Finite Group Databasebeta

← #30564778

Subgroups — #30564778

Order 15616.

2194 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459668C211yesmaximal, characteristic, complemented
2C2×#16840434C211yesmaximal, characteristic, complemented
3#21457771C214yesmaximal, complemented
4#21457755C214yesmaximal, complemented
5#21457574C214yesmaximal, complemented
6#21457550C214yesmaximal, complemented
7#21456762C211yesmaximal, characteristic, complemented
8#21456734C211yesmaximal, characteristic, complemented
9#21456583C214yesmaximal, complemented
10#21455858C211yesmaximal, characteristic, complemented
11#21455036C214yesmaximal, complemented
12#21454842C211yesmaximal, characteristic, complemented
13C61×#3103C211yesmaximal, characteristic, complemented
14C22×#11758226C2212yes—
15C2×#14519798C2218yescomplemented
16#16840494—22no—
17#16840486C2212yescomplemented
18#16840477C2212yescomplemented
19#16840458—22no—
20#16840434—22no—
21#16840434—22no—
22#16840434C2214yes—
23#16840434C2214yes—
24#16840432C2212yescomplemented
25#16840416C2212yescomplemented
26#16840349—22no—
27#16840326—21no—
28#16840319C2211yescharacteristic
29C2×#14519774—21no—
30C2×#14519774C2211yescharacteristic
31#16840272C2214yes—
32C2×#14519752—22no—
33C2×#14519752C2212yes—
34#16840245—21no—
35C2×C4×D244—21no—
36C2×C4×D244C2211yescharacteristic
37#16840138C2212yescomplemented
38#16840135C2212yescomplemented
39#16840110C2212yescomplemented
40#16840109C2212yescomplemented
41#16840097C2212yes—
42#16840088C2218yescomplemented
43#16840057C2212yescomplemented
44#16839997C2218yescomplemented
45#16839992C2218yescomplemented
46#16839972C2212yescomplemented
47#16839963C2214yescomplemented
48#16839957C2214yescomplemented
49#16839927C2218yescomplemented
50#16839918C2212yes—
51#16839915C2214yes—
52#16839908C2218yescomplemented
53#16839906C2218yescomplemented
54#16839895C2214yes—
55C4×#11758202C2214yescomplemented
56C8×D244C2214yescomplemented
57#16839827C2214yescomplemented
58#16839815C2211yescharacteristic
59#16839757C2218yescomplemented
60C22⋊C8×D61C2214yes—
61C61×#532C2211yescharacteristic
62C2×C4×Q8×C61C2211yescharacteristic
63C61×#481C2212yes—
64C61×#475C2212yescomplemented
65C61×#460C2212yescomplemented
66C61×#448C2212yes—
67C4×SD16×C61C2214yescomplemented
68C61×#431C2211yescharacteristic
69C2×#11758226—24no—
70C2×#11758226—48no—
71C2×#11758226C2318yes—
72C2×#11758224C2314yes—
73C22×D244—22no—
74C22×D244—21no—
75C22×D244—22no—
76C22×D244—21no—
77C22×D244D411yescharacteristic
78C22×D244C2312yes—
79C22×C4×D61—22no—
80C22×C4×D61C2312yes—
81#14519798—416no—
82C2×#11758202C2314yes—
83C2×#11758190C2318yes—
