Finite Group Databasebeta

← #29655080

Subgroups — #29655080

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456625C211yesmaximal, characteristic, complemented
2#21456596C211yesmaximal, characteristic, complemented
3#21451935C211yesmaximal, characteristic, complemented
4#21444822C212yesmaximal, complemented
5#21444816C212yesmaximal, complemented
6#21444788C212yesmaximal, complemented
7#21444782C212yesmaximal, complemented
8#21444323C211yesmaximal, characteristic, complemented
9C2×#16839496C211yesmaximal, characteristic, complemented
10#21444183C211yesmaximal, characteristic, complemented
11C61×#3536C211yesmaximal, characteristic
12C2×#14519756C2211yescharacteristic
13#16840103—21no—
14#16840097—21no—
15#16840090C2211yescharacteristic, complemented
16#16840073C2212yes—
17#16840073C2211yescharacteristic, complemented
18#16840006—22no—
19#16840001—21no—
20#16839989—21no—
21#16839924—21no—
22#16839917C2211yescharacteristic, complemented
23#16839913—21no—
24#16839900C2212yes—
25#16839900C2211yescharacteristic, complemented
26#16839726C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28#16839693—21no—
29#16839692C2212yescomplemented
30#16839682C2212yescomplemented
31#16839681—21no—
32#16839620—22no—
33C2×#14519684C2211yescharacteristic
34#16839499C2212yescomplemented
35#16839496C2212yes—
36#16839496C2212yes—
37#16839496C2212yescomplemented
38#16839488C2212yescomplemented
39#16839488C2212yescomplemented
40C61×#562—22no—
41C61×#554—21no—
42C61×#548—21no—
43C61×#547—21no—
44C61×#546—21no—
45C61×#546—22no—
46C61×#543—21no—
47C61×#542—21no—
48C61×#539C2212yes—
49C61×#538—21no—
50C61×#537—21no—
51C61×#537C2212yes—
52C61×#525C2211yescharacteristic
53C2×C22⋊Q8×C61C2211yescharacteristic
54C2×#121×C61C2211yescharacteristic
55C22×Dic122C2311yescharacteristic
56C23×Dic61—21no—
57C2×#11758202—21no—
58C2×#11758200—21no—
59C2×#11758200—22no—
60C2×#11758194—21no—
61C22×#92418—21no—
62#14519756—21no—
63#14519756—21no—
64#14519756—21no—
65#14519756—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758196—21no—
68C2×#11758196—22no—
69C2×#11758196—22no—
70C2×C22⋊Dic61—21no—
71#14519710—21no—
72#14519710—22no—
73#14519708—21no—
74#14519701—21no—
75#14519700—21no—
76#14519700—21no—
77#14519700—22no—
78#14519698—21no—
79#14519698—22no—
80C22⋊#92418—21no—
81C22⋊#92418—21no—
82C22⋊#92418C2311yescharacteristic
83C22⋊#92418C2311yescharacteristic
84C22⋊#92418C2312yes—
85#14519684—22no—
86#14519684—21no—
87#14519684—22no—
88#14519684C2312yes—
89C2×#41×C61—21no—
90C2×#41×C61—22no—
91C2×#41×C61D411yescharacteristic
92C22×Q8×C61D411yescharacteristic
93C22×D4×C61—21no—
94C22×D4×C61D411yescharacteristic
95C23×C4×C61C2311yescharacteristic
96C4⋊Q8×C61—21no—
97#126×C61—22no—
98#125×C61—22no—
99#124×C61—22no—
100#124×C61—21no—
101#124×C61—22no—
102#123×C61—22no—
103#123×C61—22no—
104#123×C61—22no—
105#123×C61—22no—
106#123×C61—21no—
107C22⋊Q8×C61—21no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61—22no—
