Finite Group Databasebeta

← #30564783

Subgroups — #30564783

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#21459673C211yesmaximal, characteristic, complemented
2C2×#16840434C211yesmaximal, characteristic, complemented
3#21457766C214yesmaximal, complemented
4#21457760C214yesmaximal, complemented
5#21457578C214yesmaximal, complemented
6#21457544C214yesmaximal, complemented
7#21456757C211yesmaximal, characteristic, complemented
8#21456739C211yesmaximal, characteristic, complemented
9#21456589C214yesmaximal, complemented
10#21455850C211yesmaximal, characteristic, complemented
11#21455042C214yesmaximal, complemented
12#21454842C211yesmaximal, characteristic, complemented
13C61×#3104C211yesmaximal, characteristic, complemented
14C22×#11758226C2212yes—
15C2×#14519800C2218yescomplemented
16#16840497—22no—
17#16840485C2212yescomplemented
18#16840478C2212yescomplemented
19#16840465—22no—
20#16840434—22no—
21#16840434—22no—
22#16840434C2214yes—
23#16840434C2214yes—
24#16840434C2212yescomplemented
25#16840415C2212yescomplemented
26#16840352—22no—
27#16840331—21no—
28#16840312C2211yescharacteristic
29C2×#14519774—21no—
30C2×#14519774C2211yescharacteristic
31#16840275C2214yes—
32C2×#14519752—22no—
33C2×#14519752C2212yes—
34#16840242—21no—
35C2×C4×D244—21no—
36C2×C4×D244C2211yescharacteristic
37#16840139C2212yescomplemented
38#16840136C2212yescomplemented
39#16840110C2212yescomplemented
40#16840109C2212yescomplemented
41#16840089C2218yescomplemented
42#16840080C2212yes—
43#16840047C2212yescomplemented
44#16839999C2218yescomplemented
45#16839994C2218yescomplemented
46#16839962C2214yescomplemented
47#16839962C2212yescomplemented
48#16839958C2214yescomplemented
49#16839923C2218yescomplemented
50#16839912C2214yes—
51#16839910C2218yescomplemented
52#16839910C2218yescomplemented
53#16839901C2212yes—
54#16839898C2214yes—
55C4×#11758201C2214yescomplemented
56C8×D244C2214yescomplemented
57#16839827C2214yescomplemented
58#16839815C2211yescharacteristic
59#16839757C2218yescomplemented
60C22⋊C8×D61C2214yes—
61C61×#533C2211yescharacteristic
62C2×C4×Q8×C61C2211yescharacteristic
63C61×#484C2212yes—
64C61×#477C2212yescomplemented
65C61×#462C2212yescomplemented
66C61×#451C2212yes—
67C4×Q16×C61C2214yescomplemented
68C61×#431C2211yescharacteristic
69C2×#11758226—24no—
70C2×#11758226—48no—
71C2×#11758226C2318yes—
72C2×#11758226C2314yes—
73C2×#11758222—21no—
74C22×D244—22no—
75C22×D244—21no—
76C22×D244—22no—
77C22×D244—21no—
78C22×D244D411yescharacteristic
79C22×D244C2312yes—
80C22×C4×D61—22no—
81C22×C4×D61C2312yes—
82#14519800—416no—
83C2×#11758201C2314yes—
84C2×#11758201C2318yes—
85C2×D488C2318yes—
86C2×C8×D61C2318yes—
87#14519777C2311yescharacteristic
88#14519774—22no—
89#14519774—24no—
90#14519774—24no—
91#14519774—21no—
92#14519774—21no—
93#14519774C2312yes—
94#14519774C2312yes—
95#14519774C2311yescharacteristic
96#14519772—24no—
97#14519766—24no—
