Finite Group Databasebeta

← #30558406

Subgroups — #30558406

Order 15616.

2020 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459599C211yesmaximal, characteristic, complemented
2#21457984C214yesmaximal, complemented
3#21457909C214yesmaximal, complemented
4#21457864C214yesmaximal, complemented
5#21457789C214yesmaximal, complemented
6C2×#16840132C211yesmaximal, characteristic, complemented
7#21455788C212yesmaximal, complemented
8#21455637C211yesmaximal, characteristic, complemented
9#21455512C212yesmaximal, complemented
10#21455306C211yesmaximal, characteristic, complemented
11#21455131C212yesmaximal, complemented
12#21454617C211yesmaximal, characteristic, complemented
13#21454537C211yesmaximal, characteristic, complemented
14#21454349C212yesmaximal, complemented
15C61×#3512C211yesmaximal, characteristic, complemented
16C2×#14519824C2212yes—
17C2×#14519807C2212yes—
18C2×#14519797C2211yescharacteristic
19C2×#14519784C2212yes—
20D4×Dic122C2214yes—
21#16840417C2214yes—
22#16840412—24no—
23#16840333C2212yes—
24#16840332—22no—
25#16840310—21no—
26C2×#14519777C2211yescharacteristic
27C4⋊Q8×D61—21no—
28C4×#11758222C2211yescharacteristic
29#16840132C2214yes—
30#16840132C2214yes—
31#16840103C2214yes—
32#16840103C2214yes—
33#16840097C2214yes—
34#16840092C2214yes—
35#16840066C2214yes—
36#16840059C2214yes—
37#16840018C2214yes—
38#16840006C2214yes—
39#16840001C2214yes—
40#16839989C2214yes—
41#16839972C2214yes—
42#16839967C2214yes—
43#16839922C2214yes—
44#16839917C2214yes—
45#16839905C2214yes—
46#16839900C2214yes—
47#16839872C2212yes—
48#16839867C2212yes—
49#16839862C2212yes—
50#16839860C2212yes—
51#16839834C2214yes—
52#16839834C2214yes—
53#16839802C2212yes—
54#16839798C2211yescharacteristic
55#16839786C2211yescharacteristic
56#16839770C2214yes—
57#16839770C2212yes—
58#16839761C2212yes—
59#16839761C2214yes—
60#16839758C2211yescharacteristic
61#16839756C2211yescharacteristic
62#16839756C2214yes—
63C2×#14519700C2212yes—
64C2×#14519700C2214yes—
65C2×#14519700C2212yes—
66#16839729C2212yes—
67C2×#14519688C2211yescharacteristic
68C61×#553—24no—
69C61×#553C2214yes—
70C61×#552C2214yes—
71C61×#541—21no—
72C61×#540—21no—
73C61×#537C2212yes—
74C61×#532C2211yescharacteristic
75C61×#529—22no—
76C2×#125×C61C2211yescharacteristic
77C61×#520C2212yes—
78C4×#41×C61C2211yescharacteristic
79#14519824—28no—
80#14519824—44no—
81C2×#11758222C2314yes—
82C2×#11758222C2311yescharacteristic
83C2×#11758224—22no—
84C2×#11758224—24no—
85C22×Dic122C2312yes—
86C22×Dic122C2314yes—
87C2×Q8×D61—22no—
88#14519807—44no—
89#14519797—24no—
90C2×#11758202C2311yescharacteristic
