Finite Group Databasebeta

← #30564782

Subgroups — #30564782

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#21459671C211yesmaximal, characteristic, complemented
2C2×#16840433C211yesmaximal, characteristic, complemented
3#21457770C214yesmaximal, complemented
4#21457759C214yesmaximal, complemented
5#21457579C214yesmaximal, complemented
6#21457551C214yesmaximal, complemented
7#21456762C211yesmaximal, characteristic, complemented
8#21456739C211yesmaximal, characteristic, complemented
9#21456586C214yesmaximal, complemented
10#21455859C211yesmaximal, characteristic, complemented
11#21455851C211yesmaximal, characteristic, complemented
12#21455041C214yesmaximal, complemented
13C61×#2972C211yesmaximal, characteristic, complemented
14C22×#11758222C2212yes—
15C2×#14519799C2218yescomplemented
16#16840495—22no—
17#16840482C2212yescomplemented
18#16840477C2212yescomplemented
19#16840462—22no—
20#16840433—22no—
21#16840433—22no—
22#16840433C2214yes—
23#16840433C2214yes—
24#16840433C2212yescomplemented
25#16840417C2212yescomplemented
26#16840350—22no—
27#16840329—21no—
28#16840319C2211yescharacteristic
29#16840312C2211yescharacteristic
30C2×#14519774C2211yescharacteristic
31C2×#14519773—21no—
32#16840273C2214yes—
33C2×#14519751C2212yes—
34C2×#14519750—22no—
35#16840243—21no—
36C2×C4×Dic122C2211yescharacteristic
37C2×#14519739—21no—
38#16840139C2212yescomplemented
39#16840135C2212yescomplemented
40#16840109C2212yescomplemented
41#16840109C2212yescomplemented
42#16840098C2212yes—
43#16840087C2218yescomplemented
44#16840081C2212yes—
45#16840058C2212yescomplemented
46#16840048C2212yescomplemented
47#16839998C2218yescomplemented
48#16839991C2218yescomplemented
49#16839973C2212yescomplemented
50#16839963C2214yescomplemented
51#16839963C2212yescomplemented
52#16839958C2214yescomplemented
53#16839926C2218yescomplemented
54#16839920C2212yes—
55#16839915C2214yes—
56#16839909C2218yescomplemented
57#16839909C2218yescomplemented
58#16839903C2212yes—
59#16839898C2214yes—
60C4×Dic244C2214yescomplemented
61C4×#11758190C2214yescomplemented
62#16839759C2218yescomplemented
63C2×C4×D4×C61C2211yescharacteristic
64C8×D4×C61C2214yescomplemented
65C8×D4×C61C2214yescomplemented
66C61×#431C2211yescharacteristic
67C61×#431C2211yescharacteristic
68C2×C22⋊C8×C61C2214yes—
69C2×#11758222—21no—
70C2×#11758222—24no—
71C2×#11758222—48no—
72C2×#11758222C2318yes—
73C2×#11758222C2314yes—
74C22×Dic122C2311yescharacteristic
75C22×D244—21no—
76C22×D244—21no—
77C22×D244C2312yes—
78C22×C22⋊D61—22no—
79C22×C22⋊D61—22no—
80C22×C4×D61—22no—
81#14519799—416no—
82C2×Dic244C2314yes—
83C2×#11758190C2314yes—
84C2×#11758190C2318yes—
85C2×D488C2318yes—
86#14519777C2311yescharacteristic
87#14519777C2311yescharacteristic
88#14519777C2311yescharacteristic
89#14519774—21no—
90#14519774—21no—
91#14519774C2312yes—
92#14519774C2312yes—
93#14519774C2311yescharacteristic
94#14519773—22no—
95#14519773—24no—
96#14519773—24no—
97#14519770—24no—
98#14519769—21no—
99#14519765—21no—
100#14519764—22no—
101#14519763—24no—
102#14519760—24no—
103#14519759—24no—
104#14519756—21no—
