Finite Group Databasebeta

← #6451364

Subgroups — #6451364

Order 512.

1868 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35679C211yesmaximal, characteristic
2#34254C211yesmaximal, characteristic, complemented
3#27221C214yesmaximal, complemented
4#27212C214yesmaximal, complemented
5#27200C214yesmaximal, complemented
6#27193C214yesmaximal, complemented
7#23736C211yesmaximal, characteristic, complemented
8#23632C211yesmaximal, characteristic
9#23619C211yesmaximal, characteristic, complemented
10C2×#2015C211yesmaximal, characteristic
11#21261C211yesmaximal, characteristic
12#21231C211yesmaximal, characteristic
13#21063C211yesmaximal, characteristic
14#21032C211yesmaximal, characteristic, complemented
15#21027C211yesmaximal, characteristic, complemented
16#20936C211yesmaximal, characteristic
17#20935C211yesmaximal, characteristic, complemented
18#20935C211yesmaximal, characteristic, complemented
19#20417C211yesmaximal, characteristic, complemented
20C23×Q16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2482—24no—
38#2448—22no—
39#2448—22no—
40#2448C2214yes—
41#2448C2214yes—
42#2356C2214yescomplemented
43#2355C2214yescomplemented
44#2343—24no—
45#2320—22no—
46#2317C2211yescharacteristic
47#2310C2211yescharacteristic
48#2310C2211yescharacteristic
49#2309C2211yescharacteristic
50#2309C2211yescharacteristic
51#2296C2211yescharacteristic
52#2296C2211yescharacteristic
53#2295—22no—
54C2×#391C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yescomplemented
58#2097C2212yescomplemented
59#2096C2212yescomplemented
60#2095C2212yescomplemented
61#2073C2214yes—
62#2071C2214yes—
63#2068C2214yescomplemented
64#2066C2214yescomplemented
65#2015C2214yes—
66#2015C2214yes—
67#1983C2214yes—
68#1983C2214yes—
69#1973C2212yes—
70#1973C2212yes—
71#1968C2212yes—
72#1968C2212yes—
73#1926C2214yes—
74#1925C2214yes—
75#1921C2212yescomplemented
76#1920C2212yescomplemented
77#1919C2212yescomplemented
78#1918C2212yescomplemented
79#1917C2214yescomplemented
80#1916C2214yescomplemented
81#1880C2214yes—
82#1880C2214yes—
83#1878C2214yescomplemented
84#1878C2214yescomplemented
85#1878C2214yescomplemented
86#1878C2214yescomplemented
87#1763C2214yescomplemented
88C4×#103C2214yescomplemented
89#1736C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×Q16—24no—
101C22×Q16—24no—
102C22×Q16—42no—
103C22×Q16C2314yes—
104C22×SD16—24no—
105C22×SD16—24no—
106C22×SD16—42no—
107C22×SD16C2312yes—
108C22×SD16C2312yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C2×#124—24no—
114C2×#124—21no—
115C2×#124—21no—
116C2×#124—42no—
117C2×#124C2314yes—
118#519—24no—
119#519—24no—
120#519—24no—
121#519—42no—
122C2×C4×Q8—22no—
123C2×C4×Q8—22no—
124C2×C4×Q8C2314yes—
125C2×C4×Q8C2314yes—
126C2×#117—22no—
127C2×#117—22no—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—22no—
136C22×C42C2311yescharacteristic
137#428—42no—
138#427—24no—
139C2×#107C2311yescharacteristic
140C2×#107C2311yescharacteristic
141C2×#107C2312yes—
142C2×#106C2311yescharacteristic
143C2×#106C2311yescharacteristic
144C2×#106C2312yes—
145#423—42no—
146C2×C4⋊C8C2314yes—
147#419—24no—
148#419—42no—
149#419—42no—
150#419—42no—
151#418—24no—
152#418—42no—
153C2×#103—24no—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#102—42no—
163C2×#102C2314yes—
164C2×#102C2312yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167#405—42no—
168C2×#97C2312yes—
169C2×C4×C8C2312yes—
170#399—24no—
171#399—24no—
172#397—24no—
173#397—24no—
174#391—22no—
175#391—22no—
176#391—21no—
177#391—21no—
178#391C2312yes—
179#391C2312yes—
180#391C2312yes—
181#391C2312yes—
182#391C2314yes—
183#390—22no—
184#390—22no—
185#390C2314yes—
186#390C2312yes—
187#390C2312yes—
188#389—22no—
189#389D412yes—
190#389D412yes—
191#389C2312yes—
192#389C2312yes—
193#388—24no—
194#388—24no—
195#388—24no—
196#388—24no—
197#387—24no—
198#387—24no—
199#387—24no—
200#387—24no—
201#385—22no—
202#385—22no—
203#385D414yes—
204#384C2312yes—
205#383—24no—
206#382—24no—
207#382C2312yes—
208#381—22no—
209#381—22no—
210#379—22no—
211#379—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—24no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2314yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222#376—21no—
223#376—24no—
224C2×#95—22no—
225C43—21no—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2312yes—
238#336C2312yes—
239C25—21no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8#4114yes—
253C22×Q8#4114yes—
254C22×Q8C2412yes—
255C22×Q8C2412yes—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4C2412yes—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4C2×D412yes—
268C23×C4C2411yescharacteristic
269C2×Q16—28no—
270C2×Q16—28no—
271C2×Q16—44no—
272C2×Q16—416no—
273C2×SD16—24no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—42no—
