Finite Group Databasebeta

← #7624265

Subgroups — #7624265

Order 512.

1380 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62247C211yesmaximal, characteristic, complemented
2C2×#3334C211yesmaximal, characteristic, complemented
3#33912C214yesmaximal, complemented
4#33910C214yesmaximal, complemented
5#33875C214yesmaximal, complemented
6#33873C214yesmaximal, complemented
7#33804C212yesmaximal, complemented
8#33760C212yesmaximal, complemented
9#33753C211yesmaximal, characteristic, complemented
10#33752C211yesmaximal, characteristic, complemented
11#33674C212yesmaximal, complemented
12#33646C212yesmaximal, complemented
13C2×#2235C211yesmaximal, characteristic, complemented
14C2×#2232C211yesmaximal, characteristic, complemented
15#33487C211yesmaximal, characteristic, complemented
16#3505—21no—
17C2×#547—21no—
18#3339C2212yescomplemented
19#3336C2212yescomplemented
20#3334—22no—
21#3334—22no—
22#3334C2214yes—
23#3334C2214yescomplemented
24#3301—22no—
25#3296—22no—
26#3285C2212yescomplemented
27#3280C2212yescomplemented
28#3250—21no—
29#3191—21no—
30#3187—21no—
31#3168C2212yes—
32C2×#500C2211yescharacteristic, complemented
33C2×#480—21no—
34C2×#480—21no—
35C2×#474—21no—
36#3050C2212yes—
37C2×#467C2211yescharacteristic
38C2×#466C2211yescharacteristic
39C2×#466C2211yescharacteristic
40C2×#466C2211yescharacteristic, complemented
41C2×C4×D8C2211yescharacteristic, complemented
42C2×C8×D4C2211yescharacteristic
43C22×#107C2211yescharacteristic
44C22×#107C2211yescharacteristic, complemented
45#2255C2214yescomplemented
46#2243C2214yescomplemented
47#2236C2214yescomplemented
48#2235C2212yescomplemented
49#2235C2212yescomplemented
50#2235C2214yescomplemented
51#2235C2212yescomplemented
52#2234C2214yescomplemented
53#2233C2214yescomplemented
54#2233C2212yescomplemented
55#2232C2212yes—
56#2232C2212yes—
57#2232C2212yes—
58#2232C2214yescomplemented
59#2232C2212yescomplemented
60#2228C2214yescomplemented
61#2227C2214yescomplemented
62#2226C2214yescomplemented
63#2220C2214yescomplemented
64#2218C2214yescomplemented
65#2217C2214yescomplemented
66#2215C2214yescomplemented
67#2210C2214yescomplemented
68#2187C2212yescomplemented
69#2186C2214yescomplemented
70#2185C2212yescomplemented
71#2184C2212yes—
72#2184C2212yescomplemented
73#2177C2214yescomplemented
74#2175C2214yescomplemented
75#2168C2214yescomplemented
76#2168C2214yescomplemented
77#2163C2212yescomplemented
78#2159C2214yescomplemented
79C2×#367C2212yescomplemented
80C2×#366C2212yescomplemented
81#2139C2214yescomplemented
82C2×#358C2212yescomplemented
83C2×#357C2212yescomplemented
84#2112C2212yescomplemented
85C2×C22⋊C16C2211yescharacteristic
86C2×C22⋊C16C2211yescharacteristic
87C22×#41—41no—
88C2×#135C2312yes—
89C22×Q16C2311yescharacteristic
90C22×D8C2311yescharacteristic
91C22×D8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94#553—22no—
95#553—42no—
96#552—22no—
97#551—42no—
98#548—22no—
99#547—24no—
100#547—24no—
101#547—42no—
102#547—42no—
103#546—22no—
104#539—41no—
105#538—41no—
106#534—41no—
107C2×C4⋊Q8—21no—
108#529—22no—
109#525—22no—
110C2×#123—41no—
111C2×#123—41no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—41no—
