Finite Group Databasebeta

← #7624137

Subgroups — #7624137

Order 512.

1534 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62249C211yesmaximal, characteristic, complemented
2C2×#3285C211yesmaximal, characteristic, complemented
3#33888C214yesmaximal, complemented
4#33880C214yesmaximal, complemented
5#33837C214yesmaximal, complemented
6#33835C214yesmaximal, complemented
7#33781C211yesmaximal, characteristic, complemented
8#33779C211yesmaximal, characteristic, complemented
9#33760C212yesmaximal, complemented
10#33756C212yesmaximal, complemented
11#33673C211yesmaximal, characteristic, complemented
12#33672C211yesmaximal, characteristic, complemented
13#33518C212yesmaximal, complemented
14#33516C212yesmaximal, complemented
15#33487C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C23×Q16C2211yescharacteristic
18#3506—21no—
19C2×#548—21no—
20C2×#510C2214yescomplemented
21C2×#508C2214yescomplemented
22#3342C2212yescomplemented
23#3341C2212yescomplemented
24#3304—22no—
25#3303—22no—
26#3287C2212yescomplemented
27#3285—22no—
28#3285—22no—
29#3285C2214yes—
30#3285C2214yes—
31#3283C2212yescomplemented
32#3246—21no—
33#3191—21no—
34#3191—21no—
35#3170C2211yescharacteristic
36#3169C2211yescharacteristic
37C2×#495C2211yescharacteristic
38#3050C2212yes—
39#3050C2212yes—
40C2×#467C2211yescharacteristic
41C2×#466C2211yescharacteristic
42C2×#464—21no—
43C2×#452—21no—
44C2×#448—21no—
45C2×C4×Q16C2211yescharacteristic
46C2×C8×D4C2211yescharacteristic
47#2251C2212yescomplemented
48#2250C2212yescomplemented
49#2243C2212yescomplemented
50#2243C2212yescomplemented
51#2237C2214yescomplemented
52#2235C2212yes—
53#2234C2214yescomplemented
54#2232C2212yes—
55#2227C2212yescomplemented
56#2225C2212yescomplemented
57#2220C2214yescomplemented
58#2218C2214yescomplemented
59#2217C2214yescomplemented
60#2216C2214yescomplemented
61#2211C2214yescomplemented
62#2210C2214yescomplemented
63#2209C2212yescomplemented
64#2207C2212yescomplemented
65#2189C2214yescomplemented
66#2189C2214yescomplemented
67#2188C2212yes—
68#2188C2212yes—
69#2188C2214yescomplemented
70#2187C2212yes—
71#2186C2214yescomplemented
72#2186C2214yescomplemented
73#2185C2212yes—
74#2185C2212yes—
75#2185C2214yescomplemented
76#2184C2212yes—
77#2176C2214yescomplemented
78#2175C2214yescomplemented
79#2168C2214yescomplemented
80#2168C2214yescomplemented
81#2163C2211yescharacteristic
82#2163C2211yescharacteristic
83#2141C2214yescomplemented
84#2138C2214yescomplemented
85#2112C2212yes—
86C2×C22⋊C16C2212yes—
87C22×#41—21no—
88C23×Q8—21no—
89C2×#135—22no—
90C2×#135—44no—
91C2×#135C2314yes—
92C2×#135C2312yes—
93C2×#135C2312yes—
94C22×Q16—21no—
95C22×Q16—22no—
96C22×Q16—44no—
97C22×Q16C2311yescharacteristic
98C22×Q16C2314yes—
99C22×Q16C2311yescharacteristic
100C22×SD16—21no—
101C22×SD16—21no—
102C22×D8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104C23×C8C2311yescharacteristic
105#559—42no—
106#556—22no—
107#555—22no—
108#553—42no—
109#550—22no—
110#548—24no—
111#548—24no—
112#548—42no—
113#548—42no—
114#547—22no—
115#538—41no—
116#538—41no—
117#535—41no—
118#533—21no—
119#532—21no—
120C2×#124—21no—
121C2×#124—41no—
122#525—22no—
123#525—22no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—41no—
126C2×#121—21no—
