Finite Group Databasebeta

← #6484439

Subgroups — #6484439

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic, complemented
2#34276C211yesmaximal, characteristic
3#32965C212yesmaximal
4#32964C212yesmaximal, complemented
5#29955C212yesmaximal
6#29951C212yesmaximal, complemented
7#29262C212yesmaximal, complemented
8#29255C212yesmaximal, complemented
9#29205C212yesmaximal, complemented
10#29199C212yesmaximal, complemented
11#29147C211yesmaximal, characteristic, complemented
12#29142C211yesmaximal, characteristic, complemented
13#28519C211yesmaximal, characteristic, complemented
14#28512C211yesmaximal, characteristic, complemented
15#28291C212yesmaximal, complemented
16#28289C212yesmaximal, complemented
17#21990C212yesmaximal, complemented
18#21814C212yesmaximal
19#20813C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×#519C2211yescharacteristic
23C2×#477C2212yes—
24C2×#475C2212yes—
25C2×#475C2212yes—
26C2×#476C2212yes—
27C2×#433C2212yes—
28#2868C2211yescharacteristic
29#2862C2211yescharacteristic
30#2857C2212yes—
31#2647C2212yes—
32#2628C2212yes—
33#2511—22no—
34#2492—22no—
35#2451—22no—
36#2423—22no—
37#2362—22no—
38#2360—22no—
39#2359—22no—
40#2359—22no—
41#2331C2212yes—
42#2327C2211yescharacteristic
43#2327C2211yescharacteristic
44#2326C2211yescharacteristic
45#2326C2211yescharacteristic
46#2317C2211yescharacteristic
47#2314C2212yes—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×#124C2211yescharacteristic
52C4×#124C2211yescharacteristic
53#2297C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C4×#117C2211yescharacteristic
56#2096C2212yes—
57#2096C2212yes—
58#2096C2212yes—
59#2093C2212yes—
60#2068C2214yes—
61#2066C2214yes—
62#2066C2214yes—
63#2065C2214yes—
64#2024C2212yes—
65#2024C2212yes—
66#2024C2212yes—
67#2023C2212yes—
68#1983C2214yes—
69#1981C2214yes—
70#1960C2212yes—
71#1959C2212yes—
72#1958C2212yes—
73#1956C2212yes—
74#1949C2214yes—
75#1948C2214yes—
76#1865C2214yes—
77#1865C2214yes—
78#1864C2214yes—
79#1863C2214yes—
80#1854C2214yes—
81#1853C2214yes—
82#1847C2212yes—
83#1844C2212yes—
84#1843C2212yes—
85#1836C2212yes—
86#1835C2212yes—
87#1831C2212yes—
88#1805C2212yes—
89#1805C2212yes—
90#1805C2212yes—
91#1804C2212yes—
92#1798C2212yes—
93#1797C2212yes—
94#1788C2214yes—
95#1787C2214yes—
96#1768C2212yes—
97#1767C2211yescharacteristic
98C4×C4⋊C8C2211yescharacteristic
99C22×#41C2311yescharacteristic
100C23×Q8C2311yescharacteristic
101#536—24no—
102#536Q818yescomplemented
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#124C2311yescharacteristic
109C2×#124C2311yescharacteristic
110C2×#123—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113#520—44no—
114#519—44no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117—21no—
130C2×#117—21no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#477—44no—
138#476—44no—
139#475—44no—
140#475—44no—
141#433—28no—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2314yes—
159C2×#97C2312yes—
160C2×C4×C8C2312yes—
161#400—22no—
162#400—24no—
163#399—22no—
164#398—24no—
165#395—22no—
166#393—22no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391—24no—
173#391C2311yescharacteristic
174#391C2311yescharacteristic
175#391C2311yescharacteristic
176#391C2311yescharacteristic
177#391C2311yescharacteristic
178#391C2311yescharacteristic
179#390C2311yescharacteristic
180#390C2311yescharacteristic
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389C2314yes—
187#389C2314yes—
188#388—24no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#387—24no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—24no—
199#386—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#384—22no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2312yes—
208#382C2312yes—
209#382C2312yes—
210#382C2311yescharacteristic
211#382C2311yescharacteristic
212#380—22no—
213#380C2314yes—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4C2314yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225#376C2311yescharacteristic
226#376C2311yescharacteristic
227C2×#95—22no—
228C43C2311yescharacteristic
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C2×#41—22no—
238C2×#41—42no—
239C2×#41—42no—
