Finite Group Databasebeta

← #30560135

Subgroups — #30560135

Order 15616.

1730 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459684C211yesmaximal, characteristic
2#21459624C211yesmaximal, characteristic, complemented
3#21458668C214yesmaximal, complemented
4#21458625C214yesmaximal, complemented
5#21458590C214yesmaximal, complemented
6#21458562C214yesmaximal, complemented
7#21456180C212yesmaximal, complemented
8#21456167C211yesmaximal, characteristic
9#21455680C212yesmaximal, complemented
10#21455348C212yesmaximal, complemented
11#21455293C211yesmaximal, characteristic
12#21454756C211yesmaximal, characteristic, complemented
13#21454660C212yesmaximal, complemented
14C2×#16839867C211yesmaximal, characteristic, complemented
15C61×#3078C211yesmaximal, characteristic, complemented
16C2×#14519810C2212yes—
17C22×#11758202C2211yescharacteristic
18#16840497C2212yes—
19Q8×Dic122—21no—
20#16840492C2212yes—
21#16840490—21no—
22#16840490—22no—
23#16840490C2212yes—
24#16840489C2214yes—
25#16840487C2212yes—
26#16840485—21no—
27#16840484C2214yes—
28#16840472—21no—
29#16840453—22no—
30#16840391—22no—
31#16840390C2212yes—
32#16840359—22no—
33#16840336C2212yes—
34#16840319C2211yescharacteristic
35#16840314—22no—
36#16840312C2211yescharacteristic
37#16840296—22no—
38#16840243C2212yes—
39#16840239C2212yes—
40C2×C4×Dic122C2211yescharacteristic
41#16840132C2212yes—
42#16840129C2214yes—
43#16840128C2212yes—
44#16840124C2212yes—
45#16840124C2212yes—
46#16840123C2214yes—
47#16840118C2214yes—
48#16840111C2214yes—
49#16840068C2214yes—
50#16840065C2214yes—
51#16840051C2214yes—
52#16840046C2214yes—
53#16839987C2214yes—
54#16839987C2214yes—
55#16839985C2214yes—
56#16839973C2212yes—
57#16839971C2214yes—
58#16839971C2214yes—
59#16839963C2212yes—
60#16839891C2214yes—
61#16839891C2214yes—
62#16839868C2214yes—
63#16839867C2214yes—
64#16839867C2214yes—
65#16839861C2214yes—
66#16839856C2214yes—
67#16839849C2214yes—
68C2×#14519708C2211yescharacteristic
69#16839790C2212yes—
70#16839764C2212yes—
71#16839756C2212yes—
72#16839756C2212yes—
73C2×#14519701C2211yescharacteristic
74C2×#14519700C2211yescharacteristic
75#16839720C2212yes—
76C2×#14519688C2211yescharacteristic
77C61×#533C2211yescharacteristic
78C2×C4×Q8×C61C2211yescharacteristic
79C61×#479C2214yes—
80C61×#477C2212yes—
81C61×#475C2212yes—
82C61×#428C2212yes—
83C61×#423C2211yescharacteristic
84C61×#419C2212yes—
85C2×#11758222C2312yes—
86C22×Dic122C2311yescharacteristic
87#14519810—28no—
88C2×Dic244C2314yes—
89C2×#11758202—24no—
90C2×#11758202C2314yes—
91C2×#11758190C2314yes—
92C22×#92418C2311yescharacteristic
