Finite Group Databasebeta

← #29743120

Subgroups — #29743120

Order 15616.

848 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840060C211yesmaximal, characteristic
2C2×SD16×Dic61C211yesmaximal, characteristic
3#21449683C211yesmaximal, characteristic, complemented
4#21448729C211yesmaximal, characteristic
5#21447367C211yesmaximal, characteristic, complemented
6#21447302C211yesmaximal, characteristic, complemented
7#21447196C211yesmaximal, characteristic, complemented
8#21447156C211yesmaximal, characteristic, complemented
9#21446546C211yesmaximal, characteristic, complemented
10#21446375C211yesmaximal, characteristic, complemented
11#21446155C211yesmaximal, characteristic
12#21446080C211yesmaximal, characteristic
13#21446075C211yesmaximal, characteristic, complemented
14#21446064C211yesmaximal, characteristic
15C61×#1877C211yesmaximal, characteristic
16C2×#14519768C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20#16840060—24no—
21SD16×Dic61—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839704—21no—
34#16839694—21no—
35#16839669C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839659—21no—
40#16839658C2211yescharacteristic
41#16839649—21no—
42#16839637—21no—
43#16839637—21no—
44#16839628C2211yescharacteristic
45#16839622—21no—
46C4×C4⋊Dic61C2211yescharacteristic
47C2×#14519684—21no—
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C23×Dic61—21no—
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519768—22no—
66#14519768—41no—
67Q8×Dic61—22no—
68Q8×Dic61—41no—
69D4×Dic61—22no—
70D4×Dic61—22no—
71D4×Dic61—41no—
72D4×Dic61—41no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D411yescharacteristic
78C2×C4⋊Dic61D411yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519711—22no—
92#14519710—41no—
93#14519709—22no—
94#14519708—22no—
95#14519708—22no—
96#14519706—22no—
97#14519701—22no—
98#14519701—41no—
99#14519698—22no—
100#14519698D414yes—
101C8×Dic61—22no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684D411yescharacteristic
111#14519684D411yescharacteristic
112C22×Q8×C61C2311yescharacteristic
113C22×D4×C61C2311yescharacteristic
114C23×C4×C61—21no—
115C2×SD16×C61—22no—
116C2×SD16×C61—22no—
117C2×SD16×C61Q814yes—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—22no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#103×C61—22no—
127#102×C61—22no—
128#95×C61—21no—
129#95×C61—21no—
130C24×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61C2×Q812yes—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61C2×D412yes—
141C2×D4×C61C2×Q812yes—
142C2×D4×C61#4112yes—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—24no—
146C22×C4×C61—21no—
147C22×C4×C61—81no—
148C22×C4×C61—81no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—42no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61—81no—
161C22×Dic61—81no—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166SD16×C61—28no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61C2×Q812yes—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61#4112yes—
177C22⋊C4×C61—24no—
178C42×C61—21no—
179C42×C61—21no—
180C22⋊Dic61—42no—
181C22⋊Dic61—42no—
182C4⋊Dic61—21no—
183C4⋊Dic61—22no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—41no—
191C4⋊Dic61—42no—
192C4⋊Dic61—42no—
193C4⋊Dic61—81no—
194C4⋊Dic61—81no—
195C4⋊Dic61C2×D412yes—
196#11758196—42no—
197#11758196—42no—
198#11758196—42no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D412yes—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61C22×D411yescharacteristic
218Q8×C61—22no—
219Q8×C61—22no—
220Q8×C61—41no—
221Q8×C61—41no—
222D4×C61—22no—
223D4×C61—22no—
224D4×C61—22no—
225D4×C61C22⋊Q814yes—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—84no—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×Q811yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61H(5,𝔽2)11yescharacteristic
247C2×C4×C612−1+411yescharacteristic
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—22no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—84no—
272C2×Dic61—84no—
273C2×Dic61#13512yes—
274C8×C61—22no—
275C8×C61—41no—
276#92418—22no—
277#92418—41no—
278#26378—611nomaximal, Sylow
279C22×C61—24no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61#52511yescharacteristic
287C22×C61D4211yescharacteristic
288C22×C61D4×Q811yescharacteristic
289C22×C61C2×#13511yescharacteristic
290C22×C61#57511yescharacteristic
291C22×C61#57711yescharacteristic
292C22×C61#57711yescharacteristic
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—42no—
296C4×C61—42no—
297C4×C61—81no—
298C4×C61C2×C22⋊Q811yescharacteristic
299C4×C61#53911yescharacteristic
300C4×C61#54711yescharacteristic
301C4×C61#55411yescharacteristic
302Dic61—22no—
303Dic61—41no—
304Dic61—41no—
305Dic61—42no—
306Dic61—41no—
307Dic61—81no—
308Dic61—81no—
309Dic61—81no—
310C2×#478—611no—
311C2×C4×SD16—611no—
312#2453—611no—
313#2349—611no—
314#2097—611no—
315#2085—611no—
316#2066—611no—
317#2059—611no—
318#1941—611no—
319#1916—611no—
320#1877—611no—
321#1877D6111yescharacteristic
322#1864—611no—
323#1863—611no—
324#1862—611no—
325C2×C61—44no—
326C2×C61#244911yescharacteristic