91C2×#11758202C2314yes—
92C2×#11758200C2311yescharacteristic
93C2×#11758200C2314yes—
94C2×#11758201C2311yescharacteristic
95C2×#11758199C2311yescharacteristic
96#14519784—44no—
97C2×#11758194C2312yes—
98C2×#11758194C2314yes—
99C22×#92418C2311yescharacteristic
100C22×#92418C2311yescharacteristic
101C22×#92418C2314yes—
102#14519777—21no—
103#14519777—21no—
104#14519777C2314yes—
105#14519776—24no—
106#14519775—22no—
107#14519764—22no—
108#14519759—24no—
109#14519759—24no—
110#14519758—24no—
111#14519756C2314yes—
112#14519756C2314yes—
113#14519754—24no—
114#14519754—24no—
115#14519748—24no—
116C4×Dic122C2312yes—
117C4×D244C2312yes—
118D4×Dic61—24no—
119C4×C22⋊D61—22no—
120#14519736C2311yescharacteristic
121#14519734—21no—
122C2×C4⋊Dic61—22no—
123C4⋊C4×D61—24no—
124C42×D61—21no—
125#14519710C2314yes—
126#14519710C2312yes—
127#14519708C2314yes—
128#14519708C2312yes—
129C4⋊#92418C2314yes—
130#14519701C2312yes—
131#14519700C2314yes—
132#14519700C2314yes—
133#14519700C2314yes—
134#14519700C2314yes—
135#14519700C2312yes—
136#14519700C2314yes—
137#14519700C2314yes—
138#14519700C2314yes—
139#14519698C2312yes—
140#14519697C2312yes—
141#14519697C2314yes—
142C22⋊#92418C2314yes—
143C22⋊#92418C2314yes—
144#14519688C2312yes—
145#14519688C2312yes—
146C2×#41×C61C2311yescharacteristic
147C2×#41×C61C2312yes—
148C4⋊Q8×C61D411yescharacteristic
149#127×C61D411yescharacteristic
150#126×C61—24no—
151#126×C61—24no—
152#125×C61—24no—
153#125×C61—21no—
154#125×C61—21no—
155#125×C61C2314yes—
156#124×C61D412yes—
157#123×C61—24no—
158#123×C61—24no—
159C22⋊Q8×C61—22no—
160C22⋊Q8×C61C2314yes—
161#121×C61—22no—
162#121×C61—24no—
163#121×C61—24no—
164#121×C61D414yes—
165#121×C61C2314yes—
166C4×Q8×C61—22no—
167C4×Q8×C61C2312yes—
168C4×D4×C61—22no—
169C4×D4×C61—24no—
170C4×D4×C61—22no—
171C4×D4×C61—24no—
172C4×D4×C61C2312yes—
173C4×D4×C61C2314yes—
174#117×C61—24no—
175#117×C61—21no—
176#117×C61C2314yes—
177#117×C61C2311yescharacteristic
178C2×C4⋊C4×C61—22no—
179C2×C4⋊C4×C61C2312yes—
180C2×C4⋊C4×C61C2314yes—
181C2×C42×C61—21no—
182C2×C42×C61C2311yescharacteristic
183#41×C61—24no—
184#41×C61—24no—
185#41×C61—22no—
186#41×C61—22no—
187C2×Q8×C61C2×D412yes—
188C2×Q8×C61C2411yescharacteristic
189C2×D4×C61—21no—
190C2×D4×C61—24no—
191C2×D4×C61C2×D414yes—
192C2×D4×C61C2×D412yes—
193C2×D4×C61C2×D414yes—
194C2×D4×C61C2411yescharacteristic
195C22×C4×C61—22no—
196C22×C4×C61—24no—
197C22×C4×C61—21no—
198C22×C4×C61—22no—
199C22×C4×C61C2×D412yes—
200C22×C4×C61C2414yes—
201C22×C4×C61C2411yescharacteristic
202C2×C22⋊D61—21no—
203C2×C22⋊D61—24no—
204C22×Dic61—22no—
