Finite Group Databasebeta

← #29608415

Subgroups — #29608415

Order 15616.

2018 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839664C211yesmaximal, characteristic
2C2×#16839640C212yesmaximal
3C2×#16839633C212yesmaximal, complemented
4#21444071C212yesmaximal, complemented
5#21444063C212yesmaximal, complemented
6#21444061C212yesmaximal, complemented
7#21444046C211yesmaximal, characteristic, complemented
8#21444040C212yesmaximal, complemented
9C61×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C23×C4×D61C2212yes—
12C22×C4⋊Dic61C2211yescharacteristic
13C22×#11758196C2212yes—
14C22×#11758198C2212yes—
15C22×#11758198C2212yes—
16C22×#11758198C2212yes—
17C22×#11758198C2212yes—
18C22×#11758198C2212yes—
19#16839664C418yescomplemented
20#16839640—28no—
21#16839633—28no—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2211yescharacteristic
29C22×C4⋊C4×C61C2212yes—
30C22×C42×C61C2211yescharacteristic
31C2×#95×C61C2212yes—
32C2×#95×C61C2211yescharacteristic
33C2×#95×C61C2211yescharacteristic
34C24×D61C2312yes—
35C22×D244—42no—
36C22×D244—42no—
37C22×D244C2×C414yes—
38C22×D244D414yes—
39C22×D244Q814yes—
40C23×Dic61C2312yes—
41C23×Dic61C2312yes—
42C23×Dic61C2312yes—
43C22×C4×D61—24no—
44C22×C4×D61—24no—
45C22×C4×D61—24no—
46C22×C4×D61—42no—
47C22×C4×D61—42no—
48C22×C4×D61—42no—
49C22×C4×D61—42no—
50C2×C4⋊Dic61—42no—
51C2×C4⋊Dic61C2×C414yescomplemented
52C2×#11758196—24no—
53C2×#11758196—42no—
54C2×#11758196—42no—
55C2×#11758198—24no—
56C2×#11758198—24no—
57C2×#11758198—24no—
58C2×#11758198—24no—
59C2×#11758198—24no—
60C2×#11758198—24no—
61C2×#11758198—24no—
62C2×#11758198—42no—
63C2×#11758198—42no—
64C2×#11758198—42no—
65C2×#11758198—42no—
66C2×#11758198—42no—
67C2×#11758198—42no—
68C2×#11758198C2×C418yescomplemented
69C2×#11758198C2×C418yescomplemented
70#14519684—24no—
71#14519684—22no—
72#14519684—24no—
73#14519684—42no—
74#14519684—42no—
75#14519684—42no—
76#14519684D414yescomplemented
77C23×C4×C61C2312yes—
78C23×C4×C61C2311yescharacteristic
79C23×C4×C61C2312yes—
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C2×C4⋊C4×C61—24no—
83C2×C4⋊C4×C61—24no—
84C2×C4⋊C4×C61—42no—
85C2×C42×C61—42no—
86C2×C42×C61C2×C414yescomplemented
87#95×C61—24no—
88#95×C61—41no—
89#95×C61—41no—
90C24×C61C2411yescharacteristic
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×D61—22no—
102C23×D61—22no—
103C23×D61C22×C414yes—
104C23×D61C2×D414yes—
105C23×D61C2×Q814yes—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
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×C61C22×C412yes—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×Q812yes—
133C22×C4×C61#4114yes—
134C22×C4×C61#4112yes—
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×Dic61—22no—
154C22×Dic61C22×C414yes—
155C22×Dic61C2×D414yes—
156C2×D244—44no—
157C2×D244—44no—
158C2×D244—48no—
159C2×D244—48no—
160C2×D244C4⋊C4116yescomplemented
161C2×C4×D61—48no—
162C2×C4×D61—48no—
163C2×C4×D61—48no—
164C2×C4×D61—48no—
165C2×C4×D61—48no—
166C2×C4×D61—48no—
