Finite Group Databasebeta

← #7624427

Subgroups — #7624427

Order 512.

1251 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62363C211yesmaximal, characteristic, complemented
2C2×#3380C211yesmaximal, characteristic, complemented
3#33956C214yesmaximal, complemented
4#33952C212yesmaximal, complemented
5#33947C212yesmaximal, complemented
6#33937C214yesmaximal, complemented
7#33933C212yesmaximal, complemented
8#33928C212yesmaximal, complemented
9#33804C211yesmaximal, characteristic, complemented
10#33800C212yesmaximal, complemented
11#33798C212yesmaximal, complemented
12#33792C211yesmaximal, characteristic, complemented
13#33674C212yesmaximal, complemented
14#33669C211yesmaximal, characteristic, complemented
15#33665C211yesmaximal, characteristic, complemented
16#33660C212yesmaximal, complemented
17#33498C211yesmaximal, characteristic, complemented
18#3522—21no—
19C2×#554—21no—
20#3396C2211yescharacteristic, complemented
21#3396C2212yescomplemented
22#3391C2211yescharacteristic, complemented
23Q8×Q16C2211yescharacteristic, complemented
24#3380—21no—
25#3380—22no—
26#3380—21no—
27#3380C2212yes—
28#3380C2212yes—
29#3380C2214yes—
30#3380C2211yescharacteristic, complemented
31#3380C2212yescomplemented
32#3362—21no—
33#3361—21no—
34#3361—22no—
35#3247—22no—
36#3247—21no—
37#3169C2211yescharacteristic
38#3169C2212yes—
39#3168C2212yes—
40#3168C2211yescharacteristic
41C2×#500C2211yescharacteristic
42C2×#499C2212yes—
43C2×#478—22no—
44C2×#476—21no—
45C2×C8×Q8C2211yescharacteristic
46C2×C4×D8C2212yes—
47C2×C4×D8C2211yescharacteristic
48#2255C2214yescomplemented
49#2255C2212yescomplemented
50#2253C2214yescomplemented
51#2252C2212yescomplemented
52#2244C2214yescomplemented
53#2243C2212yescomplemented
54#2242C2214yescomplemented
55#2240C2212yescomplemented
56#2236C2211yescharacteristic
57#2235C2212yes—
58#2233C2212yes—
59#2233C2211yescharacteristic
60#2232C2212yes—
61#2232C2212yes—
62#2232C2212yes—
63#2229C2214yescomplemented
64#2228C2212yes—
65#2228C2214yes—
66#2228C2214yescomplemented
67#2228C2214yescomplemented
68#2228C2214yescomplemented
69#2227C2212yescomplemented
70#2226C2212yes—
71#2226C2212yescomplemented
72#2224C2214yes—
73#2224C2214yescomplemented
74#2210C2212yes—
75#2210C2214yes—
76#2206C2214yes—
77#2206C2212yes—
78#2187C2211yescharacteristic
79#2184C2211yescharacteristic
80#2184C2212yes—
81#2183C2212yescomplemented
82#2183C2214yescomplemented
83Q8×C16C2212yescomplemented
84#2177C2214yes—
85#2177C2214yes—
86#2175C2214yes—
87C4×SD32C2212yes—
88C4×D16C2214yes—
89C4×D16C2214yes—
90C4×D16C2212yes—
91#2163C2212yes—
92#2163C2211yescharacteristic
93#2162C2211yescharacteristic
94#2162C2212yes—
95C22×D8C2311yescharacteristic
96#559—21no—
97#559—22no—
98Q82—21no—
99#554—21no—
100#554—22no—
101#554—22no—
102#554—22no—
103#554—24no—
104#554—41no—
105#554—41no—
106#554—42no—
107#550—21no—
108#548—41no—
109#547—41no—
110#547—42no—
111#542—42no—
112#542—41no—
113#532—22no—
114#532—21no—
115C2×C4⋊Q8—21no—
116#529—22no—
117#529—21no—
118C2×#125—22no—
119C2×C22⋊Q8—41no—
120C2×C22⋊Q8—42no—
121#520—41no—
122C2×C4×Q8D411yescharacteristic
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C22×C4⋊C4—41no—
126C2×SD32C2312yes—
