Finite Group Databasebeta

← #7624329

Subgroups — #7624329

Order 512.

1292 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62328C211yesmaximal, characteristic, complemented
2#61287C211yesmaximal, characteristic, complemented
3#33924C212yesmaximal, complemented
4#33921C212yesmaximal, complemented
5#33908C212yesmaximal, complemented
6#33907C212yesmaximal, complemented
7#33892C212yesmaximal, complemented
8#33891C212yesmaximal, complemented
9#33862C212yesmaximal, complemented
10#33861C212yesmaximal, complemented
11#33793C211yesmaximal, characteristic, complemented
12#33792C211yesmaximal, characteristic, complemented
13#33764C214yesmaximal, complemented
14#33666C211yesmaximal, characteristic, complemented
15#33666C211yesmaximal, characteristic, complemented
16#33650C214yesmaximal, complemented
17#33493C211yesmaximal, characteristic, complemented
18#3549—21no—
19#3511—21no—
20#3357—22no—
21#3353—22no—
22#3342C2212yescomplemented
23#3341C2212yes—
24#3341C2212yes—
25#3341C2212yescomplemented
26#3338C2212yescomplemented
27#3334C2212yescomplemented
28#3325—22no—
29#3325—22no—
30D4×D8C2212yes—
31D4×D8C2212yes—
32#3253—21no—
33#3199—22no—
34#3169C2211yescharacteristic
35#3169C2211yescharacteristic
36C2×#500C2211yescharacteristic, complemented
37#3098—22no—
38#3079—21no—
39#3061C2212yescomplemented
40#3058C2214yes—
41C4×#135C2211yescharacteristic, complemented
42#2931C2211yescharacteristic
43#2913C2212yescomplemented
44#2254C2212yescomplemented
45#2252C2212yescomplemented
46#2240C2212yes—
47#2240C2212yes—
48#2237C2214yescomplemented
49#2234C2214yescomplemented
50#2234C2214yescomplemented
51#2234C2212yescomplemented
52#2234C2212yescomplemented
53#2234C2214yescomplemented
54#2233C2214yescomplemented
55#2233C2214yescomplemented
56#2232C2214yescomplemented
57#2232C2214yescomplemented
58#2227C2211yescharacteristic, complemented
59#2227C2211yescharacteristic, complemented
60#2226C2212yescomplemented
61#2226C2212yescomplemented
62#2226C2211yescharacteristic, complemented
63#2226C2211yescharacteristic, complemented
64#2224C2212yescomplemented
65#2224C2212yescomplemented
66#2220C2214yescomplemented
67#2218C2214yescomplemented
68#2217C2214yescomplemented
69#2215C2214yescomplemented
70#2207C2211yescharacteristic
71#2207C2211yescharacteristic
72#2206C2211yescharacteristic
73#2206C2211yescharacteristic
74#2187C2214yescomplemented
75#2186C2212yescomplemented
76#2186C2212yescomplemented
77#2184C2214yescomplemented
78C4×SD32C2212yescomplemented
79C4×SD32C2212yescomplemented
80C4×D16C2212yescomplemented
81C4×D16C2212yescomplemented
82#2169C2212yescomplemented
83#2169C2212yescomplemented
84D4×C16C2212yescomplemented
85D4×C16C2212yescomplemented
86#2162C2211yescharacteristic, complemented
87#2162C2211yescharacteristic, complemented
88#2159C2214yescomplemented
89#2158C2214yescomplemented
90#2140C2214yescomplemented
91C2×#357C2214yescomplemented
92#2113C2214yescomplemented
93C2×#135C2311yescharacteristic
94C22×D8C2314yes—
95C2×#131C2312yes—
96H(3,𝔽4)—42no—
97#556—22no—
98#555—22no—
99#555—22no—
100#555—22no—
101#555—41no—
102#554—22no—
103#553—42no—
104#548—42no—
105#547—22no—
106D42—22no—
107D42—22no—
108D42—41no—
109#542—42no—
110#537—22no—
111#535—24no—
