Finite Group Databasebeta

← #30571135

Subgroups — #30571135

Order 15616.

2034 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459585C211yesmaximal, characteristic, complemented
2C2×#16840497C211yesmaximal, characteristic, complemented
3#21458860C214yesmaximal, complemented
4#21458859C214yesmaximal, complemented
5#21458610C214yesmaximal, complemented
6#21458609C214yesmaximal, complemented
7#21456805C214yesmaximal, complemented
8#21456739C211yesmaximal, characteristic, complemented
9#21456734C211yesmaximal, characteristic, complemented
10#21455907C214yesmaximal, complemented
11#21455850C211yesmaximal, characteristic, complemented
12#21454902C211yesmaximal, characteristic, complemented
13C61×#3103C211yesmaximal, characteristic, complemented
14C2×#14519824—21no—
15#16840497—22no—
16#16840497—22no—
17#16840497C2214yes—
18#16840497C2214yes—
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×#14519774C2211yescharacteristic
30C2×#14519766C2212yes—
31C2×#14519765—21no—
32C2×#14519762—22no—
33C2×C4×Dic122C2211yescharacteristic
34C2×C4×D244—21no—
35C2×#14519739C2212yes—
36#16840148C2218yescomplemented
37#16840139C2212yes—
38#16840138C2212yes—
39#16840123C2218yescomplemented
40#16840110C2212yes—
41#16840109C2212yes—
42#16840080C2212yes—
43#16840051C2218yescomplemented
44#16840047C2212yescomplemented
45#16839980C2218yescomplemented
46#16839979C2218yescomplemented
47#16839962C2212yescomplemented
48#16839958C2214yes—
49#16839957C2214yes—
50#16839956C2218yescomplemented
51#16839901C2212yes—
52#16839898C2214yes—
53#16839895C2214yes—
54#16839892C2214yescomplemented
55#16839889C2214yescomplemented
56#16839873C2218yescomplemented
57#16839871C2218yescomplemented
58C4×#11758201C2214yescomplemented
59#16839816C2214yes—
60#16839815C2211yescharacteristic
61C61×#532C2211yescharacteristic
62C2×C4×Q8×C61C2211yescharacteristic
63C61×#481C2212yes—
64C61×#475C2212yescomplemented
65C61×#460C2212yescomplemented
66C61×#448C2212yes—
67C4×SD16×C61C2214yescomplemented
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×D244—21no—
78C22×D244—21no—
79C22×D244C2312yes—
80C22×C4×D61—22no—
81C2×#11758201C2314yes—
82C2×D488C2318yes—
83C2×#11758199C2318yes—
84#14519777C2311yescharacteristic
85#14519776C2314yes—
86#14519774—21no—
87#14519774—21no—
88#14519774C2312yes—
89#14519774C2312yes—
90#14519774C2311yescharacteristic
91#14519773C2314yes—
92#14519766—24no—
93#14519766C2318yes—
94#14519765—24no—
95#14519765—24no—
96#14519765—21no—
97#14519763—24no—
98#14519762—28no—
99#14519762—28no—
100#14519756—22no—
101#14519755—24no—
102#14519754—24no—
103#14519753—22no—
104#14519751—21no—
105#14519751—22no—
106#14519751C2312yes—
107#14519750—24no—
108#14519749—24no—
109#14519749—24no—
110#14519748—22no—
111C4×Dic122D412yes—
112C4×Dic122D412yes—
113C4×Dic122C2312yes—
114C4×Dic122C2312yes—
115C4×D244—21no—
116C4×D244—24no—
