Finite Group Databasebeta

← #30619345

Subgroups — #30619345

Order 15616.

772 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462597C211yesmaximal, characteristic
2#21462580C211yesmaximal, characteristic, complemented
3#21461981C211yesmaximal, characteristic, complemented
4#21461977C211yesmaximal, characteristic
5#21461867C211yesmaximal, characteristic, complemented
6#21461866C211yesmaximal, characteristic, complemented
7#21461861C211yesmaximal, characteristic, complemented
8#21461860C211yesmaximal, characteristic, complemented
9#21461855C211yesmaximal, characteristic, complemented
10#21461854C211yesmaximal, characteristic, complemented
11#21461849C211yesmaximal, characteristic, complemented
12#21461848C211yesmaximal, characteristic, complemented
13#21461739C211yesmaximal, characteristic, complemented
14#21461541C211yesmaximal, characteristic, complemented
15#21456311C211yesmaximal, characteristic
16#16840842C2211yescharacteristic
17#16840840C2211yescharacteristic
18#16840830C2211yescharacteristic
19#16840830C2211yescharacteristic
20#16840807—22no—
21#16840805—22no—
22#16840802—22no—
23Q16×#6923—22no—
24D61⋊#108C2211yescharacteristic
25#16840772C2211yescharacteristic
26#16840761C2211yescharacteristic
27#16840761C2211yescharacteristic
28#16840760C2211yescharacteristic
29#16840760C2211yescharacteristic
30#16840755C2211yescharacteristic
31#16840755C2211yescharacteristic
32#16840755C2211yescharacteristic
33#16840755C2211yescharacteristic
34D61⋊#101C2211yescharacteristic
35D61⋊#101C2211yescharacteristic
36#16840743C2211yescharacteristic
37#16840742C2211yescharacteristic
38#16840725C2211yescharacteristic
39#16840725C2211yescharacteristic
40#16840644C2211yescharacteristic
41#16840644C2211yescharacteristic
42#16840631C2211yescharacteristic
43#16840630C2211yescharacteristic
44#16840629C2211yescharacteristic
45#16840629C2211yescharacteristic
46#16840629C2211yescharacteristic
47#16840629C2211yescharacteristic
48C2×#14519809C2211yescharacteristic
49C2×#14519806C2211yescharacteristic
50C2×#14519798C2211yescharacteristic
51C2×Q16×D61C2211yescharacteristic
52#16840337C2211yescharacteristic
53#16840332C2211yescharacteristic
54#16840156C411yescharacteristic, complemented
55#16840155C411yescharacteristic, complemented
56#16840153C411yescharacteristic, complemented
57#16840144C411yescharacteristic, complemented
58#16840143C411yescharacteristic, complemented
59#16840141C411yescharacteristic, complemented
60#16840140C411yescharacteristic, complemented
61#16839719C2211yescharacteristic
62C61×#497C411yescharacteristic, complemented
63#14519865—22no—
64#14519865—41no—
65#14519864—22no—
66#14519864—41no—
67D61⋊M16—41no—
68D61⋊M16C2311yescharacteristic
69D61⋊M16C2311yescharacteristic
70D61⋊M16C2311yescharacteristic
