Finite Group Databasebeta

← #29647782

Subgroups — #29647782

Order 15616.

990 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459615C211yesmaximal, characteristic
2#21454846C211yesmaximal, characteristic
3#21448756C211yesmaximal, characteristic
4#21444389C212yesmaximal, complemented
5#21444368C212yesmaximal, complemented
6#21444351C211yesmaximal, characteristic, complemented
7#21444340C211yesmaximal, characteristic, complemented
8#21444339C211yesmaximal, characteristic, complemented
9#21444329C212yesmaximal, complemented
10#328×D61C212yesmaximal, complemented
11C61×#1511C211yesmaximal, characteristic, complemented
12#16840470C412yescomplemented
13#16840467—22no—
14#16840467C412yescomplemented
15#16840466—22no—
16Q8×C22⋊D61C412yescomplemented
17#16840425C412yescomplemented
18#16840391—21no—
19#16840388—21no—
20#16840387—21no—
21#16840314C2212yes—
22C2×#14519756C2211yescharacteristic
23C22⋊Q8×D61C2212yes—
24#16840235—21no—
25C2×C4×C22⋊D61C2211yescharacteristic
26#16839836—22no—
27#16839831—22no—
28#16839816C2212yes—
29#16839726C2211yescharacteristic
30C22⋊C8×D61C2212yes—
31#16839656—21no—
32#16839644—21no—
33#16839638—21no—
34#16839629C2211yescharacteristic
35#16839621C2211yescharacteristic
36C4×C22⋊Dic61C2211yescharacteristic
37#16839609—21no—
38#16839607C2211yescharacteristic
39#95×D61C2211yescharacteristic
40#16839500C2212yes—
41#16839500C2212yes—
42#16839499C2212yes—
43#16839499C2212yes—
44#16839498C2212yes—
45#16839498C2212yes—
46C2×C22⋊Q8×C61C2211yescharacteristic
47C61×#407C2211yescharacteristic
48C61×#381C2211yescharacteristic
49C61×#328C2212yes—
50C61×#328C2212yes—
51C22×Dic122C2×C411yescharacteristic, complemented
52C2×Q8×D61C2×C412yes—
53C22×C22⋊D61D411yescharacteristic
54C22×C4×D61—21no—
55C22×C4×D61C2311yescharacteristic
56C2×#11758194—21no—
57C2×C8×D61—22no—
58#14519776—22no—
59#14519776C2×C412yescomplemented
60#14519775—22no—
61#14519775C2×C412yescomplemented
62#14519767—22no—
63#14519767—22no—
64#14519767—22no—
65#14519757—22no—
66#14519757C2×C412yes—
67#14519756—21no—
68#14519756—21no—
69#14519756—22no—
70#14519756—22no—
71#14519756C2×C412yescomplemented
72#14519756C2312yes—
73#14519755—22no—
74#14519755—22no—
75#14519755—22no—
76#14519754—22no—
77#14519754—22no—
78#14519754—22no—
79#14519754—22no—
80#14519753—22no—
81#14519753C2×C412yes—
82Q8×Dic61C2×C412yescomplemented
83#14519742—22no—
84#14519742C2×C412yescomplemented
85C4×C22⋊D61—22no—
86C4×C22⋊D61—22no—
87C4×C22⋊D61C2×C412yes—
88C4×C22⋊D61C2×C412yes—
89C4×C22⋊D61D412yes—
90C4×C22⋊D61D412yes—
91#14519737—22no—
92#14519737—22no—
93#14519733—22no—
94C2×C4⋊Dic61C2×C411yescharacteristic, complemented
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196D411yescharacteristic
98C4⋊C4×D61—22no—
99C4⋊C4×D61C2×C412yes—
100C2×C22⋊Dic61—21no—
101C2×C22⋊Dic61—21no—
102C2×C22⋊Dic61D411yescharacteristic
103C2×#11758198—21no—
104C2×#11758198D411yescharacteristic
105C22⋊C4×D61—22no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61—21no—
108C2×C4×Dic61C2311yescharacteristic
109#14519706—22no—
110C22⋊#92418—21no—
111C22⋊#92418C2312yes—
112#14519689—22no—
113#14519687—22no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684C2311yescharacteristic
119#14519684C2311yescharacteristic
