Finite Group Databasebeta

← #29714240

Subgroups — #29714240

Order 15616.

1278 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455612C211yesmaximal, characteristic, complemented
2C4×#14519802C211yesmaximal, characteristic, complemented
3#21450766C211yesmaximal, characteristic, complemented
4#21447026C212yesmaximal, complemented
5#21446832C212yesmaximal, complemented
6#21446764C211yesmaximal, characteristic, complemented
7#21446660C211yesmaximal, characteristic, complemented
8#21446217C212yesmaximal, complemented
9#21446029C212yesmaximal, complemented
10#21445839C211yesmaximal, characteristic, complemented
11C61×#3508C211yesmaximal, characteristic, complemented
12C2×#14519802C2211yescharacteristic
13C2×#14519773C2212yes—
14C2×#14519748—22no—
15C2×C4×D244C2211yescharacteristic
16#16840059—22no—
17#16840049—21no—
18#16840039—21no—
19#16839872—21no—
20#16839860—21no—
21C4×#11758201—21no—
22C4×#11758199—21no—
23#16839805C2211yescharacteristic
24#16839805C2212yes—
25#16839786C2211yescharacteristic
26C2×#14519704C2212yes—
27#16839766C2211yescharacteristic
28#16839766C2212yes—
29#16839763C2211yescharacteristic
30C2×#14519694C2212yes—
31C2×#14519694C2212yes—
32C4×#11758194C2211yescharacteristic
33#16839670—22no—
34#16839665C2212yes—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839521C2212yes—
37#16839521C2212yes—
38#16839521C2212yes—
39C61×#559—21no—
40C61×#559—21no—
41C61×#550—21no—
42C61×#550—21no—
43C61×#546—22no—
44C61×#546—22no—
45C61×#540—22no—
46C61×#535C2212yes—
47C61×#533C2211yescharacteristic
48C61×#532—21no—
49C61×#532—21no—
50C2×#127×C61C2211yescharacteristic
51C61×#518C2212yes—
52C4×#41×C61C2211yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—24no—
55C22×C4×D61—21no—
56#14519802—22no—
57#14519802—22no—
58C2×#11758201—21no—
59C2×#11758199—21no—
60C2×#11758194C2311yescharacteristic
61C2×#11758194C2312yes—
62C2×#11758194C2311yescharacteristic
63#14519773—24no—
64#14519773—42no—
65#14519748—48no—
66C4×D244—21no—
67C4×D244—21no—
68C4×D244—21no—
69C4×D244—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C22⋊Dic61—24no—
73C2×#11758198—21no—
74C2×C4×Dic61C2312yes—
75#14519710—22no—
76#14519710—21no—
77#14519708—22no—
78#14519708—21no—
79C4⋊#92418—21no—
80C4⋊#92418—21no—
81#14519704—42no—
82#14519701—21no—
83#14519698—21no—
84#14519697—22no—
85#14519697—22no—
86#14519697—21no—
87#14519697—21no—
88#14519694—24no—
89#14519694—42no—
90#14519688—21no—
91C4×#92418—21no—
92#14519684—21no—
93C2×#41×C61—22no—
94C2×#41×C61—22no—
95C2×#41×C61C2311yescharacteristic
96C22×D4×C61D412yes—
97C22×D4×C61D412yes—
98C22×D4×C61C2312yes—
99C4⋊Q8×C61D411yescharacteristic, complemented
100C4⋊Q8×C61D411yescharacteristic, complemented
101#127×C61—22no—
102#127×C61—21no—
103#127×C61—21no—
104#127×C61—21no—
105#127×C61—21no—
106#125×C61D412yescomplemented
107#124×C61—22no—
108#124×C61—22no—
109#124×C61—22no—
110#123×C61—24no—
111#123×C61—24no—
112C22⋊Q8×C61—21no—
113C22⋊Q8×C61—21no—
114#121×C61—22no—
115#121×C61—22no—
116#121×C61—24no—
