Finite Group Databasebeta

← #29753161

Subgroups — #29753161

Order 15616.

840 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448883C211yesmaximal, characteristic
2#21447343C211yesmaximal, characteristic, complemented
3#21447134C211yesmaximal, characteristic, complemented
4#21447053C211yesmaximal, characteristic
5#21446906C211yesmaximal, characteristic
6#21446676C211yesmaximal, characteristic, complemented
7#21446457C211yesmaximal, characteristic, complemented
8#21446439C211yesmaximal, characteristic, complemented
9#21446394C211yesmaximal, characteristic, complemented
10#21446378C211yesmaximal, characteristic, complemented
11#21446156C211yesmaximal, characteristic, complemented
12#21446156C211yesmaximal, characteristic, complemented
13#21445826C211yesmaximal, characteristic, complemented
14C4×#14519697C211yesmaximal, characteristic, complemented
15C61×#2514C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519774C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519739—21no—
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×C4×#92418C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839664C2211yescharacteristic
36#16839663—21no—
37#16839640—21no—
38#16839640—21no—
39C4×C4⋊Dic61C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839517C2211yescharacteristic
45#16839517C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C61×#401—21no—
49C61×#400—21no—
50C61×#391C2211yescharacteristic
51C61×#391C2211yescharacteristic
52C61×#391C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#387—21no—
55C61×#383—21no—
56C61×#382C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C22×D244—21no—
60C22×D244—21no—
61C22×D244C2311yescharacteristic
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C2×#11758201—22no—
66C2×#11758201—22no—
67C2×#11758201—22no—
68C2×#11758201—41no—
69C2×#11758201—41no—
70C2×#11758201—41no—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519774—22no—
76#14519774—22no—
77C4×D244—22no—
78C4×D244—22no—
79C4×D244—41no—
80C4×D244—41no—
81#14519739—44no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91#14519710—41no—
92#14519708—22no—
93C4⋊#92418—41no—
94#14519701—22no—
95#14519701—41no—
96#14519697—22no—
97#14519697—22no—
98#14519697—22no—
99#14519697—41no—
100#14519697—41no—
101#14519697—41no—
102C4×#92418—41no—
103#14519684—21no—
104C22×Q8×C61C2311yescharacteristic
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C42×C61—21no—
112C2×C42×C61C2311yescharacteristic
113C2×C42×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115#95×C61—21no—
116#95×C61—21no—
117#95×C61—21no—
118#95×C61—21no—
119#95×C61—21no—
120#95×C61—21no—
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125#95×C61D411yescharacteristic
126#95×C61D411yescharacteristic
127C23×D61—21no—
128C23×D61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—24no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—24no—
160C2×D244—42no—
161C2×D244—42no—
162C2×D244—81no—
163C2×D244—81no—
164C2×D244#4112yes—
165C2×D244#4112yes—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61—81no—
171C2×C4×D61—81no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×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—42no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—42no—
188#11758201—44no—
189#11758201—44no—
190#11758198—42no—
191#11758198—42no—
192#11758198—42no—
193#11758198—42no—
194#11758198—81no—
195#11758198—81no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61#4112yes—
200C4×Dic61—42no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418#4112yes—
213C23×C61C22×D411yescharacteristic
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228Q8×C61—41no—
229Q8×C61—41no—
230Q8×C61—41no—
231Q8×C61—41no—
