Finite Group Databasebeta

← #6345408

Subgroups — #6345408

Order 512.

1823 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2#60232C211yesmaximal, characteristic, complemented
3#60230C211yesmaximal, characteristic, complemented
4#18044C214yesmaximal, complemented
5M16×D8C214yesmaximal, complemented
6#18026C218yesmaximal, complemented
7#17950C214yesmaximal, complemented
8#17922C214yesmaximal, complemented
9#17815C211yesmaximal, characteristic, complemented
10#17490C211yesmaximal, characteristic, complemented
11#17448C211yesmaximal, characteristic, complemented
12#17324C211yesmaximal, characteristic, complemented
13C2×#532C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15#3100C412yes—
16#3100C414yes—
17#3100C412yes—
18#3100C2212yes—
19#3100C2212yes—
20#3100C2214yes—
21C2×#480C412yes—
22C2×#474C412yes—
23C2×#459C412yes—
24C2×#447C412yes—
25#2981—22no—
26#2981C2214yescomplemented
27#2980—22no—
28#2979C2214yescomplemented
29#2975C2212yescomplemented
30#2974—22no—
31#2974C2214yescomplemented
32#2973—22no—
33#2972C2212yescomplemented
34C2×C4×D8C2212yes—
35C2×C4×D8C2212yes—
36D4×M16—22no—
37D4×M16—22no—
38D4×M16C2214yes—
39D4×M16C2214yescomplemented
40#2920C2211yescharacteristic
41#2920C2211yescharacteristic
42#2920C2211yescharacteristic
43#2920C2211yescharacteristic
44C2×#431C2211yescharacteristic
45#2879—22no—
46C2×#407—22no—
47#1594C2218yescomplemented
48#1589C2218yescomplemented
49#1586C2218yescomplemented
50#1586C2218yescomplemented
51#1581C2218yescomplemented
52#1581C2218yescomplemented
53#1580C2218yescomplemented
54C8×D8C2218yescomplemented
55#1572C2214yescomplemented
56#1572C2214yescomplemented
57#1567C2212yescomplemented
58#1566C2212yescomplemented
59#1496C2214yescomplemented
60#1490C2212yescomplemented
61#1490C2212yescomplemented
62#1486C2214yes—
63#1486C2214yescomplemented
64#1484C2214yescomplemented
65#1478C2214yescomplemented
66#1454C2214yescomplemented
67#1453C2214yescomplemented
68#1451C2212yescomplemented
69C8×M16C2212yescomplemented
70C22×#41—21no—
71C23×D4C2×C411yescharacteristic
72C24×C4—21no—
73C22×D8C2×C412yes—
74C22×D8C2×C412yes—
75C22×M16—24no—
76C22×M16—24no—
77#532—21no—
78#532—21no—
79#532—21no—
80#532—21no—
81#532C2×C412yes—
82#532C2×C412yes—
83#532C2312yes—
84#532C2312yes—
85C2×C4⋊Q8C2×C411yescharacteristic
86C2×#127C2×C411yescharacteristic
87C2×#124—21no—
88C2×#121—21no—
89C2×#121C2×C412yes—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4C2×C411yescharacteristic
98C2×C4×D4C2×C412yes—
99C2×C4×D4C2×C411yescharacteristic
100C2×C4×D4C2×C412yes—
101C2×C4×D4C2311yescharacteristic
102C2×C4×D4C2312yes—
103C2×C4×D4C2311yescharacteristic
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C22×C4⋊C4C2×C411yescharacteristic
107C22×C22⋊C4—21no—
108C22×C42C2311yescharacteristic
109#480C2×C414yes—
110#480C2×C414yes—
111#480C2×C414yes—
112#480C2×C414yes—
113#474C2×C414yes—
114#474C2×C414yes—
115#474C2×C414yes—
116#474C2×C414yes—
117#459C2×C414yes—
118#459C2×C414yes—
119#459C2×C414yes—
120#459C2×C414yes—
121#447C2×C414yes—
122#447C2×C414yes—
123#447C2×C414yes—
124#447C2×C414yes—
125C4×D8C2×C414yes—
126C4×D8C2×C414yes—
127C4×D8C2×C414yes—
128C4×D8C2×C414yes—
129C4×D8C2314yes—
130C4×D8C2314yes—
131C4×D8C2314yes—
132C4×D8C2314yes—