93C22×#92418C2311yescharacteristic
94#14519777—22no—
95#14519777D411yescharacteristic
96#14519777D411yescharacteristic
97#14519777C2311yescharacteristic
98#14519776—24no—
99#14519776—22no—
100#14519776C2314yes—
101#14519775—22no—
102#14519775—24no—
103#14519775C2314yes—
104#14519774C2311yescharacteristic
105#14519772—24no—
106#14519771—24no—
107#14519769—22no—
108#14519769—22no—
109#14519769D412yes—
110#14519768C2314yes—
111#14519767—24no—
112#14519767—24no—
113#14519767—24no—
114#14519767—24no—
115#14519766C2314yes—
116#14519765—22no—
117#14519765C2312yes—
118#14519762—24no—
119#14519760—24no—
120#14519756D412yes—
121#14519756D412yes—
122#14519755—24no—
123#14519754—24no—
124#14519753—22no—
125#14519753—22no—
126#14519751—21no—
127#14519751—21no—
128Q8×Dic61—22no—
129C4×Dic122—22no—
130C4×Dic122—21no—
131C4×Dic122—21no—
132C4×Dic122—22no—
133C4×Dic122—22no—
134C4×Dic122C2312yes—
135C4×Dic122C2312yes—
136C4×Dic122C2312yes—
137C4×Dic122C2312yes—
138#14519742—22no—
139#14519742—24no—
140#14519742C2314yes—
141#14519742C2314yes—
142#14519742C2314yes—
143C4×D244—21no—
144C4×D244—21no—
145C4×D244—22no—
146#14519739C2314yes—
147C4×C22⋊D61—24no—
148#14519736C2312yes—
149#14519735—22no—
150#14519734—24no—
151#14519733—24no—
152C2×C4⋊Dic61C2311yescharacteristic
153C2×#11758196—21no—
154C4⋊C4×D61—22no—
155C2×C4×Dic61—21no—
156#14519708C2314yes—
157#14519706C2314yes—
158#14519706C2314yes—
159#14519702C2314yes—
160#14519701C2314yes—
161#14519700C2312yes—
162#14519700C2312yes—
163#14519700C2314yes—
164#14519700C2312yes—
165#14519700C2312yes—
166#14519699C2314yes—
167#14519697C2312yes—
168#14519697C2312yes—
169#14519697C2312yes—
170#14519697C2312yes—
171#14519688C2312yes—
172#14519688C2312yes—
173#14519687C2314yes—
174C8×Dic61C2314yes—
175C22×Q8×C61C2311yescharacteristic
176C2×M16×C61C2312yes—
177C4⋊Q8×C61C2311yescharacteristic
178C4⋊Q8×C61C2311yescharacteristic
179#125×C61C2312yes—
180C22⋊Q8×C61—21no—
181C22⋊Q8×C61—21no—
182C4×Q8×C61—21no—
183C4×Q8×C61—21no—
184C4×Q8×C61—21no—
185C4×Q8×C61—21no—
186C4×Q8×C61C2312yes—
187C4×Q8×C61C2312yes—
188#117×C61C2312yes—
189C2×C4⋊C4×C61—21no—
190C2×C4⋊C4×C61C2311yescharacteristic
191C2×C42×C61—21no—
192C2×C42×C61C2311yescharacteristic
193#107×C61C2314yes—
194#106×C61C2314yes—
195C4⋊C8×C61C2312yes—
196C4⋊C8×C61C2312yes—
197#103×C61C2314yes—
198#103×C61C2314yes—
199C2×Q8×C61—21no—
200C2×Q8×C61—21no—
201C2×Q8×C61—21no—
202C2×Q8×C61—22no—
203C2×Q8×C61—22no—
204C2×Q8×C61C2412yes—
205C22×C4×C61—21no—
206C22×C4×C61—21no—
207C22×C4×C61C2×D411yescharacteristic
208C22×C4×C61C2×D411yescharacteristic