84C2×#11758199C2318yes—
85C2×C8×D61C2318yes—
86#14519777C2311yescharacteristic
87#14519777C2311yescharacteristic
88#14519774—24no—
89#14519774—24no—
90#14519774—21no—
91#14519774—21no—
92#14519774C2312yes—
93#14519774C2312yes—
94#14519773—22no—
95#14519772—24no—
96#14519769—21no—
97#14519766—22no—
98#14519763—24no—
99#14519761—24no—
100#14519759—24no—
101#14519757C2314yes—
102#14519756—21no—
103#14519756C2312yes—
104#14519752—24no—
105#14519752—28no—
106#14519752—28no—
107#14519752C2318yes—
108#14519751—22no—
109#14519751—22no—
110#14519749—24no—
111#14519748C2314yes—
112#14519745—24no—
113Q8×Dic61—22no—
114C4×Dic122—21no—
115C4×Dic122C2312yes—
116C4×D244—24no—
117C4×D244—24no—
118C4×D244D412yes—
119C4×D244D412yes—
120C4×D244C2312yes—
121C4×D244C2312yes—
122D4×Dic61—24no—
123D4×Dic61—22no—
124#14519737—24no—
125#14519736—21no—
126#14519736—22no—
127#14519735C2314yes—
128C2×C4⋊Dic61—22no—
129C2×C4⋊Dic61D411yescharacteristic
130C2×#11758198—22no—
131C2×#11758198D412yes—
132#14519710C2314yes—
133C4⋊#92418C2312yes—
134#14519701C2314yes—
135#14519700C2314yes—
136#14519699C2318yes—
137#14519697C2318yes—
138C22⋊#92418C2312yes—
139#14519689C2318yes—
140C4×#92418C2312yes—
141C2×#41×C61—21no—
142C22×Q8×C61C2311yescharacteristic
143C2×SD16×C61C2314yes—
144C4⋊Q8×C61C2311yescharacteristic
145#127×C61C2311yescharacteristic
146#124×C61—21no—
147#121×C61—21no—
148#121×C61C2312yes—
149C4×Q8×C61—21no—
150C4×Q8×C61—21no—
151C4×Q8×C61C2312yes—
152C4×Q8×C61C2312yes—
153C4×D4×C61—21no—
154C4×D4×C61C2312yes—
155C2×C4⋊C4×C61—21no—
156C2×C4⋊C4×C61C2311yescharacteristic
157C2×C42×C61—21no—
158C2×C42×C61C2311yescharacteristic
159#106×C61C2314yes—
160C4⋊C8×C61C2312yes—
161#103×C61C2314yes—
162#102×C61C2314yes—
163C22⋊C8×C61C2312yes—
164C4×C8×C61C2312yes—
165C23×D61—22no—
166C23×D61C2×D412yes—
167#41×C61—44no—
168C2×Q8×C61—24no—
169C2×Q8×C61—21no—
170C2×Q8×C61—21no—
171C2×Q8×C61C2412yes—
172C2×D4×C61—22no—
173C2×D4×C61—21no—
174C2×D4×C61—21no—
175C2×D4×C61C2412yes—
176C22×C4×C61—21no—
177C22×C4×C61—21no—
178C22×C4×C61—22no—
179C22×C4×C61C2×D411yescharacteristic
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2411yescharacteristic
182C2×C22⋊D61—24no—
183C2×C22⋊D61—24no—
184C22×Dic61—24no—
185C22×Dic61C2×D412yes—
186#11758226—216no—
187#11758226—48no—
188#11758226—416no—
189#11758224—48no—
190C2×D244—24no—
191C2×D244—28no—
192C2×D244—22no—
193C2×D244—22no—
194C2×D244—22no—
195C2×D244—28no—
196C2×D244—22no—
197C2×D244—22no—
198C2×D244—44no—
199C2×D244—48no—
200C2×D244—44no—
201C2×D244—82no—
202C2×D244—84no—
203C2×D244C2×D412yes—
204C2×D244C2×D412yes—
205C2×D244C2414yes—
206C2×C4×D61—28no—
