Finite Group Databasebeta

← #7624369

Subgroups — #7624369

Order 512.

1375 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62252C211yesmaximal, characteristic, complemented
2C2×#3334C211yesmaximal, characteristic, complemented
3#33920C212yesmaximal, complemented
4#33910C214yesmaximal, complemented
5#33901C212yesmaximal, complemented
6#33897C212yesmaximal, complemented
7#33875C214yesmaximal, complemented
8#33865C212yesmaximal, complemented
9#33805C211yesmaximal, characteristic, complemented
10#33796C211yesmaximal, characteristic, complemented
11#33759C212yesmaximal, complemented
12#33754C212yesmaximal, complemented
13#33670C211yesmaximal, characteristic, complemented
14#33665C211yesmaximal, characteristic, complemented
15#33646C212yesmaximal, complemented
16C2×#2236C212yesmaximal, complemented
17#33488C211yesmaximal, characteristic, complemented
18#3520—21no—
19C2×#547—21no—
20#3342C2211yescharacteristic, complemented
21#3336C2212yescomplemented
22#3334—21no—
23#3334—22no—
24#3334—21no—
25#3334C2212yes—
26#3334C2214yes—
27#3334C2212yes—
28#3334C2211yescharacteristic, complemented
29#3302—21no—
30#3300—22no—
31#3298—21no—
32D4×Q16C2211yescharacteristic, complemented
33#3285C2212yescomplemented
34#3283C2211yescharacteristic, complemented
35#3248—21no—
36#3190—22no—
37#3170C2211yescharacteristic
38#3168C2211yescharacteristic
39C2×#500C2211yescharacteristic
40C2×#480—22no—
41C2×#474—21no—
42#3050C2212yes—
43C2×#467C2212yes—
44C2×#466C2212yes—
45C2×C4×D8C2211yescharacteristic
46C2×C8×D4C2211yescharacteristic
47C22×#107C2212yes—
48#2255C2212yescomplemented
49#2252C2212yescomplemented
50#2243C2212yescomplemented
51#2241C2212yescomplemented
52#2236C2212yes—
53#2236C2212yes—
54#2236C2212yes—
55#2236C2211yescharacteristic
56#2236C2212yes—
57#2236C2214yescomplemented
58#2236C2214yescomplemented
59#2235C2214yescomplemented
60#2234C2214yescomplemented
61#2233C2212yes—
62#2233C2211yescharacteristic
63#2232C2214yescomplemented
64#2232C2214yescomplemented
65#2228C2212yescomplemented
66#2227C2214yescomplemented
67#2225C2212yescomplemented
68#2224C2212yescomplemented
69#2224C2212yescomplemented
70#2220C2214yescomplemented
71#2220C2214yescomplemented
72#2217C2214yescomplemented
73#2215C2214yescomplemented
74#2210C2212yescomplemented
75#2208C2212yescomplemented
76#2187C2211yescharacteristic
77#2187C2212yes—
78#2186C2214yescomplemented
79#2185C2214yescomplemented
80#2184C2211yescharacteristic
81#2175C2214yescomplemented
82C4×SD32C2212yescomplemented
83C4×D16C2212yescomplemented
84#2169C2212yescomplemented
85#2168C2214yescomplemented
86D4×C16C2212yescomplemented
87#2163C2211yescharacteristic
88#2162C2211yescharacteristic
89#2159C2214yescomplemented
90C2×#366C2214yescomplemented
91#2139C2214yescomplemented
92C2×#358C2214yescomplemented
93#2112C2212yes—
94C2×C22⋊C16C2212yes—
95C22×#41—41no—
96C2×#135C2312yes—
97C22×Q16C2312yes—
98C22×D8C2311yescharacteristic
99C23×C8C2312yes—
100#556—21no—
101#554—41no—
102#552—22no—
103#551—42no—
104D4×Q8—21no—
105#548—22no—
106#547—22no—
107#547—22no—
108#547—24no—
109#547—21no—
110#547—21no—
111#547—41no—
112#547—42no—
113#547—41no—
114#546—41no—
115#542—41no—
116#539—42no—
117#533—21no—
118C2×C4⋊Q8—21no—
119#529—21no—
120#525—22no—
121C2×#123—42no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—41no—
