Finite Group Databasebeta

← #30550657

Subgroups — #30550657

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×#16840275C212yesmaximal, complemented
2#21456662C214yesmaximal, complemented
3#21456652C214yesmaximal, complemented
4#21456638C214yesmaximal, complemented
5#21456621C212yesmaximal, complemented
6#21456619C211yesmaximal, characteristic, complemented
7#21455733C214yesmaximal, complemented
8#21455430C214yesmaximal, complemented
9#21455042C214yesmaximal, complemented
10#407×D61C211yesmaximal, characteristic, complemented
11C61×#3195C211yesmaximal, characteristic, complemented
12C22×#11758226C2212yescomplemented
13C23×C4×D61C2211yescharacteristic
14C2×#14519800C2214yescomplemented
15C2×M16×D61C2212yescomplemented
16#16840275—22no—
17#16840275—22no—
18#16840275—22no—
19#16840275—22no—
20#16840275C2214yes—
21#16840275C2214yes—
22#16840275C2214yescomplemented
23#16840275C2214yescomplemented
24C2×#14519762—22no—
25C2×#14519753C2212yescomplemented
26C2×#14519752C2212yescomplemented
27C2×#14519751C2212yes—
28C2×#14519747—22no—
29C2×#14519739—22no—
30C2×#14519737—22no—
31C2×#14519735C2212yescomplemented
32#16840101C2218yescomplemented
33#16840095C2214yescomplemented
34#16840088C2218yescomplemented
35#16840081C2214yescomplemented
36#16840075C2214yescomplemented
37#16840017C2214yescomplemented
38#16840016C2214yescomplemented
39#16840015C2214yescomplemented
40#16840011C2218yescomplemented
41#16840010C2218yescomplemented
42#16839923C2218yescomplemented
43#16839912C2214yescomplemented
44#16839910C2218yescomplemented
45#16839901C2214yescomplemented
46#16839898C2214yescomplemented
47#16839797C2214yescomplemented
48#16839757C2218yescomplemented
49#16839726C2211yescharacteristic
50#16839725C2212yescomplemented
51#16839724C2214yescomplemented
52C22⋊C8×D61C2214yescomplemented
53C2×C22⋊Q8×C61C2212yes—
54C61×#483C2214yescomplemented
55C61×#470C2214yescomplemented
56C61×#451C2214yescomplemented
57C61×#407C2211yescharacteristic
58C24×D61D411yescharacteristic
59C2×#11758226—22no—
60C2×#11758226—22no—
61C2×#11758226—48no—
62C2×#11758226C2314yes—
63C2×#11758226C2314yes—
64C22×D244—22no—
65C22×D244—22no—
66C22×D244C2312yes—
67C22×C22⋊D61—22no—
68C23×Dic61D411yescharacteristic
69C22×C4×D61—22no—
70C22×C4×D61—44no—
71C22×C4×D61D412yes—
72C22×C4×D61D412yes—
73C22×C4×D61D411yescharacteristic
74C22×C4×D61D411yescharacteristic
75C22×C4×D61C2312yes—
76C22×C4×D61C2311yescharacteristic
77C22×C4×D61C2312yes—
78C22×C4×D61C2311yescharacteristic
79#14519800—48no—
80C2×#11758201C2318yescomplemented
81C2×D488C2314yescomplemented
82C2×#11758194C2312yes—
83C2×#11758189C2314yes—
84M16×D61—44no—
85C2×C8×D61C2314yes—
86#14519762—24no—
87#14519762—24no—
88#14519762—24no—
89#14519762—24no—
90#14519753—22no—
91#14519753—22no—
92#14519753C2314yes—
93#14519753C2314yes—
94#14519752—22no—
