Finite Group Databasebeta

← #29712198

Subgroups — #29712198

Order 15616.

1500 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840464C211yesmaximal, characteristic, complemented
2C22×#14519698C211yesmaximal, characteristic, complemented
3#21449169C211yesmaximal, characteristic, complemented
4#21446963C212yesmaximal, complemented
5#21446814C212yesmaximal, complemented
6#21446248C211yesmaximal, characteristic, complemented
7#21446169C212yesmaximal, complemented
8#21446138C212yesmaximal, complemented
9#21446109C211yesmaximal, characteristic, complemented
10#21445910C211yesmaximal, characteristic, complemented
11C2×C61×#447C211yesmaximal, characteristic, complemented
12C23×#92418C2211yescharacteristic
13#16840464—24no—
14#16840464—24no—
15C2×#14519770—21no—
16C2×#14519768C2211yescharacteristic, complemented
17C2×#14519761—21no—
18C2×#14519749—21no—
19C2×#14519748C2212yes—
20C2×#14519748C2211yescharacteristic, complemented
21C2×#14519746—21no—
22C2×D4×Dic61—21no—
23C2×D4×Dic61C2211yescharacteristic, complemented
24C2×C4×C22⋊D61C2211yescharacteristic
25C2×#14519735C2211yescharacteristic, complemented
26C22×C4⋊Dic61C2211yescharacteristic
27C2×#14519698—24no—
28C2×#14519698C2212yes—
29C2×#14519698C2212yes—
30C2×#14519698C2212yescomplemented
31C2×#14519689C2212yescomplemented
32#16839676—21no—
33#16839670—21no—
34#16839665C2212yes—
35#16839622C2211yescharacteristic
36#16839618—21no—
37#16839525C2212yescomplemented
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839516C2212yescomplemented
41C23×D4×C61C2211yescharacteristic
42C22×D8×C61C2212yes—
43C2×#121×C61C2211yescharacteristic
44C61×#447—24no—
45C2×#102×C61C2212yes—
46C2×C22⋊C8×C61C2211yescharacteristic
47C22×C22⋊D61—21no—
48C22×C22⋊D61—22no—
49C22×C22⋊D61D411yescharacteristic
50C23×Dic61—21no—
51C23×Dic61—21no—
52C22×C4×D61C2311yescharacteristic
53C22×#92418—22no—
54C22×#92418—42no—
55C22×#92418C2312yes—
56#14519770—28no—
57#14519768—22no—
58#14519768—22no—
59#14519761—28no—
60#14519749—28no—
61#14519748—22no—
62#14519748—24no—
63#14519748—22no—
64#14519748—42no—
65#14519746—44no—
66D4×Dic61—28no—
67D4×Dic61—22no—
68D4×Dic61—22no—
69C4×C22⋊D61—22no—
70C4×C22⋊D61—22no—
71#14519735—22no—
72#14519735—22no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61D414yescomplemented
77C2×C4⋊Dic61C2312yes—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196D411yescharacteristic
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—22no—
86C2×C22⋊Dic61—22no—
87C2×C22⋊Dic61—22no—
88C2×C22⋊Dic61D411yescharacteristic
89C2×#11758198—21no—
90C2×#11758198D411yescharacteristic
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93#14519698—24no—
94#14519698—24no—
95#14519698—42no—
96#14519689—42no—
97#14519684—21no—
98C25×C61—21no—
99C22×D4×C61—24no—
100C22×D4×C61—21no—
101C22×D4×C61—22no—
102C22×D4×C61—22no—
103C22×D4×C61—21no—
104C22×D4×C61—42no—
105C22×D4×C61C2311yescharacteristic
106C22×D4×C61C2312yes—
107C23×C4×C61C2311yescharacteristic
108C2×D8×C61—24no—
109C2×D8×C61—24no—
110C2×D8×C61—42no—
111C22×C8×C61C2312yes—
112#121×C61—22no—
113#121×C61D414yescomplemented
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C22⋊C4×C61—21no—
116#102×C61—24no—
117C22⋊C8×C61—22no—
118C24×C61—21no—
119C24×C61—22no—
