Finite Group Databasebeta

← #29779376

Subgroups — #29779376

Order 15616.

780 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450407C211yesmaximal, characteristic, complemented
2#21449665C211yesmaximal, characteristic, complemented
3#21447316C211yesmaximal, characteristic, complemented
4#21447224C211yesmaximal, characteristic, complemented
5#21447128C211yesmaximal, characteristic
6#21446781C211yesmaximal, characteristic, complemented
7#21446684C211yesmaximal, characteristic
8#21446518C211yesmaximal, characteristic, complemented
9#21446514C211yesmaximal, characteristic, complemented
10#21446513C211yesmaximal, characteristic, complemented
11#21446409C211yesmaximal, characteristic, complemented
12#21446164C211yesmaximal, characteristic
13#21445690C211yesmaximal, characteristic, complemented
14#21445643C211yesmaximal, characteristic, complemented
15C61×#2023C211yesmaximal, characteristic, complemented
16C22×#11758199C2211yescharacteristic
17C2×#14519770—21no—
18C2×#14519753—21no—
19C2×#14519739C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519699C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519687C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29#16839687—21no—
30#16839685—21no—
31#16839684—21no—
32#16839673—21no—
33#16839665C2211yescharacteristic
34#16839665C2211yescharacteristic
35#16839664C2211yescharacteristic
36#16839663C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839655—21no—
41#16839633—21no—
42#16839632C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4⋊Q8×C61C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C61×#390C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×D244C2311yescharacteristic
60C22×C4×D61—21no—
61C2×#11758199—22no—
62C2×#11758199—22no—
63C2×#11758199—41no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519770—44no—
67#14519753—44no—
68#14519739—22no—
69#14519739—41no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196D411yescharacteristic
79C2×#11758196D411yescharacteristic
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519710—22no—
93#14519706—22no—
94#14519699—22no—
95#14519699—22no—
96#14519698—22no—
97#14519697—22no—
98#14519687—22no—
99#14519687—41no—
100C4×#92418—41no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684D411yescharacteristic
106#14519684D411yescharacteristic
107C22×Q8×C61—21no—
108C22×D4×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C4⋊Q8×C61—22no—
112C4⋊Q8×C61—41no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—21no—
119C2×C42×C61C2311yescharacteristic
120#102×C61—22no—
121#102×C61—41no—
122C24×C61—21no—
123C23×D61—21no—
124C2×Q8×C61—42no—
125C2×Q8×C61—42no—
126C2×Q8×C61—81no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244#4112yes—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61—81no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—42no—
167C4⋊C4×C61#4112yes—
168C22⋊C4×C61—42no—
169C22⋊C4×C61—42no—
170C42×C61—21no—
171C42×C61—21no—
172C22⋊Dic61—42no—
173C22⋊Dic61—42no—
174C22⋊Dic61—81no—
175C22⋊Dic61—81no—
176#11758199—44no—
177#11758198—42no—
178#11758198—42no—
179#11758198—42no—
180#11758198—42no—
181#11758198—81no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—42no—
189C4⋊Dic61—42no—
190C4⋊Dic61—81no—
191#11758196—22no—
192#11758196—22no—
193#11758196—42no—
194#11758196—42no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—42no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418#4112yes—
205C2×#92418#4112yes—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61C22×D411yescharacteristic
214C22×D61—22no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221Q8×C61—84no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61AGL(1,ℤ/8)12yes—
240C2×C4×C61AGL(1,ℤ/8)12yes—
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C612−1+411yescharacteristic
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268D244—22no—
269D244—22no—
270D244—41no—
271D244—41no—
272C4×D61—84no—
273C8×C61—41no—
274C8×C61—41no—
275#92418—22no—
276#92418—22no—
277#29095—611nomaximal, Sylow
278C22×C61—81no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61#53211yescharacteristic
286C22×C61#54611yescharacteristic
287C22×C61#54811yescharacteristic
288C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C61#57611yescharacteristic
291C22×C61#57811yescharacteristic
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—42no—
302C4×C61—81no—
303C4×C61—81no—
304C4×C61#52911yescharacteristic
305C4×C61#53911yescharacteristic
306C4×C61#55111yescharacteristic
307C4×C61#55611yescharacteristic
308Dic61—41no—
309Dic61—41no—
310Dic61—41no—
311Dic61—41no—
312Dic61—81no—
313Dic61—81no—
314Dic61—81no—
315#2512—611no—
316#2452—611no—
317#2087—611no—
318#2071—611no—
319#2054—611no—
320#2023D6111yescharacteristic, complemented
321#1990—611no—
322#1970—611no—
323#1936—611no—
324#1936—611no—
325#1936—611no—
326#1921—611no—
327#1879—611no—
328#1771—611no—
329#1762—611no—
330C2×C61—161no—
331C2×C61#256011yescharacteristic
332C2×C61#317511yescharacteristic
333C2×C61#318411yescharacteristic
334C2×C61#330811yescharacteristic
335C2×C61#331811yescharacteristic
