Finite Group Databasebeta

← #6345287

Subgroups — #6345287

Order 512.

2113 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2#60144C211yesmaximal, characteristic, complemented
3C2×D4×M16C211yesmaximal, characteristic, complemented
4#18044C218yesmaximal, complemented
5#18001C218yesmaximal, complemented
6#17935C214yesmaximal, complemented
7#17907C214yesmaximal, complemented
8#17813C211yesmaximal, characteristic, complemented
9#17485C211yesmaximal, characteristic, complemented
10C2×#1486C211yesmaximal, characteristic, complemented
11#17319C211yesmaximal, characteristic, complemented
12C2×#568—21no—
13C23×M16—21no—
14C2×#532C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16#3100C412yes—
17#3100C414yes—
18#3100C412yes—
19#3100C2212yes—
20#3100C2214yes—
21#3100C2212yes—
22C2×#480C412yes—
23C2×#474C412yes—
24C2×#459C412yes—
25C2×#447C412yes—
26C2×C4×D8C2214yes—
27#2935—22no—
28#2935C2214yescomplemented
29D4×M16—22no—
30D4×M16—22no—
31D4×M16—22no—
32D4×M16C2214yes—
33D4×M16C2214yescomplemented
34D4×M16C2214yescomplemented
35#2932—22no—
36#2931C2214yescomplemented
37C2×#436C2212yescomplemented
38C2×#435—22no—
39C2×C8×D4C2212yescomplemented
40C2×#431C2211yescharacteristic
41C2×#431C2211yescharacteristic
42C2×#423C2211yescharacteristic
43C2×#423C2211yescharacteristic
44#2879—21no—
45C2×#407—21no—
46C2×C4×M16C2211yescharacteristic
47#1594C22116yescomplemented
48#1586C22116yescomplemented
49#1586C22116yescomplemented
50#1580C22116yescomplemented
51#1570C2214yescomplemented
52#1570C2214yescomplemented
53#1566C2214yescomplemented
54#1492C2214yescomplemented
55#1486C2214yes—
56#1486C2212yescomplemented
57#1486C2212yescomplemented
58#1486C2214yescomplemented
59#1480C2214yescomplemented
60C2×D4⋊C8C2214yescomplemented
61#1452C2214yescomplemented
62#1451C2214yescomplemented
63#1451C2214yescomplemented
64C22×#41—21no—
65C23×D4C2×C411yescharacteristic
66C24×C4—21no—
67C22×D8C2×C414yes—
68#568—48no—
69C2×#131—24no—
70C2×#131—24no—
71C22×M16—21no—
72C22×M16—21no—
73C22×M16—24no—
74C22×M16—24no—
75C22×M16—24no—
76C22×M16—22no—
77C22×M16—48no—
78C22×M16D411yescharacteristic
79C22×M16D411yescharacteristic
80C23×C8—22no—
81#532—21no—
82#532—21no—
83#532—21no—
84#532—21no—
85#532C2×C412yes—
86#532C2×C412yes—
87#532C2312yes—
88#532C2312yes—
89C2×C4⋊Q8C2×C411yescharacteristic
90C2×#127C2×C411yescharacteristic
91C2×#124—21no—
92C2×#121—21no—
93C2×#121C2×C412yes—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4C2×C411yescharacteristic
102C2×C4×D4C2×C412yes—
103C2×C4×D4C2×C411yescharacteristic
104C2×C4×D4C2×C412yes—
105C2×C4×D4C2311yescharacteristic
106C2×C4×D4C2311yescharacteristic
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C22×C4⋊C4C2×C411yescharacteristic
111C22×C22⋊C4—21no—
112C22×C42C2311yescharacteristic
113#480C2×C414yes—
114#480C2×C418yes—
115#480C2×C414yes—
116#474C2×C414yes—
117#474C2×C414yes—
118#474C2×C414yes—
119#474C2×C414yes—
120#459C2×C414yes—
121#459C2×C414yes—
122#459C2×C414yes—
123#459C2×C414yes—
124#447C2×C414yes—
125#447C2×C418yes—
126#447C2×C414yes—
127C4×D8C2×C418yes—
128C4×D8C2×C418yes—
129C4×D8C2318yes—
130C4×D8C2318yes—
131#436—24no—
132#436—24no—
133#436C2318yes—
134#436C2318yescomplemented
135#435—28no—
136#435—28no—
137#435—28no—
138#435—28no—
139C8×D4—24no—
140C8×D4—24no—
141C8×D4C2318yes—
142C8×D4C2318yescomplemented
143#431—21no—
144#431—21no—
145#431C2×C412yes—
146#431C2×C412yes—
147#431C2312yes—
148#431C2312yes—
149#431C2312yes—
150#431C2312yes—
151C2×#107C2×C414yes—
152#423—21no—
153#423—21no—
154#423—21no—
155#423—21no—
156#423C2312yes—
157#423C2312yes—
158#423C2312yes—
159#423C2312yes—
160C2×C4⋊C8—21no—
161C2×C4⋊C8—21no—
162C2×C4⋊C8—21no—
163C2×C4⋊C8C2×C412yes—
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2312yes—
166C2×#102C2×C414yes—
167C2×#102C2×C414yes—
168#408—24no—
169#408—24no—
170#407—24no—
171#407—24no—
