Finite Group Databasebeta

← #7623258

Subgroups — #7623258

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60682C211yesmaximal, characteristic, complemented
2C2×#3175C211yesmaximal, characteristic, complemented
3#33745C214yesmaximal, complemented
4#33744C214yesmaximal, complemented
5#33742C214yesmaximal, complemented
6#33739C214yesmaximal, complemented
7#33731C211yesmaximal, characteristic, complemented
8#33719C211yesmaximal, characteristic, complemented
9#33707C211yesmaximal, characteristic, complemented
10#33706C211yesmaximal, characteristic, complemented
11#33606C212yesmaximal, complemented
12#33605C212yesmaximal, complemented
13C4×#510C212yesmaximal, complemented
14C4×#509C212yesmaximal, complemented
15#33324C211yesmaximal, characteristic, complemented
16#3455—21no—
17C2×#542—21no—
18C2×#510C2212yes—
19C2×#509C2212yes—
20C22×M32C2212yes—
21#3177C2212yescomplemented
22#3176C2212yescomplemented
23#3175—22no—
24#3175—22no—
25#3175C2214yes—
26#3175C2214yes—
27#3175C2212yescomplemented
28#3175C2212yescomplemented
29#3174—22no—
30#3173—22no—
31#3160C2211yescharacteristic
32C2×#500C2211yescharacteristic
33#3147C2211yescharacteristic
34C2×#495C2211yescharacteristic
35#3135—21no—
36C2×#489—21no—
37#3054—21no—
38C2×#468—21no—
39#3050C2212yes—
40C2×#467C2211yescharacteristic
41C2×#466C2211yescharacteristic
42#2943—21no—
43C2×#440—21no—
44C4×#135C2212yes—
45C2×C4×Q16C2211yescharacteristic
46C2×C4×D8C2211yescharacteristic
47C2×#405C2211yescharacteristic
48#2255C2214yescomplemented
49#2254C2212yescomplemented
50#2252C2212yescomplemented
51#2251C2212yescomplemented
52#2250C2212yescomplemented
53#2249C2214yescomplemented
54#2243C2212yescomplemented
55#2243C2212yescomplemented
56#2243C2214yescomplemented
57#2242C2214yescomplemented
58#2241C2212yescomplemented
59#2240C2212yescomplemented
60#2220C2214yes—
61#2220C2214yescomplemented
62#2219C2214yescomplemented
63#2218C2214yes—
64#2217C2214yes—
65#2217C2214yescomplemented
66#2216C2214yescomplemented
67#2215C2214yes—
68#2177C2214yescomplemented
69#2176C2214yescomplemented
70#2175C2214yescomplemented
71#2175C2214yescomplemented
72C4×Q32C2214yescomplemented
73C4×SD32C2214yescomplemented
74C4×SD32C2214yescomplemented
75C4×D16C2214yescomplemented
76#2159C2212yes—
77#2159C2212yes—
78#2141C2212yes—
79#2140C2212yes—
80#2139C2212yes—
81#2139C2212yes—
82#2108C2214yescomplemented
83#2108C2214yescomplemented
84C4×M32C2211yescharacteristic
85C4×M32C2212yes—
86C4×M32C2211yescharacteristic
87C2×#135—21no—
88C2×#135C2312yes—
89C22×Q16C2311yescharacteristic
90C22×SD16—21no—
91C22×D8C2311yescharacteristic
92C22×M16D411yescharacteristic
93C23×C8C2311yescharacteristic
94#542—22no—
95#542—22no—
96#542—24no—
97#542—24no—
98#542—42no—
99#542—42no—
100#541—22no—
101#540—22no—
102#539—41no—
103#539—41no—
104#538—41no—
105#538—41no—
106#533—21no—
107#532—21no—
108C2×C4⋊Q8—21no—
109#529—42no—
110#529—42no—
111C2×#126—41no—
112C2×#124—21no—
113C2×#124—41no—
114#525—22no—
115C2×#123—41no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—41no—