232C2×C4×C61—21no—
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—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61#13512yes—
261C2×C4×C61AGL(1,ℤ/8)12yes—
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C2×Dic61—22no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274D244—22no—
275D244—22no—
276D244—22no—
277D244—22no—
278D244—48no—
279D244—41no—
280D244—41no—
281D244—41no—
282D244—41no—
283D244—84no—
284C4×D61—84no—
285C4×D61—84no—
286#92418—22no—
287#92418—41no—
288#92418—41no—
289#92418—41no—
290#27063—611nomaximal, Sylow
291C22×C61#52511yescharacteristic
292C22×C61#54611yescharacteristic
293C22×C61#55011yescharacteristic
294C22×C61C2×#13511yescharacteristic
295C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C61#57711yescharacteristic
297C22×C61#57711yescharacteristic
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
308C2×D61—81no—
309C2×D61—81no—
310C2×D61—81no—
311C2×D61—81no—
312C4×C61—22no—
313C4×C61—22no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61—41no—
317C4×C61—41no—
318C4×C61—81no—
319C4×C61—81no—
320C4×C61—81no—
321C4×C61—81no—
322C4×C61#52911yescharacteristic
323C4×C61#53811yescharacteristic
324C4×C61#55111yescharacteristic
325C4×C61#55311yescharacteristic
326Dic61—42no—
327Dic61—81no—
328#2514D6111yescharacteristic, complemented
329#2366—611no—
330#2092—611no—
331#2055—611no—
332#2041—611no—
333#2014—611no—
334#1968—611no—
335#1928—611no—
336#1925—611no—
337#1919—611no—
338#1916—611no—
339#1877—611no—
340#1877—611no—
341#1804—611no—
342C4×#102—611no—
343C2×C61#246711yescharacteristic
344C2×C61#310511yescharacteristic
345C2×C61#312311yescharacteristic
346C2×C61#330411yescharacteristic
347C2×C61#331011yescharacteristic
348C2×C61#339411yescharacteristic
349C2×C61#339711yescharacteristic
350D61—82no—
351D61—82no—
352C22×SD16—611no—
353C22×SD16—611no—
354C2×#127—611no—
355C2×C4×D4—611no—
356C2×C4×D4—611no—
357C2×C4×D4—1221no—
358C2×#107—611no—
359C2×#106—611no—
360C2×C4⋊C8—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#102—611no—
364C2×#102—611no—
365C2×#102—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×#102—611no—
369C2×C4×C8—611no—
370#401—21no—
371#400—21no—
372#391C2×D6111yescharacteristic
373#391C2×D6111yescharacteristic
374#391C2×D6111yescharacteristic
375#390—611no—
376#390—611no—
377#390—1221no—
378#389C2×D6111yescharacteristic
379#387—21no—
380#386—1221no—
381#386—1221no—
382#383—21no—
383#382C2×D6111yescharacteristic
384C4×C4⋊C4—611no—
385C4×C4⋊C4C2×D6111yescharacteristic
386C4×C4⋊C4C2×D6111yescharacteristic
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#336—611no—
392#336—611no—
393#336—611no—
394#336—611no—
395C61#2706311yescharacteristic, complemented, Sylow
396C22×Q8C22×D6111yescharacteristic
397C22×D4—611no—
398C22×D4—611no—
399C22×D4—1221no—
400C22×D4—1221no—
401C23×C4—1221no—
402C23×C4—1221no—
403C2×SD16—1222no—
404C2×SD16—1222no—
405C2×SD16—1222no—
406C2×SD16—2441no—
407C2×SD16—2441no—
408C2×SD16—2441no—
409C22×C8—611no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8—611no—
413#127—1222no—
414#127—1222no—
415C4×D4—1222no—
416C4×D4—1222no—
417C4×D4—2441no—
418C4×D4—2441no—
419C4×D4—2444no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4C22⋊D6111yescharacteristic
424C2×C4⋊C4C22⋊D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C42—21no—
435C2×C42—1221no—
436C2×C42C22×D6111yescharacteristic
437C2×C42C22×D6111yescharacteristic
438C2×C42C22×D6111yescharacteristic
439#107—2441no—
440#106—1222no—
441C4⋊C8—2441no—
442#103—1222no—
443#103—2441no—
444#102—1222no—
445#102—1222no—
446#102—1222no—
447#102—2441no—
448#102—2441no—
449#102—2441no—
450C4×C8—2441no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—1221no—
