Finite Group Databasebeta

← #30564762

Subgroups — #30564762

Order 15616.

2142 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840417C211yesmaximal, characteristic, complemented
2#21457741C214yesmaximal, complemented
3#21457730C214yesmaximal, complemented
4#21457549C214yesmaximal, complemented
5#21457537C214yesmaximal, complemented
6#21456764C211yesmaximal, characteristic, complemented
7#21456741C211yesmaximal, characteristic, complemented
8#21456596C214yesmaximal, complemented
9#21455823C211yesmaximal, characteristic, complemented
10#21455815C211yesmaximal, characteristic, complemented
11#21455035C214yesmaximal, complemented
12#21454843C211yesmaximal, characteristic, complemented
13C61×#3506C211yesmaximal, characteristic, complemented
14C22×#11758222C2212yes—
15C2×#14519797C2218yescomplemented
16#16840417—22no—
17#16840417—22no—
18#16840417C2214yes—
19#16840417C2214yes—
20C2×#14519777C2211yescharacteristic
21C2×#14519759—22no—
22C2×#14519756C2212yes—
23C2×#14519748—21no—
24C2×C4×D244C2211yescharacteristic
25C2×C4⋊C4×D61—21no—
26#16840141C2212yescomplemented
27#16840134C2212yescomplemented
28#16840114C2212yescomplemented
29#16840114C2212yescomplemented
30#16840103C2218yescomplemented
31#16840090C2212yes—
32#16840073C2212yes—
33#16840049C2212yescomplemented
34#16840039C2212yescomplemented
35#16840001C2218yescomplemented
36#16839989C2218yescomplemented
37#16839972C2214yescomplemented
38#16839967C2214yescomplemented
39#16839964C2212yescomplemented
40#16839954C2212yescomplemented
41#16839924C2218yescomplemented
42#16839922C2218yescomplemented
43#16839917C2214yes—
44#16839912C2212yes—
45#16839905C2218yescomplemented
46#16839900C2214yes—
47#16839895C2212yes—
48C4×#11758201C2214yescomplemented
49C4×#11758199C2214yescomplemented
50D4×#92418C2214yescomplemented
51D4×#92418C2214yescomplemented
52#16839815C2211yescharacteristic
53#16839815C2211yescharacteristic
54#16839756C2218yescomplemented
55C2×C22⋊#92418C2214yes—
56C61×#559—22no—
57C61×#556C2212yescomplemented
58C61×#555C2212yescomplemented
59C61×#553—22no—
60C61×#550C2212yescomplemented
61C61×#547C2212yescomplemented
62C61×#538—22no—
63C61×#535—21no—
64C61×#533C2211yescharacteristic
65C61×#532C2211yescharacteristic
66C61×#525C2214yes—
67C61×#520—21no—
68C2×C4×D4×C61C2211yescharacteristic
69C2×#11758222—24no—
70C2×#11758222—48no—
71C2×#11758222C2318yes—
72C22×Dic122C2312yes—
73C22×D244C2311yescharacteristic
74C22×C22⋊D61—22no—
75C22×C4×D61—21no—
76C22×C4×D61—22no—
77#14519797—416no—
78C2×#11758202C2318yes—
79C2×#11758200C2318yes—
80C2×#11758201C2314yes—
81C2×#11758199C2314yes—
82C22×#92418C2318yes—
83#14519777—21no—