240C2×#41C2×Q814yes—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8C2×Q814yes—
248C22×Q8C2411yescharacteristic
249C22×Q8C2411yescharacteristic
250C22×Q8C2411yescharacteristic
251C22×D4—21no—
252C22×D4C2411yescharacteristic
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C22×C8C2414yes—
261C22×C8C2414yes—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—84no—
268C4⋊Q8C2×Q814yes—
269C4⋊Q8C2×Q814yes—
270#124—22no—
271#124—22no—
272#124C2×Q814yes—
273#124C2×Q814yes—
274#123—216no—
275C22⋊Q8—216no—
276C22⋊Q8—84no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—42no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—84no—
286C4×D4—24no—
287C4×D4—42no—
288C4×D4—44no—
289C4×D4—44no—
290#117—24no—
291#117—44no—
292#117—44no—
293#117—42no—
294#117—44no—
295#117C2×Q814yes—
296#117#4114yes—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2414yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4C2×D411yescharacteristic
335C2×C22⋊C4C2×D411yescharacteristic
336C2×C22⋊C4C2×D412yes—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—24no—
345C2×C42—42no—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—44no—
358#106—44no—
359#106—44no—
360#106—44no—
361C4⋊C8—28no—
362C4⋊C8—24no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371#102—44no—
372#102—44no—
373C22⋊C8—28no—
374#97—24no—
375C4×C8—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—24no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C24C22×D411yescharacteristic
397#41—48no—
398#41—48no—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—48no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—48no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—84no—
418C2×Q8C22×Q812yes—
419C2×Q8C22×Q812yes—
420C2×Q8C22×Q812yes—
421C2×D4—24no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C22×Q812yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×Q812yes—
464C22×C4C2×#4112yes—
465C22×C4C2511yescharacteristic
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4C22⋊Q818yes—
501C22⋊C4—28no—
502C22⋊C4—28no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—82no—
510C22⋊C4—84no—
511C22⋊C4C22⋊Q818yes—
512C42—24no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—84no—
528C42C22×Q812yes—
529C42C22×Q812yes—
530C42C2×#4112yes—
531C42C2×#4112yes—
532C42H(5,𝔽2)14yes—
533C23—22no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23C2×C22⋊Q812yes—
543C23C23×D411yescharacteristic
544Q8—24no—
545Q8—24no—
546Q8—24no—
547Q8—48no—
548Q8—42no—
549Q8—42no—
550Q8—42no—
551Q8—88no—
552Q8—164no—
553D4—24no—
554D4—48no—
555D4—42no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—84no—
601C2×C4—162no—
602C2×C4—162no—
603C2×C4—162no—
604C2×C4C2×C22⋊Q814yes—
605C2×C4C2×C22⋊Q814yes—
606C2×C4C2×C22⋊Q812yes—
607C2×C4#53512yes—
608C2×C4#53512yes—
609C2×C4#57714yes—
610C2×C4C23×Q811yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C2×H(5,𝔽2)12yes—
613C2×C4#58512yes—
614C8—44no—
615C8—44no—
616C22—24no—
617C22—22no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—44no—
622C22—42no—
623C22—82no—
624C22—81no—
625C22—81no—
626C22C22×C22⋊Q811yescharacteristic
627C22C2×#53511yescharacteristic
628C22#344911yescharacteristic
629C22C2×#57712yes—
630C22#358612yes—
631C4—24no—
632C4—24no—
633C4—24no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—84no—
644C4—81no—
645C4—84no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4#350112yes—
650C4#350612yes—
651C2—42no—
652C2—82no—
653C2#3517211yescharacteristic
654C2#6049712yes—
655C2#6082011yescharacteristic
656C2#6082611yescharacteristic
657C2#6097712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35283C2semidirect
4. #32964C2semidirect
6. #29951C2semidirect
7. #29262C2semidirect
8. #29255C2semidirect
9. #29205C2semidirect
10. #29199C2semidirect
11. #29147C2semidirect
12. #29142C2semidirect
13. #28519C2semidirect
14. #28512C2semidirect
15. #28291C2semidirect
16. #28289C2semidirect
17. #21990C2semidirect
19. #20813C2semidirect
102. #536Q8semidirect

Maximal subgroups

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

subgroupindex
#352832
#342762
#329652
#329642
#299552
#299512
#292622
#292552
#292052
#291992
#291472
#291422
#285192
#285122
#282912
#282892
#219902
#218142
#208132