117#121×C61—24no—
118#121×C61—22no—
119#121×C61—22no—
120#120×C61—24no—
121C4×Q8×C61—21no—
122C4×Q8×C61—21no—
123C4×Q8×C61D411yescharacteristic, complemented
124C4×Q8×C61D411yescharacteristic, complemented
125C4×D4×C61—22no—
126C4×D4×C61—22no—
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61—22no—
130C4×D4×C61—22no—
131C4×D4×C61—22no—
132C4×D4×C61—22no—
133C4×D4×C61D411yescharacteristic, complemented
134C4×D4×C61D411yescharacteristic, complemented
135#117×C61—21no—
136#117×C61C2312yes—
137#117×C61C2311yescharacteristic
138C2×C22⋊C4×C61—24no—
139C2×C42×C61—21no—
140C2×C42×C61C2311yescharacteristic
141C24×C61C2×D414yes—
142C23×D61—21no—
143#41×C61—21no—
144#41×C61—21no—
145#41×C61—21no—
146#41×C61—21no—
147#41×C61—84no—
148#41×C61—84no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61—21no—
153C2×D4×C61—24no—
154C2×D4×C61—24no—
155C2×D4×C61—22no—
156C2×D4×C61—22no—
157C2×D4×C61—22no—
158C2×D4×C61—22no—
159C2×D4×C61—22no—
160C2×D4×C61—22no—
161C2×D4×C61—22no—
162C2×D4×C61—22no—
163C2×D4×C61—22no—
164C2×D4×C61—22no—
165C2×D4×C61—44no—
166C2×D4×C61—82no—
167C2×D4×C61C2×D411yescharacteristic
168C22×C4×C61—24no—
169C22×C4×C61—24no—
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—21no—
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2411yescharacteristic
176C2×C22⋊D61—48no—
177C2×C22⋊D61—48no—
178C2×C22⋊D61—84no—
179C22×Dic61—22no—
180C22×Dic61—21no—
181C2×D244—21no—
182C2×D244—21no—
183C2×D244—21no—
184C2×D244—21no—
185C2×D244—21no—
186C2×D244#4112yescomplemented
187C2×C4×D61—44no—
188C2×C4×D61—41no—
189C2×C4×D61—41no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61C2×D411yescharacteristic
197C4⋊C4×C61C2×D411yescharacteristic
198C4⋊C4×C61C2×D412yes—
199C4⋊C4×C61C2×D412yes—
200C22⋊C4×C61—24no—
201C22⋊C4×C61—24no—
202C22⋊C4×C61—24no—
203C22⋊C4×C61—24no—
204C22⋊C4×C61—21no—
205C22⋊C4×C61—22no—
206C22⋊C4×C61—21no—
207C22⋊C4×C61—21no—
208C42×C61—21no—
209C42×C61—21no—
210C42×C61—21no—
211C42×C61—22no—
212C42×C61C2×D411yescharacteristic
213C42×C61C2×D411yescharacteristic
214C42×C61C2×D411yescharacteristic
215C42×C61C2×D411yescharacteristic
216C22⋊Dic61—48no—
217#11758201—24no—
218#11758199—24no—
219#11758198—41no—
220#11758198—41no—
221C4⋊Dic61—24no—
222C4⋊Dic61—21no—
223C4⋊Dic61—21no—
224C4×Dic61—22no—
225C4×Dic61—22no—
226#11758194—21no—
227#11758194—22no—
228#11758194—21no—
229#11758194—22no—
230#11758194—21no—
231#11758194#4112yescomplemented
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418—22no—
235C23×C61—24no—
236C23×C61—21no—
237C23×C61—24no—
238C23×C61—24no—
239C23×C61—44no—
240C23×C61—44no—
241C23×C61C22×D411yescharacteristic
242C22×D61—22no—
243C22×D61—42no—
244C22×D61—42no—
245C22×D61—41no—
246C22×D61—41no—
247Q8×C61—21no—
248Q8×C61—21no—
249Q8×C61—42no—
250Q8×C61—42no—
251D4×C61—21no—
252D4×C61—21no—
253D4×C61—44no—
254D4×C61—41no—
255D4×C61—41no—
256D4×C61—42no—
257D4×C61—42no—
258D4×C61—42no—
259D4×C61—42no—
260D4×C61—44no—
261D4×C61—82no—
262D4×C61—82no—
263D4×C61—84no—
264D4×C61—84no—
265C2×C4×C61—22no—
