Finite Group Databasebeta

← #6345563

Subgroups — #6345563

Order 512.

1526 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60954C211yesmaximal, characteristic
2#60292C211yesmaximal, characteristic, complemented
3#60264C211yesmaximal, characteristic, complemented
4#18082C214yesmaximal, complemented
5#18081C214yesmaximal, complemented
6#18010C214yesmaximal, complemented
7#18008C214yesmaximal, complemented
8#17954C214yesmaximal, complemented
9#17927C214yesmaximal, complemented
10#17809C211yesmaximal, characteristic, complemented
11#17498C211yesmaximal, characteristic, complemented
12#17465C211yesmaximal, characteristic, complemented
13#17338C211yesmaximal, characteristic, complemented
14#3444C2211yescharacteristic
15C2×#529C2211yescharacteristic
16#3256C414yes—
17#3240C412yes—
18#3236C411yescharacteristic
19#3234C411yescharacteristic
20#3219C412yes—
21#3215C411yescharacteristic
22#3213C411yescharacteristic
23#3196C412yes—
24#3194C412yes—
25#3172C2212yes—
26#3171C2212yes—
27#3108C2212yes—
28#3105C2212yes—
29#3102C2214yes—
30#2995—22no—
31#2994C2214yes—
32#2993C2214yes—
33#2992—22no—
34#2991—21no—
35#2990—21no—
36#2989C2212yescomplemented
37#2988C2212yescomplemented
38#2986C2212yescomplemented
39#2979C2212yescomplemented
40#2976—22no—
41#2969C2214yes—
42#2966—22no—
43Q8×M16C2212yes—
44D4×M16—22no—
45D4×M16C2212yes—
46#2925—22no—
47#2921—21no—
48#2920C2211yescharacteristic
49#2920C2211yescharacteristic
50#2920C2211yescharacteristic
51C2×#431C2211yescharacteristic
52#2905—21no—
53C2×#423C2211yescharacteristic
54#1592C2214yescomplemented
55#1592C2214yescomplemented
56#1591C2214yescomplemented
57#1589C2214yescomplemented
58#1588C2214yescomplemented
59#1587C2214yescomplemented
60#1586C2214yescomplemented
61#1585C2214yescomplemented
62#1584C2214yescomplemented
63#1583C2214yescomplemented
64#1582C2214yescomplemented
65#1581C2214yescomplemented
66#1580C2214yescomplemented
67#1579C2214yescomplemented
68C8×SD16C2214yescomplemented
69C8×SD16C2214yescomplemented
70#1574C2214yes—
71#1571C2212yescomplemented
72#1570C2212yescomplemented
73#1565C2214yescomplemented
74#1495C2214yes—
75#1489C2214yescomplemented
76#1483C2212yes—
77#1483C2212yes—
78#1482C2214yes—
79#1481C2212yescomplemented
80#1480C2212yescomplemented
81#1479C2212yes—
82#1478C2212yes—
83#1453C2214yes—
84#1453C2214yescomplemented
85#1453C2214yescomplemented
86C22×M16—21no—
87C22×M16—22no—
88#533C2×C411yescharacteristic
89#533C2×C411yescharacteristic
90#533C2311yescharacteristic
91#532C2×C411yescharacteristic
92#532C2×C411yescharacteristic
93#532C2311yescharacteristic
94#529—21no—
95#529—21no—
96#529—21no—
97#529—21no—
98#529—21no—
99#529—21no—
100#529—21no—
101#529—21no—
102#529C2×C412yes—
103#529C2×C412yes—
104#529C2×C412yes—
105#529C2312yes—
106#529C2312yes—
107#529C2312yes—
108C2×#126—21no—
109C2×#125C2×C411yescharacteristic
110C2×#124C2×C411yescharacteristic
111#525—21no—
112#525—21no—
113#525C2×C412yes—
114C2×#123—21no—
115C2×C22⋊Q8C2×C411yescharacteristic
116C2×#121C2×C411yescharacteristic
117C4×#41—21no—
118C4×#41—21no—
119C4×#41C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C22×C42C2311yescharacteristic
125#501C2×C412yes—
126#501C2×C412yes—
127#497C2×C412yes—
128#496C2×C412yes—
129#491C2×C412yes—
130#490C2×C412yes—
131#489C2×C412yes—
132#489C2×C412yes—
133#485C2×C414yes—
134#484C2×C412yes—
135#483C2×C412yes—
136#482C2×C414yes—
137#481C2×C412yes—
138#480C2×C412yes—