278C2×SD16—42no—
279C2×SD16—416no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8#4114yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2411yescharacteristic
296C22×C8C2411yescharacteristic
297#124—28no—
298#124—44no—
299#124—44no—
300#124—44no—
301C4×Q8—216no—
302C4×Q8—28no—
303C4×Q8—28no—
304C4×Q8—28no—
305C4×Q8—44no—
306C4×Q8—48no—
307#117—216no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—48no—
320#117#4114yes—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—48no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—24no—
354C2×C42—21no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—24no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2412yes—
367C2×C42C2412yes—
368C2×C42C2412yes—
369#107—22no—
370#107—22no—
371#107—42no—
372#106—22no—
373#106—22no—
374#106—42no—
375C4⋊C8—44no—
376#103—24no—
377#103—24no—
378#103—28no—
379#103—42no—
380#103—42no—
381#103—42no—
382#103—42no—
383#103—44no—
384#102—28no—
385#102—44no—
386#102—42no—
387#102—42no—
388#97—42no—
389C4×C8—42no—
390#95—24no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—22no—
399#95—22no—
400#95—24no—
401#95—24no—
402#95C2×D414yes—
403#95C2×D414yes—
404C24—22no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—82no—
409C24—82no—
410C24C22×D411yescharacteristic
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—28no—
416C2×Q8—28no—
417C2×Q8—28no—
418C2×Q8—28no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—42no—
428C2×Q8—44no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—44no—
435C2×Q8C2×#4114yes—
436C2×Q8C2×#4114yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—42no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—44no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—82no—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4C2511yescharacteristic
482Q16—416no—
483SD16—48no—
484SD16—48no—
485M16—44no—
486M16—44no—
487C2×C8—28no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—44no—
502C2×C8C2×#4112yes—
503C2×C8C2×#4112yes—
504C4⋊C4—216no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—42no—
524C4⋊C4—82no—
525C4⋊C4—84no—
526C4⋊C4—82no—
527C4⋊C4C2×#4114yes—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C42—216no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—48no—
551C42—44no—
552C42—42no—
553C42—42no—
554C42—42no—
555C42—84no—
556C42—82no—
557C23—22no—
558C23—22no—
559C23—44no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—88no—
565C23—84no—
566C23—164no—
567C23—162no—
568C23—162no—
569C23#52512yes—
570C23#54612yes—
571C23#55012yes—
572C23C23×D411yescharacteristic
573Q8—24no—
574Q8—24no—
575Q8—24no—
576Q8—24no—
577Q8—42no—
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—48no—
582Q8—48no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—84no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—84no—
625C2×C4—162no—
626C2×C4#52512yes—
627C2×C4#52914yes—
628C2×C4#54014yes—
629C2×C4#54612yes—
630C2×C4#55012yes—
631C2×C4#55114yes—
632C2×C4#55114yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C8—22no—
640C8—22no—
641C8—42no—
642C8—42no—
643C8—42no—
644C22—24no—
645C22—24no—
646C22—22no—
647C22—22no—
648C22—84no—
649C22—82no—
650C22—84no—
651C22—82no—
652C22—82no—
653C22—168no—
654C22#247114yes—
655C22C2×#52511yescharacteristic
656C22C2×#54611yescharacteristic
657C22C2×#55011yescharacteristic
658C22#358412yes—
659C22#358612yes—
660C4—44no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—82no—
668C4—81no—
669C4—81no—
670C4—82no—
671C4—81no—
672C4—81no—
673C4#247114yes—
674C4C2×#52911yescharacteristic
675C4C2×#54011yescharacteristic
676C4C2×#55111yescharacteristic
677C4C2×#55111yescharacteristic
678C2—24no—
679C2—162no—
680C2C2×#247111yescharacteristic
681C2#6059512yes—
682C2#6119012yes—
683C2#6141211yescharacteristic
684C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34254C2semidirect
3. #27221C2semidirect
4. #27212C2semidirect
5. #27200C2semidirect
6. #27193C2semidirect
7. #23736C2semidirect
9. #23619C2semidirect
14. #21032C2semidirect
15. #21027C2semidirect
17. #20935C2semidirect
18. #20935C2semidirect
19. #20417C2semidirect
42. #2356C22semidirect
43. #2355C22semidirect
57. #2097C22semidirect
58. #2097C22semidirect
59. #2096C22semidirect
60. #2095C22semidirect
63. #2068C22semidirect
64. #2066C22semidirect
75. #1921C22semidirect
76. #1920C22semidirect
77. #1919C22semidirect
78. #1918C22semidirect
79. #1917C22semidirect
80. #1916C22semidirect
83. #1878C22semidirect
84. #1878C22semidirect
85. #1878C22semidirect
86. #1878C22semidirect
87. #1763C22semidirect
88. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#356792
#342542
#272212
#272122
#272002
#271932
#237362
#236322
#236192
C2×#20152
#212612
#212312
#210632
#210322
#210272
#209362
#209352
#204172