120C22×Q8×C61C2×C411yescharacteristic, complemented
121C23×C4×C61C2311yescharacteristic
122C2×M16×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61C2×C412yescomplemented
126C22⋊Q8×C61C2312yes—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
129C2×C22⋊C4×C61—21no—
130C2×C22⋊C4×C61—21no—
131C22⋊C8×C61—21no—
132C22⋊C8×C61C2312yes—
133#95×C61—21no—
134#95×C61C2311yescharacteristic
135#95×C61C2311yescharacteristic
136C24×C61C2×D411yescharacteristic
137C23×D61C2×D411yescharacteristic
138C2×Q8×C61—21no—
139C2×Q8×C61—81no—
140C2×Q8×C61C22×C412yes—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—81no—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61C22⋊C412yes—
155C2×C22⋊D61C22⋊C412yes—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163Q8×D61—82no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61—81no—
169C2×C4×D61—81no—
170C2×C4×D61C22×C412yes—
171C2×C4×D61C2×D412yes—
172C2×Dic122—21no—
173C2×Dic122—22no—
174C2×Dic122—22no—
175C2×Dic122—81no—
176C2×Dic122C22×C412yes—
177M16×C61—81no—
178C2×C8×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61C22×C412yes—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—81no—
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—42no—
189C22⋊Dic61C22⋊C412yes—
190C22⋊Dic61C22⋊C412yes—
191#11758198—22no—
192#11758198—22no—
193#11758198—42no—
194#11758198C22⋊C412yes—
195#11758198C22⋊C412yes—
196C4⋊Dic61—21no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61C22×C412yes—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—81no—
209#11758196C22⋊C412yes—
210#11758196C22⋊C412yes—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61—42no—
214C4×Dic61—81no—
215C4×Dic61C22×C412yes—
216C4×Dic61C2×D412yes—
217#11758194—81no—
218C2×#92418—22no—
219C8×D61—82no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61C2×C22⋊C411yescharacteristic
225C23×C61C2×C22⋊C411yescharacteristic
226C23×C61C22×D411yescharacteristic
227C22×D61—41no—
228C22×D61—41no—
229C22×D61C2×C22⋊C412yes—
230C22×D61#13712yes—
231C22×D61#13712yes—
232Q8×C61—42no—
233Q8×C61—81no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
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—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—81no—
249C2×C4×C61—81no—
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C22⋊D61—24no—
255C22⋊D61—24no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—81no—
269C2×Dic61—81no—
270C2×Dic61C2×C22⋊C412yes—
271C2×Dic61C2×C22⋊C412yes—
272C2×Dic61C2×C22⋊C412yes—
273C4×D61—42no—
274C4×D61—42no—
275C4×D61—82no—
276C4×D61—82no—
277C4×D61—84no—
278Dic122—42no—
279Dic122—82no—
280Dic122—82no—
281Dic122—81no—
282C8×C61—42no—
283#92418—42no—
284#17621—611nomaximal, Sylow
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61C22×C22⋊C411yescharacteristic
296C22×C61C2×#13711yescharacteristic
297C22×C61C2×#13711yescharacteristic
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61#41914yes—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—42no—
308C4×C61—42no—
309C4×C61—81no—
310C4×C61—81no—
311Dic61—22no—
312Dic61—22no—
313Dic61—22no—
314Dic61—42no—
315Dic61—42no—
316Dic61—42no—
317Dic61—81no—
318#3501—611no—
319#2973—611no—
320#2353—611no—
321#1520—612no—
322#1515—612no—
