Finite Group Databasebeta

← #6458879

Subgroups — #6458879

Order 512.

1802 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35307C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#27873C214yesmaximal, complemented
4#27870C214yesmaximal, complemented
5#27864C212yesmaximal, complemented
6#27861C212yesmaximal, complemented
7#27858C212yesmaximal, complemented
8#27855C212yesmaximal, complemented
9#23744C211yesmaximal, characteristic, complemented
10#23732C211yesmaximal, characteristic, complemented
11#23658C211yesmaximal, characteristic, complemented
12C2×#2072C211yesmaximal, characteristic, complemented
13#21064C212yesmaximal, complemented
14#21036C212yesmaximal, complemented
15#20974C212yesmaximal
16#20664C212yesmaximal
17#20336C211yesmaximal, characteristic
18C22×#131C2212yes—
19C2×#536C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C2×#428C2212yes—
23C2×#419C2212yes—
24C22×#103C2212yes—
25#2855C2212yes—
26#2855C2212yes—
27#2625C2212yes—
28#2625C2212yes—
29#2447—21no—
30#2447—21no—
31#2446—21no—
32#2446—21no—
33#2419—22no—
34#2419—22no—
35#2355—21no—
36#2355—21no—
37#2354—21no—
38#2354—21no—
39#2352—22no—
40#2352—22no—
41#2316C2211yescharacteristic
42#2314C2214yes—
43#2311C2212yes—
44#2311C2211yescharacteristic
45#2310C2211yescharacteristic
46#2296C2211yescharacteristic
47C2×#391C2211yescharacteristic
48#2285C2212yes—
49#2285C2212yes—
50#2285C2211yescharacteristic
51C2×#384C2211yescharacteristic
52#2274C2212yes—
53#2101C2212yes—
54#2101C2212yes—
55#2100C2211yescharacteristic
56#2100C2211yescharacteristic
57#2099C2211yescharacteristic
58#2099C2211yescharacteristic
59#2097C2214yes—
60#2096C2212yes—
61#2095C2212yes—
62#2084C2212yes—
63#2084C2212yes—
64#2083C2212yes—
65#2083C2212yes—
66#2072C2212yes—
67#2072C2212yes—
68#2072C2214yes—
69#1928C2214yes—
70#1928C2214yes—
71#1927C2214yes—
72#1927C2214yes—
73#1921C2214yes—
74#1920C2214yes—
75#1919C2214yes—
76#1918C2214yes—
77#1894C2214yes—
78#1894C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1809C2214yes—
82#1809C2214yes—
83#1808C2214yes—
84#1807C2214yes—
85#1736C2212yes—
86#1735C2212yes—
87#1735C2212yes—
88C2×#340C2212yes—
89#1729C2212yes—
90#1729C2214yes—
91#1728C2211yescharacteristic
92#1728C2212yes—
93C2×#336C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C2×#131—28no—
98C2×#131—28no—
99C2×#131—44no—
100C22×M16C2314yes—
101C22×M16C2312yes—
102C23×C8C2312yes—
103C23×C8C2314yes—
104#536—28no—
105C2×C4⋊Q8—21no—
106C2×#124—21no—
107C2×#123—22no—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C4×#41—28no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×D4D412yes—
114C2×C4×D4D412yes—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×#117—22no—
118C2×#117—22no—
119C2×#117C2311yescharacteristic
120C2×#117C2312yes—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#428—28no—
131#428—44no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136#419—28no—
137C2×#103—44no—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142#400—22no—
143#400—22no—
144#398—22no—
145#398—22no—
146#397—22no—
147#397—22no—
148#397—22no—
149#397—22no—
150#391—22no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#390—22no—
161#390—21no—
162#389—21no—
163#389—21no—
164#389C2312yes—
165#389C2312yes—
166#389C2311yescharacteristic
167#389C2311yescharacteristic
168#389C2311yescharacteristic
169#389C2311yescharacteristic
170#388—24no—
171#388—24no—
172#387—24no—
173#387—24no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#386—24no—
179#386—22no—
180#385—22no—
181#385—22no—
182#385—21no—
183#385—21no—
184#384—21no—
185#384—21no—
186#384C2312yes—
187#384C2312yes—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#382—22no—
191#382—22no—
192#382C2312yes—
193#381—22no—
194#381—22no—
195#381—22no—
196#380—21no—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380C2314yes—
202#380C2314yes—
203#380C2312yes—
204#380C2312yes—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2314yes—