133#436—24no—
134#436—24no—
135#436C2318yes—
136#436C2318yescomplemented
137#435—28no—
138#435—28no—
139#435—28no—
140#435—28no—
141C8×D4—24no—
142C8×D4—24no—
143C8×D4C2318yes—
144C8×D4C2318yescomplemented
145#432—21no—
146#432—21no—
147#432—21no—
148#432—21no—
149#432D412yes—
150#432D412yes—
151#432C2312yes—
152#432C2312yes—
153#431—21no—
154#431—21no—
155#431C2×C412yes—
156#431C2×C412yes—
157#431C2312yes—
158#431C2312yes—
159#431C2312yes—
160#431C2312yes—
161#431C2312yes—
162#431C2312yes—
163#431C2312yes—
164#431C2312yes—
165C2×#107C2×C412yes—
166C2×#107C2×C412yes—
167#423—21no—
168#423D412yes—
169#423C2312yes—
170#423C2312yes—
171C2×C4⋊C8C2×C412yes—
172C2×#102C2×C412yes—
173C2×#102C2×C412yes—
174C2×#102C2×C412yes—
175C2×#102C2×C412yes—
176#408—28no—
177#408—28no—
178#407—21no—
179#407—21no—
180#407—21no—
181#407—24no—
182#407—24no—
183#407—24no—
184#407—24no—
185#407D412yes—
186C2×C22⋊C8—24no—
187C2×C22⋊C8—24no—
188C2×C22⋊C8C2×C412yes—
189C4×M16—21no—
190C4×M16—21no—
191C4×M16C2312yes—
192C4×M16C2312yes—
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2311yescharacteristic
195#335C2318yescomplemented
196#334C2318yescomplemented
197D4⋊C8C2318yes—
198D4⋊C8C2318yes—
199D4⋊C8C2318yescomplemented
200D4⋊C8C2318yescomplemented
201#322C2314yescomplemented
202#322C2314yescomplemented
203#322C2314yescomplemented
204C82C2314yescomplemented
205C25—21no—
206C2×#41—22no—
207C2×#41—22no—
208C2×#41—22no—
209C2×#41—22no—
210C2×#41—84no—
211C22×Q8—21no—
212C22×D4—22no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4—44no—
219C22×D4C22×C411yescharacteristic
220C22×D4C22×C412yes—
221C22×D4C22×C412yes—
222C22×D4C22×C411yescharacteristic
223C22×D4C22×C412yes—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—21no—
229C23×C4—22no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—21no—
234C23×C4—84no—
235C23×C4C22×C411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4#4111yescharacteristic
238C2×D8—44no—
239C2×D8—44no—
240C2×D8C22×C414yes—
241C2×D8C22×C414yes—
242C2×D8C22×C414yes—
243C2×D8C22×C414yes—
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—28no—
248C2×M16—28no—
249C2×M16—88no—
250C2×M16—88no—
251C2×M16C2×D414yes—
252C22×C8—28no—
253C22×C8—28no—
254C22×C8C22×C412yes—
255C22×C8C22×C412yes—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8C22×C412yes—
259C4⋊Q8C22×C412yes—
260#127—21no—
261#127—21no—
262#127C22×C412yes—
263#127C22×C412yes—
264#124—22no—
265#124—22no—
266#124—22no—
267#124—22no—
268#121—22no—
269#121—22no—
270#121—22no—
271#121—22no—
272#121—22no—
273#121—22no—
274#121C22×C414yes—
275#121C22×C414yes—
276C4×D4—21no—
277C4×D4—22no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—22no—
281C4×D4—21no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—21no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—21no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—21no—
294C4×D4—21no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4C22×C412yes—
299C4×D4C22×C412yes—
300C4×D4C22×C412yes—
301C4×D4C22×C412yes—
302C4×D4C22×C412yes—
303C4×D4C22×C412yes—
304C4×D4C22×C414yes—
305C4×D4C2412yes—
306C4×D4C2412yes—
307C4×D4C2414yes—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4C22×C412yes—
313C2×C4⋊C4C22×C411yescharacteristic