114C2×#121—21no—
115C2×#121—21no—
116C2×#121—21no—
117#518—41no—
118C2×C4×D4—21no—
119C2×C4×D4D411yescharacteristic
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C2×SD32C2314yes—
123C2×D16C2314yes—
124C2×M32C2314yes—
125C22×C16C2312yes—
126C22×C16C2312yes—
127#500—22no—
128#500C2314yes—
129#499C2312yes—
130#495C2312yes—
131#488—22no—
132#485—22no—
133#483—22no—
134#482—22no—
135#481—22no—
136#480—24no—
137#480—24no—
138#480—24no—
139#480—24no—
140#479—22no—
141#478—22no—
142#474—24no—
143#474—24no—
144#467—21no—
145#467—21no—
146#467C2312yes—
147#467C2312yes—
148#467C2312yes—
149#466—21no—
150#466—21no—
151#466—22no—
152#466—21no—
153#466—21no—
154#466C2312yes—
155#466C2314yes—
156#466C2312yes—
157#466C2312yes—
158#466C2312yes—
159#466C2312yes—
160#465—22no—
161#465—22no—
162#465—22no—
163#464—22no—
164#463—22no—
165#452—22no—
166#450—22no—
167#449—22no—
168#448—22no—
169C4×Q16C2312yes—
170C4×SD16—22no—
171C4×D8—22no—
172C4×D8C2314yes—
173C4×D8C2312yes—
174C8×D4D412yes—
175C8×D4D412yes—
176C8×D4C2312yes—
177C8×D4C2312yes—
178#427C2312yes—
179C2×#107—21no—
180C2×#107—21no—
181C2×#107—22no—
182C2×#107—21no—
183C2×#107—21no—
184C2×#107—21no—
185C2×#107C2312yes—
186C2×#107C2314yes—
187C2×#107C2312yes—
188C2×#107C2311yescharacteristic
189C2×#107C2311yescharacteristic
190C2×#107C2311yescharacteristic
191C2×#107C2311yescharacteristic
192C2×#107C2311yescharacteristic
193C2×C4⋊C8D411yescharacteristic
194#416—22no—
195C2×#103—21no—
196C2×#102—21no—
197C2×#102—21no—
198C2×#102—21no—
199C2×#102—21no—
200C2×C22⋊C8D411yescharacteristic
201C2×C22⋊C8D411yescharacteristic
202C2×C4×C8C2311yescharacteristic
203#367C2314yes—
204#367C2314yes—
205#367C2314yes—
206#366C2314yes—
207#366C2314yes—
208#366C2314yes—
209C4⋊C16C2314yes—
210#358C2314yes—
211#358C2314yes—
212#358C2314yes—
213#357C2314yes—
214#357C2314yes—
215#357C2314yes—
216#357C2314yes—
217#357C2314yes—
218C22⋊C16C2312yes—
219C22⋊C16C2312yes—
220C22⋊C16C2312yes—
221C22⋊C16C2312yes—
222C22⋊C16C2314yes—
223#346C2314yes—
224C2×#41—22no—
225C2×#41—44no—
226C2×#41—44no—
227C2×#41—88no—
228C22×Q8—21no—
229C22×Q8—41no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—41no—
233C22×D4—41no—
234C22×D4—41no—
235C22×D4D811yescharacteristic
236C23×C4—41no—
237C23×C4—41no—
238C23×C4—41no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241#135—44no—
242C2×Q16—21no—
243C2×Q16—42no—
244C2×Q16C2412yes—
245C2×SD16—22no—
246C2×SD16—22no—
247C2×D8—21no—
248C2×D8—21no—
249C2×D8—42no—
250C2×D8—42no—
251C2×D8C2412yes—
252C2×D8C2412yes—
253C22×C8—42no—
254C22×C8—42no—
255C22×C8C2×D411yescharacteristic
256C22×C8C2×D411yescharacteristic
257C22×C8C2×D411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D412yes—
260C22×C8C2×D411yescharacteristic
261C22×C8C2411yescharacteristic
262C22×C8C2411yescharacteristic
263C22×C8C2411yescharacteristic
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—42no—
267#126—42no—
268#126—42no—
269#126—42no—
270#125—22no—
271#125—42no—
272#124—22no—
273#124—42no—
274#123—42no—
275#123—42no—
276#123—42no—
277#123—42no—
278#123—44no—
279#123—44no—
280#123—44no—
281#123—44no—
