C25—21no—
407C25—611no—
408C22×D4—24no—
409C22×D4—22no—
410C22×D4—21no—
411C22×D4—22no—
412C22×D4—42no—
413C22×D4—1222no—
414C22×D4—1222no—
415C22×D4—1228no—
416C22×D4—2444no—
417C22×D4—2444no—
418C22×D4C22×D6112yes—
419C23×C4—611no—
420C23×C4—611no—
421C23×C4—611no—
422C23×C4—611no—
423C23×C4—611no—
424C23×C4—1224no—
425C23×C4—1221no—
426C23×C4—1221no—
427C23×C4—1221no—
428C23×C4—1222no—
429C23×C4—1222no—
430C23×C4—2442no—
431C23×C4C22×D6111yescharacteristic
432C22×C8—612no—
433C22×C8C22×D6112yes—
434#121—618no—
435#121—1224no—
436#121—1224no—
437#121—1228no—
438#121—1228no—
439#121—1224no—
440C2×C4⋊C4—612no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—1224no—
444C2×C4⋊C4—1224no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—2442no—
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—1222no—
453C2×C22⋊C4—1222no—
454C2×C22⋊C4—1228no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—2444no—
459#107—1224no—
460#107—1224no—
461#102—24no—
462#102—42no—
463#102—1224no—
464#102—2442no—
465C22⋊C8—22no—
466C22⋊C8—1224no—
467C22⋊C8—1222no—
468C22⋊C8—2442no—
469#95—21no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473C24—24no—
474C24—22no—
475C24—42no—
476C24—42no—
477C24—42no—
478C24—42no—
479C24—42no—
480C24—42no—
481C24—611no—
482C24—611no—
483C24—614no—
484C24—1222no—
485C24—1222no—
486C24—1222no—
487C24—1222no—
488C24—2441no—
489C24—2441no—
490C24D4×D6111yescharacteristic
491C2×D4—24no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—48no—
498C2×D4—48no—
499C2×D4—42no—
500C2×D4—42no—
501C2×D4—1228no—
502C2×D4—1228no—
503C2×D4—2444no—
504C2×D4—2444no—
505C2×D4—2444no—
506C2×D4—24432no—
507C2×D4—2444no—
508C2×D4D4×D6114yes—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—611no—
514C22×C4—614no—
515C22×C4—611no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1222no—
522C22×C4—1224no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1222no—
529C22×C4—1222no—
530C22×C4—1222no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—2441no—
534C22×C4—2441no—
535C22×C4—2448no—
536C22×C4—2448no—
537C22×C4—2442no—
538C22×C4—2442no—
539C22×C4—2442no—
540C22×C4—2442no—
541C22×C4—2442no—
542C22×C4—2442no—
543C22×C4—2441no—
544C22×C4—2442no—
545C22×C4—2442no—
546C22×C4#1175822212yes—
547C22×C4D4×D6111yescharacteristic
548C22×C4#1175822412yes—
549C22×C4C23×D6111yescharacteristic
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—22no—
553C2×C8—1222no—
554C2×C8—1222no—
555C2×C8—1222no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—614no—
559C4⋊C4—1224no—
560C4⋊C4—1222no—
561C4⋊C4—1222no—
562C4⋊C4—1222no—
563C4⋊C4—2442no—
564C4⋊C4—2442no—
565C22⋊C4—42no—
566C22⋊C4—1228no—
567C22⋊C4—1228no—
568C22⋊C4—2442no—
569C22⋊C4—2442no—
570C22⋊C4—2442no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—24no—
576C23—48no—
577C23—48no—
578C23—48no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23—612no—
590C23—612no—
591C23—612no—
592C23—612no—
593C23—1228no—
594C23—1221no—
595C23—1221no—
596C23—1221no—
597C23—1221no—
598C23—1221no—
599C23—1221no—
600C23—1221no—
601C23—1221no—
602C23—1221no—
603C23—1221no—
604C23—1228no—
605C23—2444no—