105#14519756C2314yes—
106#14519756C2312yes—
107#14519751—24no—
108#14519751—21no—
109#14519751C2314yes—
110#14519751C2318yes—
111#14519751C2312yes—
112#14519750—24no—
113#14519750—28no—
114#14519750—28no—
115#14519748—22no—
116#14519748—22no—
117#14519746—24no—
118C4×Dic122—21no—
119C4×Dic122—21no—
120C4×Dic122—21no—
121C4×Dic122C2312yes—
122C4×Dic122C2312yes—
123C4×Dic122C2312yes—
124#14519742—22no—
125C4×D244—21no—
126C4×D244C2312yes—
127#14519739—24no—
128#14519739—24no—
129#14519739—22no—
130C4×C22⋊D61—24no—
131C4×C22⋊D61—24no—
132#14519736—21no—
133#14519736C2314yes—
134#14519735—22no—
135C2×C4⋊Dic61C2311yescharacteristic
136C2×#11758196—21no—
137C4⋊C4×D61—22no—
138C2×#11758198—22no—
139C2×C4×Dic61—21no—
140#14519711C2314yes—
141#14519709C2314yes—
142#14519702C2314yes—
143#14519702C2314yes—
144#14519702C2314yes—
145#14519699C2318yes—
146#14519699C2314yes—
147#14519699C2318yes—
148C22×D4×C61D411yescharacteristic
149C23×C4×C61C2312yes—
150C22×C8×C61C2318yes—
151C4×D4×C61D412yes—
152C4×D4×C61D412yes—
153C4×D4×C61C2312yes—
154C4×D4×C61C2312yes—
155C2×C4⋊C4×C61D411yescharacteristic
156C2×C22⋊C4×C61D412yes—
157C2×C42×C61C2311yescharacteristic
158C4⋊C8×C61C2312yes—
159C4⋊C8×C61C2312yes—
160C22⋊C8×C61C2312yes—
161C22⋊C8×C61C2312yes—
162C22⋊C8×C61C2318yes—
163C4×C8×C61C2312yes—
164C4×C8×C61C2312yes—
165C24×C61C2×D412yes—
166C23×D61—22no—
167C2×D4×C61—24no—
168C2×D4×C61C2×D412yes—
169C2×D4×C61C2×D412yes—
170C22×C4×C61—44no—
171C22×C4×C61C2×D412yes—
172C22×C4×C61C2×D411yescharacteristic
173C22×C4×C61C2×D414yes—
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2411yescharacteristic
176C22×C4×C61C2414yes—
177C2×C22⋊D61—28no—
178C2×C22⋊D61—22no—
179C2×C22⋊D61—28no—
180C2×C22⋊D61—24no—
181C2×C22⋊D61—24no—
182C2×C22⋊D61—48no—
183C2×C22⋊D61—84no—
184C22×Dic61—21no—
185C22×Dic61—21no—
186#11758222—216no—
187#11758222—48no—
188#11758222—416no—
189#11758222—48no—
190#11758222—44no—
191C2×D244—22no—
192C2×D244—22no—
193C2×D244—22no—
194C2×D244—21no—
195C2×D244—22no—
196C2×D244—22no—
197C2×D244—21no—
198C2×D244—44no—
199C2×D244—44no—
200C2×D244—82no—
201C2×D244C2414yes—
202C2×D244C2412yes—
203C2×C4×D61—28no—
204C2×C4×D61—22no—
205C2×C4×D61—24no—
206C2×C4×D61—48no—
207C2×Dic122—21no—
208C2×Dic122—24no—
209C2×Dic122—21no—
210C2×Dic122—21no—
211C2×Dic122C2412yes—
212C2×Dic122C2412yes—
213C2×C8×C61—48no—
214C2×C8×C61C2414yes—
215C2×C8×C61C2414yes—
216C4⋊C4×C61C2×D412yes—
217C4⋊C4×C61C2×D412yes—
218C22⋊C4×C61C2×D414yes—
219C22⋊C4×C61C2×D414yes—
220C42×C61C2×D411yescharacteristic
221C42×C61C2×D411yescharacteristic
222C42×C61C2411yescharacteristic
223C42×C61C2411yescharacteristic
224C22⋊Dic61—22no—
225C22⋊Dic61—24no—
226C22⋊Dic61—24no—
227#11758198—28no—
228#11758198—24no—
229#11758198—22no—
230#11758198—22no—
231C4⋊Dic61—21no—
232C4⋊Dic61—21no—
233C4⋊Dic61—21no—
234C4⋊Dic61—21no—
235C4⋊Dic61—21no—
236C4⋊Dic61C2412yes—
237C4⋊Dic61C2412yes—
238C4⋊Dic61C2412yes—
239#11758196—24no—
240#11758196—22no—
241#11758196—22no—
242#11758196—22no—
243#11758196—22no—
244C4×Dic61—22no—
245C4×Dic61—22no—
246C4×Dic61—24no—
247C4×Dic61—22no—
