Finite Group Databasebeta

← #30550647

Subgroups — #30550647

Order 15616.

2362 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840272C212yesmaximal, complemented
2#21456655C214yesmaximal, complemented
3#21456648C214yesmaximal, complemented
4#21456643C211yesmaximal, characteristic, complemented
5#21456630C214yesmaximal, complemented
6#21456625C212yesmaximal, complemented
7#21455729C214yesmaximal, complemented
8#21455426C214yesmaximal, complemented
9#21455032C214yesmaximal, complemented
10#407×D61C211yesmaximal, characteristic, complemented
11C61×#3192C211yesmaximal, characteristic, complemented
12C22×#11758224C2212yescomplemented
13C23×C4×D61C2211yescharacteristic
14C2×#14519796C2214yescomplemented
15C2×M16×D61C2212yescomplemented
16#16840272—22no—
17#16840272—22no—
18#16840272—22no—
19#16840272—22no—
20#16840272C2214yes—
21#16840272C2214yes—
22#16840272C2214yescomplemented
23#16840272C2214yescomplemented
24C2×#14519759—22no—
25C2×#14519757C2212yescomplemented
26C2×#14519756C2212yes—
27C2×#14519748C2212yescomplemented
28C2×#14519745—22no—
29C2×D4×Dic61—22no—
30C2×#14519737—22no—
31C2×#14519735C2212yescomplemented
32#16840106C2218yescomplemented
33#16840097C2214yescomplemented
34#16840093C2214yescomplemented
35#16840085C2218yescomplemented
36#16840073C2214yescomplemented
37#16840021C2214yescomplemented
38#16840020C2214yescomplemented
39#16840012C2214yescomplemented
40#16840007C2218yescomplemented
41#16840006C2218yescomplemented
42#16839925C2218yescomplemented
43#16839920C2214yescomplemented
44#16839913C2214yescomplemented
45#16839904C2218yescomplemented
46#16839900C2214yescomplemented
47#16839797C2214yescomplemented
48#16839755C2218yescomplemented
49#16839726C2211yescharacteristic
50#16839725C2212yescomplemented
51#16839724C2214yescomplemented
52C22⋊C8×D61C2214yescomplemented
53C2×#121×C61C2212yes—
54C61×#481C2214yescomplemented
55C61×#469C2214yescomplemented
56C61×#447C2214yescomplemented
57C61×#407C2211yescharacteristic
58C24×D61D411yescharacteristic
59C2×#11758224—22no—
60C2×#11758224—22no—
61C2×#11758224—48no—
62C2×#11758224C2314yes—
63C2×#11758224C2314yes—
64C22×Dic122C2312yes—
65C22×C22⋊D61—22no—
66C22×C22⋊D61—22no—
67C23×Dic61—22no—
68C23×Dic61D411yescharacteristic
69C22×C4×D61—44no—
70C22×C4×D61D412yes—
71C22×C4×D61D412yes—
72C22×C4×D61D411yescharacteristic
73C22×C4×D61D411yescharacteristic
74C22×C4×D61C2312yes—
75C22×C4×D61C2311yescharacteristic
76C22×C4×D61C2312yes—
77C22×C4×D61C2311yescharacteristic
78#14519796—48no—
79C2×Dic244C2314yescomplemented
80C2×#11758200C2318yescomplemented
81C2×#11758194C2312yes—
82C2×#11758189C2314yes—
83M16×D61—44no—
84C2×C8×D61C2314yes—
85#14519759—24no—