84#14519777—21no—
85#14519777C2312yes—
86#14519777C2312yes—
87#14519759—28no—
88#14519759—28no—
89#14519756—24no—
90#14519756C2318yes—
91#14519748—24no—
92#14519748—24no—
93C4×D244—21no—
94C4×D244—21no—
95C4×D244C2312yes—
96C4×D244C2312yes—
97C2×C4⋊Dic61—21no—
98C2×C4⋊Dic61—21no—
99C2×C4⋊Dic61C2311yescharacteristic
100C2×#11758196—22no—
101C4⋊C4×D61—24no—
102C4⋊C4×D61—24no—
103C2×C22⋊Dic61—22no—
104C2×#11758198—21no—
105#14519710C2314yes—
106#14519708C2314yes—
107C4⋊#92418C2312yes—
108C4⋊#92418C2312yes—
109#14519701C2314yes—
110#14519700C2318yes—
111#14519700C2318yes—
112#14519698C2314yes—
113#14519697C2314yes—
114#14519697C2314yes—
115C22⋊#92418C2312yes—
116C22⋊#92418C2312yes—
117C22⋊#92418C2318yes—
118C4×#92418C2312yes—
119C4×#92418C2312yes—
120C2×#41×C61—21no—
121C2×#41×C61C2314yes—
122C22×D4×C61D411yescharacteristic
123C23×C4×C61C2312yes—
124C4⋊Q8×C61C2311yescharacteristic
125C4⋊Q8×C61C2311yescharacteristic
126C4⋊Q8×C61C2311yescharacteristic
127#127×C61—22no—
128#127×C61C2311yescharacteristic
129#126×C61—24no—
130#125×C61—21no—
131#124×C61—22no—
132#124×C61—24no—
133#124×C61—21no—
134#123×C61—24no—
135#123×C61—24no—
136C22⋊Q8×C61—21no—
137C22⋊Q8×C61C2312yes—
138C22⋊Q8×C61C2314yes—
139#121×C61—22no—
140#121×C61—22no—
141#121×C61—21no—
142#121×C61—24no—
143#121×C61C2312yes—
144#121×C61C2314yes—
145#120×C61—24no—
146C4×Q8×C61—22no—
147C4×Q8×C61—21no—
148C4×Q8×C61C2312yes—
149C4×D4×C61—24no—
150C4×D4×C61—24no—
151C4×D4×C61—22no—
152C4×D4×C61—21no—
153C4×D4×C61D412yes—
154C4×D4×C61D412yes—
155C4×D4×C61C2312yes—
156C4×D4×C61C2312yes—
157C4×D4×C61C2312yes—
158#117×C61—21no—
159#117×C61—22no—
160#117×C61C2314yes—
161C2×C4⋊C4×C61—22no—
162C2×C4⋊C4×C61D411yescharacteristic
163C2×C22⋊C4×C61D412yes—
164C2×C42×C61C2311yescharacteristic
165C24×C61C2×D412yes—
166C23×D61—21no—
167#41×C61—48no—
168#41×C61—44no—
169C2×Q8×C61—21no—
170C2×Q8×C61—21no—
171C2×Q8×C61C2412yes—
172C2×D4×C61—22no—
173C2×D4×C61—24no—
174C2×D4×C61—24no—
175C2×D4×C61—24no—
176C2×D4×C61—21no—
177C2×D4×C61—21no—
178C2×D4×C61C2×D412yes—
179C2×D4×C61C2×D412yes—
180C2×D4×C61C2412yes—
181C22×C4×C61—22no—
182C22×C4×C61—24no—
183C22×C4×C61—44no—
184C22×C4×C61C2×D412yes—
185C22×C4×C61C2×D414yes—
186C22×C4×C61C2×D411yescharacteristic
187C22×C4×C61C2×D411yescharacteristic
188C22×C4×C61C2411yescharacteristic
189C22×C4×C61C2414yes—
190C2×C22⋊D61—28no—
191C2×C22⋊D61—48no—
192C22×Dic61—22no—
193C22×Dic61—21no—
194#11758222—216no—
195#11758222—48no—
196#11758222—416no—
197C2×D244—21no—
198C2×D244—24no—
199C2×D244C2412yes—
200C2×C4×D61—28no—
201C2×C4×D61—22no—
202C2×C4×D61—22no—
203C2×C4×D61—48no—
204C2×C4×D61—44no—
205C2×Dic122—22no—