248Dic244—28no—
249D488—48no—
250#11758190—28no—
251#11758190—48no—
252C23×C61—24no—
253C23×C61—42no—
254C23×C61—42no—
255C23×C61C22×D414yes—
256C23×C61C22×D411yescharacteristic
257C22×D61—22no—
258C22×D61—24no—
259C22×D61—44no—
260C22×D61—44no—
261C22×D61—44no—
262D4×C61—24no—
263D4×C61—24no—
264C2×C4×C61—28no—
265C2×C4×C61—22no—
266C2×C4×C61—42no—
267C2×C4×C61—41no—
268C2×C4×C61—44no—
269C2×C4×C61#13514yescomplemented
270C2×C4×C61C22×D414yes—
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C2511yescharacteristic
277C22⋊D61—416no—
278C22⋊D61—44no—
279C22⋊D61—84no—
280C22⋊D61—84no—
281C22⋊D61—88no—
282C2×Dic61—24no—
283C2×Dic61—22no—
284C2×Dic61—22no—
285C2×Dic61—22no—
286C2×Dic61—22no—
287C2×Dic61—22no—
288C2×Dic61—42no—
289D244—24no—
290D244—28no—
291D244—44no—
292D244—44no—
293D244—44no—
294D244—42no—
295D244—81no—
296D244—82no—
297D244—82no—
298C4×D61—416no—
299C4×D61—44no—
300C4×D61—84no—
301Dic122—24no—
302Dic122—22no—
303Dic122—24no—
304Dic122—42no—
305Dic122—81no—
306Dic122H(5,𝔽2)14yes—
307C8×C61—24no—
308C8×C61—24no—
309#61566—611nomaximal, Sylow
310C22×C61—21no—
311C22×C61—21no—
312C22×C61—24no—
313C22×C61—24no—
314C22×C61—24no—
315C22×C61—84no—
316C22×C61C2×#13512yes—
317C22×C61C23×D411yescharacteristic
318C2×D61—44no—
319C2×D61—42no—
320C2×D61—44no—
321C2×D61—42no—
322C2×D61—42no—
323C2×D61—44no—
324C2×D61—88no—
325C4×C61—21no—
326C4×C61—21no—
327C4×C61—21no—
328C4×C61—24no—
329C4×C61C2×#13512yes—
330C4×C61C2×#13512yes—
331C4×C61C2×H(5,𝔽2)12yes—
332Dic61—24no—
333Dic61—42no—
334Dic61—42no—
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#3103—611no—
348#3002—614no—
349#2972D6111yescharacteristic, complemented
350C2×C61—22no—
351C2×C61—44no—
352C2×C61#348411yescharacteristic
353C2×C61C22×#13511yescharacteristic
354C2×C61#358511yescharacteristic
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#533—611no—
372#532—611no—
373C2×#127—611no—
374C2×#127—1221no—
375#525—614no—
376C2×#121—612no—
377C2×#121—1222no—
378#520—1221no—
379C2×C4×Q8—611no—
380C2×C4×D4—1221no—
381C2×C4×D4C2×D6111yescharacteristic
382#493—612no—
383#492—612no—
384#486—612no—
385#486—612no—
386#484—612no—
387#482—618no—
388#481—612no—
389#477—612no—
390#475—612no—
391#466—618no—
392#465—618no—
393#462—612no—
394#460—612no—
395#460—614no—
396#459—614no—
397#452—618no—
398#451—612no—
399#450—614no—
400#449—618no—
401#449—618no—
402#448—612no—
403#447—614no—
404C4×Q16—614no—
405C4×SD16—614no—
406C8×D4C2×D6114yescomplemented
407C8×D4C2×D6114yescomplemented
408#431C2×D6111yescharacteristic
409#431C2×D6111yescharacteristic
410#416—618no—
411C2×C22⋊C8C2×D6114yes—
412C61#6156611yescharacteristic, complemented, Sylow
413C2×#41—618no—
414C2×#41—614no—
415C2×#41—1221no—
416C2×#41—1224no—
417C2×#41—2448no—
418C22×Q8—611no—
419C22×D4—612no—
420C22×D4—1222no—
421C22×D4—1222no—
422C22×D4—1221no—
423C22×D4—1221no—
424C22×D4D24411yescharacteristic
425C23×C4—1222no—
426C23×C4C22×D6112yes—
427#135—24416no—
428C2×Q16—614no—
429C2×SD16—614no—
430C2×SD16—618no—
431C2×D8—618no—
432C22×C8C22×D6118yes—
433C4⋊Q8—611no—
434C4⋊Q8—611no—