86#14519759—24no—
87#14519759—24no—
88#14519759—24no—
89#14519757—22no—
90#14519757—22no—
91#14519757C2314yes—
92#14519757C2314yes—
93#14519756—22no—
94#14519756—22no—
95#14519756C2314yes—
96#14519756C2314yes—
97#14519748—22no—
98#14519748—22no—
99#14519748C2314yes—
100#14519748C2314yes—
101#14519745—24no—
102#14519745—24no—
103#14519745—24no—
104#14519745—24no—
105D4×Dic61—24no—
106D4×Dic61—24no—
107D4×Dic61—24no—
108D4×Dic61—24no—
109#14519737—24no—
110#14519737—24no—
111#14519737—24no—
112#14519737—24no—
113#14519735—22no—
114#14519735—22no—
115#14519735C2314yes—
116#14519735C2314yes—
117C2×C4⋊Dic61—22no—
118C2×C4⋊Dic61C2312yes—
119C2×#11758196—22no—
120C2×C22⋊Dic61—22no—
121C2×C22⋊Dic61—22no—
122C2×#11758198—22no—
123C2×#11758198—22no—
124C2×C4×Dic61—22no—
125#14519710C2318yescomplemented
126#14519709C2314yescomplemented
127#14519702C2318yescomplemented
128#14519700C2318yescomplemented
129#14519698C2318yescomplemented
130C22⋊#92418C2314yes—
131#14519689C2318yes—
132C22×D4×C61—22no—
133C22×D4×C61C2312yes—
134C23×C4×C61C2311yescharacteristic
135C2×D8×C61C2314yescomplemented
136C2×M16×C61C2312yes—
137#121×C61—22no—
138#121×C61—22no—
139#121×C61C2314yes—
140#121×C61C2314yes—
141C2×C4⋊C4×C61C2312yes—
142C2×C22⋊C4×C61—22no—
143#106×C61C2314yescomplemented
144#102×C61C2318yescomplemented
145C22⋊C8×C61C2314yes—
146C24×C61—22no—
147C24×C61C2×D411yescharacteristic
148C23×D61—42no—
149C23×D61—44no—
150C23×D61C2×D412yes—
151C23×D61C2×D412yes—
152C23×D61C2×D411yescharacteristic
153C23×D61C2×D411yescharacteristic
154C2×D4×C61—24no—
155C2×D4×C61—24no—
156C2×D4×C61—22no—
157C2×D4×C61—44no—
158C2×D4×C61—48no—
159C2×D4×C61C2414yes—
160C22×C4×C61—22no—
161C22×C4×C61—42no—
162C22×C4×C61C2×D412yes—
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2412yes—
165C22×C4×C61C2411yescharacteristic
166C2×C22⋊D61—24no—
167C2×C22⋊D61—24no—
168C2×C22⋊D61—24no—
169C2×C22⋊D61—24no—
170C2×C22⋊D61—48no—
171C2×C22⋊D61—48no—
172C22×Dic61—22no—
173C22×Dic61—22no—
174C22×Dic61—24no—
175C22×Dic61—24no—
176C22×Dic61—42no—
177C22×Dic61—48no—
178C22×Dic61C2×D412yes—
179C22×Dic61C2×D411yescharacteristic
180C22×Dic61C2×D412yes—
181C22×Dic61C2×D411yescharacteristic
182#11758224—216no—
183#11758224—48no—
184#11758224—48no—
185#11758224—48no—
186C2×C4×D61—28no—
187C2×C4×D61—24no—
188C2×C4×D61—44no—
189C2×C4×D61—42no—
190C2×C4×D61—44no—
191C2×C4×D61—42no—
192C2×C4×D61—44no—
193C2×C4×D61—42no—
194C2×C4×D61C2×D412yes—
195C2×C4×D61C2×D412yes—
196C2×C4×D61C2×D411yescharacteristic
197C2×C4×D61C2×D411yescharacteristic
198C2×C4×D61C2×D412yes—
199C2×C4×D61C2×D412yes—
200C2×C4×D61C2×D412yes—
201C2×C4×D61C2412yes—
202C2×C4×D61C2411yescharacteristic