124C2×#121—22no—
125#520—41no—
126C2×C4×D4—21no—
127C2×C4×D4D411yescharacteristic
128C22×C4⋊C4—22no—
129C2×SD32C2312yes—
130C2×SD32C2314yes—
131C2×D16C2312yes—
132C2×M32C2314yes—
133C22×C16C2314yes—
134#500—21no—
135#500—21no—
136#500C2312yes—
137#500C2312yes—
138#500C2311yescharacteristic
139#499C2311yescharacteristic
140#495C2311yescharacteristic
141#494C2311yescharacteristic
142#488—21no—
143#487—21no—
144#485—22no—
145#485—22no—
146#483—22no—
147#480—24no—
148#480—24no—
149#480—24no—
150#480—24no—
151#480—22no—
152#479—22no—
153#475—21no—
154#474—22no—
155#474—22no—
156#474—24no—
157#474—21no—
158#467—21no—
159#467—22no—
160#467—22no—
161#467C2312yes—
162#467C2312yes—
163#467C2312yes—
164#467C2311yescharacteristic
165#467C2312yes—
166#466—22no—
167#466—22no—
168#466C2314yes—
169#466C2314yes—
170#466C2312yes—
171#466C2311yescharacteristic
172#465—21no—
173#465—22no—
174#465—22no—
175#464—22no—
176#462—21no—
177#461—21no—
178#452—22no—
179#452—22no—
180#451—22no—
181#448—22no—
182C4×Q16—21no—
183C4×Q16C2312yes—
184C4×Q16C2311yescharacteristic
185C4×SD16—21no—
186C4×D8—21no—
187C4×D8—21no—
188C4×D8C2312yes—
189C4×D8C2312yes—
190C4×D8C2311yescharacteristic
191C8×D4D411yescharacteristic
192C8×D4D412yes—
193C8×D4D411yescharacteristic
194C8×D4C2311yescharacteristic
195C8×D4C2312yes—
196C8×D4C2311yescharacteristic
197#427C2312yes—
198C2×#107—22no—
199C2×#107—22no—
200C2×#107—21no—
201C2×#107—22no—
202C2×#107C2314yes—
203C2×#107C2314yes—
204C2×#107C2312yes—
205C2×#107C2312yes—
206C2×#107C2311yescharacteristic
207C2×C4⋊C8D411yescharacteristic
208#416—22no—
209C2×#103—22no—
210C2×#102—21no—
211C2×#102—22no—
212C2×C22⋊C8D412yes—
213C2×C4×C8C2311yescharacteristic
214#367C2312yes—
215#367C2314yes—
216#366C2312yes—
217#366C2314yes—
218#366C2314yes—
219#366C2314yes—
220#366C2314yes—
221C4⋊C16C2312yes—
222C4⋊C16C2312yes—
223#358C2312yes—
224#358C2314yes—
225#358C2314yes—
226#358C2314yes—
227#358C2314yes—
228#357C2312yes—
229#357C2314yes—
230#357C2312yes—
231#357C2312yes—
232#357C2314yes—
233C22⋊C16C2311yescharacteristic
234C22⋊C16C2312yes—
235C22⋊C16C2314yes—
236C22⋊C16C2311yescharacteristic
237C22⋊C16C2314yes—
238#346C2312yes—
239C4×C16C2312yes—
240C2×#41—22no—
241C2×#41—41no—
242C2×#41—44no—
243C2×#41—42no—
244C2×#41—42no—
245C2×#41—88no—
246C22×Q8—22no—
247C22×Q8—41no—
248C22×D4—21no—
249C22×D4—42no—
250C22×D4D811yescharacteristic
251C23×C4—42no—
252C23×C4—41no—
253C23×C4C2×D412yes—
254#135—44no—
255C2×Q16—22no—
256C2×Q16—21no—
257C2×Q16—44no—
258C2×Q16C2411yescharacteristic
259C2×Q16C2412yes—
260C2×SD16—22no—
261C2×SD16—21no—
262C2×D8—21no—
263C2×D8—42no—
264C2×D8C2412yes—
265C2×D8C2411yescharacteristic
266C22×C8—44no—
267C22×C8C2×D412yes—
268C22×C8C2×D412yes—
269C22×C8C2×D412yes—
270C22×C8C2×D411yescharacteristic
271C22×C8C2411yescharacteristic
272C22×C8C2412yes—
273C22×C8C2412yes—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—41no—
279C4⋊Q8—41no—
280C4⋊Q8—41no—
281#126—42no—
282#126—42no—
283#126—41no—
284#125—21no—
285#125—42no—
286#125—41no—
287#124—21no—
288#124—41no—
289#124—42no—
290#123—42no—
291#123—42no—
292#123—41no—
293#123—44no—
294#123—44no—
295#123—44no—
296#123—44no—
297#123—42no—
298#123—42no—
299C22⋊Q8—21no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—22no—
303C22⋊Q8—22no—
304C22⋊Q8—42no—