118C2×#121—21no—
119C4×#41—22no—
120C2×C4×Q8—21no—
121C2×C4×D4—21no—
122C2×#117D411yescharacteristic
123#510—28no—
124#509—28no—
125C2×Q32C2314yes—
126C2×SD32C2314yes—
127C2×SD32C2314yes—
128C2×D16C2314yes—
129C2×M32—28no—
130C2×M32C2312yes—
131C2×M32C2314yes—
132C2×M32C2312yes—
133C22×C16C2314yes—
134#500—21no—
135#500—21no—
136#500C2312yes—
137#500C2312yes—
138#500C2311yescharacteristic
139#500C2311yescharacteristic
140#499—21no—
141#499C2312yes—
142#498—21no—
143#495—21no—
144#495—21no—
145#495—21no—
146#495C2312yes—
147#495C2312yes—
148#495C2312yes—
149#494C2311yescharacteristic
150#493C2311yescharacteristic
151#492—21no—
152#491—22no—
153#490—22no—
154#489—24no—
155#489—24no—
156#489—22no—
157#489—22no—
158#470—22no—
159#469—22no—
160#468—22no—
161#468—22no—
162#468—24no—
163#468—24no—
164#467—22no—
165#467C2314yes—
166#467C2312yes—
167#467C2312yes—
168#466—22no—
169#466C2314yes—
170#466C2312yes—
171#466C2312yes—
172#465—22no—
173#465—22no—
174#442—22no—
175#441—22no—
176#440—22no—
177#440—22no—
178#440—24no—
179#440—24no—
180C4×Q16—22no—
181C4×Q16C2314yes—
182C4×Q16C2312yes—
183C4×Q16C2312yes—
184C4×SD16—22no—
185C4×SD16—22no—
186C4×D8—22no—
187C4×D8C2314yes—
188C4×D8C2312yes—
189C4×D8C2312yes—
190#427—21no—
191#427C2312yes—
192C2×#107—21no—
193C2×#107C2311yescharacteristic
194C2×#107C2311yescharacteristic
195#416—22no—
196#416—22no—
197C2×#103—21no—
198C2×#103—21no—
199C2×#102—21no—
200C2×#102—21no—
201#405D412yes—
202#405D412yes—
203#405C2312yes—
204#405C2312yes—
205C2×#97D411yescharacteristic
206C2×#97D411yescharacteristic
207C2×C4×C8C2311yescharacteristic
208C2×C4×C8C2311yescharacteristic
209#367C2314yes—
210#367C2314yes—
211#366C2314yes—
212#366C2314yes—
213#358C2314yes—
214#358C2314yes—
215#358C2314yes—
216#358C2314yes—
217#357C2314yes—
218#357C2314yes—
219#357C2314yes—
220#357C2314yes—
221#346C2314yes—
222#346C2312yes—
223#346C2312yes—
224C4×C16C2314yes—
225C4×C16C2312yes—
226C4×C16C2312yes—
227C2×#41—22no—
228C22×Q8—21no—
229C22×Q8—41no—
230C22×D4—21no—
231C22×D4—41no—
232C23×C4—41no—
233C23×C4C2×D411yescharacteristic
234#135—24no—
235#135—24no—
236#135—44no—
237C2×Q16—21no—
238C2×Q16—21no—
239C2×Q16—24no—
240C2×Q16C2412yes—
241C2×Q16C2412yes—
242C2×SD16—22no—
243C2×SD16—22no—
244C2×SD16—22no—
245C2×SD16—22no—
246C2×SD16—44no—
247C2×D8—21no—
248C2×D8—21no—
249C2×D8—24no—
250C2×D8C2412yes—
251C2×D8C2412yes—
252C2×M16—24no—
253C2×M16C2×D412yes—
254C2×M16C2×D412yes—
255C22×C8—24no—
256C22×C8C2×D411yescharacteristic
257C22×C8C2×D412yes—
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D411yescharacteristic
260C22×C8C2412yes—
261C22×C8C2411yescharacteristic
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—41no—
268C4⋊Q8—41no—
269#127—21no—
270#126—44no—
271#126—44no—
272#126—42no—
273#126—42no—
274#126—42no—
275#126—42no—
276#125—22no—
277#125—42no—
278#125—42no—
279#124—22no—
280#124—22no—
281#124—22no—
282#124—44no—
283#124—44no—
284#124—42no—
285#124—42no—
286#124—41no—
287#124—41no—
288#123—42no—
289#123—42no—
290#123—42no—
291#123—44no—
292#123—44no—
