Finite Group Databasebeta

← #30571104

Subgroups — #30571104

Order 15616.

1970 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459585C211yesmaximal, characteristic, complemented
2C2×#16840482C211yesmaximal, characteristic, complemented
3#21458843C214yesmaximal, complemented
4#21458837C214yesmaximal, complemented
5#21458536C214yesmaximal, complemented
6#21458535C214yesmaximal, complemented
7#21456803C214yesmaximal, complemented
8#21456746C211yesmaximal, characteristic, complemented
9#21456741C211yesmaximal, characteristic, complemented
10#21455938C214yesmaximal, complemented
11#21455819C211yesmaximal, characteristic, complemented
12#21454902C211yesmaximal, characteristic, complemented
13C61×#3100C211yesmaximal, characteristic, complemented
14C2×#14519824—21no—
15#16840482—22no—
16#16840482—22no—
17#16840482C2214yes—
18#16840482C2214yes—
19#16840433C2212yescomplemented
20#16840432C2212yescomplemented
21D4×Dic122C2212yescomplemented
22#16840427C2212yescomplemented
23#16840421—22no—
24#16840420—22no—
25#16840340—21no—
26#16840339—22no—
27#16840312C2211yescharacteristic
28#16840307C2214yes—
29C2×#14519777C2211yescharacteristic
30C2×#14519770—22no—
31C2×#14519764C2212yes—
32C2×#14519756—21no—
33C2×C4×Dic122C2211yescharacteristic
34C2×#14519742C2212yes—
35C2×#14519736—21no—
36#16840150C2218yes—
37#16840135C2212yescomplemented
38#16840134C2212yescomplemented
39#16840120C2218yes—
40#16840114C2212yescomplemented
41#16840113C2212yescomplemented
42#16840075C2212yes—
43#16840056C2218yescomplemented
44#16840042C2212yes—
45#16839976C2218yes—
46#16839975C2218yes—
47#16839968C2214yescomplemented
48#16839967C2214yescomplemented
49#16839961C2218yescomplemented
50#16839957C2212yes—
51#16839903C2214yes—
52#16839900C2214yes—
53#16839896C2212yes—
54#16839892C2214yescomplemented
55#16839889C2214yescomplemented
56#16839861C2218yescomplemented
57#16839859C2218yescomplemented
58C4×#11758199C2214yes—
59#16839816C2214yes—
60#16839815C2211yescharacteristic
61C61×#532C2211yescharacteristic
62C2×C4×D4×C61C2211yescharacteristic
63C61×#480C2212yes—
64C61×#474C2212yes—
65C61×#459C2212yes—
66C61×#447C2212yes—
67C4×D8×C61C2214yes—
68C61×#431C2211yescharacteristic
69#14519824—48no—
70C2×#11758222—21no—
71C2×#11758222—22no—
72C2×#11758224—24no—
73C2×#11758224—22no—
74C22×Dic122—22no—
75C22×Dic122D411yescharacteristic
76C22×Dic122C2312yes—
77C2×Q8×D61—22no—
78C2×#11758190C2318yes—
79C2×#11758200C2318yes—
80C2×#11758199C2314yes—
81#14519777—21no—
82#14519777—21no—
83#14519777C2312yes—
84#14519777C2312yes—
85#14519777C2311yescharacteristic
86#14519776C2314yes—
87#14519774C2311yescharacteristic
88#14519773C2314yes—
89#14519770—28no—
90#14519770—28no—
91#14519765—21no—
92#14519764—24no—
93#14519764C2318yes—
94#14519763—24no—
95#14519756—24no—
96#14519756—24no—
97#14519756—22no—
98#14519755—24no—
99#14519754—24no—
100#14519753—22no—
101#14519751—21no—
102#14519751—22no—
103#14519751C2312yes—
104#14519750—24no—