111C22⋊Q8×C61—22no—
112C22⋊Q8×C61—21no—
113C22⋊Q8×C61—21no—
114C22⋊Q8×C61—21no—
115C22⋊Q8×C61—21no—
116C22⋊Q8×C61—22no—
117C22⋊Q8×C61—22no—
118C22⋊Q8×C61D411yescharacteristic
119C22⋊Q8×C61C2311yescharacteristic
120#121×C61—21no—
121#121×C61—22no—
122#121×C61—22no—
123#121×C61—22no—
124#121×C61—22no—
125#121×C61—21no—
126#121×C61—22no—
127#121×C61—21no—
128#121×C61D411yescharacteristic
129#121×C61D412yes—
130#121×C61C2311yescharacteristic
131#121×C61C2312yes—
132#120×C61—22no—
133#120×C61—21no—
134C4×Q8×C61—21no—
135C4×D4×C61—22no—
136C4×D4×C61—21no—
137C4×D4×C61—22no—
138C4×D4×C61—21no—
139C4×D4×C61—21no—
140#117×C61D411yescharacteristic
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61D411yescharacteristic
143C2×C4⋊C4×C61D411yescharacteristic
144C2×C4⋊C4×C61C2312yes—
145C2×C22⋊C4×C61—22no—
146C2×C22⋊C4×C61—21no—
147C2×C22⋊C4×C61D412yes—
148C24×C61—21no—
149C24×C61C2×D411yescharacteristic
150#41×C61—82no—
151#41×C61—82no—
152#41×C61—81no—
153#41×C61—82no—
154C2×Q8×C61—22no—
155C2×Q8×C61—21no—
156C2×Q8×C61—21no—
157C2×Q8×C61—81no—
158C2×Q8×C61C2×D411yescharacteristic
159C2×D4×C61—21no—
160C2×D4×C61—21no—
161C2×D4×C61—22no—
162C2×D4×C61—22no—
163C2×D4×C61—22no—
164C2×D4×C61—21no—
165C2×D4×C61—22no—
166C2×D4×C61—21no—
167C2×D4×C61—81no—
168C2×D4×C61—81no—
169C2×D4×C61—81no—
170C2×D4×C61C2×D411yescharacteristic
171C2×D4×C61C2×D412yes—
172C22×C4×C61—22no—
173C22×C4×C61—22no—
174C22×C4×C61—22no—
175C22×C4×C61—21no—
176C22×C4×C61—21no—
177C22×C4×C61—81no—
178C22×C4×C61C2×D412yes—
179C22×C4×C61C2×D412yes—
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2×D411yescharacteristic
183C22×C4×C61C2×D411yescharacteristic
184C22×C4×C61C2×D411yescharacteristic
185C22×C4×C61C2411yescharacteristic
186C22×Dic61—22no—
187C22×Dic61—24no—
188C22×Dic61—21no—
189C22×Dic61—21no—
190C22×Dic61—81no—
191C22×Dic61—81no—
192C2×Dic122—21no—
193C2×Dic122—21no—
194C2×Dic122—81no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61—22no—
204C4⋊C4×C61—22no—
205C4⋊C4×C61—21no—
206C4⋊C4×C61C2×D412yes—
207C4⋊C4×C61C2×D411yescharacteristic
208C4⋊C4×C61C2×D411yescharacteristic
209C22⋊C4×C61—22no—
210C22⋊C4×C61—22no—
211C22⋊C4×C61—22no—
212C22⋊C4×C61—22no—
213C22⋊C4×C61—22no—
214C22⋊C4×C61—21no—
215C22⋊C4×C61—22no—
216C22⋊C4×C61—22no—
217C22⋊C4×C61—21no—
218C22⋊C4×C61—21no—
219C22⋊C4×C61—22no—
220C22⋊C4×C61—21no—
221C22⋊C4×C61—22no—
222C22⋊C4×C61—21no—
223C42×C61—22no—
224C42×C61—21no—
225C22⋊Dic61—41no—
226C22⋊Dic61—41no—
227#11758202—81no—
228#11758200—81no—
229#11758200—82no—
230C4⋊Dic61—21no—
231C4⋊Dic61—21no—
232#11758196—41no—
233#11758196—41no—
234#11758196—82no—
235#11758196—82no—
236#11758194—81no—
237C2×#92418—21no—
238C2×#92418—22no—
239C2×#92418—21no—
240C2×#92418—81no—
241C23×C61—21no—
242C23×C61—21no—
243C23×C61—22no—
244C23×C61—22no—
245C23×C61—21no—
246C23×C61—41no—
247C23×C61—42no—
248C23×C61—81no—
249C23×C61—81no—
250C23×C61#12011yescharacteristic
251C23×C61#12011yescharacteristic
