Finite Group Databasebeta

← #29748209

Subgroups — #29748209

Order 15616.

798 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450187C211yesmaximal, characteristic, complemented
2#21448853C211yesmaximal, characteristic
3#21447135C211yesmaximal, characteristic, complemented
4#21446999C211yesmaximal, characteristic
5#21446953C211yesmaximal, characteristic, complemented
6#21446823C211yesmaximal, characteristic
7#21446769C211yesmaximal, characteristic, complemented
8#21446495C211yesmaximal, characteristic, complemented
9#21446458C211yesmaximal, characteristic, complemented
10#21446425C211yesmaximal, characteristic, complemented
11#21446411C211yesmaximal, characteristic, complemented
12#21446153C211yesmaximal, characteristic, complemented
13#21446130C211yesmaximal, characteristic, complemented
14#21445913C211yesmaximal, characteristic, complemented
15C4×#102×C61C211yesmaximal, characteristic, complemented
16C22×#11758200C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519757C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29#16839702—21no—
30#16839675—21no—
31#16839668—21no—
32#16839668C2211yescharacteristic
33#16839667C2211yescharacteristic
34#16839665C2211yescharacteristic
35#16839662—21no—
36#16839660C2211yescharacteristic
37#16839653—21no—
38#16839650—21no—
39#16839641—21no—
40#16839619C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×#11758198C2211yescharacteristic
43C4×C22⋊Dic61—21no—
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122—21no—
60C22×Dic122C2311yescharacteristic
61C22×C22⋊D61—21no—
62C22×C4×D61C2311yescharacteristic
63C2×#11758200—22no—
64C2×#11758200—41no—
65C2×#11758200—41no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519776—22no—
69#14519776—41no—
70#14519757—22no—
71#14519757—41no—
72#14519748—22no—
73#14519748—22no—
74#14519735—22no—
75#14519735—41no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198D411yescharacteristic
89C2×#11758198D411yescharacteristic
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93#14519710—22no—
94#14519706—41no—
95#14519702—22no—
96#14519702—41no—
97#14519700—22no—
98#14519698—41no—
99#14519689—41no—
100#14519689—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×C61C2311yescharacteristic
111C23×C4×C61—21no—
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C4×D4×C61—22no—
115C4×D4×C61—41no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#102×C61—22no—
123#102×C61—22no—
124C4×C8×C61—41no—
125#95×C61—21no—
126C24×C61—21no—
127C23×D61C2×D411yescharacteristic
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61#4112yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61—81no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C2×C22⋊D61—24no—
144C2×C22⋊D61—81no—
145C2×C22⋊D61—81no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—42no—
166C2×Dic122—42no—
167C2×Dic122—81no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61#4112yes—
179C4⋊C4×C61#4112yes—
180C22⋊C4×C61—42no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—81no—
184C22⋊Dic61—24no—
185C22⋊Dic61—42no—
186C22⋊Dic61—81no—
187#11758200—44no—
188#11758198—22no—
189#11758198—22no—
190#11758198—42no—
191#11758198—42no—
192#11758198—81no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—42no—
199C4⋊Dic61#4112yes—
200#11758196—42no—
201#11758196—81no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—42no—
205C4×Dic61—81no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C23×C61—22no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61C22×D411yescharacteristic
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—41no—
230D4×C61—41no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
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—84no—
246C2×C4×C61#13712yes—
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C22⋊D61—84no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
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—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61#13512yes—
275C4×D61—41no—
276C4×D61—41no—
277C4×D61—41no—
278C4×D61—41no—
279Dic122—41no—
280Dic122—41no—
281Dic122—41no—
282Dic122—41no—
283Dic122—84no—
284C8×C61—41no—
285C8×C61—41no—
286#92418—41no—
287#92418—41no—
288#26731—611nomaximal, Sylow
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61#52511yescharacteristic
297C22×C61#54611yescharacteristic
298C22×C61#54611yescharacteristic
299C22×C61C2×#13511yescharacteristic
300C22×C61C2×#13711yescharacteristic
301C22×C61#57611yescharacteristic
302C22×C61#57711yescharacteristic
303C2×D61—41no—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C4×C61—22no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—42no—
314C4×C61—81no—
315C4×C61#52911yescharacteristic
316C4×C61#53811yescharacteristic
317C4×C61#55511yescharacteristic
318C4×C61#55911yescharacteristic
319Dic61—22no—
320Dic61—41no—
321Dic61—42no—
322Dic61—41no—
323Dic61—81no—
324Dic61—81no—
325Dic61—81no—
326#2492—611no—
327#2363—611no—
328#2055—611no—
329#2033—611no—
330#2023—611no—
331#2000—611no—
332#1988—611no—
333#1933—611no—
334#1928—611no—
335#1923—611no—
336#1921—611no—
337#1877—611no—
338#1873—611no—
339#1825—611no—
340C4×#102D6111yescharacteristic, complemented
341C2×C61—82no—
342C2×C61#245911yescharacteristic
343C2×C61#310711yescharacteristic
344C2×C61#311811yescharacteristic
345C2×C61#329711yescharacteristic
