Finite Group Databasebeta

← #29729728

Subgroups — #29729728

Order 15616.

802 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839964C211yesmaximal, characteristic
2C2×#16839893C211yesmaximal, characteristic, complemented
3#21451177C211yesmaximal, characteristic
4#21447214C211yesmaximal, characteristic, complemented
5#21447200C211yesmaximal, characteristic, complemented
6#21447134C211yesmaximal, characteristic
7#21447134C211yesmaximal, characteristic
8#21446703C211yesmaximal, characteristic
9#21446676C211yesmaximal, characteristic
10#21446640C211yesmaximal, characteristic, complemented
11#21446587C211yesmaximal, characteristic, complemented
12#21446512C211yesmaximal, characteristic, complemented
13#21446498C211yesmaximal, characteristic, complemented
14#21445647C211yesmaximal, characteristic
15C61×#2568C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19#16839964—24no—
20#16839893—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35C2×#14519688C2211yescharacteristic
36C2×C4×#92418C2211yescharacteristic
37#16839697—21no—
38#16839673—21no—
39#16839664C2211yescharacteristic
40#16839664C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839633—21no—
44#16839633—21no—
45#16839628C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C61×#400—21no—
53C61×#400—21no—
54C61×#391—21no—
55C61×#391C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#382—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758201—22no—
64C2×#11758201—22no—
65C2×#11758201—22no—
66C2×#11758201—22no—
67C2×#11758201—41no—
68C2×#11758201—41no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C4×D244—22no—
74C4×D244—22no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85#14519710—41no—
86#14519708—41no—
87C4⋊#92418—22no—
88C4⋊#92418—22no—
89C4⋊#92418D414yes—
90#14519701—22no—
91#14519701—22no—
92#14519701—22no—
93#14519701—22no—
94#14519701—41no—
95#14519701—41no—
96#14519697—22no—
97#14519697—22no—
98#14519688—22no—
99#14519688—22no—
100C4×#92418—22no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105C22×Q8×C61D411yescharacteristic
106C22×Q8×C61C2311yescharacteristic
107C22×Q8×C61C2311yescharacteristic
108C4⋊Q8×C61—22no—
109C4⋊Q8×C61—22no—
110C4×Q8×C61—22no—
111C4×Q8×C61—22no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123#95×C61—21no—
124#95×C61—21no—
125#95×C61—21no—
126#95×C61—21no—
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132C23×D61—21no—
133C2×Q8×C61—22no—
134C2×Q8×C61—22no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244#4112yes—
168C2×C4×D61—42no—
169C2×C4×D61—81no—
170C2×C4×D61—81no—
171C4⋊C4×C61—22no—
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—42no—
179C4⋊C4×C61—42no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—42no—
185C42×C61C2×D412yes—
186C42×C61#4112yes—
187#11758201—44no—
188#11758201—44no—
189#11758198—42no—
190#11758198—42no—
191#11758198—42no—
192#11758198—81no—
193#11758198—81no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200#11758196—81no—
201#11758196—81no—
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×#92418C2×D412yes—
212C2×#92418C2×D412yes—
213C2×#92418#4112yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—22no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222Q8×C61—44no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61—41no—
230Q8×C61—41no—
231C2×C4×C61—21no—
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—41no—
247C2×C4×C61—41no—
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#12112yes—
259C2×C4×C61#12112yes—
260C2×C4×C61#13712yes—
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×C4×C612−1+411yescharacteristic
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276D244—22no—
277D244—22no—
278D244—41no—
279D244—41no—
280C4×D61—84no—
281#92418—22no—
282#92418—22no—
283#92418—22no—
284#92418—41no—
285#25308—611nomaximal, Sylow
286C22×C61C2×#12111yescharacteristic
287C22×C61#54811yescharacteristic
288C22×C61#55011yescharacteristic
289C22×C61C2×#13711yescharacteristic
290C22×C61#57511yescharacteristic
291C22×C61#57711yescharacteristic
292C22×C61#57711yescharacteristic
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—81no—
300C4×C61—22no—
301C4×C61—22no—
302C4×C61—22no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61—81no—
309C4×C61—81no—
310C4×C61#52911yescharacteristic
311C4×C61#54211yescharacteristic
312C4×C61#54611yescharacteristic
313C4×C61#54811yescharacteristic
314Dic61—81no—
315Dic61—81no—
316Dic61—81no—
317C2×#461—611no—
318C2×#446—611no—
319#2580—611no—
320#2568D6111yescharacteristic, complemented
321#2068—611no—
322#2066—611no—
323#2055—611no—
324#2055—611no—
325#1973—611no—
326#1968—611no—
327#1960—611no—
328#1949—611no—
329#1935—611no—
330#1933—611no—
331#1762—611no—
332C2×C61#242611yescharacteristic
333C2×C61#305611yescharacteristic
334C2×C61#305711yescharacteristic
335C2×C61#337211yescharacteristic
336C2×C61#337611yescharacteristic
337C2×C61#340011yescharacteristic
