Finite Group Databasebeta

← #6852437

Subgroups — #6852437

Order 512.

1603 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#36545C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic
3#32954C211yesmaximal, characteristic
4#32656C214yesmaximal, complemented
5#32574C214yesmaximal, complemented
6#29957C212yesmaximal
7#29298C212yesmaximal
8#29243C212yesmaximal
9#28265C212yesmaximal, complemented
10#27950C211yesmaximal, characteristic
11#27680C211yesmaximal, characteristic
12#25472C212yesmaximal, complemented
13#25063C212yesmaximal, complemented
14#24332C212yesmaximal, complemented
15Q8×C22⋊C8C211yesmaximal, characteristic
16#22044C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#475C2212yes—
22C2×#451C2211yescharacteristic
23C2×C8×Q8C2212yes—
24C2×C4×Q16C2212yes—
25#2857C2211yescharacteristic
26#2851C2212yes—
27#2584C2212yes—
28#2582—21no—
29#2582—21no—
30#2582—22no—
31#2582C2212yes—
32#2582C2214yes—
33#2582C2211yescharacteristic
34#2582C2212yes—
35#2581C2212yes—
36#2546C2211yescharacteristic
37#2473—22no—
38#2462—22no—
39#2446—21no—
40#2434—21no—
41C2×#398—22no—
42#2365—21no—
43#2360—21no—
44#2355C2211yescharacteristic
45C4×C22⋊Q8C2212yes—
46C4×C22⋊Q8C2211yescharacteristic
47C2×#391—21no—
48C2×#391C2212yes—
49C2×#389—21no—
50C2×#389C2212yes—
51C2×#384C2211yescharacteristic
52#2099C2212yes—
53#2097C2214yes—
54#2096C2212yes—
55#2088C2214yes—
56#2085C2214yes—
57#2083C2214yes—
58#2068C2214yes—
59#2068C2214yes—
60#2066C2214yes—
61#2059C2214yes—
62#2026C2214yes—
63#1990C2212yes—
64#1984C2212yes—
65#1984C2214yes—
66#1969C2214yes—
67#1948C2212yes—
68#1942C2214yes—
69#1941C2214yes—
70#1933C2214yes—
71#1928C2214yes—
72#1918C2214yes—
73#1878C2214yes—
74#1878C2212yes—
75#1872C2212yes—
76#1872C2211yescharacteristic
77#1865C2214yes—
78#1865C2214yes—
79#1863C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1852C2211yescharacteristic
85#1845C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1787C2211yescharacteristic
91#1787C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98#557—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—22no—
103C2×#125C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—44no—
110C2×C4×Q8—44no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D412yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D412yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#475—44no—
133#451Q818yescomplemented
134C8×Q8—44no—
135C4×Q16—44no—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×C4×C8C2312yes—
154C2×C4×C8C2314yes—
155#400—22no—
156#398—24no—
157#398—24no—
158#398—24no—
159#398—24no—
160#398—22no—
161#398—22no—
162#398—22no—
163#395—22no—
164#395—22no—
165#391—22no—
166#391—24no—
167#391—22no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—22no—
172#391—22no—
173#391—21no—
174#391C2314yes—
175#391C2312yes—
176#391C2311yescharacteristic
177#391C2314yes—
178#389—22no—
179#389—22no—
180#389—24no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—21no—
187#389C2314yes—
188#389C2312yes—
189#389C2312yes—
190#389C2314yes—
191#389C2312yes—
192#389C2311yescharacteristic
193#389C2312yes—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—21no—
201#385—22no—
202#385—22no—
203#385—22no—
204#385—21no—
205#384—21no—
206#384—21no—
207#384C2311yescharacteristic
208#384C2312yes—
209#384C2312yes—
210#382C2312yes—
211#380—21no—
212#380C2311yescharacteristic
213#380C2312yes—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95—22no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2×D412yes—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×Q814yes—
247C22×Q8C2411yescharacteristic
248C22×Q8C2412yes—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C2×Q16—42no—
258C2×Q16—42no—
259C2×Q16C2×Q818yescomplemented
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266#125—44no—