203C2×C4×D61C2411yescharacteristic
204C2×Dic122—22no—
205C2×Dic122—44no—
206C2×Dic122C2414yes—
207D8×C61—44no—
208M16×C61—42no—
209C2×C8×C61C2414yes—
210C4⋊C4×C61—22no—
211C4⋊C4×C61C2414yes—
212C22⋊C4×C61—24no—
213C22⋊C4×C61—24no—
214C22⋊Dic61—24no—
215C22⋊Dic61—24no—
216C22⋊Dic61—24no—
217C22⋊Dic61—24no—
218#11758200—48no—
219#11758198—24no—
220#11758198—24no—
221#11758198—24no—
222#11758198—24no—
223C4⋊Dic61—22no—
224C4⋊Dic61—24no—
225C4⋊Dic61—24no—
226C4⋊Dic61C2414yes—
227#11758196—24no—
228#11758196—24no—
229C4×Dic61—24no—
230C4×Dic61—24no—
231#11758194—42no—
232C2×#92418C2414yes—
233Dic244—44no—
234#11758189—44no—
235C8×D61—28no—
236C23×C61—24no—
237C23×C61—24no—
238C23×C61—41no—
239C23×C61—44no—
240C23×C61—44no—
241C23×C61—44no—
242C23×C61C22×D411yescharacteristic
243C23×C61C22×D412yes—
244C22×D61—24no—
245C22×D61—24no—
246C22×D61—24no—
247C22×D61—22no—
248C22×D61—41no—
249C22×D61—42no—
250C22×D61—42no—
251C22×D61—42no—
252C22×D61—42no—
253C22×D61—42no—
254C22×D61—416no—
255C22×D61C22×D412yes—
256C22×D61C22×D411yescharacteristic
257C22×D61C22×D411yescharacteristic
258D4×C61—28no—
259D4×C61—44no—
260D4×C61—48no—
261D4×C61—48no—
262C2×C4×C61—22no—
263C2×C4×C61—24no—
264C2×C4×C61—24no—
265C2×C4×C61—41no—
266C2×C4×C61—42no—
267C2×C4×C61—44no—
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D412yes—
270C2×C4×C61C2511yescharacteristic
271C22⋊D61—48no—
272C22⋊D61—48no—
273C22⋊D61—48no—
274C22⋊D61—48no—
275C2×Dic61—24no—
276C2×Dic61—22no—
277C2×Dic61—24no—
278C2×Dic61—24no—
279C2×Dic61—41no—
280C2×Dic61—42no—
281C2×Dic61—48no—
282C2×Dic61—44no—
283C2×Dic61—44no—
284C2×Dic61—48no—
285C2×Dic61C22×D412yes—
286C2×Dic61C22×D411yescharacteristic
287C2×Dic61C22×D411yescharacteristic
288C4×D61—22no—
289C4×D61—24no—
290C4×D61—22no—
291C4×D61—24no—
292C4×D61—22no—
293C4×D61—24no—
294C4×D61—42no—
295C4×D61—42no—
296C4×D61—42no—
297C4×D61H(5,𝔽2)14yes—
298Dic122—28no—
299Dic122—44no—
300C8×C61—24no—
301#92418—24no—
302#60741—611nomaximal, Sylow
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—21no—
306C22×C61—22no—
307C22×C61—44no—
308C22×C61—44no—
309C22×C61—44no—
310C22×C61—44no—
311C22×C61—88no—
312C22×C61#57514yes—
313C22×C61C23×D411yescharacteristic
314C2×D61—22no—
315C2×D61—22no—
316C2×D61—21no—
317C2×D61—22no—
318C2×D61—22no—
319C2×D61—22no—
320C2×D61—48no—
321C2×D61—44no—
322C2×D61—48no—
323C2×D61—44no—
324C2×D61#53412yes—
325C4×C61—22no—
326C4×C61—21no—
327C4×C61—44no—
328C4×C61C2×H(5,𝔽2)12yes—
329Dic61—22no—
330Dic61—21no—
331Dic61—44no—
332Dic61—44no—
333Dic61#53412yes—
334C2×#525—612no—
335#3197—614no—
336#3196—614no—
337#3195—611no—
338#3194—614no—
339#3193—612no—
340#3192D6111yescharacteristic, complemented
