Finite Group Databasebeta

← #30570399

Subgroups — #30570399

Order 15616.

1716 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460233C211yesmaximal, characteristic, complemented
2#21459622C211yesmaximal, characteristic, complemented
3#21458733C212yesmaximal, complemented
4#21458719C214yesmaximal, complemented
5#21458674C212yesmaximal, complemented
6#21458647C214yesmaximal, complemented
7#21458615C212yesmaximal, complemented
8#21458483C212yesmaximal, complemented
9#21457113C214yesmaximal, complemented
10#21457035C211yesmaximal, characteristic, complemented
11#21456951C211yesmaximal, characteristic, complemented
12#21455996C214yesmaximal, complemented
13#21455827C211yesmaximal, characteristic, complemented
14#21454916C211yesmaximal, characteristic, complemented
15C61×#3109C211yesmaximal, characteristic
16#16840512—22no—
17#16840498C2212yescomplemented
18#16840492C2212yescomplemented
19#16840482—21no—
20#16840482C2212yescomplemented
21#16840471—22no—
22#16840471C2214yes—
23#16840469—21no—
24#16840469C2212yes—
25D4×Dic122—21no—
26D4×Dic122—21no—
27D4×Dic122C2212yes—
28D4×Dic122C2212yescomplemented
29#16840393C2212yes—
30#16840388—22no—
31#16840379—22no—
32#16840379C2214yes—
33#16840333—22no—
34#16840319C2211yescharacteristic
35#16840319C2211yescharacteristic
36#16840319C2211yescharacteristic
37C2×#14519756—22no—
38#16840237C2212yes—
39#16840171C2212yescomplemented
40#16840169C2214yescomplemented
41#16840165C2211yescharacteristic, complemented
42#16840159C2214yescomplemented
43#16840158C2211yescharacteristic, complemented
44#16840131C2212yescomplemented
45#16840120C2214yescomplemented
46#16840115C2214yescomplemented
47#16840108C2211yescharacteristic, complemented
48#16840108C2211yescharacteristic, complemented
49#16840104C2214yes—
50#16840092C2212yes—
51#16840083C2214yes—
52#16840072C2214yescomplemented
53#16840062C2214yescomplemented
54#16840060C2214yescomplemented
55#16840058C2212yescomplemented
56#16840052C2212yes—
57#16840050C2214yes—
58#16840050C2214yescomplemented
59#16840048C2212yescomplemented
60#16840046C2214yescomplemented
61#16840040C2214yes—
62#16839975C2214yes—
63#16839975C2214yes—
64#16839967C2212yes—
65#16839913C2212yes—
66#16839890C2214yescomplemented
67#16839889C2212yescomplemented
68#16839886C2212yescomplemented
69#16839874C2214yes—
70#16839859C2214yes—
71#16839853C2214yes—
72C4×#11758200C2212yes—
73C4×#11758200C2212yes—
74D8×Dic61C2214yes—
75#16839820C2214yes—
76#16839815C2211yescharacteristic
77C61×#532C2211yescharacteristic
78C2×C4×D4×C61C2211yescharacteristic
79C61×#481C2212yes—
80C61×#476C2212yes—
81C61×#459C2212yes—
82C61×#447C2212yes—
83C61×#442C2214yes—
84C61×#432C2211yescharacteristic
85C22×Dic122—22no—
86C22×Dic122—22no—
87C2×#11758200C2312yes—