127C2×#121—41no—
128#520—41no—
129C2×C4×Q8—21no—
130C2×C4×D4—41no—
131C2×C4×D4D411yescharacteristic
132#510—48no—
133#508—48no—
134C2×Q32C2314yes—
135C2×Q32C2314yes—
136C2×SD32C2314yes—
137C2×SD32C2314yes—
138C2×SD32C2314yes—
139C2×D16C2314yes—
140C2×M32C2314yes—
141C22×C16C2314yes—
142#500C2311yescharacteristic
143#500C2311yescharacteristic
144#495—21no—
145#495—21no—
146#495C2312yes—
147#495C2312yes—
148#494C2311yescharacteristic
149#493C2311yescharacteristic
150#487—21no—
151#486—21no—
152#485—22no—
153#485—22no—
154#482—22no—
155#482—22no—
156#476—22no—
157#475—22no—
158#467—21no—
159#467—22no—
160#467C2312yes—
161#467C2312yes—
162#467C2312yes—
163#467C2314yes—
164#466—21no—
165#466—22no—
166#466C2312yes—
167#466C2312yes—
168#466C2312yes—
169#466C2314yes—
170#465—22no—
171#465—22no—
172#465—22no—
173#465—22no—
174#464—24no—
175#464—24no—
176#461—22no—
177#460—22no—
178#452—22no—
179#452—22no—
180#452—24no—
181#452—24no—
182#449—22no—
183#449—22no—
184#448—24no—
185#448—24no—
186C4×Q16—21no—
187C4×Q16—21no—
188C4×Q16—21no—
189C4×Q16C2312yes—
190C4×Q16C2312yes—
191C4×Q16C2312yes—
192C4×D8—21no—
193C4×D8C2312yes—
194C8×D4D412yes—
195C8×D4D412yes—
196C8×D4C2312yes—
197C8×D4C2312yes—
198#427C2312yes—
199#427C2312yes—
200C2×#107C2311yescharacteristic
201C2×C4⋊C8D411yescharacteristic
202#416—22no—
203#416—22no—
204C2×#103—21no—
205C2×#103—21no—
206C2×#102—21no—
207C2×C22⋊C8D411yescharacteristic
208C2×C22⋊C8D411yescharacteristic
209C2×C4×C8C2311yescharacteristic
210#367C2314yes—
211#366C2314yes—
212C4⋊C16C2312yes—
213C4⋊C16C2312yes—
214#358C2314yes—
215#358C2314yes—
216#358C2314yes—
217#358C2314yes—
218#358C2314yes—
219#357C2314yes—
220#357C2314yes—
221#357C2314yes—
222C22⋊C16C2312yes—
223C22⋊C16C2312yes—
224C22⋊C16C2314yes—
225C22⋊C16C2314yes—
226#346C2312yes—
227#346C2312yes—
228C2×#41—24no—
229C2×#41—22no—
230C2×#41—22no—
231C2×#41—41no—
232C2×#41—42no—
233C2×#41—44no—
234C2×#41—82no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—24no—
238C22×Q8—41no—
239C22×Q8—42no—
240C22×Q8—44no—
241C22×Q8—82no—
242C22×D4—21no—
243C22×D4—41no—
244C22×D4—41no—
245C22×D4D811yescharacteristic
246C23×C4—41no—
247C23×C4—41no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250#135—28no—
251#135—44no—
252#135—44no—
253#135—44no—
254#135—48no—
255C2×Q16—24no—
256C2×Q16—28no—
257C2×Q16—21no—
258C2×Q16—24no—
259C2×Q16—21no—
260C2×Q16—21no—
261C2×Q16—44no—
262C2×Q16—42no—
263C2×Q16—48no—
264C2×Q16—44no—
265C2×Q16C2412yes—
266C2×Q16C2412yes—
267C2×Q16C2412yes—
268C2×SD16—24no—
269C2×SD16—22no—
270C2×SD16—22no—
271C2×SD16—22no—
272C2×SD16—22no—
273C2×SD16—44no—
274C2×SD16—82no—
275C2×D8—21no—
276C2×D8—21no—
277C2×D8—42no—
278C2×D8C2412yes—
279C2×D8C2412yes—
280C22×C8—42no—
281C22×C8—42no—
282C22×C8C2×D411yescharacteristic
283C22×C8C2×D411yescharacteristic
284C22×C8C2×D412yes—
285C22×C8C2×D412yes—
286C22×C8C2×D411yescharacteristic
287C22×C8C2411yescharacteristic
288C22×C8C2412yes—
289C22×C8C2412yes—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293#127—21no—
294#127—42no—
295#126—42no—
296#126—42no—
297#125—41no—
298#124—22no—
299#124—22no—
300#124—42no—
301#124—44no—
302#124—44no—
303#124—42no—
304#124—42no—
305#124—41no—
306#124—41no—
307#124—41no—
308#123—42no—
309#123—42no—