205C22×Dic61—24no—
206Q8×D61—48no—
207#11758224—48no—
208#11758224—416no—
209#11758222—22no—
210#11758222—28no—
211#11758222—44no—
212#11758222#4114yes—
213C2×D244C2411yescharacteristic
214C2×C4×D61—21no—
215C2×C4×D61—24no—
216C2×C4×D61—22no—
217C2×Dic122—24no—
218C2×Dic122—24no—
219C2×Dic122—28no—
220C2×Dic122—24no—
221C2×Dic122—22no—
222C2×Dic122—44no—
223C2×Dic122C2414yes—
224C2×Dic122C2411yescharacteristic
225C4⋊C4×C61—24no—
226C4⋊C4×C61—24no—
227C4⋊C4×C61—22no—
228C4⋊C4×C61—24no—
229C4⋊C4×C61—22no—
230C4⋊C4×C61—24no—
231C4⋊C4×C61C2×D414yes—
232C4⋊C4×C61C2412yes—
233C4⋊C4×C61C2414yes—
234C22⋊C4×C61—24no—
235C22⋊C4×C61—21no—
236C22⋊C4×C61—24no—
237C22⋊C4×C61—22no—
238C22⋊C4×C61—24no—
239C22⋊C4×C61C2×D414yes—
240C22⋊C4×C61C2×D414yes—
241C42×C61—22no—
242C42×C61—21no—
243C42×C61—24no—
244C42×C61C2×D411yescharacteristic
245C42×C61C2×D411yescharacteristic
246C42×C61C2412yes—
247C22⋊Dic61—21no—
248C22⋊Dic61—24no—
249C22⋊Dic61—24no—
250#11758202—22no—
251#11758202—44no—
252#11758201—22no—
253#11758200—22no—
254#11758200—44no—
255#11758199—22no—
256#11758198—22no—
257C4⋊Dic61—24no—
258C4⋊Dic61—24no—
259C4⋊Dic61C2412yes—
260#11758196—24no—
261#11758196—24no—
262#11758196—22no—
263C4×Dic61—21no—
264C4×Dic61—22no—
265#11758194—24no—
266#11758194—44no—
267C2×#92418—22no—
268C2×#92418—22no—
269C2×#92418—44no—
270C2×#92418C2411yescharacteristic
271C2×#92418C2411yescharacteristic
272C2×#92418C2412yes—
273C2×#92418C2414yes—
274C23×C61—21no—
275C23×C61C22×D414yes—
276C23×C61C22×D411yescharacteristic
277C22×D61—21no—
278Q8×C61—24no—
279Q8×C61—21no—
280Q8×C61—21no—
281D4×C61—28no—
282D4×C61—24no—
283D4×C61—21no—
284D4×C61—21no—
285D4×C61—44no—
286D4×C61—42no—
287D4×C61—84no—
288C2×C4×C61—24no—
289C2×C4×C61—24no—
290C2×C4×C61—22no—
291C2×C4×C61—21no—
292C2×C4×C61—24no—
293C2×C4×C61—42no—
294C2×C4×C61—42no—
295C2×C4×C61—82no—
296C2×C4×C61—84no—
297C2×C4×C61—82no—
298C2×C4×C61C22×D412yes—
299C2×C4×C61C22×D414yes—
300C2×C4×C61C22×D412yes—
301C2×C4×C61C22×D411yescharacteristic
302C2×C4×C61C2×#4112yes—
303C2×C4×C61H(5,𝔽2)12yes—
304C2×C4×C61C2511yescharacteristic
305C22⋊D61—24no—
306C22⋊D61—48no—
307C2×Dic61—21no—
308C2×Dic61—22no—
309C2×Dic61—24no—
310C2×Dic61—48no—
311C2×Dic61—44no—
312D244—21no—
313D244C2×#4112yes—
314C4×D61—24no—
315C4×D61—48no—
316C4×D61—44no—
317Dic122—21no—
318Dic122—24no—
319Dic122—24no—
320Dic122—48no—
321Dic122—44no—
322Dic122C2×#4112yes—
323#92418—21no—
324#92418—21no—
325#92418—22no—
326#92418—24no—
327#61224—611nomaximal, Sylow
328C22×C61—22no—
329C22×C61—24no—
330C22×C61—24no—
331C22×C61—24no—
332C22×C61—42no—
