Finite Group Databasebeta

← #6793190

Subgroups — #6793190

Order 512.

1573 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#34945C211yesmaximal, characteristic
3#33163C211yesmaximal, characteristic, complemented
4#31501C214yesmaximal, complemented
5#30604C214yesmaximal, complemented
6#29926C212yesmaximal, complemented
7#29286C211yesmaximal, characteristic, complemented
8#28748C211yesmaximal, characteristic, complemented
9#28600C212yesmaximal
10#26531C212yesmaximal, complemented
11#25386C212yesmaximal
12#25323C212yesmaximal, complemented
13#24752C212yesmaximal, complemented
14#24157C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic
16#21889C212yesmaximal, complemented
17C4×#475C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#487C2212yes—
20C2×#475C2211yescharacteristic
21C2×#467C2214yes—
22C2×#462C2212yes—
23C2×#462C2212yes—
24C2×#462C2212yes—
25C2×C8×D4C2212yes—
26C22×C4⋊C8C2212yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2573—22no—
30#2571C2211yescharacteristic
31#2568C2211yescharacteristic
32#2552—21no—
33#2547C2211yescharacteristic
34#2542C2212yes—
35#2538—21no—
36#2538C2211yescharacteristic
37#2514—21no—
38#2494C2212yes—
39#2447—21no—
40#2441—22no—
41#2429C2214yes—
42#2423—22no—
43#2421C2212yes—
44#2401—22no—
45#2365C2212yes—
46#2354—21no—
47Q8×C4⋊C4—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×#124—21no—
51C4×C22⋊Q8C2212yes—
52#2295—21no—
53#2283C2212yes—
54C42×Q8C2211yescharacteristic
55C42×D4C2211yescharacteristic
56#2097C2214yes—
57#2086C2214yes—
58#2074C2214yes—
59#2068C2214yes—
60#2056C2214yes—
61#2043C2214yes—
62#2036C2214yes—
63#2035C2214yes—
64#2035C2214yes—
65#2001C2214yes—
66#1988C2212yes—
67#1988C2212yes—
68#1988C2211yescharacteristic
69#1981C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1956C2212yes—
75#1948C2211yescharacteristic
76#1947C2212yes—
77#1947C2212yes—
78#1939C2212yes—
79#1919C2212yes—
80#1892C2214yes—
81#1892C2214yes—
82#1892C2214yes—
83#1878C2214yes—
84#1865C2214yes—
85#1865C2212yes—
86#1850C2214yes—
87#1842C2211yescharacteristic
88#1805C2214yes—
89#1805C2214yes—
90#1803C2211yescharacteristic
91#1797C2212yes—
92C4×#106C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2314yes—
96C23×C8C2314yes—
97#533—28no—
98#533—44no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—22no—
105C2×#124—21no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2314yes—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2314yes—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C22×C42C2312yes—
122#487—44no—
123#475—24no—
124#467—48no—
125#462—44no—
126#462—44no—
127#462—44no—
128C8×D4—44no—
129C2×#107C2314yes—
130C2×#106C2312yes—
131C2×#106C2312yes—
132C2×C4⋊C8—44no—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2312yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C4×C8C2312yes—
150#401—22no—
151#400—22no—
152#400—22no—
153#400—22no—
154#399—24no—
155#398—24no—
156#397—22no—
157#397—22no—
158#397—22no—
159#396—22no—
160#396—24no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391C2312yes—
167#391C2314yes—
168#390—22no—
169#390—21no—
170#389—21no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—24no—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—24no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—24no—
188#385—22no—
189#384—22no—
190#384—21no—
191#384—21no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#383—22no—
197#382—21no—
198#382C2312yes—
199#382C2312yes—
200#381—22no—
201#380—22no—
202#380—22no—
203#380C2312yes—
204#380C2312yes—
205#380C2312yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226#376—22no—
227C43—21no—
228C43C2311yescharacteristic
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8C2414yes—
241C22×Q8C2411yescharacteristic
242C22×D4C2×D411yescharacteristic
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C2×Q16—28no—
247C2×Q16—44no—
248C2×Q16—44no—
249C22×C8—44no—
250C22×C8—44no—
251C22×C8—44no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—42no—
263C4⋊Q8—42no—
264C4⋊Q8#4114yes—
265#125—48no—
266#124—44no—
267C22⋊Q8—24no—
268C22⋊Q8—28no—
269C22⋊Q8—42no—
270C22⋊Q8—44no—
271C22⋊Q8—48no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C4×Q8—22no—
275C4×Q8—28no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—41no—
281C4×Q8—44no—
282C4×Q8—48no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—22no—
289#117—48no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—44no—
353#106—24no—
354#106—42no—
355C4⋊C8—22no—
356C4⋊C8—22no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C4⋊C8—44no—
360C4⋊C8—42no—
361#103—24no—
362#103—24no—
363#103—44no—
364#103—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370C22⋊C8—44no—
371C4×C8—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24C22×D412yes—
395C2×Q8—24no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—24no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—21no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8H(5,𝔽2)12yes—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4#13714yes—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—42no—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2511yescharacteristic
447Q16—88no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—88no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—81no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4C2×#4114yes—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—82no—
505C22⋊C4—84no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—21no—
513C42—21no—
514C42—41no—
515C42—48no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—44no—
520C42—44no—
521C42—42no—
522C42—42no—
523C42—81no—
524C42#13514yes—
525C42C2×#4112yes—
526C42H(5,𝔽2)12yes—
527C42H(5,𝔽2)12yes—
528C422−1+412yes—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23C2×#13714yes—
536C23C23×D411yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—48no—
542Q8—44no—
543Q8—44no—
544Q8—44no—
545Q8—44no—
546Q8—84no—
547D4—28no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—48no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4#53712yes—
600C2×C4#53712yes—
601C2×C4#54612yes—
602C2×C4#54612yes—
603C2×C4C2×#13512yes—
604C2×C4C2×#13512yes—
605C2×C4C2×#13512yes—
606C2×C4C2×#13712yes—
607C2×C4#57812yes—
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C2×C4#58511yescharacteristic
614C8—42no—
615C8—42no—
616C8—44no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22C2×#53711yescharacteristic
625C22C2×#54611yescharacteristic
626C22#348811yescharacteristic
627C22C22×#13511yescharacteristic
628C22C22×#13711yescharacteristic
629C22C2×#57811yescharacteristic
630C22#358611yescharacteristic
631C4—22no—
632C4—22no—
633C4—22no—
634C4—41no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—82no—
641C4—82no—
642C4—81no—
643C4—82no—
644C4—81no—
645C4—84no—
646C4—81no—
647C4#324814yes—
648C4#330214yes—
649C4#351711yescharacteristic
650C4#352211yescharacteristic
651C4#354311yescharacteristic
652C4#355411yescharacteristic
653C2—44no—
654C2#3890911yescharacteristic
655C2C2×#324811yescharacteristic
656C2#6092511yescharacteristic
657C2C2×#330211yescharacteristic
658C2#6115711yescharacteristic
659C2#6195411yescharacteristic
660C2#6198411yescharacteristic
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. #33163C2semidirect
4. #31501C2semidirect
5. #30604C2semidirect
6. #29926C2semidirect
7. #29286C2semidirect
8. #28748C2semidirect
10. #26531C2semidirect
12. #25323C2semidirect
13. #24752C2semidirect
14. #24157C2semidirect
16. #21889C2semidirect
17. C4×#475C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#349452
#331632
#315012
#306042
#299262
#292862
#287482
#286002
#265312
#253862
#253232
#247522
#241572
D4×C4⋊C82
#218892
C4×#4752