88C2×#11758200C2312yes—
89#14519777D411yescharacteristic
90#14519777D411yescharacteristic
91#14519777C2311yescharacteristic
92#14519777C2311yescharacteristic
93#14519777C2311yescharacteristic
94#14519777C2311yescharacteristic
95#14519776—22no—
96#14519776—22no—
97#14519776C2314yes—
98#14519776C2314yes—
99#14519775—24no—
100#14519770—24no—
101#14519770—24no—
102#14519769—21no—
103#14519769D412yes—
104#14519769C2312yes—
105#14519768—22no—
106#14519768C2314yes—
107#14519764—22no—
108#14519764—24no—
109#14519764C2314yes—
110#14519758—24no—
111#14519758—24no—
112#14519756—21no—
113#14519756—21no—
114#14519756—21no—
115#14519756—22no—
116#14519756—22no—
117#14519756—24no—
118#14519756—22no—
119#14519756—22no—
120#14519756—24no—
121#14519756D412yes—
122#14519756D412yes—
123#14519756C2312yes—
124#14519754—24no—
125#14519754—24no—
126#14519754—24no—
127#14519754—24no—
128#14519754—24no—
129#14519754—24no—
130C4×Dic122—21no—
131C4×Dic122—21no—
132C4×Dic122—21no—
133C4×Dic122C2312yes—
134C4×Dic122C2312yes—
135C4×Dic122C2312yes—
136#14519742—22no—
137#14519742C2314yes—
138D4×Dic61—22no—
139D4×Dic61—22no—
140D4×Dic61—22no—
141D4×Dic61C2314yes—
142D4×Dic61C2314yes—
143D4×Dic61C2314yes—
144#14519736—21no—
145#14519736—22no—
146#14519736C2312yes—
147#14519736C2312yes—
148C2×C4⋊Dic61—22no—
149C2×#11758196—24no—
150C2×C22⋊Dic61—24no—
151#14519711C2314yes—
152#14519710C2314yes—
153#14519709C2314yes—
154#14519708C2314yes—
155#14519708C2312yes—
156#14519708C2312yes—
157C4⋊#92418C2312yes—
158#14519706C2314yes—
159#14519706C2314yes—
160#14519702C2314yes—
161#14519702C2314yes—
162#14519700C2312yes—
163#14519700C2312yes—
164#14519698C2314yes—
165#14519698C2314yes—
166#14519698C2312yes—
167#14519698C2312yes—
168C22⋊#92418C2312yes—
169#14519687C2314yes—
170C4×#92418C2311yescharacteristic
171C4×#92418C2311yescharacteristic
172C8×Dic61C2314yes—
173C2×#41×C61—21no—
174C22×D4×C61C2311yescharacteristic
175C23×C4×C61—21no—
176C2×D8×C61C2314yes—
177C4⋊Q8×C61C2311yescharacteristic
178#127×C61C2311yescharacteristic
179#124×C61—21no—
180#121×C61—21no—
181#121×C61C2312yes—
182C4×D4×C61—21no—
183C4×D4×C61—21no—
184C4×D4×C61—21no—
185C4×D4×C61C2312yes—
186C4×D4×C61C2312yes—
187C4×D4×C61C2312yes—
188C2×C4⋊C4×C61C2311yescharacteristic
189C2×C22⋊C4×C61—21no—
190C2×C42×C61C2311yescharacteristic
191#106×C61C2314yes—
192C4⋊C8×C61C2312yes—
193#102×C61C2314yes—
194#102×C61C2314yes—
195C22⋊C8×C61C2312yes—
196#97×C61C2312yes—
197C24×C61—21no—
198#41×C61—82no—
199C2×Q8×C61—21no—
200C2×D4×C61—24no—
201C2×D4×C61—22no—
202C2×D4×C61—21no—
203C2×D4×C61—21no—
204C2×D4×C61—21no—
205C2×D4×C61C2412yes—
206C2×D4×C61C2412yes—
207C22×C4×C61—21no—
208C22×C4×C61—22no—
209C22×C4×C61—22no—
210C22×C4×C61—22no—
211C22×C4×C61—82no—