333C22×C61—41no—
334C22×C61#54612yes—
335C22×C61C23×D411yescharacteristic
336C2×D61—22no—
337C2×D61—42no—
338C4×C61—22no—
339C4×C61—22no—
340C4×C61—24no—
341C4×C61—42no—
342C4×C61—44no—
343C4×C61—41no—
344C4×C61#54612yes—
345C4×C61C22×#4111yescharacteristic
346C4×C61C2×H(5,𝔽2)11yescharacteristic
347Dic61—22no—
348Dic61—42no—
349Dic61—44no—
350#3512D6111yescharacteristic, complemented
351#3499—611no—
352#3320—614no—
353#3315—614no—
354#3312—614no—
355#3307—614no—
356C2×#491—611no—
357#3097—612no—
358#3080—611no—
359#3063—612no—
360#3039—611no—
361#3013—612no—
362#2943—611no—
363#2932—611no—
364#2898—612no—
365C2×C61—22no—
366C2×C61—44no—
367C2×C61—81no—
368C2×C61C2×#54611yescharacteristic
369C2×C61#358612yes—
370D61—42no—
371C2×2−1+4—612no—
372C2×#137—612no—
373C2×#135—611no—
374C2×#131—612no—
375#553—24no—
376#553C2×D6114yes—
377#552C2×D6114yes—
378D4×Q8—614no—
379#547—614no—
380#547—1224no—
381#541—21no—
382#540—21no—
383#537C2×D6112yes—
384#536—612no—
385#536—1222no—
386#532—1221no—
387#532C2×D6111yescharacteristic
388C2×C4⋊Q8—611no—
389C2×C4⋊Q8—1221no—
390#529—22no—
391C2×#125C2×D6111yescharacteristic
392#520C2×D6112yes—
393C4×#41—611no—
394C4×#41C2×D6111yescharacteristic
395#491—614no—
396#491—614no—
397#485—614no—
398#485—614no—
399#484—614no—
400#483—614no—
401#479—614no—
402#478—614no—
403#470—614no—
404#468—614no—
405#467—614no—
406#465—614no—
407#462—614no—
408#461—614no—
409#452—614no—
410#451—614no—
411#449—614no—
412#448—614no—
413#442—612no—
414#441—612no—
415#440—612no—
416#440—612no—
417#435—614no—
418#435—614no—
419#428—612no—
420#427—611no—
421#423—611no—
422#419—612no—
423#419—614no—
424#417—612no—
425#417—614no—
426#416—611no—
427#416—611no—
428#416—614no—
429C2×#103—612no—
430C2×#103—612no—
431C2×#103—614no—
432#408—612no—
433C2×#97—611no—
434C61#6122411yescharacteristic, complemented, Sylow
4352−1+4—1228no—
4362−1+4—2444no—
437C2×#41—611no—
438C2×#41—614no—
439C2×#41—1222no—
440C2×#41—1224no—
441C2×#41C22×D6111yescharacteristic
442C2×#41C22×D6112yes—
443C22×Q8—612no—
444C22×Q8—614no—
445C22×Q8—1222no—
446#137—2444no—
447#135—1224no—
448C2×Q16—611no—
449C2×Q16—614no—
450C2×SD16—611no—
451C2×SD16—611no—
452C2×SD16—614no—
453C2×D8—611no—
454#131—2444no—
455C2×M16—612no—
456C2×M16—614no—
457C22×C8—611no—
458C22×C8—611no—
459C22×C8—614no—
460C4⋊Q8—614no—
461C4⋊Q8—1222no—
462C4⋊Q8—1224no—
463C4⋊Q8—1221no—
464C4⋊Q8—1221no—
465C4⋊Q8C22⋊D6111yescharacteristic
466#127C22⋊D6111yescharacteristic
467#126—24no—
468#126—24no—
469#125—24no—
470#125—21no—
471#125—21no—
472#125C22×D6114yes—
473#124—1222no—
