Finite Group Databasebeta

← #29736582

Subgroups — #29736582

Order 15616.

1442 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840408C211yesmaximal, characteristic
2C2×#16840124C211yesmaximal, characteristic, complemented
3C2×#16840039C211yesmaximal, characteristic, complemented
4#21447022C212yesmaximal, complemented
5#21447001C212yesmaximal, complemented
6#21446751C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21445757C212yesmaximal, complemented
9#21445749C212yesmaximal, complemented
10#21445647C211yesmaximal, characteristic, complemented
11C61×#2563C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×D244C2211yescharacteristic
14#16840408—24no—
15#16840408—24no—
16C2×#14519765C2211yescharacteristic
17C2×#14519762—21no—
18C2×#14519756C2211yescharacteristic
19C2×#14519751C2211yescharacteristic
20C2×#14519750—21no—
21C2×#14519745—21no—
22C2×C4×D244C2211yescharacteristic
23C2×C22⋊C4×D61—21no—
24#16840124—24no—
25#16840039—24no—
26C2×#14519710C2212yescomplemented
27C2×C4⋊#92418C2211yescharacteristic, complemented
28C2×#14519700C2212yescomplemented
29C2×#14519700C2212yescomplemented
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×#14519697C2212yes—
33C2×#14519697C2212yes—
34C2×C22⋊#92418C2212yes—
35C2×C22⋊#92418C2212yescomplemented
36C2×#14519688C2211yescharacteristic, complemented
37#16839524C2212yescomplemented
38C2×C4⋊Q8×C61C2211yescharacteristic, complemented
39C2×C4×D4×C61C2211yescharacteristic
40C61×#393—21no—
41C61×#393—21no—
42C61×#390—21no—
43C61×#382—21no—
44C61×#380C2212yes—
45C61×#380C2212yescomplemented
46C4×C4⋊C4×C61C2211yescharacteristic, complemented
47C24×D61—21no—
48C2×#11758222—22no—
49C2×#11758222—24no—
50C2×#11758222—22no—
51C2×#11758222—42no—
52C22×Dic122C2311yescharacteristic
53C22×D244—21no—
54C22×D244—22no—
55C22×D244—22no—
56C22×D244—24no—
57C22×D244—42no—
58C22×D244C2311yescharacteristic
59C22×D244C2311yescharacteristic
60C22×C22⋊D61—21no—
61C22×C22⋊D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×#92418C2312yes—
65C22×#92418C2312yes—
66#14519765—22no—
67#14519765D414yes—
68#14519762—28no—
69#14519756—22no—
70#14519756—22no—
71#14519751—22no—
72#14519751—22no—
73#14519750—28no—
74#14519745—28no—
75C4×D244—22no—
76C4×D244D414yes—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C22⋊C4×D61—28no—
85#14519710—24no—
86C4⋊#92418—22no—
87#14519700—24no—
88#14519700—42no—
89#14519697—24no—
90#14519697—24no—
91#14519697—24no—
92#14519697—42no—
93C22⋊#92418—42no—
94C22⋊#92418—42no—
95#14519688—22no—
96C22×Q8×C61—21no—
97C22×D4×C61D411yescharacteristic
98C23×C4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100C4⋊Q8×C61—22no—
101C4⋊Q8×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61C2312yes—
107C2×C4⋊C4×C61C2312yes—
108C2×C22⋊C4×C61—22no—
109C2×C22⋊C4×C61—22no—
110C2×C22⋊C4×C61D411yescharacteristic
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C42×C61—21no—
113C2×C42×C61C2311yescharacteristic
114#95×C61—21no—
115#95×C61—22no—
116#95×C61—22no—
117C24×C61C2×D411yescharacteristic
118C24×C61C2×D411yescharacteristic
119C23×D61—21no—
120C23×D61—21no—
121C23×D61—24no—
122C23×D61—44no—
123C23×D61—42no—
124C23×D61—42no—
125C23×D61—82no—
126C2×Q8×C61—42no—
127C2×Q8×C61—82no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—41no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C2×C22⋊D61—24no—
147C2×C22⋊D61—24no—
148C2×C22⋊D61—44no—
149C2×C22⋊D61—82no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152#11758222—48no—
153#11758222—44no—
154#11758222—44no—
155C2×D244—21no—
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—24no—
160C2×D244—21no—
161C2×D244—48no—
162C2×D244—44no—
163C2×D244—44no—
164C2×D244—44no—
165C2×D244C2×D412yes—
166C2×D244C2×D412yes—
167C2×D244#4114yes—
168C2×C4×D61—24no—
169C2×C4×D61—24no—
170C2×C4×D61—44no—
171C2×C4×D61—82no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—21no—
175C2×Dic122C2×D412yes—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—42no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—82no—
185C22⋊C4×C61—82no—
186C42×C61—42no—
187C42×C61C2×D412yes—
188C42×C61C2×D412yes—
189C22⋊Dic61—24no—
190#11758198—24no—
191#11758198—24no—
192#11758198—24no—
193#11758198—24no—