323#1511—611no—
324#1511D6111yescharacteristic, complemented
325#1508—611no—
326#1508—611no—
327#1506—612no—
328C2×#328—612no—
329C2×C61—21no—
330C2×C61—21no—
331C2×C61—41no—
332C2×C61—41no—
333C2×C61#288311yescharacteristic
334C2×C61C2×#41911yescharacteristic
335C2×C61#290111yescharacteristic
336D61—24no—
337#554—612no—
338#554—612no—
339#554—1222no—
340#554—1222no—
341D4×Q8—612no—
342D4×Q8—612no—
343#544—1221no—
344#543—1221no—
345#543—1221no—
346#533—612no—
347C2×C22⋊Q8—611no—
348C2×C22⋊Q8—612no—
349C2×C22⋊Q8C2×D6111yescharacteristic
350#518—1221no—
351C2×C4×D4—611no—
352#435—1222no—
353#435—1222no—
354#432—612no—
355#407—611no—
356#407C2×D6111yescharacteristic
357C2×C22⋊C8—612no—
358#388—1221no—
359#387—1221no—
360#386—1221no—
361#383—611no—
362#381C2×D6111yescharacteristic
363#380—611no—
364C4×C22⋊C4—611no—
365C4×C22⋊C4—1221no—
366#376—611no—
367C2×#95—611no—
368#328—612no—
369#328—612no—
370#328—612no—
371#328—612no—
372#328—612no—
373#328—612no—
374#328C2×D6112yes—
375#328C2×D6112yes—
376C61#1762111yescharacteristic, complemented, Sylow
377C22×Q8—612no—
378C22×Q8—611no—
379C22×Q8C4×D6111yescharacteristic, complemented
380C22×D4—611no—
381C23×C4—611no—
382C23×C4—1221no—
383C23×C4C22×D6111yescharacteristic
384C2×M16—21no—
385C2×M16—1221no—
386C22×C8—1222no—
387C4⋊Q8—612no—
388C4⋊Q8—612no—
389C4⋊Q8—1222no—
390C4⋊Q8—1222no—
391#125—1222no—
392#125—1222no—
393#125—1222no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—612no—
397C22⋊Q8—612no—
398C22⋊Q8—612no—
399C22⋊Q8—612no—
400C22⋊Q8—1222no—
401C22⋊Q8—1222no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404C22⋊Q8—1222no—
405C22⋊Q8—1222no—
406C22⋊Q8—1222no—
407C22⋊Q8—1222no—
408C22⋊Q8—1222no—
409C22⋊Q8—1222no—
410C22⋊Q8—1222no—
411C22⋊Q8—1221no—
412C22⋊Q8—1221no—
413C22⋊Q8C4×D6112yescomplemented
414C22⋊Q8C22×D6112yes—
415C4×Q8—612no—
416C4×Q8—612no—
417C4×Q8—1222no—
418C4×D4—612no—
419C4×D4—612no—
420C4×D4—612no—
421C4×D4—612no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426#117—1222no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—612no—
431C2×C4⋊C4—1222no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4C4×D6111yescharacteristic, complemented
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—611no—
438C2×C22⋊C4—611no—
439C2×C22⋊C4—1222no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C42—611no—
444C2×C42—1221no—
445C2×C42—1221no—
446C4⋊C8—1222no—
447C22⋊C8—21no—
448C22⋊C8—612no—
449C22⋊C8—1222no—
450C22⋊C8—1221no—
451C22⋊C8C22×D6112yes—
452#97—1222no—
453#95—21no—
454#95—611no—
455#95—611no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95C22×D6111yescharacteristic
461#95C22×D6111yescharacteristic
462C24—611no—
463C24D4×D6111yescharacteristic
464C2×Q8—21no—
465C2×Q8—81no—
466C2×Q8—612no—
467C2×Q8—1222no—
468C2×Q8—1222no—
469C2×Q8—1221no—
470C2×Q8—4881no—
471C2×Q8—4882no—
472C2×Q8C2×C4×D6112yes—
473C2×D4—612no—
474C2×D4—612no—
475C2×D4—1222no—
476C2×D4—1222no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—81no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—612no—
486C22×C4—612no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—4881no—
494C22×C4—4881no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4C2×C4×D6111yescharacteristic