209C22×C4×C61C2411yescharacteristic
210C2×C22⋊D61—22no—
211C22×Dic61—21no—
212C22×Dic61—21no—
213#11758222—24no—
214#11758222—24no—
215C2×D244—21no—
216C2×D244—21no—
217C2×D244C2412yes—
218C2×C4×D61—22no—
219C2×C4×D61—24no—
220C2×Dic122—21no—
221C2×Dic122—22no—
222C2×Dic122—24no—
223C2×Dic122C2×D412yes—
224C2×Dic122C2×D412yes—
225C2×Dic122#4114yes—
226C2×Dic122C2412yes—
227C2×Dic122C2412yes—
228M16×C61—24no—
229C2×C8×C61C2414yes—
230C4⋊C4×C61—22no—
231C4⋊C4×C61—22no—
232C4⋊C4×C61—21no—
233C4⋊C4×C61—22no—
234C4⋊C4×C61—22no—
235C4⋊C4×C61—21no—
236C4⋊C4×C61—21no—
237C4⋊C4×C61C2412yes—
238C4⋊C4×C61C2412yes—
239C4⋊C4×C61C2412yes—
240C22⋊C4×C61—22no—
241C42×C61—22no—
242C42×C61—22no—
243C42×C61—22no—
244C42×C61C2×D411yescharacteristic
245C42×C61C2×D411yescharacteristic
246C42×C61C2411yescharacteristic
247C42×C61C2411yescharacteristic
248C22⋊Dic61C2×D414yes—
249#11758202—28no—
250#11758198—22no—
251#11758198—22no—
252#11758198—24no—
253#11758198—24no—
254#11758198—24no—
255C4⋊Dic61—21no—
256C4⋊Dic61—21no—
257C4⋊Dic61—21no—
258C4⋊Dic61—24no—
259C4⋊Dic61C2×D412yes—
260C4⋊Dic61C2×D412yes—
261C4⋊Dic61C2×D412yes—
262C4⋊Dic61C2×D412yes—
263C4⋊Dic61C2412yes—
264#11758196—22no—
265#11758196—22no—
266#11758196—24no—
267#11758196—24no—
268#11758196—24no—
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×Dic61—24no—
279C4×Dic61—24no—
280C4×Dic61C2414yes—
281C2×#92418—22no—
282C2×#92418—22no—
283C2×#92418C2412yes—
284C2×#92418C2412yes—
285Dic244—28no—
286#11758190—28no—
287C23×C61C22×D411yescharacteristic
288C22×D61—22no—
289Q8×C61—22no—
290Q8×C61—22no—
291Q8×C61—42no—
292Q8×C61—42no—
293Q8×C61—81no—
294C2×C4×C61—22no—
295C2×C4×C61—22no—
296C2×C4×C61—22no—
297C2×C4×C61—22no—
298C2×C4×C61—41no—
299C2×C4×C61—41no—
300C2×C4×C61—42no—
301C2×C4×C61—81no—
302C2×C4×C61C22×D411yescharacteristic
303C2×C4×C61C22×D411yescharacteristic
304C2×C4×C61C22×D411yescharacteristic
305C2×C4×C61C22×D411yescharacteristic
306C2×C4×C61C22×D411yescharacteristic
307C2×C4×C61C2×#4112yes—
308C2×C4×C612−1+412yes—
309C2×C4×C61C2511yescharacteristic
310C22⋊D61—44no—
311C2×Dic61—24no—
312C2×Dic61—22no—
313C2×Dic61—22no—
314C2×Dic61—22no—
315C2×Dic61—81no—
316C2×Dic61C22×D414yes—
317C2×Dic61C22×D414yes—
318D244—22no—
319D244—22no—
320D244—42no—
321D244—81no—
322C4×D61—44no—
323C4×D61—84no—
324Dic122—24no—
325Dic122—22no—
326Dic122—22no—
327Dic122—22no—
328Dic122—42no—
329Dic122—42no—
330Dic122—84no—
331Dic122C2×#4114yes—
332C8×C61—24no—
333#92418—22no—
334#92418—22no—
335#61335—611nomaximal, Sylow
336C22×C61—21no—
337C22×C61—21no—
338C22×C61#54812yes—
339C22×C61C23×D411yescharacteristic
340C2×D61—42no—