117C4×D244—24no—
118C4×D244C2312yes—
119D4×Dic61C2314yes—
120#14519739—24no—
121#14519739C2318yes—
122#14519739C2314yes—
123#14519737—24no—
124C2×C4⋊Dic61D411yescharacteristic
125C2×#11758196D412yes—
126C2×#11758198—22no—
127C2×#11758198—22no—
128C2×#11758198—22no—
129C2×C4×Dic61C2312yes—
130#14519708C2314yes—
131C4⋊#92418C2312yes—
132#14519706C2318yes—
133#14519701C2314yes—
134#14519699C2318yes—
135#14519697C2314yes—
136#14519697C2318yes—
137C22⋊#92418C2312yes—
138#14519687C2318yes—
139C4×#92418C2312yes—
140C2×#41×C61—21no—
141C22×Q8×C61C2311yescharacteristic
142C2×SD16×C61C2314yes—
143C4⋊Q8×C61C2311yescharacteristic
144#127×C61C2311yescharacteristic
145#124×C61—21no—
146#121×C61—21no—
147#121×C61C2312yes—
148C4×Q8×C61—21no—
149C4×Q8×C61—21no—
150C4×Q8×C61C2312yes—
151C4×Q8×C61C2312yes—
152C4×D4×C61—21no—
153C4×D4×C61C2312yes—
154C2×C4⋊C4×C61—21no—
155C2×C4⋊C4×C61C2311yescharacteristic
156C2×C42×C61—21no—
157C2×C42×C61C2311yescharacteristic
158#106×C61C2314yes—
159C4⋊C8×C61C2312yes—
160#103×C61C2314yes—
161#102×C61C2314yes—
162C22⋊C8×C61C2312yes—
163C4×C8×C61C2312yes—
164C23×D61—22no—
165#41×C61—44no—
166C2×Q8×C61—21no—
167C2×Q8×C61—21no—
168C2×Q8×C61—42no—
169C2×Q8×C61C2412yes—
170C2×D4×C61—22no—
171C2×D4×C61—21no—
172C2×D4×C61—21no—
173C2×D4×C61C2412yes—
174C22×C4×C61—21no—
175C22×C4×C61—21no—
176C22×C4×C61—22no—
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2411yescharacteristic
180C2×C22⋊D61—22no—
181C2×C22⋊D61—24no—
182C2×C22⋊D61—24no—
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—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—22no—
204C2×C4×D61—24no—
205C2×C4×D61—28no—
206C2×C4×D61—84no—
207C2×Dic122—24no—
208C2×Dic122—21no—
209C2×Dic122—24no—
210C2×Dic122—48no—
211C2×Dic122C2×D412yes—
212C2×Dic122C2×D412yes—
213SD16×C61—44no—
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—24no—
230#11758201—44no—
231#11758199—48no—
232#11758198—22no—
233#11758198—22no—
234#11758198—28no—
235#11758198—28no—
236#11758198—28no—
237#11758198—24no—
238C4⋊Dic61—21no—
239C4⋊Dic61C2×D412yes—
240C4⋊Dic61C2×D412yes—
241C4⋊Dic61C2412yes—
242#11758196—24no—
243#11758196C2×D414yes—
244#11758196C2×D414yes—
245C4×Dic61C2×D414yes—
246C4×Dic61C2414yes—
247C2×#92418C2414yes—
248D488—48no—
249C23×C61—22no—
250C23×C61C22×D411yescharacteristic
251C22×D61—24no—
252C22×D61—22no—
253C22×D61—44no—
254C22×D61—44no—
255C22×D61—44no—
256Q8×C61—22no—
257Q8×C61—22no—
258Q8×C61—42no—
259Q8×C61—42no—
260D4×C61—24no—
261D4×C61—44no—
262D4×C61—42no—
263D4×C61—81no—
264C2×C4×C61—22no—
265C2×C4×C61—22no—
266C2×C4×C61—22no—
267C2×C4×C61—22no—
268C2×C4×C61—41no—
269C2×C4×C61—42no—
270C2×C4×C61—42no—
271C2×C4×C61—44no—
272C2×C4×C61AGL(1,ℤ/8)14yes—
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C2511yescharacteristic
279C22⋊D61—48no—
280C22⋊D61—44no—
281C22⋊D61—84no—
282C2×Dic61—24no—