474#124C22⋊D6112yes—
475#123—24no—
476#123—24no—
477#123—1224no—
478#123—1224no—
479#123—1224no—
480C22⋊Q8—22no—
481C22⋊Q8—614no—
482C22⋊Q8—614no—
483C22⋊Q8—1224no—
484C22⋊Q8—1224no—
485C22⋊Q8C22×D6114yes—
486#121—22no—
487#121—24no—
488#121—24no—
489#121—1224no—
490#121C22⋊D6114yes—
491#121C22×D6114yes—
492C4×Q8—22no—
493C4×Q8—612no—
494C4×Q8C22×D6112yes—
495C4×D4—22no—
496C4×D4—24no—
497C4×D4—22no—
498C4×D4—24no—
499C4×D4—612no—
500C4×D4—1224no—
501C4×D4—1222no—
502C4×D4C22×D6112yes—
503C4×D4C22×D6114yes—
504#117—24no—
505#117—21no—
506#117—611no—
507#117—1221no—
508#117C22×D6111yescharacteristic
509#117C22×D6114yes—
510C2×C4⋊C4—22no—
511C2×C4⋊C4—1222no—
512C2×C4⋊C4—1224no—
513C2×C4⋊C4C22×D6112yes—
514C2×C4⋊C4C22×D6114yes—
515C2×C42—21no—
516C2×C42—1221no—
517C2×C42C22×D6111yescharacteristic
518#107—612no—
519#107—614no—
520#106—612no—
521#106—614no—
522C4⋊C8—614no—
523#103—612no—
524#103—612no—
525#103—614no—
526#103—614no—
527#103—614no—
528#103—614no—
529#103—614no—
530#103—614no—
531#103—614no—
532#102—612no—
533#102—612no—
534#102—614no—
535C22⋊C8—614no—
536C22⋊C8—614no—
537#97—612no—
538#97—612no—
539#41—24no—
540#41—24no—
541#41—22no—
542#41—22no—
543#41—614no—
544#41—1228no—
545#41—1222no—
546#41—2444no—
547#41—2448no—
548#41—24416no—
549C2×Q8—611no—
550C2×Q8—614no—
551C2×Q8—1228no—
552C2×Q8—1224no—
553C2×Q8—1222no—
554C2×Q8—1224no—
555C2×Q8—1224no—
556C2×Q8—2444no—
557C2×Q8—2448no—
558C2×Q8C2×C22⋊D6112yes—
559C2×Q8C23×D6111yescharacteristic
560C2×D4—21no—
561C2×D4—24no—
562C2×D4—611no—
563C2×D4—1221no—
564C2×D4—1224no—
565C2×D4C2×C22⋊D6114yes—
566C2×D4C2×C22⋊D6112yes—
567C2×D4C2×C22⋊D6114yes—
568C2×D4C23×D6111yescharacteristic
569C22×C4—22no—
570C22×C4—24no—
571C22×C4—21no—
572C22×C4—22no—
573C22×C4—1221no—
574C22×C4—1222no—
575C22×C4—1224no—
576C22×C4—1222no—
577C22×C4—1224no—
578C22×C4C2×C22⋊D6112yes—
579C22×C4C23×D6114yes—
580C22×C4C23×D6111yescharacteristic
581Q16—1222no—
582Q16—2444no—
583SD16—1222no—
584SD16—1222no—
585SD16—2444no—
586D8—1222no—
587M16—1224no—
588M16—2444no—
589C2×C8—611no—
590C2×C8—611no—
591C2×C8—612no—
592C2×C8—614no—
593C2×C8—1222no—
594C2×C8—1222no—
595C2×C8—2444no—
596C4⋊C4—24no—
597C4⋊C4—24no—
598C4⋊C4—22no—
599C4⋊C4—24no—
600C4⋊C4—22no—
601C4⋊C4—24no—
602C4⋊C4—612no—
603C4⋊C4—1224no—
604C4⋊C4—1224no—
605C4⋊C4—1224no—
606C4⋊C4—1224no—
607C4⋊C4—1222no—
608C4⋊C4C2×C22⋊D6114yes—
609C4⋊C4C23×D6112yes—
610C4⋊C4C23×D6114yes—
611C22⋊C4—24no—
612C22⋊C4—21no—
613C22⋊C4—24no—
614C22⋊C4—22no—
615C22⋊C4—24no—
616C22⋊C4—1221no—
617C22⋊C4—1224no—
618C22⋊C4—1224no—
619C22⋊C4—1222no—
620C22⋊C4C2×C22⋊D6114yes—