95#14519752—22no—
96#14519752C2314yes—
97#14519752C2314yes—
98#14519751—22no—
99#14519751—22no—
100#14519751C2314yes—
101#14519751C2314yes—
102#14519747—24no—
103#14519747—24no—
104#14519747—24no—
105#14519747—24no—
106#14519739—24no—
107#14519739—24no—
108#14519739—24no—
109#14519739—24no—
110#14519737—24no—
111#14519737—24no—
112#14519737—24no—
113#14519737—24no—
114#14519735—22no—
115#14519735—22no—
116#14519735C2314yes—
117#14519735C2314yes—
118C2×C4⋊Dic61C2312yes—
119C2×#11758196—22no—
120C2×#11758198—22no—
121C2×#11758198—22no—
122C2×#11758198—22no—
123C2×#11758198—22no—
124C2×C4×Dic61—22no—
125#14519710C2318yescomplemented
126#14519709C2314yescomplemented
127#14519701C2318yescomplemented
128#14519699C2318yescomplemented
129#14519697C2318yescomplemented
130C22⋊#92418C2314yes—
131#14519689C2318yes—
132C22×Q8×C61C2312yes—
133C23×C4×C61C2311yescharacteristic
134C2×Q16×C61C2314yescomplemented
135C2×M16×C61C2312yes—
136C22⋊Q8×C61—22no—
137C22⋊Q8×C61—22no—
138C22⋊Q8×C61C2314yes—
139C22⋊Q8×C61C2314yes—
140C2×C4⋊C4×C61—22no—
141C2×C4⋊C4×C61C2312yes—
142C2×C22⋊C4×C61—22no—
143#106×C61C2314yescomplemented
144#103×C61C2318yescomplemented
145C22⋊C8×C61C2314yes—
146C24×C61C2×D411yescharacteristic
147C23×D61—22no—
148C23×D61—42no—
149C23×D61—44no—
150C23×D61C2×D412yes—
151C23×D61C2×D412yes—
152C23×D61C2×D411yescharacteristic
153C23×D61C2×D411yescharacteristic
154C2×Q8×C61—22no—
155C2×Q8×C61—44no—
156C2×Q8×C61C2414yes—
157C22×C4×C61—22no—
158C22×C4×C61—22no—
159C22×C4×C61—42no—
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2412yes—
163C22×C4×C61C2411yescharacteristic
164C2×C22⋊D61—24no—
165C2×C22⋊D61—24no—
166C2×C22⋊D61—48no—
167C22×Dic61—22no—
168C22×Dic61—42no—
169C22×Dic61C2×D412yes—
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D412yes—
172C22×Dic61C2×D411yescharacteristic
173#11758226—216no—
174#11758226—48no—
175#11758226—48no—
176#11758226—48no—
177C2×D244—24no—
178C2×D244—24no—
179C2×D244—24no—
180C2×D244—24no—
181C2×D244—22no—
182C2×D244—44no—
183C2×D244—48no—
184C2×D244—48no—
185C2×D244C2414yes—
186C2×C4×D61—28no—
187C2×C4×D61—24no—
188C2×C4×D61—24no—
189C2×C4×D61—24no—
190C2×C4×D61—44no—
191C2×C4×D61—42no—
192C2×C4×D61—44no—
193C2×C4×D61—42no—
194C2×C4×D61—44no—
195C2×C4×D61—42no—
196C2×C4×D61—48no—
197C2×C4×D61C2×D412yes—
198C2×C4×D61C2×D412yes—
199C2×C4×D61C2×D411yescharacteristic
200C2×C4×D61C2×D411yescharacteristic
201C2×C4×D61C2×D412yes—
202C2×C4×D61C2×D412yes—
203C2×C4×D61C2×D412yes—
204C2×C4×D61C2412yes—
205C2×C4×D61C2411yescharacteristic
206C2×C4×D61C2411yescharacteristic
207Q16×C61—44no—
208M16×C61—42no—
209C2×C8×C61C2414yes—
210C4⋊C4×C61—24no—
211C4⋊C4×C61—24no—
212C4⋊C4×C61—22no—
213C4⋊C4×C61C2414yes—
214C22⋊C4×C61—24no—
215C22⋊C4×C61—24no—
216#11758201—48no—
217#11758198—24no—
218#11758198—24no—
219#11758198—24no—
220#11758198—24no—