71D61⋊M16C2311yescharacteristic
72C2×#11758217—41no—
73Q8×#6923—22no—
74Q8×#6923—22no—
75Q8×#6923—41no—
76Q8×#6923—41no—
77#14519856—41no—
78#14519856—41no—
79#14519856C2311yescharacteristic
80#14519856C2311yescharacteristic
81#14519856C2311yescharacteristic
82#14519856C2311yescharacteristic
83#14519848—21no—
84#14519847—21no—
85#14519846—21no—
86#14519845—21no—
87#14519842—22no—
88#14519842—41no—
89#14519841—22no—
90#14519841—41no—
91#14519840—22no—
92#14519840—22no—
93#14519840—41no—
94#14519840—41no—
95#14519838—41no—
96#14519838—41no—
97#14519836—41no—
98#14519835—41no—
99#14519831—21no—
100#14519831D412yes—
101C8×#6923—21no—
102C8×#6923D412yes—
103C2×#11758226C2311yescharacteristic
104C2×#11758224C2311yescharacteristic
105C2×Q8×D61C2311yescharacteristic
106C2×Q8×D61C2311yescharacteristic
107#14519809—41no—
108#14519809—41no—
109#14519806—41no—
110#14519806—41no—
111#14519798—21no—
112#14519798—21no—
113#14519798—21no—
114#14519798—21no—
115C2×Dic244C2×C411yescharacteristic
116C2×Dic244C2×C411yescharacteristic
117C2×#11758202C2×C411yescharacteristic
118C2×#11758202C2×C411yescharacteristic
119C2×#11758202C2×C411yescharacteristic
120C2×#11758202C2×C411yescharacteristic
121C2×#11758202C2×C411yescharacteristic
122Q16×D61—21no—
123Q16×D61—21no—
124Q16×D61—21no—
125Q16×D61—21no—
126C2×#11758190C2×C411yescharacteristic
127C2×#11758190C2×C411yescharacteristic
128C2×#11758200C2×C411yescharacteristic
129C2×#11758201C2×C411yescharacteristic
130C2×#11758199C2×C411yescharacteristic
131C2×#11758189C2311yescharacteristic
132C2×C8×D61C2311yescharacteristic
133#14519777C2×C411yescharacteristic
134#14519775C2×C411yescharacteristic
135#14519775C2×C411yescharacteristic
136#14519766C2×C411yescharacteristic
137#14519765C2×C411yescharacteristic
138#14519764C2×C411yescharacteristic
139#14519762—41no—
140#14519759—41no—
141#14519755—41no—
142#14519754—41no—
143#14519734C2311yescharacteristic
144#14519687C2×C411yescharacteristic
145C4×#92418C2×C411yescharacteristic
146C8×Dic61C2×C411yescharacteristic
147C2×Q16×C61C2×C411yescharacteristic
148C2×Q16×C61C2×C411yescharacteristic
149C2×SD16×C61C2×C411yescharacteristic
150C2×SD16×C61C2×C411yescharacteristic
151C4⋊Q8×C61C2×C411yescharacteristic
152#124×C61C2×C411yescharacteristic
153#97×C61C2×C411yescharacteristic
154C2×Q8×C61C22×C411yescharacteristic
155C2×Q8×C61C22×C411yescharacteristic
156C2×Q8×C61C22×C411yescharacteristic
157C2×D4×C61C22×C411yescharacteristic
158C2×C22⋊D61—21no—
159C22×Dic61—21no—
160#11758226—21no—
161#11758226—21no—
162#11758226—21no—
163#11758226—21no—
164Q8×D61—21no—
165Q8×D61—21no—
166Q8×D61—21no—
167Q8×D61—21no—
168Q8×D61—21no—
169Q8×D61—21no—
170Q8×D61—21no—
171Q8×D61—21no—
172#11758224—21no—
173#11758224—21no—
174#11758224—21no—
175#11758224—21no—
