Finite Group Databasebeta

← #6345552

Subgroups — #6345552

Order 512.

1507 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60952C211yesmaximal, characteristic
2#60295C211yesmaximal, characteristic
3#60250C211yesmaximal, characteristic, complemented
4#18079C214yesmaximal, complemented
5#18077C214yesmaximal, complemented
6#18024C212yesmaximal, complemented
7#18021C212yesmaximal, complemented
8#18019C212yesmaximal, complemented
9#18016C212yesmaximal, complemented
10#17953C212yesmaximal, complemented
11#17951C212yesmaximal, complemented
12#17925C212yesmaximal, complemented
13#17923C212yesmaximal, complemented
14#17808C211yesmaximal, characteristic, complemented
15#17501C211yesmaximal, characteristic, complemented
16#17462C211yesmaximal, characteristic
17#17334C211yesmaximal, characteristic, complemented
18#3444C2211yescharacteristic
19C2×#533C2211yescharacteristic
20#3255C412yes—
21#3254C412yes—
22#3239C412yes—
23#3237C412yes—
24#3218C412yes—
25#3216C412yes—
26#3196C414yes—
27#3172C2214yes—
28#3107C2212yes—
29#3107C2212yes—
30#3104C2212yes—
31#3103C2212yes—
32#2991C2212yes—
33#2990C2212yes—
34#2989—22no—
35#2986C2212yes—
36#2986C2212yes—
37#2985—22no—
38#2985C2212yes—
39#2984—22no—
40#2983C2212yes—
41#2982C2212yes—
42#2981C2212yes—
43#2978—22no—
44#2969C2214yes—
45#2967—22no—
46#2962C2214yes—
47#2960—22no—
48#2925—22no—
49#2924—22no—
50#2920C2211yescharacteristic
51#2920C2211yescharacteristic
52#2920C2211yescharacteristic
53C2×#431C2211yescharacteristic
54C2×C4×M16C2211yescharacteristic
55#1593C2214yescomplemented
56#1592C2214yescomplemented
57#1591C2214yescomplemented
58#1590C2214yescomplemented
59#1588C2214yescomplemented
60#1587C2214yescomplemented
61#1587C2214yescomplemented
62#1585C2214yescomplemented
63#1584C2214yescomplemented
64#1584C2214yescomplemented
65#1583C2214yescomplemented
66#1582C2214yescomplemented
67#1579C2214yescomplemented
68SD16⋊C8C2214yescomplemented
69C8×Q16C2214yescomplemented
70C8×SD16C2214yescomplemented
71#1574C2212yes—
72#1574C2212yes—
73#1572C2212yes—
74#1572C2212yes—
75#1565C2214yes—
76#1493C2212yes—
77#1493C2212yes—
78#1491C2212yes—
79#1491C2212yes—
80#1485C2211yescharacteristic
81#1485C2211yescharacteristic
82#1485C2212yes—
83#1485C2212yes—
84#1484C2211yescharacteristic
85#1484C2211yescharacteristic
86#1483C2212yes—
87#1483C2212yes—
88#1476C2212yes—
89#1476C2212yes—
90#1455C2212yes—
91#1455C2212yes—
92#1453C2212yes—
93#1453C2212yes—
94#1452C2211yescharacteristic
95#1452C2211yescharacteristic
96#1450C2211yescharacteristic
97#1450C2211yescharacteristic
98C22×M16—21no—
99#533—21no—
100#533—21no—
101#533—21no—
102#533—21no—
103#533C2×C412yes—
104#533C2×C412yes—
105#533C2×C411yescharacteristic
106#533C2×C411yescharacteristic
107#533C2311yescharacteristic
108#533C2312yes—
109#533C2312yes—
110#532C2×C411yescharacteristic
111#532C2×C411yescharacteristic
112#532C2311yescharacteristic
113C2×C4⋊Q8C2×C411yescharacteristic
114C2×C4⋊Q8C2×C411yescharacteristic
115#529—21no—
116#529—21no—
117#529—21no—
118#529—21no—
119#529C2×C412yes—
120#529C2312yes—
121C2×#125—21no—
122#525—21no—
123#525—21no—
124#525C2×C412yes—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8C2×C412yes—
127C4×#41—21no—
128C4×#41—21no—
129C4×#41C2312yes—
130C2×C4×Q8—21no—
131C2×C4×Q8C2312yes—
132C2×#117—21no—
133C22×C42C2311yescharacteristic
134#501C2×C414yes—
135#497C2×C414yes—
136#491C2×C414yes—
137#489C2×C414yes—
138#485C2×C414yes—
139#485C2×C414yes—
140#484C2×C414yes—