314C2×C4⋊C4C22×C412yes—
315C2×C4⋊C4C22×C411yescharacteristic
316C2×C4⋊C4C22×C412yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329C2×C42C22×C411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42#4111yescharacteristic
334C2×C42#4111yescharacteristic
335C2×C42#4111yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#107C22×C414yes—
340#107C22×C414yes—
341#107C22×C414yes—
342#107C22×C414yes—
343C4⋊C8—22no—
344C4⋊C8—22no—
345C4⋊C8—22no—
346C4⋊C8—22no—
347C4⋊C8C22×C414yes—
348C4⋊C8C22×C414yes—
349C4⋊C8C2×D414yes—
350C4⋊C8C2×D414yes—
351C4⋊C8C2414yes—
352C4⋊C8C2414yes—
353#102C22×C414yes—
354#102C22×C414yes—
355#102C22×C414yes—
356#102C22×C414yes—
357#102C22×C414yes—
358#102C22×C414yes—
359#102C22×C414yes—
360#102C22×C414yes—
361C22⋊C8—22no—
362C22⋊C8—22no—
363C22⋊C8—22no—
364C22⋊C8—22no—
365C22⋊C8—22no—
366C22⋊C8—22no—
367C22⋊C8—28no—
368C22⋊C8—28no—
369C22⋊C8—28no—
370C22⋊C8—28no—
371C22⋊C8C22×C414yes—
372C22⋊C8C22×C414yes—
373C22⋊C8C2×D414yes—
374C22⋊C8C2×D414yes—
375#97—22no—
376#97—22no—
377#97—22no—
378#97C2×D414yes—
379C4×C8—22no—
380C4×C8—22no—
381C4×C8C22×C412yes—
382C4×C8C22×C412yes—
383C4×C8C2414yes—
384C4×C8C2412yes—
385C4×C8C2412yes—
386C4×C8C2414yes—
387C24—21no—
388C24—22no—
389C24—22no—
390C24—21no—
391C24—22no—
392C24—44no—
393C24—82no—
394C24—82no—
395C24C4×D411yescharacteristic
396#41—84no—
397#41—84no—
398#41—82no—
399#41—84no—
400#41—82no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—82no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4—84no—
429C2×D4C23×C412yes—
430C2×D4C23×C414yes—
431C2×D4C23×C412yes—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—82no—
452C22×C4—84no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—84no—
456C22×C4—82no—
457C22×C4—84no—
458C22×C4—81no—
459C22×C4—82no—
460C22×C4C4×D411yescharacteristic
461C22×C4C4×D411yescharacteristic
462C22×C4C4×D411yescharacteristic
463C22×C4C4×D411yescharacteristic
464C22×C4C4×D411yescharacteristic
465C22×C4C4×D411yescharacteristic
466C22×C4C4×D411yescharacteristic
467C22×C4C4×D411yescharacteristic
468C22×C4C4×D411yescharacteristic
469C22×C4C23×C411yescharacteristic
470C22×C4C23×C411yescharacteristic
471C22×C4C23×C411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4111yescharacteristic
476C22×C4C2×#4111yescharacteristic
477C22×C4C2×#4111yescharacteristic
478D8—28no—
479D8—28no—
480D8—44no—
481D8—44no—
482M16—28no—
483M16—44no—
484M16—44no—
485M16—88no—
486M16—88no—
487M16—82no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—88no—
494C2×C8—88no—
495C2×C8C23×C414yes—
496C2×C8C23×C414yes—
497C2×C8C22×D418yes—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4C23×C412yes—
506C4⋊C4C23×C412yes—
507C4⋊C4C23×C414yes—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—24no—
520C42C4×D411yescharacteristic
521C42C4×D411yescharacteristic
522C42C4×D411yescharacteristic
523C42C4×D411yescharacteristic
524C42C4×D411yescharacteristic
525C42C4×D411yescharacteristic
526C42C23×C411yescharacteristic
527C42C23×C411yescharacteristic
528C42C23×C411yescharacteristic
529C42C22×D411yescharacteristic
530C42C22×D411yescharacteristic
531C42C22×D411yescharacteristic
532C42C2×#4111yescharacteristic
533C42C2×#4111yescharacteristic
534C42C2×#4111yescharacteristic
