Finite Group Databasebeta

← #29737154

Subgroups — #29737154

Order 15616.

986 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840060C211yesmaximal, characteristic, complemented
2#21449526C211yesmaximal, characteristic, complemented
3#21448649C211yesmaximal, characteristic
4#21447355C211yesmaximal, characteristic, complemented
5#21447212C211yesmaximal, characteristic, complemented
6#21447196C211yesmaximal, characteristic, complemented
7#21446968C211yesmaximal, characteristic, complemented
8#21446068C211yesmaximal, characteristic, complemented
9#21446064C211yesmaximal, characteristic, complemented
10#21446018C211yesmaximal, characteristic, complemented
11#21445997C211yesmaximal, characteristic, complemented
12#21445848C211yesmaximal, characteristic, complemented
13#21445817C211yesmaximal, characteristic, complemented
14#21445759C211yesmaximal, characteristic
15C2×C61×#416C211yesmaximal, characteristic
16C2×#14519768C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19#16840060—24no—
20C2×#14519711C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×C22⋊#92418C2211yescharacteristic
31#16839698—21no—
32#16839685—21no—
33#16839669C2211yescharacteristic
34#16839669C2211yescharacteristic
35#16839661C2211yescharacteristic
36#16839660—21no—
37#16839658C2211yescharacteristic
38#16839646—21no—
39#16839642—21no—
40#16839642—21no—
41#16839631C2211yescharacteristic
42#16839625—21no—
43#16839622C2211yescharacteristic
44#16839622C2211yescharacteristic
45C4×C22⋊Dic61C2211yescharacteristic
46C2×#14519684—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×#41×C61C2211yescharacteristic
54C23×C8×C61C2211yescharacteristic
55C2×#117×C61C2211yescharacteristic
56C61×#416—24no—
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C23×Dic61—21no—
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519768—22no—
66#14519768—41no—
67D4×Dic61—22no—
68D4×Dic61—22no—
69D4×Dic61—41no—
70D4×Dic61—41no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61D411yescharacteristic
86C2×C22⋊Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519711—22no—
90#14519711—41no—
91#14519710—41no—
92#14519708—22no—
93#14519706—22no—
94#14519706—41no—
95#14519701—41no—
96#14519701—41no—
97#14519698—22no—
98#14519698—22no—
99C22⋊#92418—22no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C2×#41×C61—22no—
111C2×#41×C61—22no—
112C2×#41×C61—22no—
113C2×#41×C61—41no—
114C2×#41×C61—41no—
115C2×#41×C61Q814yes—
116C22×Q8×C61C2311yescharacteristic
117C22×D4×C61—21no—
118C22×D4×C61C2311yescharacteristic
119C23×C4×C61—21no—
120C23×C4×C61C2311yescharacteristic
121C22×C8×C61—22no—
122C22×C8×C61—22no—
123C22×C8×C61—41no—
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126#117×C61—22no—
127#117×C61—41no—
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C22⋊C4×C61—21no—
131C2×C42×C61—21no—
132#103×C61—22no—
133#103×C61—22no—
134#102×C61—22no—
135#102×C61D414yes—
136C24×C61—21no—
137C24×C61C2×D411yescharacteristic
138#41×C61—28no—
139#41×C61—44no—
140#41×C61—44no—
141#41×C61—44no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×Q812yes—
148C2×D4×C61—24no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—42no—
153C2×D4×C61—42no—
154C2×D4×C61C2×D412yes—
155C2×D4×C61C2×Q812yes—
156C2×D4×C61#4112yes—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—24no—
164C22×C4×C61—21no—
165C22×C4×C61—42no—
166C22×C4×C61—42no—
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×Q812yes—
169C22×C4×C61#4112yes—
170C22×C4×C61C2411yescharacteristic
171C22×Dic61—21no—
172C22×Dic61—21no—
173C22×Dic61—21no—
174C22×Dic61—21no—
175C22×Dic61—42no—
176C22×Dic61—42no—
177C22×Dic61—42no—
178C22×Dic61—42no—
179C22×Dic61—81no—
180C22×Dic61—81no—
181C22×Dic61C2×D411yescharacteristic
182C22×Dic61C2×D411yescharacteristic
183C22×Dic61C2×D411yescharacteristic
184C22×Dic61C2×D411yescharacteristic
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—44no—
191C2×C8×C61C2×D412yes—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61C2×D412yes—
196C22⋊C4×C61—42no—
197C42×C61—42no—
198C22⋊Dic61—22no—
199C22⋊Dic61—41no—
200C22⋊Dic61—42no—
201C22⋊Dic61—42no—
202C22⋊Dic61—81no—
203C22⋊Dic61—81no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—42no—
213C4⋊Dic61—42no—
214#11758196—42no—
215#11758196—42no—
216#11758196—81no—
