Finite Group Databasebeta

← #29696718

Subgroups — #29696718

Order 15616.

1004 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459397C211yesmaximal, characteristic
2M16×Dic122C211yesmaximal, characteristic, complemented
3#21448293C211yesmaximal, characteristic, complemented
4#21446798C212yesmaximal, complemented
5#21446797C211yesmaximal, characteristic, complemented
6#21446761C212yesmaximal, complemented
7#21445840C211yesmaximal, characteristic
8#21445838C212yesmaximal
9#21445628C212yesmaximal
10C4×#14519703C211yesmaximal, characteristic
11C61×#1986C211yesmaximal, characteristic, complemented
12#16840492C411yescharacteristic, complemented
13#16840489C412yescomplemented
14#16840487C411yescharacteristic, complemented
15#16840473C411yescharacteristic, complemented
16#16840470—21no—
17#16840468C412yescomplemented
18#16840466—22no—
19#16840427C411yescharacteristic, complemented
20#16840425—21no—
21#16840316—21no—
22#16840316—22no—
23#16840315C2212yes—
24C4×#11758222C2211yescharacteristic
25#16840230C2212yes—
26C2×C4×Dic122C2211yescharacteristic
27C4×Q8×D61—21no—
28#16839892—21no—
29#16839890—22no—
30C8×Dic122—21no—
31#16839786C2211yescharacteristic
32#16839785C2212yes—
33C2×#14519705C2212yes—
34C2×#14519703C2211yescharacteristic
35C2×#14519703C2211yescharacteristic
36M16×Dic61C2212yes—
37#16839667—21no—
38#16839658—21no—
39#16839644—22no—
40C4×#11758196C2212yes—
41#16839608C2212yes—
42C42×Dic61C2211yescharacteristic
43#16839523C2211yescharacteristic
44#16839523C2211yescharacteristic
45#16839522C2212yes—
46#16839521C2212yes—
47#16839521C2212yes—
48C61×#533C2211yescharacteristic
49C4×M16×C61C2211yescharacteristic
50C4×C4⋊C4×C61C2211yescharacteristic
51C61×#339C2212yes—
52C61×#339C2212yes—
53C2×#11758222C2311yescharacteristic
54C22×Dic122D411yescharacteristic
55C2×Q8×D61—21no—
56C2×#11758194C2312yes—
57#14519776—22no—
58#14519776—21no—
59#14519776C2×C412yes—
60#14519776C2×C412yes—
61#14519775—21no—
62#14519775—22no—
63#14519768C2×C412yes—
64#14519768C2×C412yes—
65#14519767—22no—
66#14519767—22no—
67#14519767—22no—
68#14519757—21no—
69#14519757—22no—
70#14519755—22no—
71#14519755—22no—
72#14519755—22no—
73#14519754—22no—
74#14519754—22no—
75#14519753—21no—
76#14519753—22no—
77Q8×Dic61—21no—
78C4×Dic122—22no—
79C4×Dic122—22no—
80C4×Dic122C2×C411yescharacteristic
81C4×Dic122C2×C411yescharacteristic
82C4×Dic122C2×C412yes—
83C4×Dic122C2×C411yescharacteristic
84C4×Dic122C2×C411yescharacteristic
85C4×Dic122D411yescharacteristic
86C4×Dic122D411yescharacteristic
87C4×Dic122D412yes—
88#14519742—21no—
89#14519742—22no—
90#14519742C2×C412yes—
91#14519742C2×C412yes—
92C4×D244C2×C411yescharacteristic
93C4×D244C2×C411yescharacteristic
94#14519739C2×C412yes—
95#14519739C2×C412yes—
96C4×C22⋊D61—21no—
97C4×C22⋊D61—21no—
98#14519737—22no—
99#14519737—22no—
100#14519736C2311yescharacteristic
101#14519734—21no—
102#14519734—22no—
103C2×C4⋊Dic61D411yescharacteristic
104C2×#11758196—22no—
105C2×#11758196D412yes—
106C4⋊C4×D61—21no—
107C4⋊C4×D61—22no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C2×C4×Dic61C2312yes—
113C42×D61—21no—
114C42×D61—22no—
115C4⋊#92418—21no—
116C4⋊#92418—21no—
117#14519706—22no—
118#14519706—22no—
119#14519705—24no—
120#14519703—22no—
121#14519703Q814yescomplemented
122#14519687—22no—
123C8×Dic61—22no—
124#14519684—22no—
125#14519684—22no—
126#14519684—22no—
127C2×M16×C61C2311yescharacteristic
128C2×M16×C61C2311yescharacteristic