120C24×C61—44no—
121C24×C61—42no—
122C24×C61—42no—
123C24×C61—42no—
124C24×C61—82no—
125C24×C61C2×D411yescharacteristic
126C23×D61C2×D411yescharacteristic
127C2×D4×C61—28no—
128C2×D4×C61—24no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—21no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61—48no—
138C2×D4×C61—44no—
139C2×D4×C61—44no—
140C2×D4×C61—42no—
141C2×D4×C61—42no—
142C2×D4×C61—42no—
143C2×D4×C61C2×D412yes—
144C2×D4×C61#4114yes—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—24no—
156C2×C22⋊D61—44no—
157C2×C22⋊D61—44no—
158C2×C22⋊D61—82no—
159C2×C22⋊D61—82no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—24no—
163C22×Dic61—24no—
164C22×Dic61—22no—
165C22×Dic61—82no—
166C22×Dic61—82no—
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174D8×C61—48no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C2×D412yes—
180C22⋊C4×C61—24no—
181C22⋊Dic61—24no—
182C22⋊Dic61—24no—
183C22⋊Dic61—22no—
184C22⋊Dic61—44no—
185C22⋊Dic61—44no—
186C22⋊Dic61—42no—
187C22⋊Dic61—82no—
188#11758198—24no—
189#11758198—22no—
190C4⋊Dic61—24no—
191C4⋊Dic61—24no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61C2×D412yes—
197C4⋊Dic61C2×D412yes—
198#11758196—24no—
199#11758196—22no—
200C4×Dic61—24no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—22no—
206C2×#92418—44no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—24no—
211C23×C61—48no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61—42no—
215C23×C61—41no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61—41no—
219C23×C61—42no—
220C23×C61—42no—
221C23×C61—42no—
222C23×C61—88no—
223C23×C61—88no—
224C23×C61—84no—
225C23×C61—84no—
226C23×C61#12012yes—
227C23×C61#13512yes—
228C23×C61AGL(1,ℤ/8)12yes—
229C23×C61C22×D411yescharacteristic
230C22×D61—21no—
231C22×D61—22no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—22no—
235D4×C61—24no—
236D4×C61—24no—
237D4×C61—24no—
238D4×C61—48no—
239D4×C61—48no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—42no—
243D4×C61—42no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—24no—
248C2×C4×C61—22no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61#12012yes—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4112yes—
255C2×C4×C61H(5,𝔽2)12yes—
256C22⋊D61—44no—
257C22⋊D61—84no—
258C22⋊D61—88no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—84no—
274C2×Dic61—84no—
275C4×D61—41no—
276C4×D61—41no—
277C4×D61—42no—
278C8×C61—42no—
279#92418—42no—
280#23963—611nomaximal, Sylow
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—24no—
284C22×C61—24no—
285C22×C61—44no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—44no—
290C22×C61—44no—
291C22×C61—816no—
292C22×C61—82no—
293C22×C61—82no—
294C22×C61—82no—
295C22×C61—82no—
296C22×C61—82no—
297C22×C61—82no—
298C22×C61#44914yes—
299C22×C61C2×#12011yescharacteristic
300C22×C61#54612yes—
301C22×C61C2×#13511yescharacteristic
302C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C61#57612yes—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—42no—
308C2×D61—42no—
309C4×C61—22no—
310C4×C61—42no—
311C4×C61#52511yescharacteristic
312C4×C61#53511yescharacteristic
313C4×C61#54612yes—
314Dic61—41no—
315Dic61—41no—
316Dic61—41no—
317Dic61—42no—
318Dic61—42no—
