Finite Group Databasebeta

← #6451259

Subgroups — #6451259

Order 512.

1781 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35557C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic, complemented
3#27514C214yesmaximal, complemented
4#27509C214yesmaximal, complemented
5#27497C214yesmaximal, complemented
6#27486C214yesmaximal, complemented
7#23737C211yesmaximal, characteristic, complemented
8C2×#2093C211yesmaximal, characteristic, complemented
9#23660C211yesmaximal, characteristic, complemented
10#23546C211yesmaximal, characteristic
11#21235C211yesmaximal, characteristic
12#21171C211yesmaximal, characteristic, complemented
13#21067C211yesmaximal, characteristic, complemented
14#21063C211yesmaximal, characteristic, complemented
15#21058C211yesmaximal, characteristic, complemented
16#21027C211yesmaximal, characteristic, complemented
17#20929C211yesmaximal, characteristic
18#20879C211yesmaximal, characteristic, complemented
19#20455C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C2×#518C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C2×#405C2211yescharacteristic
35#2512C2214yescomplemented
36#2511C2214yescomplemented
37#2497—22no—
38#2496—22no—
39#2481—24no—
40#2357C2214yes—
41#2354C2214yes—
42#2342—24no—
43#2319—22no—
44#2317C2211yescharacteristic
45#2316C2211yescharacteristic
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48#2309C2211yescharacteristic
49#2297C2211yescharacteristic
50#2296C2211yescharacteristic
51#2293—22no—
52C2×#390C2211yescharacteristic
53C2×#384C2211yescharacteristic
54#2274C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yescomplemented
58#2095C2212yescomplemented
59#2093C2214yes—
60#2093C2214yes—
61#2093C2212yescomplemented
62#2092C2212yescomplemented
63#2085C2214yescomplemented
64#2083C2214yescomplemented
65#2055C2214yes—
66#2054C2214yes—
67#1973C2212yes—
68#1972C2212yes—
69#1970C2212yes—
70#1968C2212yes—
71#1948C2214yescomplemented
72#1947C2214yescomplemented
73#1928C2212yescomplemented
74#1928C2214yescomplemented
75#1927C2212yescomplemented
76#1926C2214yes—
77#1926C2212yescomplemented
78#1925C2214yes—
79#1925C2212yescomplemented
80#1925C2214yescomplemented
81#1917C2214yescomplemented
82#1916C2214yescomplemented
83#1879C2214yes—
84#1877C2214yes—
85#1864C2214yes—
86#1862C2214yes—
87#1776C2214yescomplemented
88C4×#107C2214yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C2×AGL(1,ℤ/8)—24no—
100C2×AGL(1,ℤ/8)—24no—
101C2×AGL(1,ℤ/8)—42no—
102C22×SD16C2312yes—
103C22×D8C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C2×#126—24no—
109#518—24no—
110#518—24no—
111C2×C4×D4—22no—
112C2×C4×D4C2314yes—
113C2×#117—24no—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—22no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—22no—
124C22×C22⋊C4—22no—
125C22×C42C2311yescharacteristic
126#428—42no—
127#428—42no—
128#427—24no—
129C2×#107—24no—
130C2×#107—42no—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×#107C2312yes—
134C2×#107C2312yes—
135C2×#107C2314yes—
136C2×#107C2312yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140#423—24no—
141C2×C4⋊C8C2314yes—
142#418—24no—
143#417—42no—
144#416—24no—
145#416—24no—
146C2×#103C2311yescharacteristic
147C2×#103C2311yescharacteristic
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#102—24no—
151C2×#102—42no—
152C2×#102C2314yes—
153C2×#102C2311yescharacteristic
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2311yescharacteristic
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160#405—24no—
161C2×#97C2312yes—
162C2×C4×C8C2312yes—
163#400—24no—
164#400—24no—
165#397—24no—
166#396—24no—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#390—22no—
171#390—21no—
172#390—21no—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#390C2314yes—
177#390C2312yes—
178#390C2312yes—
179#389—22no—
180#389—22no—
181#389D412yes—
182#389D412yes—
183#389C2314yes—
184#388—24no—
185#388—24no—
186#388—24no—
187#388—24no—
188#388—24no—
189#387—24no—
190#387—24no—
191#386—22no—
192#386—22no—
193#386—22no—
194#385—22no—
195#385D414yes—
196#384—21no—
197#384—21no—
198#384C2312yes—
199#384C2314yes—
200#382C2312yes—
201#381—24no—
202#380—22no—
203#380—22no—
204#380—24no—
205#380—22no—
206#379—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2314yes—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—24no—
220#376—21no—
221C43—21no—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#336C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2314yes—
233C25—21no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C22×Q8C2411yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4#4114yes—
249C22×D4C2411yescharacteristic
250C22×D4C2412yes—
251C23×C4—24no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258AGL(1,ℤ/8)—416no—
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—42no—
262C2×D8—24no—
263C2×D8—24no—