621C22⋊C4C2×C22⋊D6114yes—
622C42—22no—
623C42—21no—
624C42—24no—
625C42—1221no—
626C42—1222no—
627C42C2×C22⋊D6111yescharacteristic
628C42C2×C22⋊D6111yescharacteristic
629C42C23×D6112yes—
630C23—21no—
631C23—1221no—
632C23C22×C22⋊D6114yes—
633C23C22×C22⋊D6111yescharacteristic
634Q8—24no—
635Q8—21no—
636Q8—21no—
637Q8—612no—
638Q8—1221no—
639Q8—1224no—
640Q8—1224no—
641Q8—2448no—
642Q8—2444no—
643D4—28no—
644D4—21no—
645D4—21no—
646D4—24no—
647D4—44no—
648D4—42no—
649D4—84no—
650D4—612no—
651D4—1224no—
652D4—1221no—
653D4—2448no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—22no—
657C2×C4—21no—
658C2×C4—24no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—82no—
662C2×C4—84no—
663C2×C4—82no—
664C2×C4—1221no—
665C2×C4—1224no—
666C2×C4—1222no—
667C2×C4—1224no—
668C2×C4—2448no—
669C2×C4—2444no—
670C2×C4—2448no—
671C2×C4—2444no—
672C2×C4C22×C22⋊D6112yes—
673C2×C4C22×C22⋊D6114yes—
674C2×C4C22×C22⋊D6112yes—
675C2×C4C22×C22⋊D6111yescharacteristic
676C2×C4#41×D6112yes—
677C2×C4#1451982312yes—
678C2×C4C24×D6111yescharacteristic
679C8—1221no—
680C8—1221no—
681C8—1222no—
682C8—1224no—
683C22—22no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—41no—
688C22—42no—
689C22—1222no—
690C22—2442no—
691C22#1684040012yes—
692C22C23×C22⋊D6111yescharacteristic
693C4—24no—
694C4—22no—
695C4—22no—
696C4—41no—
697C4—42no—
698C4—44no—
699C4—1222no—
700C4—2444no—
701C4—2442no—
702C4#1684040012yes—
703C4C2×#41×D6111yescharacteristic
704C4C2×#1451982311yescharacteristic
705C2—22no—
706C2—44no—
707C2—81no—
708C2—2442no—
709C2C2×#1684040011yescharacteristic
710C2#2146060512yes—
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3512C2
socleC22×C61C23×D4
hypercenter#3512D61
hypocenterC61#61224
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #532 ◁ #3512

Complements

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

NH ≅ G/Ntype
1. #21459599C2semidirect
2. #21457984C2semidirect
3. #21457909C2semidirect
4. #21457864C2semidirect
5. #21457789C2semidirect
6. C2×#16840132C2semidirect
7. #21455788C2semidirect
8. #21455637C2semidirect
9. #21455512C2semidirect
10. #21455306C2semidirect
11. #21455131C2semidirect
12. #21454617C2semidirect
13. #21454537C2semidirect
14. #21454349C2semidirect
15. C61×#3512C2semidirect
350. #3512D61semidirect
434. C61#61224semidirect

Maximal subgroups

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

subgroupindex
#214595992
#214579842
#214579092
#214578642
#214577892
C2×#168401322
#214557882
#214556372
#214555122
#214553062
#214551312
#214546172
#214545372
#214543492
C61×#35122
#6122461