Finite Group Databasebeta

← #7624524

Subgroups — #7624524

Order 512.

1344 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62417C211yesmaximal, characteristic
2Q16×#41C211yesmaximal, characteristic, complemented
3#33967C212yesmaximal, complemented
4#33942C212yesmaximal, complemented
5#33900C212yesmaximal, complemented
6#33881C214yesmaximal, complemented
7#33844C212yesmaximal, complemented
8#33832C214yesmaximal, complemented
9#33820C212yesmaximal, complemented
10#33815C211yesmaximal, characteristic, complemented
11#33782C212yesmaximal, complemented
12#33774C211yesmaximal, characteristic, complemented
13#33672C212yesmaximal, complemented
14#33664C211yesmaximal, characteristic, complemented
15#33581C212yesmaximal, complemented
16C2×#2207C211yesmaximal, characteristic, complemented
17#33503C211yesmaximal, characteristic, complemented
18#3508—21no—
19Q8×#41—21no—
20C2×#510C2214yescomplemented
21C22×SD32C2212yes—
22#3396—21no—
23#3396C2212yes—
24#3395C2211yescharacteristic, complemented
25Q8×Q16C2211yescharacteristic, complemented
26#3363—21no—
27#3338C2212yes—
28#3338C2211yescharacteristic, complemented
29#3299—22no—
30#3299—21no—
31#3287—21no—
32#3287C2212yes—
33D4×Q16C2212yes—
34D4×Q16C2211yescharacteristic, complemented
35#3280—22no—
36#3280C2214yes—
37#3208—21no—
38#3207—22no—
39#3170C2211yescharacteristic
40#3170C2211yescharacteristic
41#3170C2212yescomplemented
42#3169C2212yes—
43#3147C2212yes—
44C2×#493C2211yescharacteristic
45#3104—21no—
46#3104—22no—
47C8×#41C2211yescharacteristic
48C2×C4×Q16C2211yescharacteristic
49C2×C4×Q16C2212yescomplemented
50#2251C2214yescomplemented
51#2248C2212yes—
52#2243C2214yescomplemented
53#2240C2212yes—
54#2236C2212yes—
55#2232C2214yescomplemented
56#2227C2212yescomplemented
57#2227C2212yescomplemented
58#2227C2211yescharacteristic, complemented
59#2227C2211yescharacteristic, complemented
60#2224C2214yescomplemented
61#2220C2214yescomplemented
62#2220C2214yescomplemented
63#2216C2214yes—
64#2211C2214yescomplemented
65#2211C2214yescomplemented
66#2209C2212yescomplemented
67#2209C2212yescomplemented
68#2209C2211yescharacteristic, complemented
69#2209C2211yescharacteristic, complemented
70#2208C2214yescomplemented
71#2207C2212yes—
72#2207C2212yes—
73#2207C2212yes—
74#2207C2212yes—
75#2207C2212yes—
76#2207C2212yes—
77#2189C2214yescomplemented
78#2189C2214yescomplemented
79#2188C2212yes—
80#2187C2214yes—
81#2185C2214yescomplemented
82#2183C2212yescomplemented
83#2175C2214yescomplemented
84#2175C2214yescomplemented
85C4×Q32C2212yescomplemented
86C4×Q32C2212yescomplemented
87#2169C2212yescomplemented
88#2168C2214yes—
89#2163C2212yescomplemented
90#2162C2211yescharacteristic
91#2150C2212yescomplemented
92C2×C4⋊C16C2211yescharacteristic
93#2139C2214yescomplemented
94C2×#357C2212yes—
95C2×#135—22no—
96C2×#135C2314yes—
97C22×Q16C2311yescharacteristic
98C22×Q16C2312yes—
99C22×D8—21no—
100C22×D8C2312yes—
101#559—22no—
102#559—41no—
103Q82—21no—
104#557—21no—
105#557—41no—
106#554—22no—
107#554—21no—
108#554—42no—
109#554—41no—
110#550—22no—
111#550—41no—
112D4×Q8—22no—
113D4×Q8—21no—
114#546—24no—
115#546—42no—
116#540—42no—
117#535—41no—
118#533—21no—
119#533—22no—
120#533—21no—
121#533—41no—
122#533—42no—
123#532—22no—
124#532—22no—
125C2×#127—21no—
126#518—41no—
127C4×#41—41no—
128C4×#41D411yescharacteristic
129C2×C4×Q8—21no—
130C2×C4×Q8—22no—
131#510—48no—
132C2×Q32C2312yes—