176C2×D244—21no—
177C2×D244C22×C411yescharacteristic
178C2×C4×D61—21no—
179C2×C4×D61—21no—
180C2×C4×D61—21no—
181C2×C4×D61C2411yescharacteristic
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122C22×C411yescharacteristic
185C2×Dic122C22×C411yescharacteristic
186C2×Dic122C22×C411yescharacteristic
187C22⋊#6923—21no—
188C22⋊#6923—21no—
189C22⋊#6923—21no—
190C22⋊#6923—21no—
191#11758217—21no—
192#11758217—21no—
193#11758217—21no—
194#11758217—21no—
195#11758217—42no—
196C2×#92420—42no—
197C4⋊#6923—21no—
198C4⋊#6923—21no—
199C4⋊#6923—21no—
200C4⋊#6923—21no—
201C4⋊#6923—42no—
202C4⋊#6923—42no—
203C4×#6923—21no—
204C4×#6923—21no—
205C4×#6923—21no—
206C4×#6923—42no—
207C4×#6923—42no—
208C4×#6923C2×D412yes—
209#11758213—21no—
210#11758213—21no—
211#11758213—21no—
212#11758213—21no—
213#11758213—42no—
214D61⋊C8—21no—
215D61⋊C8—21no—
216D61⋊C8—21no—
217D61⋊C8—42no—
218D61⋊C8C2×D412yes—
219Q16×C61—21no—
220Q16×C61—21no—
221Q16×C61—41no—
222SD16×C61—21no—
223SD16×C61—21no—
224SD16×C61—41no—
225C2×C8×C61C22×C411yescharacteristic
226C2×C8×C61C22×C411yescharacteristic
227C4⋊C4×C61—41no—
228C22⋊C4×C61—41no—
229C42×C61C22×C411yescharacteristic
230C22⋊Dic61—41no—
231#11758202—21no—
232#11758202—21no—
233#11758202—21no—
234#11758202—21no—
235#11758202—21no—
236#11758202—21no—
237#11758202—41no—
238#11758202—41no—
239#11758201—41no—
240#11758200—41no—
241#11758199—21no—
242#11758199—21no—
243#11758198—21no—
244#11758198—41no—
245#11758198—41no—
246C4⋊Dic61—41no—
247#11758196—21no—
248#11758196—41no—
249#11758196—41no—
250C4×Dic61C22×C411yescharacteristic
251C2×#92418C22×C411yescharacteristic
252C2×#92418C22×C411yescharacteristic
253Dic244—21no—
254Dic244—21no—
255Dic244—41no—
256#11758190—21no—
257#11758190—21no—
258#11758190—41no—
259#11758189—21no—
260#11758189—21no—
261C8×D61C2×D411yescharacteristic
262C8×D61C2×D411yescharacteristic
263C8×D61#4111yescharacteristic
264C8×D61#4111yescharacteristic
265C23×C61—21no—
266C22×D61—21no—
267C22×D61H(5,𝔽2)11yescharacteristic
268C2×#6923—22no—
269C2×#6923—41no—
270C2×#6923—41no—
271C2×#6923—41no—
272Q8×C61—21no—
273Q8×C61—21no—
274Q8×C61—21no—
275Q8×C61—21no—
276Q8×C61—21no—
277Q8×C61—21no—
278D4×C61—21no—
279D4×C61—21no—
280C2×C4×C61—21no—
281C2×C4×C61—21no—
282C2×C4×C61—21no—
283C2×C4×C61—21no—
284C2×C4×C61C23×C411yescharacteristic
285C22⋊D61—41no—
286C22⋊D61—41no—
287C2×Dic61—21no—
288C2×Dic61—21no—
289C2×Dic61—21no—
290C2×Dic61—21no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61H(5,𝔽2)11yescharacteristic
294D244—21no—
295D244—21no—
296D244—41no—
297D244—41no—
298C4×D61—41no—
299C4×D61—41no—
300C4×D61—41no—
301C4×D61—41no—
302C4×D61—41no—
303C4×D61—41no—
304C4×D61C22×D411yescharacteristic
305C4×D61C2×#4111yescharacteristic
306C4×D61H(5,𝔽2)11yescharacteristic