129C4⋊Q8×C61C2×C411yescharacteristic
130C4⋊Q8×C61C2×C411yescharacteristic
131#125×C61C2×C412yes—
132C22⋊Q8×C61—21no—
133C22⋊Q8×C61—21no—
134C4×Q8×C61—21no—
135C4×Q8×C61—21no—
136#117×C61C2312yes—
137C2×C4⋊C4×C61—21no—
138C2×C42×C61C2312yes—
139C2×C42×C61C2311yescharacteristic
140#97×C61—21no—
141C4×C8×C61—21no—
142#95×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C22×C4×C61—22no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61C2×D412yes—
153C22×Dic61C2×D412yes—
154Q8×D61—44no—
155#11758222—21no—
156#11758222—21no—
157#11758222C2×Q812yes—
158#11758222#4112yes—
159C2×D244C22×C411yescharacteristic
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—22no—
163C2×C4×D61—21no—
164C2×C4×D61—22no—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—42no—
168C2×Dic122C22⋊C411yescharacteristic
169C2×Dic122C22⋊C411yescharacteristic
170C2×Dic122C22⋊C412yes—
171C2×Dic122C22×C411yescharacteristic
172M16×C61—21no—
173M16×C61—21no—
174M16×C61—21no—
175M16×C61C2×Q812yescomplemented
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61C22×C412yes—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—22no—
186C42×C61—22no—
187C42×C61—22no—
188C42×C61—22no—
189C42×C61C22×C411yescharacteristic
190C42×C61C22×C411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C22⋊Dic61—21no—
194#11758198—21no—
195#11758198—21no—
196#11758198—22no—
197#11758198—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61C22⋊C411yescharacteristic
201C4⋊Dic61C22⋊C411yescharacteristic
202C4⋊Dic61C22⋊C412yes—
203C4⋊Dic61C22×C411yescharacteristic
204C4⋊Dic61C22×C411yescharacteristic
205#11758196—21no—
206#11758196—21no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—44no—
214#11758196C22⋊C412yes—
215#11758196C22⋊C412yes—
216#11758196C22⋊C412yes—
217#11758196C22⋊C412yes—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61—22no—
223C4×Dic61—22no—
224C4×Dic61—21no—
225C4×Dic61—21no—
226C4×Dic61—42no—
227C4×Dic61—42no—
228C4×Dic61C22×C412yes—
229C4×Dic61C22×C412yes—
230C4×Dic61C2×D412yes—
231C4×Dic61C2×D412yes—
232C4×Dic61C2×Q812yescomplemented
233#11758194—22no—
234#11758194—22no—
235C2×#92418—22no—
236C23×C61C22×D411yescharacteristic
237C22×D61—21no—
238Q8×C61—42no—
239Q8×C61—81no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—41no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×Q811yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C612−1+411yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C22⋊D61—22no—
258C22⋊D61—22no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61C2×C22⋊C412yes—
271C2×Dic61C2×C22⋊C412yes—
272C2×Dic61C2×C22⋊C412yes—
273C2×Dic61C2×C22⋊C412yes—
274D244—21no—
275D244C4×Q812yes—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—44no—
279C4×D61—42no—
280C4×D61—82no—
281C4×D61—81no—
282Dic122—21no—
283Dic122—22no—
284Dic122—22no—
285Dic122—42no—
286Dic122—44no—
287Dic122—82no—
288Dic122C4×Q812yes—
289C8×C61—22no—
290C8×C61—41no—
291#92418—42no—
292#22529—611nomaximal, Sylow
293C22×C61—21no—
294C22×C61—21no—
295C22×C61C22×C22⋊C411yescharacteristic
296C22×C61#54811yescharacteristic
297C22×C61D4×Q811yescharacteristic
298C2×D61—22no—
299C2×D61—41no—
300C2×D61—41no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61—82no—
307C4×C61#42112yes—
308C4×C61C2×C4×Q811yescharacteristic
309C4×C61#51911yescharacteristic
310C4×C61#54811yescharacteristic
311C4×C61D4×Q811yescharacteristic
312Dic61—22no—