221#11758198—24no—
222#11758198—24no—
223#11758198—24no—
224#11758198—24no—
225C4⋊Dic61—22no—
226C4⋊Dic61C2414yes—
227#11758196—24no—
228#11758196—24no—
229C4×Dic61—24no—
230C4×Dic61—24no—
231#11758194—42no—
232C2×#92418C2414yes—
233D488—44no—
234#11758189—44no—
235C8×D61—28no—
236C23×C61—24no—
237C23×C61—41no—
238C23×C61C22×D411yescharacteristic
239C23×C61C22×D412yes—
240C22×D61—24no—
241C22×D61—24no—
242C22×D61—24no—
243C22×D61—22no—
244C22×D61—24no—
245C22×D61—41no—
246C22×D61—42no—
247C22×D61—42no—
248C22×D61—42no—
249C22×D61—42no—
250C22×D61—42no—
251C22×D61—416no—
252C22×D61—44no—
253C22×D61—44no—
254C22×D61—44no—
255C22×D61C22×D412yes—
256C22×D61C22×D411yescharacteristic
257C22×D61C22×D411yescharacteristic
258Q8×C61—28no—
259Q8×C61—44no—
260C2×C4×C61—22no—
261C2×C4×C61—24no—
262C2×C4×C61—24no—
263C2×C4×C61—24no—
264C2×C4×C61—41no—
265C2×C4×C61—42no—
266C2×C4×C61—44no—
267C2×C4×C61—44no—
268C2×C4×C61C22×D412yes—
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C2511yescharacteristic
271C22⋊D61—48no—
272C22⋊D61—48no—
273C2×Dic61—24no—
274C2×Dic61—22no—
275C2×Dic61—24no—
276C2×Dic61—41no—
277C2×Dic61—42no—
278C2×Dic61—44no—
279C2×Dic61C22×D412yes—
280C2×Dic61C22×D411yescharacteristic
281C2×Dic61C22×D411yescharacteristic
282D244—28no—
283D244—44no—
284D244—48no—
285D244—48no—
286D244—48no—
287D244—48no—
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×D61—48no—
298C4×D61—48no—
299C4×D61H(5,𝔽2)14yes—
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#57514yes—
309C22×C61C23×D411yescharacteristic
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—21no—
313C2×D61—22no—
314C2×D61—22no—
315C2×D61—22no—
316C2×D61—48no—
317C2×D61—44no—
318C2×D61—48no—
319C2×D61—44no—
320C2×D61—44no—
321C2×D61—44no—
322C2×D61—44no—
323C2×D61—88no—
324C2×D61#53412yes—
325C4×C61—22no—
326C4×C61—21no—
327C4×C61—44no—
328C4×C61—44no—
329C4×C61C2×H(5,𝔽2)12yes—
330Dic61—22no—
331Dic61—21no—
332Dic61—44no—
333Dic61#53412yes—
334C2×#525—612no—
335#3197—614no—
336#3196—614no—
337#3195D6111yescharacteristic, complemented
338#3194—614no—
339#3193—612no—
340#3192—611no—
341#3091—614no—
342#3054—614no—
343#3002—614no—
344C2×#407—611no—
345C2×C61—24no—
346C2×C61—41no—
347C2×C61C2×#53411yescharacteristic
348C2×C61C2×#57512yes—
349D61—22no—
350D61—42no—
351D61—41no—
352D61—84no—
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⋊Q8C2×D6112yes—
368C2×#121—612no—
369C2×#121—612no—
370C2×#120—1222no—
371C2×C4×D4—1222no—
372C2×C4×D4—1222no—
373C2×#117—612no—
374#485—618no—
375#484—614no—
376#483C2×D6114yescomplemented
377#482—618no—
378#481—614no—
379#480—614no—
380#470—614no—
381#470C2×D6114yescomplemented
382#469—614no—
383#469—614no—
384#468—618no—
385#468—618no—
386#452—618no—
387#451C2×D6114yescomplemented
388#450—614no—
389#449—618no—
390#448—614no—
391#447—614no—
