Finite Group Databasebeta

← #30571106

Subgroups — #30571106

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×#16840486C211yesmaximal, characteristic, complemented
3#21458844C214yesmaximal, complemented
4#21458838C214yesmaximal, complemented
5#21458542C214yesmaximal, complemented
6#21458541C214yesmaximal, complemented
7#21456805C214yesmaximal, complemented
8#21456744C211yesmaximal, characteristic, complemented
9#21455936C214yesmaximal, complemented
10#21455820C211yesmaximal, characteristic, complemented
11#21455815C211yesmaximal, characteristic, complemented
12#21454902C211yesmaximal, characteristic, complemented
13C61×#3206C211yesmaximal, characteristic, complemented
14C2×#14519824—21no—
15#16840486—22no—
16#16840486—22no—
17#16840486C2214yes—
18#16840486C2214yes—
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×#14519772—22no—
30C2×#14519766C2212yes—
31C2×#14519757—21no—
32C2×Q8×Dic61C2212yes—
33C2×C4×Dic122C2211yescharacteristic
34C2×C4×D244C2211yescharacteristic
35C2×#14519735—21no—
36#16840148C2218yes—
37#16840136C2212yescomplemented
38#16840123C2218yes—
39#16840112C2212yescomplemented
40#16840076C2212yes—
41#16840073C2212yes—
42#16840055C2218yescomplemented
43#16840043C2212yes—
44#16840039C2212yes—
45#16839980C2218yes—
46#16839979C2218yes—
47#16839964C2214yescomplemented
48#16839960C2218yescomplemented
49#16839958C2212yes—
50#16839954C2212yes—
51#16839901C2214yes—
52#16839898C2212yes—
53#16839895C2212yes—
54#16839892C2214yescomplemented
55#16839889C2214yescomplemented
56#16839865C2218yescomplemented
57#16839863C2218yescomplemented
58C4×D488C2214yes—
59C4×#11758199C2214yes—
60#16839816C2214yes—
61#16839815C2211yescharacteristic
62C61×#532C2211yescharacteristic
63C2×C4⋊Q8×C61C2211yescharacteristic
64C61×#492C2212yescomplemented
65C61×#487C2212yescomplemented
66C61×#461C2214yescomplemented
67C61×#448C2214yes—
68C61×#431C2211yescharacteristic
69#14519824—48no—
70C2×#11758222—21no—
71C2×#11758222—22no—
72C2×#11758224—24no—
73C2×#11758224—22no—
74C22×Dic122—22no—
75C22×Dic122D411yescharacteristic
76C2×Q8×D61—22no—
77C22×D244C2311yescharacteristic
78C22×C4×D61—21no—
79C2×#11758201C2318yes—
80C2×D488C2314yes—
81C2×#11758199C2314yes—
82#14519777C2311yescharacteristic
83#14519776C2314yes—
84#14519774C2311yescharacteristic
85#14519773C2314yes—
86#14519772—28no—
87#14519772—28no—
88#14519766—24no—
89#14519766C2318yes—
90#14519765—21no—
91#14519763—24no—
92#14519757—24no—
93#14519757—24no—
94#14519756—22no—
95#14519755—24no—
96#14519754—24no—
97#14519753—22no—
98#14519751—21no—
99#14519751—22no—
100#14519751C2312yes—
101#14519750—24no—
102#14519749—24no—
103#14519749—24no—
104#14519748—22no—
105Q8×Dic61—24no—
106Q8×Dic61C2318yes—
107C4×Dic122D412yes—
108C4×Dic122D412yes—
109C4×Dic122C2312yes—
110C4×Dic122C2312yes—
111C4×D244—21no—
112C4×D244—21no—
113C4×D244—21no—
114C4×D244C2312yes—
115C4×D244C2312yes—
116C4×D244C2312yes—
117D4×Dic61C2314yes—
118#14519739C2314yes—
119#14519737—24no—
120#14519735—24no—
121#14519735—24no—
122C2×C4⋊Dic61—22no—
123C2×C4⋊Dic61D411yescharacteristic
124C2×C4⋊Dic61C2311yescharacteristic
125C2×#11758196D412yes—
126C2×#11758198—21no—
127C2×#11758198—22no—
128C2×#11758198—22no—
129C2×C4×Dic61—22no—
130C2×C4×Dic61C2312yes—
131#14519711C2314yes—
132#14519710C2314yes—
133C4⋊#92418C2312yes—
