Finite Group Databasebeta

← #7623574

Subgroups — #7623574

Order 512.

1297 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60717C211yesmaximal, characteristic, complemented
2#60714C211yesmaximal, characteristic, complemented
3#33826C212yesmaximal, complemented
4#33825C212yesmaximal, complemented
5#33806C212yesmaximal, complemented
6#33800C212yesmaximal, complemented
7#33783C212yesmaximal, complemented
8#33780C212yesmaximal, complemented
9#33764C214yesmaximal, complemented
10#33746C212yesmaximal, complemented
11#33737C212yesmaximal, complemented
12#33683C212yesmaximal, complemented
13#33682C212yesmaximal, complemented
14#33679C214yesmaximal, complemented
15#33421C211yesmaximal, characteristic, complemented
16#3512—21no—
17#3508—21no—
18#3184—21no—
19#3184—21no—
20#3183—21no—
21#3183—21no—
22#3177C2211yescharacteristic
23#3177C2211yescharacteristic, complemented
24#3176—21no—
25#3176—21no—
26#3176C2212yes—
27#3176C2212yes—
28#3176C2211yescharacteristic
29#3176C2211yescharacteristic, complemented
30#3174—21no—
31#3174—21no—
32#3169C2212yes—
33#3169C2212yes—
34#3168C2212yes—
35#3168C2212yescomplemented
36#3160C2211yescharacteristic
37C2×#499C2211yescharacteristic, complemented
38#3147C2211yescharacteristic, complemented
39C2×#493C2211yescharacteristic
40#3142—21no—
41#3138—21no—
42#3061C2212yes—
43#3060—21no—
44#3058—21no—
45#3058C2212yes—
46#2952—21no—
47#2951C2212yescomplemented
48#2948—21no—
49#2948C2212yes—
50C4×#131C2211yescharacteristic
51#2255C2212yescomplemented
52#2255C2212yescomplemented
53#2250C2212yes—
54#2250C2212yes—
55#2244C2212yes—
56#2244C2212yes—
57#2243C2212yescomplemented
58#2243C2212yescomplemented
59#2237C2214yes—
60#2234C2214yes—
61#2233C2214yescomplemented
62#2232C2214yescomplemented
63#2229C2214yescomplemented
64#2228C2214yescomplemented
65#2228C2214yescomplemented
66#2228C2214yescomplemented
67#2220C2214yescomplemented
68#2218C2214yescomplemented
69#2217C2214yescomplemented
70#2215C2214yescomplemented
71#2208C2214yescomplemented
72#2207C2214yes—
73#2206C2214yes—
74#2206C2214yescomplemented
75#2187C2214yes—
76#2186C2214yescomplemented
77#2186C2214yescomplemented
78#2184C2214yes—
79#2177C2214yescomplemented
80#2175C2214yescomplemented
81C4×SD32C2214yescomplemented
82C4×D16C2214yescomplemented
83#2164C2212yes—
84#2164C2212yescomplemented
85#2163C2212yes—
86#2163C2212yescomplemented
87#2151C2212yescomplemented
88#2113C2212yes—
89C4×M32C2211yescharacteristic, complemented
90C2×#346C2211yescharacteristic, complemented
91C2×#135—21no—
92C2×#135C2312yes—
93C22×D8—21no—
94C22×D8C2312yes—
95C2×#131D411yescharacteristic
96C2×#131C2311yescharacteristic
97#559—42no—
98#553—42no—
99#553—42no—
100#553—42no—
101#552—42no—
102#550—42no—
103#546—42no—
104#546—42no—
105#541—21no—
106#541—21no—
107#540—21no—
108#540—21no—
109#540—22no—
110#540—22no—
111#537—22no—
112#535—22no—
113#533—21no—
114#532—22no—
115#532—22no—
116#532—21no—
117C2×#127—21no—
118#529—22no—
119#529—22no—
120C2×#125—21no—
121#520—22no—
122#518—22no—
123C4×#41D411yescharacteristic
124C2×SD32C2314yes—
125C2×SD32C2314yes—