336C2×C61#336611yescharacteristic
337C2×C61#336711yescharacteristic
338D61—82no—
339C22×D8—611no—
340C2×C4⋊Q8C2×D6111yescharacteristic
341C2×#126—1221no—
342C2×C22⋊Q8—1221no—
343C2×C4×D4—611no—
344C2×#107—611no—
345C2×C4⋊C8—611no—
346C2×#102—611no—
347C2×#102—611no—
348C2×#102—611no—
349C2×#102—611no—
350C2×#102C2×D6111yescharacteristic
351C2×#102C2×D6111yescharacteristic
352C2×#97—611no—
353C2×#97—611no—
354C2×C4×C8—611no—
355#397—1221no—
356#396—1221no—
357#396—1221no—
358#393—1221no—
359#390—611no—
360#390—611no—
361#390—611no—
362#390—611no—
363#390C2×D6111yescharacteristic
364#389—611no—
365#389—611no—
366#389—611no—
367#388—1221no—
368#385—1221no—
369#384—611no—
370C4×C4⋊C4—611no—
371C4×C4⋊C4—611no—
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#340—611no—
378#340C2×D6111yescharacteristic
379#340C2×D6111yescharacteristic
380#336—611no—
381#336C2×D6111yescharacteristic
382C61#2909511yescharacteristic, complemented, Sylow
383C22×Q8—21no—
384C22×D4—611no—
385C22×D4C22×D6111yescharacteristic
386C23×C4—1221no—
387C2×D8—1222no—
388C2×D8—1222no—
389C2×D8—2441no—
390C22×C8—611no—
391C22×C8—611no—
392C22×C8C22×D6111yescharacteristic
393C22×C8C22×D6111yescharacteristic
394C4⋊Q8—22no—
395C4⋊Q8—41no—
396#126—2444no—
397C22⋊Q8—2444no—
398C4×D4—1222no—
399C4×D4—2441no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C4⋊C4C22×D6111yescharacteristic
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—1221no—
428C2×C42C22×D6111yescharacteristic
429#107—1222no—
430C4⋊C8—1222no—
431#102—22no—
432#102—41no—
433#102—1222no—
434#102—1222no—
435#102—1222no—
436#102—1222no—
437#97—1222no—
438#97—2441no—
439C4×C8—2441no—
440#95—611no—
441#95—611no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446C24—21no—
447C24—1221no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—81no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—612no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—4881no—
474C22×C4—2442no—
475C22×C4—2442no—
476C22×C4D4×D6111yescharacteristic
477C22×C4D4×D6111yescharacteristic
478C22×C4C23×D6111yescharacteristic
479D8—2444no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—612no—
487C2×C8—612no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—4881no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4—2442no—
511C4⋊C4#1175822412yes—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—4881no—
515C22⋊C4—4881no—
516C22⋊C4—4881no—
517C22⋊C4—2442no—
518C22⋊C4—2442no—
519C22⋊C4—2442no—
520C22⋊C4—2442no—
521C22⋊C4—2442no—
522C22⋊C4—2442no—
523C42—21no—
524C42—21no—
525C42—1221no—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42—2442no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—1222no—
538C23—2441no—
539C23—2441no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23C2×D4×D6111yescharacteristic
545Q8—84no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—1222no—
551D4—1222no—
552D4—2441no—
553D4—2441no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—4884no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4#1451980112yes—
590C2×C4#1451980412yes—
591C2×C4#41×D6111yescharacteristic
592C2×C4#41×D6111yescharacteristic
593C2×C4#41×D6111yescharacteristic
594C2×C4C2×#1175822411yescharacteristic
595C2×C4#1451982211yescharacteristic
596C2×C4#1451982511yescharacteristic
597C8—41no—
598C8—41no—
599C8—1222no—
600C8—1222no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22—4881no—
612C22—2442no—
613C22—2442no—
614C22—2442no—
615C22#1684031111yescharacteristic
616C22#1684039911yescharacteristic
617C22#1684042211yescharacteristic
618C22C2×#1451980111yescharacteristic
619C22C2×#1451980411yescharacteristic
620C22#1684056611yescharacteristic
621C22#1684057811yescharacteristic
622C4—22no—
623C4—22no—
624C4—42no—
625C4—81no—
626C4—81no—
627C4—4881no—
628C4—4881no—
629C4—4881no—
630C4—2441no—
631C4—2441no—
632C4—2441no—
633C4—2441no—
634C4#1684029111yescharacteristic
635C4#1684035811yescharacteristic
636C4#1684044211yescharacteristic
637C4#1684048311yescharacteristic
638C2—161no—
639C2—4882no—
640C2#2145096611yescharacteristic
641C2#2145645711yescharacteristic
642C2#2145655611yescharacteristic
643C2#2145781411yescharacteristic
644C2#2145796411yescharacteristic
645C2#2145852511yescharacteristic
646C2#2145853611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2023C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenter#2023D61
hypocenterC61#29095
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. #21450407C2semidirect
2. #21449665C2semidirect
3. #21447316C2semidirect
4. #21447224C2semidirect
6. #21446781C2semidirect
8. #21446518C2semidirect
9. #21446514C2semidirect
10. #21446513C2semidirect
11. #21446409C2semidirect
13. #21445690C2semidirect
14. #21445643C2semidirect
15. C61×#2023C2semidirect
320. #2023D61semidirect
382. C61#29095semidirect

Maximal subgroups

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

subgroupindex
#214504072
#214496652
#214473162
#214472242
#214471282
#214467812
#214466842
#214465182
#214465142
#214465132
#214464092
#214461642
#214456902
#214456432
C61×#20232
#2909561