98#14519765—22no—
99#14519765—21no—
100#14519762—24no—
101#14519762—24no—
102#14519753C2314yes—
103#14519752—22no—
104#14519752—22no—
105#14519752—24no—
106#14519752—24no—
107#14519752—28no—
108#14519752—28no—
109#14519752C2314yes—
110#14519752C2318yes—
111#14519751—21no—
112#14519751C2312yes—
113#14519747—24no—
114C4×Dic122—22no—
115C4×D244—21no—
116C4×D244—24no—
117C4×D244—24no—
118C4×D244—22no—
119C4×D244D412yes—
120C4×D244D412yes—
121C4×D244C2312yes—
122C4×D244C2312yes—
123C4×D244C2312yes—
124#14519739—24no—
125#14519737—24no—
126#14519735—22no—
127#14519735C2314yes—
128C2×C4⋊Dic61D411yescharacteristic
129C4⋊C4×D61—22no—
130C2×#11758198—22no—
131C2×#11758198D412yes—
132#14519708C2314yes—
133C4⋊#92418C2312yes—
134#14519701C2314yes—
135#14519699C2318yes—
136#14519697C2314yes—
137#14519697C2318yes—
138C22⋊#92418C2312yes—
139#14519689C2318yes—
140C4×#92418C2312yes—
141C22×Q8×C61C2311yescharacteristic
142C2×Q16×C61C2314yes—
143C4⋊Q8×C61C2311yescharacteristic
144C4⋊Q8×C61C2311yescharacteristic
145#125×C61—21no—
146C22⋊Q8×C61—21no—
147C22⋊Q8×C61C2312yes—
148C4×Q8×C61—21no—
149C4×Q8×C61—21no—
150C4×Q8×C61—21no—
151C4×Q8×C61C2312yes—
152C4×Q8×C61C2312yes—
153C4×Q8×C61C2312yes—
154#117×C61—21no—
155C2×C4⋊C4×C61—21no—
156C2×C4⋊C4×C61C2311yescharacteristic
157C2×C42×C61—21no—
158C2×C42×C61C2311yescharacteristic
159#107×C61C2314yes—
160C4⋊C8×C61C2312yes—
161#103×C61C2314yes—
162#103×C61C2314yes—
163C22⋊C8×C61C2312yes—
164C4×C8×C61C2312yes—
165C23×D61—22no—
166C23×D61C2×D412yes—
167C2×Q8×C61—24no—
168C2×Q8×C61—21no—
169C2×Q8×C61—21no—
170C2×Q8×C61C2412yes—
171C2×Q8×C61C2412yes—
172C22×C4×C61—21no—
173C22×C4×C61—21no—
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2×D411yescharacteristic
176C22×C4×C61C2411yescharacteristic
177C2×C22⋊D61—22no—
178C22×Dic61C2×D412yes—
179#11758226—216no—
180#11758226—48no—
181#11758226—48no—
182#11758226—416no—
183#11758222—44no—
184C2×D244—24no—
185C2×D244—28no—
186C2×D244—22no—
187C2×D244—22no—
188C2×D244—22no—
189C2×D244—21no—
190C2×D244—28no—
191C2×D244—24no—
192C2×D244—24no—
193C2×D244—22no—
194C2×D244—22no—
195C2×D244—21no—
196C2×D244—44no—
197C2×D244—48no—
198C2×D244—44no—
199C2×D244—82no—
200C2×D244—84no—
201C2×D244C2×D412yes—
202C2×D244C2×D412yes—
203C2×D244C2414yes—
204C2×D244C2412yes—
205C2×C4×D61—22no—
206C2×C4×D61—28no—
207C2×C4×D61—24no—
208C2×C4×D61—44no—
209C2×C4×D61—48no—
210C2×C4×D61C2×D414yes—
211C2×C4×D61C2414yes—
212C2×Dic122—21no—
213Q16×C61—28no—
214C2×C8×C61C2414yes—
215C4⋊C4×C61—22no—
216C4⋊C4×C61—22no—
217C4⋊C4×C61—21no—
218C4⋊C4×C61—21no—
219C4⋊C4×C61—21no—
220C4⋊C4×C61—21no—
221C4⋊C4×C61—22no—
222C4⋊C4×C61—22no—
223C4⋊C4×C61C2412yes—
224C4⋊C4×C61C2412yes—
225C22⋊C4×C61—22no—
226C42×C61—22no—
227C42×C61—22no—
228C42×C61—22no—
229C42×C61C2×D411yescharacteristic
230C42×C61C2×D411yescharacteristic
231C42×C61C2411yescharacteristic
232C42×C61C2411yescharacteristic