126C2×D16C2314yes—
127C2×D16C2314yes—
128C2×M32C2312yes—
129C22×C16C2312yes—
130#500C2312yes—
131#499—21no—
132#499—21no—
133#499—21no—
134#499C2312yes—
135#499C2311yescharacteristic
136#499C2311yescharacteristic
137#499C2312yes—
138#498—21no—
139#495—21no—
140#495C2312yes—
141#494C2311yescharacteristic
142#493—21no—
143#493—21no—
144#493C2312yes—
145#493C2312yes—
146#493C2311yescharacteristic
147#492—21no—
148#491—21no—
149#491—21no—
150#490—22no—
151#490—22no—
152#490—21no—
153#490—21no—
154#488—22no—
155#488—22no—
156#486—22no—
157#486—22no—
158#470—21no—
159#470—21no—
160#469—22no—
161#469—22no—
162#469—21no—
163#469—22no—
164#469—22no—
165#469—21no—
166#468—22no—
167#468—22no—
168#467C2312yes—
169#467C2312yes—
170#466—22no—
171#466—22no—
172#466C2312yes—
173#466C2312yes—
174#466C2314yes—
175#466C2314yes—
176#465—22no—
177#465—22no—
178#442—22no—
179#442—22no—
180#442—21no—
181#442—22no—
182#442—22no—
183#442—21no—
184#441—21no—
185#441—21no—
186#440—22no—
187#440—22no—
188C4×Q16C2312yes—
189C4×Q16C2312yes—
190C4×SD16—22no—
191C4×SD16—22no—
192C4×D8—22no—
193C4×D8—22no—
194C4×D8C2312yes—
195C4×D8C2312yes—
196C4×D8C2314yes—
197C4×D8C2314yes—
198#427—21no—
199#427C2312yes—
200C2×#107C2312yes—
201C2×#106—21no—
202#418—22no—
203#418—22no—
204#417—22no—
205#417—22no—
206#405D411yescharacteristic
207#405D411yescharacteristic
208#405D411yescharacteristic
209#405C2311yescharacteristic
210#405C2311yescharacteristic
211#405C2311yescharacteristic
212C4×M16D411yescharacteristic
213C4×M16D411yescharacteristic
214C4×M16D411yescharacteristic
215C2×C4×C8C2311yescharacteristic
216C2×C4×C8C2311yescharacteristic
217C2×C4×C8C2311yescharacteristic
218#367C2314yes—
219#367C2314yes—
220#366C2314yes—
221#366C2314yes—
222C4⋊C16C2314yes—
223C4⋊C16C2314yes—
224#358C2314yes—
225#358C2314yes—
226#357C2314yes—
227#357C2314yes—
228#357C2314yes—
229#357C2314yes—
230#357C2314yes—
231#357C2314yes—
232C22⋊C16C2314yes—
233C22⋊C16C2314yes—
234#346C2312yes—
235#346C2312yes—
236#346C2312yes—
237C4×C16C2312yes—
238C2×#41—22no—
239C2×#41—42no—
240C2×#41—42no—
241C2×#41C2×D411yescharacteristic
242C22×D4—22no—
243C22×D4—42no—
244#135—44no—
245#135—44no—
246C2×Q16—21no—
247C2×Q16C2412yes—
248C2×SD16—22no—
249C2×SD16—22no—
250C2×D8—22no—
251C2×D8—22no—
252C2×D8—22no—
253C2×D8—21no—
254C2×D8—44no—
255C2×D8—44no—
256C2×D8C2414yes—
257C2×D8C2412yes—
258#131—42no—
259#131—42no—
260C2×M16C2×D411yescharacteristic
261C2×M16C2×D411yescharacteristic
262C2×M16C2×D411yescharacteristic
263C2×M16C2×D411yescharacteristic
264C2×M16C2×D411yescharacteristic
265C2×M16C2×D411yescharacteristic
266C22×C8C2×D411yescharacteristic
267C22×C8C2×D411yescharacteristic
268C22×C8C2×D411yescharacteristic
269C22×C8C2411yescharacteristic
270C22×C8C2411yescharacteristic
271C22×C8C2411yescharacteristic
272C4⋊Q8—22no—
273C4⋊Q8—21no—
274#127—22no—
275#127—22no—
276#127—21no—
277#127—42no—
278#127—42no—
279#126—42no—
280#126—42no—
281#126—42no—
282#126—42no—
283#125—22no—
284#125—21no—
285#125—21no—
286#125—22no—
287#125—42no—
288#125—42no—
289#125—42no—
290#124—22no—