305C22⋊Q8—42no—
306C22⋊Q8—41no—
307C22⋊Q8—42no—
308C22⋊Q8—44no—
309C22⋊Q8—42no—
310C22⋊Q8—42no—
311C22⋊Q8—41no—
312C22⋊Q8—42no—
313#121—24no—
314#121—24no—
315#121—21no—
316#121—22no—
317#121—42no—
318#121—41no—
319#121—42no—
320#121—42no—
321#120—42no—
322C4×Q8—21no—
323C4×Q8—22no—
324C4×Q8—41no—
325C4×Q8—41no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—21no—
329C4×D4—41no—
330C4×D4—42no—
331C4×D4—42no—
332C4×D4—41no—
333C4×D4—42no—
334C4×D4—41no—
335C4×D4—41no—
336C4×D4D811yescharacteristic
337C4×D4D812yes—
338C4×D4D811yescharacteristic
339C4×D4C2×D411yescharacteristic
340C4×D4C2×D411yescharacteristic
341C4×D4C2×D412yes—
342#117—22no—
343#117—41no—
344#117—42no—
345#117—41no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—41no—
356C2×C4⋊C4—41no—
357C2×C4⋊C4D811yescharacteristic
358C2×C22⋊C4—42no—
359C2×C22⋊C4—41no—
360C2×C22⋊C4—42no—
361C2×C22⋊C4D812yes—
362C2×C42C2×D411yescharacteristic
363SD32—44no—
364SD32—42no—
365D16—42no—
366M32—44no—
367C2×C16—44no—
368C2×C16C2412yes—
369C2×C16C2412yes—
370C2×C16C2414yes—
371#107—21no—
372#107—22no—
373#107—24no—
374#107—24no—
375#107—22no—
376#107—22no—
377#107—22no—
378#107—21no—
379#107C2411yescharacteristic
380#107C2412yes—
381#107C2411yescharacteristic
382#107C2412yes—
383#107C2414yes—
384#106—22no—
385#106—21no—
386C4⋊C8C2×D411yescharacteristic
387C4⋊C8C2×D411yescharacteristic
388C4⋊C8C2×D412yes—
389#103—21no—
390#103—22no—
391#103—22no—
392#103—22no—
393#103—22no—
394#103—22no—
395#103—21no—
396#102—22no—
397#102—22no—
398#102—24no—
399#102—24no—
400#102—21no—
401#102—22no—
402#102—21no—
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×D812yes—
413#41—82no—
414#41—82no—
415#41—816no—
416#41—88no—
417#41—84no—
418#41—88no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—41no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—84no—
426C2×Q8—82no—
427C2×Q8—82no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—41no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—41no—
439C2×D4—88no—
440C2×D4C2×D812yes—
441C2×D4C2×D811yescharacteristic
442C2×D4C2×D811yescharacteristic
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—84no—
456C22×C4—88no—
457C22×C4C2×D811yescharacteristic
458C22×C4C2×D811yescharacteristic
459C22×C4C2×D812yes—
460C22×C4C2×D812yes—
461C22×C4C2×D812yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465Q16—22no—
466Q16—24no—
467Q16—81no—
468Q16—82no—
469SD16—81no—
470SD16—82no—
471D8—22no—
472D8—22no—
473D8—22no—
474D8—42no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—41no—
482C2×C8C22×D411yescharacteristic
483C2×C8C22×D411yescharacteristic
484C2×C8C22×D411yescharacteristic
485C2×C8C22×D412yes—
486C2×C8C22×D412yes—
487C2×C8C2511yescharacteristic
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—41no—
499C4⋊C4—42no—
500C4⋊C4—41no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—42no—
509C4⋊C4—41no—
510C4⋊C4—42no—
511C4⋊C4—41no—
512C4⋊C4C2×D811yescharacteristic
513C4⋊C4C2×D812yes—
514C4⋊C4C2×D811yescharacteristic
515C22⋊C4—41no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—42no—
529C22⋊C4—41no—
530C22⋊C4C2×D812yes—
531C22⋊C4C2×D812yes—
532C22⋊C4C2×D812yes—
533C22⋊C4C2×D812yes—
534C42—42no—
535C42—42no—
536C42—41no—
537C42C2×D811yescharacteristic
538C42C2×D811yescharacteristic