341#3091—614no—
342#3054—614no—
343#3002—614no—
344C2×#407—611no—
345C2×C61—24no—
346C2×C61—41no—
347C2×C61—84no—
348C2×C61C2×#53411yescharacteristic
349C2×C61C2×#57512yes—
350D61—22no—
351D61—42no—
352D61—41no—
353C22×#41—612no—
354C24×C4—611no—
355C2×#135—614no—
356C22×M16—612no—
357#525—614no—
358#525—614no—
359#525—614no—
360#525—614no—
361#525—1222no—
362#525—1222no—
363#525—1222no—
364#525—1222no—
365C2×#123—1222no—
366C2×C22⋊Q8—612no—
367C2×C22⋊Q8—612no—
368C2×#121—612no—
369C2×#121C2×D6112yes—
370C2×#120—1222no—
371C2×C4×D4—1222no—
372C2×C4×D4—1222no—
373C2×#117—612no—
374#485—618no—
375#484—614no—
376#483—614no—
377#482—618no—
378#481C2×D6114yescomplemented
379#480—614no—
380#470—614no—
381#470—614no—
382#469—614no—
383#469C2×D6114yescomplemented
384#468—618no—
385#468—618no—
386#452—618no—
387#451—614no—
388#450—614no—
389#449—618no—
390#448—614no—
391#447C2×D6114yescomplemented
392#427—614no—
393#416—618no—
394#407—611no—
395#407—612no—
396#407—614no—
397#407C2×D6111yescharacteristic
398C2×C22⋊C8—614no—
399C61#6074111yescharacteristic, complemented, Sylow
400C25—611no—
401C2×#41—614no—
402C2×#41—614no—
403C2×#41—1222no—
404C2×#41—1222no—
405C2×#41—2448no—
406C22×Q8—612no—
407C22×D4—22no—
408C22×D4—1222no—
409C22×D4—1222no—
410C22×D4C22×D6112yes—
411C23×C4—611no—
412C23×C4—612no—
413C23×C4—611no—
414C23×C4—612no—
415C23×C4—612no—
416C23×C4—611no—
417C23×C4—611no—
418C23×C4—612no—
419C23×C4—611no—
420C23×C4—1222no—
421C23×C4—2444no—
422C23×C4C22×D6111yescharacteristic
423#135—2448no—
424C2×Q16—614no—
425C2×SD16—618no—
426C2×D8C22×D6114yescomplemented
427C2×M16—612no—
428C2×M16—614no—
429C2×M16—2444no—
430C2×M16C22×D6112yes—
431C22×C8—614no—
432#123—1224no—
433#123—1224no—
434#123—1224no—
435#123—1224no—
436C22⋊Q8—614no—
437C22⋊Q8—614no—
438C22⋊Q8—614no—
439C22⋊Q8—614no—
440C22⋊Q8—1222no—
441C22⋊Q8—1222no—
442C22⋊Q8—1222no—
443C22⋊Q8—1222no—
444#121—22no—
445#121—22no—
446#121—614no—
447#121—614no—
448#121—1222no—
449#121—1222no—
450#121C22×D6114yes—
451#121C22×D6114yes—
452#120—1224no—
453#120—1224no—
454#120—1224no—
455#120—1224no—
456C4×D4—1224no—
457C4×D4—1224no—
458C4×D4—1224no—
459C4×D4—1224no—
460C4×D4—1224no—
461C4×D4—1224no—
462C4×D4—1224no—
463C4×D4—1224no—
464#117—614no—
465#117—614no—
466#117—1222no—
467#117—1222no—
468C2×C4⋊C4—612no—
469C2×C4⋊C4—1222no—
470C2×C4⋊C4—1222no—
471C2×C4⋊C4C22×D6112yes—
472C2×C22⋊C4—22no—
473C2×C22⋊C4—1222no—
474C2×C22⋊C4—1222no—
475C2×C22⋊C4—1222no—
476C2×C22⋊C4—1222no—
477C2×C42—1222no—
478#107—618no—
479#106—614no—
480#106C22×D6114yescomplemented
481#103—618no—
482#103—618no—
483#102—618no—
484#102C22×D6118yescomplemented