217#11758196—81no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61—21no—
231C23×C61—22no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61—21no—
236C23×C61—21no—
237C23×C61—41no—
238C23×C61—41no—
239C23×C61—41no—
240C23×C61—41no—
241C23×C61C22⋊Q812yes—
242C23×C61C22×D411yescharacteristic
243Q8×C61—22no—
244Q8×C61—22no—
245Q8×C61—41no—
246Q8×C61—41no—
247D4×C61—22no—
248D4×C61—22no—
249D4×C61—22no—
250D4×C61—48no—
251D4×C61C22⋊Q814yes—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—48no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61C22⋊Q812yes—
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×Q811yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61H(5,𝔽2)11yescharacteristic
274C2×C4×C61H(5,𝔽2)11yescharacteristic
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—21no—
285C2×Dic61—21no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—41no—
296C2×Dic61—41no—
297C2×Dic61—41no—
298C2×Dic61—84no—
299C2×Dic61—84no—
300C2×Dic61#13512yes—
301C8×C61—22no—
302C8×C61—41no—
303#92418—41no—
304#92418—41no—
305#25956—611nomaximal, Sylow
306C22×C61—22no—
307C22×C61—22no—
308C22×C61—22no—
309C22×C61—24no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61—42no—
313C22×C61—41no—
314C22×C61—41no—
315C22×C61—42no—
316C22×C61—42no—
317C22×C61—41no—
318C22×C61—41no—
319C22×C61—42no—
320C22×C61C2×C22⋊Q811yescharacteristic
321C22×C61D4211yescharacteristic
322C22×C61#54611yescharacteristic
323C22×C61C2×#13511yescharacteristic
324C22×C61#57511yescharacteristic
325C22×C61#57611yescharacteristic
326C22×C61#57811yescharacteristic
327C4×C61—22no—
328C4×C61—42no—
329C4×C61—42no—
330C4×C61—81no—
331C4×C61C2×C22⋊Q811yescharacteristic
332C4×C61#54311yescharacteristic
333C4×C61#54611yescharacteristic
334C4×C61#55111yescharacteristic
335Dic61—22no—
336Dic61—41no—
337Dic61—41no—
338Dic61—41no—
339Dic61—81no—
340Dic61—81no—
341Dic61—81no—
342Dic61—81no—
343C2×#478—611no—
344C2×#416D6111yescharacteristic
345#2441—611no—
346#2339—611no—
347#2095—611no—
348#2068—611no—
349#2066—611no—
350#2026—611no—
351#1862—611no—
352#1862—611no—
353#1854—611no—
354#1850—611no—
355#1808—611no—
356#1802—611no—
357#1788—611no—
358C2×C61—42no—
359C2×C61—44no—
360C2×C61#243711yescharacteristic
361C2×C61#307611yescharacteristic
362C2×C61#308311yescharacteristic
363C2×C61#326811yescharacteristic
364C2×C61#328311yescharacteristic
365C2×C61#329711yescharacteristic
366C2×C61#331511yescharacteristic
367C22×#41C2×D6111yescharacteristic
368C23×C8C2×D6111yescharacteristic
369C2×#125—611no—
370C2×C4×D4—611no—
371C2×C4×D4—611no—
372C2×#117C2×D6111yescharacteristic
373#478—1224no—
374C2×#107—611no—
375C2×#107—611no—
376C2×#107—611no—
377C2×#106—611no—
378C2×C4⋊C8—611no—
379C2×C4⋊C8—611no—
380#416—24no—
381C2×#103—611no—
382C2×#103—611no—
383C2×#103C2×D6111yescharacteristic
384C2×#103C2×D6111yescharacteristic
385C2×#102—611no—
386C2×#102—611no—
387C2×#102C2×D6111yescharacteristic
388C2×#102C2×D6111yescharacteristic
389C2×C22⋊C8—611no—
390#399—1221no—
391#396—1221no—
392#391—611no—
393#391—611no—
394#389—611no—
395#389—611no—
396#389—1221no—
397#387—1221no—
398#386—1221no—
399#386—1221no—
400#384—611no—
401#381—1221no—
402#380—611no—
403#380—611no—
404C4×C22⋊C4—611no—
405C2×#95—1221no—
406#340—611no—
407#340—611no—
408#340—611no—
409#340—611no—
410#336—611no—
411#336—611no—
412C61#2595611yescharacteristic, complemented, Sylow
413C2×#41—22no—
414C2×#41—22no—
415C2×#41—22no—
416C2×#41—41no—
417C2×#41—41no—
418C2×#41Dic12214yes—
419C22×Q8C22×D6111yescharacteristic
420C22×D4—21no—
421C22×D4C22×D6111yescharacteristic
422C23×C4—21no—
423C23×C4—1221no—
424C23×C4—1221no—
425C23×C4C22×D6111yescharacteristic
426C22×C8—22no—
427C22×C8—22no—
428C22×C8—41no—
429C22×C8—611no—
430C22×C8—611no—
431C22×C8C22×D6111yescharacteristic
432C22×C8C22×D6111yescharacteristic
433#125—1222no—
434#125—2441no—
435C4×D4—1222no—
436C4×D4—1222no—
437C4×D4—2441no—
438C4×D4—2441no—
439#117—22no—
440#117—41no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4C22×D6111yescharacteristic
452C2×C4⋊C4C22×D6111yescharacteristic
453C2×C22⋊C4—21no—
454C2×C22⋊C4—611no—
455C2×C22⋊C4—611no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C42—21no—
461C2×C42—611no—
462C2×C42—611no—
463#107—1222no—
464#107—2441no—
465#107—2441no—