435C4⋊Q8—611no—
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—1221no—
446#124—1222no—
447#124—1224no—
448#124—1221no—
449#123—1224no—
450#123—1224no—
451C22⋊Q8—612no—
452C22⋊Q8—614no—
453C22⋊Q8—1221no—
454#121—618no—
455#121—612no—
456#121—614no—
457#121—1222no—
458#121—1222no—
459#121—1224no—
460#121—1221no—
461#121—1228no—
462#121—1228no—
463#121—1224no—
464#120—1224no—
465C4×Q8—612no—
466C4×Q8—612no—
467C4×Q8—612no—
468C4×Q8—1222no—
469C4×Q8—1221no—
470C4×Q8—1221no—
471C4×Q8—1221no—
472C4×D4—612no—
473C4×D4—1224no—
474C4×D4—1224no—
475C4×D4—1224no—
476C4×D4—1224no—
477C4×D4—1222no—
478C4×D4—1221no—
479C4×D4D24412yes—
480C4×D4D24412yes—
481C4×D4C22×D6112yes—
482C4×D4C22×D6112yes—
483#117—614no—
484#117—1221no—
485#117—1222no—
486C2×C4⋊C4—611no—
487C2×C4⋊C4—1221no—
488C2×C4⋊C4—1222no—
489C2×C4⋊C4D24411yescharacteristic
490C2×C22⋊C4—1222no—
491C2×C22⋊C4D24412yes—
492C2×C42—1221no—
493C2×C42C22×D6111yescharacteristic
494#107—614no—
495#106—614no—
496C4⋊C8C22×D6112yes—
497C4⋊C8C22×D6112yes—
498#103—614no—
499#103—614no—
500#103—614no—
501#102—618no—
502#102—618no—
503#102—614no—
504C22⋊C8C22×D6112yes—
505C22⋊C8C22×D6112yes—
506C22⋊C8C22×D6118yes—
507C4×C8C22×D6112yes—
508C4×C8C22×D6112yes—
509C24—1222no—
510C24C2×D24412yes—
511#41—12216no—
512#41—2448no—
513#41—2448no—
514#41—24416no—
515#41—2444no—
516C2×Q8—612no—
517C2×Q8—612no—
518C2×Q8—1224no—
519C2×Q8—1221no—
520C2×Q8—1221no—
521C2×Q8—1221no—
522C2×D4—24no—
523C2×D4—614no—
524C2×D4—612no—
525C2×D4—1224no—
526C2×D4—1224no—
527C2×D4—1228no—
528C2×D4—1222no—
529C2×D4—1228no—
530C2×D4—1221no—
531C2×D4—1222no—
532C2×D4—1222no—
533C2×D4—1221no—
534C2×D4—1222no—
535C2×D4—1222no—
536C2×D4—1222no—
537C2×D4—4882no—
538C2×D4—4884no—
539C2×D4—2444no—
540C2×D4—2448no—
541C2×D4—2444no—
542C2×D4C2×D24412yes—
543C2×D4C2×D24412yes—
544C22×C4—44no—
545C22×C4—1221no—
546C22×C4—1221no—
547C22×C4—1224no—
548C22×C4—1222no—
549C22×C4—1228no—
550C22×C4—2448no—
551C22×C4C2×D24411yescharacteristic
552C22×C4C2×D24412yes—
553C22×C4C2×D24414yes—
554C22×C4C2×D24411yescharacteristic
555C22×C4C23×D6111yescharacteristic
556C22×C4C23×D6114yes—
557Q16—1228no—
558SD16—1228no—
559SD16—2448no—
560D8—2448no—
561C2×C8—48no—
562C2×C8C23×D6114yes—
563C2×C8C23×D6114yes—
564C4⋊C4—612no—
565C4⋊C4—612no—
566C4⋊C4—612no—
567C4⋊C4—1221no—
568C4⋊C4—1221no—
569C4⋊C4—1224no—
570C4⋊C4—1222no—
571C4⋊C4—1222no—
572C4⋊C4—1222no—
573C4⋊C4—1222no—
574C4⋊C4—1221no—
575C4⋊C4—1221no—
576C4⋊C4—1221no—
577C4⋊C4C2×D24412yes—
578C4⋊C4C2×D24412yes—
579C22⋊C4—1224no—
580C22⋊C4—1222no—
581C22⋊C4—1224no—
582C22⋊C4—1228no—
583C22⋊C4—1224no—
584C22⋊C4—1222no—
585C22⋊C4—1222no—
586C22⋊C4C2×D24414yes—
587C22⋊C4C2×D24414yes—
588C42—1222no—
589C42—1222no—
590C42—1224no—
591C42—1222no—
592C42C2×D24411yescharacteristic
593C42C2×D24411yescharacteristic
594C42C23×D6111yescharacteristic
595C42C23×D6111yescharacteristic
596C23—24no—
597C23—42no—
598C23—42no—
599C23—1222no—
600C23—1224no—
601C23—2444no—
602C23—2444no—
603C23—2444no—
604C23C22×D24414yes—
605C23C22×D24411yescharacteristic
606Q8—614no—
607Q8—1224no—