307C4×D61H(5,𝔽2)11yescharacteristic
308Dic122—21no—
309Dic122—21no—
310Dic122—21no—
311Dic122—21no—
312Dic122—21no—
313Dic122—21no—
314Dic122—41no—
315Dic122—41no—
316Dic122—41no—
317Dic122—41no—
318#92420—22no—
319#92420—41no—
320#92420—41no—
321#92420—41no—
322C8×C61—21no—
323C8×C61C4×D411yescharacteristic
324C8×C61C4×D411yescharacteristic
325#92418—21no—
326#92418C4×D411yescharacteristic
327#92418C4×D411yescharacteristic
328#22285—611nomaximal, Sylow
329C22×C61—41no—
330C22×C61—81no—
331C22×C61#51811yescharacteristic
332C2×D61—41no—
333C2×D61—41no—
334C2×D61—81no—
335C2×D61#53511yescharacteristic
336C2×D61#54011yescharacteristic
337#6923—42no—
338#6923—81no—
339#6923—81no—
340#6923—81no—
341C4×C61—41no—
342C4×C61—41no—
343C4×C61—41no—
344C4×C61—41no—
345C4×C61C2×C4×D411yescharacteristic
346C4×C61#51811yescharacteristic
347Dic61—41no—
348Dic61—41no—
349Dic61—41no—
350Dic61—41no—
351Dic61#53511yescharacteristic
352Dic61#54011yescharacteristic
353#3156—611no—
354#2957—611no—
355#2950—611no—
356#1979—611no—
357#1976—611no—
358#1912—611no—
359#1912—611no—
360#1909—611no—
361#1909—611no—
362#1905—611no—
363#1905—611no—
364#1902—611no—
365#1902—611no—
366#1833—611no—
367#1747—611no—
368C2×C61—21no—
369C2×C61—81no—
370C2×C61#232811yescharacteristic
371D61—21no—
372D61—21no—
373D61—81no—
374C2×#137—611no—
375C2×#137—611no—
376C2×#135—611no—
377C22×Q16—611no—
378#568—611no—
379#568—611no—
380#536—611no—
381#536—611no—
382#519—611no—
383#519—611no—
384#497—611no—
385#497—611no—
386#497—611no—
387#497#692311yescharacteristic, complemented
388#495—611no—
389#495—611no—
390#494—611no—
391#494—611no—
392#444—1222no—
393#443—1222no—
394#441—1222no—
395C4×Q16—1222no—
396C2×#108—611no—
397#427—611no—
398#421—611no—
399#421—611no—
400#421—611no—
401#421—611no—
402#419—611no—
403#419—611no—
404#419—611no—
405#419—611no—
406C2×#101—611no—
407C2×#101—611no—
408#410—611no—
409#410—611no—
410#405—611no—
411#405—611no—
412#405—611no—
413#344—611no—
414#344—611no—
415#338—611no—
416H(3,ℤ/4)—611no—
417H(3,ℤ/4)—611no—
418H(3,ℤ/4)—611no—
419H(3,ℤ/4)—611no—
420H(3,ℤ/4)—611no—
421C61#2228511yescharacteristic, complemented, Sylow
422C2×#41—611no—
423C2×#41—611no—
424C22×Q8—611no—
425C22×Q8—611no—
426#137—2441no—
427#137—2441no—
428#137—2441no—
429#137—2441no—
430#135—1221no—
431#135—1221no—
432#135—1221no—
433#135—1221no—
434C2×Q16—611no—
435C2×Q16—611no—
436C2×Q16—611no—
437C2×Q16—611no—
438C2×Q16—611no—
439C2×Q16—611no—
440C2×Q16—611no—
441C2×Q16—1221no—
442C2×Q16—1221no—
443C2×Q16—1221no—
444C2×Q16—1221no—
445C2×Q16C2×#692311yescharacteristic
446C2×Q16C2×#692311yescharacteristic
447C2×SD16—611no—
448C2×SD16—611no—
449C2×SD16—611no—
450C2×SD16—611no—
451C2×SD16C2×#692311yescharacteristic
452C2×SD16C2×#692311yescharacteristic