291#124—42no—
292#124—42no—
293#124—42no—
294#123—42no—
295#123—42no—
296#123—44no—
297#123—44no—
298#123—42no—
299#123—42no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—42no—
303C22⋊Q8—42no—
304#121—22no—
305#121—22no—
306#121—24no—
307#121—24no—
308#121—42no—
309#121—42no—
310#121—42no—
311#121—42no—
312#121—42no—
313#121—42no—
314#121—44no—
315#121—44no—
316#121—42no—
317#121—42no—
318#120—42no—
319#120—42no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×Q8—42no—
323C4×Q8D812yes—
324C4×D4—24no—
325C4×D4—24no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—42no—
329C4×D4—42no—
330C4×D4—44no—
331C4×D4—42no—
332C4×D4—42no—
333C4×D4—42no—
334C4×D4—42no—
335C4×D4—42no—
336C4×D4D812yes—
337C4×D4D812yes—
338C4×D4D812yes—
339#117—22no—
340#117—42no—
341#117—42no—
342#117—42no—
343#117C2×D411yescharacteristic
344#117C2×D411yescharacteristic
345#117C2×D411yescharacteristic
346C2×C4⋊C4—22no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—42no—
349C2×C22⋊C4—44no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353SD32—44no—
354SD32—44no—
355D16—44no—
356D16—44no—
357M32—42no—
358C2×C16—42no—
359C2×C16C2414yes—
360C2×C16C2414yes—
361#107—22no—
362#107—22no—
363#107C2412yes—
364#107C2412yes—
365#107C2414yes—
366#106—22no—
367#106—21no—
368#106—21no—
369#106—22no—
370#106—22no—
371#103—22no—
372#103—22no—
373#103—22no—
374#103—22no—
375#102—22no—
376#102—22no—
377#102—22no—
378#102—22no—
379#102—24no—
380#102—24no—
381#102—24no—
382#102—24no—
383#97C2×D411yescharacteristic
384#97C2×D411yescharacteristic
385#97C2×D411yescharacteristic
386#97C2×D411yescharacteristic
387#97C2×D411yescharacteristic
388#97C2×D411yescharacteristic
389C4×C8C2×D411yescharacteristic
390C4×C8C2×D411yescharacteristic
391C4×C8C2×D411yescharacteristic
392C4×C8C2×D411yescharacteristic
393C4×C8C2×D411yescharacteristic
394C4×C8C2×D411yescharacteristic
395C4×C8C2411yescharacteristic
396C4×C8C2411yescharacteristic
397C4×C8C2411yescharacteristic
398C4×C8C2411yescharacteristic
399C24—42no—
400#41—41no—
401#41—41no—
402#41—44no—
403#41—44no—
404#41—164no—
405#41—164no—
406C2×Q8—22no—
407C2×Q8—42no—
408C2×Q8C2×D811yescharacteristic
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—164no—
424C2×D4C2×D811yescharacteristic
425C2×D4C2×D811yescharacteristic
426C2×D4C2×D811yescharacteristic
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4C2×D812yes—
434C22×C4C2×D812yes—
435C22×C4C2×D812yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439Q16—24no—
440Q16—42no—
441SD16—44no—
442SD16—44no—
443D8—24no—
444D8—24no—
445D8—24no—
446D8—44no—
447D8—44no—
448D8—42no—
449D8—44no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—41no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8C22×D411yescharacteristic
463C2×C8C22×D411yescharacteristic
464C2×C8C22×D411yescharacteristic
465C2×C8C22×D411yescharacteristic
466C2×C8C22×D411yescharacteristic
467C2×C8C22×D411yescharacteristic
468C2×C8C22×D411yescharacteristic
469C2×C8C2511yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4C2×D812yes—
486C4⋊C4C2×D812yes—