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×Q8C22×D6112yes—
407C22×D4—612no—
408C22×D4—1222no—
409C22×D4—1222no—
410C22×D4—1222no—
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×Q16C22×D6114yescomplemented
425C2×SD16—618no—
426C2×D8—614no—
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—22no—
437C22⋊Q8—22no—
438C22⋊Q8—614no—
439C22⋊Q8—614no—
440C22⋊Q8—1222no—
441C22⋊Q8—1222no—
442C22⋊Q8C22×D6114yes—
443C22⋊Q8C22×D6114yes—
444#121—614no—
445#121—614no—
446#121—614no—
447#121—614no—
448#121—1222no—
449#121—1222no—
450#121—1222no—
451#121—1222no—
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—22no—
469C2×C4⋊C4—612no—
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#103C22×D6118yescomplemented
483#102—618no—
484#102—618no—
485C22⋊C8—614no—
486C22⋊C8—618no—
487C22⋊C8C22×D6114yes—
488C24—612no—
489C24—612no—
490C24—611no—
491C24—611no—
492C24—1222no—
493C24—2444no—
494C24—2442no—
495C24D4×D6111yescharacteristic
496#41—12216no—
497#41—2448no—
498#41—2448no—
499#41—2448no—
500C2×Q8—22no—
501C2×Q8—44no—
502C2×Q8C23×D6114yes—
503C2×D4—614no—
504C2×D4—1224no—
505C2×D4—1224no—
506C2×D4—1224no—
507C2×D4—1224no—
508C2×D4—1224no—
509C2×D4—1224no—
510C2×D4—1222no—
511C2×D4—2444no—
512C2×D4—2448no—
513C2×D4—2448no—
514C2×D4—2448no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—42no—
518C22×C4—611no—
519C22×C4—612no—
520C22×C4—611no—
521C22×C4—612no—
522C22×C4—611no—
523C22×C4—612no—
524C22×C4—612no—
525C22×C4—611no—
526C22×C4—611no—
527C22×C4—612no—
528C22×C4—611no—
529C22×C4—612no—
530C22×C4—612no—
531C22×C4—612no—
532C22×C4—1224no—
533C22×C4—1228no—
534C22×C4—1224no—
535C22×C4—1224no—
536C22×C4—1222no—
537C22×C4—2442no—
538C22×C4—2444no—
539C22×C4—2444no—
540C22×C4—2442no—
541C22×C4—2444no—
542C22×C4—2442no—
543C22×C4—2448no—
544C22×C4—2442no—
545C22×C4D4×D6112yes—
546C22×C4D4×D6111yescharacteristic
547C22×C4C23×D6112yes—
548C22×C4C23×D6111yescharacteristic
549Q16—44no—
550SD16—2448no—
551D8—2444no—
552M16—42no—
553M16—2444no—
554M16—2442no—
555C2×C8—614no—
556C2×C8—1228no—
557C2×C8C23×D6114yes—
558C4⋊C4—24no—
559C4⋊C4—24no—
560C4⋊C4—22no—
561C4⋊C4—614no—
562C4⋊C4—1224no—
563C4⋊C4—1224no—
564C4⋊C4—1222no—
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—41no—
580C23—611no—
581C23—611no—
582C23—612no—
583C23—1224no—
584C23—1222no—
585C23—1224no—
586C23—1224no—
587C23—1224no—
588C23—2442no—
589C23—2441no—
590C23—2442no—
591C23—2442no—
592C23—2442no—
593C23—2442no—
594C23—2444no—
595C23—2444no—
596C23—24416no—
597C23—2444no—
598C23C2×D4×D6111yescharacteristic
599C23C2×D4×D6112yes—
600Q8—28no—
601Q8—44no—
602D4—1228no—
603D4—2444no—
604D4—2448no—
605D4—2448no—
606D4—2448no—
607D4—2448no—
608D4—2448no—
609D4—2448no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—611no—
619C2×C4—614no—
620C2×C4—612no—
621C2×C4—611no—
622C2×C4—1222no—