139#479C2×C414yes—
140#479C2×C414yes—
141#478C2×C414yes—
142#478C2×C414yes—
143#470C2×C414yes—
144#469C2×C414yes—
145#468C2×C414yes—
146#468C2×C414yes—
147#464C2×C414yes—
148#464C2×C414yes—
149#463C2×C414yes—
150#463C2×C414yes—
151#452C2×C414yes—
152#451C2×C412yes—
153#450C2×C412yes—
154#449C2×C414yes—
155#448C2×C412yes—
156#447C2×C412yes—
157#446—22no—
158#446—24no—
159#446C2314yes—
160#446C2314yes—
161C8×Q8—22no—
162C8×Q8C2314yes—
163C8×Q8C2314yes—
164C4×Q16C2314yes—
165C4×SD16C2314yes—
166C4×SD16C2314yes—
167C4×D8C2314yes—
168#436—24no—
169#436—22no—
170#436—24no—
171#436C2314yes—
172#436C2314yes—
173#435—24no—
174#435—24no—
175#435—24no—
176#435—24no—
177#435—24no—
178C8×D4—22no—
179C8×D4—24no—
180C8×D4—24no—
181C8×D4C2314yes—
182C8×D4C2314yes—
183#433—24no—
184#433—24no—
185#433—24no—
186#433—24no—
187#432—21no—
188#432—21no—
189#432—22no—
190#432C2×C411yescharacteristic
191#432C2×C411yescharacteristic
192#432C2×C412yes—
193#432D412yes—
194#432D412yes—
195#431—21no—
196#431—21no—
197#431—22no—
198#431C2312yes—
199#431C2312yes—
200#431C2312yes—
201#431C2312yes—
202#431C2312yes—
203#431C2312yes—
204#431C2312yes—
205#431C2312yes—
206#424—24no—
207#424—24no—
208#423—22no—
209#423—21no—
210#423—21no—
211#423—21no—
212#423—22no—
213#423C2×C412yes—
214#423D411yescharacteristic
215#423D411yescharacteristic
216#423C2312yes—
217#423C2312yes—
218C2×C4⋊C8—21no—
219C2×C4⋊C8—22no—
220C2×C4⋊C8C2312yes—
221C2×C4⋊C8C2311yescharacteristic
222C2×C4⋊C8C2311yescharacteristic
223#408—24no—
224#407—21no—
225#407—22no—
226#407C2×C412yes—
227#407D412yes—
228C2×C22⋊C8—21no—
229#405—24no—
230C4×M16—22no—
231C4×M16C2312yes—
232C4×M16C2312yes—
233C4×M16C2312yes—
234C2×C4×C8—21no—
235#335C2314yes—
236#335C2314yes—
237#334C2314yes—
238#334C2314yes—
239Q8⋊C8C2314yes—
240Q8⋊C8C2314yes—
241Q8⋊C8C2314yes—
242Q8⋊C8C2314yes—
243D4⋊C8C2314yes—
244D4⋊C8C2314yes—
245D4⋊C8C2314yes—
246D4⋊C8C2314yes—
247#322C2314yes—
248#322C2314yes—
249#322C2314yes—
250C82C2314yes—
251C2×#41—21no—
252C2×#41—21no—
253C2×#41C22×C412yes—
254C22×Q8C22×C411yescharacteristic
255C22×D4—21no—
256C22×D4C22×C411yescharacteristic
257C23×C4—21no—
258C23×C4C22×C411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4#4111yescharacteristic
261C2×Q16C22×C414yes—
262C2×SD16C22×C414yes—
263C2×SD16C22×C414yes—
264C2×D8C22×C414yes—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—24no—
269C2×M16—24no—
270C2×M16—84no—
271C2×M16—82no—
272C2×M16C22×C414yes—
273C2×M16C2×D412yes—
274C2×M16C2×D412yes—
275C22×C8—22no—
276C22×C8—21no—
277C22×C8—21no—
278C22×C8—24no—
279C22×C8—24no—
280C4⋊Q8C22×C411yescharacteristic
281C4⋊Q8C22×C411yescharacteristic
282C4⋊Q8C22×C411yescharacteristic
283#127C22×C411yescharacteristic
284#126—22no—
285#126—22no—
286#126—22no—
287#126—22no—
288#126—22no—
289#126—22no—
290#125—21no—
291#125—21no—
292#125—21no—
293#125—21no—
294#125C22×C412yes—
295#125C22×C412yes—
296#125C22×C412yes—
297#124—21no—
298#124—21no—
299#124—21no—
300#124—21no—
301#124C22×C412yes—
302#124C22×C412yes—
303#124C22×C412yes—
304#123—22no—
305#123—22no—
306#123—22no—
307#123—22no—
308#123—22no—
309#123—22no—
310#123—22no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C22⋊Q8—21no—
317C22⋊Q8C22×C412yes—
318C22⋊Q8C22×C412yes—
319C22⋊Q8C22×C412yes—
320C22⋊Q8C22×C412yes—
321#121—21no—
322#121—21no—
323#121—21no—
324#121—21no—
325#121—21no—
326#121—21no—
327#121C22×C412yes—
328#121C22×C412yes—
329#121C22×C412yes—