606C23—2444no—
607C23—2444no—
608C23—2444no—
609C23—2444no—
610C23#1451975912yes—
611C23D8×D6112yes—
612C23#1451980612yes—
613C23C2×D4×D6111yescharacteristic
614D4—24no—
615D4—24no—
616D4—48no—
617D4—42no—
618D4—42no—
619D4—24416no—
620D4—24416no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—612no—
632C2×C4—1221no—
633C2×C4—1221no—
634C2×C4—1224no—
635C2×C4—1224no—
636C2×C4—1221no—
637C2×C4—1224no—
638C2×C4—1221no—
639C2×C4—1221no—
640C2×C4—4884no—
641C2×C4—4884no—
642C2×C4—2444no—
643C2×C4—2444no—
644C2×C4—2444no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4—2441no—
651C2×C4—2448no—
652C2×C4—2442no—
653C2×C4—2442no—
654C2×C4—2442no—
655C2×C4—2442no—
656C2×C4#1451975912yes—
657C2×C4C2×D4×D6112yes—
658C2×C4C2×#1175822411yescharacteristic
659C2×C4C2×#1175822211yescharacteristic
660C2×C4#1451982212yes—
661C8—42no—
662C8—2442no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—816no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—614no—
678C22—1222no—
679C22—1222no—
680C22—1222no—
681C22—1222no—
682C22—1222no—
683C22—1222no—
684C22—1222no—
685C22—1222no—
686C22—1222no—
687C22—24416no—
688C22—2441no—
689C22—2441no—
690C22—2441no—
691C22—2441no—
692C22#1684007414yescomplemented
693C22#120×D6111yescharacteristic
694C22C2×#1451975911yescharacteristic
695C22#1684032211yescharacteristic
696C22C2×D8×D6111yescharacteristic
697C22AGL(1,ℤ/8)×D6111yescharacteristic
698C22C2×#1451980611yescharacteristic
699C22#1684055811yescharacteristic
700C4—22no—
701C4—81no—
702C4—81no—
703C4—1222no—
704C4—4881no—
705C4—4881no—
706C4—2444no—
707C4—2441no—
708C4#1684039912yes—
709C4#1684040212yes—
710C2—24no—
711C2—84no—
712C2—1224no—
713C2—2442no—
714C2#2144914411yescharacteristic
715C2#447×D6111yescharacteristic
716C2#2145507411yescharacteristic
717C2C2×#1684007411yescharacteristic
718C2#2145575711yescharacteristic
719C2#2145656811yescharacteristic
720C2#2145662211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1786C2
socleC23×C61C22×D4
prime radicalC22×#14519748C2
hypercenter#1786D61
hypocenterC61#24002
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. C2×#16840074C2semidirect
3. #380×D61C2semidirect
4. #21447000C2semidirect
5. #21446999C2semidirect
6. #21446133C2semidirect
7. #21446009C2semidirect
8. #21446006C2semidirect
9. #21445750C2semidirect
10. #21445728C2semidirect
11. C61×#1786C2semidirect
20. C2×C4⋊C4×D61C22semidirect
21. C2×C4⋊C4×D61C22semidirect
25. C2×#14519711C22semidirect
26. C2×#14519710C22semidirect
27. C2×#14519698C22semidirect
28. C2×#14519698C22semidirect
30. C2×#14519689C22semidirect
31. C2×#14519689C22semidirect
35. #16839619C22semidirect
36. #16839619C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
41. C2×#102×C61C22semidirect
42. C2×#102×C61C22semidirect
317. C22×C61#480semidirect
359. #1786D61semidirect
390. C2×#102C2×D61semidirect
391. C2×#102C2×D61semidirect
405. C61#24002semidirect
692. C22#16840074semidirect

Maximal subgroups

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

subgroupindex
C22×#145197482
C2×#168400742
#380×D612
#214470002
#214469992
#214461332
#214460092
#214460062
#214457502
#214457282
C61×#17862
#2400261