Finite Group Databasebeta

← #30615853

Subgroups — #30615853

Order 15616.

898 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462300C211yesmaximal, characteristic
2#21462256C211yesmaximal, characteristic
3#21461883C212yesmaximal, complemented
4#21461882C212yesmaximal, complemented
5#21461881C212yesmaximal, complemented
6#21461880C212yesmaximal, complemented
7#21461508C211yesmaximal, characteristic
8#21461508C211yesmaximal, characteristic
9C2×#16840636C211yesmaximal, characteristic
10#21461484C211yesmaximal, characteristic
11#427×D61C211yesmaximal, characteristic
12C2×D61⋊M16C2211yescharacteristic
13C2×#14519856C2211yescharacteristic
14C22×C4⋊#6923C2211yescharacteristic
15C22×C4⋊#6923C2211yescharacteristic
16#16840713C2212yes—
17#16840713C2212yes—
18#16840709—22no—
19#16840708—21no—
20#16840706—21no—
21C4⋊C4×#6923C2212yes—
22C4⋊C4×#6923C2212yes—
23C4×C4⋊#6923—21no—
24C4×C4⋊#6923—21no—
25C22⋊C4×#6923—22no—
26#16840636C2212yes—
27#16840636C2212yes—
28#16840635C2212yes—
29#16840635C2212yes—
30#16840635C2212yes—
31#16840635C2212yes—
32(C2×#6923)⋊C8C2212yes—
33(C2×#6923)⋊C8C2212yes—
34C22×C8×D61C2211yescharacteristic
35#117×D61C2211yescharacteristic
36#117×D61C2211yescharacteristic
37#16839800C411yescharacteristic, complemented
38#16839799C411yescharacteristic, complemented
39#16839799C411yescharacteristic, complemented
40#16839798C411yescharacteristic, complemented
41#16839798C411yescharacteristic, complemented
42#16839797C411yescharacteristic, complemented
43#16839797C411yescharacteristic, complemented
44#107×D61C2211yescharacteristic
45#107×D61C2211yescharacteristic
46#106×D61C2211yescharacteristic
47#106×D61C2211yescharacteristic
48C61×#427C411yescharacteristic, complemented
49C23×#6923—21no—
50C23×#6923—21no—
51C2×#11758213—21no—
52D61⋊M16—22no—
53D61⋊M16—22no—
54C2×#11758217—21no—
55C2×D61⋊C8C2312yes—
56#14519856—22no—
57#14519856—22no—
58C2×C4⋊#6923—21no—
59C2×C4⋊#6923—22no—
60C2×C4⋊#6923—22no—
61C2×C4⋊#6923—22no—
62C2×C4⋊#6923—21no—
63C2×C4⋊#6923—22no—
64C2×C4⋊#6923—22no—
65C2×C4⋊#6923D411yescharacteristic
66C2×C4⋊#6923D411yescharacteristic
67C2×C4⋊#6923Q814yes—
68C2×C4⋊#6923C2312yes—
69C2×C4⋊#6923C2312yes—
70C2×C22⋊#6923D412yes—
71C2×C4×#6923—22no—
72C2×C4×#6923—22no—
73C2×C4×#6923C2312yes—
74(C2×#6923)⋊C4—22no—
75(C2×#6923)⋊C4—22no—
76(C2×#6923)⋊C4—22no—
77(C2×#6923)⋊C4—22no—
78C22×C4×D61C2311yescharacteristic
79C22×#92418C2×C411yescharacteristic
80C2×C8×D61—22no—
81C2×C8×D61—22no—
82C2×C8×D61C2×C411yescharacteristic
83C2×C8×D61C2×C411yescharacteristic
84C2×C8×D61C2311yescharacteristic
85C2×C8×D61C2311yescharacteristic
86#14519736C2×C411yescharacteristic
87#14519736C2×C411yescharacteristic
88#14519735C2×C411yescharacteristic
89#14519735C2×C411yescharacteristic
90#14519735C2×C411yescharacteristic
91#14519735C2×C411yescharacteristic
92#14519734—21no—
93#14519734—21no—
94#14519733—21no—
95#14519733—21no—
96C4⋊C4×D61C2311yescharacteristic
97C4⋊C4×D61C2311yescharacteristic
98C4⋊C4×D61C2311yescharacteristic
99C4⋊C4×D61C2311yescharacteristic
100C22⋊C4×D61—21no—
101C22⋊C4×D61—21no—
102C42×D61—21no—
103C42×D61—21no—
104#14519711C2×C411yescharacteristic, complemented
105#14519711C2×C411yescharacteristic, complemented
106#14519710C2×C411yescharacteristic, complemented
107#14519710C2×C411yescharacteristic, complemented
108#14519710C2×C411yescharacteristic, complemented
109#14519710C2×C411yescharacteristic, complemented
110#14519709C2×C411yescharacteristic, complemented
111#14519709C2×C411yescharacteristic, complemented
112#14519708C2×C411yescharacteristic, complemented