133C2×Q32C2314yes—
134C2×Q32C2312yes—
135C2×SD32—44no—
136C2×SD32C2314yes—
137C2×SD32C2314yes—
138C2×SD32C2314yes—
139C2×SD32C2314yes—
140C2×M32C2314yes—
141C22×C16C2312yes—
142#500C2312yes—
143#500C2311yescharacteristic
144#500C2311yescharacteristic
145#495—22no—
146#495C2314yes—
147#494C2311yescharacteristic
148#494C2312yes—
149#494C2311yescharacteristic
150#493—21no—
151#493—21no—
152#493C2312yes—
153#493C2312yes—
154#492—22no—
155#487—21no—
156#487—22no—
157#487—21no—
158#486—22no—
159#484—22no—
160#484—22no—
161#484—22no—
162#477—22no—
163#477—21no—
164#477—21no—
165#477—22no—
166#467—21no—
167#467—21no—
168#467—22no—
169#467C2312yes—
170#467C2312yes—
171#467C2312yes—
172#467C2312yes—
173#466—22no—
174#466C2314yes—
175#463—24no—
176#463—24no—
177#462—22no—
178#462—21no—
179#462—21no—
180#462—22no—
181#460—22no—
182#460—22no—
183#451—22no—
184#451—22no—
185#451—22no—
186#450—24no—
187#449—24no—
188#449—24no—
189C8×Q8D411yescharacteristic
190C8×Q8C2311yescharacteristic
191C4×Q16—21no—
192C4×Q16—22no—
193C4×Q16—21no—
194C4×Q16—22no—
195C4×Q16—21no—
196C4×Q16—22no—
197C4×Q16—21no—
198C4×Q16C2312yes—
199C4×Q16C2311yescharacteristic
200C4×Q16C2311yescharacteristic
201C4×Q16C2312yes—
202C4×Q16C2312yes—
203C4×Q16C2312yes—
204C4×Q16C2311yescharacteristic
205C4×Q16C2311yescharacteristic
206C4×D8—22no—
207C4×D8C2314yes—
208C8×D4D412yes—
209C8×D4D411yescharacteristic
210C8×D4C2312yes—
211C8×D4C2311yescharacteristic
212#431D412yes—
213#431D411yescharacteristic
214C2×#107C2311yescharacteristic
215C2×#107C2312yes—
216C2×#103—21no—
217C2×#103—22no—
218C2×C4×C8C2311yescharacteristic
219C2×C4×C8C2312yes—
220#366C2312yes—
221#366C2312yes—
222#366C2314yes—
223C4⋊C16C2312yes—
224C4⋊C16C2312yes—
225C4⋊C16C2311yescharacteristic
226C4⋊C16C2314yes—
227C4⋊C16C2311yescharacteristic
228#358C2312yes—
229#358C2314yes—
230#358C2312yes—
231#358C2312yes—
232#358C2312yes—
233#358C2314yes—
234#357C2314yes—
235#357C2314yes—
236#357C2314yes—
237#357C2314yes—
238C22⋊C16C2312yes—
239C22⋊C16C2314yes—
240#346C2314yes—
241C4×C16C2312yes—
242C2×#41—24no—
243C2×#41—42no—
244C2×#41D811yescharacteristic
245C22×Q8—22no—
246C22×Q8—21no—
247C22×D4—22no—
248C22×D4—41no—
249C22×D4—41no—
250#135—44no—
251#135—44no—
252#135—84no—
253C2×Q16—22no—
254C2×Q16—24no—
255C2×Q16—22no—
256C2×Q16—21no—
257C2×Q16—21no—
258C2×Q16—22no—
259C2×Q16—21no—
260C2×Q16—41no—
261C2×Q16—42no—
262C2×Q16C2411yescharacteristic
263C2×Q16C2412yes—
264C2×Q16C2412yes—
265C2×Q16C2411yescharacteristic
266C2×SD16—24no—
267C2×SD16—24no—
268C2×D8—22no—
269C2×D8—22no—
270C2×D8—22no—
271C2×D8—42no—
272C2×D8—42no—
273C2×D8—82no—
274C2×D8C2414yes—
275C22×C8C2×D411yescharacteristic
276C22×C8C2×D412yes—
277C22×C8C2411yescharacteristic
278C22×C8C2412yes—
279C4⋊Q8—21no—
280C4⋊Q8—22no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—22no—
284C4⋊Q8—21no—
285C4⋊Q8—41no—
286C4⋊Q8—42no—
287C4⋊Q8—41no—
288C4⋊Q8—42no—
289#127—22no—
290#127—22no—
291#127—42no—
292#127—41no—
293#127—41no—
294#125—41no—
295#125—42no—
296#125—41no—
297#125—42no—
298#125—41no—
299#125—42no—
300#125—41no—
301#124—24no—
302#124—42no—
303#124—42no—
304#123—44no—
305#123—44no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310C22⋊Q8—41no—
311C22⋊Q8—42no—
312C22⋊Q8—41no—
313C22⋊Q8—42no—
314C22⋊Q8—42no—
315C22⋊Q8—42no—
316C22⋊Q8—42no—
317C22⋊Q8—42no—
318C22⋊Q8—42no—
319#121—24no—
320#121—42no—
321#121—42no—
322#121—42no—