283C2×Dic61—22no—
284C2×Dic61—42no—
285C2×Dic61—48no—
286C2×Dic61C22×D414yes—
287C2×Dic61C22×D414yes—
288D244—24no—
289D244—28no—
290D244—44no—
291D244—44no—
292D244—44no—
293D244—42no—
294D244—81no—
295D244—82no—
296D244—82no—
297C4×D61—48no—
298C4×D61—44no—
299C4×D61—88no—
300Dic122—24no—
301Dic122—24no—
302Dic122—48no—
303Dic122—42no—
304C8×C61—24no—
305#92418—24no—
306#61891—611nomaximal, Sylow
307C22×C61—21no—
308C22×C61—21no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61C2×AGL(1,ℤ/8)12yes—
313C22×C61C23×D411yescharacteristic
314C2×D61—44no—
315C2×D61—42no—
316C2×D61—44no—
317C2×D61—42no—
318C2×D61—42no—
319C2×D61—44no—
320C2×D61—88no—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61—21no—
324C4×C61—42no—
325C4×C61—42no—
326C4×C61—42no—
327C4×C61C2×AGL(1,ℤ/8)12yes—
328C4×C61C2×AGL(1,ℤ/8)12yes—
329C4×C61#58512yes—
330Dic61—24no—
331Dic61—24no—
332Dic61—42no—
333#3497—611no—
334C2×#559—611no—
335#3399—614no—
336#3399—614no—
337#3373—614no—
338#3373—614no—
339#3213—614no—
340#3205—611no—
341#3205—611no—
342#3109—614no—
343#3103—611no—
344#3103D6111yescharacteristic, complemented
345#2983—611no—
346C2×C61—22no—
347C2×C61—82no—
348C2×C61#348711yescharacteristic
349C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
350C2×C61#358511yescharacteristic
351D61—82no—
352D61—84no—
353C2×2−1+4—1221no—
354#559—614no—
355#559—614no—
356#559—1222no—
357#559—1222no—
358#550—612no—
359#550—612no—
360D4×Q8—612no—
361D4×Q8—612no—
362#548—1222no—
363#548—1222no—
364#537—1222no—
365#537—1221no—
366#532—614no—
367#532—611no—
368#532C2×D6111yescharacteristic
369C2×#127—611no—
370C2×#124—612no—
371C2×#124—1221no—
372C2×#123—1222no—
373C2×C4×Q8—611no—
374C2×C4×Q8C2×D6111yescharacteristic
375C2×C4×D4—612no—
376C2×C4×D4—1221no—
377#496—618no—
378#493—612no—
379#493—612no—
380#489—618no—
381#486—612no—
382#486—612no—
383#481—612no—
384#481C2×D6112yes—
385#476—618no—
386#475—612no—
387#475C2×D6112yescomplemented
388#463—618no—
389#463—618no—
390#460—612no—
391#460C2×D6112yescomplemented
392#459—614no—
393#459—614no—
394#459—618no—
395#448—612no—
396#448C2×D6112yes—
397#447—614no—
398#447—614no—
399#446—614no—
400#446—614no—
401#442—618no—
402#442—618no—
403C4×SD16—614no—
404C4×SD16C2×D6114yescomplemented
405#432—614no—
406#431—611no—
407#431C2×D6111yescharacteristic
408C61#6189111yescharacteristic, complemented, Sylow
4092−1+4—2448no—
410C2×#41—21no—
411C2×#41—1222no—
412C2×#41—1224no—
413C2×#41—1222no—
414C2×#41—1221no—
415C22×Q8—611no—
416C22×Q8—1222no—
417C22×Q8—1222no—
418C22×Q8C22×D6111yescharacteristic
419C22×D4—612no—
420C22×D4—1221no—
421C22×D4—1221no—
422C23×C4—1222no—
423C2×SD16—614no—
424C2×SD16C22×D6114yes—
425C2×D8—618no—
426C2×D8—618no—
427C4⋊Q8—614no—
428C4⋊Q8—611no—
429C4⋊Q8C22×D6111yescharacteristic
430#127—614no—
431#127—612no—
432#127—612no—
433#127—611no—
434#127—1221no—
