Finite Group Databasebeta

← #7623428

Subgroups — #7623428

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60707C211yesmaximal, characteristic
2C2×#3164C211yesmaximal, characteristic, complemented
3#33810C214yesmaximal, complemented
4#33807C214yesmaximal, complemented
5#33805C212yesmaximal, complemented
6#33800C212yesmaximal, complemented
7#33798C212yesmaximal, complemented
8#33793C212yesmaximal, complemented
9#33730C212yesmaximal, complemented
10#33702C212yesmaximal, complemented
11#33648C211yesmaximal, characteristic, complemented
12#33646C211yesmaximal, characteristic, complemented
13#33645C211yesmaximal, characteristic, complemented
14C2×#2233C211yesmaximal, characteristic, complemented
15#33634C212yesmaximal, complemented
16#33631C212yesmaximal, complemented
17#33422C211yesmaximal, characteristic, complemented
18#3494—21no—
19C2×#544—21no—
20#3180—22no—
21#3179—22no—
22#3176C2212yes—
23#3175C2212yes—
24#3169C2212yes—
25#3168C2212yes—
26#3164—21no—
27#3164—21no—
28#3164—22no—
29#3164C2211yescharacteristic
30#3164C2212yes—
31#3164C2212yes—
32#3164C2214yescomplemented
33C2×#501—22no—
34C2×#500C2212yescomplemented
35C2×#499C2212yescomplemented
36#3155C2211yescharacteristic
37#3136—21no—
38#3055—21no—
39#3050C2211yescharacteristic
40C2×#466C2211yescharacteristic
41#2951C2211yescharacteristic
42#2948C2211yescharacteristic
43#2946—21no—
44C2×#428—21no—
45C2×#427C2211yescharacteristic
46C22×#107C2211yescharacteristic
47C2×#405C2211yescharacteristic
48#2255C2214yescomplemented
49#2255C2214yescomplemented
50#2254C2214yescomplemented
51#2253C2214yescomplemented
52#2244C2214yescomplemented
53#2243C2214yescomplemented
54#2242C2214yescomplemented
55#2240C2214yescomplemented
56#2237C2212yescomplemented
57#2236C2212yescomplemented
58#2236C2211yescharacteristic, complemented
59#2235C2211yescharacteristic, complemented
60#2235C2212yescomplemented
61#2234C2214yes—
62#2234C2212yescomplemented
63#2233C2212yes—
64#2233C2212yes—
65#2233C2212yes—
66#2233C2214yes—
67#2233C2211yescharacteristic, complemented
68#2233C2212yescomplemented
69#2232C2212yes—
70#2232C2211yescharacteristic, complemented
71#2232C2212yescomplemented
72#2229C2214yescomplemented
73#2228C2214yescomplemented
74#2228C2214yescomplemented
75#2228C2214yescomplemented
76#2226C2214yescomplemented
77#2226C2214yescomplemented
78#2225C2214yescomplemented
79#2224C2214yescomplemented
80#2164C2214yescomplemented
81#2163C2212yescomplemented
82#2162C2212yescomplemented
83#2159C2212yes—
84#2159C2212yes—
85#2158C2212yes—
86C2×#367C2212yes—
87#2151C2212yes—
88#2140C2212yes—
89#2139C2212yes—
90#2138C2212yes—
91C2×#357C2212yes—
92#2112C2211yescharacteristic
93C2×C22⋊C16C2211yescharacteristic
94#2108C2212yes—
95C2×AGL(1,ℤ/8)—21no—
96C2×#135C2311yescharacteristic
97C22×D8C2311yescharacteristic
98C22×M16D411yescharacteristic
99C23×C8C2311yescharacteristic
100#555—41no—
101#553—41no—
102#553—41no—
103#551—42no—
104#547—41no—
105#546—42no—
106#544—21no—
107#544—22no—
108#544—22no—
109#544—24no—
110#544—42no—
111#544—41no—
112#544—41no—
113#542—22no—