127C2×D16C2312yes—
128C2×D16C2314yes—
129#500—21no—
130#500—21no—
131#500C2312yes—
132#500C2312yes—
133#500C2312yes—
134#500C2311yescharacteristic
135#499—22no—
136#499—22no—
137#499C2314yes—
138#499C2314yes—
139#499C2312yes—
140#499C2311yescharacteristic
141#495C2312yes—
142#495C2311yescharacteristic
143#493C2311yescharacteristic
144#493C2312yes—
145#488—22no—
146#488—21no—
147#486—22no—
148#486—21no—
149#478—24no—
150#478—24no—
151#478—24no—
152#478—24no—
153#478—22no—
154#478—22no—
155#478—22no—
156#477—21no—
157#477—22no—
158#476—21no—
159#476—22no—
160#476—22no—
161#476—22no—
162#476—24no—
163#467C2311yescharacteristic
164#466—22no—
165#466—21no—
166#466C2312yes—
167#466C2311yescharacteristic
168#465—22no—
169#465—21no—
170#463—22no—
171#463—22no—
172#463—22no—
173#460—21no—
174#460—22no—
175#459—21no—
176#459—22no—
177C8×Q8D411yescharacteristic
178C8×Q8D412yes—
179C8×Q8D411yescharacteristic
180C8×Q8C2311yescharacteristic
181C8×Q8C2312yes—
182C8×Q8C2311yescharacteristic
183C4×Q16C2311yescharacteristic
184C4×Q16C2312yes—
185C4×SD16—22no—
186C4×SD16—21no—
187C4×D8—22no—
188C4×D8—21no—
189C4×D8—22no—
190C4×D8—21no—
191C4×D8—22no—
192C4×D8—21no—
193C4×D8C2314yes—
194C4×D8C2312yes—
195C4×D8C2314yes—
196C4×D8C2312yes—
197C4×D8C2311yescharacteristic
198C4×D8C2312yes—
199C4×D8C2312yes—
200C4×D8C2312yes—
201C4×D8C2312yes—
202C2×#107—21no—
203C2×#107C2311yescharacteristic
204C2×#107C2312yes—
205C2×#106—22no—
206C2×C4⋊C8D412yes—
207C2×C4⋊C8D411yescharacteristic
208C2×#102—21no—
209C2×#102—22no—
210C2×C4×C8C2312yes—
211C2×C4×C8C2311yescharacteristic
212#367C2312yes—
213#367C2314yes—
214#367C2314yes—
215#366C2314yes—
216#366C2314yes—
217#366C2312yes—
218#366C2314yes—
219C4⋊C16C2314yes—
220C4⋊C16C2312yes—
221C4⋊C16C2312yes—
222C4⋊C16C2314yes—
223#358C2312yes—
224#358C2314yes—
225#357C2312yes—
226#357C2314yes—
227#357C2312yes—
228#357C2314yes—
229#357C2314yes—
230#357C2312yes—
231#357C2314yes—
232#357C2314yes—
233C22⋊C16C2311yescharacteristic
234C22⋊C16C2312yes—
235C22⋊C16C2311yescharacteristic
236#346C2314yes—
237#346C2312yes—
238C4×C16C2314yes—
239C4×C16C2312yes—
240C2×#41—41no—
241C2×#41—41no—
242C2×#41—42no—
243C22×Q8—41no—
244C22×Q8D811yescharacteristic
245C22×D4—21no—
246C23×C4—41no—
247C23×C4—42no—
248C2×Q16C2411yescharacteristic
249C2×SD16—22no—
250C2×SD16—21no—
251C2×D8—24no—
252C2×D8—21no—
253C2×D8—22no—
254C2×D8—21no—
255C2×D8C2412yes—
256C2×D8C2411yescharacteristic
257C2×D8C2412yes—
258C22×C8C2×D412yes—
259C22×C8C2×D411yescharacteristic
260C22×C8C2411yescharacteristic
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—21no—
265C4⋊Q8—41no—
266C4⋊Q8—42no—
267C4⋊Q8—41no—
268C4⋊Q8—41no—
269#127—22no—
270#127—21no—
271#126—42no—
272#126—41no—
273#125—24no—
274#125—24no—
275#125—22no—
276#125—21no—
277#125—42no—
278#125—42no—
279#124—22no—
280#124—21no—
281#124—41no—
282#124—42no—
283#124—42no—
284#124—41no—
285#123—41no—
286#123—42no—
287#123—42no—
288#123—42no—
289#123—42no—
290C22⋊Q8—21no—
291C22⋊Q8—42no—
292C22⋊Q8—42no—
293C22⋊Q8—44no—
294C22⋊Q8—41no—
295C22⋊Q8—42no—
296C22⋊Q8—44no—
297C22⋊Q8—44no—
298C22⋊Q8—44no—
299C22⋊Q8—44no—
300C22⋊Q8—42no—
301C22⋊Q8—42no—
302C22⋊Q8—42no—
303#121—22no—
304#121—21no—
