Finite Group Databasebeta

← #29611085

Subgroups — #29611085

Order 15616.

1494 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839663C211yesmaximal, characteristic
2#21444072C211yesmaximal, characteristic, complemented
3#21444067C212yesmaximal, complemented
4#21444067C212yesmaximal, complemented
5#21444065C212yesmaximal, complemented
6#21444061C212yesmaximal, complemented
7#21444059C212yesmaximal
8#21444044C212yesmaximal, complemented
9C61×#1441C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C22×C4⋊Dic61C2211yescharacteristic
12C22×#11758198C2212yes—
13C22×#11758198C2212yes—
14C22×#11758198C2212yes—
15C22×#11758198C2212yes—
16C22×#11758198C2212yes—
17C22×#11758198C2212yes—
18C22×C4×Dic61C2211yescharacteristic
19#16839663C418yescomplemented
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2211yescharacteristic
27C22×C4⋊C4×C61C2211yescharacteristic
28C22×C4⋊C4×C61C2211yescharacteristic
29C2×#95×C61C2212yes—
30C2×#95×C61C2212yes—
31C2×#95×C61C2211yescharacteristic
32C24×D61C2312yes—
33C22×D244—22no—
34C22×D244—22no—
35C22×D244—41no—
36C22×D244—41no—
37C22×D244—41no—
38C22×D244—41no—
39C22×D244C2×C414yes—
40C23×Dic61C2312yes—
41C23×Dic61C2312yes—
42C23×Dic61C2312yes—
43C2×C4⋊Dic61—42no—
44C2×C4⋊Dic61—41no—
45C2×#11758198—42no—
46C2×#11758198—42no—
47C2×#11758198—42no—
48C2×#11758198—42no—
49C2×#11758198—42no—
50C2×#11758198—42no—
51C2×#11758198—42no—
52C2×#11758198—42no—
53C2×#11758198—42no—
54C2×#11758198—42no—
55C2×#11758198—42no—
56C2×#11758198—42no—
57C2×#11758198—42no—
58C2×#11758198—42no—
59C2×#11758198—42no—
60C2×#11758198—42no—
61C2×#11758198C2×C418yes—
62C2×#11758198C2×C418yes—
63C2×C4×Dic61—41no—
64C2×C4×Dic61—41no—
65C2×C4×Dic61C2×C414yes—
66#14519684—24no—
67#14519684—42no—
68#14519684—42no—
69#14519684—42no—
70#14519684—42no—
71#14519684—41no—
72#14519684—42no—
73C23×C4×C61C2312yes—
74C23×C4×C61C2312yes—
75C23×C4×C61C2311yescharacteristic
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C2×C4⋊C4×C61—41no—
79C2×C4⋊C4×C61—41no—
80C2×C4⋊C4×C61—41no—
81C2×C4⋊C4×C61—42no—
82C2×C4⋊C4×C61C2×C414yes—
83#95×C61—24no—
84#95×C61—41no—
85#95×C61—42no—
86C24×C61C2411yescharacteristic
87C23×D61—22no—
88C23×D61—22no—
89C23×D61—22no—
90C23×D61—22no—
91C23×D61—22no—
92C23×D61—22no—
93C23×D61—22no—
94C23×D61—22no—
95C23×D61—22no—
96C23×D61—22no—
97C23×D61—22no—
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61C22×C414yes—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C22×C412yes—
131C22×C4×C61#4112yes—
132C22×C4×C61#4112yes—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61C22×C414yes—
154C2×D244—28no—
155C2×D244—44no—
156C2×D244—44no—
157C2×D244—44no—
158C2×D244—44no—
159C2×D244—44no—
160C2×D244—44no—
161C4⋊C4×C61—44no—
162C4⋊C4×C61—44no—
163#11758198—48no—
164#11758198—48no—
165#11758198—48no—
166#11758198—48no—
167#11758198—48no—
168#11758198—48no—
169C4⋊Dic61—44no—
170C4×Dic61—44no—
171C23×C61C23×C411yescharacteristic
172C23×C61C2×#4111yescharacteristic
173C23×C61C2×#4111yescharacteristic
174C23×C61H(5,𝔽2)11yescharacteristic
175C23×C61H(5,𝔽2)11yescharacteristic
176C23×C61H(5,𝔽2)11yescharacteristic
177C23×C61H(5,𝔽2)11yescharacteristic
178C23×C61H(5,𝔽2)12yes—
179C23×C61H(5,𝔽2)12yes—
180C23×C61H(5,𝔽2)12yes—
181C23×C612−1+412yes—
182C22×D61—24no—
183C22×D61—24no—
184C22×D61—24no—
185C22×D61—24no—
186C22×D61—24no—
187C22×D61—24no—
188C22×D61—24no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61—42no—
216C22×D61—42no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61#11714yes—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269D244—416no—