319Dic61—82no—
320C2×#553—611no—
321C2×#447D6111yescharacteristic, complemented
322C22×#102—611no—
323#2408—611no—
324#2025—612no—
325#1999—612no—
326#1893—611no—
327#1879—612no—
328#1875—612no—
329#1869—611no—
330#1824—611no—
331C2×C61—24no—
332C2×C61—82no—
333C2×C61—84no—
334C2×C61#240411yescharacteristic
335C2×C61C2×#44911yescharacteristic
336C2×C61#300811yescharacteristic
337C2×C61#329712yes—
338C2×C61#331812yes—
339D61—81no—
340C23×D4C2×D6111yescharacteristic
341C22×D8C2×D6112yes—
342C23×C8—611no—
343#553—1224no—
344#553—1224no—
345C2×#126—1221no—
346C2×#125—611no—
347C2×#123—1221no—
348C2×#121—611no—
349C2×#121—612no—
350C2×#121—1221no—
351C2×#121C2×D6111yescharacteristic
352C2×#120—1221no—
353C2×C4×D4—611no—
354C2×C4×D4—611no—
355C2×C4×D4—1221no—
356C2×#117—611no—
357C22×C4⋊C4—611no—
358#447—24no—
359C2×#102—612no—
360C2×#102—612no—
361C2×#102—612no—
362C2×#102—1224no—
363C2×#102C2×D6112yes—
364C2×C22⋊C8—612no—
365C2×C22⋊C8C2×D6111yescharacteristic
366#394—1221no—
367#392—1221no—
368#390—612no—
369#380—611no—
370#379—1221no—
371#340—612no—
372#340—612no—
373#340—612no—
374#336—612no—
375C61#2396311yescharacteristic, complemented, Sylow
376C25—21no—
377C22×D4—24no—
378C22×D4—21no—
379C22×D4—22no—
380C22×D4—21no—
381C22×D4—22no—
382C22×D4—42no—
383C22×D4—611no—
384C22×D4—1221no—
385C22×D4—1222no—
386C22×D4C22×D6112yes—
387C22×D4C22×D6111yescharacteristic
388C23×C4—611no—
389C23×C4—1221no—
390C23×C4—1221no—
391C23×C4C22×D6111yescharacteristic
392C2×D8—24no—
393C2×D8—24no—
394C2×D8—42no—
395C22×C8—612no—
396C22×C8—1222no—
397C22×C8—2442no—
398C22×C8C22×D6112yes—
399#126—1228no—
400#125—1222no—
401#125—1222no—
402#123—1228no—
403#121—22no—
404#121—1222no—
405#121—1228no—
406#121—1222no—
407#121—1224no—
408#121—2442no—
409#121C22⋊D6114yescomplemented
410#120—2444no—
411C4×D4—1222no—
412C4×D4—1228no—
413C4×D4—1222no—
414C4×D4—1222no—
415C4×D4—1222no—
416#117—1222no—
417#117—1222no—
418C2×C4⋊C4—614no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—612no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1222no—
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—611no—
430C2×C22⋊C4—611no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1222no—
435C2×C22⋊C4—1222no—
436C2×C22⋊C4—1222no—
437C2×C22⋊C4—1221no—
438C2×C42—611no—
439C2×C42—1221no—
440#102—24no—
441#102—1224no—
442#102—1224no—
443#102—2442no—
444C22⋊C8—22no—
445C22⋊C8—2442no—
446#95—1221no—
447C24—21no—
448C24—22no—
449C24—44no—
450C24—42no—
451C24—42no—
452C24—42no—
453C24—82no—
454C24—611no—
455C24D4×D6111yescharacteristic
456C2×D4—28no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—48no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—1224no—
473C2×D4—1222no—
474C2×D4—1222no—
475C2×D4—1222no—
476C2×D4—4882no—
477C2×D4—4882no—
478C2×D4—2444no—
479C2×D4—2444no—
480C2×D4#1175822414yes—
481C2×D4C2×C22⋊D6112yes—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—1224no—
489C22×C4—1224no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—4882no—
498C22×C4—4882no—
499C22×C4D4×D6112yes—
500C22×C4D4×D6111yescharacteristic
501C22×C4C2×C22⋊D6112yes—
502C22×C4C23×D6111yescharacteristic
503D8—48no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—2444no—
511C4⋊C4—21no—
512C4⋊C4—612no—