305#121—41no—
306#121—42no—
307#121—41no—
308#121—42no—
309C4×Q8—22no—
310C4×Q8—21no—
311C4×Q8—41no—
312C4×Q8D811yescharacteristic
313C4×Q8D812yes—
314C4×Q8D811yescharacteristic
315C4×Q8C2×D411yescharacteristic
316C4×Q8C2×D411yescharacteristic
317C4×Q8C2×D412yes—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—22no—
323C4×D4—21no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—41no—
328C4×D4—42no—
329C4×D4—41no—
330C4×D4—42no—
331C4×D4—42no—
332C4×D4—41no—
333C4×D4—42no—
334C4×D4—41no—
335C4×D4—42no—
336C4×D4—41no—
337#117—42no—
338#117—41no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—42no—
342C2×C4⋊C4—41no—
343C2×C4⋊C4—42no—
344C2×C4⋊C4—42no—
345C2×C4⋊C4—42no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—41no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4D811yescharacteristic
350C2×C4⋊C4D812yes—
351C2×C22⋊C4—41no—
352C2×C22⋊C4—42no—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355SD32—24no—
356D16—24no—
357D16—28no—
358C2×C16C2412yes—
359C2×C16C2412yes—
360C2×C16C2414yes—
361#107—22no—
362#107—22no—
363#107—21no—
364#107—22no—
365#107—22no—
366#107—21no—
367#107C2414yes—
368#107C2412yes—
369#107C2411yescharacteristic
370#107C2412yes—
371#107C2411yescharacteristic
372#106—24no—
373#106—24no—
374#106—22no—
375#106—21no—
376C4⋊C8C2×D411yescharacteristic
377C4⋊C8C2×D412yes—
378C4⋊C8C2×D411yescharacteristic
379C4⋊C8C2×D412yes—
380C4⋊C8C2×D412yes—
381C4⋊C8C2×D412yes—
382C4⋊C8C2×D412yes—
383#103—21no—
384#103—22no—
385#103—21no—
386#102—24no—
387#102—24no—
388#102—22no—
389#102—22no—
390#102—21no—
391#102—22no—
392#102—22no—
393#102—21no—
394#102—22no—
395#102—22no—
396#102—22no—
397C4×C8C2×D412yes—
398C4×C8C2×D411yescharacteristic
399C4×C8C2×D412yes—
400C4×C8C2×D411yescharacteristic
401C4×C8C2412yes—
402C4×C8C2411yescharacteristic
403C4×C8C2411yescharacteristic
404C4×C8C2412yes—
405C24—41no—
406#41—84no—
407#41—164no—
408#41—162no—
409C2×Q8—21no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—41no—
414C2×Q8—42no—
415C2×Q8—162no—
416C2×Q8C2×D811yescharacteristic
417C2×Q8C2×D811yescharacteristic
418C2×Q8C2×D812yes—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—42no—
424C2×D4—41no—
425C2×D4—42no—
426C2×D4—41no—
427C2×D4—42no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—42no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—84no—
443C22×C4—164no—
444C22×C4C2×D811yescharacteristic
445C22×C4C2×D812yes—
446C22×C4C2×D811yescharacteristic
447C22×C4C2×D812yes—
448C22×C4C22×D411yescharacteristic
449Q16—22no—
450SD16—82no—
451SD16—81no—
452D8—24no—
453D8—22no—
454D8—22no—
455D8—24no—
456D8—82no—
457D8—81no—
458D82−1+414yes—
459C2×C8—22no—
460C2×C8—42no—
461C2×C8—41no—
462C2×C8C22×D411yescharacteristic
463C2×C8C22×D412yes—
464C2×C8C22×D411yescharacteristic
465C2×C8C22×D412yes—
466C2×C8C22×D411yescharacteristic
467C2×C8C2511yescharacteristic
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—41no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—41no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—42no—
490C4⋊C4C2×D811yescharacteristic
491C4⋊C4C2×D811yescharacteristic
492C4⋊C4C2×D812yes—
493C4⋊C4C2×D812yes—
494C4⋊C4C2×D812yes—
495C4⋊C4C2×D812yes—
496C4⋊C4C2×D812yes—