462#95C22⋊D6111yescharacteristic
463#95C22⋊D6111yescharacteristic
464C24—1221no—
465C24—1221no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×D4—612no—
473C2×D4—612no—
474C2×D4—1224no—
475C2×D4—1224no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—4881no—
487C2×D4—4881no—
488C2×D4—2442no—
489C2×D4—2442no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—4881no—
497C22×C4—4881no—
498C22×C4—2442no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4—2442no—
502C22×C4C2×C22⋊D6111yescharacteristic
503C22×C4C2×C22⋊D6111yescharacteristic
504C22×C4C2×C22⋊D6111yescharacteristic
505C22×C4C2×C22⋊D6111yescharacteristic
506C22×C4C2×C22⋊D6111yescharacteristic
507C22×C4C2×C22⋊D6111yescharacteristic
508C22×C4C23×D6111yescharacteristic
509SD16—2444no—
510SD16—2444no—
511C2×C8—612no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—42no—
532C4⋊C4—612no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C22⋊C4—4881no—
537C22⋊C4—4881no—
538C22⋊C4—2442no—
539C22⋊C4—2442no—
540C22⋊C4—2442no—
541C22⋊C4—2442no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—42no—
549C42—2442no—
550C23—1222no—
551C23—1222no—
552C23—2441no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23—2441no—
564C23C22×C22⋊D6111yescharacteristic
565Q8—41no—
566Q8—41no—
567Q8—41no—
568Q8—41no—
569D4—1222no—
570D4—1222no—
571D4—1222no—
572D4—1222no—
573D4—4884no—
574D4—2448no—
575D4—2441no—
576D4—2441no—
577D4—2441no—
578D4—2441no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—1222no—
608C2×C4—4884no—
609C2×C4—4884no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4#1451979712yes—
616C2×C4#1451980212yes—
617C2×C4#41×D6111yescharacteristic
618C2×C4#41×D6111yescharacteristic
619C2×C4#41×D6111yescharacteristic
620C2×C4#41×D6111yescharacteristic
621C2×C4#1451982311yescharacteristic
622C2×C4#1451982311yescharacteristic
623C8—1222no—
624C8—2441no—
625C8—2441no—
626C8—2441no—
627C22—4881no—
628C22—4881no—
629C22—4881no—
630C22—4881no—
631C22—4881no—
632C22—4881no—
633C22—4881no—
634C22—4881no—
635C22—2442no—
636C22—2442no—
637C22—2442no—
638C22—2442no—
639C22—2442no—
640C22—2442no—
641C22#1684026911yescharacteristic
642C22#1684040011yescharacteristic
643C22#1684043111yescharacteristic
644C22C2×#1451979711yescharacteristic
645C22C2×#1451980211yescharacteristic
646C22#1684057011yescharacteristic
647C22#1684057011yescharacteristic
648C4—22no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—4881no—
659C4—2442no—
660C4#1684029511yescharacteristic
661C4#1684035111yescharacteristic
662C4#1684043711yescharacteristic
663C4#1684046311yescharacteristic
664C2—4882no—
665C2—4882no—
666C2#2144986611yescharacteristic
667C2#2145586211yescharacteristic
668C2#2145606611yescharacteristic
669C2#2145775211yescharacteristic
670C2#2145783411yescharacteristic
671C2#2145881511yescharacteristic
672C2#2145883911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2514C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#2514D61
hypocenterC61#27063
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2514

Complements

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

NH ≅ G/Ntype
2. #21447343C2semidirect
3. #21447134C2semidirect
6. #21446676C2semidirect
7. #21446457C2semidirect
8. #21446439C2semidirect
9. #21446394C2semidirect
10. #21446378C2semidirect
11. #21446156C2semidirect
12. #21446156C2semidirect
13. #21445826C2semidirect
14. C4×#14519697C2semidirect
15. C61×#2514C2semidirect
328. #2514D61semidirect
395. C61#27063semidirect

Maximal subgroups

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

subgroupindex
#214488832
#214473432
#214471342
#214470532
#214469062
#214466762
#214464572
#214464392
#214463942
#214463782
#214461562
#214458262
C4×#145196972
C61×#25142
#2706361