330#121C22×C412yes—
331#120—22no—
332C4×Q8—21no—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8—21no—
338C4×Q8C22×C412yes—
339C4×Q8C22×C412yes—
340C4×Q8C2412yes—
341C4×Q8C2412yes—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4—21no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4C22×C412yes—
359C4×D4C22×C412yes—
360C4×D4C2412yes—
361C4×D4C2412yes—
362#117—21no—
363#117—21no—
364#117—22no—
365#117—22no—
366#117—22no—
367#117—22no—
368#117—22no—
369#117—22no—
370#117—22no—
371#117C22×C412yes—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4C22×C411yescharacteristic
377C2×C4⋊C4C22×C411yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C42—21no—
384C2×C42—22no—
385C2×C42C22×C411yescharacteristic
386C2×C42C22×C411yescharacteristic
387C2×C42C22×C411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42#4111yescharacteristic
392C2×C42#4111yescharacteristic
393C2×C42#4111yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#107C22×C414yes—
398#107C22×C414yes—
399#106C22×C414yes—
400#106C22×C414yes—
401C4⋊C8—24no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405C4⋊C8—21no—
406C4⋊C8—22no—
407C4⋊C8—22no—
408C4⋊C8—22no—
409C4⋊C8—21no—
410C4⋊C8—22no—
411C4⋊C8—22no—
412C4⋊C8C22×C412yes—
413C4⋊C8C22×C414yes—
414C4⋊C8C22×C412yes—
415C4⋊C8C2×D412yes—
416C4⋊C8C2×D412yes—
417C4⋊C8C2×D412yes—
418C4⋊C8C2×D412yes—
419C4⋊C8C2412yes—
420C4⋊C8C2414yes—
421C4⋊C8C2412yes—
422#103C22×C414yes—
423#103C22×C414yes—
424#103C22×C414yes—
425#103C22×C414yes—
426#102C22×C414yes—
427#102C22×C414yes—
428#102C22×C414yes—
429#102C22×C414yes—
430C22⋊C8—22no—
431C22⋊C8—22no—
432C22⋊C8—22no—
433C22⋊C8—22no—
434C22⋊C8—24no—
435C22⋊C8—24no—
436C22⋊C8—24no—
437C22⋊C8—24no—
438C22⋊C8C22×C412yes—
439C22⋊C8C22×C414yes—
440C22⋊C8C22×C412yes—
441C22⋊C8C2×D414yes—
442C22⋊C8C2×D414yes—
443#97—24no—
444#97—24no—
445#97—22no—
446#97C22×C412yes—
447#97C22×C412yes—
448#97C2×D414yes—
449C4×C8—24no—
450C4×C8—24no—
451C4×C8—22no—
452C4×C8—22no—
453C4×C8C2414yes—
454C4×C8C2414yes—
455C4×C8C2414yes—
456C24—21no—
457C24C4×D411yescharacteristic
458#41—44no—
459#41—81no—
460#41—81no—
461#41—82no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—42no—
466C2×Q8C23×C412yes—
467C2×Q8C23×C412yes—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—42no—
477C2×D4—82no—
478C2×D4C23×C412yes—
479C2×D4C23×C412yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—41no—
489C22×C4—42no—
490C22×C4—41no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—81no—
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×C4C4×D411yescharacteristic
503C22×C4C23×C411yescharacteristic
504C22×C4C23×C411yescharacteristic
505C22×C4C23×C411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C2×#4111yescharacteristic
510C22×C4C2×#4111yescharacteristic
511C22×C4C2×#4111yescharacteristic
512Q16—44no—
513SD16—44no—
514SD16—44no—
515D8—44no—
516M16—42no—
517M16—44no—
518M16—42no—
519M16—44no—
520M16—84no—
521M16—84no—
522M16—82no—
523M16—82no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—84no—
529C2×C8—84no—
530C2×C8—81no—
531C2×C8—81no—
532C2×C8—82no—
533C2×C8C23×C414yes—
534C2×C8C23×C414yes—
535C2×C8C22×D414yes—
536C2×C8C22×D414yes—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4C23×C412yes—
558C4⋊C4C23×C412yes—
559C4⋊C4C23×C412yes—
560C4⋊C4C23×C412yes—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—22no—
