Finite Group Databasebeta

← #29736593

Subgroups — #29736593

Order 15616.

1462 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840408C211yesmaximal, characteristic
2C2×#16840124C211yesmaximal, characteristic, complemented
3C2×#16840049C211yesmaximal, characteristic, complemented
4#21447015C212yesmaximal, complemented
5#21447008C212yesmaximal, complemented
6#21446751C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21445757C212yesmaximal, complemented
9#21445749C212yesmaximal, complemented
10#21445647C211yesmaximal, characteristic, complemented
11C61×#2547C211yesmaximal, 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#16840049—24no—
26C2×#14519708C2212yescomplemented
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×#124×C61—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C61×#397—21no—
42C61×#397—21no—
43C61×#391—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#14519708—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—
102#124×C61—44no—
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61C2312yes—
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—22no—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C42×C61—21no—
114C2×C42×C61C2311yescharacteristic
115#95×C61—21no—
116#95×C61—22no—
117#95×C61—22no—
118C24×C61C2×D411yescharacteristic
119C24×C61C2×D411yescharacteristic
120C23×D61—21no—
121C23×D61—21no—
122C23×D61—24no—
123C23×D61—44no—
124C23×D61—42no—
125C23×D61—42no—
126C23×D61—82no—
127C2×Q8×C61—42no—
128C2×Q8×C61—42no—
129C2×Q8×C61—42no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—41no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—24no—
149C2×C22⋊D61—24no—
150C2×C22⋊D61—44no—
151C2×C22⋊D61—82no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154#11758222—48no—
155#11758222—44no—
156#11758222—44no—
157C2×D244—21no—
158C2×D244—22no—
159C2×D244—22no—
160C2×D244—22no—
161C2×D244—24no—
162C2×D244—21no—
163C2×D244—48no—
164C2×D244—44no—
165C2×D244—44no—
166C2×D244—44no—
167C2×D244C2×D412yes—
168C2×D244C2×D412yes—
169C2×D244#4114yes—
170C2×C4×D61—24no—
171C2×C4×D61—24no—
172C2×C4×D61—44no—
173C2×C4×D61—82no—
174C2×Dic122—22no—
175C2×Dic122—22no—
176C2×Dic122—21no—
177C2×Dic122C2×D412yes—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—42no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—44no—
187C22⋊C4×C61—44no—
188C42×C61—42no—
189C42×C61C2×D412yes—
190C42×C61C2×D412yes—
191C22⋊Dic61—24no—
192#11758198—24no—
193#11758198—24no—
194#11758198—24no—
195#11758198—24no—
196C4⋊Dic61—21no—
197C4⋊Dic61C2×D412yes—
198#11758196—24no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C23×C61—21no—
206C23×C61—22no—
207C23×C61—21no—
208C23×C61—22no—
209C23×C61—21no—
210C23×C61—22no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—22no—
214C23×C61AGL(1,ℤ/8)12yes—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—24no—
218C22×D61—22no—
219C22×D61—48no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—42no—
223C22×D61—48no—
224C22×D61—42no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—42no—
228C22×D61—84no—
229C22×D61—88no—
230C22×D61—88no—
231Q8×C61—84no—
232D4×C61—44no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
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—42no—
251C2×C4×C61—42no—
252C2×C4×C61#12012yes—
253C2×C4×C61#12012yes—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C2×#4112yes—
257C2×C4×C61H(5,𝔽2)12yes—
258C22⋊D61—48no—
259C22⋊D61—84no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264D244—22no—
265D244—24no—
266D244—24no—
267D244—48no—
268D244—41no—
269D244—41no—
270D244#12414yescomplemented
271C4×D61—48no—
272C4×D61—84no—
273Dic122—24no—
