Finite Group Databasebeta

← #7623977

Subgroups — #7623977

Order 512.

1350 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#61413C211yesmaximal, characteristic, complemented
2#61411C211yesmaximal, characteristic, complemented
3#33969C214yesmaximal, complemented
4#33968C214yesmaximal, complemented
5#33967C214yesmaximal, complemented
6#33965C214yesmaximal, complemented
7#33722C211yesmaximal, characteristic
8#33721C211yesmaximal, characteristic, complemented
9C2×#2251C211yesmaximal, characteristic, complemented
10#33588C211yesmaximal, characteristic, complemented
11#33588C211yesmaximal, characteristic, complemented
12#33587C211yesmaximal, characteristic, complemented
13#33586C211yesmaximal, characteristic, complemented
14#33584C211yesmaximal, characteristic, complemented
15#33583C211yesmaximal, characteristic, complemented
16#33475C211yesmaximal, characteristic, complemented
17#33458C211yesmaximal, characteristic, complemented
18#33457C211yesmaximal, characteristic, complemented
19#33451C211yesmaximal, characteristic, complemented
20#3510—21no—
21#3509—21no—
22C2×#510C2211yescharacteristic
23C2×#510C2211yescharacteristic
24C2×#510C2211yescharacteristic
25C2×#509C2211yescharacteristic
26C2×#508C2211yescharacteristic
27C22×Q32C2211yescharacteristic
28C22×Q32C2211yescharacteristic
29C22×SD32C2211yescharacteristic
30C22×SD32C2211yescharacteristic
31#3401—24no—
32#3398—24no—
33#3396C2214yes—
34#3395C2214yescomplemented
35#3393C2214yes—
36#3393C2214yescomplemented
37#3156C2211yescharacteristic
38#3156C2211yescharacteristic
39#3156C2211yescharacteristic
40#3155C2211yescharacteristic
41C2×#495C2211yescharacteristic
42C2×#494C2211yescharacteristic
43#3044—21no—
44#3044—21no—
45#3043—21no—
46#3042—21no—
47#3040—21no—
48#3039—21no—
49#2960C2211yescharacteristic
50#2950C2211yescharacteristic
51#2950C2211yescharacteristic
52#2950C2211yescharacteristic
53#2948C2211yescharacteristic
54C4×#135C2211yescharacteristic
55C4×#135C2211yescharacteristic
56#2251C2212yes—
57#2251C2214yes—
58#2251C2214yes—
59#2251C2212yescomplemented
60#2251C2212yescomplemented
61#2250C2212yes—
62#2249C2212yes—
63#2249C2212yescomplemented
64#2248C2212yes—
65#2247C2212yescomplemented
66#2211C2214yes—
67#2211C2214yes—
68#2211C2214yescomplemented
69#2211C2214yescomplemented
70#2210C2214yes—
71#2210C2214yes—
72#2209C2214yescomplemented
73#2209C2214yescomplemented
74#2209C2214yescomplemented
75#2208C2214yes—
76#2208C2214yescomplemented
77#2207C2214yes—
78#2183C2214yescomplemented
79#2183C2214yescomplemented
80#2177C2212yescomplemented
81#2176C2212yescomplemented
82#2176C2214yescomplemented
83#2175C2214yes—
84#2175C2212yescomplemented
85#2175C2212yescomplemented
86C4×Q32C2214yescomplemented
87C4×SD32C2214yes—
88#2158C2211yescharacteristic
89#2151C2211yescharacteristic
90#2151C2211yescharacteristic
91#2150C2211yescharacteristic
92#2141C2211yescharacteristic
93#2141C2211yescharacteristic
94#2141C2211yescharacteristic
95#2140C2211yescharacteristic
96#2138C2211yescharacteristic
97#2138C2211yescharacteristic
98#2108C2211yescharacteristic
99#2108C2211yescharacteristic
100C2×#346C2211yescharacteristic
101C2×#137—21no—
102C2×#137—21no—
103C2×#137—21no—
104C2×AGL(1,ℤ/8)—21no—
105C2×#135C2311yescharacteristic