435#127—1221no—
436#127C22×D6111yescharacteristic
437#124—21no—
438#124—618no—
439#124—1224no—
440#124—1224no—
441#124—1224no—
442#124—1224no—
443#124—1221no—
444#123—1228no—
445#123—1228no—
446C22⋊Q8—1224no—
447C22⋊Q8—1224no—
448C22⋊Q8—1222no—
449C22⋊Q8—1222no—
450#121—21no—
451#121—612no—
452#121—1224no—
453#121—1222no—
454#121—1222no—
455#121—1224no—
456#121—1224no—
457#121—1221no—
458#121C22×D6112yes—
459C4×Q8—21no—
460C4×Q8—21no—
461C4×Q8—612no—
462C4×Q8—612no—
463C4×Q8—612no—
464C4×Q8—612no—
465C4×Q8C22×D6112yes—
466C4×Q8C22×D6112yes—
467C4×D4—21no—
468C4×D4—614no—
469C4×D4—618no—
470C4×D4—614no—
471C4×D4—612no—
472C4×D4—1224no—
473C4×D4—1224no—
474C4×D4—1224no—
475C4×D4—1224no—
476C4×D4—1221no—
477C4×D4C22×D6112yes—
478C2×C4⋊C4—21no—
479C2×C4⋊C4—611no—
480C2×C4⋊C4—612no—
481C2×C4⋊C4C22×D6111yescharacteristic
482C2×C22⋊C4—1222no—
483C2×C22⋊C4—1222no—
484C2×C22⋊C4—1222no—
485C2×C42—21no—
486C2×C42—612no—
487C2×C42C22×D6111yescharacteristic
488#106—614no—
489#106C22×D6114yes—
490C4⋊C8—618no—
491C4⋊C8—612no—
492C4⋊C8C22×D6112yes—
493#103—614no—
494#103C22×D6114yes—
495#102—618no—
496#102—614no—
497#102—618no—
498#102C22×D6114yes—
499C22⋊C8—612no—
500C22⋊C8C22×D6112yes—
501#97—618no—
502C4×C8—612no—
503C4×C8C22×D6112yes—
504C24—1222no—
505#41—44no—
506#41—2448no—
507#41—24416no—
508#41—2448no—
509#41—2444no—
510C2×Q8—21no—
511C2×Q8—21no—
512C2×Q8—42no—
513C2×Q8—612no—
514C2×Q8—612no—
515C2×Q8—1224no—
516C2×Q8—1221no—
517C2×Q8—1224no—
518C2×Q8—2448no—
519C2×Q8—2448no—
520C2×Q8C23×D6112yes—
521C2×D4—22no—
522C2×D4—21no—
523C2×D4—21no—
524C2×D4—614no—
525C2×D4—612no—
526C2×D4—1224no—
527C2×D4—1224no—
528C2×D4—1222no—
529C2×D4—1221no—
530C2×D4—1222no—
531C2×D4—1222no—
532C2×D4—1222no—
533C2×D4—1222no—
534C2×D4—1222no—
535C2×D4—1221no—
536C2×D4—4882no—
537C2×D4—2444no—
538C2×D4—2444no—
539C2×D4C23×D6112yes—
540C22×C4—21no—
541C22×C4—21no—
542C22×C4—22no—
543C22×C4—612no—
544C22×C4—612no—
545C22×C4—1228no—
546C22×C4—1224no—
547C22×C4—1224no—
548C22×C4—1222no—
549C22×C4—4884no—
550C22×C4D4×D6111yescharacteristic
551C22×C4D4×D6111yescharacteristic
552C22×C4C23×D6111yescharacteristic
553SD16—44no—
554SD16—2444no—
555D8—2448no—
556D8—2448no—
557C2×C8—614no—
558C2×C8C23×D6114yes—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—614no—
564C4⋊C4—614no—
565C4⋊C4—612no—
566C4⋊C4—612no—
567C4⋊C4—612no—
568C4⋊C4—1224no—
569C4⋊C4—1221no—
570C4⋊C4C23×D6112yes—
571C4⋊C4C23×D6112yes—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—1228no—
575C22⋊C4—1228no—
576C22⋊C4—1228no—
577C22⋊C4—1224no—
578C22⋊C4—1224no—
579C22⋊C4—1222no—
580C22⋊C4—1222no—
581C42—22no—
582C42—22no—
583C42—614no—
584C42—614no—
585C42D4×D6111yescharacteristic
586C42D4×D6111yescharacteristic
587C42C23×D6111yescharacteristic
588C42C23×D6111yescharacteristic
589C23—22no—
590C23—1224no—
591C23—1222no—
592C23—2444no—
593C23—2444no—
