Finite Group Databasebeta

← #7510501

Subgroups — #7510501

Order 512.

1764 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35662C211yesmaximal, characteristic, complemented
2#35338C211yesmaximal, characteristic
3#32624C214yesmaximal, complemented
4#32618C214yesmaximal, complemented
5#32609C214yesmaximal, complemented
6#32603C214yesmaximal, complemented
7#23736C211yesmaximal, characteristic, complemented
8#23632C211yesmaximal, characteristic
9#23614C211yesmaximal, characteristic
10#23542C211yesmaximal, characteristic, complemented
11#23506C211yesmaximal, characteristic
12C2×#2015C211yesmaximal, characteristic
13#21304C211yesmaximal, characteristic, complemented
14#21231C211yesmaximal, characteristic
15#21088C211yesmaximal, characteristic, complemented
16#21061C211yesmaximal, characteristic, complemented
17#21032C211yesmaximal, characteristic, complemented
18#20936C211yesmaximal, characteristic
19#20935C211yesmaximal, characteristic, complemented
20C23×Q16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2861—22no—
38#2640—22no—
39#2592C2214yes—
40#2592C2214yes—
41#2582C2214yescomplemented
42#2581C2214yescomplemented
43#2561—24no—
44#2453—22no—
45#2451—22no—
46#2438—24no—
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2310C2211yescharacteristic
50#2310C2211yescharacteristic
51#2310C2211yescharacteristic
52#2309C2211yescharacteristic
53#2309C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#391C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yescomplemented
58#2097C2212yescomplemented
59#2096C2212yescomplemented
60#2095C2212yescomplemented
61#2073C2214yes—
62#2071C2214yes—
63#2067C2214yes—
64#2065C2214yes—
65#2056C2214yescomplemented
66#2056C2214yescomplemented
67#2042C2214yes—
68#2042C2214yes—
69#2015C2214yes—
70#2015C2214yes—
71#1985C2214yescomplemented
72#1984C2214yescomplemented
73#1973C2212yes—
74#1973C2212yes—
75#1968C2212yes—
76#1968C2212yes—
77#1935C2214yescomplemented
78#1933C2214yescomplemented
79#1928C2214yescomplemented
80#1927C2214yescomplemented
81#1921C2212yescomplemented
82#1920C2212yescomplemented
83#1919C2212yescomplemented
84#1918C2212yescomplemented
85#1880C2214yes—
86#1880C2214yes—
87#1878C2214yescomplemented
88#1878C2214yescomplemented
89#1736C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×Q16—24no—
101C22×Q16—42no—
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
113#533—42no—
114#533—42no—
115C2×C4⋊Q8D412yes—
116C2×#125D412yes—
117C2×#124—21no—
118C2×#124—21no—
119C2×#124—42no—
120C2×#124—42no—
121C2×#124C2314yes—
122C2×C22⋊Q8D414yes—
123#519—42no—
124#519—42no—
125C2×C4×Q8—22no—
126C2×C4×Q8C2314yes—
127C2×C4×Q8C2314yes—
128C2×#117—22no—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—22no—
137C22×C42C2311yescharacteristic
138#428—42no—
139#427—24no—
140C2×#107C2311yescharacteristic
141C2×#107C2311yescharacteristic
142C2×#107C2312yes—
143C2×#106C2311yescharacteristic
144C2×#106C2311yescharacteristic
145C2×#106C2312yes—
146#424—42no—
147C2×C4⋊C8C2314yes—
148#419—24no—
149#419—42no—
150#419—42no—
151#419—42no—
152#418—42no—
153#418—42no—
154C2×#103—24no—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#102—42no—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168#405—42no—
169C2×#97C2312yes—
170C2×C4×C8C2312yes—
171#400—24no—
172#398—24no—
173#398—24no—
174#397—24no—
175#397—24no—
176#395—24no—
177#393—24no—
178#393—24no—
179#391—24no—
180#391—22no—
181#391—22no—
182#391—22no—
183#391—21no—
184#391—21no—
185#391C2312yes—
186#391C2312yes—
187#391C2314yes—
188#391C2312yes—
189#391C2312yes—
190#391C2314yes—
191#390—22no—
192#390—22no—
193#390C2312yes—
194#390C2312yes—
195#390C2314yes—
196#389—22no—
197#389—22no—
198#389—24no—
199#389C2314yes—
200#389C2312yes—
201#389C2312yes—
202#388—24no—
203#388—24no—
204#388—24no—
205#388—24no—
206#387—24no—
207#387—24no—
208#385—22no—
209#385—22no—
210#384C2312yes—
211#382C2312yes—
212#381—22no—
213#381—22no—
214#381—22no—
215#380—22no—
216#379—22no—
217#379—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C22⋊C4—22no—
223#376—21no—
224C2×#95—22no—
225C43—21no—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
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—22no—
253C22×Q8C2×D414yes—
254C22×Q8#4114yes—
255C22×Q8C2412yes—
256C22×Q8C2412yes—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4C2412yes—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4C2×D412yes—
269C23×C4C2411yescharacteristic
270C2×Q16—28no—
271C2×Q16—44no—
272C2×Q16—44no—
273C2×Q16—416no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—42no—
279C2×SD16—42no—