293#123—42no—
294#123—42no—
295C22⋊Q8—24no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—42no—
299C22⋊Q8—41no—
300C22⋊Q8—41no—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303C22⋊Q8—42no—
304C22⋊Q8—42no—
305#121—24no—
306#121—22no—
307#121—22no—
308#121—42no—
309#121—41no—
310#121—41no—
311#121—42no—
312#121—42no—
313#120—42no—
314C4×Q8—24no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—42no—
318C4×Q8—41no—
319C4×Q8—41no—
320C4×D4—24no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—42no—
324C4×D4—42no—
325C4×D4—42no—
326C4×D4—42no—
327C4×D4—42no—
328C4×D4—41no—
329C4×D4—41no—
330#117—22no—
331#117—42no—
332#117D812yes—
333#117D812yes—
334#117C2×D412yes—
335#117C2×D412yes—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—41no—
339C2×C4⋊C4—41no—
340C2×C4⋊C4—41no—
341C2×C4⋊C4D811yescharacteristic
342C2×C4⋊C4D811yescharacteristic
343C2×C22⋊C4—41no—
344C2×C22⋊C4—41no—
345C2×C22⋊C4—41no—
346C2×C22⋊C4—41no—
347C2×C22⋊C4—41no—
348C2×C22⋊C4D811yescharacteristic
349C2×C22⋊C4D811yescharacteristic
350C2×C42—41no—
351C2×C42—42no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354Q32—28no—
355SD32—28no—
356SD32—28no—
357D16—28no—
358M32—24no—
359M32—28no—
360M32#4118yes—
361C2×C16—28no—
362C2×C16C2414yes—
363C2×C16C2414yes—
364#107—22no—
365#107—22no—
366#107—21no—
367#107—21no—
368#107—21no—
369#107—21no—
370#107C2412yes—
371#107C2412yes—
372#107C2412yes—
373#107C2412yes—
374#106—22no—
375#106—22no—
376#103—24no—
377#103—24no—
378#103—22no—
379#103—22no—
380#103—22no—
381#103—22no—
382#102—24no—
383#102—24no—
384#102—22no—
385#102—22no—
386#102—22no—
387#102—22no—
388#97C2×D412yes—
389#97C2×D412yes—
390#97C2×D412yes—
391#97C2×D412yes—
392C4×C8C2×D412yes—
393C4×C8C2×D411yescharacteristic
394C4×C8C2×D411yescharacteristic
395C4×C8C2412yes—
396C4×C8C2411yescharacteristic
397C4×C8C2411yescharacteristic
398C24—41no—
399C24C2×D811yescharacteristic
400#41—44no—
401#41—44no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—41no—
405C2×Q8—41no—
406C2×Q8—41no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—162no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—42no—
423C2×D4—162no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—162no—
434C22×C4C2×D811yescharacteristic
435C22×C4C2×D812yes—
436C22×C4C2×D811yescharacteristic
437C22×C4C2×D811yescharacteristic
438C22×C4C2×D811yescharacteristic
439C22×C4C2×D811yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442Q16—22no—
443Q16—24no—
444Q16—22no—
445Q16—42no—
446Q16—42no—
447SD16—44no—
448SD16—44no—
449SD16—44no—
450SD16—44no—
451D8—22no—
452D8—24no—
453D8—22no—
454D8—42no—
455D8—42no—
456M16—24no—
457M16—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8#13514yes—
464C2×C8C22×D412yes—
465C2×C8C22×D412yes—
466C2×C8C22×D411yescharacteristic
467C2×C8C22×D411yescharacteristic
468C2×C8C22×D411yescharacteristic
469C2×C8C2×#4114yes—
470C2×C8C2511yescharacteristic
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4C2×D812yes—
492C4⋊C4C2×D812yes—
493C4⋊C4C2×D812yes—
494C4⋊C4C2×D812yes—