206C2×Dic122—44no—
207C2×Dic122C2414yes—
208C4⋊C4×C61—21no—
209C4⋊C4×C61—24no—
210C4⋊C4×C61—21no—
211C4⋊C4×C61—22no—
212C4⋊C4×C61—21no—
213C4⋊C4×C61—21no—
214C4⋊C4×C61—22no—
215C4⋊C4×C61C2×D412yes—
216C4⋊C4×C61C2×D412yes—
217C4⋊C4×C61C2412yes—
218C4⋊C4×C61C2412yes—
219C22⋊C4×C61—22no—
220C22⋊C4×C61—24no—
221C22⋊C4×C61—24no—
222C22⋊C4×C61—24no—
223C22⋊C4×C61—22no—
224C22⋊C4×C61—22no—
225C22⋊C4×C61C2×D414yes—
226C22⋊C4×C61C2×D414yes—
227C42×C61—24no—
228C42×C61—22no—
229C42×C61C2×D411yescharacteristic
230C42×C61C2×D411yescharacteristic
231C42×C61C2411yescharacteristic
232C42×C61C2411yescharacteristic
233C22⋊Dic61—28no—
234#11758202—48no—
235#11758201—28no—
236#11758200—48no—
237#11758199—28no—
238#11758198—22no—
239#11758198—22no—
240C4⋊Dic61—22no—
241C4⋊Dic61—22no—
242C4⋊Dic61—22no—
243C4⋊Dic61—22no—
244C4⋊Dic61—21no—
245C4⋊Dic61C2412yes—
246#11758196—28no—
247C2×#92418—48no—
248C2×#92418C2414yes—
249C2×#92418C2414yes—
250C23×C61—24no—
251C23×C61—22no—
252C23×C61—42no—
253C23×C61—42no—
254C23×C61C22×D414yes—
255C23×C61C22×D411yescharacteristic
256C22×D61—24no—
257C22×D61—22no—
258C22×D61—42no—
259C22×D61—42no—
260Q8×C61—24no—
261Q8×C61—42no—
262Q8×C61—81no—
263D4×C61—24no—
264D4×C61—24no—
265D4×C61—24no—
266D4×C61—44no—
267D4×C61—42no—
268D4×C61—81no—
269D4×C61—84no—
270D4×C61—84no—
271C2×C4×C61—28no—
272C2×C4×C61—22no—
273C2×C4×C61—22no—
274C2×C4×C61—22no—
275C2×C4×C61—22no—
276C2×C4×C61—42no—
277C2×C4×C61—44no—
278C2×C4×C61—41no—
279C2×C4×C61—44no—
280C2×C4×C61—84no—
281C2×C4×C61#13514yescomplemented
282C2×C4×C61C22×D414yes—
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C22×D411yescharacteristic
285C2×C4×C61C22×D411yescharacteristic
286C2×C4×C61C22×D411yescharacteristic
287C2×C4×C61C22×D411yescharacteristic
288C2×C4×C61C2511yescharacteristic
289C22⋊D61—416no—
290C2×Dic61—22no—
291C2×Dic61—24no—
292C2×Dic61—44no—
293C2×Dic61—42no—
294D244—24no—
295D244—22no—
296D244H(5,𝔽2)14yes—
297C4×D61—416no—
298C4×D61—44no—
299C4×D61—44no—
300Dic122—28no—
301Dic122—44no—
302#92418—24no—
303#92418—24no—
304#61565—611nomaximal, Sylow
305C22×C61—21no—
306C22×C61—21no—
307C22×C61—24no—
308C22×C61—24no—
309C22×C61—24no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61—42no—
313C22×C61—84no—
314C22×C61C2×#13512yes—
315C22×C61C23×D411yescharacteristic
316C2×D61—24no—
317C2×D61—48no—
318C2×D61—42no—
319C2×D61—42no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61—24no—
324C4×C61—42no—
325C4×C61—42no—
326C4×C61—42no—
327C4×C61C2×#13512yes—
328C4×C61C2×#13512yes—
329C4×C61C2×H(5,𝔽2)12yes—
330Dic61—44no—
331Dic61—42no—
332#3506D6111yescharacteristic, complemented