341C2×D61—42no—
342C2×D61—42no—
343C4×C61—21no—
344C4×C61—21no—
345C4×C61—21no—
346C4×C61—21no—
347C4×C61—42no—
348C4×C61—42no—
349C4×C61—42no—
350C4×C61—42no—
351C4×C61#54812yes—
352C4×C61C22×#4111yescharacteristic
353C4×C61C2×2−1+411yescharacteristic
354Dic61—24no—
355Dic61—24no—
356Dic61—24no—
357Dic61—42no—
358Dic61—42no—
359#3507—611no—
360#3502—611no—
361#3377—614no—
362#3374—614no—
363#3371—614no—
364#3369—614no—
365#3137—612no—
366#3135—611no—
367#3084—612no—
368#3078D6111yescharacteristic, complemented
369#3044—612no—
370#3037—611no—
371#2961—611no—
372#2947—612no—
373C2×#441—611no—
374C2×C61—22no—
375C2×C61C2×#54811yescharacteristic
376C2×C61#358611yescharacteristic
377C2×C61#358611yescharacteristic
378D61—82no—
379C2×#137—612no—
380C22×Q16—611no—
381#559—612no—
382Q82—612no—
383Q82—1221no—
384#557—612no—
385#557—614no—
386#557—612no—
387#557—1222no—
388#557—1221no—
389#556—614no—
390#556—1221no—
391#554—1221no—
392#552—1222no—
393#544—612no—
394#544—1222no—
395#539—1222no—
396#536—612no—
397#533—611no—
398#533—1222no—
399#533C2×D6111yescharacteristic
400#532—611no—
401#529—1222no—
402#520—612no—
403#519—612no—
404C2×C4×Q8—611no—
405C2×C4×Q8C2×D6111yescharacteristic
406#491—612no—
407#491—614no—
408#490—612no—
409#489—612no—
410#489—612no—
411#489—614no—
412#488—614no—
413#487—614no—
414#479—614no—
415#479C2×D6114yes—
416#478—614no—
417#477C2×D6112yes—
418#476—614no—
419#475—614no—
420#475C2×D6112yes—
421#464—614no—
422#464—614no—
423#464—614no—
424#462—612no—
425#462—614no—
426#462—614no—
427#460—612no—
428#446—614no—
429#446—614no—
430#441—614no—
431#441—614no—
432#441—614no—
433#440—614no—
434C4×Q16—614no—
435C4×SD16—614no—
436#428C2×D6112yes—
437C2×#106—611no—
438#424—612no—
439#423C2×D6111yescharacteristic
440#419C2×D6112yes—
441#417—612no—
442#416—612no—
443#416—612no—
444C2×#103—611no—
445C2×#103—611no—
446#405—612no—
447C2×#97—611no—
448C61#6133511yescharacteristic, complemented, Sylow
449C2×#41—612no—
450C22×Q8—611no—
451C22×Q8C22×D6111yescharacteristic
452#137—1228no—
453C2×Q16—614no—
454C2×Q16—614no—
455C2×Q16—1224no—
456C2×SD16—614no—
457C2×M16C22×D6112yes—
458C22×C8—611no—
459C22×C8—611no—
460C4⋊Q8—614no—
461C4⋊Q8—614no—
462C4⋊Q8—611no—
463C4⋊Q8—611no—
464C4⋊Q8—611no—
465C4⋊Q8—1222no—
466C4⋊Q8—1222no—
467C4⋊Q8—1224no—
468C4⋊Q8—1224no—
469C4⋊Q8—1222no—
470C4⋊Q8C22×D6111yescharacteristic
471C4⋊Q8C22×D6111yescharacteristic
472#127—611no—
473#126—1224no—
474#126—1224no—
475#125—614no—
476#125—612no—
477#125—1222no—
478#125—1222no—
479#125—1224no—
480#125—1224no—
481#125—1224no—
482#125—1224no—
483#125C22×D6112yes—