310#123—42no—
311#123—42no—
312C22⋊Q8—24no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—42no—
317C22⋊Q8—41no—
318C22⋊Q8—44no—
319C22⋊Q8—44no—
320#121—24no—
321#121—22no—
322#121—22no—
323#121—22no—
324#121—44no—
325#121—44no—
326#121—42no—
327#121—42no—
328#121—42no—
329#121—41no—
330#121—42no—
331#121—42no—
332#120—42no—
333#120—42no—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8—42no—
338C4×Q8—41no—
339C4×Q8—41no—
340C4×Q8—41no—
341C4×D4—22no—
342C4×D4—42no—
343C4×D4—42no—
344C4×D4—42no—
345C4×D4—42no—
346C4×D4—42no—
347C4×D4—44no—
348C4×D4—44no—
349C4×D4—41no—
350C4×D4D812yes—
351C4×D4D812yes—
352C4×D4C2×D412yes—
353C4×D4C2×D412yes—
354#117—22no—
355#117—22no—
356#117—41no—
357#117—42no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—41no—
360C2×C4⋊C4—41no—
361C2×C4⋊C4—42no—
362C2×C4⋊C4D811yescharacteristic
363C2×C22⋊C4—41no—
364C2×C22⋊C4—41no—
365C2×C22⋊C4—41no—
366C2×C22⋊C4—41no—
367C2×C22⋊C4D811yescharacteristic
368C2×C22⋊C4D811yescharacteristic
369C2×C42—41no—
370C2×C42C2×D411yescharacteristic
371Q32—28no—
372Q32—44no—
373SD32—28no—
374SD32—44no—
375SD32—44no—
376D16—44no—
377M32—44no—
378C2×C16—44no—
379C2×C16C2414yes—
380C2×C16C2414yes—
381#107—21no—
382#107—21no—
383#107—21no—
384#107C2412yes—
385#107C2412yes—
386#107C2412yes—
387#106—22no—
388#106—22no—
389C4⋊C8C2×D412yes—
390C4⋊C8C2×D412yes—
391#103—22no—
392#103—22no—
393#103—24no—
394#103—22no—
395#103—22no—
396#103—22no—
397#103—22no—
398#102—24no—
399#102—22no—
400#102—22no—
401#102—22no—
402#102—22no—
403C22⋊C8C2×D412yes—
404C22⋊C8C2×D412yes—
405C22⋊C8C2×D412yes—
406C22⋊C8C2×D412yes—
407C4×C8C2×D411yescharacteristic
408C4×C8C2×D411yescharacteristic
409C4×C8C2411yescharacteristic
410C4×C8C2411yescharacteristic
411C24—41no—
412C24C2×D811yescharacteristic
413C24C2×D811yescharacteristic
414#41—48no—
415#41—84no—
416#41—84no—
417#41—82no—
418#41—82no—
419#41—82no—
420#41—82no—
421#41—88no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—42no—
427C2×Q8—48no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—41no—
431C2×Q8—41no—
432C2×Q8—44no—
433C2×Q8—41no—
434C2×Q8—41no—
435C2×Q8—84no—
436C2×Q8—84no—
437C2×Q8—88no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—84no—
454C2×D4—84no—
455C2×D4C2×D812yes—
456C2×D4C2×D812yes—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4C2×D811yescharacteristic
472C22×C4C2×D811yescharacteristic
473C22×C4C2×D811yescharacteristic
474C22×C4C2×D811yescharacteristic
475C22×C4C2×D812yes—
476C22×C4C2×D812yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480Q16—24no—
481Q16—22no—
482Q16—24no—
483Q16—24no—
484Q16—48no—
485Q16—42no—
486Q16—81no—
487Q16H(5,𝔽2)14yes—
488SD16—48no—
489SD16—82no—
490SD16—82no—
491SD16—82no—
492SD16—82no—
493D8—24no—
494D8—24no—
495D8—42no—
496D8—81no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—41no—
501C2×C8—42no—
502C2×C8—41no—
503C2×C8—42no—
504C2×C8—41no—
505C2×C8C22×D411yescharacteristic
506C2×C8C22×D411yescharacteristic
507C2×C8C22×D411yescharacteristic
508C2×C8C22×D412yes—
509C2×C8C22×D412yes—
510C2×C8C2511yescharacteristic
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—41no—
522C4⋊C4—42no—
523C4⋊C4—41no—
524C4⋊C4—42no—
525C4⋊C4—41no—
526C4⋊C4C2×D812yes—