212C22×C4×C61C2×D411yescharacteristic
213C22×C4×C61C2×D411yescharacteristic
214C22×C4×C61C2411yescharacteristic
215C22×Dic61—24no—
216C22×Dic61—24no—
217C22×Dic61—24no—
218C2×Dic122—21no—
219C2×Dic122—21no—
220C2×Dic122—21no—
221C2×Dic122—24no—
222C2×Dic122—24no—
223C2×Dic122—84no—
224C2×Dic122—84no—
225C2×Dic122C2×D412yes—
226C2×Dic122C2×D412yes—
227C2×Dic122C2412yes—
228D8×C61—28no—
229C2×C8×C61C2414yes—
230C4⋊C4×C61—21no—
231C4⋊C4×C61—21no—
232C4⋊C4×C61C2412yes—
233C4⋊C4×C61C2412yes—
234C22⋊C4×C61—22no—
235C22⋊C4×C61—22no—
236C22⋊C4×C61—22no—
237C22⋊C4×C61—22no—
238C42×C61C2×D411yescharacteristic
239C42×C61C2×D411yescharacteristic
240C42×C61C2411yescharacteristic
241C42×C61C2411yescharacteristic
242C22⋊Dic61—22no—
243C22⋊Dic61—22no—
244C22⋊Dic61—24no—
245C22⋊Dic61—24no—
246C22⋊Dic61—24no—
247C22⋊Dic61—24no—
248C22⋊Dic61—24no—
249C22⋊Dic61C2×D414yes—
250#11758200—24no—
251#11758200—24no—
252C4⋊Dic61—21no—
253C4⋊Dic61—21no—
254C4⋊Dic61—21no—
255C4⋊Dic61—21no—
256C4⋊Dic61—21no—
257C4⋊Dic61—24no—
258C4⋊Dic61C2×D412yes—
259C4⋊Dic61C2×D412yes—
260C4⋊Dic61C2×D412yes—
261C4⋊Dic61C2×D412yes—
262C4⋊Dic61C2412yes—
263C4⋊Dic61C2412yes—
264C4⋊Dic61C2412yes—
265#11758196—22no—
266#11758196—22no—
267#11758196—22no—
268#11758196—22no—
269#11758196—24no—
270#11758196—24no—
271#11758196—24no—
272#11758196—24no—
273#11758196C2×D414yes—
274#11758196C2×D414yes—
275C4×Dic61—22no—
276C4×Dic61—22no—
277C4×Dic61—24no—
278C4×Dic61C2414yes—
279C2×#92418C2412yes—
280C2×#92418C2412yes—
281C23×C61—24no—
282C23×C61—22no—
283C23×C61—22no—
284C23×C61—81no—
285C23×C61—81no—
286C23×C61C22×D411yescharacteristic
287Q8×C61—81no—
288D4×C61—24no—
289D4×C61—22no—
290D4×C61—24no—
291D4×C61—42no—
292D4×C61—81no—
293D4×C61—82no—
294D4×C612−1+414yes—
295C2×C4×C61—22no—
296C2×C4×C61—22no—
297C2×C4×C61—22no—
298C2×C4×C61—22no—
299C2×C4×C61—41no—
300C2×C4×C61—44no—
301C2×C4×C61—44no—
302C2×C4×C61—82no—
303C2×C4×C61—81no—
304C2×C4×C61C22×D411yescharacteristic
305C2×C4×C61C22×D411yescharacteristic
306C2×C4×C61C22×D411yescharacteristic
307C2×C4×C61C22×D411yescharacteristic
308C2×C4×C61C22×D411yescharacteristic
309C2×C4×C61C2511yescharacteristic
310C2×Dic61—22no—
311C2×Dic61—22no—
312C2×Dic61—22no—
313C2×Dic61—22no—
314C2×Dic61—84no—
315C2×Dic61—84no—
316C2×Dic61—84no—
317C2×Dic61C22×D414yes—
318C2×Dic61C22×D414yes—
319Dic122—24no—
320Dic122—22no—
321Dic122—22no—
322Dic122—42no—
323Dic122—42no—
324Dic122—42no—
325Dic122—84no—
326Dic122—84no—
327Dic122—81no—
328C8×C61—24no—
329#92418—22no—
330#92418—22no—
331#61850—611nomaximal, Sylow
332C22×C61—21no—
333C22×C61—21no—
334C22×C61—24no—
335C22×C61—21no—
336C22×C61—42no—
337C22×C61—42no—
338C22×C61—42no—
339C22×C61—42no—
340C22×C61—42no—
341C22×C61—84no—
342C22×C61C23×D411yescharacteristic