113#14519708C2×C411yescharacteristic, complemented
114#14519708C2×C411yescharacteristic, complemented
115#14519708C2×C411yescharacteristic, complemented
116C22×C8×C61C2×C411yescharacteristic
117#117×C61C2×C411yescharacteristic
118#117×C61C2×C411yescharacteristic
119#107×C61C2×C411yescharacteristic, complemented
120#107×C61C2×C411yescharacteristic, complemented
121#106×C61C2×C411yescharacteristic, complemented
122#106×C61C2×C411yescharacteristic, complemented
123C23×D61C2×D411yescharacteristic
124C22×#6923—22no—
125C22×#6923—24no—
126C22×#6923—22no—
127C22×#6923—42no—
128C22×#6923—81no—
129C22×#6923—81no—
130C22×#6923—81no—
131C22×#6923—81no—
132C22×#6923C2×D412yes—
133C22×#6923C2×D412yes—
134C22×C4×C61C22×C411yescharacteristic
135C22×Dic61#4111yescharacteristic
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61C22×C411yescharacteristic
141C2×C4×D61C2×D411yescharacteristic
142C2×C4×D61C2×Q812yes—
143C2×C4×D61#4112yes—
144C2×C4×D61#4112yes—
145C2×C4×D61#4112yes—
146C2×C4×D61#4111yescharacteristic
147C2×C4×D61C2411yescharacteristic
148C22⋊#6923—24no—
149#11758217—42no—
150C2×#92420—22no—
151C4⋊#6923—24no—
152C4⋊#6923—22no—
153C4⋊#6923—22no—
154C4⋊#6923—22no—
155C4⋊#6923—22no—
156C4⋊#6923—22no—
157C4⋊#6923—42no—
158C4⋊#6923—82no—
159C4⋊#6923—82no—
160C4⋊#6923C2×Q814yes—
161C4×#6923—22no—
162C4×#6923—22no—
163C4×#6923—82no—
164C4×#6923—82no—
165#11758213—42no—
166D61⋊C8—22no—
167D61⋊C8—22no—
168C2×C8×C61—22no—
169C2×C8×C61C22×C411yescharacteristic
170C2×C8×C61C22×C411yescharacteristic
171C4⋊C4×C61C22×C411yescharacteristic
172C4⋊C4×C61C22×C411yescharacteristic
173C4⋊C4×C61C22×C411yescharacteristic
174C4⋊C4×C61C22×C411yescharacteristic
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C22⋊Dic61—21no—
180C22⋊Dic61—21no—
181#11758198—21no—
182#11758198—21no—
183#11758198—21no—
184#11758198—21no—
185C4⋊Dic61C22×C411yescharacteristic
186C4⋊Dic61C22×C411yescharacteristic
187C4⋊Dic61C22×C411yescharacteristic
188C4⋊Dic61C22×C411yescharacteristic
189#11758196—21no—
190#11758196—21no—
191#11758196—21no—
192#11758196—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C2×#92418—22no—
200C2×#92418C22×C411yescharacteristic
201C2×#92418C22×C411yescharacteristic
202C8×D61—22no—
203C8×D61—22no—
204C8×D61—21no—
205C8×D61—21no—
206C8×D61—21no—
207C8×D61—21no—
208C23×C61C4×D411yescharacteristic
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61C4×D411yescharacteristic
214C22×D61C22⋊Q812yes—
215C22×D61#12312yes—
216C22×D61C22×D411yescharacteristic
217C2×#6923—24no—
218C2×#6923—22no—
219C2×#6923—22no—
220C2×#6923—22no—
221C2×#6923—22no—
222C2×#6923—22no—
223C2×#6923—22no—
224C2×#6923—42no—
225C2×#6923—42no—
226C2×#6923—42no—
227C2×#6923—42no—
228C2×#6923—42no—
229C2×#6923—84no—
230C2×#6923—84no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61#11712yes—
236C2×C4×C61C4×D411yescharacteristic
237C2×C4×C61C4×Q812yes—
238C2×C4×C61C23×C411yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61C4×D411yescharacteristic
244C2×Dic61#12512yes—
245C2×Dic61#12612yes—
246C2×Dic61C2×#4111yescharacteristic
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—41no—
252C4×D61—41no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61#11712yes—
256C4×D61C4×Q812yes—
257C4×D61C22⋊Q812yes—
258C4×D61#12312yes—
259C4×D61#12512yes—
260C4×D61#12612yes—
261C4×D61C22×Q811yescharacteristic
262C4×D61C2×#4111yescharacteristic
263C4×D61C2×#4111yescharacteristic
264C4×D61C2×#4111yescharacteristic
265#92420—42no—
266C8×C61—21no—
267C8×C61—21no—
268#92418—21no—
269#92418—21no—
270#21039—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—21no—