487C4⋊C4C2×D812yes—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4C2×D812yes—
502C22⋊C4C2×D812yes—
503C22⋊C4C2×D812yes—
504C42—42no—
505C42—42no—
506C42—42no—
507C42C2×D812yes—
508C42C2×D812yes—
509C42C2×D812yes—
510C42C22×D411yescharacteristic
511C42C22×D411yescharacteristic
512C42C22×D411yescharacteristic
513C42C22×D411yescharacteristic
514C16—24no—
515C16—24no—
516C23—42no—
517C23—44no—
518C23—84no—
519C23—164no—
520C23C22×D811yescharacteristic
521C23C22×D811yescharacteristic
522C23C22×D811yescharacteristic
523Q8—41no—
524Q8—42no—
525Q8—42no—
526Q8—84no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—44no—
531D4—42no—
532D4—44no—
533D4—42no—
534D4—84no—
535D4—84no—
536D4—84no—
537D4—84no—
538D4—162no—
539D4—164no—
540D4—162no—
541D4—162no—
542D4—164no—
543D4—164no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—84no—
558C2×C4—162no—
559C2×C4—162no—
560C2×C4—164no—
561C2×C4—164no—
562C2×C4—162no—
563C2×C4C22×D811yescharacteristic
564C2×C4C22×D812yes—
565C2×C4C22×D812yes—
566C2×C4C22×D812yes—
567C2×C4C22×D811yescharacteristic
568C2×C4C22×D811yescharacteristic
569C2×C4C22×D812yes—
570C2×C4C23×D411yescharacteristic
571C8—21no—
572C8—21no—
573C8—21no—
574C8—21no—
575C8—21no—
576C8—21no—
577C8#57512yes—
578C8#58512yes—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—82no—
586C22—84no—
587C22—82no—
588C22—84no—
589C22—84no—
590C22—82no—
591C22—324no—
592C22C23×D811yescharacteristic
593C4—21no—
594C4—21no—
595C4—21no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—84no—
604C4#344911yescharacteristic
605C4C2×#57511yescharacteristic
606C2—41no—
607C2—41no—
608C2—41no—
609C2—162no—
610C2—164no—
611C2#6094011yescharacteristic
612C2#6263612yes—
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60717C2semidirect
2. #60714C2semidirect
3. #33826C2semidirect
4. #33825C2semidirect
5. #33806C2semidirect
6. #33800C2semidirect
7. #33783C2semidirect
8. #33780C2semidirect
9. #33764C2semidirect
10. #33746C2semidirect
11. #33737C2semidirect
12. #33683C2semidirect
13. #33682C2semidirect
14. #33679C2semidirect
15. #33421C2semidirect
23. #3177C22semidirect
29. #3176C22semidirect
35. #3168C22semidirect
37. C2×#499C22semidirect
38. #3147C22semidirect
47. #2951C22semidirect
51. #2255C22semidirect
52. #2255C22semidirect
57. #2243C22semidirect
58. #2243C22semidirect
61. #2233C22semidirect
62. #2232C22semidirect
63. #2229C22semidirect
64. #2228C22semidirect
65. #2228C22semidirect
66. #2228C22semidirect
67. #2220C22semidirect
68. #2218C22semidirect
69. #2217C22semidirect
70. #2215C22semidirect
71. #2208C22semidirect
74. #2206C22semidirect
76. #2186C22semidirect
77. #2186C22semidirect
79. #2177C22semidirect
80. #2175C22semidirect
81. C4×SD32C22semidirect
82. C4×D16C22semidirect
84. #2164C22semidirect
86. #2163C22semidirect
87. #2151C22semidirect
89. C4×M32C22semidirect
90. C2×#346C22semidirect

Maximal subgroups

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

subgroupindex
#607172
#607142
#338262
#338252
#338062
#338002
#337832
#337802
#337642
#337462
#337372
#336832
#336822
#336792
#334212