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—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—22no—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C22⋊C4—22no—
579C22⋊C4—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42C4×D411yescharacteristic
586C42C4×D411yescharacteristic
587C42C4×D411yescharacteristic
588C42C4×D411yescharacteristic
589C42C4×D411yescharacteristic
590C42C4×D411yescharacteristic
591C42C23×C411yescharacteristic
592C42C23×C411yescharacteristic
593C42C23×C411yescharacteristic
594C42C22×D411yescharacteristic
595C42C22×D411yescharacteristic
596C42C22×D411yescharacteristic
597C42C2×#4111yescharacteristic
598C42C2×#4111yescharacteristic
599C42C2×#4111yescharacteristic
600C42C2511yescharacteristic
601C23—22no—
602C23—22no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23—42no—
607C23—81no—
608C23—81no—
609C23C2×C4×D411yescharacteristic
610C23C2×C4×D411yescharacteristic
611C23C2×C4×D411yescharacteristic
612Q8—24no—
613Q8—24no—
614Q8—42no—
615Q8—81no—
616Q8—81no—
617D4—24no—
618D4—24no—
619D4—42no—
620D4—44no—
621D4—44no—
622D4—81no—
623D4—81no—
624D4—82no—
625D4—82no—
626D4—82no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—82no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4C2×C4×D411yescharacteristic
665C2×C4C2×C4×D411yescharacteristic
666C2×C4C2×C4×D411yescharacteristic
667C2×C4C2×C4×D411yescharacteristic
668C2×C4C2×C4×D411yescharacteristic
669C2×C4C2×C4×D411yescharacteristic
670C2×C4C2×C4×D411yescharacteristic
671C2×C4C2×C4×D411yescharacteristic
672C2×C4C2×C4×D411yescharacteristic
673C2×C4C24×C411yescharacteristic
674C2×C4C23×D411yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C8—24no—
677C8—24no—
678C8—24no—
679C8—24no—
680C8—44no—
681C8—44no—
682C8—42no—
683C8—42no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—82no—
700C22—162no—
701C22C22×C4×D411yescharacteristic
702C4—21no—
703C4—21no—
704C4—21no—
705C4—21no—
706C4—21no—
707C4—21no—
708C4—21no—
709C4—21no—
710C4—21no—
711C4—21no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4#357312yes—
719C4#358511yescharacteristic
720C4#358611yescharacteristic
721C2—41no—
722C2—41no—
723C2—41no—
724C2—82no—
725C2—82no—
726C2#6014211yescharacteristic
727C2#6017011yescharacteristic
728C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#60954C2
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
2. #60292C2semidirect
3. #60264C2semidirect
4. #18082C2semidirect
5. #18081C2semidirect
6. #18010C2semidirect
7. #18008C2semidirect
8. #17954C2semidirect
9. #17927C2semidirect
10. #17809C2semidirect
11. #17498C2semidirect
12. #17465C2semidirect
13. #17338C2semidirect
36. #2989C22semidirect
37. #2988C22semidirect
38. #2986C22semidirect
39. #2979C22semidirect
54. #1592C22semidirect
55. #1592C22semidirect
56. #1591C22semidirect
57. #1589C22semidirect
58. #1588C22semidirect
59. #1587C22semidirect
60. #1586C22semidirect
61. #1585C22semidirect
62. #1584C22semidirect
63. #1583C22semidirect
64. #1582C22semidirect
65. #1581C22semidirect
66. #1580C22semidirect
67. #1579C22semidirect
68. C8×SD16C22semidirect
69. C8×SD16C22semidirect
71. #1571C22semidirect
72. #1570C22semidirect
73. #1565C22semidirect
75. #1489C22semidirect
79. #1481C22semidirect
80. #1480C22semidirect
84. #1453C22semidirect
85. #1453C22semidirect

Maximal subgroups

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

subgroupindex
#609542
#602922
#602642
#180822
#180812
#180102
#180082
#179542
#179272
#178092
#174982
#174652
#173382