194C4⋊Dic61—21no—
195C4⋊Dic61C2×D412yes—
196#11758196—24no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61—21no—
206C23×C61—22no—
207C23×C61—21no—
208C23×C61—22no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—22no—
212C23×C61#13712yes—
213C23×C61C22×D411yescharacteristic
214C22×D61—22no—
215C22×D61—24no—
216C22×D61—22no—
217C22×D61—48no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—42no—
221C22×D61—48no—
222C22×D61—42no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—42no—
226C22×D61—84no—
227C22×D61—88no—
228C22×D61—88no—
229Q8×C61—84no—
230D4×C61—44no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61#12012yes—
251C2×C4×C61#12012yes—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4112yes—
255C2×C4×C61H(5,𝔽2)12yes—
256C22⋊D61—48no—
257C22⋊D61—84no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262D244—22no—
263D244—24no—
264D244—24no—
265D244—48no—
266D244—41no—
267D244—41no—
268D244#12414yes—
269C4×D61—48no—
270C4×D61—84no—
271Dic122—24no—
272Dic122—41no—
273Dic122—41no—
274#92418—42no—
275#92418—42no—
276#25897—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—24no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—42no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61C2×#12011yescharacteristic
288C22×C61#54612yes—
289C22×C61C2×#13711yescharacteristic
290C22×C61#57511yescharacteristic
291C22×C61#57612yes—
292C2×D61—24no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—44no—
296C2×D61—44no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—816no—
301C2×D61—82no—
302C2×D61—82no—
303C4×C61—22no—
304C4×C61—22no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—82no—
308C4×C61—82no—
309C4×C61C2×#12411yescharacteristic
310C4×C61#53511yescharacteristic
311C4×C61#54612yes—
312Dic61—42no—
313Dic61—42no—
314C2×#546—611no—
315C2×#489—611no—
316C2×#474—611no—
317#2563D6111yescharacteristic, complemented
318#2036—612no—
319#2033—612no—
320#1983—611no—
321#1955—611no—
322#1788—612no—
323#1786—612no—
324#1762—611no—
325C2×C61—42no—
326C2×C61—81no—
327C2×C61#243511yescharacteristic
328C2×C61#300711yescharacteristic
329C2×C61#300811yescharacteristic
330C2×C61#331412yes—
331C2×C61#331912yes—
332D61—44no—
333D61—82no—
334C22×#41—611no—
335C23×D4—611no—
336#546—1224no—
337#546—1224no—
338C2×C4⋊Q8C2×D6111yescharacteristic, complemented
339C2×#124—611no—
340C2×#123—1221no—
341C2×C22⋊Q8—611no—
342C2×#121—611no—
343C2×#121—1221no—
344C2×#120—1221no—
345C2×C4×D4—611no—
346C2×C4×D4C2×D6111yescharacteristic
347C22×C22⋊C4—1221no—
348#489—1224no—
349#474—1224no—
350C2×#107—612no—
351C2×C4⋊C8—611no—
352C2×#103—612no—
353C2×#103—612no—
354C2×#102—612no—
355C2×#102—612no—
356C2×#102—612no—
357C2×#102—612no—
358C2×C22⋊C8—612no—
359C2×C22⋊C8—612no—
360C2×#97—611no—
361#393—21no—
362#393—21no—
363#390—21no—
364#382—21no—
365#380C2×D6112yes—
366#380C2×D6112yescomplemented
367C4×C4⋊C4C2×D6111yescharacteristic, complemented
368#340—612no—
369C61#2589711yescharacteristic, complemented, Sylow
370C25—1221no—
371C2×#41—1222no—
372C2×#41—1222no—
373C2×#41—1224no—
374C2×#41—2442no—
375C22×Q8—21no—
376C22×Q8—611no—
377C22×D4—611no—
378C22×D4—611no—
379C22×D4—1222no—
380C22×D4—1222no—
381C22×D4—1224no—
382C22×D4—1221no—
383C22×D4—1221no—
384C22×D4—1221no—
385C22×D4—2442no—
386C22×D4C22⋊D6111yescharacteristic
387C23×C4—1221no—
388C23×C4—1221no—
389C23×C4C22×D6111yescharacteristic
390C23×C4C22×D6111yescharacteristic
391C22×C8—612no—
392C22×C8—612no—
393C4⋊Q8—22no—
394C4⋊Q8—22no—
395#124—614no—
396#124—1222no—
397#123—1228no—
398C22⋊Q8—1222no—
399C22⋊Q8—1222no—
400#121—1228no—
401#121—1222no—
402#121—1222no—
403#120—1228no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—614no—
407C4×D4—1222no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4C22⋊D6111yescharacteristic
412C2×C4⋊C4C22×D6112yes—
413C2×C4⋊C4C22×D6112yes—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—1228no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4C22⋊D6111yescharacteristic
423C2×C22⋊C4C22⋊D6111yescharacteristic
424C2×C42—21no—
425C2×C42C22×D6111yescharacteristic
426#107—1224no—
427C4⋊C8—1222no—
428#103—1224no—
429#103—2442no—
430#102—1224no—
431#102—1224no—
432#102—1224no—
433#102—2442no—