207C2×C4×D61—44no—
208C2×C4×D61—48no—
209C2×C4×D61C2×D414yes—
210C2×C4×D61C2414yes—
211C2×Dic122—21no—
212C2×Dic122C2412yes—
213SD16×C61—28no—
214C2×C8×C61C2414yes—
215C4⋊C4×C61—22no—
216C4⋊C4×C61—22no—
217C4⋊C4×C61—21no—
218C4⋊C4×C61—21no—
219C4⋊C4×C61C2412yes—
220C4⋊C4×C61C2412yes—
221C22⋊C4×C61—22no—
222C22⋊C4×C61—22no—
223C42×C61—22no—
224C42×C61—22no—
225C42×C61C2×D411yescharacteristic
226C42×C61C2×D411yescharacteristic
227C42×C61C2411yescharacteristic
228C42×C61C2411yescharacteristic
229C22⋊Dic61—22no—
230C22⋊Dic61—24no—
231#11758202—28no—
232#11758199—48no—
233#11758198—24no—
234#11758198—24no—
235#11758198—28no—
236#11758198C2×D414yes—
237#11758198C2×D414yes—
238C4⋊Dic61—21no—
239C4⋊Dic61—21no—
240C4⋊Dic61—21no—
241C4⋊Dic61—24no—
242C4⋊Dic61C2×D412yes—
243C4⋊Dic61C2×D412yes—
244C4⋊Dic61C2412yes—
245#11758196—22no—
246#11758196—22no—
247C4×Dic61—22no—
248C4×Dic61—24no—
249C2×#92418C2414yes—
250#11758190—48no—
251C8×D61—48no—
252C23×C61—22no—
253C23×C61C22×D411yescharacteristic
254C22×D61—24no—
255C22×D61—24no—
256C22×D61—42no—
257C22×D61—42no—
258C22×D61—44no—
259C22×D61—44no—
260C22×D61—44no—
261C22×D61C22×D414yes—
262Q8×C61—24no—
263Q8×C61—22no—
264Q8×C61—42no—
265Q8×C61H(5,𝔽2)14yes—
266D4×C61—24no—
267D4×C61—44no—
268D4×C61—42no—
269D4×C61—81no—
270C2×C4×C61—24no—
271C2×C4×C61—22no—
272C2×C4×C61—22no—
273C2×C4×C61—22no—
274C2×C4×C61—22no—
275C2×C4×C61—41no—
276C2×C4×C61—42no—
277C2×C4×C61—44no—
278C2×C4×C61#13514yescomplemented
279C2×C4×C61C22×D411yescharacteristic
280C2×C4×C61C22×D411yescharacteristic
281C2×C4×C61C22×D411yescharacteristic
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C2511yescharacteristic
285C22⋊D61—84no—
286C22⋊D61—84no—
287C2×Dic61—22no—
288C2×Dic61—22no—
289C2×Dic61—42no—
290C2×Dic61—84no—
291C2×Dic61C22×D414yes—
292D244—24no—
293D244—24no—
294D244—28no—
295D244—416no—
296D244—44no—
297D244—44no—
298D244—44no—
299D244—82no—
300D244—82no—
301D244—88no—
302C4×D61—28no—
303C4×D61—44no—
304C4×D61—416no—
305Dic122—24no—
306Dic122—81no—
307C8×C61—24no—
308#92418—24no—
309#61566—611nomaximal, Sylow
310C22×C61—21no—
311C22×C61—21no—
312C22×C61—42no—
313C22×C61—42no—
314C22×C61—42no—
315C22×C61C2×#13512yes—
316C22×C61C23×D411yescharacteristic
317C2×D61—24no—
318C2×D61—24no—
319C2×D61—24no—
320C2×D61—44no—
321C2×D61—44no—
322C2×D61—44no—
323C2×D61—84no—
324C2×D61—88no—
325C4×C61—21no—
326C4×C61—21no—
327C4×C61—24no—
328C4×C61—21no—
329C4×C61—42no—
330C4×C61—42no—
331C4×C61C2×#13512yes—
332C4×C61C2×#13512yes—
333C4×C61C2×H(5,𝔽2)12yes—
334Dic61—24no—
335Dic61—42no—
336Dic61—42no—
337#3506—611no—
338C2×#550—611no—
339#3305—614no—