343C4×C61—21no—
344C4×C61—21no—
345C4×C61—21no—
346C4×C61—21no—
347C4×C61—42no—
348C4×C61—42no—
349C4×C61#57712yes—
350C4×C61C2×2−1+412yes—
351Dic61—24no—
352Dic61—24no—
353Dic61—42no—
354Dic61—42no—
355Dic61—42no—
356Dic61—42no—
357#3550—611no—
358#3501—611no—
359#3385—612no—
360#3384—614no—
361Q8×SD16—612no—
362#3376—614no—
363#3373—612no—
364#3361—612no—
365#3244—614no—
366#3237—611no—
367#3228—611no—
368#3118—614no—
369#3109D6111yescharacteristic
370#3101—611no—
371#2985—611no—
372C2×C61—41no—
373C2×C61—44no—
374C2×C61—82no—
375C2×C61#348611yescharacteristic
376C2×C61C2×#57711yescharacteristic
377C2×C61#358511yescharacteristic
378#563—1222no—
379#559—612no—
380Q82—612no—
381#556—612no—
382#556—1221no—
383#554—612no—
384#554—614no—
385#554—1222no—
386#554—1221no—
387D4×Q8—612no—
388D4×Q8—612no—
389D4×Q8—1221no—
390D4×Q8—1221no—
391#544—612no—
392#543—1222no—
393#542—614no—
394#542—1222no—
395#536—1222no—
396#533—611no—
397#533—611no—
398#533—611no—
399#532C2×D6111yescharacteristic
400C2×C22⋊Q8—1222no—
401#518—612no—
402C2×C4×D4C2×D6111yescharacteristic
403#501—614no—
404#501—612no—
405#500—611no—
406#498—611no—
407#498—614no—
408#491—612no—
409#489—614no—
410#488—614no—
411#486—611no—
412#486—611no—
413#485—614no—
414#483—612no—
415#482—614no—
416#481C2×D6112yes—
417#479—614no—
418#478—614no—
419#478—614no—
420#477—612no—
421#476—612no—
422#476—614no—
423#476—614no—
424#476C2×D6112yes—
425#475—614no—
426#475—612no—
427#474—614no—
428#463—614no—
429#463—614no—
430#461—612no—
431#459C2×D6112yes—
432#450—612no—
433#447C2×D6112yes—
434#446—612no—
435#446—614no—
436C8×Q8—612no—
437#442—614no—
438#442C2×D6114yes—
439#440—614no—
440C4×SD16—612no—
441C4×SD16—612no—
442C4×SD16—614no—
443C4×D8—614no—
444#433—614no—
445#432C2×D6111yescharacteristic
446#431—611no—
447C61#6185011yescharacteristic, complemented, Sylow
448C2×#41—21no—
449C22×Q8—1222no—
450C22×Q8—1222no—
451C22×D4C22×D6111yescharacteristic
452C23×C4—21no—
453C2×SD16—612no—
454C2×SD16—612no—
455C2×D8C22×D6114yes—
456C4⋊Q8—614no—
457C4⋊Q8—614no—
458C4⋊Q8—611no—
459C4⋊Q8—611no—
460C4⋊Q8—611no—
461C4⋊Q8—611no—
462C4⋊Q8—611no—
463C4⋊Q8—611no—
464C4⋊Q8—1222no—
465C4⋊Q8—1222no—
466C4⋊Q8—1224no—
467C4⋊Q8C22×D6111yescharacteristic
468#127C22×D6111yescharacteristic
469#126—1224no—
470#126—1224no—
471#125—614no—
472#125—612no—
473#125—612no—
474#125—1222no—
475#125—1221no—
476#124—21no—
477#124—614no—
478#124—1224no—
479#124—1222no—
480#123—1224no—
481#123—1224no—
482C22⋊Q8—612no—
483C22⋊Q8—612no—
484C22⋊Q8—612no—
485C22⋊Q8—1224no—
486C22⋊Q8—1224no—
487C22⋊Q8—1224no—
488C22⋊Q8—1224no—
489C22⋊Q8—1224no—
490C22⋊Q8—1224no—
491C22⋊Q8—1224no—
492C22⋊Q8—1222no—
493C22⋊Q8—1224no—
494C22⋊Q8—1222no—
495C22⋊Q8—1222no—
496C22⋊Q8—1222no—
497C22⋊Q8—1221no—