266C2×C4×C61—21no—
267C2×C4×C61—21no—
268C2×C4×C61—21no—
269C2×C4×C61—22no—
270C2×C4×C61—22no—
271C2×C4×C61—41no—
272C2×C4×C61—42no—
273C2×C4×C61—41no—
274C2×C4×C61—41no—
275C2×C4×C61—42no—
276C2×C4×C61—84no—
277C2×C4×C61—84no—
278C2×C4×C61#12011yescharacteristic
279C2×C4×C61#12011yescharacteristic
280C2×C4×C61#12011yescharacteristic
281C2×C4×C61#12011yescharacteristic
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C2×#4111yescharacteristic
285C2×C4×C61C2×#4111yescharacteristic
286C2×C4×C61H(5,𝔽2)11yescharacteristic
287C2×C4×C612−1+411yescharacteristic
288C22⋊D61—816no—
289C2×Dic61—24no—
290C2×Dic61—42no—
291C2×Dic61—42no—
292C2×Dic61—42no—
293C2×Dic61—41no—
294D244—22no—
295D244—22no—
296D244—41no—
297D244—41no—
298C4×D61—82no—
299C4×D61—82no—
300#92418—22no—
301#92418—41no—
302#92418—42no—
303#24144—611nomaximal, Sylow
304C22×C61—21no—
305C22×C61—21no—
306C22×C61—41no—
307C22×C61—44no—
308C22×C61—41no—
309C22×C61—44no—
310C22×C61—41no—
311C22×C61—44no—
312C22×C61—84no—
313C22×C61—84no—
314C22×C61C2×#12011yescharacteristic
315C22×C61#54611yescharacteristic
316C22×C61#54711yescharacteristic
317C2×D61—44no—
318C2×D61—42no—
319C2×D61—82no—
320C2×D61—81no—
321C2×D61—81no—
322C4×C61—21no—
323C4×C61—21no—
324C4×C61—21no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61—42no—
328C4×C61—41no—
329C4×C61—42no—
330C4×C61#45312yes—
331C4×C61#52511yescharacteristic
332C4×C61#53711yescharacteristic
333C4×C61#54811yescharacteristic
334C4×C61#55011yescharacteristic
335Dic61—44no—
336Dic61—81no—
337#3508D6111yescharacteristic, complemented
338#3077—611no—
339C4×AGL(1,ℤ/8)—611no—
340#2541—611no—
341#2037—612no—
342#2002—612no—
343#1986—611no—
344#1965—611no—
345#1887—612no—
346#1856—612no—
347#1806—611no—
348C2×C61—21no—
349C2×C61—21no—
350C2×C61—81no—
351C2×C61—84no—
352C2×C61#240711yescharacteristic
353C2×C61C2×#45311yescharacteristic
354C2×C61#301911yescharacteristic
355D61—82no—
356C2×AGL(1,ℤ/8)—611no—
357#559—21no—
358#559—21no—
359#550—21no—
360#550—21no—
361#546—22no—
362#546—22no—
363#540—22no—
364#535C2×D6112yes—
365#533C2×D6111yescharacteristic
366#532—21no—
367#532—21no—
368C2×#127—612no—
369C2×#127C2×D6111yescharacteristic
370C2×#121—1222no—
371#518C2×D6112yes—
372C4×#41C2×D6111yescharacteristic
373C2×C4×D4—611no—
374#478—1222no—
375#476—1221no—
376#474—1221no—
377#442—1221no—
378#440—1221no—
379C4×SD16—1221no—
380C4×D8—1221no—
381#428—612no—
382#428—611no—
383#423—611no—
384C2×#104—612no—
385#418—612no—
386#418—611no—
387#417—611no—
388C2×#100—612no—
389C2×#100—612no—
390C4×M16—611no—
391#392—1222no—
392#390—612no—
393C4×C4⋊C4—611no—
394#339—612no—
395#339—612no—
396#339—612no—
397C61#2414411yescharacteristic, complemented, Sylow
398C2×#41—22no—
399C2×#41—22no—
400C2×#41C22×D6111yescharacteristic
401C22×D4—611no—
402C22×D4—1224no—
403C22×D4C22⋊D6112yes—
404C22×D4C22⋊D6112yes—
405C22×D4C22×D6112yes—
406C23×C4—1221no—
407AGL(1,ℤ/8)—1222no—
408AGL(1,ℤ/8)—1222no—
409C2×SD16—1221no—
410C2×D8—1221no—
411C2×M16—611no—
412C2×M16—611no—