274Dic122—41no—
275Dic122—41no—
276#92418—42no—
277#92418—42no—
278#25898—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—24no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—42no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61C2×#12011yescharacteristic
290C22×C61#54612yes—
291C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C61#57511yescharacteristic
293C22×C61#57712yes—
294C2×D61—24no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—44no—
298C2×D61—44no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—816no—
303C2×D61—82no—
304C2×D61—82no—
305C4×C61—22no—
306C4×C61—22no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61—82no—
310C4×C61—82no—
311C4×C61C2×#12411yescharacteristic
312C4×C61#53511yescharacteristic
313C4×C61#54612yes—
314Dic61—42no—
315Dic61—42no—
316C2×#546—611no—
317C2×#489—611no—
318C2×#476—611no—
319#2547D6111yescharacteristic, complemented
320#2035—612no—
321#2034—612no—
322#1983—611no—
323#1955—611no—
324#1788—612no—
325#1786—612no—
326#1762—611no—
327C2×C61—42no—
328C2×C61—81no—
329C2×C61#243511yescharacteristic
330C2×C61#300611yescharacteristic
331C2×C61#300911yescharacteristic
332C2×C61#330612yes—
333C2×C61#331112yes—
334D61—44no—
335D61—82no—
336C22×#41—611no—
337C23×D4—611no—
338#546—1224no—
339#546—1224no—
340C2×C4⋊Q8C2×D6111yescharacteristic, complemented
341C2×#124—21no—
342C2×#124—611no—
343C2×#123—1221no—
344C2×C22⋊Q8—611no—
345C2×#121—611no—
346C2×#121—1221no—
347C2×#120—1221no—
348C2×C4×D4—611no—
349C2×C4×D4C2×D6111yescharacteristic
350C22×C22⋊C4—1221no—
351#489—1224no—
352#476—1224no—
353C2×#106—612no—
354C2×C4⋊C8—611no—
355C2×#103—612no—
356C2×#103—612no—
357C2×#102—612no—
358C2×#102—612no—
359C2×#102—612no—
360C2×#102—612no—
361C2×C22⋊C8—612no—
362C2×C22⋊C8—612no—
363C2×#97—611no—
364#397—21no—
365#397—21no—
366#391—21no—
367#380C2×D6112yes—
368#380C2×D6112yescomplemented
369C4×C4⋊C4C2×D6111yescharacteristic, complemented
370#340—612no—
371C61#2589811yescharacteristic, complemented, Sylow
372C25—1221no—
373C2×#41—1222no—
374C2×#41—1222no—
375C2×#41—1224no—
376C2×#41—2442no—
377C22×Q8—21no—
378C22×Q8—611no—
379C22×D4—611no—
380C22×D4—611no—
381C22×D4—1222no—
382C22×D4—1222no—
383C22×D4—1224no—
384C22×D4—1221no—
385C22×D4—1221no—
386C22×D4—1221no—
387C22×D4—2442no—
388C22×D4C22⋊D6111yescharacteristic
389C23×C4—1221no—
390C23×C4—1221no—
391C23×C4C22×D6111yescharacteristic
392C23×C4C22×D6111yescharacteristic
393C22×C8—612no—
394C22×C8—612no—
395C4⋊Q8—22no—
396C4⋊Q8—22no—
397#124—44no—
398#124—614no—
399#124—1222no—
400#123—1228no—
401C22⋊Q8—1222no—
402C22⋊Q8—1222no—
403#121—1228no—
404#121—1222no—
405#121—1222no—
406#120—1228no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—614no—
410C4×D4—1222no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22⋊D6111yescharacteristic
415C2×C4⋊C4C22×D6112yes—
416C2×C4⋊C4C22×D6112yes—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—1228no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4C22⋊D6111yescharacteristic
426C2×C22⋊C4C22⋊D6111yescharacteristic
427C2×C42—21no—
428C2×C42C22×D6111yescharacteristic
429#106—1224no—
430C4⋊C8—1222no—
431#103—1224no—
432#103—2442no—
433#102—1224no—
434#102—1224no—
435#102—1224no—
436#102—2442no—
437C22⋊C8—2442no—
438C22⋊C8—2442no—
439#97—1222no—
440#95—21no—
441#95—22no—
442#95—22no—
443C24—1224no—
444C24—1221no—
445C24—1221no—
446C24—4882no—
447C24—2444no—
448C24—2442no—
449C24—2442no—
450C24C2×C22⋊D6111yescharacteristic
451C24C2×C22⋊D6111yescharacteristic
452#41—2444no—
453#41—2448no—
454#41—2444no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—612no—
459C2×Q8—1221no—
460C2×Q8—1222no—