594C23—2444no—
595C23C2×D4×D6111yescharacteristic
596Q8—22no—
597Q8—22no—
598Q8—42no—
599Q8—42no—
600Q8—1224no—
601Q8—1224no—
602Q8—2448no—
603Q8—2442no—
604D4—24no—
605D4—44no—
606D4—42no—
607D4—81no—
608D4—1224no—
609D4—1228no—
610D4—4881no—
611D4—4882no—
612D4—4882no—
613D4—4884no—
614D4—2444no—
615D4—2448no—
616D4—2444no—
617D4—2444no—
618D4—2444no—
619D4—2442no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—614no—
629C2×C4—614no—
630C2×C4—1224no—
631C2×C4—1222no—
632C2×C4—4888no—
633C2×C4—2448no—
634C2×C4—2442no—
635C2×C4—2448no—
636C2×C4—2444no—
637C2×C4#1451980414yes—
638C2×C4C2×D4×D6111yescharacteristic
639C2×C4C2×D4×D6111yescharacteristic
640C2×C4C2×D4×D6111yescharacteristic
641C2×C4C2×D4×D6111yescharacteristic
642C2×C4C2×D4×D6111yescharacteristic
643C2×C4C24×D6111yescharacteristic
644C8—24no—
645C8—1224no—
646C22—21no—
647C22—21no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—4888no—
652C22—2444no—
653C22—2442no—
654C22—2442no—
655C22—2444no—
656C22—2444no—
657C22—2442no—
658C22C2×#1451980412yes—
659C22C22×D4×D6111yescharacteristic
660C4—21no—
661C4—21no—
662C4—21no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—1224no—
667C4—1224no—
668C4—2442no—
669C4C2×#1451980412yes—
670C4C2×#1451980412yes—
671C4#1684059812yes—
672C2—22no—
673C2—82no—
674C2—4884no—
675C2—4882no—
676C2#2145947311yescharacteristic
677C2C22×#1451980411yescharacteristic
678C2#2146059611yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3103C2
socleC22×C61C23×D4
hypercenter#3103D61
hypocenterC61#61891
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. #21459585C2semidirect
2. C2×#16840497C2semidirect
3. #21458860C2semidirect
4. #21458859C2semidirect
5. #21458610C2semidirect
6. #21458609C2semidirect
7. #21456805C2semidirect
8. #21456739C2semidirect
9. #21456734C2semidirect
10. #21455907C2semidirect
11. #21455850C2semidirect
12. #21454902C2semidirect
13. C61×#3103C2semidirect
19. #16840433C22semidirect
20. #16840432C22semidirect
21. D4×Dic122C22semidirect
22. #16840427C22semidirect
36. #16840148C22semidirect
39. #16840123C22semidirect
43. #16840051C22semidirect
44. #16840047C22semidirect
45. #16839980C22semidirect
46. #16839979C22semidirect
47. #16839962C22semidirect
50. #16839956C22semidirect
54. #16839892C22semidirect
55. #16839889C22semidirect
56. #16839873C22semidirect
57. #16839871C22semidirect
58. C4×#11758201C22semidirect
64. C61×#475C22semidirect
65. C61×#460C22semidirect
67. C4×SD16×C61C22semidirect
344. #3103D61semidirect
387. #475C2×D61semidirect
391. #460C2×D61semidirect
404. C4×SD16C2×D61semidirect
408. C61#61891semidirect

Maximal subgroups

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

subgroupindex
#214595852
C2×#168404972
#214588602
#214588592
#214586102
#214586092
#214568052
#214567392
#214567342
#214559072
#214558502
#214549022
C61×#31032
#6189161