498C22×C4C2×C4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500C22×C4D4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503M16—81no—
504M16—4881no—
505C2×C8—22no—
506C2×C8—1222no—
507C2×C8—4882no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—612no—
512C4⋊C4—612no—
513C4⋊C4—612no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1221no—
524C4⋊C4—1222no—
525C4⋊C4—4881no—
526C4⋊C4C2×C4×D6112yes—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—81no—
530C22⋊C4—612no—
531C22⋊C4—612no—
532C22⋊C4—612no—
533C22⋊C4—612no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—2442no—
539C22⋊C4—2442no—
540C42—612no—
541C42—612no—
542C42—1222no—
543C42—1222no—
544C42—4881no—
545C42—2442no—
546C23—21no—
547C23—21no—
548C23—41no—
549C23—41no—
550C23—612no—
551C23—612no—
552C23—612no—
553C23—2441no—
554C23—2441no—
555C23C22⋊C4×D6111yescharacteristic
556C23C22⋊C4×D6111yescharacteristic
557C23C2×D4×D6111yescharacteristic
558Q8—42no—
559Q8—81no—
560Q8—4881no—
561Q8—4882no—
562Q8—4882no—
563Q8—2442no—
564D4—1224no—
565D4—1224no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—612no—
583C2×C4—612no—
584C2×C4—612no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—4882no—
591C2×C4—4882no—
592C2×C4—4884no—
593C2×C4—4881no—
594C2×C4—4881no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4C22⋊C4×D6111yescharacteristic
605C2×C4C22⋊C4×D6111yescharacteristic
606C2×C4C22×C4×D6111yescharacteristic
607C2×C4C2×D4×D6111yescharacteristic
608C8—42no—
609C8—2442no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—81no—
619C22—81no—
620C22—614no—
621C22—1222no—
622C22—1222no—
623C22—2442no—
624C22—2442no—
625C22C2×C22⋊C4×D6111yescharacteristic
626C22#137×D6111yescharacteristic
627C22#137×D6111yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—42no—
633C4—42no—
634C4—81no—
635C4—81no—
636C4—1222no—
637C4—1222no—
638C4—1222no—
639C4—4881no—
640C4—2442no—
641C4—2442no—
642C4—2442no—
643C2—21no—
644C2—21no—
645C2—41no—
646C2—41no—
647C2—1224no—
648C2#2145427711yescharacteristic
649C2#419×D6111yescharacteristic
650C2#2145436511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1511C2
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61D4
cosocleC22×C4×C61C24
hypercenter#1511D61
hypocenterC61#17621
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
4. #21444389C2semidirect
5. #21444368C2semidirect
6. #21444351C2semidirect
7. #21444340C2semidirect
8. #21444339C2semidirect
9. #21444329C2semidirect
10. #328×D61C2semidirect
11. C61×#1511C2semidirect
12. #16840470C4semidirect
14. #16840467C4semidirect
16. Q8×C22⋊D61C4semidirect
17. #16840425C4semidirect
51. C22×Dic122C2×C4semidirect
59. #14519776C2×C4semidirect
61. #14519775C2×C4semidirect
71. #14519756C2×C4semidirect
82. Q8×Dic61C2×C4semidirect
84. #14519742C2×C4semidirect
94. C2×C4⋊Dic61C2×C4semidirect
120. C22×Q8×C61C2×C4semidirect
125. C22⋊Q8×C61C2×C4semidirect
128. C2×C4⋊C4×C61C2×C4semidirect
324. #1511D61semidirect
376. C61#17621semidirect
379. C22×Q8C4×D61semidirect
413. C22⋊Q8C4×D61semidirect
434. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214596152
#214548462
#214487562
#214443892
#214443682
#214443512
#214443402
#214443392
#214443292
#328×D612
C61×#15112
#1762161