Finite Group Databasebeta

← #6800684

Subgroups — #6800684

Order 512.

1797 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#127C211yesmaximal, characteristic
2#34951C211yesmaximal, characteristic, complemented
3#33196C211yesmaximal, characteristic, complemented
4#32428C214yesmaximal, complemented
5#31636C214yesmaximal, complemented
6#29983C212yesmaximal, complemented
7#29141C212yesmaximal, complemented
8#28717C211yesmaximal, characteristic, complemented
9#28400C212yesmaximal, complemented
10#28252C211yesmaximal, characteristic
11#28160C212yesmaximal
12#27727C212yesmaximal, complemented
13#25929C212yesmaximal, complemented
14#24330C212yesmaximal, complemented
15#22821C211yesmaximal, characteristic, complemented
16#22147C212yesmaximal
17C4×#460C211yesmaximal, characteristic
18C2×D4×Q8—21no—
19C2×D4×Q8—21no—
20C2×D4×Q8C2212yes—
21C2×#487C2212yes—
22C2×#475C2212yes—
23C2×#460C2211yescharacteristic
24C2×#460C2212yes—
25C2×#461C2212yes—
26C2×C4×SD16C2212yes—
27#2868C2212yes—
28#2868C2211yescharacteristic
29#2585—22no—
30#2582C2211yescharacteristic
31#2568C2211yescharacteristic
32#2552C2211yescharacteristic
33#2546C2211yescharacteristic
34#2542—21no—
35#2542—22no—
36#2542—22no—
37#2542—21no—
38#2542C2212yes—
39#2542C2214yes—
40#2511C2212yes—
41#2509—21no—
42#2453—21no—
43#2446C2212yes—
44#2402—22no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2212yes—
48C4×#127C2211yescharacteristic
49C4×#125—21no—
50#2295—21no—
51#2295C2212yes—
52C42×Q8C2211yescharacteristic
53C42×Q8C2211yescharacteristic
54#2099C2214yes—
55#2096C2214yes—
56#2072C2214yes—
57#2071C2214yes—
58#2058C2214yes—
59#2055C2214yes—
60#2024C2214yes—
61#2024C2214yes—
62#2024C2214yes—
63#2013C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1968C2214yes—
69#1956C2214yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1948C2212yes—
73#1934C2214yes—
74#1926C2212yes—
75#1916C2214yes—
76#1880C2212yes—
77#1877C2214yes—
78#1877C2214yes—
79#1877C2214yes—
80#1864C2214yes—
81#1842C2211yescharacteristic
82#1834C2214yes—
83#1831C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1803C2211yescharacteristic
90#1802C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#102C2212yes—
95C23×Q8—22no—
96C22×SD16C2312yes—
97C22×SD16C2314yes—
98D4×Q8—44no—
99D4×Q8—44no—
100D4×Q8—48no—
101D4×Q8—48no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2314yes—
110C2×#127C2311yescharacteristic
111C2×#125—21no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—21no—
126C2×C4×D4—22no—
127C2×C4×D4—21no—
128C2×C4×D4—22no—
129C2×C4×D4C2312yes—
130C2×C4×D4C2314yes—
131#487—44no—
132#475—44no—
133#461—44no—
134#460—24no—
135#460—28no—
136C4×SD16—44no—
137C2×#106C2314yes—
138C2×#106C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×C4×C8C2312yes—
154C2×C4×C8C2314yes—
155#400—22no—
156#400—22no—
157#400—22no—
158#400—22no—
159#399—24no—
160#399—24no—
161#399—24no—
162#398—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#391—21no—
167#391—21no—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#391C2312yes—
172#391C2312yes—
173#390—22no—
174#390—21no—
175#390—21no—
176#390—22no—
177#390C2312yes—
178#390C2314yes—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389—21no—
187#389C2312yes—
188#389C2312yes—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#384—22no—
196#384—21no—
197#384—21no—
198#384—22no—
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2312yes—
204#384C2314yes—
205#382—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223#376—22no—
224C43—21no—
225C43C2312yes—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237#336C2314yes—
238C22×Q8—24no—
239C22×Q8—22no—
240C22×Q8—48no—
241C22×Q8—48no—
242C22×Q8—48no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2412yes—
245C22×Q8C2411yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2412yes—
249C23×C4—24no—
250C23×C4—22no—
251C2×SD16—24no—
252C2×SD16—24no—
253C2×SD16—28no—
254C2×SD16—42no—