215C4×C22⋊C4—24no—
216C2×#95—21no—
217C2×#95—21no—
218#340C2312yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237C2×#41—24no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41#4118yes—
241C22×Q8—21no—
242C22×Q8—24no—
243C22×Q8—24no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2412yes—
248C22×D4C2×D412yes—
249C22×D4C2×D411yescharacteristic
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259#131—432no—
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—28no—
263C2×M16—28no—
264C2×M16—44no—
265C2×M16#4118yes—
266C22×C8—28no—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8—24no—
270C22×C8—28no—
271C22×C8—44no—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2414yes—
275C4⋊Q8—44no—
276#124—44no—
277#123—48no—
278C22⋊Q8—28no—
279C22⋊Q8—28no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×D4—28no—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—24no—
286#117—28no—
287#117—24no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—22no—
292#117—22no—
293#117—84no—
294#117—84no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—28no—
343C2×C42—24no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—28no—
357#107—44no—
358#106—28no—
359#106—44no—
360#103—28no—
361#103—28no—
362#103—44no—
363#103—44no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386C24C22×D412yes—
387C24C22×D411yescharacteristic
388#41—216no—
389#41—216no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8#12314yes—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4#12318yes—
407C2×D4#12314yes—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4118yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C42−1+412yes—
454C22×C4C2511yescharacteristic
455M16—216no—
456M16—416no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—416no—
466C2×C8—48no—
467C2×C8C2×#4118yes—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—82no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—22no—
522C42—42no—
523C42—44no—
524C42—42no—
525C42—84no—
526C42—84no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23C2×#12314yes—
538C23C2×#12312yes—
539C23#53212yes—
540C23#53712yes—
541C23C23×D411yescharacteristic
542Q8—28no—
543Q8—44no—
544Q8—44no—
545Q8—84no—
546D4—216no—
547D4—28no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—162no—
601C2×C4—164no—
602C2×C4C2×#12314yes—
603C2×C4C2×#12312yes—
604C2×C4#53212yes—
605C2×C4#53712yes—
606C2×C4#55118yes—
607C2×C4#55218yes—
608C2×C4#57712yes—
609C2×C4#57812yes—
610C2×C4C22×#4112yes—
611C2×C4C22×#4112yes—
612C2×C4C2×H(5,𝔽2)11yescharacteristic
613C2×C4C2×2−1+411yescharacteristic
614C8—28no—
615C8—42no—
616C8—42no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22#249814yes—
631C22C22×#12311yescharacteristic
632C22C2×#53211yescharacteristic
633C22C2×#53711yescharacteristic
634C22C2×#57711yescharacteristic
635C22C2×#57811yescharacteristic
636C22#358511yescharacteristic
637C22#358611yescharacteristic
638C4—22no—
639C4—22no—
640C4—24no—
641C4—42no—
642C4—41no—
643C4—42no—
644C4—41no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—81no—
649C4—81no—
650C4—82no—
651C4—81no—
652C4—84no—
653C4—81no—
654C4#249814yes—
655C4C2×#55112yes—
656C4C2×#55212yes—
657C2—24no—
658C2—44no—
659C2C2×#249811yescharacteristic
660C2#6052211yescharacteristic
661C2#6052611yescharacteristic
662C2#6072311yescharacteristic
663C2#6072711yescharacteristic
664C2#6092411yescharacteristic
665C2#6092511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27873C2semidirect
4. #27870C2semidirect
5. #27864C2semidirect
6. #27861C2semidirect
7. #27858C2semidirect
8. #27855C2semidirect
9. #23744C2semidirect
10. #23732C2semidirect
11. #23658C2semidirect
12. C2×#2072C2semidirect
13. #21064C2semidirect
14. #21036C2semidirect

Maximal subgroups

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

subgroupindex
#353072
#341882
#278732
#278702
#278642
#278612
#278582
#278552
#237442
#237322
#236582
C2×#20722
#210642
#210362
#209742
#206642
#203362