112#535—42no—
113#535—42no—
114#533—41no—
115#532—21no—
116#532—21no—
117#532—41no—
118#532D411yescharacteristic
119C2×C4⋊Q8—21no—
120#520—22no—
121C4×#41—21no—
122C2×SD32C2312yes—
123C2×SD32C2312yes—
124C2×D16C2312yes—
125C2×D16C2312yes—
126C2×M32C2314yes—
127C22×C16C2314yes—
128#500—21no—
129#500—21no—
130#500C2312yes—
131#500C2312yes—
132#500C2311yescharacteristic
133#500C2311yescharacteristic
134#493C2311yescharacteristic
135#493C2311yescharacteristic
136#486—21no—
137#486—21no—
138#485—24no—
139#484—24no—
140#482—24no—
141#482—24no—
142#482—24no—
143#480—24no—
144#477—22no—
145#476—22no—
146#476—22no—
147#476—22no—
148#475—22no—
149#474—22no—
150#469—24no—
151#469—24no—
152#468—24no—
153#467C2314yes—
154#466—21no—
155#466—21no—
156#466C2314yes—
157#466C2314yes—
158#466C2314yes—
159#466C2312yes—
160#466C2312yes—
161#459—22no—
162#459—22no—
163#447—24no—
164#447—24no—
165C4×Q16C2311yescharacteristic
166C4×Q16C2311yescharacteristic
167C4×SD16—21no—
168C4×SD16—21no—
169C4×D8—21no—
170C4×D8—21no—
171C4×D8C2311yescharacteristic
172C4×D8C2312yes—
173C4×D8C2312yes—
174C4×D8C2311yescharacteristic
175#436D412yes—
176#436D412yes—
177C8×D4C2312yes—
178C8×D4C2312yes—
179#428—22no—
180#428—24no—
181#427C2314yes—
182#427C2314yes—
183#427C2311yescharacteristic
184C2×#107—21no—
185C2×#107C2314yes—
186C2×#107C2312yes—
187#423D411yescharacteristic
188#418—24no—
189#417—22no—
190#416—21no—
191#408D412yes—
192C2×C4×C8C2311yescharacteristic
193#367C2312yes—
194#367C2314yes—
195#367C2312yes—
196#367C2314yes—
197#366C2312yes—
198#366C2314yes—
199#366C2312yes—
200#366C2314yes—
201C4⋊C16C2311yescharacteristic
202C4⋊C16C2311yescharacteristic
203C4⋊C16C2311yescharacteristic
204C4⋊C16C2311yescharacteristic
205#358C2312yes—
206#358C2312yes—
207#357C2312yes—
208#357C2312yes—
209#357C2314yes—
210#357C2314yes—
211#357C2312yes—
212#357C2312yes—
213#357C2314yes—
214#357C2314yes—
215#357C2312yes—
216#357C2312yes—
217C22⋊C16C2312yes—
218C22⋊C16C2312yes—
219C22⋊C16C2314yes—
220C22⋊C16C2314yes—
221C4×C16C2312yes—
222C4×C16C2312yes—
223C2×#41—21no—
224C2×#41—42no—
225C2×#41—41no—
226C2×#41—41no—
227C2×#41C2×D412yes—
228C22×Q8—41no—
229C22×D4—24no—
230C22×D4—42no—
231C22×D4—42no—
232#135—41no—
233#135—41no—
234C2×Q16C2411yescharacteristic
235C2×SD16—21no—
236C2×D8—24no—
237C2×D8—21no—
238C2×D8—21no—
239C2×D8—44no—
240C2×D8—44no—
241C2×D8C2412yes—
242C2×D8C2412yes—
243C2×D8C2411yescharacteristic
244#131—42no—
245#131—42no—
246C2×M16C2×D412yes—
247C2×M16C2×D414yes—
248C22×C8C2×D411yescharacteristic
249C22×C8C2411yescharacteristic
250C22×C8C2414yes—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—41no—
256C4⋊Q8—41no—
257C4⋊Q8D811yescharacteristic
258#127—21no—
259#127—21no—
260#127—44no—
261#127D811yescharacteristic
262#126—44no—
263#125—42no—
264#124—44no—
265#124—42no—
266#124—44no—
267#124—44no—
268#124—44no—
269#124—41no—
270#124—41no—
271#124D812yes—
272#123—44no—
273C22⋊Q8—24no—
274C22⋊Q8—42no—
275C22⋊Q8—42no—
276C22⋊Q8—44no—
277#121—22no—
278#121—22no—
279#121—24no—
280#121—24no—
281#121—24no—
282#121—42no—
283#121—42no—
284#121—42no—
285#121—42no—
286#121—44no—
287#121—41no—
288#121—41no—
289#121—42no—
290#121—42no—
291#121D812yes—
292#121D812yes—