105#14519749—24no—
106#14519749—24no—
107#14519748—22no—
108C4×Dic122D412yes—
109C4×Dic122D412yes—
110C4×Dic122C2312yes—
111C4×Dic122C2312yes—
112#14519742—24no—
113#14519742C2318yes—
114C4×D244—21no—
115C4×D244C2312yes—
116D4×Dic61C2314yes—
117#14519739C2314yes—
118#14519737—24no—
119#14519736—24no—
120#14519736—24no—
121C2×C4⋊Dic61—21no—
122C2×C4⋊Dic61—21no—
123C2×C4⋊Dic61D411yescharacteristic
124C2×#11758196—22no—
125C2×#11758196D412yes—
126C2×C22⋊Dic61—22no—
127C2×C22⋊Dic61—22no—
128C2×C4×Dic61—22no—
129C2×C4×Dic61C2312yes—
130#14519710C2314yes—
131C4⋊#92418C2312yes—
132#14519706C2318yes—
133#14519702C2318yes—
134#14519700C2318yes—
135#14519698C2314yes—
136#14519697C2314yes—
137C22⋊#92418C2312yes—
138#14519687C2318yes—
139C4×#92418C2312yes—
140C2×#41×C61—21no—
141C22×D4×C61C2311yescharacteristic
142C23×C4×C61—21no—
143C2×D8×C61C2314yes—
144C4⋊Q8×C61C2311yescharacteristic
145#127×C61C2311yescharacteristic
146#124×C61—21no—
147#121×C61—21no—
148#121×C61C2312yes—
149C4×D4×C61—21no—
150C4×D4×C61—21no—
151C4×D4×C61—21no—
152C4×D4×C61C2312yes—
153C4×D4×C61C2312yes—
154C4×D4×C61C2312yes—
155C2×C4⋊C4×C61C2311yescharacteristic
156C2×C22⋊C4×C61—21no—
157C2×C42×C61C2311yescharacteristic
158#107×C61C2314yes—
159C4⋊C8×C61C2312yes—
160#102×C61C2314yes—
161#102×C61C2314yes—
162C22⋊C8×C61C2312yes—
163C4×C8×C61C2312yes—
164C24×C61—21no—
165#41×C61—44no—
166C2×Q8×C61—21no—
167C2×D4×C61—22no—
168C2×D4×C61—21no—
169C2×D4×C61—21no—
170C2×D4×C61—21no—
171C2×D4×C61—42no—
172C2×D4×C61C2412yes—
173C2×D4×C61C2412yes—
174C22×C4×C61—21no—
175C22×C4×C61—22no—
176C22×C4×C61—22no—
177C22×C4×C61—22no—
178C22×C4×C61—44no—
179C22×C4×C61C2×D411yescharacteristic
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2411yescharacteristic
182C2×C22⋊D61—22no—
183C2×C22⋊D61—24no—
184C2×C22⋊D61—24no—
185C22×Dic61—22no—
186C22×Dic61—24no—
187C22×Dic61C2×D412yes—
188C22×Dic61C2×D412yes—
189Q8×D61—48no—
190#11758224—48no—
191#11758224—416no—
192#11758222—48no—
193#11758222—44no—
194C2×D244—21no—
195C2×D244—21no—
196C2×D244C2412yes—
197C2×C4×D61—22no—
198C2×C4×D61—24no—
199C2×Dic122—24no—
200C2×Dic122—21no—
201C2×Dic122—22no—
202C2×Dic122—24no—
203C2×Dic122—44no—
204C2×Dic122—48no—
205C2×Dic122C2×D412yes—
206C2×Dic122C2×D412yes—
207C2×Dic122C2414yes—
208D8×C61—44no—
209C2×C8×C61C2414yes—
210C4⋊C4×C61—21no—
211C4⋊C4×C61—21no—
212C4⋊C4×C61C2412yes—
213C4⋊C4×C61C2412yes—
214C22⋊C4×C61—22no—
215C22⋊C4×C61—22no—
216C22⋊C4×C61—22no—
217C22⋊C4×C61—22no—
218C42×C61C2×D411yescharacteristic
219C42×C61C2×D411yescharacteristic
220C42×C61C2411yescharacteristic
221C42×C61C2411yescharacteristic
222C22⋊Dic61—28no—
223C22⋊Dic61—28no—
224C22⋊Dic61—24no—
225#11758200—48no—
226#11758199—44no—
227#11758198—22no—
228#11758198—22no—
229#11758198—24no—
230C4⋊Dic61—22no—
231C4⋊Dic61—22no—
232C4⋊Dic61—22no—
233C4⋊Dic61—22no—
234C4⋊Dic61—21no—
235C4⋊Dic61C2×D412yes—
236C4⋊Dic61C2×D412yes—
237C4⋊Dic61C2412yes—
238#11758196—28no—
239#11758196—24no—