134#14519706C2318yes—
135#14519701C2318yes—
136#14519699C2314yes—
137#14519699C2314yes—
138#14519698C2314yes—
139#14519697C2314yes—
140C22⋊#92418C2312yes—
141#14519687C2318yes—
142C4×#92418C2312yes—
143C2×#41×C61—21no—
144C22×Q8×C61C2312yes—
145C2×SD16×C61C2318yes—
146C4⋊Q8×C61—21no—
147C4⋊Q8×C61—21no—
148C4⋊Q8×C61C2312yes—
149C4⋊Q8×C61C2312yes—
150C4⋊Q8×C61C2311yescharacteristic
151#127×C61C2311yescharacteristic
152#124×C61—21no—
153#121×C61—21no—
154#121×C61C2312yes—
155C4×D4×C61—21no—
156C4×D4×C61C2312yes—
157C2×C4⋊C4×C61—21no—
158C2×C4⋊C4×C61—21no—
159C2×C42×C61C2311yescharacteristic
160C4⋊C8×C61C2312yes—
161#103×C61C2318yes—
162C22⋊C8×C61C2312yes—
163C4×C8×C61C2312yes—
164C23×D61—21no—
165#41×C61—44no—
166C2×Q8×C61—21no—
167C2×Q8×C61—22no—
168C2×Q8×C61—44no—
169C2×Q8×C61C2414yes—
170C2×D4×C61—22no—
171C2×D4×C61—21no—
172C2×D4×C61—21no—
173C2×D4×C61C2412yes—
174C22×C4×C61—22no—
175C22×C4×C61—22no—
176C22×C4×C61C2×D411yescharacteristic
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2411yescharacteristic
179C2×C22⋊D61—22no—
180C2×C22⋊D61—24no—
181C2×C22⋊D61—24no—
182C22×Dic61—21no—
183C22×Dic61—24no—
184C22×Dic61C2×D412yes—
185C22×Dic61C2×D412yes—
186Q8×D61—48no—
187#11758224—48no—
188#11758224—416no—
189#11758222—48no—
190#11758222—44no—
191C2×D244—21no—
192C2×D244—21no—
193C2×D244—21no—
194C2×D244—42no—
195C2×D244C2412yes—
196C2×D244C2412yes—
197C2×C4×D61—22no—
198C2×C4×D61—22no—
199C2×C4×D61—22no—
200C2×C4×D61—24no—
201C2×C4×D61—44no—
202C2×Dic122—24no—
203C2×Dic122—21no—
204C2×Dic122—24no—
205C2×Dic122—48no—
206C2×Dic122C2×D412yes—
207C2×Dic122C2×D412yes—
208SD16×C61—48no—
209C2×C8×C61C2414yes—
210C4⋊C4×C61—22no—
211C4⋊C4×C61—22no—
212C4⋊C4×C61—22no—
213C4⋊C4×C61—22no—
214C4⋊C4×C61—21no—
215C4⋊C4×C61C2412yes—
216C22⋊C4×C61—22no—
217C22⋊C4×C61—22no—
218C42×C61C2×D411yescharacteristic
219C42×C61C2×D411yescharacteristic
220C42×C61C2411yescharacteristic
221C42×C61C2411yescharacteristic
222C22⋊Dic61—24no—
223#11758201—48no—
224#11758199—44no—
225#11758198—22no—
226#11758198—22no—
227#11758198—22no—
228#11758198—22no—
229#11758198—28no—
230#11758198—28no—
231#11758198—24no—
232C4⋊Dic61—21no—
233C4⋊Dic61—21no—
234C4⋊Dic61—28no—
235C4⋊Dic61C2×D412yes—
236C4⋊Dic61C2×D412yes—
237C4⋊Dic61C2412yes—
238C4⋊Dic61C2412yes—
239#11758196—24no—
240#11758196C2×D414yes—
241#11758196C2×D414yes—
242C4×Dic61—28no—
243C4×Dic61C2×D414yes—
244C4×Dic61C2414yes—
245C2×#92418C2414yes—
246D488—44no—
247C23×C61—22no—
248C23×C61C22×D411yescharacteristic
249C22×D61—22no—
250C22×D61—22no—
251C22×D61—42no—
252C22×D61—42no—
253C22×D61—42no—
254Q8×C61—28no—
255Q8×C61—44no—
256Q8×C61—42no—
257D4×C61—24no—
258D4×C61—44no—
259D4×C61—42no—
260D4×C61—81no—
261C2×C4×C61—22no—
262C2×C4×C61—24no—
263C2×C4×C61—22no—
264C2×C4×C61—41no—
265C2×C4×C61—44no—
266C2×C4×C61—44no—
267C2×C4×C61AGL(1,ℤ/8)14yes—
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C2511yescharacteristic
274C22⋊D61—48no—
275C22⋊D61—44no—
276C22⋊D61—84no—
277C2×Dic61—24no—
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—42no—
281C2×Dic61—48no—
282C2×Dic61—42no—
283C2×Dic61C22×D414yes—
284C2×Dic61C22×D414yes—
285D244—22no—
286D244—24no—
287D244—22no—
288D244—42no—
289D244—42no—
290D244—81no—
291C4×D61—48no—
292C4×D61—44no—
293C4×D61—44no—
294C4×D61—44no—
