Finite Group Databasebeta

← #6627611

Subgroups — #6627611

Order 512.

1679 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2C4×D4×Q8C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic
4#33032C212yesmaximal
5#29355C212yesmaximal, complemented
6#29276C212yesmaximal, complemented
7#29219C211yesmaximal, characteristic, complemented
8#28283C211yesmaximal, characteristic, complemented
9#27573C214yesmaximal
10#27145C214yesmaximal, complemented
11#24996C212yesmaximal, complemented
12#24330C212yesmaximal
13#23068C212yesmaximal, complemented
14#22825C211yesmaximal, characteristic, complemented
15#22762C212yesmaximal
16#21613C212yesmaximal
17C4×#461C211yesmaximal, characteristic
18C2×D4×Q8C2211yescharacteristic
19C22×C4×Q8—21no—
20C2×#475C2212yes—
21C2×#474C2212yes—
22C2×#461C2211yescharacteristic
23C2×C4×SD16C2212yes—
24#2868C2211yescharacteristic
25#2862C2212yes—
26#2582C2212yes—
27#2571—21no—
28#2570—22no—
29#2568—21no—
30#2511C2214yes—
31#2446—22no—
32#2365—21no—
33#2365C2212yes—
34#2365C2211yescharacteristic
35#2360—22no—
36#2360—21no—
37#2355—21no—
38#2355C2212yes—
39Q8×C4⋊C4C2211yescharacteristic
40D4×C4⋊C4C2212yes—
41D4×C4⋊C4C2211yescharacteristic
42Q8×C22⋊C4—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8—21no—
48C4×C22⋊Q8—22no—
49C42×Q8C2211yescharacteristic
50C42×D4C2212yes—
51C42×D4C2211yescharacteristic
52#2100C2214yes—
53#2099C2214yes—
54#2072C2214yes—
55#2066C2214yes—
56#2058C2214yes—
57#2024C2214yes—
58#1990C2212yes—
59#1988C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1984C2212yes—
63#1981C2212yes—
64#1981C2211yescharacteristic
65#1948C2211yescharacteristic
66#1948C2212yes—
67#1947C2212yes—
68#1947C2212yes—
69#1935C2214yes—
70#1935C2214yes—
71#1928C2214yes—
72#1928C2214yes—
73#1927C2212yes—
74#1921C2214yes—
75#1920C2214yes—
76#1920C2214yes—
77#1917C2214yes—
78#1877C2214yes—
79#1877C2214yes—
80#1877C2212yes—
81#1864C2214yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1831C2212yes—
85#1803C2212yes—
86#1802C2212yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1762C2214yes—
94C4×#103C2212yes—
95C23×Q8—21no—
96C22×SD16C2312yes—
97D4×Q8—24no—
98D4×Q8—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—48no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C42—22no—
126C22×C42—21no—
127#475—28no—
128#474—28no—
129#461—24no—
130C4×SD16—28no—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#400—24no—
149#400—24no—
150#398—24no—
151#391—22no—
152#391—21no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391—21no—
157#391—22no—
158#391—22no—
159#391C2312yes—
160#391C2314yes—
161#391C2314yes—
162#389—22no—
163#389—21no—
164#389—22no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389C2314yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#387—24no—
175#387—24no—
176#387—24no—
177#387—22no—
178#387—22no—
179#387—22no—
180#385—22no—
181#385—22no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#384C2312yes—
190#382—22no—
191#382—22no—
192#382—22no—
193#380—21no—
194#380—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D412yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C43C2312yes—
221C43C2311yescharacteristic
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—28no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—44no—
240C22×Q8—44no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2412yes—
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—21no—
249C2×SD16—24no—
250C2×SD16—24no—
251C2×SD16—24no—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264C4⋊Q8#4114yes—
265C22⋊Q8—28no—
266C22⋊Q8—216no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—48no—
274C4×Q8—44no—
275C4×Q8—48no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—84no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—22no—
282C4×D4—24no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—42no—
286C4×D4—41no—
287C4×D4#4114yes—
288C4×D4#4114yes—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—48no—
338C2×C42—44no—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—28no—
353#106—24no—
354#106—28no—
355C4⋊C8—22no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358C4⋊C8—22no—
359C4⋊C8—42no—
360#103—24no—
361#103—24no—
362#103—28no—
363#103—42no—
364#103—44no—
365#102—24no—
366#102—28no—
367#97—44no—
368C4×C8—24no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C2×Q8—28no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—432no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8#13714yes—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4C2×#4112yes—
413C2×D4C2×#4112yes—
414C2×D42−1+412yes—
415C22×C4—28no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—81no—
437C22×C4—82no—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2511yescharacteristic
449SD16—48no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—81no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4C2×#4114yes—
497C4⋊C4C2×#4112yes—
498C4⋊C4C2×#4112yes—
499C22⋊C4—28no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42—41no—
516C42—42no—
517C42—44no—
518C42—44no—
519C42—44no—
520C42—42no—
521C42—42no—
522C42—44no—
523C42—42no—
524C42—84no—
525C42AGL(1,ℤ/8)14yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C42C2×#4112yes—
529C422−1+412yes—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C23×D411yescharacteristic
538Q8—28no—
539Q8—416no—
540Q8—48no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—88no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4#55614yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—416no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—84no—
598C2×C4—88no—
599C2×C4#52512yes—
600C2×C4#52512yes—
601C2×C4#54812yes—
602C2×C4#54812yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C2×#13714yes—
607C2×C4C2×#13712yes—
608C2×C4#57712yes—
609C2×C4C22×#4111yescharacteristic
610C2×C4C22×#4111yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C2×C4C2×2−1+411yescharacteristic
614C2×C4#58511yescharacteristic
615C8—42no—
616C8—44no—
617C8—42no—
618C22—24no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22C2×#52511yescharacteristic
626C22C2×#54811yescharacteristic
627C22#348911yescharacteristic
628C22C22×AGL(1,ℤ/8)11yescharacteristic
629C22C22×#13711yescharacteristic
630C22C2×#57711yescharacteristic
631C22#358611yescharacteristic
632C4—24no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—41no—
643C4—84no—
644C4—81no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4#311014yes—
649C4#337314yes—
650C4C2×#55611yescharacteristic
651C4#348011yescharacteristic
652C4#350711yescharacteristic
653C4#352311yescharacteristic
654C2—44no—
655C2#3657911yescharacteristic
656C2C2×#311011yescharacteristic
657C2#6059911yescharacteristic
658C2C2×#337311yescharacteristic
659C2#6132211yescharacteristic
660C2#6210911yescharacteristic
661C2#6214711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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
1. #36598C2semidirect
5. #29355C2semidirect
6. #29276C2semidirect
7. #29219C2semidirect
8. #28283C2semidirect
10. #27145C2semidirect
11. #24996C2semidirect
13. #23068C2semidirect
14. #22825C2semidirect

Maximal subgroups

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

subgroupindex
#365982
C4×D4×Q82
#331952
#330322
#293552
#292762
#292192
#282832
#275732
#271452
#249962
#243302
#230682
#228252
#227622
#216132
C4×#4612