527C4⋊C4C2×D812yes—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4C2×D812yes—
544C22⋊C4C2×D812yes—
545C22⋊C4C2×D812yes—
546C22⋊C4C2×D812yes—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42C2×D811yescharacteristic
553C42C2×D811yescharacteristic
554C42C22×D411yescharacteristic
555C42C22×D411yescharacteristic
556C16—24no—
557C16—24no—
558C23—22no—
559C23—22no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—82no—
567C23—82no—
568C23—82no—
569C23C22×D812yes—
570C23C22×D812yes—
571C23C22×D811yescharacteristic
572Q8—24no—
573Q8—44no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—42no—
579Q8—88no—
580Q8—82no—
581Q8—82no—
582Q8—161no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—82no—
590D4—82no—
591D4—88no—
592D4—88no—
593D4—84no—
594D4—161no—
595D4—162no—
596D4—162no—
597D4—162no—
598D4—162no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—88no—
619C2×C4—84no—
620C2×C4—88no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—162no—
624C2×C4—162no—
625C2×C4C22×D811yescharacteristic
626C2×C4C22×D811yescharacteristic
627C2×C4C22×D811yescharacteristic
628C2×C4C22×D812yes—
629C2×C4C22×D812yes—
630C2×C4C22×D811yescharacteristic
631C2×C4C22×D811yescharacteristic
632C2×C4C23×D411yescharacteristic
633C8—21no—
634C8—21no—
635C8—21no—
636C8—22no—
637C8—22no—
638C8C2×H(5,𝔽2)12yes—
639C22—21no—
640C22—21no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—44no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—164no—
656C22#341412yes—
657C22C23×D811yescharacteristic
658C4—21no—
659C4—22no—
660C4—22no—
661C4—21no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—44no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4#348411yescharacteristic
672C4#358511yescharacteristic
673C2—22no—
674C2—42no—
675C2—42no—
676C2—162no—
677C2—162no—
678C2#6155911yescharacteristic
679C2C2×#341411yescharacteristic
680C2#6263711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62249C2semidirect
2. C2×#3285C2semidirect
3. #33888C2semidirect
4. #33880C2semidirect
5. #33837C2semidirect
6. #33835C2semidirect
7. #33781C2semidirect
8. #33779C2semidirect
9. #33760C2semidirect
10. #33756C2semidirect
11. #33673C2semidirect
12. #33672C2semidirect
13. #33518C2semidirect
14. #33516C2semidirect
15. #33487C2semidirect
20. C2×#510C22semidirect
21. C2×#508C22semidirect
22. #3342C22semidirect
23. #3341C22semidirect
26. #3287C22semidirect
31. #3283C22semidirect
47. #2251C22semidirect
48. #2250C22semidirect
49. #2243C22semidirect
50. #2243C22semidirect
51. #2237C22semidirect
53. #2234C22semidirect
55. #2227C22semidirect
56. #2225C22semidirect
57. #2220C22semidirect
58. #2218C22semidirect
59. #2217C22semidirect
60. #2216C22semidirect
61. #2211C22semidirect
62. #2210C22semidirect
63. #2209C22semidirect
64. #2207C22semidirect
65. #2189C22semidirect
66. #2189C22semidirect
69. #2188C22semidirect
71. #2186C22semidirect
72. #2186C22semidirect
75. #2185C22semidirect
77. #2176C22semidirect
78. #2175C22semidirect
79. #2168C22semidirect
80. #2168C22semidirect
83. #2141C22semidirect
84. #2138C22semidirect

Maximal subgroups

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

subgroupindex
#622492
C2×#32852
#338882
#338802
#338372
#338352
#337812
#337792
#337602
#337562
#336732
#336722
#335182
#335162
#334872