338C2×C61#340111yescharacteristic
339D61—82no—
340C22×SD16—611no—
341C22×SD16—611no—
342C2×C4⋊Q8C2×D6111yescharacteristic
343C2×C4×Q8C2×D6111yescharacteristic
344C2×C4×D4—611no—
345#461—1224no—
346#446—1224no—
347C2×#107—611no—
348C2×#106—611no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8—611no—
351C2×C4⋊C8—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#97—611no—
361C2×#97—611no—
362C2×C4×C8—611no—
363#400—21no—
364#400—21no—
365#399—1221no—
366#393—1221no—
367#391—21no—
368#391C2×D6111yescharacteristic
369#391C2×D6111yescharacteristic
370#391C2×D6111yescharacteristic
371#391C2×D6111yescharacteristic
372#390—611no—
373#390—611no—
374#389—611no—
375#389—611no—
376#385—1221no—
377#385—1221no—
378#382—21no—
379#382—611no—
380C4×C4⋊C4C2×D6111yescharacteristic
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385C61#2530811yescharacteristic, complemented, Sylow
386C22×Q8C22⋊D6111yescharacteristic
387C22×Q8C22×D6111yescharacteristic
388C22×Q8C22×D6111yescharacteristic
389C22×D4—611no—
390C23×C4—1221no—
391C2×SD16—1222no—
392C2×SD16—1222no—
393C2×SD16—1222no—
394C2×SD16—1222no—
395C2×SD16—2441no—
396C2×SD16—2441no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—611no—
401C4⋊Q8—22no—
402C4⋊Q8—22no—
403C4×Q8—22no—
404C4×Q8—22no—
405C4×D4—1222no—
406C4×D4—1222no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22⋊D6111yescharacteristic
415C2×C4⋊C4C22⋊D6111yescharacteristic
416C2×C4⋊C4C22⋊D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—21no—
425C2×C42C22×D6111yescharacteristic
426C2×C42C22×D6111yescharacteristic
427C2×C42C22×D6111yescharacteristic
428#107—2441no—
429#106—2441no—
430C4⋊C8—614no—
431C4⋊C8—1222no—
432C4⋊C8—1222no—
433#103—1222no—
434#103—1222no—
435#103—1222no—
436#103—1222no—
437#103—2441no—
438#103—2441no—
439#102—1222no—
440#102—1222no—
441#97—1222no—
442#97—1222no—
443C4×C8—1222no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457C24—1221no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—41no—
473C2×D4—612no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—4881no—
487C22×C4—4881no—
488C22×C4—2442no—
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C2×C22⋊D6111yescharacteristic
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496SD16—2444no—
497SD16—2444no—
498C2×C8—612no—
499C2×C8—612no—
500C2×C8—612no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—4881no—
526C4⋊C4—4881no—
527C22⋊C4—4881no—
528C22⋊C4—4881no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C22⋊C4—2442no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—42no—
537C42D4×D6112yes—
538C42#1175822412yes—
539C23—1222no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23C22×C22⋊D6111yescharacteristic
547Q8—44no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556D4—1222no—
557D4—1222no—
558D4—2441no—
559D4—2441no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
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—4884no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4#1451974912yes—
598C2×C4#1451974912yes—
599C2×C4#1451980812yes—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4#41×D6111yescharacteristic
603C2×C4C2×#1175822411yescharacteristic
604C2×C4#1451982311yescharacteristic
605C2×C4#1451982411yescharacteristic
606C8—1222no—
607C8—1222no—
608C8—1222no—
609C8—2441no—
610C22—4881no—
611C22—4881no—
612C22—4881no—
613C22—4881no—
614C22—2442no—
615C22—2442no—
616C22—2442no—
617C22C2×#1451974911yescharacteristic
618C22#1684041911yescharacteristic
619C22#1684043111yescharacteristic
620C22C2×#1451980811yescharacteristic
621C22#1684056011yescharacteristic
622C22#1684057011yescharacteristic
623C22#1684057011yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—4881no—
635C4—4881no—
636C4—4881no—
637C4#1684029111yescharacteristic
638C4#1684038111yescharacteristic
639C4#1684040511yescharacteristic
640C4#1684042011yescharacteristic
641C2—4882no—
642C2#2144938811yescharacteristic
643C2#2145544511yescharacteristic
644C2#2145546011yescharacteristic
645C2#2145859611yescharacteristic
646C2#2145864011yescharacteristic
647C2#2145886811yescharacteristic
648C2#2145888511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2568C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2568D61
hypocenterC61#25308
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2568

Complements

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

NH ≅ G/Ntype
2. C2×#16839893C2semidirect
4. #21447214C2semidirect
5. #21447200C2semidirect
10. #21446640C2semidirect
11. #21446587C2semidirect
12. #21446512C2semidirect
13. #21446498C2semidirect
15. C61×#2568C2semidirect
320. #2568D61semidirect
385. C61#25308semidirect

Maximal subgroups

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

subgroupindex
C2×#168399642
C2×#168398932
#214511772
#214472142
#214472002
#214471342
#214467032
#214466762
#214466402
#214465872
#214465122
#214464982
#214456472
C61×#25682
#2530861