453C2×D8—611no—
454#131—1222no—
455#131—1222no—
456#131—2441no—
457#131—2441no—
458C2×M16—611no—
459C2×M16—611no—
460C2×M16—611no—
461C2×M16—611no—
462C2×M16—611no—
463C2×M16—2441no—
464C2×M16—2441no—
465C22×C8—611no—
466C4⋊Q8—611no—
467C4⋊Q8—611no—
468C4⋊Q8—611no—
469C4⋊Q8C2×#692311yescharacteristic
470#124—611no—
471#124—611no—
472#124—611no—
473#124C2×#692311yescharacteristic
474#123—2441no—
475#123—2441no—
476C22⋊Q8—2441no—
477C22⋊Q8—2441no—
478C4×Q8—1222no—
479C4×Q8—1222no—
480C4×Q8—2441no—
481C4×Q8—2441no—
482#117—611no—
483#117—611no—
484#117—611no—
485#117—611no—
486#117—611no—
487#117—2441no—
488#117—2441no—
489#108—1221no—
490#108—1221no—
491#107—1221no—
492#106—1221no—
493#104—1222no—
494#104—1222no—
495#104—2441no—
496#104—2441no—
497#103—1222no—
498#103—1222no—
499#103—2441no—
500#103—2441no—
501#101—2441no—
502#101—2441no—
503#99—2441no—
504#99—2441no—
505#97—612no—
506#97—611no—
507#97—1221no—
508#97C2×#692311yescharacteristic
509C4×C8—612no—
510C4×C8—611no—
511C4×C8—611no—
512C4×C8—1221no—
513#41—1221no—
514#41—1221no—
515#41—1221no—
516#41—1221no—
517#41—1221no—
518#41—1221no—
519#41—1221no—
520#41—1221no—
521C2×Q8—611no—
522C2×Q8—611no—
523C2×Q8—611no—
524C2×Q8—1221no—
525C2×Q8—1221no—
526C2×Q8—1221no—
527C2×Q8—1221no—
528C2×Q8—1221no—
529C2×Q8—1221no—
530C2×Q8—1221no—
531C2×Q8—1221no—
532C2×Q8—1221no—
533C2×Q8—1221no—
534C2×Q8C22×#692311yescharacteristic
535C2×Q8C22×#692311yescharacteristic
536C2×Q8C22×#692311yescharacteristic
537C2×D4—611no—
538C2×D4—1221no—
539C2×D4—1221no—
540C2×D4C22×#692311yescharacteristic
541C22×C4—611no—
542C22×C4—1221no—
543C22×C4—1221no—
544C22×C4—1221no—
545C22×C4—1221no—
546Q16—21no—
547Q16—21no—
548Q16—41no—
549Q16—1221no—
550Q16—1221no—
551Q16—1221no—
552Q16—1221no—
553Q16—1221no—
554Q16—1221no—
555Q16—1221no—
556Q16—1221no—
557Q16—2441no—
558Q16—2441no—
559Q16—2441no—
560SD16—21no—
561SD16—21no—
562SD16—41no—
563SD16—1221no—
564SD16—1221no—
565SD16—2441no—
566SD16—2441no—
567SD16—2441no—
568D8—1221no—
569D8—1221no—
570M16—1221no—
571M16—1221no—
572M16—1221no—
573M16—1221no—
574M16—1221no—
575M16—1221no—
576M16—1221no—
577M16—1221no—
578M16—1221no—
579M16—1221no—
580M16—2442no—
581M16—2442no—
582C2×C8—611no—
583C2×C8—611no—
584C2×C8—611no—
585C2×C8—611no—
586C2×C8—611no—
587C2×C8—612no—
588C2×C8—611no—
589C2×C8—1221no—
590C2×C8—1221no—
591C2×C8—1221no—
592C2×C8—2442no—
593C2×C8—2442no—
594C2×C8C22×#692311yescharacteristic
595C2×C8C22×#692311yescharacteristic
596C4⋊C4—41no—
597C4⋊C4—1221no—
598C4⋊C4—1221no—
599C4⋊C4—1221no—
600C4⋊C4—1221no—
601C4⋊C4—1221no—
602C4⋊C4—2441no—
603C4⋊C4—2441no—
604C4⋊C4—2441no—
605C4⋊C4—2442no—
606C4⋊C4—2442no—
607C22⋊C4—41no—
608C22⋊C4—1221no—
609C22⋊C4—1221no—
610C22⋊C4—1221no—
611C22⋊C4—1221no—