167C2×C4×D61—48no—
168C4⋊C4×C61—48no—
169C42×C61—44no—
170#11758198—216no—
171#11758198—48no—
172#11758198—48no—
173#11758198—48no—
174#11758198—48no—
175C4⋊Dic61—44no—
176#11758196—48no—
177C23×C61C23×C411yescharacteristic
178C23×C61C22×D411yescharacteristic
179C23×C61C22×D411yescharacteristic
180C23×C61C22×Q811yescharacteristic
181C23×C61C2×#4111yescharacteristic
182C23×C61C2×#4112yes—
183C23×C61H(5,𝔽2)12yes—
184C23×C61H(5,𝔽2)12yes—
185C23×C61H(5,𝔽2)12yes—
186C23×C61H(5,𝔽2)11yescharacteristic
187C23×C612−1+411yescharacteristic
188C22×D61—24no—
189C22×D61—24no—
190C22×D61—24no—
191C22×D61—24no—
192C22×D61—24no—
193C22×D61—24no—
194C22×D61—24no—
195C22×D61—24no—
196C22×D61—24no—
197C22×D61—24no—
198C22×D61—24no—
199C22×D61—24no—
200C22×D61—24no—
201C22×D61—24no—
202C22×D61—24no—
203C22×D61—24no—
204C22×D61—24no—
205C22×D61—24no—
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—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61C2×C4⋊C418yes—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
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—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61C2×C4⋊C414yes—
251C2×C4×C61C4×D414yes—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—24no—
257C2×Dic61—24no—
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—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273D244—416no—
274C4×D61—432no—
275#14438—611nomaximal, Sylow
276C22×C61C22×C4⋊C411yescharacteristic
277C22×C61C2×C4×D411yescharacteristic
278C22×C61C4×#4111yescharacteristic
279C22×C61#51811yescharacteristic
280C22×C61#51811yescharacteristic
281C22×C61#51811yescharacteristic
282C22×C61#52011yescharacteristic
283C22×C61#53412yes—
284C22×C61#53512yes—
285C22×C61#53812yes—
286C22×C61#54012yes—
287C22×C61#54212yes—
288C22×C61#54312yes—
289C22×C61D4211yescharacteristic
290C22×C61#54612yes—
291C22×C61#54612yes—
292C22×C61#54711yescharacteristic
293C22×C61D4×Q811yescharacteristic
294C22×C61#55312yes—
295C22×C61#55412yes—
296C22×C61#55411yescharacteristic
297C22×C61#56012yes—
298C22×C61#56212yes—
299C2×D61—28no—
300C2×D61—28no—
301C2×D61—28no—
302C2×D61—44no—
303C2×D61—44no—
304C2×D61—44no—
305C2×D61—44no—
306C2×D61—44no—
307C2×D61—44no—
308C2×D61—44no—
309C2×D61—44no—
310C2×D61—44no—
311C2×D61—44no—
312C2×D61—44no—
313C2×D61—44no—
314C4×C61—24no—
315C4×C61—24no—
316C4×C61—42no—
317C4×C61—44no—
318C4×C61—82no—
319Dic61—44no—
320Dic61—44no—
321Dic61—82no—
322C2×#390—611no—
323C2×#386—612no—
324C2×#385—612no—
325#1441—612no—
326#1439—612no—
327#1439—612no—
328C4⋊#95D6111yescharacteristic, complemented
329#1434—611no—
330C22⋊#95—612no—
331C2×C61#231511yescharacteristic
332C2×C61#232811yescharacteristic
333C2×C61#233711yescharacteristic
334C2×C61D4×C4⋊C411yescharacteristic
335C2×C61#234911yescharacteristic
336C2×C61#236411yescharacteristic
337C2×C61#237711yescharacteristic
338C2×C61#267812yes—
339C2×C61#269212yes—
340C2×C61#270112yes—
341C2×C61#278012yes—
342D61—48no—
343C23×D4—611no—