340#3304—614no—
341#3287—614no—
342#3284—614no—
343#3208—611no—
344#3205—611no—
345#3188—614no—
346#3104—611no—
347#3103D6111yescharacteristic, complemented
348#3002—614no—
349#2972—611no—
350C2×C61—22no—
351C2×C61—82no—
352C2×C61#348411yescharacteristic
353C2×C61C22×#13511yescharacteristic
354C2×C61#358511yescharacteristic
355D61—44no—
356D61—84no—
357C22×#41—612no—
358C2×#135—618no—
359#559—1222no—
360#556—612no—
361#555—612no—
362#553—1222no—
363#550—612no—
364#550—614no—
365#550—614no—
366#550—1222no—
367#550—1222no—
368#547—612no—
369#538—1222no—
370#535—1221no—
371#533—611no—
372#532C2×D6111yescharacteristic
373C2×#127—611no—
374C2×#127—1221no—
375#525—614no—
376C2×#121—612no—
377C2×#121—1222no—
378#520—1221no—
379C2×C4×Q8C2×D6111yescharacteristic
380C2×C4×D4—611no—
381C2×C4×D4—1221no—
382#493—612no—
383#492—612no—
384#486—612no—
385#486—612no—
386#484—612no—
387#482—618no—
388#481C2×D6112yes—
389#477—612no—
390#475C2×D6112yescomplemented
391#466—618no—
392#465—618no—
393#462—612no—
394#460—614no—
395#460C2×D6112yescomplemented
396#459—614no—
397#452—618no—
398#451—612no—
399#450—614no—
400#449—618no—
401#449—618no—
402#448C2×D6112yes—
403#447—614no—
404C4×Q16—614no—
405C4×SD16C2×D6114yescomplemented
406C8×D4—614no—
407C8×D4—614no—
408#431—611no—
409#431C2×D6111yescharacteristic
410#416—618no—
411C2×C22⋊C8—614no—
412C61#6156611yescharacteristic, complemented, Sylow
413C2×#41—21no—
414C2×#41—618no—
415C2×#41—614no—
416C2×#41—1224no—
417C2×#41—2448no—
418C22×Q8C22×D6111yescharacteristic
419C22×D4—611no—
420C22×D4—612no—
421C22×D4—1222no—
422C22×D4—1222no—
423C22×D4—1221no—
424C22×D4—1221no—
425C23×C4—612no—
426C23×C4—1222no—
427#135—24416no—
428C2×Q16—614no—
429C2×SD16—618no—
430C2×SD16C22×D6114yes—
431C2×D8—618no—
432C22×C8—618no—
433C4⋊Q8—611no—
434C4⋊Q8—611no—
435C4⋊Q8C22×D6111yescharacteristic
436#127—612no—
437#127—612no—
438#127—1224no—
439#127—1224no—
440#127—1222no—
441#127—1221no—
442#127—1221no—
443#127C22×D6111yescharacteristic
444#126—1224no—
445#125—1221no—
446#124—21no—
447#124—1224no—
448#124—1222no—
449#123—1224no—
450#123—1224no—
451C22⋊Q8—612no—
452C22⋊Q8—614no—
453C22⋊Q8—1221no—
454#121—21no—
455#121—614no—
456#121—618no—
457#121—1228no—
458#121—1228no—
459#121—1224no—
460#121—1224no—
461#121—1222no—
462#121—1222no—
463#121C22×D6112yes—
464#120—1224no—
465C4×Q8—21no—
466C4×Q8—21no—
467C4×Q8—612no—
468C4×Q8—1222no—
469C4×Q8—1221no—
470C4×Q8C22×D6112yes—
471C4×Q8C22×D6112yes—
472C4×D4—21no—
473C4×D4—612no—
474C4×D4—612no—
475C4×D4—612no—
476C4×D4—612no—
477C4×D4—1224no—
478C4×D4—1224no—
479C4×D4—1224no—
480C4×D4—1222no—
481C4×D4—1224no—