612C22⋊C4—1221no—
613C22⋊C4—2441no—
614C22⋊C4—2441no—
615C22⋊C4—2441no—
616C42—612no—
617C42—611no—
618C42—1221no—
619C42—1221no—
620C42—1221no—
621C42—2442no—
622C42—2442no—
623C42C22×#692311yescharacteristic
624C23—21no—
625C23—611no—
626C23—1221no—
627Q8—21no—
628Q8—21no—
629Q8—21no—
630Q8—21no—
631Q8—21no—
632Q8—21no—
633Q8—1221no—
634Q8—1221no—
635Q8—1221no—
636Q8—1221no—
637Q8—1221no—
638Q8—1221no—
639Q8—2441no—
640Q8—2441no—
641Q8—2441no—
642Q8—2441no—
643D4—21no—
644D4—21no—
645D4—1221no—
646D4—1221no—
647D4—2441no—
648D4—2441no—
649D4—2441no—
650D4—2441no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—611no—
656C2×C4—611no—
657C2×C4—611no—
658C2×C4—611no—
659C2×C4—611no—
660C2×C4—1222no—
661C2×C4—1221no—
662C2×C4—1221no—
663C2×C4—1221no—
664C2×C4—1221no—
665C2×C4—2441no—
666C2×C4—2441no—
667C2×C4—2441no—
668C2×C4—2441no—
669C2×C4—2441no—
670C2×C4—2441no—
671C2×C4—2441no—
672C2×C4—2441no—
673C2×C4—2441no—
674C2×C4—2441no—
675C2×C4—2441no—
676C2×C4C23×#692311yescharacteristic
677C8—21no—
678C8—611no—
679C8—611no—
680C8—1222no—
681C8—1221no—
682C8—2441no—
683C8—2441no—
684C8—2441no—
685C8D4×#692311yescharacteristic
686C8D4×#692311yescharacteristic
687C22—41no—
688C22—81no—
689C22—611no—
690C22—611no—
691C22—4881no—
692C22—2441no—
693C22—2441no—
694C22#1684082911yescharacteristic
695C4—41no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—611no—
700C4—611no—
701C4—4881no—
702C4—4881no—
703C4—4881no—
704C4—2441no—
705C4—2442no—
706C4—2441no—
707C4—2441no—
708C4—2441no—
709C4C2×D4×#692311yescharacteristic
710C4#1684082911yescharacteristic
711C2—21no—
712C2—81no—
713C2—1221no—
714C2—1221no—
715C2—4881no—
716C2#2146231111yescharacteristic
subgroupG/H
centerC2#21462311
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#497C4
socleC2×C61#2328
prime radicalC2×#14519798C22
cosocleC2×C4×D61C24
hypercenter#497#6923
hypocenterC61#22285
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #21462580C2semidirect
3. #21461981C2semidirect
5. #21461867C2semidirect
6. #21461866C2semidirect
7. #21461861C2semidirect
8. #21461860C2semidirect
9. #21461855C2semidirect
10. #21461854C2semidirect
11. #21461849C2semidirect
12. #21461848C2semidirect
13. #21461739C2semidirect
14. #21461541C2semidirect
54. #16840156C4semidirect
55. #16840155C4semidirect
56. #16840153C4semidirect
57. #16840144C4semidirect
58. #16840143C4semidirect
59. #16840141C4semidirect
60. #16840140C4semidirect
62. C61×#497C4semidirect
387. #497#6923semidirect
421. C61#22285semidirect

Maximal subgroups

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

subgroupindex
#214625972
#214625802
#214619812
#214619772
#214618672
#214618662
#214618612
#214618602
#214618552
#214618542
#214618492
#214618482
#214617392
#214615412
#214563112
#2228561