Finite Group Databasebeta

← #29748152

Subgroups — #29748152

Order 15616.

790 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
1#21450174C211yesmaximal, characteristic
2#21448829C211yesmaximal, characteristic, complemented
3#21447134C211yesmaximal, characteristic, complemented
4#21447021C211yesmaximal, characteristic, complemented
5#21446821C211yesmaximal, characteristic
6#21446777C211yesmaximal, characteristic, complemented
7#21446500C211yesmaximal, characteristic, complemented
8#21446440C211yesmaximal, characteristic, complemented
9#21446426C211yesmaximal, characteristic, complemented
10#21446410C211yesmaximal, characteristic, complemented
11#21446172C211yesmaximal, characteristic, complemented
12#21446135C211yesmaximal, characteristic
13#21445904C211yesmaximal, characteristic, complemented
14#103×Dic61C211yesmaximal, characteristic, complemented
15C61×#2026C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519753C2211yescharacteristic
18C2×#14519752C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×C4⋊C4×D61C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839688—21no—
31#16839679—21no—
32#16839664C2211yescharacteristic
33#16839663—21no—
34#16839663C2211yescharacteristic
35#16839659—21no—
36#16839658C2211yescharacteristic
37#16839652—21no—
38#16839644—21no—
39#16839636—21no—
40#16839619C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C4×#11758198C2211yescharacteristic
44#16839609—21no—
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#125×C61C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C61×#391C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×D244—21no—
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61C2311yescharacteristic
63C2×#11758201—22no—
64C2×#11758201—41no—
65C2×#11758201—41no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519753—22no—
69#14519753—22no—
70#14519752—22no—
71#14519752—41no—
72Q8×Dic61—22no—
73Q8×Dic61—41no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C4⋊C4×D61—22no—
82C4⋊C4×D61—41no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198D411yescharacteristic
91C2×#11758198D411yescharacteristic
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61C2311yescharacteristic
96#14519711—22no—
97C4⋊#92418—41no—
98#14519701—22no—
99#14519699—22no—
100#14519699—22no—
101#14519697—41no—
102#14519689—41no—
103#14519689—41no—
104C8×Dic61—41no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684D411yescharacteristic
110#14519684D411yescharacteristic
111C22×Q8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114#125×C61—22no—
115#125×C61—41no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122#103×C61—22no—
123#103×C61—41no—
124#95×C61—21no—
125#95×C61—21no—
126C23×D61—21no—
127C23×D61C2×D411yescharacteristic
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61#4112yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—42no—
154C2×D244—42no—
155C2×D244—81no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—81no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61#4112yes—
179C42×C61—21no—
180C42×C61—21no—
181#11758201—44no—
182#11758198—24no—
183#11758198—22no—
184#11758198—22no—
185#11758198—42no—
186#11758198—42no—
187#11758198—81no—
188#11758198—81no—
189#11758198—81no—
190#11758198—81no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—42no—
196C4⋊Dic61#4112yes—
197C4⋊Dic61#4112yes—
198#11758196—24no—
199#11758196—42no—
200#11758196—81no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—42no—
204C4×Dic61—81no—
205C4×Dic61—81no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C23×C61C22×D411yescharacteristic
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61#13512yes—
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—41no—
230Q8×C61—41no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61#13712yes—
273D244—41no—
274D244—41no—
275D244—41no—
276D244—41no—
277D244—84no—
278C4×D61—41no—
279C4×D61—41no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—84no—
283C8×C61—41no—
284C8×C61—41no—
285#92418—41no—
286#92418—41no—
287#26727—611nomaximal, Sylow
288C22×C61#52511yescharacteristic
289C22×C61#54611yescharacteristic
290C22×C61#54611yescharacteristic
291C22×C61C2×#13511yescharacteristic
292C22×C61C2×#13711yescharacteristic
293C22×C61#57611yescharacteristic
294C22×C61#57711yescharacteristic
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—41no—
301C2×D61—41no—
302C2×D61—81no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
308C2×D61—81no—
309C4×C61—41no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—81no—
314C4×C61—81no—
315C4×C61#52911yescharacteristic
316C4×C61#53811yescharacteristic
317C4×C61#55511yescharacteristic
318C4×C61#55911yescharacteristic
319Dic61—22no—
320Dic61—42no—
321Dic61—41no—
322Dic61—42no—
323Dic61—41no—
324Dic61—81no—
325#2490—611no—
326#2360—611no—
327#2055—611no—
328#2036—611no—
329#2026D6111yescharacteristic, complemented
330#2000—611no—
331#1989—611no—
332#1934—611no—
333#1925—611no—
334#1923—611no—
335#1921—611no—
336#1880—611no—
337#1874—611no—
338#1823—611no—
339C4×#103—611no—
340C2×C61#245911yescharacteristic
341C2×C61#310811yescharacteristic
342C2×C61#311911yescharacteristic
343C2×C61#329611yescharacteristic
344C2×C61#329711yescharacteristic
345C2×C61#332011yescharacteristic
346C2×C61#333311yescharacteristic
347D61—42no—
348D61—161no—
349C22×SD16—611no—
350C2×#125C2×D6111yescharacteristic
351C2×C22⋊Q8—611no—
352C2×#121—611no—