270#14667—611nomaximal, Sylow
271C22×C61C2×#11711yescharacteristic
272C22×C61#51812yes—
273C22×C61#51811yescharacteristic
274C22×C61#51811yescharacteristic
275C22×C61#52012yes—
276C22×C61#53811yescharacteristic
277C22×C61#53811yescharacteristic
278C22×C61#53811yescharacteristic
279C22×C61#53811yescharacteristic
280C22×C61#54012yes—
281C22×C61#54012yes—
282C22×C61#54212yes—
283C22×C61#54212yes—
284C22×C61#56012yes—
285C22×C61#56011yescharacteristic
286C22×C61#56012yes—
287C22×C61#56011yescharacteristic
288C22×C61#56012yes—
289C22×C61H(3,𝔽4)11yescharacteristic
290C22×C61#56212yes—
291C22×C61#56212yes—
292C22×C61#56212yes—
293C22×C61#56311yescharacteristic
294C2×D61—44no—
295C2×D61—44no—
296C2×D61—44no—
297C2×D61—44no—
298C2×D61—44no—
299C2×D61—44no—
300C2×D61—44no—
301C2×D61—82no—
302C2×D61—82no—
303C2×D61—82no—
304C2×D61—82no—
305C2×D61—82no—
306C2×D61—82no—
307C2×D61—82no—
308C2×D61—82no—
309C4×C61—24no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—82no—
313C4×C61—82no—
314Dic61—44no—
315Dic61—82no—
316Dic61—82no—
317C2×#390—611no—
318#1441—611no—
319#1441D6111yescharacteristic, complemented
320#1440—612no—
321#1440—612no—
322#1439—612no—
323#1439—612no—
324#1438—612no—
325#1434—612no—
326C2×C61#231211yescharacteristic
327C2×C61#231912yes—
328C2×C61#237712yes—
329C2×C61#237911yescharacteristic
330C2×C61#238011yescharacteristic
331C2×C61#282811yescharacteristic
332C2×C61#282811yescharacteristic
333C2×C61#282912yes—
334C2×C61#282912yes—
335C2×C61#283011yescharacteristic
336C2×C61#283011yescharacteristic
337D61—84no—
338C23×D4—611no—
339C22×C4⋊C4—611no—
340C22×C4⋊C4C2×D6111yescharacteristic
341C22×C4⋊C4C2×D6111yescharacteristic
342C22×C22⋊C4—612no—
343C22×C22⋊C4—612no—
344C22×C22⋊C4—612no—
345C22×C22⋊C4—612no—
346C22×C22⋊C4—612no—
347C22×C22⋊C4—612no—
348C22×C42—611no—
349#390—618no—
350C2×#95—612no—
351C2×#95—612no—
352C2×#95—612no—
353C2×#95—612no—
354C2×#95—612no—
355C2×#95—612no—
356C2×#95—611no—
357C2×#95C2×D6112yes—
358C2×#95C2×D6112yes—
359C2×#95C2×D6111yescharacteristic
360C61#1466711yescharacteristic, complemented, Sylow
361C25—612no—
362C22×D4—614no—
363C22×D4—1222no—
364C22×D4—1222no—
365C22×D4—2441no—
366C22×D4—2441no—
367C22×D4—2441no—
368C22×D4—2441no—
369C23×C4—612no—
370C23×C4—612no—
371C23×C4—612no—
372C23×C4C22×D6112yes—
373C23×C4C22×D6112yes—
374C23×C4C22×D6111yescharacteristic
375C23×C4C22×D6111yescharacteristic
376C23×C4C22×D6111yescharacteristic
377C2×C4⋊C4—41no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4—42no—
380C2×C4⋊C4—41no—
381C2×C4⋊C4—2441no—
382C2×C4⋊C4—2442no—
383C2×C4⋊C4C4×D6114yes—
384C2×C22⋊C4—618no—
385C2×C22⋊C4—618no—
386C2×C22⋊C4—2442no—
387C2×C22⋊C4—2442no—
388C2×C22⋊C4—2442no—
389C2×C22⋊C4—2442no—
390C2×C22⋊C4—2442no—
391C2×C22⋊C4—2442no—
392C2×C22⋊C4—2442no—
393C2×C22⋊C4—2442no—
394C2×C22⋊C4—2442no—
395C2×C22⋊C4—2442no—
396C2×C22⋊C4—2442no—
397C2×C22⋊C4—2442no—
398C2×C22⋊C4—2442no—
399C2×C22⋊C4—2442no—
400C2×C22⋊C4—2442no—
401C2×C22⋊C4—2442no—
402C2×C42—614no—
403C2×C42—2441no—
404C2×C42—2441no—
405#95—24no—
406#95—42no—
407#95—41no—
408#95—1224no—
409#95—2441no—
410#95—2442no—
411#95—2442no—
412#95—2442no—
413#95—2442no—
414#95—2442no—
415C24—614no—
416C24—1222no—
417C24—1222no—
418C24—1222no—
419C24—1222no—
420C24—1222no—
421C24—1222no—
422C24—1222no—
423C24—1222no—
424C24—1222no—
425C24—1222no—
426C24—1222no—
427C24—1222no—
428C24—1222no—
429C24—1222no—
430C24C23×D6111yescharacteristic
431C2×D4—1228no—