233#11758201—28no—
234#11758201—48no—
235#11758198—24no—
236#11758198—24no—
237#11758198—22no—
238#11758198—22no—
239#11758198—28no—
240#11758198—24no—
241#11758198C2×D414yes—
242#11758198C2×D414yes—
243C4⋊Dic61—21no—
244C4⋊Dic61C2×D412yes—
245C4⋊Dic61C2×D412yes—
246C4⋊Dic61C2412yes—
247#11758196—24no—
248C4×Dic61—24no—
249C2×#92418C2414yes—
250D488—48no—
251C8×D61—48no—
252C23×C61C22×D411yescharacteristic
253C22×D61—24no—
254C22×D61—22no—
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—24no—
265Q8×C61—81no—
266Q8×C61H(5,𝔽2)14yes—
267C2×C4×C61—24no—
268C2×C4×C61—22no—
269C2×C4×C61—22no—
270C2×C4×C61—22no—
271C2×C4×C61—22no—
272C2×C4×C61—22no—
273C2×C4×C61—41no—
274C2×C4×C61—42no—
275C2×C4×C61#13514yescomplemented
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C22×D411yescharacteristic
279C2×C4×C61C22×D411yescharacteristic
280C2×C4×C61C22×D411yescharacteristic
281C2×C4×C61C2511yescharacteristic
282C22⋊D61—44no—
283C2×Dic61—22no—
284C2×Dic61—42no—
285C2×Dic61C22×D414yes—
286D244—24no—
287D244—24no—
288D244—24no—
289D244—28no—
290D244—416no—
291D244—44no—
292D244—44no—
293D244—44no—
294D244—42no—
295D244—81no—
296D244—82no—
297D244—82no—
298D244—84no—
299D244—88no—
300D244—84no—
301C4×D61—28no—
302C4×D61—44no—
303C4×D61—416no—
304C4×D61—44no—
305C4×D61—84no—
306Dic122—42no—
307C8×C61—24no—
308#92418—24no—
309#61566—611nomaximal, Sylow
310C22×C61—21no—
311C22×C61—21no—
312C22×C61C2×#13512yes—
313C22×C61C23×D411yescharacteristic
314C2×D61—24no—
315C2×D61—24no—
316C2×D61—24no—
317C2×D61—44no—
318C2×D61—42no—
319C2×D61—44no—
320C2×D61—42no—
321C2×D61—42no—
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×C61—42no—
332C4×C61C2×#13512yes—
333C4×C61C2×#13512yes—
334C4×C61C2×H(5,𝔽2)12yes—
335Dic61—24no—
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#3104D6111yescharacteristic, complemented
347#3103—611no—
348#3002—614no—
349#2972—611no—
350C2×C61—22no—
351C2×C61#348411yescharacteristic
352C2×C61C22×#13511yescharacteristic
353C2×C61#358511yescharacteristic
354D61—44no—
355D61—82no—
356D61—84no—
357C22×#41—612no—
358C2×#135—618no—
359#559—1222no—
360#556—612no—
361#555—612no—
362#553—1222no—
363#550—614no—
364#550—614no—
365#550—612no—
366#550—1222no—
367#550—1222no—
368#547—612no—
369#538—1222no—
370#535—1221no—
371#533C2×D6111yescharacteristic
372#532—611no—
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#484C2×D6112yes—
387#482—618no—
388#481—612no—
389#477C2×D6112yescomplemented
390#475—612no—
391#466—618no—
392#465—618no—
393#462C2×D6112yescomplemented
394#460—612no—
395#460—614no—
396#459—614no—
397#452—618no—
398#451C2×D6112yes—
399#450—614no—
400#449—618no—
401#449—618no—
402#448—612no—
403#447—614no—
404C4×Q16C2×D6114yescomplemented
405C4×SD16—614no—
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—618no—
414C2×#41—614no—
415C2×#41—1224no—
416C2×#41—1221no—