295Dic122—24no—
296Dic122—24no—
297Dic122—48no—
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×C61C2×AGL(1,ℤ/8)12yes—
308C22×C61C23×D411yescharacteristic
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—42no—
315C2×D61—84no—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—21no—
319C4×C61—44no—
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—42no—
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#3206D6111yescharacteristic, complemented
337#3112—614no—
338#3100—611no—
339#3100—611no—
340#2983—611no—
341C2×C61—22no—
342C2×C61—82no—
343C2×C61#348711yescharacteristic
344C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
345C2×C61#358611yescharacteristic
346D61—82no—
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⋊Q8C2×D6111yescharacteristic
365C2×#126—1222no—
366C2×#124—612no—
367C2×C22⋊Q8—1221no—
368C2×C4×Q8—611no—
369C2×C4×Q8—612no—
370C2×C4×D4—611no—
371C2×#117—1221no—
372#496—618no—
373#492—612no—
374#492C2×D6112yescomplemented
375#489—618no—
376#487—612no—
377#487C2×D6112yescomplemented
378#480—612no—
379#480—612no—
380#477—618no—
381#474—612no—
382#474—612no—
383#463—618no—
384#463—618no—
385#461—614no—
386#461C2×D6114yescomplemented
387#460—618no—
388#459—612no—
389#459—612no—
390#448—614no—
391#448C2×D6114yes—
392#447—612no—
393#447—612no—
394#446—614no—
395#446—614no—
396#440—618no—
397#440—618no—
398C4×D8—614no—
399C4×D8—614no—
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—1222no—
412C22×Q8—1222no—
413C22×Q8C22×D6112yes—
414C22×D4—611no—
415C23×C4—1221no—
416C2×SD16—618no—
417C2×SD16C22×D6118yes—
418C2×D8—614no—
419C2×D8—614no—
420C4⋊Q8—21no—
421C4⋊Q8—21no—
422C4⋊Q8—614no—
423C4⋊Q8—611no—
424C4⋊Q8C22×D6112yes—
425C4⋊Q8C22×D6112yes—
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—1224no—
440C22⋊Q8—1224no—
441C22⋊Q8—1222no—
442C22⋊Q8—1222no—
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—614no—
460C4×D4—614no—
461C4×D4—612no—
462C4×D4—612no—
463C4×D4—612no—
464C4×D4—1224no—
465C4×D4—1221no—
466C4×D4—1221no—
467C4×D4—1221no—
468C4×D4C22×D6112yes—
469#117—1224no—
470#117—1224no—
471C2×C4⋊C4—21no—
472C2×C4⋊C4—21no—
473C2×C4⋊C4—611no—
474C2×C4⋊C4—612no—
475C2×C4⋊C4—611no—
476C2×C4⋊C4—1222no—
477C2×C22⋊C4—1222no—
478C2×C22⋊C4—1222no—
479C2×C22⋊C4—1221no—
480C2×C42—612no—
481C2×C42—1222no—
482C2×C42C22×D6111yescharacteristic
483#107—614no—
484#107—614no—
485C4⋊C8—618no—
486C4⋊C8—612no—
487C4⋊C8C22×D6112yes—
488#103—618no—
489#103C22×D6118yes—
490#102—614no—
491#102—614no—
492#102—614no—
493#102—614no—
494C22⋊C8—612no—
495C22⋊C8C22×D6112yes—
496#97—618no—
497C4×C8—612no—
498C4×C8C22×D6112yes—
499C24—1221no—
500#41—44no—
501#41—2448no—
502#41—24416no—
503#41—2448no—
504#41—2444no—
505C2×Q8—22no—
506C2×Q8—21no—
507C2×Q8—44no—
508C2×Q8—612no—
509C2×Q8—612no—
510C2×Q8—1224no—
511C2×Q8—1221no—
512C2×Q8—1224no—
513C2×Q8—2448no—
514C2×Q8—2448no—
515C2×Q8C23×D6114yes—
516C2×D4—22no—
517C2×D4—21no—
518C2×D4—21no—
519C2×D4—612no—
520C2×D4—612no—
521C2×D4—1224no—
522C2×D4—1224no—
523C2×D4—1222no—
524C2×D4—1221no—
525C2×D4—1221no—
526C2×D4—1221no—
527C2×D4—2442no—
528C2×D4C23×D6112yes—
529C22×C4—22no—
530C22×C4—22no—
531C22×C4—612no—
532C22×C4—612no—
533C22×C4—1224no—
534C22×C4—1224no—
535C22×C4—1221no—
536C22×C4—1222no—
537C22×C4—1222no—
538C22×C4—1222no—