240#11758196C2×D414yes—
241#11758196C2×D414yes—
242C4×Dic61—28no—
243C4×Dic61C2×D414yes—
244C4×Dic61C2414yes—
245C2×#92418C2414yes—
246#11758190—48no—
247C23×C61—22no—
248C23×C61—22no—
249C23×C61—42no—
250C23×C61—42no—
251C23×C61—42no—
252C23×C61C22×D411yescharacteristic
253C22×D61—22no—
254Q8×C61—42no—
255D4×C61—22no—
256D4×C61—24no—
257D4×C61—22no—
258D4×C61—42no—
259D4×C61—44no—
260D4×C61—42no—
261D4×C61—81no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—22no—
265C2×C4×C61—41no—
266C2×C4×C61—42no—
267C2×C4×C61—44no—
268C2×C4×C61—44no—
269C2×C4×C61—44no—
270C2×C4×C61AGL(1,ℤ/8)14yes—
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—48no—
278C22⋊D61—44no—
279C22⋊D61—84no—
280C2×Dic61—24no—
281C2×Dic61—24no—
282C2×Dic61—22no—
283C2×Dic61—42no—
284C2×Dic61—48no—
285C2×Dic61—44no—
286C2×Dic61C22×D414yes—
287C2×Dic61C22×D414yes—
288D244—24no—
289D244—42no—
290D244—81no—
291C4×D61—48no—
292C4×D61—44no—
293Dic122—24no—
294Dic122—24no—
295Dic122—28no—
296Dic122—48no—
297Dic122—44no—
298Dic122—42no—
299C8×C61—24no—
300#92418—24no—
301#61888—611nomaximal, Sylow
302C22×C61—21no—
303C22×C61—21no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61—84no—
311C22×C61C2×AGL(1,ℤ/8)12yes—
312C22×C61C23×D411yescharacteristic
313C2×D61—42no—
314C2×D61—42no—
315C2×D61—42no—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—21no—
319C4×C61—42no—
320C4×C61—42no—
321C4×C61C2×AGL(1,ℤ/8)12yes—
322C4×C61C2×AGL(1,ℤ/8)12yes—
323C4×C61#58512yes—
324Dic61—24no—
325Dic61—24no—
326Dic61—44no—
327Dic61—42no—
328#3497—611no—
329C2×#556—611no—
330#3397—614no—
331#3397—614no—
332#3367—614no—
333#3367—614no—
334#3213—614no—
335#3206—611no—
336#3206—611no—
337#3112—614no—
338#3100—611no—
339#3100D6111yescharacteristic, complemented
340#2983—611no—
341C2×C61—22no—
342C2×C61—82no—
343C2×C61—82no—
344C2×C61#348711yescharacteristic
345C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
346C2×C61#358611yescharacteristic
347D61—82no—
348C2×2−1+4—1221no—
349#556—614no—
350#556—614no—
351#556—1222no—
352#556—1222no—
353#550—612no—
354#550—612no—
355D4×Q8—612no—
356D4×Q8—612no—
357#548—1222no—
358#548—1222no—
359#537—1222no—
360#537—1221no—
361#532—614no—
362#532—611no—
363#532C2×D6111yescharacteristic
364C2×C4⋊Q8—611no—
365C2×#126—1222no—
366C2×#124—612no—
367C2×C22⋊Q8—1221no—
368C2×C4×Q8—611no—
369C2×C4×Q8—612no—
370C2×C4×D4C2×D6111yescharacteristic
371C2×#117—1221no—
372#496—618no—
373#492—612no—
374#492—612no—
375#489—618no—
376#487—612no—
377#487—612no—
378#480—612no—
379#480C2×D6112yes—
380#477—618no—
381#474—612no—
382#474C2×D6112yes—
383#463—618no—
384#463—618no—
385#461—614no—
386#461—614no—
387#460—618no—
388#459—612no—
389#459C2×D6112yes—
390#448—614no—
391#448—614no—
392#447—612no—
393#447C2×D6112yes—
394#446—614no—
395#446—614no—
396#440—618no—
397#440—618no—
398C4×D8—614no—
399C4×D8C2×D6114yes—
400#432—614no—
401#431—611no—
402#431C2×D6111yescharacteristic
403C61#6188811yescharacteristic, complemented, Sylow