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×Q16C22×D6114yes—
429C2×SD16—614no—
430C2×SD16—618no—
431C2×D8—618no—
432C22×C8—618no—
433C4⋊Q8—611no—
434C4⋊Q8C22×D6111yescharacteristic
435C4⋊Q8C22×D6111yescharacteristic
436#127—612no—
437#127—612no—
438#127—611no—
439#127—1224no—
440#127—1224no—
441#127—1222no—
442#127—1221no—
443#127—1221no—
444#126—1224no—
445#125—21no—
446#124—1222no—
447#124—1224no—
448#124—1221no—
449#123—1224no—
450#123—1224no—
451C22⋊Q8—21no—
452C22⋊Q8—614no—
453C22⋊Q8C22×D6112yes—
454#121—612no—
455#121—618no—
456#121—614no—
457#121—1228no—
458#121—1228no—
459#121—1224no—
460#121—1224no—
461#121—1222no—
462#121—1222no—
463#121—1221no—
464#120—1224no—
465C4×Q8—21no—
466C4×Q8—21no—
467C4×Q8—21no—
468C4×Q8—1222no—
469C4×Q8C22×D6112yes—
470C4×Q8C22×D6112yes—
471C4×Q8C22×D6112yes—
472C4×D4—612no—
473C4×D4—612no—
474C4×D4—612no—
475C4×D4—612no—
476C4×D4—612no—
477C4×D4—1224no—
478C4×D4—1224no—
479C4×D4—1222no—
480C4×D4—1224no—
481C4×D4—1224no—
482C4×D4—1221no—
483#117—21no—
484#117—614no—
485#117—1222no—
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#107C22×D6114yes—
495#106—614no—
496C4⋊C8—612no—
497C4⋊C8C22×D6112yes—
498#103—614no—
499#103C22×D6114yes—
500#103C22×D6114yes—
501#102—618no—
502#102—618no—
503#102—614no—
504C22⋊C8—612no—
505C22⋊C8—618no—
506C22⋊C8C22×D6112yes—
507C4×C8—612no—
508C4×C8C22×D6112yes—
509C24—612no—
510C24—1222no—
511#41—12216no—
512#41—2448no—
513#41—2448no—
514#41—24416no—
515#41—2444no—
516C2×Q8—24no—
517C2×Q8—21no—
518C2×Q8—21no—
519C2×Q8—1221no—
520C2×Q8C23×D6112yes—
521C2×Q8C23×D6112yes—
522C2×D4—612no—
523C2×D4—612no—
524C2×D4—614no—
525C2×D4—612no—
526C2×D4—1224no—
527C2×D4—1224no—
528C2×D4—1228no—
529C2×D4—1228no—
530C2×D4—1222no—
531C2×D4—1221no—
532C2×D4—1222no—
533C2×D4—1222no—
534C2×D4—1221no—
535C2×D4—1222no—
536C2×D4—1222no—
537C2×D4—1222no—
538C2×D4—1224no—
539C2×D4—4882no—
540C2×D4—4884no—
541C2×D4—2444no—
542C2×D4—2448no—
543C2×D4—2444no—
544C22×C4—21no—
545C22×C4—21no—
546C22×C4—612no—
547C22×C4—614no—
548C22×C4—614no—
549C22×C4—1224no—
550C22×C4—1228no—
551C22×C4—1222no—
552C22×C4—2444no—
553C22×C4—2448no—
554C22×C4D4×D6111yescharacteristic
555C22×C4D4×D6111yescharacteristic
556C22×C4C23×D6111yescharacteristic
557Q16—28no—
558SD16—1228no—
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—21no—
569C4⋊C4—22no—
570C4⋊C4—21no—
571C4⋊C4—22no—
572C4⋊C4—612no—
573C4⋊C4—612no—
574C4⋊C4—612no—
575C4⋊C4—1224no—
576C4⋊C4—1221no—
577C4⋊C4C23×D6112yes—
578C4⋊C4C23×D6112yes—
579C22⋊C4—22no—
580C22⋊C4—614no—
581C22⋊C4—614no—
582C22⋊C4—1228no—
583C22⋊C4—1224no—
584C22⋊C4—1224no—
585C22⋊C4—1224no—
586C22⋊C4—1222no—
587C22⋊C4—1222no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—1224no—
592C42D4×D6111yescharacteristic
593C42D4×D6111yescharacteristic
594C42C23×D6111yescharacteristic
595C42C23×D6111yescharacteristic
596C23—614no—
597C23—1224no—
598C23—1222no—
599C23—1224no—
600C23—2442no—