280C2×SD16—416no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8#4114yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2411yescharacteristic
297C22×C8C2411yescharacteristic
298C4⋊Q8—44no—
299#125—44no—
300#124—44no—
301#124—44no—
302#124—44no—
303#124—44no—
304C22⋊Q8—48no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8—48no—
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—22no—
320#117—22no—
321#117—22no—
322#117—48no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2×D414yes—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—48no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—24no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2412yes—
369C2×C42C2412yes—
370#107—22no—
371#107—22no—
372#107—42no—
373#106—22no—
374#106—22no—
375#106—42no—
376C4⋊C8—44no—
377#103—28no—
378#103—24no—
379#103—24no—
380#103—42no—
381#103—42no—
382#103—42no—
383#103—42no—
384#103—44no—
385#102—44no—
386#102—42no—
387#102—42no—
388#102—44no—
389#97—42no—
390C4×C8—42no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405C24—22no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—82no—
410C24—82no—
411C24C22×D411yescharacteristic
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—28no—
415C2×Q8—28no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—44no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8C2×#4114yes—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—44no—
453C2×D4—44no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—82no—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D414yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C42−1+412yes—
485C22×C4C2511yescharacteristic
486Q16—416no—
487SD16—48no—
488SD16—48no—
489M16—44no—
490M16—44no—
491C2×C8—28no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—44no—
506C2×C8C2×#4112yes—
507C2×C8C2×#4112yes—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—48no—
529C4⋊C4—42no—
530C4⋊C4—82no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—48no—
554C42—42no—
555C42—42no—
556C42—42no—
557C42—42no—
558C42—82no—
559C42—82no—
560C23—22no—
561C23—22no—
562C23—44no—
563C23—44no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—88no—
568C23—84no—
569C23—164no—
570C23—162no—
571C23—162no—
572C23#53412yes—
573C23#54812yes—
574C23#55012yes—
575C23C23×D411yescharacteristic
576Q8—24no—
577Q8—24no—
578Q8—42no—
579Q8—48no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—88no—
586Q8—84no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—84no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
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—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—162no—
630C2×C4#53412yes—
631C2×C4#53814yes—
632C2×C4#54214yes—
633C2×C4#54812yes—
634C2×C4#55012yes—
635C2×C4#55114yes—
636C2×C4#56214yes—
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×2−1+411yescharacteristic
643C8—22no—
644C8—22no—
645C8—42no—
646C8—42no—
647C8—42no—
648C22—24no—
649C22—24no—
650C22—22no—
651C22—22no—
652C22—84no—
653C22—84no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—168no—
658C22#273014yes—
659C22C2×#53411yescharacteristic
660C22C2×#54811yescharacteristic
661C22C2×#55011yescharacteristic
662C22#358412yes—
663C22#358612yes—
664C4—42no—
665C4—42no—
666C4—44no—
667C4—44no—
668C4—44no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—81no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4—81no—
677C4#273014yes—
678C4C2×#53811yescharacteristic
679C4C2×#54211yescharacteristic
680C4C2×#55111yescharacteristic
681C4C2×#56211yescharacteristic
682C2—24no—
683C2—162no—
684C2C2×#273011yescharacteristic
685C2#6076112yes—
686C2#6134412yes—
687C2#6141211yescharacteristic
688C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35662C2semidirect
3. #32624C2semidirect
4. #32618C2semidirect
5. #32609C2semidirect
6. #32603C2semidirect
7. #23736C2semidirect
10. #23542C2semidirect
13. #21304C2semidirect
15. #21088C2semidirect
16. #21061C2semidirect
17. #21032C2semidirect
19. #20935C2semidirect
41. #2582C22semidirect
42. #2581C22semidirect
57. #2097C22semidirect
58. #2097C22semidirect
59. #2096C22semidirect
60. #2095C22semidirect
65. #2056C22semidirect
66. #2056C22semidirect
71. #1985C22semidirect
72. #1984C22semidirect
77. #1935C22semidirect
78. #1933C22semidirect
79. #1928C22semidirect
80. #1927C22semidirect
81. #1921C22semidirect
82. #1920C22semidirect
83. #1919C22semidirect
84. #1918C22semidirect
87. #1878C22semidirect
88. #1878C22semidirect

Maximal subgroups

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

subgroupindex
#356622
#353382
#326242
#326182
#326092
#326032
#237362
#236322
#236142
#235422
#235062
C2×#20152
#213042
#212312
#210882
#210612
#210322
#209362
#209352