482C4×D4C22×D6112yes—
483#117—614no—
484#117—1222no—
485#117—1221no—
486C2×C4⋊C4—21no—
487C2×C4⋊C4—611no—
488C2×C4⋊C4—1222no—
489C2×C4⋊C4C22×D6111yescharacteristic
490C2×C22⋊C4—612no—
491C2×C22⋊C4—1222no—
492C2×C42—21no—
493C2×C42C22×D6111yescharacteristic
494#107—614no—
495#106C22×D6114yes—
496C4⋊C8—612no—
497C4⋊C8C22×D6112yes—
498#103—614no—
499#103—614no—
500#103C22×D6114yes—
501#102—618no—
502#102—618no—
503#102C22×D6114yes—
504C22⋊C8—612no—
505C22⋊C8—618no—
506C22⋊C8C22×D6112yes—
507C4×C8—612no—
508C4×C8C22×D6112yes—
509C24—612no—
510C24—1222no—
511#41—44no—
512#41—12216no—
513#41—2448no—
514#41—2448no—
515#41—24416no—
516C2×Q8—24no—
517C2×Q8—21no—
518C2×Q8—21no—
519C2×Q8—612no—
520C2×Q8—1221no—
521C2×Q8C23×D6112yes—
522C2×D4—22no—
523C2×D4—21no—
524C2×D4—21no—
525C2×D4—612no—
526C2×D4—612no—
527C2×D4—614no—
528C2×D4—1228no—
529C2×D4—1228no—
530C2×D4—1224no—
531C2×D4—1224no—
532C2×D4—1222no—
533C2×D4—1222no—
534C2×D4—1222no—
535C2×D4—1222no—
536C2×D4—1222no—
537C2×D4—1224no—
538C2×D4—4882no—
539C2×D4—4884no—
540C2×D4—2444no—
541C2×D4—2448no—
542C2×D4—2444no—
543C2×D4C23×D6112yes—
544C22×C4—21no—
545C22×C4—21no—
546C22×C4—22no—
547C22×C4—612no—
548C22×C4—614no—
549C22×C4—614no—
550C22×C4—1228no—
551C22×C4—1224no—
552C22×C4—2444no—
553C22×C4—2448no—
554C22×C4D4×D6111yescharacteristic
555C22×C4D4×D6111yescharacteristic
556C22×C4C23×D6111yescharacteristic
557Q16—1228no—
558SD16—28no—
559SD16—2448no—
560D8—2448no—
561C2×C8—614no—
562C2×C8—2448no—
563C2×C8C23×D6114yes—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—612no—
569C4⋊C4—612no—
570C4⋊C4—612no—
571C4⋊C4—1224no—
572C4⋊C4—1221no—
573C4⋊C4—1222no—
574C4⋊C4—1222no—
575C4⋊C4—1221no—
576C4⋊C4—1221no—
577C4⋊C4C23×D6112yes—
578C4⋊C4C23×D6112yes—
579C22⋊C4—22no—
580C22⋊C4—22no—
581C22⋊C4—614no—
582C22⋊C4—614no—
583C22⋊C4—1228no—
584C22⋊C4—1224no—
585C22⋊C4—1224no—
586C22⋊C4—1224no—
587C22⋊C4—1222no—
588C42—22no—
589C42—22no—
590C42—1224no—
591C42—1222no—
592C42D4×D6111yescharacteristic
593C42D4×D6111yescharacteristic
594C42C23×D6111yescharacteristic
595C42C23×D6111yescharacteristic
596C23—22no—
597C23—614no—
598C23—1224no—
599C23—1224no—
600C23—2442no—
601C23—2442no—
602C23—2444no—
603C23—2444no—
604C23—2444no—
605C23C2×D4×D6111yescharacteristic
606Q8—24no—
607Q8—22no—
608Q8—42no—
609Q8—1224no—
610Q8—4881no—
611Q8#1451982314yes—
612D4—24no—
613D4—44no—
614D4—42no—
615D4—81no—
616D4—1224no—
617D4—1224no—
618D4—1228no—
619D4—4882no—
620D4—4882no—
621D4—4888no—
622D4—4884no—
623D4—4884no—
624D4—24416no—
625D4—2444no—
626D4—2444no—
627D4—2444no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—614no—
637C2×C4—1228no—