413C2×M16—612no—
414C4⋊Q8C22⋊D6111yescharacteristic, complemented
415C4⋊Q8C22⋊D6111yescharacteristic, complemented
416#127—22no—
417#127—21no—
418#127—21no—
419#127—21no—
420#127—21no—
421#127—1224no—
422#127—2442no—
423#125C22⋊D6112yescomplemented
424#124—22no—
425#124—22no—
426#124—22no—
427#123—24no—
428#123—24no—
429C22⋊Q8—21no—
430C22⋊Q8—21no—
431#121—22no—
432#121—22no—
433#121—24no—
434#121—24no—
435#121—22no—
436#121—22no—
437#121—2448no—
438#120—24no—
439C4×Q8—21no—
440C4×Q8—21no—
441C4×Q8C22⋊D6111yescharacteristic, complemented
442C4×Q8C22⋊D6111yescharacteristic, complemented
443C4×D4—22no—
444C4×D4—22no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4—22no—
448C4×D4—22no—
449C4×D4—22no—
450C4×D4—22no—
451C4×D4—1221no—
452C4×D4—1221no—
453C4×D4—1221no—
454C4×D4—1221no—
455C4×D4C22⋊D6111yescharacteristic, complemented
456C4×D4C22⋊D6111yescharacteristic, complemented
457#117—21no—
458#117C22×D6111yescharacteristic
459#117C22×D6112yes—
460C2×C4⋊C4—611no—
461C2×C4⋊C4—1221no—
462C2×C22⋊C4—24no—
463C2×C22⋊C4—1224no—
464C2×C22⋊C4—1221no—
465C2×C42—21no—
466C2×C42—612no—
467C2×C42C22×D6111yescharacteristic
468#107—1221no—
469#107—1222no—
470#106—1221no—
471#106—1222no—
472C4⋊C8—1221no—
473C4⋊C8—1221no—
474#104—2442no—
475#103—1221no—
476#102—1221no—
477#102—1221no—
478#102—1221no—
479#102—1222no—
480#102—1222no—
481#100—1224no—
482#100—2442no—
483#97—1221no—
484C4×C8—1221no—
485#95—1221no—
486C24—1221no—
487C24C2×C22⋊D6114yes—
488#41—21no—
489#41—21no—
490#41—21no—
491#41—21no—
492#41—84no—
493#41—84no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8C2×C22⋊D6111yescharacteristic
497C2×D4—21no—
498C2×D4—24no—
499C2×D4—24no—
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—22no—
505C2×D4—22no—
506C2×D4—22no—
507C2×D4—22no—
508C2×D4—22no—
509C2×D4—22no—
510C2×D4—44no—
511C2×D4—82no—
512C2×D4—612no—
513C2×D4—1221no—
514C2×D4—1221no—
515C2×D4—1221no—
516C2×D4—1221no—
517C2×D4—1221no—
518C2×D4—4884no—
519C2×D4—2448no—
520C2×D4—2448no—
521C2×D4C2×C22⋊D6111yescharacteristic
522C22×C4—24no—
523C22×C4—24no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—1222no—
528C22×C4—1221no—
529C22×C4—2444no—
530C22×C4—2441no—
531C22×C4—2441no—
532C22×C4C2×C22⋊D6111yescharacteristic
533C22×C4C2×C22⋊D6111yescharacteristic
534C22×C4C23×D6111yescharacteristic
535SD16—1224no—
536D8—1224no—
537M16—612no—
538M16—1221no—
539M16—1222no—
540M16—1221no—
541M16—1222no—
542M16—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1222no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—1224no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4C2×C22⋊D6111yescharacteristic
556C4⋊C4C2×C22⋊D6112yes—
557C4⋊C4C2×C22⋊D6111yescharacteristic
558C4⋊C4C2×C22⋊D6112yes—
559C22⋊C4—24no—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C22⋊C4—21no—
564C22⋊C4—22no—
565C22⋊C4—21no—
566C22⋊C4—21no—
567C22⋊C4—2448no—
568C22⋊C4—2441no—
569C22⋊C4—2441no—
570C42—21no—
571C42—22no—
572C42—21no—
573C42—21no—
574C42—1222no—
575C42—1222no—
576C42C2×C22⋊D6111yescharacteristic