608Q8—1222no—
609Q8—1224no—
610Q8—4881no—
611Q8—2442no—
612D4—24no—
613D4—24no—
614D4—1224no—
615D4—1228no—
616D4—4881no—
617D4—4882no—
618D4—4882no—
619D4—4884no—
620D4—4888no—
621D4—4884no—
622D4—2444no—
623D4—24416no—
624D4—2444no—
625D4—2444no—
626D4—2444no—
627D4—2442no—
628C2×C4—28no—
629C2×C4—22no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—1224no—
634C2×C4—1222no—
635C2×C4—1222no—
636C2×C4—1222no—
637C2×C4—1222no—
638C2×C4—1222no—
639C2×C4—4884no—
640C2×C4—24416no—
641C2×C4—2442no—
642C2×C4—2444no—
643C2×C4#1451979914yescomplemented
644C2×C4C22×D24414yes—
645C2×C4C22×D24411yescharacteristic
646C2×C4C22×D24411yescharacteristic
647C2×C4C22×D24411yescharacteristic
648C2×C4C22×D24411yescharacteristic
649C2×C4C22×D24411yescharacteristic
650C2×C4C24×D6111yescharacteristic
651C8—24no—
652C8—24no—
653C22—21no—
654C22—21no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—84no—
659C22—4888no—
660C22—2442no—
661C22—2444no—
662C22—2444no—
663C22—2442no—
664C22—2442no—
665C22—2444no—
666C22C2×#1451979912yes—
667C22C23×D24411yescharacteristic
668C4—21no—
669C4—21no—
670C4—21no—
671C4—24no—
672C4—1224no—
673C4—2442no—
674C4—2442no—
675C4—2442no—
676C4—2442no—
677C4C2×#1451979912yes—
678C4C2×#1451979912yes—
679C4H(5,𝔽2)×D6112yes—
680C2—22no—
681C2—44no—
682C2—4884no—
683C2—4882no—
684C2#2145945511yescharacteristic
685C2C22×#1451979911yescharacteristic
686C2#2146060011yescharacteristic
subgroupG/H
centerC22C23×D244
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#2972C2
socleC22×C61C23×D4
hypercenter#2972D61
hypocenterC61#61566
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #21459671C2semidirect
2. C2×#16840433C2semidirect
3. #21457770C2semidirect
4. #21457759C2semidirect
5. #21457579C2semidirect
6. #21457551C2semidirect
7. #21456762C2semidirect
8. #21456739C2semidirect
9. #21456586C2semidirect
10. #21455859C2semidirect
11. #21455851C2semidirect
12. #21455041C2semidirect
13. C61×#2972C2semidirect
15. C2×#14519799C22semidirect
17. #16840482C22semidirect
18. #16840477C22semidirect
24. #16840433C22semidirect
25. #16840417C22semidirect
38. #16840139C22semidirect
39. #16840135C22semidirect
40. #16840109C22semidirect
41. #16840109C22semidirect
43. #16840087C22semidirect
45. #16840058C22semidirect
46. #16840048C22semidirect
47. #16839998C22semidirect
48. #16839991C22semidirect
49. #16839973C22semidirect
50. #16839963C22semidirect
51. #16839963C22semidirect
52. #16839958C22semidirect
53. #16839926C22semidirect
56. #16839909C22semidirect
57. #16839909C22semidirect
60. C4×Dic244C22semidirect
61. C4×#11758190C22semidirect
62. #16839759C22semidirect
64. C8×D4×C61C22semidirect
65. C8×D4×C61C22semidirect
269. C2×C4×C61#135semidirect
349. #2972D61semidirect
406. C8×D4C2×D61semidirect
407. C8×D4C2×D61semidirect
412. C61#61566semidirect
643. C2×C4#14519799semidirect

Maximal subgroups

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

subgroupindex
#214596712
C2×#168404332
#214577702
#214577592
#214575792
#214575512
#214567622
#214567392
#214565862
#214558592
#214558512
#214550412
C61×#29722
#6156661