106C2×#135C2311yescharacteristic
107C22×Q16—21no—
108C22×Q16C2311yescharacteristic
109C22×Q16C2311yescharacteristic
110C22×Q16C2311yescharacteristic
111C22×SD16—21no—
112C22×D8C2311yescharacteristic
113#559—24no—
114#557—24no—
115#556—24no—
116#556—24no—
117#552—44no—
118#551—44no—
119#542—42no—
120#541—42no—
121#536—21no—
122#536—21no—
123#536—21no—
124#535—21no—
125#533—42no—
126#533D411yescharacteristic
127C2×C4⋊Q8—21no—
128#529—42no—
129C2×#124—21no—
130#519—21no—
131#519—21no—
132#519—21no—
133#518—21no—
134C4×#41—21no—
135C4×#41—21no—
136#510—42no—
137#510—42no—
138#510—42no—
139#509—42no—
140#508—24no—
141C2×Q32—24no—
142C2×Q32—42no—
143C2×Q32C2312yes—
144C2×Q32C2312yes—
145C2×Q32C2311yescharacteristic
146C2×Q32C2314yes—
147C2×Q32C2312yes—
148C2×Q32C2312yes—
149C2×Q32C2312yes—
150C2×SD32—24no—
151C2×SD32—42no—
152C2×SD32C2312yes—
153C2×SD32C2312yes—
154C2×SD32C2311yescharacteristic
155C2×SD32C2311yescharacteristic
156C2×SD32C2314yes—
157C2×SD32C2312yes—
158C2×SD32C2312yes—
159C2×SD32C2312yes—
160C2×SD32C2312yes—
161C2×D16C2311yescharacteristic
162C2×D16C2312yes—
163C2×M32C2311yescharacteristic
164C2×M32C2311yescharacteristic
165C22×C16C2311yescharacteristic
166C22×C16C2311yescharacteristic
167#497—22no—
168#497—22no—
169#497—22no—
170#496—22no—
171#495—22no—
172#495C2312yes—
173#495C2314yes—
174#495C2312yes—
175#495C2312yes—
176#495C2312yes—
177#494—22no—
178#494C2312yes—
179#494C2314yes—
180#494C2312yes—
181#494C2312yes—
182#493C2312yes—
183#464—24no—
184#464—24no—
185#464—24no—
186#464—24no—
187#463—24no—
188#463—24no—
189#462—24no—
190#462—24no—
191#462—24no—
192#461—24no—
193#461—24no—
194#460—24no—
195#446D414yes—
196C8×Q8C2314yes—
197#442—22no—
198#441—22no—
199#441—22no—
200#441—22no—
201C4×Q16C2312yes—
202C4×Q16C2314yes—
203C4×Q16C2314yes—
204C4×Q16C2314yes—
205C4×Q16C2312yes—
206C4×SD16—22no—
207C4×SD16—22no—
208C4×D8C2314yes—
209C4×D8C2312yes—
210C4×D8C2312yes—
211#427C2311yescharacteristic
212#427C2311yescharacteristic
213#424D411yescharacteristic
214#424D411yescharacteristic
215#423D411yescharacteristic
216#419—21no—
217#419—21no—
218#419—21no—
219#418—21no—
220#416—21no—
221#416—21no—
222#405C2311yescharacteristic
223#405C2311yescharacteristic
224C2×C4×C8C2311yescharacteristic
225#367C2312yes—
226#366C2312yes—
227C4⋊C16C2314yes—
228C4⋊C16C2314yes—
229C4⋊C16C2314yes—
230#358C2312yes—
231#358C2312yes—
232#358C2314yes—
233#358C2314yes—
234#358C2314yes—
235#357C2312yes—
236#357C2312yes—
237#357C2314yes—
238#346C2312yes—
239#346C2312yes—
240#346C2312yes—
241#346C2312yes—
242C4×C16C2312yes—
243C4×C16C2312yes—
244C2×#41—21no—
245C2×#41—21no—
246C22×Q8—21no—
247C22×Q8—21no—
248C22×Q8—21no—
249C22×D4—21no—
250#137—82no—
251#137—82no—
252#137—82no—
253AGL(1,ℤ/8)—82no—
254#135—22no—
255#135—22no—
256#135—22no—
257#135—22no—
258C2×Q16—21no—
259C2×Q16—21no—
260C2×Q16—21no—
261C2×Q16—24no—
262C2×Q16—22no—
263C2×Q16—22no—
264C2×Q16—22no—
265C2×Q16—22no—
266C2×Q16—22no—
267C2×Q16—22no—
268C2×Q16—22no—
269C2×Q16—22no—
270C2×Q16—22no—
271C2×Q16—82no—
272C2×Q16C2412yes—
273C2×Q16C2412yes—
274C2×Q16C2412yes—
275C2×Q16C2411yescharacteristic
276C2×Q16C2411yescharacteristic
277C2×SD16—21no—
278C2×SD16—21no—
279C2×SD16—24no—
280C2×SD16—22no—
