Finite Group Databasebeta

← #29748052

Subgroups — #29748052

Order 15616.

832 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450156C211yesmaximal, characteristic, complemented
2#21448839C211yesmaximal, characteristic
3#21447135C211yesmaximal, characteristic, complemented
4#21446963C211yesmaximal, characteristic, complemented
5#21446822C211yesmaximal, characteristic
6#21446775C211yesmaximal, characteristic, complemented
7#21446496C211yesmaximal, characteristic, complemented
8#21446459C211yesmaximal, characteristic, complemented
9#21446428C211yesmaximal, characteristic, complemented
10#21446392C211yesmaximal, characteristic, complemented
11#21446151C211yesmaximal, characteristic, complemented
12#21446124C211yesmaximal, characteristic, complemented
13#21445907C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#2033C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519756C2211yescharacteristic
19C2×D4×Dic61—21no—
20C2×D4×Dic61C2211yescharacteristic
21C22×C4⋊Dic61C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839704—21no—
30#16839690—21no—
31#16839667C2211yescharacteristic
32#16839667C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839655—21no—
36#16839655—21no—
37#16839642—21no—
38#16839637—21no—
39#16839626—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×C22⋊Dic61C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#121×C61C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×C22⋊C8×C61C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C23×Dic61—21no—
61C23×Dic61—21no—
62C2×#11758200—22no—
63C2×#11758200—41no—
64C2×#11758200—41no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519768—22no—
68#14519768—41no—
69#14519756—22no—
70#14519756—41no—
71D4×Dic61—22no—
72D4×Dic61—41no—
73D4×Dic61—44no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—41no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61D411yescharacteristic
89C2×C22⋊Dic61D411yescharacteristic
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519706—41no—
95#14519702—22no—
96#14519702—41no—
97#14519700—22no—
98#14519698—22no—
99C22⋊#92418—41no—
100C8×Dic61—41no—
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
110C22×D4×C61—21no—
111C22×D4×C61C2311yescharacteristic
112C23×C4×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115#121×C61—22no—
116#121×C61—22no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122#107×C61—22no—
123#102×C61—22no—
124#102×C61—41no—
125C22⋊C8×C61—41no—
126#95×C61—21no—
127C24×C61—21no—
128C24×C61C2×D411yescharacteristic
129C2×D4×C61—24no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—42no—
136C2×D4×C61—42no—
137C2×D4×C61#4112yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—42no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61—42no—
157C22×Dic61—81no—
158C22×Dic61—81no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61#4112yes—
181C22⋊C4×C61—24no—
182C22⋊C4×C61—42no—
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185C22⋊Dic61—42no—
186C22⋊Dic61—42no—
187C22⋊Dic61—42no—
188C22⋊Dic61—81no—
189#11758200—44no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—42no—
195C4⋊Dic61—42no—
196C4⋊Dic61—42no—
197C4⋊Dic61#4112yes—
198C4⋊Dic61#4112yes—
199#11758196—42no—
200#11758196—42no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—22no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61#13512yes—
227C23×C61C22×D411yescharacteristic
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—48no—
231D4×C61—41no—
232D4×C61—41no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
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—21no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—84no—
279C2×Dic61—84no—
280C2×Dic61AGL(1,ℤ/8)12yes—
281Dic122—41no—
282Dic122—41no—
283Dic122—41no—
284Dic122—41no—
285C8×C61—41no—
286C8×C61—41no—
287#92418—41no—
288#92418—41no—
289#26721—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61#52511yescharacteristic
305C22×C61#54611yescharacteristic
306C22×C61#54611yescharacteristic
307C22×C61C2×#13511yescharacteristic
308C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
309C22×C61#57711yescharacteristic
310C22×C61#57811yescharacteristic
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—41no—
321Dic61—42no—
322Dic61—41no—
323Dic61—41no—
324Dic61—42no—
325Dic61—81no—
326Dic61—81no—
327#2489—611no—
328#2361—611no—
329#2055—611no—
330#2033D6111yescharacteristic
331#2025—611no—
332#2000—611no—
333#1989—611no—
334#1933—611no—
335#1928—611no—
336#1923—611no—
337#1918—611no—
338#1877—611no—
339#1872—611no—
340#1824—611no—
341C4×#102—611no—
342C2×C61—42no—
343C2×C61—82no—
344C2×C61#245911yescharacteristic
345C2×C61#310511yescharacteristic
346C2×C61#312011yescharacteristic
347C2×C61#330011yescharacteristic
348C2×C61#330111yescharacteristic
349C2×C61#331011yescharacteristic
350C2×C61#333011yescharacteristic
351C22×SD16—611no—
352C2×#125—611no—
353C2×C22⋊Q8—611no—
354C2×#121C2×D6111yescharacteristic
355C2×C4×D4—611no—
356C2×C4×D4—1221no—
357C22×C4⋊C4—611no—