172#407—24no—
173C2×C22⋊C8—21no—
174C2×C22⋊C8—22no—
175C2×C22⋊C8—22no—
176C2×C22⋊C8C2×C412yes—
177C4×M16D412yes—
178C4×M16D412yes—
179C4×M16C2312yes—
180C4×M16C2312yes—
181C2×#97D412yes—
182C2×C4×C8C2312yes—
183C2×C4×C8C2312yes—
184C2×C4×C8C2312yes—
185#334C23116yescomplemented
186D4⋊C8C2318yes—
187D4⋊C8C2318yes—
188D4⋊C8C23116yescomplemented
189#322C2318yescomplemented
190#322C2318yescomplemented
191C25—21no—
192C2×#41—22no—
193C2×#41—22no—
194C2×#41—22no—
195C2×#41—22no—
196C2×#41—44no—
197C2×#41—44no—
198C22×Q8—21no—
199C22×D4—22no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4—44no—
206C22×D4C22×C411yescharacteristic
207C22×D4C22×C412yes—
208C22×D4C22×C412yes—
209C22×D4C22×C411yescharacteristic
210C22×D4C22×C412yes—
211C23×C4—21no—
212C23×C4—22no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—22no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4C22×C411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4#4111yescharacteristic
226C2×D8—48no—
227C2×D8C22×C418yes—
228C2×D8C22×C418yes—
229#131—416no—
230#131—416no—
231C2×M16—24no—
232C2×M16—24no—
233C2×M16—22no—
234C2×M16—22no—
235C2×M16—22no—
236C2×M16—22no—
237C2×M16—28no—
238C2×M16—28no—
239C2×M16—48no—
240C2×M16—416no—
241C2×M16—416no—
242C2×M16—416no—
243C2×M16—44no—
244C2×M16—44no—
245C2×M16C2×D412yes—
246C2×M16C2×D412yes—
247C2×M16C2×D412yes—
248C2×M16C2×D412yes—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—28no—
253C22×C8—28no—
254C22×C8—48no—
255C22×C8C22×C414yes—
256C22×C8C2×D412yes—
257C22×C8C2×D412yes—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8C22×C412yes—
261C4⋊Q8C22×C412yes—
262#127—21no—
263#127—21no—
264#127C22×C412yes—
265#127C22×C412yes—
266#124—22no—
267#124—22no—
268#124—22no—
269#124—22no—
270#121—22no—
271#121—22no—
272#121—22no—
273#121—22no—
274#121—22no—
275#121—22no—
276#121C22×C414yes—
277#121C22×C414yes—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—22no—
281C4×D4—21no—
282C4×D4—22no—
283C4×D4—21no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—21no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—21no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—22no—
300C4×D4C22×C412yes—
301C4×D4C22×C412yes—
302C4×D4C22×C412yes—
303C4×D4C22×C412yes—
304C4×D4C22×C412yes—
305C4×D4C22×C412yes—
306C4×D4C22×C414yes—
307C4×D4C2412yes—
308C4×D4C2412yes—
309C4×D4C2414yes—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4C22×C412yes—
315C2×C4⋊C4C22×C411yescharacteristic
316C2×C4⋊C4C22×C411yescharacteristic
317C2×C4⋊C4C22×C412yes—
318C2×C4⋊C4C22×C412yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C42C22×C411yescharacteristic
330C2×C42C22×C411yescharacteristic
331C2×C42C22×C411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42#4111yescharacteristic
336C2×C42#4111yescharacteristic
337C2×C42#4111yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107C22×C418yes—
342#107C22×C418yes—
343C4⋊C8—22no—
344C4⋊C8—22no—
345C4⋊C8—22no—
346C4⋊C8—22no—
347C4⋊C8—22no—
348C4⋊C8—22no—
349C4⋊C8—22no—
350C4⋊C8—22no—
351C4⋊C8C22×C414yes—
352C4⋊C8C22×C414yes—
353C4⋊C8C2414yes—
354C4⋊C8C2414yes—
355#102C22×C418yes—
356#102C22×C418yes—
357#102C22×C418yes—
358#102C22×C418yes—
359C22⋊C8—22no—
360C22⋊C8—22no—
361C22⋊C8—28no—
362C22⋊C8—28no—
363C22⋊C8—28no—
364C22⋊C8—28no—
365C22⋊C8C22×C414yes—
366C22⋊C8C22×C414yes—
367#97C2×D414yes—
368#97C2×D414yes—
369C4×C8—22no—
370C4×C8C22×C414yes—
371C4×C8C2×D414yes—
372C4×C8C2414yes—
373C4×C8C2414yes—
374C4×C8C2414yes—
375C24—21no—
376C24—21no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—44no—
381C24—44no—
382C24—82no—
383C24C4×D411yescharacteristic
384#41—44no—
385#41—44no—
386#41—44no—
387#41—44no—
388#41—48no—
389#41—48no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—82no—
417C2×D4C23×C412yes—
418C2×D4C23×C414yes—
419C2×D4C23×C412yes—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—81no—
