Finite Group Databasebeta

← #6848665

Subgroups — #6848665

Order 512.

1621 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic
2#34929C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#32198C214yesmaximal, complemented
5#31488C214yesmaximal, complemented
6#29688C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#28600C212yesmaximal
9#28283C211yesmaximal, characteristic, complemented
10#27628C212yesmaximal, complemented
11#26444C212yesmaximal, complemented
12#25341C212yesmaximal
13#25276C212yesmaximal, complemented
14#24838C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic
16#21885C212yesmaximal, complemented
17C4×#475C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#533C2212yes—
20C2×#487C2212yes—
21C2×#475C2211yescharacteristic
22C2×#475C2212yes—
23C2×#475C2212yes—
24C2×C8×D4C2212yes—
25C22×C4⋊C8C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2568—21no—
29#2556—22no—
30#2551C2211yescharacteristic
31#2547—21no—
32#2547C2212yes—
33#2546C2211yescharacteristic
34#2542C2211yescharacteristic
35#2538C2211yescharacteristic
36#2506—21no—
37#2499—22no—
38#2496—22no—
39#2496C2214yes—
40#2492C2212yes—
41#2452—21no—
42#2429C2212yes—
43#2396—22no—
44#2365—21no—
45#2365C2212yes—
46C4×C4⋊Q8—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C22⋊Q8C2212yes—
50#2293—21no—
51#2285C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2088C2214yes—
55#2077C2214yes—
56#2072C2214yes—
57#2066C2214yes—
58#2066C2214yes—
59#2058C2214yes—
60#2043C2214yes—
61#2035C2214yes—
62#2035C2214yes—
63#2035C2214yes—
64#1990C2212yes—
65#1988C2211yescharacteristic
66#1988C2212yes—
67#1987C2214yes—
68#1981C2214yes—
69#1981C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2211yescharacteristic
73#1948C2212yes—
74#1939C2212yes—
75#1927C2214yes—
76#1920C2214yes—
77#1919C2212yes—
78#1892C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1865C2212yes—
82#1850C2214yes—
83#1842C2211yescharacteristic
84#1807C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91#1797C2212yes—
92C4×#106C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C23×C8C2314yes—
96#556—48no—
97#533—28no—
98#533—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#126—24no—
108C2×#125—22no—
109C2×#124—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C2×#117—22no—
127C22×C42C2312yes—
128#487—44no—
129#475—24no—
130#475—44no—
131#475—44no—
132C8×D4—44no—
133C2×#106C2312yes—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8—44no—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8C2312yes—
153#400—24no—
154#400—22no—
155#399—22no—
156#398—22no—
157#398—22no—
158#397—24no—
159#397—24no—
160#396—22no—
161#396—22no—
162#395—22no—
163#393—22no—
164#391—22no—
165#391—22no—
166#391—21no—
167#391—22no—
168#391—22no—
169#391C2312yes—
170#391C2312yes—
171#390—21no—
172#390—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389C2312yes—
183#389C2314yes—
184#389C2312yes—
185#388—22no—
186#388—22no—
187#387—22no—
188#387—24no—
189#387—24no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#382—22no—
199#380—22no—
200#380—22no—
201#380C2312yes—
202#380C2314yes—
203#380C2314yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C43—21no—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8C2412yes—
241C22×Q8C2411yescharacteristic
242C22×D4C2×D411yescharacteristic
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C22×C8—44no—
247C22×C8—44no—
248C22×C8—44no—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265#126—416no—
266#125—48no—
267#124—48no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—42no—
272C22⋊Q8—42no—
273C22⋊Q8—42no—
274C22⋊Q8—48no—
275C22⋊Q8—48no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—48no—
283C4×Q8—41no—
284C4×Q8—48no—
285C4×Q8—44no—
286C4×Q8#4114yes—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—42no—
290C4×D4SD1618yes—
291#117—48no—
292#117—48no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#106—24no—
359#106—24no—
360#106—44no—
361#106—44no—
362C4⋊C8—22no—
363C4⋊C8—22no—
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367C4⋊C8—42no—
368#103—24no—
369#103—44no—
370#103—42no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375C22⋊C8—44no—
376C4×C8—24no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—22no—
390#95—24no—
391#95—24no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24C22×D412yes—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8—82no—
416C2×Q8C2×#4112yes—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4C2×SD1614yes—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4C2×SD1618yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2511yescharacteristic
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—88no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4C2×SD1614yes—
501C4⋊C4C2×#4112yes—
502C4⋊C42−1+412yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—82no—
511C22⋊C4C2×SD1618yes—
512C42—24no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—21no—
520C42—21no—
521C42—41no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—48no—
526C42—44no—
527C42—42no—
528C42—42no—
529C42—81no—
530C42C2×SD1614yes—
531C42C2×#4112yes—
532C42H(5,𝔽2)12yes—
533C42H(5,𝔽2)12yes—
534C422−1+412yes—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23C22×SD1614yes—
542C23C23×D411yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—41no—
546Q8—41no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—42no—
551Q8—88no—
552Q8—88no—
553Q8—84no—
554D4—28no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—216no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—84no—
605C2×C4#53712yes—
606C2×C4#53712yes—
607C2×C4#54612yes—
608C2×C4#54612yes—
609C2×C4C22×SD1614yes—
610C2×C4C22×SD1612yes—
611C2×C4C22×SD1612yes—
612C2×C4C22×SD1612yes—
613C2×C4#57512yes—
614C2×C4#57812yes—
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C2×C4#58511yescharacteristic
621C8—42no—
622C8—44no—
623C8—42no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22C2×#53711yescharacteristic
632C22C2×#54611yescharacteristic
633C22#348911yescharacteristic
634C22C23×SD1611yescharacteristic
635C22C2×#57511yescharacteristic
636C22C2×#57811yescharacteristic
637C22#358611yescharacteristic
638C4—22no—
639C4—24no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—84no—
652C4—82no—
653C4—81no—
654C4#324614yes—
655C4#329814yes—
656C4#350711yescharacteristic
657C4#351111yescharacteristic
658C4#354611yescharacteristic
659C4#355011yescharacteristic
660C2—44no—
661C2#3946811yescharacteristic
662C2C2×#324611yescharacteristic
663C2#6092511yescharacteristic
664C2C2×#329811yescharacteristic
665C2#6116511yescharacteristic
666C2#6209911yescharacteristic
667C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33196C2semidirect
4. #32198C2semidirect
5. #31488C2semidirect
6. #29688C2semidirect
7. #28717C2semidirect
9. #28283C2semidirect
10. #27628C2semidirect
11. #26444C2semidirect
13. #25276C2semidirect
14. #24838C2semidirect
16. #21885C2semidirect
17. C4×#475C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#349292
#331962
#321982
#314882
#296882
#287172
#286002
#282832
#276282
#264442
#253412
#252762
#248382
D4×C4⋊C82
#218852
C4×#4752