313Dic61—22no—
314Dic61—22no—
315Dic61—22no—
316Dic61—22no—
317Dic61—42no—
318Dic61—42no—
319Dic61—41no—
320Dic61—41no—
321Q8×#41—611no—
322Q8×M16—611no—
323#2295—611no—
324#1995—612no—
325#1994—611no—
326#1986—612no—
327#1986D6111yescharacteristic, complemented
328#1806—611no—
329#1806—612no—
330#1759—612no—
331C4×#104—611no—
332C2×C61—21no—
333C2×C61—21no—
334C2×C61Q8×C22⋊C411yescharacteristic
335C2×C61C2×#42111yescharacteristic
336C2×C61#290111yescharacteristic
337D61—42no—
338Q82—611no—
339#557—611no—
340#557—612no—
341#554—611no—
342#554—612no—
343#554—1222no—
344#554—1221no—
345D4×Q8—611no—
346D4×Q8—1221no—
347#533—612no—
348#533—1222no—
349#533—1221no—
350#533C2×D6111yescharacteristic
351C4×#41—612no—
352C4×#41—611no—
353C2×C4×Q8—611no—
354C2×C4×Q8—1221no—
355#446—1221no—
356#446—1222no—
357C8×Q8—1221no—
358#423—612no—
359#423—611no—
360C2×#104—611no—
361C2×#104—611no—
362C2×#104—612no—
363C4×M16—612no—
364C4×M16C2×D6111yescharacteristic
365#391—1221no—
366#389—1221no—
367#387—1222no—
368C4×C4⋊C4—612no—
369C4×C4⋊C4C2×D6111yescharacteristic
370#376—612no—
371C43—611no—
372#339—611no—
373#339—611no—
374#339—612no—
375#339—612no—
376#339—612no—
377#339C2×D6112yes—
378#339C2×D6112yes—
379C61#2252911yescharacteristic, complemented, Sylow
380C2×#41—611no—
381C22×Q8—611no—
382C22×Q8—1221no—
383C2×M16—612no—
384C2×M16C22×D6111yescharacteristic
385C2×M16C22×D6111yescharacteristic
386C4⋊Q8—612no—
387C4⋊Q8—612no—
388C4⋊Q8—1221no—
389C4⋊Q8—1222no—
390C4⋊Q8—1222no—
391C4⋊Q8—1221no—
392C4⋊Q8C4×D6111yescharacteristic
393C4⋊Q8C4×D6111yescharacteristic
394#125—612no—
395#125—612no—
396#125—1222no—
397#125—1222no—
398#125—1222no—
399#125C4×D6112yes—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404C22⋊Q8—1222no—
405C22⋊Q8—1222no—
406C22⋊Q8—1222no—
407C22⋊Q8—1222no—
408C22⋊Q8—1221no—
409C22⋊Q8—1221no—
410C22⋊Q8—1222no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—611no—
414C4×Q8—611no—
415C4×Q8—612no—
416C4×Q8—611no—
417C4×Q8—611no—
418C4×Q8—612no—
419C4×Q8—612no—
420C4×Q8—612no—
421C4×Q8—611no—
422C4×Q8—611no—
423C4×Q8—1222no—
424C4×Q8—1222no—
425C4×Q8—1222no—
426C4×Q8—1221no—
427C4×Q8—1221no—
428C4×D4—612no—
429C4×D4—612no—
430C4×D4—611no—
431C4×D4—611no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—1221no—
435C4×D4—1221no—
436#117—611no—
437#117—1222no—
438#117—1221no—
439#117C22×D6112yes—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—612no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—1222no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1222no—
446C2×C42—612no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—1222no—
450C2×C42—1221no—
451C2×C42—1221no—
452C2×C42—1221no—
453C2×C42C22×D6112yes—
454C2×C42C22×D6111yescharacteristic
455C4⋊C8—1222no—
456C4⋊C8—1222no—
457C4⋊C8—1221no—
458C4⋊C8—1221no—
459#104—614no—
460#104—1224no—
461#104—1222no—
462#97—21no—
463#97—1222no—
464C4×C8—21no—
465C4×C8—1222no—
466#95—21no—
467#95—1222no—
468#95—1222no—
469#95—1222no—
470#41—612no—
471#41—612no—
472#41—1221no—
473#41—1221no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—611no—
477C2×Q8—611no—
478C2×Q8—612no—
479C2×Q8—611no—
480C2×Q8—1222no—
481C2×Q8—1222no—
482C2×Q8—2442no—
483C2×Q8—2444no—
484C2×D4—611no—
485C2×D4—1221no—
486C22×C4—22no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4D4×D6111yescharacteristic
497C22×C4D4×D6111yescharacteristic
498C22×C4C23×D6111yescharacteristic
499M16—21no—