539C22×C4—2444no—
540C22×C4D4×D6111yescharacteristic
541C22×C4D4×D6111yescharacteristic
542C22×C4C23×D6111yescharacteristic
543SD16—48no—
544SD16—2448no—
545D8—2444no—
546D8—2444no—
547C2×C8—614no—
548C2×C8C23×D6114yes—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—21no—
554C4⋊C4—614no—
555C4⋊C4—614no—
556C4⋊C4—612no—
557C4⋊C4—612no—
558C4⋊C4—612no—
559C4⋊C4—612no—
560C4⋊C4—1228no—
561C4⋊C4—1224no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4C23×D6112yes—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—1228no—
568C22⋊C4—1228no—
569C22⋊C4—1224no—
570C22⋊C4—1224no—
571C22⋊C4—1222no—
572C22⋊C4—1222no—
573C22⋊C4—1222no—
574C22⋊C4—1222no—
575C42—614no—
576C42—614no—
577C42—1228no—
578C42D4×D6111yescharacteristic
579C42D4×D6111yescharacteristic
580C42C23×D6111yescharacteristic
581C42C23×D6111yescharacteristic
582C23—22no—
583C23—1222no—
584C23—1222no—
585C23—2442no—
586C23—2442no—
587C23—2442no—
588C23C2×D4×D6111yescharacteristic
589Q8—28no—
590Q8—44no—
591Q8—42no—
592Q8—1224no—
593Q8—1224no—
594Q8—2448no—
595Q8—2442no—
596D4—24no—
597D4—44no—
598D4—42no—
599D4—81no—
600D4—1222no—
601D4—1222no—
602D4—1224no—
603D4—4881no—
604D4—4884no—
605D4—2442no—
606D4—2448no—
607D4—2444no—
608D4—2442no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—614no—
616C2×C4—614no—
617C2×C4—1224no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—2448no—
621C2×C4—2442no—
622C2×C4—2448no—
623C2×C4—2442no—
624C2×C4—2444no—
625C2×C4—2444no—
626C2×C4—2444no—
627C2×C4#1451980414yes—
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—4884no—
642C22—2442no—
643C22—2442no—
644C22—2442no—
645C22—2442no—
646C22—2442no—
647C22—2442no—
648C22C2×#1451980412yes—
649C22C22×D4×D6111yescharacteristic
650C4—21no—
651C4—21no—
652C4—21no—
653C4—44no—
654C4—42no—
655C4—1224no—
656C4—1224no—
657C4—2442no—
658C4—2442no—
659C4C2×#1451980412yes—
660C4C2×#1451980412yes—
661C4#1684059712yes—
662C2—22no—
663C2—82no—
664C2—4882no—
665C2—4882no—
666C2#2145947511yescharacteristic
667C2C22×#1451980411yescharacteristic
668C2#2146060211yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3206C2
socleC22×C61C23×D4
hypercenter#3206D61
hypocenterC61#61888
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
1. #21459585C2semidirect
2. C2×#16840486C2semidirect
3. #21458844C2semidirect
4. #21458838C2semidirect
5. #21458542C2semidirect
6. #21458541C2semidirect
7. #21456805C2semidirect
8. #21456744C2semidirect
9. #21455936C2semidirect
10. #21455820C2semidirect
11. #21455815C2semidirect
12. #21454902C2semidirect
13. C61×#3206C2semidirect
19. #16840433C22semidirect
20. #16840432C22semidirect
21. D4×Dic122C22semidirect
22. #16840427C22semidirect
37. #16840136C22semidirect
39. #16840112C22semidirect
42. #16840055C22semidirect
47. #16839964C22semidirect
48. #16839960C22semidirect
54. #16839892C22semidirect
55. #16839889C22semidirect
56. #16839865C22semidirect
57. #16839863C22semidirect
64. C61×#492C22semidirect
65. C61×#487C22semidirect
66. C61×#461C22semidirect
336. #3206D61semidirect
374. #492C2×D61semidirect
377. #487C2×D61semidirect
386. #461C2×D61semidirect
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×#168404862
#214588442
#214588382
#214585422
#214585412
#214568052
#214567442
#214559362
#214558202
#214558152
#214549022
C61×#32062
#6188861