323#121—44no—
324#121—44no—
325#120—44no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—21no—
331C4×Q8—22no—
332C4×Q8—21no—
333C4×Q8—22no—
334C4×Q8—41no—
335C4×Q8—41no—
336C4×Q8—42no—
337C4×Q8—41no—
338C4×Q8—42no—
339C4×Q8—41no—
340C4×Q8—42no—
341C4×Q8—41no—
342C4×Q8—41no—
343C4×Q8D811yescharacteristic
344C4×Q8C2×D411yescharacteristic
345C4×D4—24no—
346C4×D4—42no—
347C4×D4—41no—
348C4×D4—42no—
349C4×D4—41no—
350C4×D4—42no—
351C4×D4—42no—
352C4×D4—42no—
353C4×D4—42no—
354C4×D4—42no—
355C4×D4D812yes—
356C4×D4D811yescharacteristic
357C4×D4C2×D412yes—
358C4×D4C2×D411yescharacteristic
359#117—42no—
360#117—41no—
361#117—41no—
362#117D811yescharacteristic
363#117D812yes—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—42no—
367C2×C4⋊C4—41no—
368C2×C22⋊C4—44no—
369C2×C42—42no—
370C2×C42—41no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D412yes—
373Q32—24no—
374Q32—42no—
375Q32—44no—
376SD32—28no—
377SD32—44no—
378SD32—44no—
379SD32—44no—
380M32—44no—
381C2×C16—42no—
382C2×C16C2412yes—
383C2×C16C2412yes—
384C2×C16C2414yes—
385#107—22no—
386#107—21no—
387#107—21no—
388#107—22no—
389#107—21no—
390#107C2411yescharacteristic
391#107C2411yescharacteristic
392#107C2412yes—
393#107C2412yes—
394C4⋊C8C2×D411yescharacteristic
395C4⋊C8C2×D412yes—
396C4⋊C8C2×D411yescharacteristic
397C4⋊C8C2×D412yes—
398#103—21no—
399#103—22no—
400#103—21no—
401#103—22no—
402#103—21no—
403#103—21no—
404#103—22no—
405#103—22no—
406#103—22no—
407#103—22no—
408#103—22no—
409#103—22no—
410#102—24no—
411#102—24no—
412C22⋊C8C2×D412yes—
413C22⋊C8C2×D412yes—
414C22⋊C8C2×D412yes—
415C4×C8C2×D412yes—
416C4×C8C2×D411yescharacteristic
417C4×C8C2×D412yes—
418C4×C8C2×D411yescharacteristic
419C4×C8C2411yescharacteristic
420C4×C8C2412yes—
421C4×C8C2411yescharacteristic
422C24—42no—
423#41—24no—
424#41—84no—
425#41—84no—
426#41—164no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8—41no—
435C2×Q8—41no—
436C2×Q8—42no—
437C2×Q8—41no—
438C2×Q8—81no—
439C2×Q8—82no—
440C2×Q8C2×D811yescharacteristic
441C2×D4—24no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4—84no—
452C2×D4—162no—
453C2×D4C2×D812yes—
454C2×D4C2×D811yescharacteristic
455C22×C4—42no—
456C22×C4—41no—
457C22×C4—42no—
458C22×C4—41no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4C2×D812yes—
462C22×C4C2×D812yes—
463C22×C4C2×D811yescharacteristic
464C22×C4C2×D811yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467Q16—21no—
468Q16—22no—
469Q16—22no—
470Q16—21no—
471Q16—22no—
472Q16—21no—
473Q16—22no—
474Q16—42no—
475Q16—44no—
476Q16—42no—
477Q16—81no—
478Q16—82no—
479Q16H(5,𝔽2)12yes—
480SD16—84no—
481SD16—84no—
482D8—24no—
483D8—24no—
484D8—44no—
485D8—82no—
486D8—82no—
487C2×C8—21no—
488C2×C8—22no—
489C2×C8—21no—
490C2×C8—22no—
491C2×C8—41no—
492C2×C8—42no—
493C2×C8C22×D411yescharacteristic
494C2×C8C22×D412yes—
495C2×C8C22×D411yescharacteristic
496C2×C8C22×D412yes—
497C2×C8C22×D411yescharacteristic
498C2×C8C2511yescharacteristic
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—41no—
508C4⋊C4—42no—
509C4⋊C4—41no—
510C4⋊C4—42no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—42no—
515C4⋊C4—41no—
516C4⋊C4—42no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—41no—
526C4⋊C4C2×D811yescharacteristic
527C4⋊C4C2×D811yescharacteristic
528C4⋊C4C2×D812yes—
529C4⋊C4C2×D812yes—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4C2×D812yes—
539C22⋊C4C2×D812yes—