601C23—2442no—
602C23—2444no—
603C23—2444no—
604C23—2444no—
605C23C2×D4×D6111yescharacteristic
606Q8—24no—
607Q8—22no—
608Q8—24no—
609Q8—81no—
610Q8—2442no—
611Q8#1451982314yes—
612D4—1224no—
613D4—1224no—
614D4—1224no—
615D4—1228no—
616D4—4881no—
617D4—4882no—
618D4—4882no—
619D4—4884no—
620D4—4888no—
621D4—4884no—
622D4—24416no—
623D4—2444no—
624D4—2444no—
625D4—2444no—
626D4—2444no—
627D4—2442no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—614no—
637C2×C4—1228no—
638C2×C4—1222no—
639C2×C4—4884no—
640C2×C4—2444no—
641C2×C4—24416no—
642C2×C4—2442no—
643C2×C4—2444no—
644C2×C4#1451980014yescomplemented
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—1224no—
656C22—1224no—
657C22—1224no—
658C22—4888no—
659C22—4884no—
660C22—2442no—
661C22—2444no—
662C22—2444no—
663C22—2442no—
664C22—2442no—
665C22—2444no—
666C22C2×#1451980012yes—
667C22C22×D4×D6111yescharacteristic
668C4—21no—
669C4—21no—
670C4—24no—
671C4—21no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—1224no—
676C4—2442no—
677C4C2×#1451980012yes—
678C4C2×#1451980012yes—
679C4C2×#1451982312yes—
680C2—22no—
681C2—4884no—
682C2—4882no—
683C2—2444no—
684C2#2145945211yescharacteristic
685C2C22×#1451980011yescharacteristic
686C2#2146059611yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3104C2
socleC22×C61C23×D4
hypercenter#3104D61
hypocenterC61#61566
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
1. #21459673C2semidirect
2. C2×#16840434C2semidirect
3. #21457766C2semidirect
4. #21457760C2semidirect
5. #21457578C2semidirect
6. #21457544C2semidirect
7. #21456757C2semidirect
8. #21456739C2semidirect
9. #21456589C2semidirect
10. #21455850C2semidirect
11. #21455042C2semidirect
12. #21454842C2semidirect
13. C61×#3104C2semidirect
15. C2×#14519800C22semidirect
17. #16840485C22semidirect
18. #16840478C22semidirect
24. #16840434C22semidirect
25. #16840415C22semidirect
37. #16840139C22semidirect
38. #16840136C22semidirect
39. #16840110C22semidirect
40. #16840109C22semidirect
41. #16840089C22semidirect
43. #16840047C22semidirect
44. #16839999C22semidirect
45. #16839994C22semidirect
46. #16839962C22semidirect
47. #16839962C22semidirect
48. #16839958C22semidirect
49. #16839923C22semidirect
51. #16839910C22semidirect
52. #16839910C22semidirect
55. C4×#11758201C22semidirect
56. C8×D244C22semidirect
57. #16839827C22semidirect
59. #16839757C22semidirect
64. C61×#477C22semidirect
65. C61×#462C22semidirect
67. C4×Q16×C61C22semidirect
275. C2×C4×C61#135semidirect
346. #3104D61semidirect
389. #477C2×D61semidirect
393. #462C2×D61semidirect
404. C4×Q16C2×D61semidirect
412. C61#61566semidirect
644. C2×C4#14519800semidirect

Maximal subgroups

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

subgroupindex
#214596732
C2×#168404342
#214577662
#214577602
#214575782
#214575442
#214567572
#214567392
#214565892
#214558502
#214550422
#214548422
C61×#31042
#6156661