282#123—42no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—41no—
287C22⋊Q8—41no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C22⋊Q8—42no—
291#121—22no—
292#121—22no—
293#121—22no—
294#121—24no—
295#121—22no—
296#121—22no—
297#121—42no—
298#121—42no—
299#121—41no—
300#121—41no—
301#121—41no—
302#121—42no—
303#121—41no—
304#121—42no—
305#120—42no—
306#120—42no—
307C4×Q8—22no—
308C4×D4—24no—
309C4×D4—22no—
310C4×D4—42no—
311C4×D4—42no—
312C4×D4—42no—
313C4×D4—42no—
314C4×D4—42no—
315C4×D4—42no—
316C4×D4D812yes—
317C4×D4D812yes—
318C4×D4C2×D412yes—
319C4×D4C2×D412yes—
320#117—22no—
321#117—42no—
322#117—42no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—41no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4—41no—
334C2×C4⋊C4—41no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—41no—
337C2×C4⋊C4—41no—
338C2×C4⋊C4D811yescharacteristic
339C2×C22⋊C4—41no—
340C2×C22⋊C4—41no—
341C2×C22⋊C4—41no—
342C2×C22⋊C4—41no—
343C2×C22⋊C4—41no—
344C2×C22⋊C4—42no—
345C2×C22⋊C4D811yescharacteristic
346C2×C22⋊C4D811yescharacteristic
347C2×C42C2×D411yescharacteristic
348SD32—44no—
349D16—44no—
350M32—44no—
351C2×C16—42no—
352C2×C16—42no—
353C2×C16C2414yes—
354C2×C16C2414yes—
355#107—21no—
356#107—21no—
357#107—21no—
358#107—22no—
359#107—24no—
360#107—22no—
361#107—24no—
362#107—21no—
363#107—21no—
364#107C2412yes—
365#107C2412yes—
366#107C2412yes—
367#107C2412yes—
368#107C2412yes—
369#106—22no—
370#106—22no—
371C4⋊C8C2×D412yes—
372C4⋊C8C2×D412yes—
373#103—22no—
374#103—22no—
375#103—22no—
376#103—22no—
377#102—22no—
378#102—24no—
379#102—22no—
380#102—22no—
381#102—24no—
382#102—22no—
383#102—22no—
384#102—22no—
385C22⋊C8C2×D412yes—
386C22⋊C8C2×D412yes—
387C22⋊C8C2×D412yes—
388C22⋊C8C2×D412yes—
389C4×C8C2×D412yes—
390C4×C8C2412yes—
391C24—41no—
392C24—41no—
393C24C2×D811yescharacteristic
394C24C2×D811yescharacteristic
395#41—82no—
396#41—82no—
397#41—816no—
398#41—816no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—41no—
402C2×Q8—84no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4—84no—
422C2×D4—84no—
423C2×D4—84no—
424C2×D4C2×D812yes—
425C2×D4C2×D812yes—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4C2×D811yescharacteristic
446C22×C4C2×D811yescharacteristic
447C22×C4C2×D811yescharacteristic
448C22×C4C2×D811yescharacteristic
449C22×C4C2×D811yescharacteristic
450C22×C4C2×D811yescharacteristic
451C22×C4C2×D812yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456Q16—24no—
457Q16—81no—
458SD16—82no—
459SD16—82no—
460D8—24no—
461D8—24no—
462D8—42no—
463D8—81no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—41no—
469C2×C8—42no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8C22×D411yescharacteristic
475C2×C8C22×D412yes—
476C2×C8C22×D411yescharacteristic
477C2×C8C22×D412yes—
478C2×C8C22×D411yescharacteristic
479C2×C8C2511yescharacteristic
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4C2×D812yes—
501C4⋊C4C2×D812yes—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4C2×D812yes—
517C22⋊C4C2×D812yes—
518C22⋊C4C2×D812yes—
519C22⋊C4C2×D812yes—
520C42—42no—
521C42—42no—
522C42C2×D812yes—
523C42C22×D412yes—
524C16—24no—