281C2×SD16—22no—
282C2×SD16—22no—
283C2×SD16—22no—
284C2×SD16—82no—
285C2×D8—21no—
286C2×D8—22no—
287C2×D8—22no—
288C2×D8—22no—
289C2×D8C2412yes—
290C2×D8C2411yescharacteristic
291C2×D8C2411yescharacteristic
292C2×M16C2×D411yescharacteristic
293C2×M16C2×D411yescharacteristic
294C22×C8C2×D411yescharacteristic
295C22×C8C2411yescharacteristic
296C4⋊Q8—24no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—41no—
301C4⋊Q8—41no—
302C4⋊Q8D812yes—
303#127—22no—
304#126—44no—
305#126—44no—
306#126—44no—
307#126—44no—
308#125—44no—
309#125—44no—
310#125—44no—
311#125D812yes—
312#124—24no—
313#124—22no—
314#124—22no—
315#124—22no—
316#124—22no—
317#124—44no—
318#124—41no—
319#124—41no—
320#123—44no—
321#123—44no—
322#123—42no—
323#123—42no—
324#123—42no—
325C22⋊Q8—44no—
326C22⋊Q8—44no—
327C22⋊Q8—42no—
328C22⋊Q8—42no—
329C22⋊Q8—42no—
330C22⋊Q8D814yes—
331#121—42no—
332#120—42no—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×Q8—24no—
336C4×Q8—24no—
337C4×Q8—24no—
338C4×Q8—44no—
339C4×Q8—42no—
340C4×Q8—42no—
341C4×Q8—42no—
342C4×Q8C2×D414yes—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—24no—
346C4×D4—42no—
347C4×D4—42no—
348C4×D4—42no—
349C4×D4—44no—
350#117—21no—
351#117—21no—
352#117—44no—
353#117—44no—
354#117—42no—
355#117—42no—
356#117—42no—
357#117—41no—
358#117—41no—
359#117C2×D411yescharacteristic
360#117C2×D411yescharacteristic
361C2×C4⋊C4—42no—
362C2×C22⋊C4—42no—
363C2×C22⋊C4—42no—
364C2×C42—41no—
365C2×C42—41no—
366C2×C42C2×D411yescharacteristic
367Q32—24no—
368Q32—24no—
369Q32—22no—
370Q32—44no—
371Q32—42no—
372Q32—42no—
373Q32—42no—
374SD32—24no—
375SD32—24no—
376SD32—22no—
377SD32—22no—
378SD32—44no—
379SD32—42no—
380SD32—42no—
381SD32—42no—
382SD32—42no—
383D16—22no—
384D16—42no—
385M32—22no—
386M32—22no—
387C2×C16—22no—
388C2×C16#4114yes—
389C2×C16C2412yes—
390C2×C16C2412yes—
391C2×C16C2412yes—
392C2×C16C2411yescharacteristic
393C2×C16C2411yescharacteristic
394#107C2412yes—
395#107C2412yes—
396#106—21no—
397#106—21no—
398C4⋊C8C2×D414yes—
399C4⋊C8C2×D414yes—
400C4⋊C8C2×D414yes—
401#103—22no—
402#103—22no—
403#103—24no—
404#103—24no—
405#103—24no—
406#102—22no—
407#102—22no—
408#102—24no—
409#97C2×D412yes—
410#97C2×D412yes—
411C4×C8C2×D412yes—
412C4×C8C2412yes—
413C4×C8C2412yes—
414C4×C8C2412yes—
415C24—41no—
416#41—42no—
417#41—42no—
418#41—42no—
419#41—42no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—41no—
426C2×Q8—41no—
427C2×Q8—41no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8C2×D814yes—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—21no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—42no—
441C2×D4—41no—
442C2×D4—44no—
443C2×D4—42no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4C2×D812yes—
453C22×C4C22×D411yescharacteristic
454Q16—22no—
455Q16—21no—
456Q16—21no—
457Q16—22no—
458Q16—22no—
459Q16—22no—
460Q16—21no—
461Q16—22no—
462Q16—22no—
463Q16—21no—
464Q16—42no—
465Q16—42no—
466Q16—42no—
467Q16—84no—
468Q16—82no—
469Q16—82no—
470Q16—82no—
471SD16—42no—
472SD16—42no—
473SD16—84no—
474SD16—82no—
475SD16—82no—
476SD16—82no—
477SD16—82no—
478D8—21no—
479D8—21no—
480D8—22no—
481D8—21no—
482D8—22no—
483D8—21no—
484D8—42no—
485D8—82no—
486M16—22no—
487M16—22no—
488C2×C8—22no—
489C2×C8C22×D412yes—