353C2×C4×Q8—611no—
354C22×C4⋊C4—611no—
355C2×#107—611no—
356C2×C4⋊C8—611no—
357C2×#103—611no—
358C2×#103C2×D6111yescharacteristic
359C2×#103C2×D6111yescharacteristic
360C2×#102—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×C22⋊C8—611no—
364C2×C22⋊C8—611no—
365C2×C4×C8—611no—
366#397—1221no—
367#394—1221no—
368#391C2×D6111yescharacteristic
369#390—611no—
370#390—611no—
371#390—1221no—
372#389—611no—
373#389—1221no—
374#388—1221no—
375#387—1221no—
376#385—1221no—
377#380—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4—611no—
380C4×C22⋊C4—611no—
381C4×C22⋊C4—1221no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340C2×D6111yescharacteristic
389#340C2×D6111yescharacteristic
390#336—611no—
391#336C2×D6111yescharacteristic
392C61#2672711yescharacteristic, complemented, Sylow
393C22×Q8C22×D6111yescharacteristic
394C22×D4—611no—
395C22×D4—1221no—
396C23×C4—611no—
397C23×C4—1221no—
398C2×SD16—1222no—
399C2×SD16—2441no—
400C2×SD16—2441no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405#125—22no—
406#125—41no—
407C22⋊Q8—1222no—
408C22⋊Q8—1222no—
409#121—1222no—
410#121—2441no—
411C4×Q8—1222no—
412C4×Q8—2441no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—2441no—
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C22⋊C4—611no—
428C2×C22⋊C4—611no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C42—611no—
437C2×C42—1221no—
438C2×C42—1221no—
439C2×C42—1221no—
440C2×C42C22×D6111yescharacteristic
441#107—1222no—
442C4⋊C8—2441no—
443#103—22no—
444#103—41no—
445#103—1222no—
446#102—1222no—
447#102—1222no—
448#102—2441no—
449C22⋊C8—2441no—
450C22⋊C8—2441no—
451C4×C8—2441no—
452#95—21no—
453#95—21no—
454#95—611no—
455#95—611no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460C24—611no—
461C24—1221no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8#1175822612yes—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—4881no—
475C2×D4—2442no—
476C2×D4—2442no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—4881no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4D4×D6111yescharacteristic
497C22×C4D4×D6111yescharacteristic
498C22×C4C23×D6111yescharacteristic
499SD16—2444no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—612no—
520C4⋊C4—612no—
521C4⋊C4—1224no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—4881no—
527C4⋊C4—2442no—
528C4⋊C4—2442no—
529C4⋊C4#1175822412yes—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1224no—
533C22⋊C4—4881no—
534C22⋊C4—4881no—
535C22⋊C4—4881no—
536C22⋊C4—4881no—
537C22⋊C4—2442no—
538C22⋊C4—2442no—
539C42—21no—
540C42—21no—
541C42—1221no—
542C42—1221no—
543C42—4881no—
544C42—4881no—
545C42—2442no—
546C23—612no—
547C23—1221no—
548C23—1221no—
549C23—1221no—
550C23—1221no—
551C23—1221no—
552C23—1221no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23—2441no—
560C23C2×D4×D6111yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—41no—
564Q8—41no—
565D4—4884no—
566D4—2441no—
567D4—2441no—
568D4—2441no—
569D4—2441no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—612no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1222no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1222no—
598C2×C4—1221no—
599C2×C4—4884no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4#41×D6111yescharacteristic
612C2×C4#41×D6111yescharacteristic
613C2×C4C2×#1175822411yescharacteristic
614C2×C4C2×#1175822611yescharacteristic
615C2×C4#1451982211yescharacteristic
616C2×C4#1451982311yescharacteristic
617C8—41no—
618C8—41no—
619C8—2441no—
620C8—2441no—
621C22—1222no—
622C22—1222no—
623C22—1222no—
624C22—4881no—
625C22—4881no—
626C22—4881no—
627C22—4881no—
628C22—4881no—
629C22—4881no—
630C22—4881no—
631C22—2441no—
632C22—2441no—
633C22—2441no—
634C22—2441no—
635C22#1684027511yescharacteristic
636C22#1684039911yescharacteristic
637C22#1684040511yescharacteristic
638C22#135×D6111yescharacteristic
639C22#137×D6111yescharacteristic
640C22#1684056411yescharacteristic
641C22#1684057111yescharacteristic
642C4—42no—
643C4—41no—
644C4—41no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—1222no—
649C4—4881no—
650C4—2441no—
651C4—2442no—
652C4—2441no—
653C4—2442no—
654C4#1684029811yescharacteristic
655C4#1684035211yescharacteristic
656C4#1684047411yescharacteristic
657C4#1684049411yescharacteristic
658C2—9761no—
659C2—2442no—
660C2#2144975911yescharacteristic
661C2#2145590411yescharacteristic
662C2#2145602511yescharacteristic
663C2#2145764711yescharacteristic
664C2#2145766811yescharacteristic
665C2#2145798911yescharacteristic
666C2#2145815011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2026C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#2026D61
hypocenterC61#26727
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2026

Complements

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

NH ≅ G/Ntype
2. #21448829C2semidirect
3. #21447134C2semidirect
4. #21447021C2semidirect
6. #21446777C2semidirect
7. #21446500C2semidirect
8. #21446440C2semidirect
9. #21446426C2semidirect
10. #21446410C2semidirect
11. #21446172C2semidirect
13. #21445904C2semidirect
14. #103×Dic61C2semidirect
15. C61×#2026C2semidirect
329. #2026D61semidirect
392. C61#26727semidirect

Maximal subgroups

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

subgroupindex
#214501742
#214488292
#214471342
#214470212
#214468212
#214467772
#214465002
#214464402
#214464262
#214464102
#214461722
#214461352
#214459042
#103×Dic612
C61×#20262
#2672761