513C4⋊C4—612no—
514C4⋊C4—1224no—
515C4⋊C4—1224no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1224no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4C2×C22⋊D6112yes—
523C22⋊C4—24no—
524C22⋊C4—1224no—
525C22⋊C4—1224no—
526C22⋊C4—1224no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—4882no—
530C22⋊C4—2444no—
531C22⋊C4—2444no—
532C22⋊C4—2442no—
533C42—1224no—
534C42—1221no—
535C42—1221no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—48no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—88no—
552C23—88no—
553C23—84no—
554C23—84no—
555C23—1221no—
556C23—1221no—
557C23—1221no—
558C23—1222no—
559C23—1222no—
560C23#1451974512yes—
561C23#1451979612yes—
562C23#1451980112yes—
563C23C2×D4×D6111yescharacteristic
564D4—24no—
565D4—24no—
566D4—24no—
567D4—48no—
568D4—48no—
569D4—41no—
570D4—41no—
571D4—42no—
572D4—42no—
573D4—4888no—
574D4—4884no—
575D4—2444no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—1221no—
584C2×C4—1222no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1222no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—4884no—
593C2×C4—4884no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2442no—
602C2×C4#1451974512yes—
603C2×C4C2×D4×D6111yescharacteristic
604C2×C4C22×C22⋊D6111yescharacteristic
605C2×C4C2×#1175822412yes—
606C2×C4#1451982212yes—
607C8—42no—
608C8—2442no—
609C22—24no—
610C22—24no—
611C22—22no—
612C22—22no—
613C22—44no—
614C22—44no—
615C22—42no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—816no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—2442no—
627C22—2442no—
628C22—2441no—
629C22—2441no—
630C22—2441no—
631C22#1683990414yes—
632C22C2×#1451974511yescharacteristic
633C22#1684039912yes—
634C22C2×#1451979611yescharacteristic
635C22C2×#1451980111yescharacteristic
636C22#1684056612yes—
637C4—22no—
638C4—42no—
639C4—4882no—
640C4—2441no—
641C4—2441no—
642C4—2441no—
643C4—2442no—
644C4—2442no—
645C4#1684026811yescharacteristic
646C4#1684032711yescharacteristic
647C4#1684040612yes—
648C2—24no—
649C2—84no—
650C2—82no—
651C2—4881no—
652C2#2144913511yescharacteristic
653C2C2×#1683990411yescharacteristic
654C2#2145508611yescharacteristic
655C2#2145767312yes—
656C2#2145796412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#447C2
socleC23×C61C22×D4
hypercenterC2×#447D61
hypocenterC61#23963
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. C2×#16840464C2semidirect
2. C22×#14519698C2semidirect
3. #21449169C2semidirect
4. #21446963C2semidirect
5. #21446814C2semidirect
6. #21446248C2semidirect
7. #21446169C2semidirect
8. #21446138C2semidirect
9. #21446109C2semidirect
10. #21445910C2semidirect
11. C2×C61×#447C2semidirect
16. C2×#14519768C22semidirect
20. C2×#14519748C22semidirect
23. C2×D4×Dic61C22semidirect
25. C2×#14519735C22semidirect
30. C2×#14519698C22semidirect
31. C2×#14519689C22semidirect
37. #16839525C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839516C22semidirect
76. C2×C4⋊Dic61D4semidirect
113. #121×C61D4semidirect
321. C2×#447D61semidirect
375. C61#23963semidirect
409. #121C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404642
C22×#145196982
#214491692
#214469632
#214468142
#214462482
#214461692
#214461382
#214461092
#214459102
C2×C61×#4472
#2396361