346C2×C61#330411yescharacteristic
347C2×C61#331411yescharacteristic
348C2×C61#332911yescharacteristic
349D61—81no—
350C22×SD16—611no—
351C2×C4⋊Q8—611no—
352C2×C22⋊Q8—611no—
353C2×#121—611no—
354C2×C4×D4C2×D6111yescharacteristic
355C2×#117—611no—
356C2×#107—611no—
357C2×C4⋊C8—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#102—611no—
362C2×#102C2×D6111yescharacteristic
363C2×#102C2×D6111yescharacteristic
364C2×C22⋊C8—611no—
365C2×C22⋊C8—611no—
366C2×C4×C8C2×D6111yescharacteristic
367#400—1221no—
368#393—1221no—
369#391—611no—
370#391—611no—
371#391—1221no—
372#390—611no—
373#390—1221no—
374#389—611no—
375#388—1221no—
376#388—1221no—
377#386—1221no—
378#380—611no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4C2×D6111yescharacteristic
381C4×C22⋊C4—611no—
382C4×C22⋊C4—1221no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340C2×D6111yescharacteristic
390#340C2×D6111yescharacteristic
391#336—611no—
392#336—611no—
393C61#2673111yescharacteristic, complemented, Sylow
394C22×Q8—611no—
395C22×Q8—1221no—
396C22×D4—1221no—
397C22×D4C22×D6111yescharacteristic
398C23×C4—21no—
399C23×C4—611no—
400C2×SD16—1222no—
401C2×SD16—2441no—
402C2×SD16—2441no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407C4⋊Q8—1222no—
408C4⋊Q8—2441no—
409C22⋊Q8—1222no—
410C22⋊Q8—2441no—
411#121—1222no—
412#121—1222no—
413C4×D4—22no—
414C4×D4—41no—
415#117—1222no—
416#117—2441no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C22⋊C4—21no—
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×C42—21no—
436C2×C42—611no—
437C2×C42—1221no—
438C2×C42—1221no—
439C2×C42C22×D6111yescharacteristic
440#107—1222no—
441C4⋊C8—2441no—
442#103—1222no—
443#103—1222no—
444#103—2441no—
445#102—22no—
446#102—22no—
447#102—2441no—
448C22⋊C8—2441no—
449C22⋊C8—2441no—
450C4×C8—41no—
451#95—21no—
452#95—611no—
453#95—611no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461C24—21no—
462C24—611no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—4881no—
470C2×Q8—2442no—
471C2×Q8—2442no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—1224no—
478C2×D4—4881no—
479C2×D4—4881no—
480C2×D4#1175822212yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—81no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4D4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503SD16—2444no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—612no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—4881no—
527C4⋊C4—2442no—
528C4⋊C4—2442no—
529C4⋊C4#1175822212yes—
530C4⋊C4#1175822212yes—
531C22⋊C4—42no—
532C22⋊C4—1222no—
533C22⋊C4—1224no—
534C22⋊C4—1222no—
535C22⋊C4—4881no—
536C22⋊C4—4881no—
537C22⋊C4—2442no—
538C22⋊C4—2442no—
539C22⋊C4—2442no—
540C42—21no—
541C42—21no—
542C42—81no—
543C42—1221no—
544C42—1221no—
545C42—4881no—
546C42—2442no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—1221no—
555C23—1221no—
556C23—1221no—
557C23—1221no—
558C23—1221no—
559C23—1221no—
560C23—1221no—
561C23C2×D4×D6111yescharacteristic
562Q8—4884no—
563Q8—2441no—
564Q8—2441no—
565Q8—2441no—
566Q8—2441no—
567D4—22no—
568D4—22no—
569D4—41no—
570D4—41no—
571D4—4884no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—612no—
588C2×C4—1221no—
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—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
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—2441no—
612C2×C4#1451980612yes—
613C2×C4#41×D6111yescharacteristic
614C2×C4C2×#1175822211yescharacteristic
615C2×C4C2×#1175822211yescharacteristic
616C2×C4C2×#1175822211yescharacteristic
617C2×C4#1451982211yescharacteristic
618C2×C4#1451982311yescharacteristic
619C8—41no—
620C8—41no—
621C8—2441no—
622C8—2441no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—2441no—
631C22—2441no—
632C22—2441no—
633C22—2441no—
634C22—2441no—
635C22—2441no—
636C22—2441no—
637C22#1684027111yescharacteristic
638C22#1684040211yescharacteristic
639C22#1684040811yescharacteristic
640C22#135×D6111yescharacteristic
641C22C2×#1451980611yescharacteristic
642C22#1684056311yescharacteristic
643C22#1684056911yescharacteristic
644C4—22no—
645C4—42no—
646C4—41no—
647C4—42no—
648C4—81no—
649C4—1222no—
650C4—4881no—
651C4—4881no—
652C4—4881no—
653C4—2441no—
654C4—2442no—
655C4—2441no—
656C4#1684029611yescharacteristic
657C4#1684034511yescharacteristic
658C4#1684047511yescharacteristic
659C4#1684049511yescharacteristic
660C2—82no—
661C2—4881no—
662C2#2144975611yescharacteristic
663C2#2145588611yescharacteristic
664C2#2145599911yescharacteristic
665C2#2145767411yescharacteristic
666C2#2145775411yescharacteristic
667C2#2145789611yescharacteristic
668C2#2145811511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenterC4×#102D61
hypocenterC61#26731
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
1. #21450187C2semidirect
3. #21447135C2semidirect
5. #21446953C2semidirect
7. #21446769C2semidirect
8. #21446495C2semidirect
9. #21446458C2semidirect
10. #21446425C2semidirect
11. #21446411C2semidirect
12. #21446153C2semidirect
13. #21446130C2semidirect
14. #21445913C2semidirect
15. C4×#102×C61C2semidirect
340. C4×#102D61semidirect
393. C61#26731semidirect

Maximal subgroups

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

subgroupindex
#214501872
#214488532
#214471352
#214469992
#214469532
#214468232
#214467692
#214464952
#214464582
#214464252
#214464112
#214461532
#214461302
#214459132
C4×#102×C612
#2673161