484#124—614no—
485#124—612no—
486#124—1222no—
487#123—1224no—
488#123—1224no—
489C22⋊Q8—21no—
490C22⋊Q8—21no—
491C22⋊Q8—612no—
492C22⋊Q8—612no—
493C22⋊Q8—1224no—
494C22⋊Q8—1224no—
495C22⋊Q8—1222no—
496C22⋊Q8—1222no—
497#121—1221no—
498#121—1221no—
499C4×Q8—21no—
500C4×Q8—21no—
501C4×Q8—21no—
502C4×Q8—21no—
503C4×Q8—612no—
504C4×Q8—612no—
505C4×Q8—614no—
506C4×Q8—614no—
507C4×Q8—614no—
508C4×Q8—612no—
509C4×Q8—612no—
510C4×Q8—1222no—
511C4×Q8—1222no—
512C4×Q8—1222no—
513C4×Q8—1224no—
514C4×Q8—1222no—
515C4×Q8—1222no—
516C4×Q8—1221no—
517C4×Q8—1221no—
518C4×Q8C22×D6112yes—
519C4×Q8C22×D6112yes—
520C4×D4—614no—
521C4×D4—1222no—
522C4×D4—1224no—
523C4×D4—1221no—
524C4×D4—1221no—
525#117—612no—
526#117—1224no—
527#117—1224no—
528#117—1222no—
529#117C22×D6112yes—
530C2×C4⋊C4—21no—
531C2×C4⋊C4—611no—
532C2×C4⋊C4—1222no—
533C2×C4⋊C4—1221no—
534C2×C4⋊C4C22×D6111yescharacteristic
535C2×C42—21no—
536C2×C42—1221no—
537C2×C42C22×D6111yescharacteristic
538#107C22×D6114yes—
539#106—614no—
540#106C22×D6114yes—
541C4⋊C8—614no—
542C4⋊C8—614no—
543C4⋊C8C22×D6112yes—
544C4⋊C8C22×D6112yes—
545#103—614no—
546#103—614no—
547#103—614no—
548#103—612no—
549#103—612no—
550#103—612no—
551#103—612no—
552#103C22×D6114yes—
553#103C22×D6114yes—
554#102—614no—
555#102—612no—
556#102—612no—
557#102—612no—
558#102—612no—
559#97—614no—
560#97—612no—
561#97—612no—
562C4×C8—614no—
563#41—1224no—
564#41—1224no—
565C2×Q8—21no—
566C2×Q8—21no—
567C2×Q8—21no—
568C2×Q8—22no—
569C2×Q8—22no—
570C2×Q8—612no—
571C2×Q8—612no—
572C2×Q8—614no—
573C2×Q8—612no—
574C2×Q8—612no—
575C2×Q8—1224no—
576C2×Q8—1221no—
577C2×Q8—1222no—
578C2×Q8C23×D6112yes—
579C2×D4—612no—
580C2×D4—1222no—
581C2×D4—1221no—
582C2×D4—1221no—
583C22×C4—21no—
584C22×C4—21no—
585C22×C4—1224no—
586C22×C4—1221no—
587C22×C4—1221no—
588C22×C4—1222no—
589C22×C4D4×D6111yescharacteristic
590C22×C4D4×D6111yescharacteristic
591C22×C4C23×D6111yescharacteristic
592Q16—1228no—
593Q16—1228no—
594SD16—1228no—
595M16—24no—
596C2×C8—612no—
597C2×C8—612no—
598C2×C8—1222no—
599C2×C8—1222no—
600C2×C8C23×D6114yes—
601C4⋊C4—22no—
602C4⋊C4—22no—
603C4⋊C4—21no—
604C4⋊C4—22no—
605C4⋊C4—21no—
606C4⋊C4—21no—
607C4⋊C4—22no—
608C4⋊C4—612no—
609C4⋊C4—614no—
610C4⋊C4—614no—
611C4⋊C4—612no—
612C4⋊C4—612no—
613C4⋊C4—612no—
614C4⋊C4—612no—
615C4⋊C4—1224no—
616C4⋊C4—1224no—
617C4⋊C4—1224no—
618C4⋊C4—1224no—
619C4⋊C4—1224no—
620C4⋊C4—1224no—
621C4⋊C4—1224no—
622C4⋊C4—1224no—
623C4⋊C4—1221no—
624C4⋊C4—1222no—
625C4⋊C4—1222no—
626C4⋊C4—1221no—
627C4⋊C4—1221no—
628C4⋊C4C23×D6112yes—
629C4⋊C4C23×D6112yes—
630C4⋊C4C23×D6112yes—
631C22⋊C4—22no—