434C22⋊C8—2442no—
435C22⋊C8—2442no—
436#97—1222no—
437#95—21no—
438#95—22no—
439#95—22no—
440C24—1224no—
441C24—1221no—
442C24—1221no—
443C24—4882no—
444C24—2444no—
445C24—2442no—
446C24—2442no—
447C24C2×C22⋊D6111yescharacteristic
448C24C2×C22⋊D6111yescharacteristic
449#41—2444no—
450#41—2448no—
451#41—2444no—
452C2×Q8—42no—
453C2×Q8—82no—
454C2×Q8—612no—
455C2×Q8—1221no—
456C2×Q8—1222no—
457C2×Q8—1222no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—41no—
461C2×D4—612no—
462C2×D4—612no—
463C2×D4—614no—
464C2×D4—1224no—
465C2×D4—1224no—
466C2×D4—1224no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—4882no—
473C2×D4—2444no—
474C2×D4—2444no—
475C2×D4—2448no—
476C2×D4—2444no—
477C2×D4—2444no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—1224no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1224no—
492C22×C4—4882no—
493C22×C4—2444no—
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C23×D6111yescharacteristic
499C2×C8—1222no—
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—1222no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—612no—
512C4⋊C4—1224no—
513C4⋊C4—1221no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—1224no—
519C22⋊C4—1224no—
520C22⋊C4—1224no—
521C22⋊C4—1224no—
522C22⋊C4—1224no—
523C42—42no—
524C42D4×D6112yes—
525C42D4×D6112yes—
526C23—21no—
527C23—21no—
528C23—22no—
529C23—22no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—22no—
535C23—1222no—
536C23—1224no—
537C23—1222no—
538C23—4888no—
539C23—4888no—
540C23—4884no—
541C23—2448no—
542C23—2448no—
543C23—2441no—
544C23—2441no—
545C23—2442no—
546C23—2442no—
547C23—2441no—
548C23—2441no—
549C23—2442no—
550C23#1451980812yes—
551C23C22×C22⋊D6111yescharacteristic
552Q8—84no—
553Q8—1224no—
554Q8—2441no—
555Q8—2441no—
556D4—44no—
557D4—614no—
558D4—1222no—
559D4—1224no—
560D4—1224no—
561D4—4884no—
562D4—2448no—
563D4—2448no—
564D4—2441no—
565D4—2441no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—4884no—
588C2×C4—2448no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4#1451974512yes—
592C2×C4#1451974512yes—
593C2×C4C2×D4×D6111yescharacteristic
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4#41×D6112yes—
596C2×C4#1451982312yes—
597C8—2442no—
598C8—2442no—
599C22—22no—
600C22—24no—
601C22—42no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—41no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—1224no—
610C22—48816no—
611C22—4882no—
612C22—4882no—
613C22—2442no—
614C22—2442no—
615C22—2442no—
616C22—2444no—
617C22—2442no—
618C22—2444no—
619C22—2442no—
620C22C2×#1451974511yescharacteristic
621C22#1684040012yes—
622C22C2×#1451980811yescharacteristic
623C22#1684056011yescharacteristic
624C22#1684056512yes—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—82no—
630C4—82no—
631C4—2442no—
632C4—2442no—
633C4#124×D6111yescharacteristic
634C4#1684033011yescharacteristic
635C4#1684040512yes—
636C2—42no—
637C2—81no—
638C2—4882no—
639C2—2444no—
640C2#2144946711yescharacteristic
641C2#2145507711yescharacteristic
642C2#2145508511yescharacteristic
643C2#2145789412yes—
644C2#2145796912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2563C2
socleC23×C61C22×D4
prime radicalC2×#16840408C2
hypercenter#2563D61
hypocenterC61#25897
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
2. C2×#16840124C2semidirect
3. C2×#16840039C2semidirect
4. #21447022C2semidirect
5. #21447001C2semidirect
6. #21446751C2semidirect
7. #21446618C2semidirect
8. #21445757C2semidirect
9. #21445749C2semidirect
10. #21445647C2semidirect
11. C61×#2563C2semidirect
26. C2×#14519710C22semidirect
27. C2×C4⋊#92418C22semidirect
28. C2×#14519700C22semidirect
29. C2×#14519700C22semidirect
35. C2×C22⋊#92418C22semidirect
36. C2×#14519688C22semidirect
37. #16839524C22semidirect
38. C2×C4⋊Q8×C61C22semidirect
45. C61×#380C22semidirect
46. C4×C4⋊C4×C61C22semidirect
317. #2563D61semidirect
338. C2×C4⋊Q8C2×D61semidirect
366. #380C2×D61semidirect
367. C4×C4⋊C4C2×D61semidirect
369. C61#25897semidirect

Maximal subgroups

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

subgroupindex
C2×#168404082
C2×#168401242
C2×#168400392
#214470222
#214470012
#214467512
#214466182
#214457572
#214457492
#214456472
C61×#25632
#2589761