273C22×C61#38712yes—
274C22×C61C2×C4×D411yescharacteristic
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61#38712yes—
282C2×D61C2×C22⋊Q811yescharacteristic
283C2×D61C2×#12311yescharacteristic
284C2×D61#57611yescharacteristic
285C2×D61#57711yescharacteristic
286C2×D61#57711yescharacteristic
287C2×D61#57811yescharacteristic
288#6923—44no—
289#6923—42no—
290#6923—42no—
291#6923—82no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61#38712yes—
297C4×C61C2×#11711yescharacteristic
298C4×C61C2×C4×Q811yescharacteristic
299Dic61—41no—
300Dic61—41no—
301Dic61—41no—
302Dic61—41no—
303Dic61#38712yes—
304Dic61C2×#12511yescharacteristic
305Dic61C2×#12611yescharacteristic
306C2×#427—611no—
307#2317—611no—
308#2274—611no—
309#1921—612no—
310#1920—612no—
311#1919—612no—
312#1918—612no—
313#1736—611no—
314#1736—611no—
315C2×#340—611no—
316#1729—611no—
317C2×C61—22no—
318C2×C61C2×#38711yescharacteristic
319C2×C61#295111yescharacteristic
320C2×C61#295211yescharacteristic
321D61—22no—
322D61#308111yescharacteristic
323D61#308211yescharacteristic
324D61#309611yescharacteristic
325D61#309711yescharacteristic
326C22×M16—611no—
327C23×C8—611no—
328C2×#117—611no—
329C2×#117—611no—
330C2×#117—611no—
331C22×C4⋊C4—611no—
332C22×C4⋊C4—611no—
333#427—611no—
334#427—611no—
335#427—611no—
336#427—611no—
337#427—611no—
338#427—611no—
339#427—611no—
340#427#692311yescharacteristic, complemented
341C2×#107—611no—
342C2×#107—611no—
343C2×#106—611no—
344C2×#106—611no—
345#389—612no—
346#389—612no—
347#385—1222no—
348#384—1221no—
349#382—1221no—
350C4×C4⋊C4—612no—
351C4×C4⋊C4—612no—
352C4×C4⋊C4—1221no—
353C4×C4⋊C4—1221no—
354C4×C22⋊C4—1222no—
355#340—612no—
356#340—612no—
357#340—612no—
358#340—612no—
359#340—612no—
360#340—612no—
361#336—612no—
362#336—612no—
363C61#2103911yescharacteristic, complemented, Sylow
364C23×C4—611no—
365C23×C4—1221no—
366C23×C4—1221no—
367C2×M16—1221no—
368C2×M16—1222no—
369C2×M16—1222no—
370C2×M16—1221no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8—611no—
374C22×C8—611no—
375C22×C8—611no—
376C22×C8—612no—
377C22×C8—1222no—
378C22×C8—1222no—
379C22×C8C2×#692311yescharacteristic
380#117—611no—
381#117—611no—
382#117—611no—
383#117—611no—
384#117—611no—
385#117—611no—
386#117—1221no—
387#117—1221no—
388#117—1221no—
389#117—1221no—
390#117—1222no—
391#117—1222no—
392#117C2×#692311yescharacteristic
393#117C2×#692311yescharacteristic
394C2×C4⋊C4—614no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—612no—
403C2×C4⋊C4—1222no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1222no—
407C2×C4⋊C4—1222no—
408C2×C4⋊C4—1222no—
409C2×C4⋊C4—1221no—
410C2×C22⋊C4—612no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C42—612no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42—1222no—
417C2×C42—1222no—
418#107—611no—
419#107—611no—
420#107—611no—
421#107—611no—
422#107—611no—
423#107—611no—
424#107C2×#692311yescharacteristic, complemented
425#107C2×#692311yescharacteristic, complemented
426#106—611no—
427#106—611no—
428#106—611no—
429#106—611no—
430#106—611no—
431#106—611no—
432#106C2×#692311yescharacteristic, complemented
433#106C2×#692311yescharacteristic, complemented
434#95—1222no—
435#95—1222no—
436#95—1222no—
437#95—1222no—
438C24—611no—
439C22×C4—611no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1224no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—4881no—
458C22×C4—4881no—
459C22×C4—4881no—
460C22×C4—4881no—
461C22×C4—2442no—
462C22×C4C22×#692311yescharacteristic
463M16—2442no—
464M16—2442no—
465C2×C8—22no—
466C2×C8—611no—