4042−1+4—2448no—
405C2×#41—21no—
406C2×#41—1222no—
407C2×#41—1224no—
408C2×#41—1222no—
409C2×#41—1221no—
410C22×Q8—611no—
411C22×Q8—612no—
412C22×Q8—1222no—
413C22×Q8—1222no—
414C22×D4C22×D6111yescharacteristic
415C23×C4—21no—
416C2×SD16—618no—
417C2×SD16—618no—
418C2×D8—614no—
419C2×D8C22×D6114yes—
420C4⋊Q8—614no—
421C4⋊Q8—611no—
422C4⋊Q8—612no—
423C4⋊Q8—612no—
424C4⋊Q8—1221no—
425C4⋊Q8—1221no—
426C4⋊Q8C22×D6111yescharacteristic
427#127—614no—
428#127—611no—
429#127C22×D6111yescharacteristic
430#126—1228no—
431#126—1228no—
432#124—21no—
433#124—618no—
434#124—1224no—
435#124—1224no—
436#124—1221no—
437C22⋊Q8—1224no—
438C22⋊Q8—1224no—
439C22⋊Q8—1222no—
440C22⋊Q8—1222no—
441C22⋊Q8—1224no—
442C22⋊Q8—1224no—
443#121—21no—
444#121—612no—
445#121—1224no—
446#121—1222no—
447#121—1222no—
448#121—1224no—
449#121—1224no—
450#121—1221no—
451#121C22×D6112yes—
452C4×Q8—612no—
453C4×Q8—612no—
454C4×Q8—612no—
455C4×Q8—612no—
456C4×Q8—618no—
457C4×Q8—1224no—
458C4×D4—21no—
459C4×D4—21no—
460C4×D4—21no—
461C4×D4—614no—
462C4×D4—614no—
463C4×D4—612no—
464C4×D4—1224no—
465C4×D4—1221no—
466C4×D4C22×D6112yes—
467C4×D4C22×D6112yes—
468C4×D4C22×D6112yes—
469#117—1224no—
470#117—1224no—
471C2×C4⋊C4—611no—
472C2×C4⋊C4—612no—
473C2×C4⋊C4—1222no—
474C2×C4⋊C4—1221no—
475C2×C4⋊C4—1221no—
476C2×C4⋊C4C22×D6111yescharacteristic
477C2×C22⋊C4—21no—
478C2×C22⋊C4—1222no—
479C2×C22⋊C4—1222no—
480C2×C42—612no—
481C2×C42—1222no—
482C2×C42C22×D6111yescharacteristic
483#107—614no—
484#107C22×D6114yes—
485C4⋊C8—618no—
486C4⋊C8—612no—
487C4⋊C8C22×D6112yes—
488#103—618no—
489#103—618no—
490#102—614no—
491#102—614no—
492#102C22×D6114yes—
493#102C22×D6114yes—
494C22⋊C8—612no—
495C22⋊C8C22×D6112yes—
496#97—618no—
497C4×C8—612no—
498C4×C8C22×D6112yes—
499C24—21no—
500#41—44no—
501#41—2448no—
502#41—24416no—
503#41—2448no—
504#41—2444no—
505C2×Q8—21no—
506C2×Q8—612no—
507C2×Q8—612no—
508C2×Q8—614no—
509C2×Q8—1224no—
510C2×Q8—1222no—
511C2×Q8—1221no—
512C2×Q8—1224no—
513C2×Q8—2444no—
514C2×Q8—2448no—
515C2×Q8—2448no—
516C2×D4—22no—
517C2×D4—21no—
518C2×D4—21no—
519C2×D4—21no—
520C2×D4—42no—
521C2×D4—612no—
522C2×D4—1224no—
523C2×D4—1224no—
524C2×D4—1222no—
525C2×D4—1221no—
526C2×D4—1221no—
527C2×D4C23×D6112yes—
528C2×D4C23×D6112yes—
529C22×C4—21no—
530C22×C4—22no—
531C22×C4—22no—
532C22×C4—22no—
533C22×C4—44no—
534C22×C4—612no—
535C22×C4—612no—
536C22×C4—1224no—
537C22×C4—1224no—
538C22×C4—1222no—
539C22×C4—1222no—
540C22×C4D4×D6111yescharacteristic
541C22×C4D4×D6111yescharacteristic
542C22×C4C23×D6111yescharacteristic
543SD16—2448no—
544SD16—2448no—
545D8—44no—
546D8—2444no—
547C2×C8—614no—
548C2×C8C23×D6114yes—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—614no—
552C4⋊C4—614no—
553C4⋊C4—612no—
554C4⋊C4—612no—
555C4⋊C4—612no—
556C4⋊C4—1228no—
557C4⋊C4—1224no—
558C4⋊C4—1222no—
559C4⋊C4—1222no—
560C4⋊C4—1222no—
561C4⋊C4—1222no—
562C4⋊C4—1221no—
563C4⋊C4C23×D6112yes—