344C24×C4—612no—
345C22×C4⋊C4—611no—
346C22×C4⋊C4—612no—
347C22×C4⋊C4C2×D6112yes—
348C22×C22⋊C4—612no—
349C22×C22⋊C4—612no—
350C22×C22⋊C4—612no—
351C22×C22⋊C4—612no—
352C22×C22⋊C4—612no—
353C22×C42C2×D6111yescharacteristic
354#390—618no—
355#386—1228no—
356#385—1228no—
357C2×#95—611no—
358C2×#95—612no—
359C2×#95—611no—
360C2×#95—612no—
361C2×#95—612no—
362C2×#95—612no—
363C2×#95—612no—
364C2×#95C2×D6112yes—
365C2×#95C2×D6111yescharacteristic
366C2×#95C2×D6111yescharacteristic
367C61#1443811yescharacteristic, complemented, Sylow
368C25—612no—
369C22×D4—614no—
370C22×D4—614no—
371C22×D4—614no—
372C22×D4—2442no—
373C22×D4—2442no—
374C23×C4—612no—
375C23×C4—612no—
376C23×C4—612no—
377C23×C4—1224no—
378C23×C4—1224no—
379C23×C4—1224no—
380C23×C4—2442no—
381C23×C4—2442no—
382C23×C4—2442no—
383C23×C4—2442no—
384C23×C4C22×D6112yes—
385C23×C4C22×D6111yescharacteristic
386C23×C4C22×D6112yes—
387C23×C4C22×D6111yescharacteristic
388C23×C4C22×D6111yescharacteristic
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—42no—
392C2×C4⋊C4—614no—
393C2×C4⋊C4—1224no—
394C2×C4⋊C4—2442no—
395C2×C4⋊C4—2442no—
396C2×C4⋊C4—2442no—
397C2×C22⋊C4—618no—
398C2×C22⋊C4—618no—
399C2×C22⋊C4—1224no—
400C2×C22⋊C4—1224no—
401C2×C22⋊C4—1224no—
402C2×C22⋊C4—1224no—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1224no—
406C2×C22⋊C4—2442no—
407C2×C22⋊C4—2442no—
408C2×C22⋊C4—2442no—
409C2×C22⋊C4—2442no—
410C2×C22⋊C4—2442no—
411C2×C22⋊C4—2442no—
412C2×C42—42no—
413C2×C42C4×D6114yescomplemented
414#95—24no—
415#95—41no—
416#95—41no—
417#95—614no—
418#95—1224no—
419#95—1224no—
420#95—1222no—
421#95—2442no—
422#95—2442no—
423#95—2442no—
424C24—614no—
425C24—614no—
426C24—614no—
427C24—1222no—
428C24—1222no—
429C24—1222no—
430C24—1222no—
431C24—1222no—
432C24—1222no—
433C24—1222no—
434C24—1222no—
435C24—1222no—
436C24—1222no—
437C24—1222no—
438C24—1222no—
439C24C23×D6111yescharacteristic
440C2×D4—6116no—
441C2×D4—2444no—
442C2×D4—2444no—
443C2×D4—2448no—
444C2×D4—2448no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—614no—
467C22×C4—614no—
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×C4—1222no—
486C22×C4—1222no—
487C22×C4—2448no—
488C22×C4—2448no—
489C22×C4—2448no—
490C22×C4—2448no—
491C22×C4—2448no—
492C22×C4—2448no—
493C22×C4—2448no—
494C22×C4C2×C4×D6112yes—
495C22×C4C2×C4×D6112yes—
496C22×C4C2×C4×D6112yes—
497C22×C4#1175822214yes—
498C22×C4D4×D6112yes—
499C22×C4D4×D6112yes—
500C22×C4Q8×D6112yes—
501C22×C4#1175822612yes—
502C4⋊C4—48no—
503C4⋊C4—2444no—
504C4⋊C4—2448no—
505C22⋊C4—12216no—
506C22⋊C4—2448no—
507C22⋊C4—2448no—
508C22⋊C4—2448no—
509C22⋊C4—2448no—
510C42—44no—
511C23—618no—
512C23—1224no—
513C23—1224no—
514C23—1224no—
515C23—1224no—
516C23—1224no—
517C23—1224no—
518C23—1224no—
519C23—1224no—
520C23—1224no—
521C23—1224no—
522C23—1224no—
523C23—1224no—
524C23—1224no—
525C23—1224no—
526C23—1224no—
527C23—1224no—
528C23—1224no—
529C23—1224no—
530C23—2442no—
531C23—2442no—
532C23—2442no—
533C23—2442no—
534C23—2442no—