255C2×SD16—44no—
256C2×SD16—44no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—42no—
268C4⋊Q8—44no—
269C4⋊Q8—48no—
270C4⋊Q8—48no—
271C4⋊Q8—44no—
272#127—22no—
273#127#4114yes—
274#125—44no—
275C22⋊Q8—48no—
276C22⋊Q8—416no—
277C22⋊Q8—416no—
278C22⋊Q8—416no—
279C22⋊Q8—48no—
280C22⋊Q8—48no—
281C4×Q8—24no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—24no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—24no—
291C4×Q8—41no—
292C4×Q8SD1618yescomplemented
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—48no—
299C4×D4—48no—
300C4×D4—44no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#106—42no—
364#106—44no—
365C4⋊C8—22no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—22no—
369C4⋊C8—42no—
370#103—42no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375#102—28no—
376#102—24no—
377#102—42no—
378#102—42no—
379C4×C8—24no—
380C4×C8—44no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—44no—
416C2×Q8—48no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—832no—
420C2×Q8C2×SD1614yes—
421C2×Q82−1+412yes—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4C2×#4114yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2511yescharacteristic
459SD16—48no—
460SD16—88no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—82no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4C2×SD1618yes—
510C4⋊C4C2×SD1614yes—
511C4⋊C4AGL(1,ℤ/8)14yes—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—21no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—41no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—44no—
534C42—42no—
535C42—42no—
536C42—81no—
537C42C2×SD1618yes—
538C42C2×SD1614yes—
539C42C2×#4112yes—
540C42H(5,𝔽2)12yes—
541C42H(5,𝔽2)12yes—
542C422−1+412yes—
543C23—24no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23C23×D411yescharacteristic
551Q8—28no—
552Q8—22no—
553Q8—22no—
554Q8—41no—
555Q8—41no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—816no—
561Q8—88no—
562D4—24no—
563D4—24no—
564D4—48no—
565D4—48no—
566D4—42no—
567D4—42no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—816no—
611C2×C4—88no—
612C2×C4#53512yes—
613C2×C4#53512yes—
614C2×C4#54812yes—
615C2×C4#54812yes—
616C2×C4C22×SD1612yes—
617C2×C4C22×SD1614yes—
618C2×C4C22×SD1614yes—
619C2×C4C22×SD1612yes—
620C2×C4C22×SD1612yes—
621C2×C4C2×AGL(1,ℤ/8)14yes—
622C2×C4C2×AGL(1,ℤ/8)12yes—
623C2×C4#57812yes—
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×2−1+411yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C2×C4#58511yescharacteristic
630C8—44no—
631C8—42no—
632C8—42no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22C2×#53511yescharacteristic
641C22C2×#54811yescharacteristic
642C22#348911yescharacteristic
643C22C23×SD1611yescharacteristic
644C22C22×AGL(1,ℤ/8)11yescharacteristic
645C22C2×#57811yescharacteristic
646C22#358611yescharacteristic
647C4—24no—
648C4—22no—
649C4—22no—
650C4—24no—
651C4—22no—
652C4—24no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—84no—
657C4—82no—
658C4—81no—
659C4—82no—
660C4—81no—
661C4—81no—
662C4—82no—
663C4#320814yes—
664C4#335914yes—
665C4#351411yescharacteristic
666C4#352011yescharacteristic
667C4#354711yescharacteristic
668C4#355211yescharacteristic
669C2—84no—
670C2#3912411yescharacteristic
671C2C2×#320811yescharacteristic
672C2#6083711yescharacteristic
673C2C2×#335911yescharacteristic
674C2#6131711yescharacteristic
675C2#6209911yescharacteristic
676C2#6213211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34951C2semidirect
3. #33196C2semidirect
4. #32428C2semidirect
5. #31636C2semidirect
6. #29983C2semidirect
7. #29141C2semidirect
8. #28717C2semidirect
9. #28400C2semidirect
12. #27727C2semidirect
13. #25929C2semidirect
14. #24330C2semidirect
15. #22821C2semidirect
292. C4×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
Q8×#1272
#349512
#331962
#324282
#316362
#299832
#291412
#287172
#284002
#282522
#281602
#277272
#259292
#243302
#228212
#221472
C4×#4602