525C16—24no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—161no—
538C23—161no—
539C23#50914yes—
540C23C22×D812yes—
541C23C22×D811yescharacteristic
542C23C22×D811yescharacteristic
543C23C22×D811yescharacteristic
544Q8—42no—
545Q8—42no—
546Q8—88no—
547Q8—161no—
548D4—24no—
549D4—24no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—82no—
555D4—88no—
556D4—84no—
557D4—84no—
558D4—84no—
559D4—84no—
560D4—161no—
561D4—162no—
562D4—162no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—88no—
585C2×C4—88no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—82no—
592C2×C4—162no—
593C2×C4—162no—
594C2×C4—161no—
595C2×C4—161no—
596C2×C4C22×D812yes—
597C2×C4C22×D812yes—
598C2×C4C22×D811yescharacteristic
599C2×C4C22×D812yes—
600C2×C4C22×D811yescharacteristic
601C2×C4C22×D811yescharacteristic
602C2×C4C23×D411yescharacteristic
603C8—22no—
604C8—21no—
605C8—21no—
606C8—22no—
607C8—21no—
608C8#58512yes—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—44no—
619C22—44no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—164no—
627C22—322no—
628C22C2×#50912yes—
629C22C2×#50912yes—
630C22C2×#50912yes—
631C22C23×D811yescharacteristic
632C4—22no—
633C4—22no—
634C4—21no—
635C4—22no—
636C4—21no—
637C4—21no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4#348511yescharacteristic
645C4#358611yescharacteristic
646C2—22no—
647C2—22no—
648C2—22no—
649C2—42no—
650C2—162no—
651C2—162no—
652C2#6169411yescharacteristic
653C2C22×#50911yescharacteristic
654C2#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. #62247C2semidirect
2. C2×#3334C2semidirect
3. #33912C2semidirect
4. #33910C2semidirect
5. #33875C2semidirect
6. #33873C2semidirect
7. #33804C2semidirect
8. #33760C2semidirect
9. #33753C2semidirect
10. #33752C2semidirect
11. #33674C2semidirect
12. #33646C2semidirect
13. C2×#2235C2semidirect
14. C2×#2232C2semidirect
15. #33487C2semidirect
18. #3339C22semidirect
19. #3336C22semidirect
23. #3334C22semidirect
26. #3285C22semidirect
27. #3280C22semidirect
32. C2×#500C22semidirect
40. C2×#466C22semidirect
41. C2×C4×D8C22semidirect
44. C22×#107C22semidirect
45. #2255C22semidirect
46. #2243C22semidirect
47. #2236C22semidirect
48. #2235C22semidirect
49. #2235C22semidirect
50. #2235C22semidirect
51. #2235C22semidirect
52. #2234C22semidirect
53. #2233C22semidirect
54. #2233C22semidirect
58. #2232C22semidirect
59. #2232C22semidirect
60. #2228C22semidirect
61. #2227C22semidirect
62. #2226C22semidirect
63. #2220C22semidirect
64. #2218C22semidirect
65. #2217C22semidirect
66. #2215C22semidirect
67. #2210C22semidirect
68. #2187C22semidirect
69. #2186C22semidirect
70. #2185C22semidirect
72. #2184C22semidirect
73. #2177C22semidirect
74. #2175C22semidirect
75. #2168C22semidirect
76. #2168C22semidirect
77. #2163C22semidirect
78. #2159C22semidirect
79. C2×#367C22semidirect
80. C2×#366C22semidirect
81. #2139C22semidirect
82. C2×#358C22semidirect
83. C2×#357C22semidirect
84. #2112C22semidirect

Maximal subgroups

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

subgroupindex
#622472
C2×#33342
#339122
#339102
#338752
#338732
#338042
#337602
#337532
#337522
#336742
#336462
C2×#22352
C2×#22322
#334872