452C22×C4C4×D411yescharacteristic
453C22×C4C4×D411yescharacteristic
454C22×C4C4×D411yescharacteristic
455C22×C4C4×D411yescharacteristic
456C22×C4C4×D411yescharacteristic
457C22×C4C4×D411yescharacteristic
458C22×C4C4×D411yescharacteristic
459C22×C4C4×D411yescharacteristic
460C22×C4C4×D411yescharacteristic
461C22×C4AGL(1,ℤ/8)14yes—
462C22×C4AGL(1,ℤ/8)14yes—
463C22×C4C23×C411yescharacteristic
464C22×C4C23×C411yescharacteristic
465C22×C4C23×C411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4111yescharacteristic
470C22×C4C2×#4111yescharacteristic
471C22×C4C2×#4111yescharacteristic
472D8—216no—
473D8—48no—
474M16—24no—
475M16—24no—
476M16—24no—
477M16—24no—
478M16—416no—
479M16—416no—
480M16—44no—
481M16—44no—
482M16—44no—
483M16—44no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—44no—
490C2×C8—416no—
491C2×C8—416no—
492C2×C8—44no—
493C2×C8—44no—
494C2×C8—82no—
495C2×C8C23×C418yes—
496C2×C8C22×D414yes—
497C2×C8C22×D414yes—
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—24no—
540C23—41no—
541C23—41no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—82no—
553C23—82no—
554C23—82no—
555C23—82no—
556C23—816no—
557C23#44214yes—
558C23C2×C4×D411yescharacteristic
559C23C2×C4×D411yescharacteristic
560C23C2×C4×D411yescharacteristic
561Q8—44no—
562Q8—44no—
563D4—24no—
564D4—22no—
565D4—28no—
566D4—22no—
567D4—22no—
568D4—44no—
569D4—44no—
570D4—42no—
571D4—44no—
572D4—48no—
573D4—48no—
574D4—44no—
575D4—44no—
576D4—82no—
577D4—82no—
578D4#56814yescomplemented
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4#44214yes—
613C2×C4#44214yes—
614C2×C4#44214yes—
615C2×C4C2×C4×D411yescharacteristic
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×AGL(1,ℤ/8)12yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4C2×AGL(1,ℤ/8)12yes—
630C2×C4C24×C411yescharacteristic
631C2×C4C23×D411yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C8—28no—
634C8—24no—
635C8—24no—
636C8—44no—
637C8—44no—
638C8—44no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—24no—
646C22—44no—
647C22—48no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—44no—
654C22—42no—
655C22—44no—
656C22—88no—
657C22—88no—
658C22—84no—
659C22—164no—
660C22C2×#44212yes—
661C22C2×#44212yes—
662C22C2×#44212yes—
663C22C22×C4×D411yescharacteristic
664C4—21no—
665C4—21no—
666C4—21no—
667C4—21no—
668C4—21no—
669C4—21no—
670C4—21no—
671C4—42no—
672C4—42no—
673C4—44no—
674C4C2×#44212yes—
675C4C2×#44212yes—
676C4C2×#44212yes—
677C4C2×#56812yes—
678C4C22×AGL(1,ℤ/8)11yescharacteristic
679C4C22×AGL(1,ℤ/8)11yescharacteristic
680C2—22no—
681C2—22no—
682C2—22no—
683C2—44no—
684C2—82no—
685C2—84no—
686C2#6014111yescharacteristic
687C2C22×#44211yescharacteristic
688C2#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 ◁ C2×C4×M16 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60144C2semidirect
3. C2×D4×M16C2semidirect
4. #18044C2semidirect
5. #18001C2semidirect
6. #17935C2semidirect
7. #17907C2semidirect
8. #17813C2semidirect
9. #17485C2semidirect
10. C2×#1486C2semidirect
11. #17319C2semidirect
28. #2935C22semidirect
33. D4×M16C22semidirect
34. D4×M16C22semidirect
36. #2931C22semidirect
37. C2×#436C22semidirect
39. C2×C8×D4C22semidirect
47. #1594C22semidirect
48. #1586C22semidirect
49. #1586C22semidirect
50. #1580C22semidirect
51. #1570C22semidirect
52. #1570C22semidirect
53. #1566C22semidirect
54. #1492C22semidirect
56. #1486C22semidirect
57. #1486C22semidirect
58. #1486C22semidirect
59. #1480C22semidirect
60. C2×D4⋊C8C22semidirect
61. #1452C22semidirect
62. #1451C22semidirect
63. #1451C22semidirect
134. #436C23semidirect
142. C8×D4C23semidirect
185. #334C23semidirect
188. D4⋊C8C23semidirect
189. #322C23semidirect
190. #322C23semidirect
578. D4#568semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
#601442
C2×D4×M162
#180442
#180012
#179352
#179072
#178132
#174852
C2×#14862
#173192