141#481C2×C414yes—
142#477C2×C414yes—
143#475C2×C414yes—
144#475C2×C412yes—
145#475C2×C412yes—
146#474C2×C412yes—
147#474C2×C412yes—
148#470C2×C414yes—
149#470C2×C414yes—
150#468C2×C414yes—
151#468C2×C414yes—
152#462C2×C414yes—
153#462C2×C412yes—
154#462C2×C412yes—
155#461C2×C412yes—
156#461C2×C412yes—
157#460C2×C414yes—
158#452C2×C414yes—
159#452C2×C414yes—
160#451C2×C414yes—
161#448C2×C414yes—
162#446—24no—
163#446—22no—
164#446—22no—
165#446—24no—
166#446—24no—
167#446—24no—
168#446C2314yes—
169#446C2314yes—
170#446C2314yes—
171C8×Q8—24no—
172C8×Q8—22no—
173C8×Q8—22no—
174C8×Q8C2314yes—
175C8×Q8C2314yes—
176C8×Q8C2314yes—
177C4×Q16C2314yes—
178C4×Q16C2314yes—
179C4×SD16C2314yes—
180C4×SD16C2314yes—
181#436—24no—
182#436—24no—
183#436C2314yes—
184#435—24no—
185#435—24no—
186#435—24no—
187C8×D4C2314yes—
188#433—24no—
189#433—24no—
190#433—24no—
191#433—24no—
192#432—21no—
193#432—22no—
194#432—24no—
195#432C2×C412yes—
196#432C2×C412yes—
197#432D412yes—
198#432D412yes—
199#432C2312yes—
200#431—21no—
201#431—21no—
202#431—21no—
203#431—22no—
204#431—24no—
205#431C2312yes—
206#431C2311yescharacteristic
207#431C2311yescharacteristic
208#431C2312yes—
209#431C2312yes—
210#431C2311yescharacteristic
211#431C2311yescharacteristic
212#431C2312yes—
213#431C2311yescharacteristic
214#431C2311yescharacteristic
215#424—24no—
216#423—22no—
217#423C2×C412yes—
218#423D412yes—
219#423C2312yes—
220C2×C4⋊C8—22no—
221C2×C4⋊C8C2312yes—
222#408—24no—
223#407—21no—
224#407C2×C412yes—
225#407D412yes—
226C2×C22⋊C8—21no—
227#405—24no—
228#405—24no—
229C4×M16—21no—
230C4×M16—21no—
231C4×M16—21no—
232C4×M16C2311yescharacteristic
233C4×M16C2311yescharacteristic
234C4×M16C2311yescharacteristic
235C4×M16C2311yescharacteristic
236C4×M16C2312yes—
237C4×M16C2312yes—
238C2×#97—21no—
239C2×C4×C8—21no—
240C2×C4×C8C2311yescharacteristic
241C2×C4×C8C2311yescharacteristic
242#335C2314yes—
243#335C2314yes—
244#334C2314yes—
245#334C2314yes—
246Q8⋊C8C2314yes—
247Q8⋊C8C2312yes—
248Q8⋊C8C2314yes—
249Q8⋊C8C2314yes—
250Q8⋊C8C2312yes—
251Q8⋊C8C2314yes—
252Q8⋊C8C2312yes—
253Q8⋊C8C2312yes—
254D4⋊C8C2312yes—
255D4⋊C8C2312yes—
256D4⋊C8C2312yes—
257D4⋊C8C2312yes—
258#322C2312yes—
259#322C2312yes—
260#322C2312yes—
261#322C2312yes—
262#322C2312yes—
263#322C2312yes—
264C82C2312yes—
265C82C2312yes—
266C2×#41—21no—
267C2×#41—21no—
268C2×#41C22×C412yes—
269C22×Q8—21no—
270C22×Q8C22×C412yes—
271C23×C4C22×C411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4#4111yescharacteristic
274C2×Q16C22×C414yes—
275C2×Q16C22×C414yes—
276C2×SD16C22×C414yes—
277C2×SD16C22×C414yes—
278C2×M16—21no—
279C2×M16—21no—
280C2×M16—21no—
281C2×M16—21no—
282C2×M16—22no—
283C2×M16—24no—
284C2×M16—44no—
285C2×M16C22×C414yes—
286C2×M16C2×D414yes—
287C22×C8—21no—
288C22×C8—22no—
289C22×C8—21no—
290C22×C8—24no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8C22×C412yes—
296C4⋊Q8C22×C412yes—
297C4⋊Q8C22×C411yescharacteristic
298C4⋊Q8C22×C412yes—
299C4⋊Q8C22×C412yes—
300C4⋊Q8C22×C411yescharacteristic
301C4⋊Q8C22×C411yescharacteristic
302#127C22×C411yescharacteristic
303#126—22no—
304#126—22no—
305#125—22no—
306#125—22no—
307#125—21no—
308#125—22no—
309#125—22no—
310#125—21no—
311#125C22×C412yes—
312#124—21no—
313#124—21no—
314#124C22×C412yes—
315#123—22no—
316#123—22no—
317#123—22no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—21no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8—21no—
328C22⋊Q8C22×C414yes—
329C22⋊Q8C22×C414yes—