638C2×C4—1222no—
639C2×C4—1222no—
640C2×C4—4884no—
641C2×C4—2444no—
642C2×C4—24416no—
643C2×C4—2442no—
644C2×C4#1451979814yescomplemented
645C2×C4C2×D4×D6111yescharacteristic
646C2×C4C2×D4×D6111yescharacteristic
647C2×C4C2×D4×D6111yescharacteristic
648C2×C4C2×D4×D6111yescharacteristic
649C2×C4C2×D4×D6111yescharacteristic
650C2×C4C24×D6111yescharacteristic
651C8—24no—
652C8—1224no—
653C22—21no—
654C22—21no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—1224no—
659C22—1224no—
660C22—1224no—
661C22—4888no—
662C22—4884no—
663C22—2444no—
664C22—2444no—
665C22—2444no—
666C22C2×#1451979812yes—
667C22C22×D4×D6111yescharacteristic
668C4—21no—
669C4—21no—
670C4—24no—
671C4—21no—
672C4—42no—
673C4—42no—
674C4—1224no—
675C4—2442no—
676C4—2442no—
677C4C2×#1451979812yes—
678C4C2×#1451979812yes—
679C4C2×#1451982312yes—
680C2—22no—
681C2—82no—
682C2—4884no—
683C2—2444no—
684C2#2145945211yescharacteristic
685C2C22×#1451979811yescharacteristic
686C2#2146059611yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3103C2
socleC22×C61C23×D4
hypercenter#3103D61
hypocenterC61#61566
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. #21459668C2semidirect
2. C2×#16840434C2semidirect
3. #21457771C2semidirect
4. #21457755C2semidirect
5. #21457574C2semidirect
6. #21457550C2semidirect
7. #21456762C2semidirect
8. #21456734C2semidirect
9. #21456583C2semidirect
10. #21455858C2semidirect
11. #21455036C2semidirect
12. #21454842C2semidirect
13. C61×#3103C2semidirect
15. C2×#14519798C22semidirect
17. #16840486C22semidirect
18. #16840477C22semidirect
24. #16840432C22semidirect
25. #16840416C22semidirect
37. #16840138C22semidirect
38. #16840135C22semidirect
39. #16840110C22semidirect
40. #16840109C22semidirect
42. #16840088C22semidirect
43. #16840057C22semidirect
44. #16839997C22semidirect
45. #16839992C22semidirect
46. #16839972C22semidirect
47. #16839963C22semidirect
48. #16839957C22semidirect
49. #16839927C22semidirect
52. #16839908C22semidirect
53. #16839906C22semidirect
55. C4×#11758202C22semidirect
56. C8×D244C22semidirect
57. #16839827C22semidirect
59. #16839757C22semidirect
64. C61×#475C22semidirect
65. C61×#460C22semidirect
67. C4×SD16×C61C22semidirect
278. C2×C4×C61#135semidirect
347. #3103D61semidirect
390. #475C2×D61semidirect
395. #460C2×D61semidirect
405. C4×SD16C2×D61semidirect
412. C61#61566semidirect
644. C2×C4#14519798semidirect

Maximal subgroups

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

subgroupindex
#214596682
C2×#168404342
#214577712
#214577552
#214575742
#214575502
#214567622
#214567342
#214565832
#214558582
#214550362
#214548422
C61×#31032
#6156661