333C2×#547—611no—
334#3303—614no—
335#3302—614no—
336#3284—614no—
337#3283—614no—
338#3209—611no—
339#3206—611no—
340#3190—614no—
341#3101—611no—
342#3100—611no—
343#3002—614no—
344#2972—611no—
345C2×C61—22no—
346C2×C61—44no—
347C2×C61—82no—
348C2×C61#348411yescharacteristic
349C2×C61C22×#13511yescharacteristic
350C2×C61#358611yescharacteristic
351D61—44no—
352C22×#41—612no—
353C2×#135—618no—
354#559—22no—
355#556C2×D6112yescomplemented
356#555C2×D6112yescomplemented
357#553—22no—
358#550C2×D6112yescomplemented
359#547—614no—
360#547—614no—
361#547—1222no—
362#547—1222no—
363#547C2×D6112yescomplemented
364#538—22no—
365#535—21no—
366#533C2×D6111yescharacteristic
367#532C2×D6111yescharacteristic
368C2×C4⋊Q8—611no—
369#525C2×D6114yes—
370C2×#123—1222no—
371C2×C22⋊Q8—612no—
372C2×#121—1221no—
373#520—21no—
374C2×C4×D4—611no—
375C2×C4×D4C2×D6111yescharacteristic
376C22×C4⋊C4—1221no—
377#494—612no—
378#492—612no—
379#487—612no—
380#487—612no—
381#485—618no—
382#483—612no—
383#480—612no—
384#476—612no—
385#474—612no—
386#467—618no—
387#465—618no—
388#462—614no—
389#461—612no—
390#461—614no—
391#459—612no—
392#452—618no—
393#452—618no—
394#451—614no—
395#450—612no—
396#449—618no—
397#448—614no—
398#447—612no—
399C4×SD16—614no—
400C4×D8—614no—
401C8×D4—614no—
402C8×D4—614no—
403#431—611no—
404#431—611no—
405#416—618no—
406C2×C22⋊C8—614no—
407C61#6156511yescharacteristic, complemented, Sylow
408C2×#41—21no—
409C2×#41—618no—
410C2×#41—1224no—
411C2×#41—2448no—
412C2×#41C22×D6114yes—
413C22×Q8—612no—
414C22×D4—611no—
415C22×D4—1222no—
416C22×D4C22⋊D6111yescharacteristic
417C23×C4—1221no—
418C23×C4—1222no—
419C23×C4C22×D6112yes—
420#135—24416no—
421C2×Q16—618no—
422C2×SD16—614no—
423C2×SD16—618no—
424C2×D8—614no—
425C22×C8—618no—
426C4⋊Q8—612no—
427C4⋊Q8—612no—
428C4⋊Q8—1221no—
429C4⋊Q8—1221no—
430C4⋊Q8C22×D6111yescharacteristic
431C4⋊Q8C22×D6111yescharacteristic
432C4⋊Q8C22×D6111yescharacteristic
433#127—22no—
434#127C22×D6111yescharacteristic
435#126—24no—
436#125—21no—
437#124—22no—
438#124—24no—
439#124—21no—
440#123—24no—
441#123—24no—
442#123—1228no—
443#123—1228no—
444C22⋊Q8—21no—
445C22⋊Q8—618no—
446C22⋊Q8—1224no—
447C22⋊Q8C22×D6112yes—
448C22⋊Q8C22×D6114yes—
449#121—22no—
450#121—22no—
451#121—21no—
452#121—24no—
453#121—1224no—
454#121—1224no—
455#121C22×D6112yes—
456#121C22×D6114yes—
457#120—24no—
458C4×Q8—22no—
459C4×Q8—21no—
460C4×Q8C22×D6112yes—
461C4×D4—24no—
462C4×D4—24no—
463C4×D4—22no—
464C4×D4—21no—
465C4×D4—612no—
466C4×D4—612no—
467C4×D4—1221no—
468C4×D4—1221no—
469C4×D4C22⋊D6112yes—
470C4×D4C22⋊D6112yes—
471C4×D4C22×D6112yes—
472C4×D4C22×D6112yes—
473C4×D4C22×D6112yes—