498C22⋊Q8—1221no—
499C22⋊Q8—1221no—
500#121—21no—
501#121C22×D6112yes—
502C4×Q8—612no—
503C4×Q8—612no—
504C4×Q8—614no—
505C4×Q8—612no—
506C4×Q8—1222no—
507C4×Q8—1221no—
508C4×Q8—1221no—
509C4×Q8—1221no—
510C4×D4—21no—
511C4×D4—21no—
512C4×D4—21no—
513C4×D4—614no—
514C4×D4—614no—
515C4×D4—614no—
516C4×D4—1222no—
517C4×D4—1222no—
518C4×D4—1222no—
519C4×D4C22×D6112yes—
520C4×D4C22×D6112yes—
521C4×D4C22×D6112yes—
522#117—612no—
523#117—612no—
524#117—1222no—
525#117—1221no—
526C2×C4⋊C4—1224no—
527C2×C4⋊C4—1222no—
528C2×C4⋊C4C22×D6111yescharacteristic
529C2×C22⋊C4—21no—
530C2×C22⋊C4—1224no—
531C2×C42C22×D6111yescharacteristic
532#107—614no—
533#107—614no—
534#106—614no—
535#106—614no—
536#106—612no—
537#106—612no—
538#106C22×D6114yes—
539C4⋊C8—614no—
540C4⋊C8—614no—
541C4⋊C8—612no—
542C4⋊C8C22×D6112yes—
543#103—614no—
544#103—614no—
545#103—612no—
546#103—612no—
547#102—612no—
548#102—614no—
549#102—612no—
550#102—614no—
551#102C22×D6114yes—
552#102C22×D6114yes—
553C22⋊C8—612no—
554C22⋊C8C22×D6112yes—
555#97—614no—
556#97C22×D6112yes—
557C4×C8—614no—
558C4×C8—611no—
559C4×C8—611no—
560C24—21no—
561#41—82no—
562C2×Q8—21no—
563C2×Q8—612no—
564C2×Q8—612no—
565C2×Q8—612no—
566C2×Q8—1224no—
567C2×Q8—1224no—
568C2×Q8—1221no—
569C2×Q8—1221no—
570C2×Q8—1221no—
571C2×Q8—4884no—
572C2×Q8—4884no—
573C2×D4—24no—
574C2×D4—22no—
575C2×D4—21no—
576C2×D4—21no—
577C2×D4—21no—
578C2×D4C23×D6112yes—
579C2×D4C23×D6112yes—
580C22×C4—21no—
581C22×C4—22no—
582C22×C4—22no—
583C22×C4—22no—
584C22×C4—82no—
585C22×C4—1224no—
586C22×C4—1224no—
587C22×C4—1224no—
588C22×C4D4×D6111yescharacteristic
589C22×C4D4×D6111yescharacteristic
590C22×C4C23×D6111yescharacteristic
591SD16—1224no—
592SD16—1224no—
593D8—28no—
594C2×C8—612no—
595C2×C8—612no—
596C2×C8C23×D6114yes—
597C4⋊C4—21no—
598C4⋊C4—21no—
599C4⋊C4—614no—
600C4⋊C4—612no—
601C4⋊C4—614no—
602C4⋊C4—612no—
603C4⋊C4—612no—
604C4⋊C4—612no—
605C4⋊C4—612no—
606C4⋊C4—612no—
607C4⋊C4—612no—
608C4⋊C4—1224no—
609C4⋊C4—1224no—
610C4⋊C4—1224no—
611C4⋊C4—1224no—
612C4⋊C4—1224no—
613C4⋊C4—1222no—
614C4⋊C4—1221no—
615C4⋊C4—1222no—
616C4⋊C4—1222no—
617C4⋊C4—1222no—
618C4⋊C4—1221no—
619C4⋊C4—1221no—
620C4⋊C4—1221no—
621C4⋊C4—1221no—
622C4⋊C4C23×D6112yes—
623C4⋊C4C23×D6112yes—
624C22⋊C4—22no—
625C22⋊C4—22no—
626C22⋊C4—22no—
627C22⋊C4—22no—
628C22⋊C4—614no—
629C22⋊C4—1224no—
630C22⋊C4—1224no—
631C22⋊C4—1224no—
632C22⋊C4—1224no—
633C22⋊C4—1224no—
634C22⋊C4—1222no—
635C22⋊C4—1222no—
636C42—614no—
637C42—1224no—
638C42—1222no—
639C42—1222no—
640C42D4×D6111yescharacteristic
641C42D4×D6111yescharacteristic
642C42C23×D6111yescharacteristic
643C42C23×D6111yescharacteristic
644C23—24no—
645C23—22no—
646C23—22no—
647C23—81no—
648C23—81no—
649C23C2×D4×D6111yescharacteristic
650Q8—81no—