623C2×C4—1224no—
624C2×C4—1222no—
625C2×C4—1224no—
626C2×C4—1222no—
627C2×C4—1224no—
628C2×C4—1222no—
629C2×C4—1224no—
630C2×C4—1224no—
631C2×C4—2442no—
632C2×C4—2442no—
633C2×C4—2442no—
634C2×C4—2448no—
635C2×C4—2448no—
636C2×C4—2442no—
637C2×C4—2441no—
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—612no—
650C22—1222no—
651C22—1222no—
652C22—1222no—
653C22—1222no—
654C22—1222no—
655C22—1221no—
656C22—4888no—
657C22—2448no—
658C22—2448no—
659C22—2444no—
660C22—2444no—
661C22—2444no—
662C22—2444no—
663C22—2444no—
664C22#1684056114yes—
665C22C22×D4×D6111yescharacteristic
666C4—21no—
667C4—22no—
668C4—44no—
669C4—44no—
670C4—612no—
671C4—1221no—
672C4—1222no—
673C4—2444no—
674C4H(5,𝔽2)×D6112yes—
675C2—24no—
676C2—41no—
677C2—1222no—
678C2—4884no—
679C2—2441no—
680C2—2442no—
681C2#534×D6111yescharacteristic
682C2C2×#1684056112yes—
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3195C2
socleC22×C61C23×D4
hypercenter#3195D61
hypocenterC61#60741
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
1. C2×#16840275C2semidirect
2. #21456662C2semidirect
3. #21456652C2semidirect
4. #21456638C2semidirect
5. #21456621C2semidirect
6. #21456619C2semidirect
7. #21455733C2semidirect
8. #21455430C2semidirect
9. #21455042C2semidirect
10. #407×D61C2semidirect
11. C61×#3195C2semidirect
12. C22×#11758226C22semidirect
14. C2×#14519800C22semidirect
15. C2×M16×D61C22semidirect
22. #16840275C22semidirect
23. #16840275C22semidirect
25. C2×#14519753C22semidirect
26. C2×#14519752C22semidirect
31. C2×#14519735C22semidirect
32. #16840101C22semidirect
33. #16840095C22semidirect
34. #16840088C22semidirect
35. #16840081C22semidirect
36. #16840075C22semidirect
37. #16840017C22semidirect
38. #16840016C22semidirect
39. #16840015C22semidirect
40. #16840011C22semidirect
41. #16840010C22semidirect
42. #16839923C22semidirect
43. #16839912C22semidirect
44. #16839910C22semidirect
45. #16839901C22semidirect
46. #16839898C22semidirect
47. #16839797C22semidirect
48. #16839757C22semidirect
50. #16839725C22semidirect
51. #16839724C22semidirect
52. C22⋊C8×D61C22semidirect
54. C61×#483C22semidirect
55. C61×#470C22semidirect
56. C61×#451C22semidirect
80. C2×#11758201C23semidirect
81. C2×D488C23semidirect
125. #14519710C23semidirect
126. #14519709C23semidirect
127. #14519701C23semidirect
128. #14519699C23semidirect
129. #14519697C23semidirect
134. C2×Q16×C61C23semidirect
143. #106×C61C23semidirect
144. #103×C61C23semidirect
337. #3195D61semidirect
376. #483C2×D61semidirect
381. #470C2×D61semidirect
387. #451C2×D61semidirect
399. C61#60741semidirect
424. C2×Q16C22×D61semidirect
480. #106C22×D61semidirect
482. #103C22×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402752
#214566622
#214566522
#214566382
#214566212
#214566192
#214557332
#214554302
#214550422
#407×D612
C61×#31952
#6074161