358C2×#107C2×D6111yescharacteristic
359C2×C4⋊C8—611no—
360C2×#103—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#102—611no—
364C2×#102C2×D6111yescharacteristic
365C2×#102C2×D6111yescharacteristic
366C2×C22⋊C8—611no—
367C2×C22⋊C8C2×D6111yescharacteristic
368C2×C4×C8—611no—
369#400—1221no—
370#397—1221no—
371#391—611no—
372#391—611no—
373#390—611no—
374#389—611no—
375#388—1221no—
376#388—1221no—
377#386—1221no—
378#385—1221no—
379#382—1221no—
380#380C2×D6111yescharacteristic
381C4×C4⋊C4—611no—
382C4×C4⋊C4—611no—
383C4×C22⋊C4—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340C2×D6111yescharacteristic
392#336—611no—
393#336—611no—
394C61#2672111yescharacteristic, complemented, Sylow
395C22×Q8—611no—
396C22×D4—21no—
397C22×D4C22×D6111yescharacteristic
398C23×C4—1221no—
399C23×C4—1221no—
400C23×C4C22×D6111yescharacteristic
401C2×SD16—1222no—
402C2×SD16—2441no—
403C2×SD16—2441no—
404C22×C8—611no—
405C22×C8—611no—
406C22×C8C22×D6111yescharacteristic
407C22×C8C22×D6111yescharacteristic
408#125—1222no—
409#125—2441no—
410C22⋊Q8—1222no—
411C22⋊Q8—2441no—
412#121—22no—
413#121—22no—
414C4×D4—1222no—
415C4×D4—2441no—
416C4×D4—2444no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1222no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—2441no—
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—611no—
433C2×C22⋊C4—611no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—1221no—
441C2×C42—1221no—
442#107—22no—
443C4⋊C8—2441no—
444#103—1222no—
445#103—1222no—
446#103—2441no—
447#102—22no—
448#102—41no—
449#102—1222no—
450C22⋊C8—41no—
451C22⋊C8—2441no—
452C4×C8—2441no—
453#95—21no—
454#95—611no—
455#95—611no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463C24—21no—
464C24D4×D6111yescharacteristic
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×D4—24no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4#1175822412yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—4881no—
498C22×C4—4881no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4D4×D6111yescharacteristic
504C22×C4C23×D6111yescharacteristic
505SD16—2444no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—612no—
524C4⋊C4—612no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—2442no—
530C4⋊C4—2442no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C4⋊C4—2442no—
534C4⋊C4#1175822412yes—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—4881no—
540C22⋊C4—2442no—
541C22⋊C4—2442no—
542C22⋊C4—2442no—
543C42—1221no—
544C42—1221no—
545C42—1221no—
546C42—1221no—
547C42—2442no—
548C42—2442no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23#1451979612yes—
563C23C2×D4×D6111yescharacteristic
564Q8—2441no—
565Q8—2441no—
566Q8—2441no—
567Q8—2441no—
568D4—22no—
569D4—22no—
570D4—48no—
571D4—41no—
572D4—41no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
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—41no—
589C2×C4—612no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1222no—
601C2×C4—1221no—
602C2×C4—1222no—
603C2×C4—4884no—
604C2×C4—4884no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4#41×D6111yescharacteristic
616C2×C4#41×D6111yescharacteristic
617C2×C4C2×#1175822411yescharacteristic
618C2×C4C2×#1175822411yescharacteristic
619C2×C4#1451982211yescharacteristic
620C2×C4#1451982311yescharacteristic
621C8—41no—
622C8—41no—
623C8—2441no—
624C8—2441no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22#1684026811yescharacteristic
640C22#1684040111yescharacteristic
641C22#1684040411yescharacteristic
642C22C2×#1451979611yescharacteristic
643C22AGL(1,ℤ/8)×D6111yescharacteristic
644C22#1684056911yescharacteristic
645C22#1684057811yescharacteristic
646C4—41no—
647C4—42no—
648C4—81no—
649C4—81no—
650C4—1222no—
651C4—4881no—
652C4—4881no—
653C4—2441no—
654C4—2441no—
655C4—2442no—
656C4—2441no—
657C4—2442no—
658C4#1684028911yescharacteristic
659C4#1684034911yescharacteristic
660C4#1684047611yescharacteristic
661C4#1684049411yescharacteristic
662C2—42no—
663C2—82no—
664C2#2144974911yescharacteristic
665C2#2145586111yescharacteristic
666C2#2145602611yescharacteristic
667C2#2145770411yescharacteristic
668C2#2145772711yescharacteristic
669C2#2145784311yescharacteristic
670C2#2145811911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2033C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#2033D61
hypocenterC61#26721
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
1. #21450156C2semidirect
3. #21447135C2semidirect
4. #21446963C2semidirect
6. #21446775C2semidirect
7. #21446496C2semidirect
8. #21446459C2semidirect
9. #21446428C2semidirect
10. #21446392C2semidirect
11. #21446151C2semidirect
12. #21446124C2semidirect
13. #21445907C2semidirect
14. #102×Dic61C2semidirect
394. C61#26721semidirect

Maximal subgroups

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

subgroupindex
#214501562
#214488392
#214471352
#214469632
#214468222
#214467752
#214464962
#214464592
#214464282
#214463922
#214461512
#214461242
#214459072
#102×Dic612
C61×#20332
#2672161