500M16—21no—
501M16—21no—
502M16—1222no—
503M16—1222no—
504M16Q8×D6112yescomplemented
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—1222no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—42no—
513C4⋊C4—612no—
514C4⋊C4—611no—
515C4⋊C4—612no—
516C4⋊C4—612no—
517C4⋊C4—612no—
518C4⋊C4—612no—
519C4⋊C4—611no—
520C4⋊C4—611no—
521C4⋊C4—611no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—2444no—
533C4⋊C4C2×C4×D6112yes—
534C4⋊C4C2×C4×D6112yes—
535C22⋊C4—22no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1221no—
539C22⋊C4—1221no—
540C22⋊C4—1221no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—612no—
545C42—612no—
546C42—612no—
547C42—612no—
548C42—612no—
549C42—1222no—
550C42—1222no—
551C42—1221no—
552C42—1221no—
553C42—1221no—
554C42—1221no—
555C42—1221no—
556C42—1221no—
557C42—2442no—
558C42—2442no—
559C42C2×C4×D6111yescharacteristic
560C42C2×C4×D6111yescharacteristic
561C42D4×D6111yescharacteristic
562C42D4×D6111yescharacteristic
563C23—1221no—
564C23C2×D4×D6111yescharacteristic
565Q8—42no—
566Q8—81no—
567Q8—612no—
568Q8—1221no—
569Q8—1222no—
570Q8—1222no—
571Q8—4882no—
572Q8—2442no—
573Q8—2444no—
574D4—612no—
575D4—1222no—
576D4—1222no—
577D4—1221no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—1221no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1222no—
596C2×C4—4881no—
597C2×C4—4882no—
598C2×C4—2444no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2442no—
606C2×C4C22⋊C4×D6111yescharacteristic
607C2×C4C22⋊C4×D6111yescharacteristic
608C2×C4C22⋊C4×D6111yescharacteristic
609C2×C4C22⋊C4×D6111yescharacteristic
610C2×C4C22×C4×D6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4C2×Q8×D6111yescharacteristic
613C2×C4#41×D6111yescharacteristic
614C2×C4#1451982411yescharacteristic
615C2×C4#1451982511yescharacteristic
616C8—22no—
617C8—41no—
618C8—2442no—
619C22—21no—
620C22—21no—
621C22—1222no—
622C22—2441no—
623C22—2441no—
624C22C2×C22⋊C4×D6111yescharacteristic
625C22#1684042011yescharacteristic
626C22#1684042711yescharacteristic
627C4—21no—
628C4—21no—
629C4—21no—
630C4—42no—
631C4—42no—
632C4—82no—
633C4—1222no—
634C4—1222no—
635C4—1222no—
636C4—1222no—
637C4—1222no—
638C4—2442no—
639C4—2442no—
640C4—2441no—
641C4—2441no—
642C4#1683977712yes—
643C4C4×Q8×D6111yescharacteristic
644C4#1684023811yescharacteristic
645C4#1684042211yescharacteristic
646C4#1684042511yescharacteristic
647C2—21no—
648C2—21no—
649C2—2442no—
650C2#2144865311yescharacteristic
651C2C2×#1683977711yescharacteristic
652C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1986C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758222C23
cosocleC22×C4×C61C24
hypercenter#1986D61
hypocenterC61#22529
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
2. M16×Dic122C2semidirect
3. #21448293C2semidirect
4. #21446798C2semidirect
5. #21446797C2semidirect
6. #21446761C2semidirect
11. C61×#1986C2semidirect
12. #16840492C4semidirect
13. #16840489C4semidirect
14. #16840487C4semidirect
15. #16840473C4semidirect
17. #16840468C4semidirect
19. #16840427C4semidirect
121. #14519703Q8semidirect
175. M16×C61C2×Q8semidirect
232. C4×Dic61C2×Q8semidirect
327. #1986D61semidirect
379. C61#22529semidirect
504. M16Q8×D61semidirect

Maximal subgroups

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

subgroupindex
#214593972
M16×Dic1222
#214482932
#214467982
#214467972
#214467612
#214458402
#214458382
#214456282
C4×#145197032
C61×#19862
#2252961