264C2×D8—42no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16#4114yes—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8#4114yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2411yescharacteristic
280C22×C8C2411yescharacteristic
281C22×C8C2412yes—
282#126—88no—
283C4×D4—216no—
284C4×D4—28no—
285C4×D4—28no—
286#117—22no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—88no—
296#117#4114yes—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—84no—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—216no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—84no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352#107—24no—
353#107—24no—
354#107—24no—
355#107—44no—
356#107—42no—
357#107—42no—
358#106—24no—
359#106—42no—
360#106—42no—
361C4⋊C8—28no—
362#103—22no—
363#103—22no—
364#103—24no—
365#103—42no—
366#102—28no—
367#102—24no—
368#102—24no—
369#102—22no—
370#102—22no—
371#102—24no—
372#102—44no—
373#102—42no—
374#97—24no—
375C4×C8—24no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95C2×D414yes—
387#95C2×D414yes—
388C24—22no—
389C24—24no—
390C24—21no—
391C24—42no—
392C24—42no—
393C24—82no—
394C24—82no—
395C24C22×D411yescharacteristic
396#41—44no—
397#41—44no—
398#41—44no—
399#41—44no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—28no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C2×#4114yes—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—216no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C42−1+412yes—
462C22×C4C2511yescharacteristic
463SD16—48no—
464D8—48no—
465M16—28no—
466M16—44no—
467C2×C8—28no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—44no—
481C2×C8C2×#4114yes—
482C2×C8C2×#4112yes—
483C2×C8C2×#4112yes—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—28no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—48no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4C2×#4114yes—
509C22⋊C4—216no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—48no—
517C22⋊C4—44no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42—82no—
536C42—84no—
537C42—84no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—48no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—88no—
552C23—164no—
553C23—162no—
554C23—162no—
555C23#52512yes—
556C23#54712yes—
557C23#55012yes—
558C23C23×D411yescharacteristic
559Q8—41no—
560Q8—41no—
561Q8—41no—
562Q8—41no—
563D4—24no—
564D4—24no—
565D4—42no—
566D4—41no—
567D4—42no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—48no—
572D4—84no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—816no—
605C2×C4—84no—
606C2×C4—162no—
607C2×C4—162no—
608C2×C4—162no—
609C2×C4#52512yes—
610C2×C4#52914yes—
611C2×C4#54214yes—
612C2×C4#54712yes—
613C2×C4#55012yes—
614C2×C4#55114yes—
615C2×C4#55214yes—
616C2×C4#57614yes—
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C8—22no—
624C8—22no—
625C8—24no—
626C8—42no—
627C8—42no—
628C22—24no—
629C22—24no—
630C22—22no—
631C22—22no—
632C22—44no—
633C22—48no—
634C22—84no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—81no—
639C22—82no—
640C22—81no—
641C22—81no—
642C22—168no—
643C22#247914yes—
644C22C2×#52511yescharacteristic
645C22C2×#54711yescharacteristic
646C22C2×#55011yescharacteristic
647C22C2×#57612yes—
648C22#358612yes—
649C4—24no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—81no—
655C4—82no—
656C4—81no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—81no—
661C4#247914yes—
662C4C2×#52911yescharacteristic
663C4C2×#54211yescharacteristic
664C4C2×#55111yescharacteristic
665C4C2×#55211yescharacteristic
666C2—24no—
667C2—84no—
668C2—161no—
669C2C2×#247911yescharacteristic
670C2#6059612yes—
671C2#6127012yes—
672C2#6140911yescharacteristic
673C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
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
1. #35557C2semidirect
2. #34255C2semidirect
3. #27514C2semidirect
4. #27509C2semidirect
5. #27497C2semidirect
6. #27486C2semidirect
7. #23737C2semidirect
8. C2×#2093C2semidirect
9. #23660C2semidirect
12. #21171C2semidirect
13. #21067C2semidirect
14. #21063C2semidirect
15. #21058C2semidirect
16. #21027C2semidirect
18. #20879C2semidirect
19. #20455C2semidirect
35. #2512C22semidirect
36. #2511C22semidirect
57. #2097C22semidirect
58. #2095C22semidirect
61. #2093C22semidirect
62. #2092C22semidirect
63. #2085C22semidirect
64. #2083C22semidirect
71. #1948C22semidirect
72. #1947C22semidirect
73. #1928C22semidirect
74. #1928C22semidirect
75. #1927C22semidirect
77. #1926C22semidirect
79. #1925C22semidirect
80. #1925C22semidirect
81. #1917C22semidirect
82. #1916C22semidirect
87. #1776C22semidirect
88. C4×#107C22semidirect

Maximal subgroups

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

subgroupindex
#355572
#342552
#275142
#275092
#274972
#274862
#237372
C2×#20932
#236602
#235462
#212352
#211712
#210672
#210632
#210582
#210272
#209292
#208792
#204552