474#117—21no—
475#117—22no—
476#117C22×D6114yes—
477C2×C4⋊C4—22no—
478C2×C4⋊C4—611no—
479C2×C4⋊C4—1221no—
480C2×C4⋊C4—1222no—
481C2×C4⋊C4—1221no—
482C2×C4⋊C4—1224no—
483C2×C4⋊C4—1224no—
484C2×C4⋊C4C22⋊D6111yescharacteristic
485C2×C22⋊C4—1222no—
486C2×C22⋊C4—1221no—
487C2×C22⋊C4C22⋊D6112yes—
488C2×C42C22×D6111yescharacteristic
489#107—614no—
490#106—614no—
491C4⋊C8—612no—
492C4⋊C8—612no—
493#103—614no—
494#103—618no—
495#103—618no—
496#102—614no—
497#102—614no—
498#102—614no—
499C22⋊C8—612no—
500C22⋊C8—612no—
501C22⋊C8—618no—
502C4×C8—612no—
503C4×C8—612no—
504C24—1221no—
505C24C2×C22⋊D6112yes—
506#41—48no—
507#41—44no—
508#41—12216no—
509#41—2448no—
510#41—24416no—
511C2×Q8—21no—
512C2×Q8—21no—
513C2×Q8—614no—
514C2×Q8—1222no—
515C2×Q8—2444no—
516C2×Q8C23×D6112yes—
517C2×D4—22no—
518C2×D4—24no—
519C2×D4—24no—
520C2×D4—21no—
521C2×D4—21no—
522C2×D4—24no—
523C2×D4—612no—
524C2×D4—1224no—
525C2×D4—1228no—
526C2×D4—1221no—
527C2×D4—2448no—
528C2×D4C2×C22⋊D6112yes—
529C2×D4C2×C22⋊D6112yes—
530C2×D4C23×D6112yes—
531C22×C4—22no—
532C22×C4—24no—
533C22×C4—44no—
534C22×C4—1221no—
535C22×C4—1222no—
536C22×C4—1222no—
537C22×C4—1222no—
538C22×C4—1228no—
539C22×C4—2448no—
540C22×C4—2444no—
541C22×C4C2×C22⋊D6111yescharacteristic
542C22×C4C2×C22⋊D6112yes—
543C22×C4C2×C22⋊D6114yes—
544C22×C4C2×C22⋊D6111yescharacteristic
545C22×C4C23×D6111yescharacteristic
546C22×C4C23×D6114yes—
547Q16—2448no—
548SD16—1228no—
549SD16—2448no—
550D8—1228no—
551C2×C8—614no—
552C2×C8—614no—
553C2×C8—2448no—
554C4⋊C4—21no—
555C4⋊C4—24no—
556C4⋊C4—21no—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—612no—
562C4⋊C4—1221no—
563C4⋊C4—1228no—
564C4⋊C4—1222no—
565C4⋊C4—1222no—
566C4⋊C4—1222no—
567C4⋊C4—1222no—
568C4⋊C4C2×C22⋊D6112yes—
569C4⋊C4C2×C22⋊D6112yes—
570C4⋊C4C23×D6112yes—
571C4⋊C4C23×D6112yes—
572C22⋊C4—22no—
573C22⋊C4—24no—
574C22⋊C4—24no—
575C22⋊C4—24no—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C22⋊C4—1228no—
579C22⋊C4—1222no—
580C22⋊C4—1222no—
581C22⋊C4C2×C22⋊D6114yes—
582C22⋊C4C2×C22⋊D6114yes—
583C42—24no—
584C42—22no—
585C42C2×C22⋊D6111yescharacteristic
586C42C2×C22⋊D6111yescharacteristic
587C42C23×D6111yescharacteristic
588C42C23×D6111yescharacteristic
589C23—24no—
590C23—22no—
591C23—42no—
592C23—42no—
593C23—1224no—
594C23—1222no—
595C23—2442no—
596C23—2442no—
597C23C22×C22⋊D6114yes—
598C23C22×C22⋊D6111yescharacteristic
599Q8—24no—
600Q8—42no—
601Q8—81no—
602Q8—1228no—
603Q8—2444no—
604D4—24no—
605D4—24no—
606D4—24no—
607D4—44no—
608D4—42no—
609D4—81no—
610D4—84no—
611D4—84no—
612D4—614no—
613D4—1224no—
614D4—1222no—
615D4—24416no—
616C2×C4—28no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—84no—