252C23×C61C22×D411yescharacteristic
253Q8×C61—42no—
254Q8×C61—41no—
255Q8×C61—81no—
256Q8×C61—82no—
257D4×C61—42no—
258D4×C61—41no—
259D4×C61—42no—
260D4×C61—42no—
261D4×C61—41no—
262D4×C61—42no—
263D4×C61—41no—
264D4×C61—42no—
265D4×C61—82no—
266D4×C61—81no—
267D4×C61—82no—
268D4×C61—82no—
269C2×C4×C61—21no—
270C2×C4×C61—22no—
271C2×C4×C61—21no—
272C2×C4×C61—22no—
273C2×C4×C61—21no—
274C2×C4×C61—22no—
275C2×C4×C61—22no—
276C2×C4×C61—22no—
277C2×C4×C61—41no—
278C2×C4×C61—41no—
279C2×C4×C61—41no—
280C2×C4×C61—41no—
281C2×C4×C61—41no—
282C2×C4×C61—41no—
283C2×C4×C61—42no—
284C2×C4×C61—42no—
285C2×C4×C61—81no—
286C2×C4×C61—82no—
287C2×C4×C61—82no—
288C2×C4×C61—82no—
289C2×C4×C61#12011yescharacteristic
290C2×C4×C61#12011yescharacteristic
291C2×C4×C61C22×D411yescharacteristic
292C2×C4×C61C22×D411yescharacteristic
293C2×Dic61—24no—
294C2×Dic61—42no—
295C2×Dic61—42no—
296C2×Dic61—42no—
297C2×Dic61—41no—
298C2×Dic61—41no—
299C2×Dic61—41no—
300C2×Dic61—41no—
301C2×Dic61—84no—
302Dic122—42no—
303Dic122—41no—
304Dic122—41no—
305#92418—41no—
306#92418—41no—
307#92418—42no—
308#18425—611nomaximal, Sylow
309C22×C61—21no—
310C22×C61—21no—
311C22×C61—21no—
312C22×C61—21no—
313C22×C61—21no—
314C22×C61—21no—
315C22×C61—21no—
316C22×C61—42no—
317C22×C61—42no—
318C22×C61—41no—
319C22×C61—42no—
320C22×C61—41no—
321C22×C61—42no—
322C22×C61—41no—
323C22×C61—42no—
324C22×C61—162no—
325C22×C61#45612yes—
326C22×C61C2×#12011yescharacteristic
327C22×C61#57611yescharacteristic
328C22×C61#57711yescharacteristic
329C4×C61—21no—
330C4×C61—21no—
331C4×C61—21no—
332C4×C61—21no—
333C4×C61—42no—
334C4×C61—42no—
335C4×C61—41no—
336C4×C61—41no—
337C4×C61—41no—
338C4×C61—42no—
339Dic61—44no—
340Dic61—81no—
341Dic61—81no—
342#3536D6111yescharacteristic
343#3193—611no—
344#3190—611no—
345#2659—611no—
346#1610—612no—
347#1609—612no—
348#1604—612no—
349#1603—612no—
350#1504—611no—
351C2×#327—611no—
352#1472—611no—
353C2×C61—21no—
354C2×C61—22no—
355C2×C61—21no—
356C2×C61—41no—
357C2×C61—81no—
358C2×C61—82no—
359C2×C61#301211yescharacteristic
360C2×C61C2×#45611yescharacteristic
361C2×C61#302611yescharacteristic
362#562—22no—
363#554—21no—
364#548—21no—
365#547—21no—
366#546—21no—
367#546—22no—
368#543—21no—
369#542—21no—
370#539C2×D6112yes—
371#538—21no—
372#537—21no—
373#537C2×D6112yes—
374#525C2×D6111yescharacteristic
375C2×C22⋊Q8—611no—
376C2×C22⋊Q8C2×D6111yescharacteristic
377C2×#121C2×D6111yescharacteristic
378#485—1221no—
379#484—1221no—
380#483—611no—
381#480—612no—
382#480—611no—
383#468—1222no—
384#467—1221no—
385#465—1221no—
386#452—1221no—
387#451—611no—
388#450—1221no—
389#448—612no—
390#448—611no—
391#407—611no—
392C2×C22⋊C8—611no—
393#398—612no—
394#398—1221no—
395#395—612no—
396#395—1221no—
397#380—1222no—
398C2×#95—611no—
399#328—612no—
400#327—612no—
401#327—612no—
402#327—612no—
403#324—612no—
404#324—612no—
405C61#1842511yescharacteristic, complemented, Sylow
406C2×#41—21no—
407C2×#41—22no—
408C2×#41C22⋊D6111yescharacteristic