293#120—44no—
294#120—44no—
295#120—44no—
296#120—44no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—42no—
300C4×Q8—42no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—42no—
306C4×D4—44no—
307C4×D4—42no—
308C4×D4—41no—
309C4×D4—41no—
310C4×D4—41no—
311C4×D4—41no—
312C4×D4C2×D412yes—
313C4×D4C2×D412yes—
314#117—21no—
315#117—24no—
316#117—24no—
317#117—42no—
318#117—42no—
319#117—42no—
320#117—41no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—41no—
324C2×C42—41no—
325C2×C42C2×D411yescharacteristic
326SD32—42no—
327SD32—42no—
328D16—42no—
329D16—42no—
330M32—44no—
331C2×C16—44no—
332C2×C16C2412yes—
333C2×C16C2412yes—
334C2×C16C2412yes—
335C2×C16C2412yes—
336#107—22no—
337#107—22no—
338#107—22no—
339#107—24no—
340#107—21no—
341#107—21no—
342#107C2411yescharacteristic
343#107C2414yes—
344#107C2411yescharacteristic
345#107C2412yes—
346#107C2412yes—
347#106—21no—
348#106—24no—
349#106—24no—
350C4⋊C8C2×D412yes—
351C4⋊C8C2×D412yes—
352#103—21no—
353#103—21no—
354#103—24no—
355#102—21no—
356#102—21no—
357#102—24no—
358#102—22no—
359#102—22no—
360#102—24no—
361#102—24no—
362#102—22no—
363#102—22no—
364C22⋊C8C2×D414yes—
365C22⋊C8C2×D414yes—
366C4×C8C2×D411yescharacteristic
367C4×C8C2×D411yescharacteristic
368C4×C8C2411yescharacteristic
369C4×C8C2411yescharacteristic
370C24—44no—
371#41—42no—
372#41—42no—
373#41—42no—
374#41—42no—
375#41—162no—
376#41—162no—
377#41—164no—
378C2×Q8—21no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—162no—
384C2×Q8C2×D812yes—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—41no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—44no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—44no—
399C2×D4—84no—
400C2×D4—84no—
401C2×D4—164no—
402C2×D4—164no—
403C2×D4C2×D812yes—
404C2×D4C2×D812yes—
405C2×D4C2×D812yes—
406C2×D4C2×D814yes—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4C2×D811yescharacteristic
415C22×C4C2×D811yescharacteristic
416C22×C4C22×D414yes—
417C22×C4C22×D411yescharacteristic
418Q16—21no—
419Q16—21no—
420SD16—42no—
421D8—21no—
422D8—21no—
423D8—22no—
424D8—22no—
425D8—22no—
426D8—22no—
427D8—81no—
428D8—81no—
429D8—84no—
430M16—44no—
431M16—42no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—41no—
437C2×C8C22×D411yescharacteristic
438C2×C8C22×D411yescharacteristic
439C2×C8C22×D411yescharacteristic
440C2×C8C22×D414yes—
441C2×C8C2511yescharacteristic
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—24no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—44no—
456C4⋊C4C2×D812yes—
457C4⋊C4C2×D812yes—
458C22⋊C4—41no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4C2×D814yes—
472C22⋊C4C2×D814yes—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—44no—
477C42—44no—
478C42C2×D811yescharacteristic
479C42C2×D811yescharacteristic
480C42C22×D411yescharacteristic
481C42C22×D411yescharacteristic
482C16—22no—
483C16—22no—
484C16—22no—
485C16—22no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—164no—
490C23—164no—
491C23—164no—
492C23C22×D814yes—
493C23C22×D811yescharacteristic
494Q8—42no—
495Q8—41no—
496Q8—41no—
497Q8—84no—
498Q8—162no—
499Q8—161no—
500Q8—161no—
501D4—24no—
502D4—24no—
503D4—42no—
504D4—44no—
505D4—42no—
506D4—42no—