632C22⋊C4—614no—
633C22⋊C4—1224no—
634C22⋊C4—1224no—
635C22⋊C4—1224no—
636C22⋊C4—1222no—
637C22⋊C4—1222no—
638C42—22no—
639C42—22no—
640C42—22no—
641C42—614no—
642C42—1224no—
643C42—1224no—
644C42—1224no—
645C42—1222no—
646C42—1222no—
647C42D4×D6111yescharacteristic
648C42D4×D6111yescharacteristic
649C42C23×D6111yescharacteristic
650C42C23×D6111yescharacteristic
651C23—1222no—
652C23C2×D4×D6111yescharacteristic
653Q8—22no—
654Q8—22no—
655Q8—42no—
656Q8—42no—
657Q8—81no—
658Q8—614no—
659Q8—1222no—
660Q8—1224no—
661Q8—1222no—
662Q8—1222no—
663Q8—4884no—
664Q8—2442no—
665Q8—2442no—
666D4—1222no—
667D4—1222no—
668D4—4881no—
669D4—2442no—
670D4—2444no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—22no—
674C2×C4—22no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—81no—
679C2×C4—614no—
680C2×C4—614no—
681C2×C4—1222no—
682C2×C4—1224no—
683C2×C4—1222no—
684C2×C4—1222no—
685C2×C4—4884no—
686C2×C4—4881no—
687C2×C4—2444no—
688C2×C4C2×D4×D6111yescharacteristic
689C2×C4C2×D4×D6111yescharacteristic
690C2×C4C2×D4×D6111yescharacteristic
691C2×C4C2×D4×D6111yescharacteristic
692C2×C4C2×D4×D6111yescharacteristic
693C2×C4#41×D6112yes—
694C2×C4#1451982512yes—
695C2×C4C24×D6111yescharacteristic
696C8—24no—
697C8—1222no—
698C8—1222no—
699C22—21no—
700C22—21no—
701C22—2442no—
702C22—2442no—
703C22—2442no—
704C22#1684042212yes—
705C22C22×D4×D6111yescharacteristic
706C4—21no—
707C4—21no—
708C4—21no—
709C4—21no—
710C4—42no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—1224no—
715C4—1224no—
716C4—1224no—
717C4—2442no—
718C4—2442no—
719C4#1684042212yes—
720C4C2×#41×D6111yescharacteristic
721C4C2×#1451982511yescharacteristic
722C2—22no—
723C2—4882no—
724C2C2×#1684042211yescharacteristic
725C2#2146060211yescharacteristic
726C2#2146060311yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3078C2
socleC22×C61C23×D4
prime radical#16840312C22
hypercenter#3078D61
hypocenterC61#61335
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
2. #21459624C2semidirect
3. #21458668C2semidirect
4. #21458625C2semidirect
5. #21458590C2semidirect
6. #21458562C2semidirect
7. #21456180C2semidirect
9. #21455680C2semidirect
10. #21455348C2semidirect
12. #21454756C2semidirect
13. #21454660C2semidirect
14. C2×#16839867C2semidirect
15. C61×#3078C2semidirect
368. #3078D61semidirect
448. C61#61335semidirect

Maximal subgroups

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

subgroupindex
#214596842
#214596242
#214586682
#214586252
#214585902
#214585622
#214561802
#214561672
#214556802
#214553482
#214552932
#214547562
#214546602
C2×#168398672
C61×#30782
#6133561