409C22×Q8—611no—
410C22×Q8C22⋊D6111yescharacteristic
411C22×D4—21no—
412C22×D4C22⋊D6111yescharacteristic
413C23×C4—1221no—
414C23×C4C22×D6111yescharacteristic
415C2×Q16—1221no—
416C2×SD16—1222no—
417C2×SD16—1221no—
418C2×M16—1221no—
419C22×C8—1221no—
420C4⋊Q8—21no—
421#126—22no—
422#125—22no—
423#124—22no—
424#124—21no—
425#124—22no—
426#123—22no—
427#123—22no—
428#123—22no—
429#123—21no—
430#123—22no—
431C22⋊Q8—21no—
432C22⋊Q8—22no—
433C22⋊Q8—22no—
434C22⋊Q8—22no—
435C22⋊Q8—22no—
436C22⋊Q8—21no—
437C22⋊Q8—21no—
438C22⋊Q8—21no—
439C22⋊Q8—21no—
440C22⋊Q8—22no—
441C22⋊Q8—22no—
442C22⋊Q8—1221no—
443C22⋊Q8—1221no—
444C22⋊Q8—1221no—
445C22⋊Q8—1221no—
446C22⋊Q8C22⋊D6111yescharacteristic
447C22⋊Q8C22×D6111yescharacteristic
448#121—21no—
449#121—22no—
450#121—22no—
451#121—22no—
452#121—22no—
453#121—21no—
454#121—21no—
455#121—22no—
456#121C22⋊D6112yes—
457#121C22⋊D6111yescharacteristic
458#121C22×D6112yes—
459#121C22×D6111yescharacteristic
460#120—22no—
461#120—21no—
462C4×Q8—21no—
463C4×D4—22no—
464C4×D4—21no—
465C4×D4—21no—
466C4×D4—21no—
467C4×D4—22no—
468#117C22⋊D6111yescharacteristic
469C2×C4⋊C4—21no—
470C2×C4⋊C4—611no—
471C2×C4⋊C4—1222no—
472C2×C4⋊C4—1222no—
473C2×C4⋊C4—1221no—
474C2×C4⋊C4C22⋊D6111yescharacteristic
475C2×C4⋊C4C22⋊D6111yescharacteristic
476C2×C4⋊C4C22×D6112yes—
477C2×C22⋊C4—22no—
478C2×C22⋊C4—21no—
479C2×C22⋊C4—1221no—
480C2×C22⋊C4C22⋊D6112yes—
481#107—1221no—
482#107—1222no—
483#106—1221no—
484#103—1221no—
485#103—1221no—
486#103—1222no—
487#103—1221no—
488#102—1222no—
489#102—1221no—
490C22⋊C8—611no—
491C22⋊C8—611no—
492C22⋊C8—612no—
493C22⋊C8—1221no—
494C22⋊C8—1221no—
495#95—612no—
496#95—1221no—
497#95—1222no—
498#95—1222no—
499C24—21no—
500C24C2×C22⋊D6111yescharacteristic
501#41—82no—
502#41—82no—
503#41—81no—
504#41—82no—
505C2×Q8—22no—
506C2×Q8—21no—
507C2×Q8—21no—
508C2×Q8—81no—
509C2×Q8—1221no—
510C2×Q8—1221no—
511C2×Q8—4881no—
512C2×Q8C2×C22⋊D6111yescharacteristic
513C2×D4—21no—
514C2×D4—22no—
515C2×D4—21no—
516C2×D4—22no—
517C2×D4—22no—
518C2×D4—21no—
519C2×D4—22no—
520C2×D4—21no—
521C2×D4—81no—
522C2×D4—81no—
523C2×D4—81no—
524C2×D4C2×C22⋊D6111yescharacteristic
525C2×D4C2×C22⋊D6112yes—
526C22×C4—22no—
527C22×C4—22no—
528C22×C4—22no—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—81no—
532C22×C4—1224no—
533C22×C4—1222no—
534C22×C4—1221no—
535C22×C4—1221no—
536C22×C4—4881no—
537C22×C4—4881no—
538C22×C4C2×C22⋊D6112yes—
539C22×C4C2×C22⋊D6112yes—
540C22×C4C2×C22⋊D6111yescharacteristic
541C22×C4C2×C22⋊D6111yescharacteristic
542C22×C4C2×C22⋊D6111yescharacteristic
543C22×C4C2×C22⋊D6111yescharacteristic
544C22×C4C2×C22⋊D6111yescharacteristic
545C22×C4C23×D6111yescharacteristic
546Q16—4881no—
547SD16—4881no—
548SD16—4882no—
549M16—4881no—
550C2×C8—1221no—
551C2×C8—1222no—
552C2×C8—1221no—
553C2×C8—4881no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—1221no—
566C4⋊C4—1221no—
567C4⋊C4—4882no—
568C4⋊C4—4882no—
569C4⋊C4—2441no—
570C4⋊C4—2441no—