577C42C2×C22⋊D6111yescharacteristic
578C42C2×C22⋊D6111yescharacteristic
579C42C2×C22⋊D6111yescharacteristic
580C23—24no—
581C23—21no—
582C23—24no—
583C23—24no—
584C23—44no—
585C23—44no—
586C23—1222no—
587C23—2442no—
588C23—2442no—
589C23—2441no—
590C23—2441no—
591C23C22×C22⋊D6111yescharacteristic
592Q8—21no—
593Q8—21no—
594Q8—42no—
595Q8—42no—
596D4—21no—
597D4—21no—
598D4—44no—
599D4—41no—
600D4—41no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—44no—
606D4—82no—
607D4—82no—
608D4—84no—
609D4—84no—
610D4—1222no—
611D4—1222no—
612D4—48816no—
613D4—2441no—
614D4—2441no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—1224no—
629C2×C4—4882no—
630C2×C4—4882no—
631C2×C4—2442no—
632C2×C4—2441no—
633C2×C4—2442no—
634C2×C4—2442no—
635C2×C4#1451974611yescharacteristic
636C2×C4#1451974611yescharacteristic
637C2×C4#1451974611yescharacteristic
638C2×C4#1451974611yescharacteristic
639C2×C4C22×C22⋊D6111yescharacteristic
640C2×C4C22×C22⋊D6111yescharacteristic
641C2×C4#41×D6111yescharacteristic
642C2×C4#41×D6111yescharacteristic
643C2×C4#1451982311yescharacteristic
644C2×C4#1451982411yescharacteristic
645C8—1222no—
646C8—2441no—
647C8—2442no—
648C22—21no—
649C22—21no—
650C22—41no—
651C22—44no—
652C22—41no—
653C22—44no—
654C22—41no—
655C22—44no—
656C22—84no—
657C22—84no—
658C22—4882no—
659C22—4881no—
660C22—4881no—
661C22—2444no—
662C22—2442no—
663C22C2×#1451974611yescharacteristic
664C22#1684040011yescharacteristic
665C22#1684041111yescharacteristic
666C4—21no—
667C4—21no—
668C4—21no—
669C4—41no—
670C4—42no—
671C4—41no—
672C4—42no—
673C4—41no—
674C4—4881no—
675C4—2444no—
676C4#1683993012yes—
677C4#1684026911yescharacteristic
678C4#1684034011yescharacteristic
679C4#1684041911yescharacteristic
680C4#1684043111yescharacteristic
681C2—21no—
682C2—21no—
683C2—84no—
684C2—81no—
685C2—4882no—
686C2#2144915811yescharacteristic
687C2C2×#1683993011yescharacteristic
688C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3508C2
socleC22×C61C2×#120
hypercenter#3508D61
hypocenterC61#24144
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #21455612C2semidirect
2. C4×#14519802C2semidirect
3. #21450766C2semidirect
4. #21447026C2semidirect
5. #21446832C2semidirect
6. #21446764C2semidirect
7. #21446660C2semidirect
8. #21446217C2semidirect
9. #21446029C2semidirect
10. #21445839C2semidirect
11. C61×#3508C2semidirect
99. C4⋊Q8×C61D4semidirect
100. C4⋊Q8×C61D4semidirect
106. #125×C61D4semidirect
123. C4×Q8×C61D4semidirect
124. C4×Q8×C61D4semidirect
133. C4×D4×C61D4semidirect
134. C4×D4×C61D4semidirect
186. C2×D244#41semidirect
231. #11758194#41semidirect
337. #3508D61semidirect
397. C61#24144semidirect
414. C4⋊Q8C22⋊D61semidirect
415. C4⋊Q8C22⋊D61semidirect
423. #125C22⋊D61semidirect
441. C4×Q8C22⋊D61semidirect
442. C4×Q8C22⋊D61semidirect
455. C4×D4C22⋊D61semidirect
456. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214556122
C4×#145198022
#214507662
#214470262
#214468322
#214467642
#214466602
#214462172
#214460292
#214458392
C61×#35082
#2414461