507D4—41no—
508D4—42no—
509D4—42no—
510D4—41no—
511D4—82no—
512D4—82no—
513D4—82no—
514D4—164no—
515D4—161no—
516D4—161no—
517D4—164no—
518D4—164no—
519D4—162no—
520D4—164no—
521D4—162no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—82no—
535C2×C4—84no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—84no—
539C2×C4—162no—
540C2×C4—164no—
541C2×C4C22×D814yes—
542C2×C4C22×D811yescharacteristic
543C2×C4C22×D811yescharacteristic
544C2×C4C22×D811yescharacteristic
545C2×C4C22×D811yescharacteristic
546C2×C4C22×D811yescharacteristic
547C2×C4C23×D411yescharacteristic
548C8—21no—
549C8—21no—
550C8—21no—
551C8—24no—
552C8#58511yescharacteristic
553C8#58511yescharacteristic
554C22—21no—
555C22—24no—
556C22—21no—
557C22—24no—
558C22—21no—
559C22—24no—
560C22—81no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—81no—
565C22—82no—
566C22—81no—
567C22—82no—
568C22—82no—
569C22—328no—
570C22C23×D811yescharacteristic
571C4—21no—
572C4—21no—
573C4—21no—
574C4—24no—
575C4—81no—
576C4—82no—
577C4—84no—
578C4—81no—
579C4—82no—
580C4—81no—
581C4#341512yes—
582C4#341612yes—
583C4#348511yescharacteristic
584C4#358611yescharacteristic
585C2—41no—
586C2—44no—
587C2—161no—
588C2—162no—
589C2—162no—
590C2#6169411yescharacteristic
591C2C2×#341511yescharacteristic
592C2C2×#341611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #532 ◁ #2931 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62328C2semidirect
2. #61287C2semidirect
3. #33924C2semidirect
4. #33921C2semidirect
5. #33908C2semidirect
6. #33907C2semidirect
7. #33892C2semidirect
8. #33891C2semidirect
9. #33862C2semidirect
10. #33861C2semidirect
11. #33793C2semidirect
12. #33792C2semidirect
13. #33764C2semidirect
14. #33666C2semidirect
15. #33666C2semidirect
16. #33650C2semidirect
17. #33493C2semidirect
22. #3342C22semidirect
25. #3341C22semidirect
26. #3338C22semidirect
27. #3334C22semidirect
36. C2×#500C22semidirect
39. #3061C22semidirect
41. C4×#135C22semidirect
43. #2913C22semidirect
44. #2254C22semidirect
45. #2252C22semidirect
48. #2237C22semidirect
49. #2234C22semidirect
50. #2234C22semidirect
51. #2234C22semidirect
52. #2234C22semidirect
53. #2234C22semidirect
54. #2233C22semidirect
55. #2233C22semidirect
56. #2232C22semidirect
57. #2232C22semidirect
58. #2227C22semidirect
59. #2227C22semidirect
60. #2226C22semidirect
61. #2226C22semidirect
62. #2226C22semidirect
63. #2226C22semidirect
64. #2224C22semidirect
65. #2224C22semidirect
66. #2220C22semidirect
67. #2218C22semidirect
68. #2217C22semidirect
69. #2215C22semidirect
74. #2187C22semidirect
75. #2186C22semidirect
76. #2186C22semidirect
77. #2184C22semidirect
78. C4×SD32C22semidirect
79. C4×SD32C22semidirect
80. C4×D16C22semidirect
81. C4×D16C22semidirect
82. #2169C22semidirect
83. #2169C22semidirect
84. D4×C16C22semidirect
85. D4×C16C22semidirect
86. #2162C22semidirect
87. #2162C22semidirect
88. #2159C22semidirect
89. #2158C22semidirect
90. #2140C22semidirect
91. C2×#357C22semidirect
92. #2113C22semidirect

Maximal subgroups

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

subgroupindex
#623282
#612872
#339242
#339212
#339082
#339072
#338922
#338912
#338622
#338612
#337932
#337922
#337642
#336662
#336502
#334932