651Q8—1224no—
652Q8—1222no—
653Q8—1222no—
654Q8—4881no—
655Q8—4884no—
656Q8—4884no—
657Q8—2442no—
658Q8—2442no—
659Q8—2442no—
660D4—24no—
661D4—22no—
662D4—24no—
663D4—42no—
664D4—81no—
665D4—82no—
666D4#1451982414yes—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—41no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—82no—
675C2×C4—81no—
676C2×C4—614no—
677C2×C4—614no—
678C2×C4—1222no—
679C2×C4—1222no—
680C2×C4—1222no—
681C2×C4—1222no—
682C2×C4—4884no—
683C2×C4—4884no—
684C2×C4—4884no—
685C2×C4C2×D4×D6111yescharacteristic
686C2×C4C2×D4×D6111yescharacteristic
687C2×C4C2×D4×D6111yescharacteristic
688C2×C4C2×D4×D6111yescharacteristic
689C2×C4C2×D4×D6111yescharacteristic
690C2×C4C24×D6111yescharacteristic
691C8—24no—
692C8—1222no—
693C8—1222no—
694C22—21no—
695C22—24no—
696C22—21no—
697C22—21no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22—84no—
704C22C22×D4×D6111yescharacteristic
705C4—21no—
706C4—21no—
707C4—21no—
708C4—21no—
709C4—42no—
710C4—42no—
711C4—1224no—
712C4—1224no—
713C4—2442no—
714C4—2442no—
715C4—2442no—
716C4—2442no—
717C4#1684057212yes—
718C4C2×#1451982412yes—
719C2—41no—
720C2—44no—
721C2—82no—
722C2#2145946611yescharacteristic
723C2C2×#1684057211yescharacteristic
724C2#2146059511yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
Fitting, prime radicalC61×#3109C2
socleC22×C61C23×D4
hypercenter#3109D61
hypocenterC61#61850
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
1. #21460233C2semidirect
2. #21459622C2semidirect
3. #21458733C2semidirect
4. #21458719C2semidirect
5. #21458674C2semidirect
6. #21458647C2semidirect
7. #21458615C2semidirect
8. #21458483C2semidirect
9. #21457113C2semidirect
10. #21457035C2semidirect
11. #21456951C2semidirect
12. #21455996C2semidirect
13. #21455827C2semidirect
14. #21454916C2semidirect
17. #16840498C22semidirect
18. #16840492C22semidirect
20. #16840482C22semidirect
28. D4×Dic122C22semidirect
39. #16840171C22semidirect
40. #16840169C22semidirect
41. #16840165C22semidirect
42. #16840159C22semidirect
43. #16840158C22semidirect
44. #16840131C22semidirect
45. #16840120C22semidirect
46. #16840115C22semidirect
47. #16840108C22semidirect
48. #16840108C22semidirect
52. #16840072C22semidirect
53. #16840062C22semidirect
54. #16840060C22semidirect
55. #16840058C22semidirect
58. #16840050C22semidirect
59. #16840048C22semidirect
60. #16840046C22semidirect
66. #16839890C22semidirect
67. #16839889C22semidirect
68. #16839886C22semidirect
447. C61#61850semidirect

Maximal subgroups

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

subgroupindex
#214602332
#214596222
#214587332
#214587192
#214586742
#214586472
#214586152
#214584832
#214571132
#214570352
#214569512
#214559962
#214558272
#214549162
C61×#31092
#6185061