626C2×C4—1222no—
627C2×C4—1224no—
628C2×C4—24416no—
629C2×C4—2444no—
630C2×C4—2442no—
631C2×C4—2444no—
632C2×C4—2444no—
633C2×C4#1451979714yescomplemented
634C2×C4C22×C22⋊D6114yes—
635C2×C4C22×C22⋊D6111yescharacteristic
636C2×C4C22×C22⋊D6111yescharacteristic
637C2×C4C22×C22⋊D6111yescharacteristic
638C2×C4C22×C22⋊D6111yescharacteristic
639C2×C4C22×C22⋊D6111yescharacteristic
640C2×C4C24×D6111yescharacteristic
641C8—1224no—
642C8—1224no—
643C22—21no—
644C22—21no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—84no—
652C22—1224no—
653C22—2448no—
654C22—2442no—
655C22—2442no—
656C22C2×#1451979712yes—
657C22C23×C22⋊D6111yescharacteristic
658C4—21no—
659C4—21no—
660C4—21no—
661C4—24no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—2442no—
666C4—2444no—
667C4C2×#1451979712yes—
668C4C2×#1451979712yes—
669C4H(5,𝔽2)×D6112yes—
670C2—22no—
671C2—44no—
672C2—82no—
673C2—2444no—
674C2#2145945011yescharacteristic
675C2C22×#1451979711yescharacteristic
676C2#2146060511yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3506C2
socleC22×C61C23×D4
hypercenter#3506D61
hypocenterC61#61565
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. C2×#16840417C2semidirect
2. #21457741C2semidirect
3. #21457730C2semidirect
4. #21457549C2semidirect
5. #21457537C2semidirect
6. #21456764C2semidirect
7. #21456741C2semidirect
8. #21456596C2semidirect
9. #21455823C2semidirect
10. #21455815C2semidirect
11. #21455035C2semidirect
12. #21454843C2semidirect
13. C61×#3506C2semidirect
15. C2×#14519797C22semidirect
26. #16840141C22semidirect
27. #16840134C22semidirect
28. #16840114C22semidirect
29. #16840114C22semidirect
30. #16840103C22semidirect
33. #16840049C22semidirect
34. #16840039C22semidirect
35. #16840001C22semidirect
36. #16839989C22semidirect
37. #16839972C22semidirect
38. #16839967C22semidirect
39. #16839964C22semidirect
40. #16839954C22semidirect
41. #16839924C22semidirect
42. #16839922C22semidirect
45. #16839905C22semidirect
48. C4×#11758201C22semidirect
49. C4×#11758199C22semidirect
50. D4×#92418C22semidirect
51. D4×#92418C22semidirect
54. #16839756C22semidirect
57. C61×#556C22semidirect
58. C61×#555C22semidirect
60. C61×#550C22semidirect
61. C61×#547C22semidirect
281. C2×C4×C61#135semidirect
332. #3506D61semidirect
355. #556C2×D61semidirect
356. #555C2×D61semidirect
358. #550C2×D61semidirect
363. #547C2×D61semidirect
407. C61#61565semidirect
633. C2×C4#14519797semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
#214577412
#214577302
#214575492
#214575372
#214567642
#214567412
#214565962
#214558232
#214558152
#214550352
#214548432
C61×#35062
#6156561