461C2×Q8—1222no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—41no—
465C2×D4—612no—
466C2×D4—612no—
467C2×D4—614no—
468C2×D4—1224no—
469C2×D4—1224no—
470C2×D4—1224no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1222no—
474C2×D4—1222no—
475C2×D4—1222no—
476C2×D4—4882no—
477C2×D4—2444no—
478C2×D4—2444no—
479C2×D4—2448no—
480C2×D4—2444no—
481C2×D4—2444no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—1224no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1224no—
496C22×C4—4882no—
497C22×C4—2444no—
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C2×C22⋊D6111yescharacteristic
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C2×C22⋊D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503C2×C8—1222no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—1222no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—42no—
515C4⋊C4—612no—
516C4⋊C4—1224no—
517C4⋊C4—1221no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—1224no—
523C22⋊C4—1224no—
524C22⋊C4—1224no—
525C22⋊C4—1224no—
526C22⋊C4—1224no—
527C42—42no—
528C42D4×D6112yes—
529C42D4×D6112yes—
530C23—21no—
531C23—21no—
532C23—22no—
533C23—22no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—22no—
539C23—1222no—
540C23—1224no—
541C23—1222no—
542C23—4888no—
543C23—4888no—
544C23—4884no—
545C23—2448no—
546C23—2448no—
547C23—2441no—
548C23—2441no—
549C23—2442no—
550C23—2442no—
551C23—2441no—
552C23—2441no—
553C23—2442no—
554C23#1451980312yes—
555C23C22×C22⋊D6111yescharacteristic
556Q8—84no—
557Q8—1224no—
558Q8—2441no—
559Q8—2441no—
560D4—44no—
561D4—614no—
562D4—1222no—
563D4—1224no—
564D4—1224no—
565D4—4884no—
566D4—2448no—
567D4—2448no—
568D4—2441no—
569D4—2441no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—4884no—
592C2×C4—2448no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4#1451974512yes—
596C2×C4#1451974512yes—
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4#41×D6112yes—
600C2×C4#1451982312yes—
601C8—2442no—
602C8—2442no—
603C22—24no—
604C22—22no—
605C22—42no—
606C22—41no—
607C22—41no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—1224no—
614C22—48816no—
615C22—4882no—
616C22—4882no—
617C22—2442no—
618C22—2442no—
619C22—2442no—
620C22—2444no—
621C22—2444no—
622C22—2442no—
623C22—2442no—
624C22C2×#1451974511yescharacteristic
625C22#1684040012yes—
626C22C2×#1451980311yescharacteristic
627C22#1684056011yescharacteristic
628C22#1684057012yes—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—82no—
634C4—82no—
635C4—2442no—
636C4—2442no—
637C4#124×D6111yescharacteristic
638C4#1684033011yescharacteristic
639C4#1684040512yes—
640C2—42no—
641C2—81no—
642C2—4882no—
643C2—2444no—
644C2#2144946711yescharacteristic
645C2#2145506911yescharacteristic
646C2#2145509411yescharacteristic
647C2#2145777412yes—
648C2#2145784912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2547C2
socleC23×C61C22×D4
prime radicalC2×#16840408C2
hypercenter#2547D61
hypocenterC61#25898
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
2. C2×#16840124C2semidirect
3. C2×#16840049C2semidirect
4. #21447015C2semidirect
5. #21447008C2semidirect
6. #21446751C2semidirect
7. #21446618C2semidirect
8. #21445757C2semidirect
9. #21445749C2semidirect
10. #21445647C2semidirect
11. C61×#2547C2semidirect
26. C2×#14519708C22semidirect
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
270. D244#124semidirect
319. #2547D61semidirect
340. C2×C4⋊Q8C2×D61semidirect
368. #380C2×D61semidirect
369. C4×C4⋊C4C2×D61semidirect
371. C61#25898semidirect

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×#168400492
#214470152
#214470082
#214467512
#214466182
#214457572
#214457492
#214456472
C61×#25472
#2589861