539C42C22×D411yescharacteristic
540C42C22×D411yescharacteristic
541C16—22no—
542C16—24no—
543C16—22no—
544C23—24no—
545C23—24no—
546C23—42no—
547C23—41no—
548C23—42no—
549C23—82no—
550C23—82no—
551C23—82no—
552C23C22×D812yes—
553C23C22×D812yes—
554C23C22×D811yescharacteristic
555Q8—41no—
556Q8—42no—
557Q8—42no—
558Q8—41no—
559Q8—84no—
560Q8—84no—
561Q8—82no—
562Q8—162no—
563D4—24no—
564D4—22no—
565D4—41no—
566D4—42no—
567D4—42no—
568D4—41no—
569D4—82no—
570D4—88no—
571D4—88no—
572D4—84no—
573D4—82no—
574D4—162no—
575D4#50914yes—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—84no—
595C2×C4—82no—
596C2×C4—84no—
597C2×C4—88no—
598C2×C4—88no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—82no—
602C2×C4—162no—
603C2×C4—162no—
604C2×C4C22×D811yescharacteristic
605C2×C4C22×D812yes—
606C2×C4C22×D811yescharacteristic
607C2×C4C22×D811yescharacteristic
608C2×C4C22×D811yescharacteristic
609C2×C4C22×D812yes—
610C2×C4C22×D811yescharacteristic
611C2×C4C23×D411yescharacteristic
612C8—21no—
613C8—21no—
614C8—21no—
615C8—22no—
616C8—22no—
617C8#58512yes—
618C22—21no—
619C22—22no—
620C22—21no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—48no—
626C22—82no—
627C22—81no—
628C22—81no—
629C22—82no—
630C22—81no—
631C22—82no—
632C22—164no—
633C22C2×#50914yes—
634C22#341712yes—
635C22C23×D811yescharacteristic
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—22no—
641C4—22no—
642C4—81no—
643C4—82no—
644C4—84no—
645C4—82no—
646C4—81no—
647C4—82no—
648C4—81no—
649C4C2×#50912yes—
650C4#348511yescharacteristic
651C4#358611yescharacteristic
652C2—22no—
653C2—24no—
654C2—42no—
655C2—161no—
656C2—162no—
657C2#6169411yescharacteristic
658C2C22×#50911yescharacteristic
659C2C2×#341711yescharacteristic
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. #62252C2semidirect
2. C2×#3334C2semidirect
3. #33920C2semidirect
4. #33910C2semidirect
5. #33901C2semidirect
6. #33897C2semidirect
7. #33875C2semidirect
8. #33865C2semidirect
9. #33805C2semidirect
10. #33796C2semidirect
11. #33759C2semidirect
12. #33754C2semidirect
13. #33670C2semidirect
14. #33665C2semidirect
15. #33646C2semidirect
16. C2×#2236C2semidirect
17. #33488C2semidirect
20. #3342C22semidirect
21. #3336C22semidirect
28. #3334C22semidirect
32. D4×Q16C22semidirect
33. #3285C22semidirect
34. #3283C22semidirect
48. #2255C22semidirect
49. #2252C22semidirect
50. #2243C22semidirect
51. #2241C22semidirect
57. #2236C22semidirect
58. #2236C22semidirect
59. #2235C22semidirect
60. #2234C22semidirect
63. #2232C22semidirect
64. #2232C22semidirect
65. #2228C22semidirect
66. #2227C22semidirect
67. #2225C22semidirect
68. #2224C22semidirect
69. #2224C22semidirect
70. #2220C22semidirect
71. #2220C22semidirect
72. #2217C22semidirect
73. #2215C22semidirect
74. #2210C22semidirect
75. #2208C22semidirect
78. #2186C22semidirect
79. #2185C22semidirect
81. #2175C22semidirect
82. C4×SD32C22semidirect
83. C4×D16C22semidirect
84. #2169C22semidirect
85. #2168C22semidirect
86. D4×C16C22semidirect
89. #2159C22semidirect
90. C2×#366C22semidirect
91. #2139C22semidirect
92. C2×#358C22semidirect

Maximal subgroups

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

subgroupindex
#622522
C2×#33342
#339202
#339102
#339012
#338972
#338752
#338652
#338052
#337962
#337592
#337542
#336702
#336652
#336462
C2×#22362
#334882