330C22⋊Q8C22×C412yes—
331C22⋊Q8C22×C412yes—
332#121—21no—
333#121—21no—
334#121—21no—
335#121—21no—
336#121C22×C412yes—
337#121C22×C412yes—
338#120—22no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8—21no—
342C4×Q8—21no—
343C4×Q8—21no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—21no—
348C4×Q8—22no—
349C4×Q8C22×C414yes—
350C4×Q8C22×C412yes—
351C4×Q8C2414yes—
352C4×Q8C2412yes—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4C22×C412yes—
364C4×D4C2412yes—
365#117—22no—
366#117—22no—
367#117—21no—
368#117—22no—
369#117—22no—
370#117—21no—
371#117—22no—
372#117—22no—
373#117—22no—
374#117C22×C412yes—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4C22×C412yes—
381C2×C22⋊C4—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42C22×C411yescharacteristic
385C2×C42C22×C411yescharacteristic
386C2×C42C22×C411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42#4111yescharacteristic
391C2×C42#4111yescharacteristic
392C2×C42#4111yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396#107C22×C414yes—
397#107C22×C414yes—
398#106C22×C414yes—
399#106C22×C414yes—
400C4⋊C8—24no—
401C4⋊C8—24no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406C4⋊C8—22no—
407C4⋊C8—22no—
408C4⋊C8C22×C414yes—
409C4⋊C8C22×C412yes—
410C4⋊C8C22×C412yes—
411C4⋊C8C2×D412yes—
412C4⋊C8C2×D412yes—
413C4⋊C8C2×D414yes—
414C4⋊C8C2412yes—
415C4⋊C8C2412yes—
416C4⋊C8C2414yes—
417#103C22×C414yes—
418#103C22×C414yes—
419#103C22×C414yes—
420#103C22×C414yes—
421#103C22×C414yes—
422#103C22×C414yes—
423#102C22×C414yes—
424#102C22×C414yes—
425C22⋊C8—21no—
426C22⋊C8—21no—
427C22⋊C8—22no—
428C22⋊C8—22no—
429C22⋊C8—22no—
430C22⋊C8—24no—
431C22⋊C8—24no—
432C22⋊C8C22×C414yes—
433C22⋊C8C22×C412yes—
434C22⋊C8C22×C412yes—
435C22⋊C8C2×D414yes—
436C22⋊C8C2×D412yes—
437C22⋊C8C2×D412yes—
438#97—24no—
439#97—24no—
440#97—24no—
441#97—22no—
442#97—21no—
443#97—21no—
444#97—21no—
445#97—21no—
446#97C22×C414yes—
447#97C2×D414yes—
448C4×C8—24no—
449C4×C8—24no—
450C4×C8—24no—
451C4×C8—21no—
452C4×C8—22no—
453C4×C8—21no—
454C4×C8—22no—
455C4×C8—21no—
456C4×C8—21no—
457C4×C8C2411yescharacteristic
458C4×C8C2411yescharacteristic
459C4×C8C2411yescharacteristic
460C4×C8C2411yescharacteristic
461C4×C8C2412yes—
462C4×C8C2412yes—
463C4×C8C2412yes—
464C4×C8C2412yes—
465C24C4×D411yescharacteristic
466#41—42no—
467#41—42no—
468#41—82no—
469#41—82no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—22no—
475C2×Q8—44no—
476C2×Q8—82no—
477C2×Q8C23×C412yes—
478C2×Q8C23×C414yes—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4C23×C412yes—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4C4×D411yescharacteristic
494C22×C4C4×D411yescharacteristic
495C22×C4C4×D411yescharacteristic
496C22×C4C4×D411yescharacteristic
497C22×C4C4×D411yescharacteristic
498C22×C4C4×D411yescharacteristic
499C22×C4C4×D411yescharacteristic
500C22×C4C4×D411yescharacteristic
501C22×C4C4×D411yescharacteristic
502C22×C4C23×C411yescharacteristic
503C22×C4C23×C411yescharacteristic
504C22×C4C23×C411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C2×#4111yescharacteristic
509C22×C4C2×#4111yescharacteristic
510C22×C4C2×#4111yescharacteristic
511Q16—44no—
512Q16—44no—
513SD16—44no—
514SD16—44no—
515M16—44no—
516M16—44no—
517M16—42no—
518M16—42no—
519M16—42no—
520M16—42no—
521M16—84no—
522M16—82no—
523C2×C8—24no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—24no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—44no—
532C2×C8—42no—