485C22⋊C8—614no—
486C22⋊C8—618no—
487C22⋊C8C22×D6114yes—
488C24—22no—
489C24—612no—
490C24—612no—
491C24—611no—
492C24—611no—
493C24—2444no—
494C24—2442no—
495C24D4×D6111yescharacteristic
496#41—12216no—
497#41—2448no—
498#41—2448no—
499#41—2448no—
500C2×Q8—614no—
501C2×Q8—1222no—
502C2×Q8—2444no—
503C2×D4—24no—
504C2×D4—24no—
505C2×D4—22no—
506C2×D4—44no—
507C2×D4—48no—
508C2×D4—1224no—
509C2×D4—1224no—
510C2×D4—1224no—
511C2×D4—1224no—
512C2×D4—2448no—
513C2×D4—2448no—
514C2×D4C23×D6114yes—
515C22×C4—22no—
516C22×C4—42no—
517C22×C4—611no—
518C22×C4—612no—
519C22×C4—611no—
520C22×C4—612no—
521C22×C4—611no—
522C22×C4—612no—
523C22×C4—612no—
524C22×C4—611no—
525C22×C4—611no—
526C22×C4—612no—
527C22×C4—611no—
528C22×C4—612no—
529C22×C4—612no—
530C22×C4—612no—
531C22×C4—1224no—
532C22×C4—1228no—
533C22×C4—1224no—
534C22×C4—1224no—
535C22×C4—1222no—
536C22×C4—1222no—
537C22×C4—2442no—
538C22×C4—2444no—
539C22×C4—2444no—
540C22×C4—2442no—
541C22×C4—2444no—
542C22×C4—2442no—
543C22×C4—2442no—
544C22×C4—2448no—
545C22×C4D4×D6112yes—
546C22×C4D4×D6111yescharacteristic
547C22×C4C23×D6112yes—
548C22×C4C23×D6111yescharacteristic
549Q16—2444no—
550SD16—2448no—
551D8—44no—
552M16—42no—
553M16—2444no—
554M16—2442no—
555C2×C8—614no—
556C2×C8—1228no—
557C2×C8C23×D6114yes—
558C4⋊C4—22no—
559C4⋊C4—614no—
560C4⋊C4—1224no—
561C4⋊C4—1224no—
562C4⋊C4—1222no—
563C4⋊C4—1224no—
564C4⋊C4—1224no—
565C4⋊C4C23×D6114yes—
566C22⋊C4—24no—
567C22⋊C4—24no—
568C22⋊C4—1224no—
569C22⋊C4—1224no—
570C22⋊C4—1224no—
571C22⋊C4—1224no—
572C22⋊C4—1224no—
573C22⋊C4—1224no—
574C22⋊C4—1224no—
575C22⋊C4—1224no—
576C42—1224no—
577C42—1224no—
578C23—24no—
579C23—24no—
580C23—41no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—611no—
585C23—611no—
586C23—612no—
587C23—1224no—
588C23—1222no—
589C23—1224no—
590C23—1224no—
591C23—2442no—
592C23—2441no—
593C23—2442no—
594C23—2442no—
595C23—2442no—
596C23—2442no—
597C23—24416no—
598C23C2×D4×D6111yescharacteristic
599C23C2×D4×D6112yes—
600Q8—1228no—
601Q8—2444no—
602D4—28no—
603D4—44no—
604D4—48no—
605D4—48no—
606D4—2448no—
607D4—2448no—
608D4—2448no—
609D4—2448no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—611no—
617C2×C4—614no—
618C2×C4—612no—
619C2×C4—611no—
620C2×C4—1222no—
621C2×C4—1224no—
622C2×C4—1222no—
623C2×C4—1224no—
624C2×C4—1222no—
625C2×C4—1224no—
626C2×C4—1222no—
627C2×C4—1224no—
628C2×C4—1224no—
629C2×C4—1224no—
630C2×C4—2442no—
631C2×C4—2442no—
632C2×C4—2442no—
633C2×C4—2448no—
634C2×C4—2442no—
635C2×C4—2441no—
636C2×C4—2448no—
637C2×C4—2444no—
638C2×C4—2444no—
639C2×C4C2×D4×D6111yescharacteristic
640C2×C4C2×D4×D6112yes—
641C2×C4C24×D6111yescharacteristic
642C8—24no—