564C4⋊C4C23×D6112yes—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—1224no—
570C22⋊C4—1228no—
571C22⋊C4—1228no—
572C22⋊C4—1224no—
573C22⋊C4—1222no—
574C22⋊C4—1222no—
575C42—614no—
576C42—614no—
577C42—1228no—
578C42D4×D6111yescharacteristic
579C42D4×D6111yescharacteristic
580C42C23×D6111yescharacteristic
581C42C23×D6111yescharacteristic
582C23—22no—
583C23—22no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—1222no—
588C23C2×D4×D6111yescharacteristic
589Q8—42no—
590Q8—1224no—
591Q8—1224no—
592Q8—1228no—
593Q8—2444no—
594Q8—2448no—
595Q8—2442no—
596D4—22no—
597D4—24no—
598D4—22no—
599D4—42no—
600D4—44no—
601D4—42no—
602D4—81no—
603D4—1224no—
604D4—4881no—
605D4—4884no—
606D4—2448no—
607D4—2444no—
608D4—2442no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—614no—
618C2×C4—614no—
619C2×C4—1224no—
620C2×C4—1224no—
621C2×C4—1222no—
622C2×C4—2448no—
623C2×C4—2442no—
624C2×C4—2448no—
625C2×C4—2444no—
626C2×C4—2444no—
627C2×C4#1451980114yes—
628C2×C4C2×D4×D6111yescharacteristic
629C2×C4C2×D4×D6111yescharacteristic
630C2×C4C2×D4×D6111yescharacteristic
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4C2×D4×D6111yescharacteristic
633C2×C4C24×D6111yescharacteristic
634C8—24no—
635C8—1224no—
636C22—21no—
637C22—21no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—84no—
645C22—2442no—
646C22—2442no—
647C22—2442no—
648C22C2×#1451980112yes—
649C22C22×D4×D6111yescharacteristic
650C4—21no—
651C4—21no—
652C4—21no—
653C4—42no—
654C4—42no—
655C4—1224no—
656C4—1224no—
657C4—2444no—
658C4—2442no—
659C4C2×#1451980112yes—
660C4C2×#1451980112yes—
661C4#1684059812yes—
662C2—22no—
663C2—82no—
664C2—82no—
665C2—4882no—
666C2#2145947311yescharacteristic
667C2C22×#1451980111yescharacteristic
668C2#2146060311yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3100C2
socleC22×C61C23×D4
hypercenter#3100D61
hypocenterC61#61888
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #21459585C2semidirect
2. C2×#16840482C2semidirect
3. #21458843C2semidirect
4. #21458837C2semidirect
5. #21458536C2semidirect
6. #21458535C2semidirect
7. #21456803C2semidirect
8. #21456746C2semidirect
9. #21456741C2semidirect
10. #21455938C2semidirect
11. #21455819C2semidirect
12. #21454902C2semidirect
13. C61×#3100C2semidirect
19. #16840433C22semidirect
20. #16840432C22semidirect
21. D4×Dic122C22semidirect
22. #16840427C22semidirect
37. #16840135C22semidirect
38. #16840134C22semidirect
40. #16840114C22semidirect
41. #16840113C22semidirect
43. #16840056C22semidirect
47. #16839968C22semidirect
48. #16839967C22semidirect
49. #16839961C22semidirect
54. #16839892C22semidirect
55. #16839889C22semidirect
56. #16839861C22semidirect
57. #16839859C22semidirect
339. #3100D61semidirect
403. C61#61888semidirect

Maximal subgroups

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

subgroupindex
#214595852
C2×#168404822
#214588432
#214588372
#214585362
#214585352
#214568032
#214567462
#214567412
#214559382
#214558192
#214549022
C61×#31002
#6188861