540C22⋊C4C2×D812yes—
541C42—41no—
542C42—42no—
543C42—41no—
544C42—42no—
545C42—41no—
546C42—42no—
547C42—41no—
548C42—42no—
549C42C2×D812yes—
550C42C2×D811yescharacteristic
551C42C2×D812yes—
552C42C2×D811yescharacteristic
553C42C22×D411yescharacteristic
554C42C22×D412yes—
555C42C22×D411yescharacteristic
556C16—24no—
557C16—22no—
558C16—22no—
559C23—44no—
560C23—84no—
561C23—162no—
562C23—162no—
563C23C22×D812yes—
564C23C22×D811yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—42no—
569Q8—42no—
570Q8—41no—
571Q8—42no—
572Q8—41no—
573Q8—42no—
574Q8—84no—
575Q8—82no—
576Q8—82no—
577Q8—162no—
578Q8—161no—
579D4—24no—
580D4—22no—
581D4—44no—
582D4—44no—
583D4—84no—
584D4—84no—
585D4—84no—
586D4—162no—
587D4—162no—
588D4—164no—
589D4—164no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—82no—
606C2×C4—84no—
607C2×C4—164no—
608C2×C4—164no—
609C2×C4—161no—
610C2×C4—162no—
611C2×C4#51014yescomplemented
612C2×C4C22×D812yes—
613C2×C4C22×D811yescharacteristic
614C2×C4C22×D812yes—
615C2×C4C22×D811yescharacteristic
616C2×C4C22×D812yes—
617C2×C4C22×D811yescharacteristic
618C2×C4C22×D811yescharacteristic
619C2×C4C22×D811yescharacteristic
620C2×C4C23×D411yescharacteristic
621C8—21no—
622C8—21no—
623C8—22no—
624C8—21no—
625C8—22no—
626C8C2×H(5,𝔽2)11yescharacteristic
627C8#58511yescharacteristic
628C22—21no—
629C22—21no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—324no—
637C22C2×#51012yes—
638C22C23×D811yescharacteristic
639C4—22no—
640C4—21no—
641C4—22no—
642C4—21no—
643C4—22no—
644C4—21no—
645C4—42no—
646C4—81no—
647C4—82no—
648C4—81no—
649C4—82no—
650C4—81no—
651C4—82no—
652C4—82no—
653C4C2×#51012yes—
654C4C2×#51012yes—
655C4#348811yescharacteristic
656C4#358511yescharacteristic
657C2—22no—
658C2—42no—
659C2—164no—
660C2#6194711yescharacteristic
661C2C22×#51011yescharacteristic
662C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#62417C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q16×#41C2semidirect
3. #33967C2semidirect
4. #33942C2semidirect
5. #33900C2semidirect
6. #33881C2semidirect
7. #33844C2semidirect
8. #33832C2semidirect
9. #33820C2semidirect
10. #33815C2semidirect
11. #33782C2semidirect
12. #33774C2semidirect
13. #33672C2semidirect
14. #33664C2semidirect
15. #33581C2semidirect
16. C2×#2207C2semidirect
17. #33503C2semidirect
20. C2×#510C22semidirect
24. #3395C22semidirect
25. Q8×Q16C22semidirect
28. #3338C22semidirect
34. D4×Q16C22semidirect
41. #3170C22semidirect
49. C2×C4×Q16C22semidirect
50. #2251C22semidirect
52. #2243C22semidirect
55. #2232C22semidirect
56. #2227C22semidirect
57. #2227C22semidirect
58. #2227C22semidirect
59. #2227C22semidirect
60. #2224C22semidirect
61. #2220C22semidirect
62. #2220C22semidirect
64. #2211C22semidirect
65. #2211C22semidirect
66. #2209C22semidirect
67. #2209C22semidirect
68. #2209C22semidirect
69. #2209C22semidirect
70. #2208C22semidirect
77. #2189C22semidirect
78. #2189C22semidirect
81. #2185C22semidirect
82. #2183C22semidirect
83. #2175C22semidirect
84. #2175C22semidirect
85. C4×Q32C22semidirect
86. C4×Q32C22semidirect
87. #2169C22semidirect
89. #2163C22semidirect
91. #2150C22semidirect
93. #2139C22semidirect
611. C2×C4#510semidirect

Maximal subgroups

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

subgroupindex
#624172
Q16×#412
#339672
#339422
#339002
#338812
#338442
#338322
#338202
#338152
#337822
#337742
#336722
#336642
#335812
C2×#22072
#335032