643C8—1224no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—21no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—88no—
653C22—612no—
654C22—1222no—
655C22—1222no—
656C22—1222no—
657C22—1222no—
658C22—1222no—
659C22—1221no—
660C22—2448no—
661C22—2448no—
662C22—2444no—
663C22—2444no—
664C22#1684055814yes—
665C22C22×D4×D6111yescharacteristic
666C4—21no—
667C4—22no—
668C4—44no—
669C4—612no—
670C4—1221no—
671C4—1222no—
672C4—2444no—
673C4—2444no—
674C4H(5,𝔽2)×D6112yes—
675C2—24no—
676C2—41no—
677C2—84no—
678C2—1222no—
679C2—2441no—
680C2—2442no—
681C2#534×D6111yescharacteristic
682C2C2×#1684055812yes—
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3192C2
socleC22×C61C23×D4
hypercenter#3192D61
hypocenterC61#60741
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3192

Complements

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

NH ≅ G/Ntype
1. C2×#16840272C2semidirect
2. #21456655C2semidirect
3. #21456648C2semidirect
4. #21456643C2semidirect
5. #21456630C2semidirect
6. #21456625C2semidirect
7. #21455729C2semidirect
8. #21455426C2semidirect
9. #21455032C2semidirect
10. #407×D61C2semidirect
11. C61×#3192C2semidirect
12. C22×#11758224C22semidirect
14. C2×#14519796C22semidirect
15. C2×M16×D61C22semidirect
22. #16840272C22semidirect
23. #16840272C22semidirect
25. C2×#14519757C22semidirect
27. C2×#14519748C22semidirect
31. C2×#14519735C22semidirect
32. #16840106C22semidirect
33. #16840097C22semidirect
34. #16840093C22semidirect
35. #16840085C22semidirect
36. #16840073C22semidirect
37. #16840021C22semidirect
38. #16840020C22semidirect
39. #16840012C22semidirect
40. #16840007C22semidirect
41. #16840006C22semidirect
42. #16839925C22semidirect
43. #16839920C22semidirect
44. #16839913C22semidirect
45. #16839904C22semidirect
46. #16839900C22semidirect
47. #16839797C22semidirect
48. #16839755C22semidirect
50. #16839725C22semidirect
51. #16839724C22semidirect
52. C22⋊C8×D61C22semidirect
54. C61×#481C22semidirect
55. C61×#469C22semidirect
56. C61×#447C22semidirect
79. C2×Dic244C23semidirect
80. C2×#11758200C23semidirect
125. #14519710C23semidirect
126. #14519709C23semidirect
127. #14519702C23semidirect
128. #14519700C23semidirect
129. #14519698C23semidirect
135. C2×D8×C61C23semidirect
143. #106×C61C23semidirect
144. #102×C61C23semidirect
340. #3192D61semidirect
378. #481C2×D61semidirect
383. #469C2×D61semidirect
391. #447C2×D61semidirect
399. C61#60741semidirect
426. C2×D8C22×D61semidirect
480. #106C22×D61semidirect
484. #102C22×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402722
#214566552
#214566482
#214566432
#214566302
#214566252
#214557292
#214554262
#214550322
#407×D612
C61×#31922
#6074161