535C23—2442no—
536C23—2442no—
537C23—2442no—
538C23—2442no—
539C23—2442no—
540C23—2442no—
541C23—2442no—
542C23—2442no—
543C23—2442no—
544C23—2442no—
545C23—2442no—
546C23C22×C4×D6111yescharacteristic
547C23C2×D4×D6111yescharacteristic
548C23C2×D4×D6111yescharacteristic
549C23C2×Q8×D6111yescharacteristic
550C23C2×#1175822212yes—
551C23C2×#1175822611yescharacteristic
552C23#1451982212yes—
553C23#1451982211yescharacteristic
554C23#1451982312yes—
555C23#1451982312yes—
556C23#1451982511yescharacteristic
557D4—24416no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
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—1224no—
586C2×C4—1224no—
587C2×C4—1224no—
588C2×C4—1224no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—24432no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4C4⋊C4×D6114yes—
608C2×C4#1451973914yes—
609C22—1228no—
610C22—1228no—
611C22—1228no—
612C22—2444no—
613C22—2444no—
614C22—2444no—
615C22—2444no—
616C22—2444no—
617C22—2444no—
618C22—2444no—
619C22—2444no—
620C22—2444no—
621C22—2444no—
622C22—2444no—
623C22—2444no—
624C22C2×C4⋊C4×D6111yescharacteristic
625C22C2×#1451973911yescharacteristic
626C22C4×#1175822211yescharacteristic
627C22#1684023511yescharacteristic
628C22#1684023611yescharacteristic
629C22#1684023611yescharacteristic
630C22#1684024311yescharacteristic
631C22#1684032012yes—
632C22#1684033012yes—
633C22#1684034512yes—
634C22#1684036812yes—
635C22#1684038412yes—
636C22#1684038712yes—
637C22#1684039711yescharacteristic
638C22#1684040212yes—
639C22#1684040812yes—
640C22#1684041511yescharacteristic
641C22#1684042711yescharacteristic
642C22#1684046012yes—
643C22#1684047212yes—
644C22#1684047311yescharacteristic
645C22#1684050012yes—
646C22#1684050812yes—
647C4—24no—
648C4—24no—
649C4—42no—
650C4—44no—
651C4—82no—
652C4—4882no—
653C4—2444no—
654C4—2444no—
655C2—2448no—
656C2#2144843211yescharacteristic
657C2#2144854211yescharacteristic
658C2#2144863211yescharacteristic
659C2#2144871911yescharacteristic
660C2#2144873011yescharacteristic
661C2#2144886611yescharacteristic
662C2#2144897311yescharacteristic
663C2#2145214112yes—
664C2#2145231012yes—
665C2#2145242712yes—
666C2#2145346112yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×C4⋊#95C2
socle, cosocleC24×C61C24
prime radicalC23×D244C22
hypercenterC4⋊#95D61
hypocenterC61#14438
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#16839633C2semidirect
4. #21444071C2semidirect
5. #21444063C2semidirect
6. #21444061C2semidirect
7. #21444046C2semidirect
8. #21444040C2semidirect
9. C61×C4⋊#95C2semidirect
19. #16839664C4semidirect
51. C2×C4⋊Dic61C2×C4semidirect
68. C2×#11758198C2×C4semidirect
69. C2×#11758198C2×C4semidirect
76. #14519684D4semidirect
86. C2×C42×C61C2×C4semidirect
160. C2×D244C4⋊C4semidirect
328. C4⋊#95D61semidirect
367. C61#14438semidirect
413. C2×C42C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396642
C2×#168396402
C2×#168396332
#214440712
#214440632
#214440612
#214440462
#214440402
C61×C4⋊#952
#1443861