467C2×C8—611no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8C22×#692311yescharacteristic
479C2×C8C22×#692311yescharacteristic
480C4⋊C4—614no—
481C4⋊C4—611no—
482C4⋊C4—611no—
483C4⋊C4—611no—
484C4⋊C4—611no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1224no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—4882no—
496C4⋊C4—4882no—
497C4⋊C4—2442no—
498C4⋊C4C22×#692311yescharacteristic
499C4⋊C4C22×#692311yescharacteristic
500C4⋊C4C22×#692311yescharacteristic
501C4⋊C4C22×#692311yescharacteristic
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—1221no—
505C22⋊C4—1221no—
506C22⋊C4—1221no—
507C22⋊C4—1221no—
508C22⋊C4—1224no—
509C22⋊C4—1221no—
510C22⋊C4—1221no—
511C42—21no—
512C42—21no—
513C42—1221no—
514C42—1221no—
515C42—1221no—
516C42—1221no—
517C42—1222no—
518C42—1222no—
519C42—1221no—
520C42—1221no—
521C42—4882no—
522C42—4882no—
523C23—611no—
524C23—612no—
525C23—612no—
526C23—611no—
527C23—1221no—
528C23—1221no—
529C23—1221no—
530C23—1221no—
531C23D4×#692311yescharacteristic
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—612no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—612no—
548C2×C4—611no—
549C2×C4—611no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1224no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—4884no—
562C2×C4—4884no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4#1451985612yes—
577C2×C4D4×#692311yescharacteristic
578C2×C4Q8×#692312yes—
579C2×C4C23×#692311yescharacteristic
580C8—21no—
581C8—21no—
582C8—1221no—
583C8—1221no—
584C8—2442no—
585C22—21no—
586C22—21no—
587C22—611no—
588C22—611no—
589C22—611no—
590C22—611no—
591C22—611no—
592C22—611no—
593C22—612no—
594C22—1222no—
595C22—1222no—
596C22—1222no—
597C22—1222no—
598C22—1221no—
599C22—1221no—
600C22#1684071112yes—
601C22C2×D4×#692311yescharacteristic
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—611no—
607C4—611no—
608C4—612no—
609C4—4882no—
610C4—2442no—
611C4—2442no—
612C4—2444no—
613C4—2441no—
614C4—2441no—
615C4—2441no—
616C4—2441no—
617C4#1684071112yes—
618C4C2×#1451985611yescharacteristic
619C4C2×Q8×#692311yescharacteristic
620C2—22no—
621C2—611no—
622C2—611no—
623C2—611no—
624C2—611no—
625C2—1222no—
626C2C2×#1684071111yescharacteristic
627C2#2146258111yescharacteristic
628C2#2146258211yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#427C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#427#6923
hypocenterC61#21039
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #21461883C2semidirect
4. #21461882C2semidirect
5. #21461881C2semidirect
6. #21461880C2semidirect
37. #16839800C4semidirect
38. #16839799C4semidirect
39. #16839799C4semidirect
40. #16839798C4semidirect
41. #16839798C4semidirect
42. #16839797C4semidirect
43. #16839797C4semidirect
48. C61×#427C4semidirect
104. #14519711C2×C4semidirect
105. #14519711C2×C4semidirect
106. #14519710C2×C4semidirect
107. #14519710C2×C4semidirect
108. #14519710C2×C4semidirect
109. #14519710C2×C4semidirect
110. #14519709C2×C4semidirect
111. #14519709C2×C4semidirect
112. #14519708C2×C4semidirect
113. #14519708C2×C4semidirect
114. #14519708C2×C4semidirect
115. #14519708C2×C4semidirect
119. #107×C61C2×C4semidirect
120. #107×C61C2×C4semidirect
121. #106×C61C2×C4semidirect
122. #106×C61C2×C4semidirect
340. #427#6923semidirect
363. C61#21039semidirect
424. #107C2×#6923semidirect
425. #107C2×#6923semidirect
432. #106C2×#6923semidirect
433. #106C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623002
#214622562
#214618832
#214618822
#214618812
#214618802
#214615082
C2×#168406362
#214614842
#427×D612
#2103961