432C2×D4—2444no—
433C2×D4—2444no—
434C2×D4—2444no—
435C2×D4—2444no—
436C2×D4—2444no—
437C2×D4—2444no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—614no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4C2×C4×D6112yes—
486C22×C4C2×C4×D6112yes—
487C22×C4C2×C4×D6112yes—
488C22×C4#1175822412yes—
489C22×C4#1175822612yes—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—2444no—
493C22⋊C4—2448no—
494C22⋊C4—2448no—
495C22⋊C4—2448no—
496C22⋊C4—2448no—
497C22⋊C4—2448no—
498C22⋊C4—2448no—
499C42—2444no—
500C23—1224no—
501C23—1224no—
502C23—1224no—
503C23—1224no—
504C23—1224no—
505C23—1224no—
506C23—1224no—
507C23—2442no—
508C23—2442no—
509C23—2442no—
510C23—2442no—
511C23—2442no—
512C23—2442no—
513C23—2442no—
514C23—2442no—
515C23—2442no—
516C23—2442no—
517C23—2442no—
518C23—2442no—
519C23—2442no—
520C23—2442no—
521C23—2442no—
522C23—2442no—
523C23—2442no—
524C23—2442no—
525C23—2442no—
526C23—2442no—
527C23—2442no—
528C23—2442no—
529C23—2442no—
530C23—2442no—
531C23—2442no—
532C23—2442no—
533C23—2442no—
534C23—2442no—
535C23C22×C4×D6111yescharacteristic
536C23C2×#1175822411yescharacteristic
537C23C2×#1175822611yescharacteristic
538C23#1451982212yes—
539C23#1451982211yescharacteristic
540C23#1451982211yescharacteristic
541C23#1451982312yes—
542C23#1451982311yescharacteristic
543C23#1451982311yescharacteristic
544C23#1451982312yes—
545C23#1451982512yes—
546D4—24416no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—1224no—
578C2×C4—1224no—
579C2×C4—1224no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4#1451973514yes—
599C22—4882no—
600C22—4882no—
601C22—4882no—
602C22—4882no—
603C22—4882no—
604C22—4882no—
605C22—4882no—
606C22—4882no—
607C22—2444no—
608C22—2444no—
609C22—2444no—
610C22—2444no—
611C22—2444no—
612C22—2444no—
613C22—2444no—
614C22C2×#1451973511yescharacteristic
615C22#1684023412yes—
616C22#1684023511yescharacteristic
617C22#1684023611yescharacteristic
618C22#1684024412yes—
619C22#1684034911yescharacteristic
620C22#1684034911yescharacteristic
621C22#1684035211yescharacteristic
622C22#1684035211yescharacteristic
623C22#1684036512yes—
624C22#1684036812yes—
625C22#1684038512yes—
626C22#1684038612yes—
627C22#1684050012yes—
628C22#1684050111yescharacteristic
629C22#1684050212yes—
630C22#1684050211yescharacteristic
631C22#1684050312yes—
632C22#1684050411yescharacteristic
633C22#1684050812yes—
634C22#1684050812yes—
635C22#1684051012yes—
636C22#1684051111yescharacteristic
637C4—24no—
638C4—42no—
639C4—42no—
640C4—82no—
641C4—82no—
642C4—4882no—
643C4—4882no—
644C4—2444no—
645C2—4884no—
646C2#2144841611yescharacteristic
647C2#2144845812yes—
648C2#2144897412yes—
649C2#2144899111yescharacteristic
650C2#2144899411yescharacteristic
651C2#2145394211yescharacteristic
652C2#2145394311yescharacteristic
653C2#2145395312yes—
654C2#2145395712yes—
655C2#2145396611yescharacteristic
656C2#2145396711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1441C2
socle, cosocleC24×C61C24
prime radicalC23×D244C22
hypercenter#1441D61
hypocenterC61#14667
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
2. #21444072C2semidirect
3. #21444067C2semidirect
4. #21444067C2semidirect
5. #21444065C2semidirect
6. #21444061C2semidirect
8. #21444044C2semidirect
9. C61×#1441C2semidirect
19. #16839663C4semidirect
319. #1441D61semidirect
360. C61#14667semidirect

Maximal subgroups

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

subgroupindex
C2×#168396632
#214440722
#214440672
#214440652
#214440612
#214440592
#214440442
C61×#14412
#1466761