535C42C2511yescharacteristic
536C23—24no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—41no—
542C23—41no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—81no—
550C23—82no—
551C23—81no—
552C23—82no—
553C23—81no—
554C23—81no—
555C23—82no—
556C23—82no—
557C23—816no—
558C23C2×C4×D411yescharacteristic
559C23C2×C4×D411yescharacteristic
560C23C2×C4×D411yescharacteristic
561Q8—82no—
562Q8—82no—
563D4—24no—
564D4—22no—
565D4—24no—
566D4—24no—
567D4—22no—
568D4—22no—
569D4—44no—
570D4—44no—
571D4—42no—
572D4—44no—
573D4—82no—
574D4—82no—
575D4—82no—
576D4—82no—
577D4—84no—
578D4—84no—
579D4#56814yes—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—81no—
614C2×C4—82no—
615C2×C4—81no—
616C2×C4C2×C4×D411yescharacteristic
617C2×C4C2×C4×D411yescharacteristic
618C2×C4C2×C4×D411yescharacteristic
619C2×C4C2×C4×D411yescharacteristic
620C2×C4C2×C4×D411yescharacteristic
621C2×C4C2×C4×D411yescharacteristic
622C2×C4C2×C4×D411yescharacteristic
623C2×C4C2×C4×D411yescharacteristic
624C2×C4C2×C4×D411yescharacteristic
625C2×C4#57614yes—
626C2×C4C24×C411yescharacteristic
627C2×C4C23×D411yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C8—24no—
630C8—24no—
631C8—28no—
632C8—44no—
633C8—44no—
634C8—44no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—24no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—44no—
649C22—44no—
650C22—42no—
651C22—42no—
652C22—44no—
653C22—88no—
654C22—84no—
655C22—88no—
656C22—84no—
657C22—164no—
658C22#294812yes—
659C22C22×C4×D411yescharacteristic
660C4—21no—
661C4—21no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—21no—
667C4—21no—
668C4—21no—
669C4—42no—
670C4—42no—
671C4—44no—
672C4#294812yes—
673C4C2×#56812yes—
674C4C2×#57612yes—
675C2—22no—
676C2—41no—
677C2—41no—
678C2—44no—
679C2—82no—
680C2—84no—
681C2#6014111yescharacteristic
682C2C2×#294811yescharacteristic
683C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radicalC2×#3100C2
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. #60232C2semidirect
3. #60230C2semidirect
4. #18044C2semidirect
5. M16×D8C2semidirect
6. #18026C2semidirect
7. #17950C2semidirect
8. #17922C2semidirect
9. #17815C2semidirect
10. #17490C2semidirect
11. #17448C2semidirect
12. #17324C2semidirect
26. #2981C22semidirect
28. #2979C22semidirect
29. #2975C22semidirect
31. #2974C22semidirect
33. #2972C22semidirect
39. D4×M16C22semidirect
47. #1594C22semidirect
48. #1589C22semidirect
49. #1586C22semidirect
50. #1586C22semidirect
51. #1581C22semidirect
52. #1581C22semidirect
53. #1580C22semidirect
54. C8×D8C22semidirect
55. #1572C22semidirect
56. #1572C22semidirect
57. #1567C22semidirect
58. #1566C22semidirect
59. #1496C22semidirect
60. #1490C22semidirect
61. #1490C22semidirect
63. #1486C22semidirect
64. #1484C22semidirect
65. #1478C22semidirect
66. #1454C22semidirect
67. #1453C22semidirect
68. #1451C22semidirect
69. C8×M16C22semidirect
136. #436C23semidirect
144. C8×D4C23semidirect
195. #335C23semidirect
196. #334C23semidirect
199. D4⋊C8C23semidirect
200. D4⋊C8C23semidirect
201. #322C23semidirect
202. #322C23semidirect
203. #322C23semidirect
204. C82C23semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
#602322
#602302
#180442
M16×D82
#180262
#179502
#179222
#178152
#174902
#174482
#173242