571C4⋊C4C2×C22⋊D6111yescharacteristic
572C4⋊C4C2×C22⋊D6112yes—
573C4⋊C4C2×C22⋊D6111yescharacteristic
574C22⋊C4—22no—
575C22⋊C4—22no—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C22⋊C4—22no—
579C22⋊C4—21no—
580C22⋊C4—21no—
581C22⋊C4—22no—
582C22⋊C4—22no—
583C22⋊C4—21no—
584C22⋊C4—22no—
585C22⋊C4—21no—
586C22⋊C4—22no—
587C22⋊C4—21no—
588C22⋊C4—2441no—
589C22⋊C4—2441no—
590C42—22no—
591C42—21no—
592C23—22no—
593C23—21no—
594C23—21no—
595C23—22no—
596C23—21no—
597C23—41no—
598C23—42no—
599C23—81no—
600C23—81no—
601C23#1451974611yescharacteristic
602C23#1451974611yescharacteristic
603C23C22×C22⋊D6111yescharacteristic
604Q8—41no—
605Q8—42no—
606Q8—81no—
607Q8—82no—
608Q8—2442no—
609Q8—2441no—
610Q8—2441no—
611D4—42no—
612D4—41no—
613D4—42no—
614D4—42no—
615D4—42no—
616D4—41no—
617D4—42no—
618D4—41no—
619D4—82no—
620D4—81no—
621D4—82no—
622D4—82no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—81no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—1224no—
644C2×C4—4884no—
645C2×C4—2442no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4—2442no—
651C2×C4—2442no—
652C2×C4#1451974611yescharacteristic
653C2×C4#1451974611yescharacteristic
654C2×C4C22×C22⋊D6111yescharacteristic
655C2×C4C22×C22⋊D6111yescharacteristic
656C8—2441no—
657C8—2441no—
658C8—2442no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—42no—
667C22—42no—
668C22—41no—
669C22—42no—
670C22—41no—
671C22—42no—
672C22—41no—
673C22—42no—
674C22—162no—
675C22#1683994512yes—
676C22C2×#1451974611yescharacteristic
677C22#1684056511yescharacteristic
678C22#1684057311yescharacteristic
679C4—21no—
680C4—21no—
681C4—21no—
682C4—21no—
683C4—42no—
684C4—42no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—42no—
689C4—4881no—
690C4—4881no—
691C4—2444no—
692C2—21no—
693C2—22no—
694C2—21no—
695C2—41no—
696C2—82no—
697C2—81no—
698C2#2145512311yescharacteristic
699C2C2×#1683994511yescharacteristic
700C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3536C2
socleC22×C61C2×#120
hypercenter#3536D61
hypocenterC61#18425
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3536

Complements

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

NH ≅ G/Ntype
1. #21456625C2semidirect
2. #21456596C2semidirect
3. #21451935C2semidirect
4. #21444822C2semidirect
5. #21444816C2semidirect
6. #21444788C2semidirect
7. #21444782C2semidirect
8. #21444323C2semidirect
9. C2×#16839496C2semidirect
10. #21444183C2semidirect
15. #16840090C22semidirect
17. #16840073C22semidirect
22. #16839917C22semidirect
25. #16839900C22semidirect
29. #16839692C22semidirect
30. #16839682C22semidirect
34. #16839499C22semidirect
37. #16839496C22semidirect
38. #16839488C22semidirect
39. #16839488C22semidirect
405. C61#18425semidirect

Maximal subgroups

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

subgroupindex
#214566252
#214565962
#214519352
#214448222
#214448162
#214447882
#214447822
#214443232
C2×#168394962
#214441832
C61×#35362
#1842561