490C2×C8C22×D412yes—
491C2×C8C22×D412yes—
492C2×C8C22×D411yescharacteristic
493C2×C8C2×#4112yes—
494C2×C8H(5,𝔽2)12yes—
495C2×C8C2511yescharacteristic
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4C2×D814yes—
509C4⋊C4C2×D814yes—
510C4⋊C4C2×D812yes—
511C4⋊C4C2×D812yes—
512C4⋊C4C2×D814yes—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4C2×D812yes—
525C22⋊C4C2×D812yes—
526C42—41no—
527C42—41no—
528C42—42no—
529C42—42no—
530C42—44no—
531C42—44no—
532C42—44no—
533C42C2×D811yescharacteristic
534C42C2×D811yescharacteristic
535C42C22×D412yes—
536C42C22×D412yes—
537C42C22×D412yes—
538C16—22no—
539C16—22no—
540C16—22no—
541C16C2×#4112yes—
542C16C2×#4112yes—
543C23—41no—
544C23—42no—
545C23—41no—
546C23—82no—
547C23—162no—
548C23C22×D811yescharacteristic
549Q8—44no—
550Q8—41no—
551Q8—41no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—41no—
557Q8—42no—
558Q8—42no—
559Q8—41no—
560Q8—82no—
561Q8—82no—
562Q8—82no—
563D4—42no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—42no—
568D4—41no—
569D4—82no—
570D4—82no—
571D4—82no—
572D4—164no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—162no—
588C2×C4—164no—
589C2×C4#55012yes—
590C2×C4C22×D814yes—
591C2×C4C22×D812yes—
592C2×C4C22×D812yes—
593C2×C4C22×D814yes—
594C2×C4C22×D811yescharacteristic
595C2×C4#57612yes—
596C2×C4C23×D411yescharacteristic
597C8—22no—
598C8—22no—
599C8—22no—
600C8#55012yes—
601C8C22×#4111yescharacteristic
602C8C2×H(5,𝔽2)11yescharacteristic
603C22—22no—
604C22—82no—
605C22—82no—
606C22—84no—
607C22—81no—
608C22—81no—
609C22—82no—
610C22—322no—
611C22#339112yes—
612C22C23×D811yescharacteristic
613C4—22no—
614C4—22no—
615C4—24no—
616C4—24no—
617C4—82no—
618C4—82no—
619C4—81no—
620C4—81no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4#339112yes—
625C4C2×#55011yescharacteristic
626C4C2×#57611yescharacteristic
627C2—22no—
628C2—161no—
629C2—161no—
630C2—162no—
631C2C2×#339111yescharacteristic
632C2#6263712yes—
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#33722C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #533 ◁ #2960 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #61413C2semidirect
2. #61411C2semidirect
3. #33969C2semidirect
4. #33968C2semidirect
5. #33967C2semidirect
6. #33965C2semidirect
8. #33721C2semidirect
9. C2×#2251C2semidirect
10. #33588C2semidirect
11. #33588C2semidirect
12. #33587C2semidirect
13. #33586C2semidirect
14. #33584C2semidirect
15. #33583C2semidirect
16. #33475C2semidirect
17. #33458C2semidirect
18. #33457C2semidirect
19. #33451C2semidirect
34. #3395C22semidirect
36. #3393C22semidirect
59. #2251C22semidirect
60. #2251C22semidirect
63. #2249C22semidirect
65. #2247C22semidirect
68. #2211C22semidirect
69. #2211C22semidirect
72. #2209C22semidirect
73. #2209C22semidirect
74. #2209C22semidirect
76. #2208C22semidirect
78. #2183C22semidirect
79. #2183C22semidirect
80. #2177C22semidirect
81. #2176C22semidirect
82. #2176C22semidirect
84. #2175C22semidirect
85. #2175C22semidirect
86. C4×Q32C22semidirect

Maximal subgroups

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

subgroupindex
#614132
#614112
#339692
#339682
#339672
#339652
#337222
#337212
C2×#22512
#335882
#335872
#335862
#335842
#335832
#334752
#334582
#334572
#334512