267#125—48no—
268C22⋊Q8—22no—
269C22⋊Q8—44no—
270C22⋊Q8C2×Q814yes—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—42no—
279C4×Q8—42no—
280C4×Q8—42no—
281C4×Q8—42no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—44no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4—84no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—44no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351#107—44no—
352#107—44no—
353#107—42no—
354#106—44no—
355#106—44no—
356C4⋊C8—24no—
357C4⋊C8—44no—
358#103—42no—
359#103—44no—
360#103—44no—
361#103—42no—
362#103—42no—
363#103—44no—
364#103C2×Q818yescomplemented
365C22⋊C8—22no—
366C22⋊C8—24no—
367C22⋊C8C2×Q814yes—
368C4×C8—24no—
369C4×C8—44no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—21no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391C24C22×D411yescharacteristic
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—42no—
409C2×Q8—48no—
410C2×Q8—416no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—48no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8#13714yes—
418C2×Q8C22×Q814yes—
419C2×Q8C22×Q812yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—42no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4#13514yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×Q812yes—
462C22×C42−1+412yes—
463C22×C4C2511yescharacteristic
464Q16—48no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8C22×Q814yes—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—81no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—84no—
511C4⋊C4—82no—
512C4⋊C4—84no—
513C4⋊C4C22×Q812yes—
514C22⋊C4—24no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C42—22no—
518C42—22no—
519C42—24no—
520C42—22no—
521C42—24no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—44no—
526C42—44no—
527C42—42no—
528C42—42no—
529C42—82no—
530C42H(5,𝔽2)14yes—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23D4×Q812yes—
536C23C2×#13512yes—
537C23#57812yes—
538C23C23×D411yescharacteristic
539Q8—28no—
540Q8—24no—
541Q8—24no—
542Q8—24no—
543Q8—48no—
544Q8—48no—
545Q8—44no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—84no—
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—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—88no—
606C2×C4#53412yes—
607C2×C4#53412yes—
608C2×C4D4×Q812yes—
609C2×C4C2×#13512yes—
610C2×C4C2×#13512yes—
611C2×C4C2×#13714yes—
612C2×C4C2×#13712yes—
613C2×C4C23×Q811yescharacteristic
614C2×C4C2×H(5,𝔽2)12yes—
615C2×C4C2×2−1+411yescharacteristic
616C2×C4#58512yes—
617C8—24no—
618C8—42no—
619C8—44no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22#338314yes—
627C22C2×#53411yescharacteristic
628C22C2×D4×Q811yescharacteristic
629C22#348711yescharacteristic
630C22C22×#13511yescharacteristic
631C22C22×#13711yescharacteristic
632C22C2×#57811yescharacteristic
633C22#358511yescharacteristic
634C4—22no—
635C4—24no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—41no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—42no—
644C4—42no—
645C4—82no—
646C4—81no—
647C4—82no—
648C4—82no—
649C4—84no—
650C4—81no—
651C4#319014yes—
652C4#350112yes—
653C4#354112yes—
654C2—24no—
655C2#3959511yescharacteristic
656C2C2×#319011yescharacteristic
657C2#6076411yescharacteristic
658C2C2×#338311yescharacteristic
659C2Q8×#13711yescharacteristic
660C2#6188511yescharacteristic
661C2#6190011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32656C2semidirect
5. #32574C2semidirect
9. #28265C2semidirect
12. #25472C2semidirect
13. #25063C2semidirect
14. #24332C2semidirect
133. #451Q8semidirect
259. C2×Q16C2×Q8semidirect
364. #103C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365452
C2×#25822
#329542
#326562
#325742
#299572
#292982
#292432
#282652
#279502
#276802
#254722
#250632
#243322
Q8×C22⋊C82
#220442
C4×#4512