533C2×C8—84no—
534C2×C8C23×C414yes—
535C2×C8C23×C414yes—
536C2×C8C22×D414yes—
537C2×C8C22×D414yes—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—24no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—24no—
558C4⋊C4—22no—
559C4⋊C4C23×C414yes—
560C4⋊C4C23×C412yes—
561C4⋊C4C23×C412yes—
562C22⋊C4—24no—
563C22⋊C4—24no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C42—22no—
574C42—24no—
575C42—22no—
576C42—22no—
577C42—24no—
578C42C4×D411yescharacteristic
579C42C4×D411yescharacteristic
580C42C4×D411yescharacteristic
581C42C4×D411yescharacteristic
582C42C4×D411yescharacteristic
583C42C4×D411yescharacteristic
584C42C23×C411yescharacteristic
585C42C23×C411yescharacteristic
586C42C23×C411yescharacteristic
587C42C22×D411yescharacteristic
588C42C22×D411yescharacteristic
589C42C22×D411yescharacteristic
590C42C2×#4111yescharacteristic
591C42C2×#4111yescharacteristic
592C42C2×#4111yescharacteristic
593C42C2511yescharacteristic
594C23—22no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23C2×C4×D411yescharacteristic
599C23C2×C4×D411yescharacteristic
600C23C2×C4×D411yescharacteristic
601Q8—24no—
602Q8—24no—
603Q8—22no—
604Q8—22no—
605Q8—44no—
606Q8—44no—
607Q8—82no—
608Q8—81no—
609Q8—81no—
610D4—22no—
611D4—22no—
612D4—81no—
613D4—81no—
614D4—82no—
615D4—82no—
616D4—82no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4C2×C4×D411yescharacteristic
647C2×C4C2×C4×D411yescharacteristic
648C2×C4C2×C4×D411yescharacteristic
649C2×C4C2×C4×D411yescharacteristic
650C2×C4C2×C4×D411yescharacteristic
651C2×C4C2×C4×D411yescharacteristic
652C2×C4C2×C4×D411yescharacteristic
653C2×C4C2×C4×D411yescharacteristic
654C2×C4C2×C4×D411yescharacteristic
655C2×C4C24×C411yescharacteristic
656C2×C4C23×D411yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C8—22no—
659C8—22no—
660C8—24no—
661C8—24no—
662C8—24no—
663C8—24no—
664C8—41no—
665C8—41no—
666C8—42no—
667C8—42no—
668C8—44no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—82no—
682C22C22×C4×D411yescharacteristic
683C4—21no—
684C4—21no—
685C4—21no—
686C4—21no—
687C4—21no—
688C4—21no—
689C4—21no—
690C4—21no—
691C4—21no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—44no—
696C4—42no—
697C4#295612yes—
698C4#357312yes—
699C4#358612yes—
700C2—41no—
701C2—41no—
702C2—41no—
703C2—82no—
704C2#6014211yescharacteristic
705C2#6016911yescharacteristic
706C2C2×#295611yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#3444C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #2920 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #60250C2semidirect
4. #18079C2semidirect
5. #18077C2semidirect
6. #18024C2semidirect
7. #18021C2semidirect
8. #18019C2semidirect
9. #18016C2semidirect
10. #17953C2semidirect
11. #17951C2semidirect
12. #17925C2semidirect
13. #17923C2semidirect
14. #17808C2semidirect
15. #17501C2semidirect
17. #17334C2semidirect
55. #1593C22semidirect
56. #1592C22semidirect
57. #1591C22semidirect
58. #1590C22semidirect
59. #1588C22semidirect
60. #1587C22semidirect
61. #1587C22semidirect
62. #1585C22semidirect
63. #1584C22semidirect
64. #1584C22semidirect
65. #1583C22semidirect
66. #1582C22semidirect
67. #1579C22semidirect
68. SD16⋊C8C22semidirect
69. C8×Q16C22semidirect
70. C8×SD16C22semidirect

Maximal subgroups

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

subgroupindex
#609522
#602952
#602502
#180792
#180772
#180242
#180212
#180192
#180162
#179532
#179512
#179252
#179232
#178082
#175012
#174622
#173342