497C22⋊C4—41no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—42no—
506C22⋊C4—41no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—41no—
513C42—42no—
514C42—41no—
515C42C2×D812yes—
516C42C2×D811yescharacteristic
517C42C2×D812yes—
518C42C2×D811yescharacteristic
519C42C22×D412yes—
520C42C22×D411yescharacteristic
521C42C22×D412yes—
522C42C22×D411yescharacteristic
523C16—22no—
524C16—24no—
525C16—22no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—82no—
531C23—82no—
532C23C22×D811yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—42no—
536Q8—41no—
537Q8—41no—
538Q8—162no—
539Q8—162no—
540Q8—162no—
541Q8—161no—
542D4—24no—
543D4—44no—
544D4—42no—
545D4—41no—
546D4—42no—
547D4—41no—
548D4—42no—
549D4—84no—
550D4—82no—
551D4—84no—
552D4—82no—
553D4—162no—
554D4—161no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—82no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—82no—
573C2×C4—164no—
574C2×C4—164no—
575C2×C4—162no—
576C2×C4—162no—
577C2×C4—162no—
578C2×C4C22×D812yes—
579C2×C4C22×D811yescharacteristic
580C2×C4C22×D812yes—
581C2×C4C22×D811yescharacteristic
582C2×C4C22×D811yescharacteristic
583C2×C4C22×D811yescharacteristic
584C2×C4C22×D812yes—
585C2×C4C22×D812yes—
586C2×C4C22×D811yescharacteristic
587C2×C4C23×D411yescharacteristic
588C8—22no—
589C8—21no—
590C8—22no—
591C8—21no—
592C8—21no—
593C8C2×2−1+412yes—
594C22—21no—
595C22—21no—
596C22—44no—
597C22—88no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—81no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22—82no—
606C22#341512yes—
607C22C23×D811yescharacteristic
608C4—21no—
609C4—21no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—21no—
615C4—21no—
616C4—81no—
617C4—81no—
618C4—82no—
619C4—84no—
620C4—81no—
621C4—82no—
622C4#341412yes—
623C4#348611yescharacteristic
624C4#358611yescharacteristic
625C2—22no—
626C2—84no—
627C2—162no—
628C2—161no—
629C2#6181711yescharacteristic
630C2C2×#341411yescharacteristic
631C2C2×#341511yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62363C2semidirect
2. C2×#3380C2semidirect
3. #33956C2semidirect
4. #33952C2semidirect
5. #33947C2semidirect
6. #33937C2semidirect
7. #33933C2semidirect
8. #33928C2semidirect
9. #33804C2semidirect
10. #33800C2semidirect
11. #33798C2semidirect
12. #33792C2semidirect
13. #33674C2semidirect
14. #33669C2semidirect
15. #33665C2semidirect
16. #33660C2semidirect
17. #33498C2semidirect
20. #3396C22semidirect
21. #3396C22semidirect
22. #3391C22semidirect
23. Q8×Q16C22semidirect
30. #3380C22semidirect
31. #3380C22semidirect
48. #2255C22semidirect
49. #2255C22semidirect
50. #2253C22semidirect
51. #2252C22semidirect
52. #2244C22semidirect
53. #2243C22semidirect
54. #2242C22semidirect
55. #2240C22semidirect
63. #2229C22semidirect
66. #2228C22semidirect
67. #2228C22semidirect
68. #2228C22semidirect
69. #2227C22semidirect
71. #2226C22semidirect
73. #2224C22semidirect
81. #2183C22semidirect
82. #2183C22semidirect
83. Q8×C16C22semidirect

Maximal subgroups

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

subgroupindex
#623632
C2×#33802
#339562
#339522
#339472
#339372
#339332
#339282
#338042
#338002
#337982
#337922
#336742
#336692
#336652
#336602
#334982