466#106—1222no—
467C4⋊C8—1222no—
468C4⋊C8—2441no—
469#103—22no—
470#103—22no—
471#103—2441no—
472#103—2441no—
473#102—22no—
474#102—1222no—
475#102—1222no—
476#102C22⋊D6114yes—
477C22⋊C8—1222no—
478#95—611no—
479#95—611no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487#95—1221no—
488C24—21no—
489C24D4×D6111yescharacteristic
490#41—28no—
491#41—44no—
492#41—44no—
493#41—44no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8C2×Dic12212yes—
500C2×D4—24no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—42no—
505C2×D4—42no—
506C2×D4C2×Dic12212yes—
507C2×D4#1175822212yes—
508C2×D4C2×C22⋊D6112yes—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—24no—
516C22×C4—21no—
517C22×C4—42no—
518C22×C4—42no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—611no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—4881no—
528C22×C4—4881no—
529C22×C4—2442no—
530C22×C4—2442no—
531C22×C4—2442no—
532C22×C4—2442no—
533C22×C4C2×Dic12212yes—
534C22×C4D4×D6111yescharacteristic
535C22×C4#1175822412yes—
536C22×C4C23×D6111yescharacteristic
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—44no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C2×C8C2×C22⋊D6112yes—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4—4881no—
562C4⋊C4—4881no—
563C4⋊C4—2442no—
564C4⋊C4—2442no—
565C4⋊C4—2442no—
566C4⋊C4—2442no—
567C4⋊C4C2×C22⋊D6112yes—
568C22⋊C4—42no—
569C22⋊C4—1222no—
570C22⋊C4—4881no—
571C22⋊C4—4881no—
572C22⋊C4—2441no—
573C22⋊C4—2442no—
574C22⋊C4—2442no—
575C42—42no—
576C42—1221no—
577C42—1221no—
578C42—1221no—
579C42—1221no—
580C23—22no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—22no—
587C23—22no—
588C23—21no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23#1451975412yes—
594C23C2×D4×D6111yescharacteristic
595Q8—22no—
596Q8—22no—
597Q8—41no—
598Q8—41no—
599D4—22no—
600D4—22no—
601D4—22no—
602D4—48no—
603D4#1451975614yes—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—48no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—612no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—1221no—
631C2×C4—1221no—
632C2×C4—4884no—
633C2×C4—4884no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4#1451975412yes—
647C2×C4C22×C22⋊D6111yescharacteristic
648C2×C4C22×Dic12211yescharacteristic
649C2×C4C2×#1175822411yescharacteristic
650C2×C4C2×#1175822211yescharacteristic
651C2×C4#1451982211yescharacteristic
652C2×C4#1451982211yescharacteristic
653C8—22no—
654C8—41no—
655C8—2441no—
656C8—2441no—
657C22—22no—
658C22—24no—
659C22—22no—
660C22—22no—
661C22—42no—
662C22—41no—
663C22—42no—
664C22—41no—
665C22—42no—
666C22—42no—
667C22—41no—
668C22—42no—
669C22—41no—
670C22—42no—
671C22C2×#1451975411yescharacteristic
672C22D4×C22⋊D6111yescharacteristic
673C22#1684040211yescharacteristic
674C22#135×D6111yescharacteristic
675C22#1684055911yescharacteristic
676C22#1684056411yescharacteristic
677C22#1684057611yescharacteristic
678C4—22no—
679C4—42no—
680C4—42no—
681C4—81no—
682C4—1222no—
683C4—4881no—
684C4—4881no—
685C4—4881no—
686C4—4881no—
687C4—2441no—
688C4—2441no—
689C4—2441no—
690C4C2×#1451975611yescharacteristic
691C4#1684038811yescharacteristic
692C4#1684040611yescharacteristic
693C4#1684043811yescharacteristic
694C2—42no—
695C2—44no—
696C2#2144948011yescharacteristic
697C2#2145560311yescharacteristic
698C2#2145566811yescharacteristic
699C2#2145735911yescharacteristic
700C2#2145753511yescharacteristic
701C2#2145767011yescharacteristic
702C2#2145791111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#416C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
hypercenterC2×#416D61
hypocenterC61#25956
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
1. C2×#16840060C2semidirect
2. #21449526C2semidirect
4. #21447355C2semidirect
5. #21447212C2semidirect
6. #21447196C2semidirect
7. #21446968C2semidirect
8. #21446068C2semidirect
9. #21446064C2semidirect
10. #21446018C2semidirect
11. #21445997C2semidirect
12. #21445848C2semidirect
13. #21445817C2semidirect
412. C61#25956semidirect

Maximal subgroups

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

subgroupindex
C2×#168400602
#214495262
#214486492
#214473552
#214472122
#214471962
#214469682
#214460682
#214460642
#214460182
#214459972
#214458482
#214458172
#214457592
C2×C61×#4162
#2595661