495C22⋊C4—44no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4C2×D812yes—
512C22⋊C4C2×D812yes—
513C22⋊C4C2×D812yes—
514C22⋊C4C2×D812yes—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—44no—
519C42C2×D812yes—
520C42C2×D811yescharacteristic
521C42C2×D811yescharacteristic
522C42C22×D412yes—
523C42C22×D411yescharacteristic
524C42C22×D411yescharacteristic
525C16—24no—
526C16C2×#4118yes—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23C22×D812yes—
536C23C22×D811yescharacteristic
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—82no—
542Q8—161no—
543Q8—161no—
544Q8—162no—
545Q8—162no—
546D4—42no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—84no—
551D4—82no—
552D4—161no—
553D4—161no—
554D4—162no—
555D4—162no—
556D4—164no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—84no—
575C2×C4—162no—
576C2×C4—162no—
577C2×C4—164no—
578C2×C4#53212yes—
579C2×C4C22×D812yes—
580C2×C4C22×D812yes—
581C2×C4C22×D811yescharacteristic
582C2×C4C22×D811yescharacteristic
583C2×C4C22×D811yescharacteristic
584C2×C4C22×D811yescharacteristic
585C2×C4C22×D812yes—
586C2×C4C22×D811yescharacteristic
587C2×C4C2×#13512yes—
588C2×C4C23×D411yescharacteristic
589C8—22no—
590C8—22no—
591C8#53212yes—
592C8C2×#13512yes—
593C8C2×#13512yes—
594C8C22×#4112yes—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—164no—
608C22#316912yes—
609C22C23×D811yescharacteristic
610C4—22no—
611C4—21no—
612C4—21no—
613C4—22no—
614C4—21no—
615C4—21no—
616C4—22no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4#316912yes—
624C4C2×#53211yescharacteristic
625C4C22×#13511yescharacteristic
626C2—22no—
627C2—42no—
628C2—162no—
629C2—162no—
630C2C2×#316911yescharacteristic
631C2#6263611yescharacteristic
632C2#6263711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60682C2semidirect
2. C2×#3175C2semidirect
3. #33745C2semidirect
4. #33744C2semidirect
5. #33742C2semidirect
6. #33739C2semidirect
7. #33731C2semidirect
8. #33719C2semidirect
9. #33707C2semidirect
10. #33706C2semidirect
11. #33606C2semidirect
12. #33605C2semidirect
13. C4×#510C2semidirect
14. C4×#509C2semidirect
15. #33324C2semidirect
21. #3177C22semidirect
22. #3176C22semidirect
27. #3175C22semidirect
28. #3175C22semidirect
48. #2255C22semidirect
49. #2254C22semidirect
50. #2252C22semidirect
51. #2251C22semidirect
52. #2250C22semidirect
53. #2249C22semidirect
54. #2243C22semidirect
55. #2243C22semidirect
56. #2243C22semidirect
57. #2242C22semidirect
58. #2241C22semidirect
59. #2240C22semidirect
61. #2220C22semidirect
62. #2219C22semidirect
65. #2217C22semidirect
66. #2216C22semidirect
68. #2177C22semidirect
69. #2176C22semidirect
70. #2175C22semidirect
71. #2175C22semidirect
72. C4×Q32C22semidirect
73. C4×SD32C22semidirect
74. C4×SD32C22semidirect
75. C4×D16C22semidirect
82. #2108C22semidirect
83. #2108C22semidirect

Maximal subgroups

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

subgroupindex
#606822
C2×#31752
#337452
#337442
#337422
#337392
#337312
#337192
#337072
#337062
#336062
#336052
C4×#5102
C4×#5092
#333242