327C2×C61#310211yescharacteristic
328C2×C61#312311yescharacteristic
329C2×C61#326711yescharacteristic
330C2×C61#328211yescharacteristic
331C2×C61#338111yescharacteristic
332C2×C61#338811yescharacteristic
333C22×SD16C2×D6111yescharacteristic
334C2×#125—611no—
335C2×C4×Q8—611no—
336C2×C4×D4—611no—
337C2×C4×D4—611no—
338C2×C4×D4C2×D6111yescharacteristic
339#478—1224no—
340C4×SD16—1224no—
341C2×#107—611no—
342C2×#107—611no—
343C2×#106—611no—
344C2×#106—611no—
345C2×#106—611no—
346C2×C4⋊C8—611no—
347C2×#103—611no—
348C2×#103—611no—
349C2×#103C2×D6111yescharacteristic
350C2×#102—611no—
351C2×#102—611no—
352C2×#102C2×D6111yescharacteristic
353C2×C4×C8—611no—
354#400—1221no—
355#398—1221no—
356#391—611no—
357#391C2×D6111yescharacteristic
358#389—611no—
359#389—611no—
360#389—611no—
361#389—611no—
362#389—1221no—
363#387—1221no—
364#385—1221no—
365#385—1221no—
366#382—611no—
367#380—1221no—
368C4×C4⋊C4—611no—
369C2×#95—1221no—
370#340—611no—
371#340—611no—
372#340—611no—
373#340C2×D6111yescharacteristic
374#336—611no—
375#336—611no—
376#336—611no—
377#336C2×D6111yescharacteristic
378C61#2637811yescharacteristic, complemented, Sylow
379C22×Q8C22×D6111yescharacteristic
380C22×D4C22×D6111yescharacteristic
381C23×C4—21no—
382C23×C4—1221no—
383C23×C4—1221no—
384C2×SD16—22no—
385C2×SD16—22no—
386C2×SD16Dic12214yes—
387C22×C8—611no—
388C22×C8—611no—
389C22×C8C22×D6111yescharacteristic
390C22×C8C22×D6111yescharacteristic
391#125—1222no—
392#125—2441no—
393C4×Q8—1222no—
394C4×Q8—2441no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—1222no—
398C4×D4—1222no—
399C4×D4—2441no—
400C4×D4—2441no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—1221no—
422C2×C42C22×D6111yescharacteristic
423#107—1222no—
424#107—2441no—
425#106—1222no—
426#106—1222no—
427#106—1222no—
428C4⋊C8—1222no—
429#103—22no—
430#103—1222no—
431#103—2441no—
432#102—22no—
433#102—614no—
434#102—1222no—
435C4×C8—1222no—
436#95—21no—
437#95—21no—
438#95—611no—
439#95—611no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448C24—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8C2×Dic12212yes—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4C2×Dic12212yes—
459C2×D4D4×D6112yes—
460C2×D4#1175822412yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—4881no—
476C22×C4—4881no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484SD16—28no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—612no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8C2×Dic12212yes—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—612no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—4881no—
510C4⋊C4—4881no—
511C4⋊C4—2441no—
512C4⋊C4—2442no—
513C4⋊C4—2442no—
514C4⋊C4—2442no—
515C4⋊C4—2442no—
516C4⋊C4—2442no—
517C4⋊C4#1175822412yes—
518C22⋊C4—24no—
519C22⋊C4—2442no—
520C22⋊C4—2442no—
521C42—21no—
522C42—21no—
523C42—1221no—
524C42—1221no—
525C42—1221no—
526C42—1221no—
527C42—2442no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23C2×D4×D6111yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540D4—22no—
541D4—22no—
542D4—22no—
543D4#1451975414yes—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—84no—
560C2×C4—612no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1222no—
573C2×C4—4884no—
574C2×C4—4884no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4C2×D4×D6111yescharacteristic
587C2×C4C22×Dic12211yescharacteristic
588C2×C4C2×#1175822411yescharacteristic
589C2×C4C2×#1175822411yescharacteristic
590C2×C4#1451982311yescharacteristic
591C2×C4#1451982411yescharacteristic
592C8—22no—
593C8—41no—
594C8—1222no—
595C8—2441no—
596C22—24no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22#1684026811yescharacteristic
604C22#1684039611yescharacteristic
605C22D4×Dic12211yescharacteristic
606C22#135×D6111yescharacteristic
607C22#1684056111yescharacteristic
608C22#1684056811yescharacteristic
609C22#1684057211yescharacteristic
610C4—42no—
611C4—41no—
612C4—42no—
613C4—41no—
614C4—81no—
615C4—1222no—
616C4—4881no—
617C4—4881no—
618C4—4881no—
619C4—2441no—
620C4—2441no—
621C4—2441no—
622C4—2442no—
623C4C2×#1451975411yescharacteristic
624C4#1684035611yescharacteristic
625C4#1684041211yescharacteristic
626C4#1684046911yescharacteristic
627C2—44no—
628C2#2144963411yescharacteristic
629C2#2145583811yescharacteristic
630C2#2145606511yescharacteristic
631C2#2145735011yescharacteristic
632C2#2145753211yescharacteristic
633C2#2145869511yescharacteristic
634C2#2145876511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1877C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1877D61
hypocenterC61#26378
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
3. #21449683C2semidirect
5. #21447367C2semidirect
6. #21447302C2semidirect
7. #21447196C2semidirect
8. #21447156C2semidirect
9. #21446546C2semidirect
10. #21446375C2semidirect
13. #21446075C2semidirect
378. C61#26378semidirect

Maximal subgroups

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

subgroupindex
C2×#168400602
C2×SD16×Dic612
#214496832
#214487292
#214473672
#214473022
#214471962
#214471562
#214465462
#214463752
#214461552
#214460802
#214460752
#214460642
C61×#18772
#2637861