114#540—22no—
115#535—21no—
116#532—22no—
117C2×C4⋊Q8—22no—
118C2×C4⋊Q8—41no—
119#529—22no—
120C2×#125—22no—
121C2×#125—41no—
122#525—21no—
123C2×C22⋊Q8—42no—
124C2×#121—21no—
125#520—21no—
126#518—21no—
127C2×#117—21no—
128C2×#117D411yescharacteristic
129C22×C4⋊C4—21no—
130C2×M32C2312yes—
131C22×C16C2312yes—
132#501—24no—
133#501—24no—
134#501—24no—
135#501—24no—
136#501—22no—
137#500—22no—
138#500—22no—
139#500C2314yes—
140#500C2314yes—
141#500C2312yes—
142#499—22no—
143#499—22no—
144#499C2314yes—
145#499C2314yes—
146#499C2312yes—
147#496—22no—
148#495C2312yes—
149#493C2312yes—
150#490—22no—
151#489—22no—
152#488—22no—
153#486—22no—
154#469—22no—
155#468—22no—
156#467C2311yescharacteristic
157#467C2311yescharacteristic
158#466—22no—
159#466—21no—
160#466—21no—
161#466C2312yes—
162#466C2311yescharacteristic
163#466C2312yes—
164#466C2311yescharacteristic
165#465—22no—
166#465—21no—
167#465—21no—
168#442—22no—
169#442—22no—
170#440—22no—
171#440—22no—
172C4×Q16C2312yes—
173C4×SD16—22no—
174C4×D8—22no—
175C4×D8C2312yes—
176C4×D8C2312yes—
177C4×D8C2312yes—
178#428—22no—
179#428—22no—
180#428—24no—
181#428—21no—
182#427—21no—
183#427—21no—
184#427C2312yes—
185#427C2311yescharacteristic
186#427C2312yes—
187C2×#107—22no—
188C2×#107—21no—
189C2×#107—21no—
190C2×#107—21no—
191C2×#107C2311yescharacteristic
192C2×#107C2312yes—
193C2×#107C2311yescharacteristic
194C2×#107C2312yes—
195C2×#107C2312yes—
196C2×#107C2311yescharacteristic
197C2×#106—22no—
198C2×#106—21no—
199#418—21no—
200#417—21no—
201#416—21no—
202C2×#102—21no—
203#405D411yescharacteristic
204#405D412yes—
205#405D411yescharacteristic
206#405C2311yescharacteristic
207#405C2311yescharacteristic
208#405C2312yes—
209C2×#97D412yes—
210C2×C4×C8C2312yes—
211#367C2314yes—
212#367C2314yes—
213#367C2312yes—
214#367C2312yes—
215#367C2314yes—
216#367C2314yes—
217#366C2312yes—
218#366C2312yes—
219#366C2314yes—
220#366C2314yes—
221C4⋊C16C2314yes—
222C4⋊C16C2314yes—
223#358C2312yes—
224#358C2312yes—
225#358C2314yes—
226#357C2314yes—
227#357C2314yes—
228#357C2312yes—
229#357C2312yes—
230#357C2314yes—
231#357C2314yes—
232#357C2314yes—
233C22⋊C16C2311yescharacteristic
234C22⋊C16C2311yescharacteristic
235C22⋊C16C2312yes—
236C22⋊C16C2312yes—
237C22⋊C16C2312yes—
238#346C2314yes—
239C4×C16C2314yes—
240C2×#41—21no—
241C2×#41—42no—
242C22×Q8—42no—
243C22×D4—21no—
244C22×D4—41no—
245C23×C4—41no—
246C23×C4C2×D411yescharacteristic
247AGL(1,ℤ/8)—44no—
248#135—24no—
249C2×Q16C2411yescharacteristic
250C2×SD16—21no—
251C2×SD16—22no—
252C2×D8—21no—
253C2×D8—22no—
254C2×D8—24no—
255C2×D8C2412yes—
256C2×D8C2411yescharacteristic
257C2×M16—24no—
258C2×M16C2×D411yescharacteristic
259C2×M16C2×D412yes—
260C2×M16C2×D411yescharacteristic
261C22×C8—24no—
262C22×C8C2×D411yescharacteristic
263C22×C8C2×D411yescharacteristic
264C22×C8C2×D411yescharacteristic
265C22×C8C2×D412yes—
266C22×C8C2411yescharacteristic
267C22×C8C2411yescharacteristic
268C22×C8C2411yescharacteristic
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—22no—
272C4⋊Q8—42no—
273C4⋊Q8—42no—
274C4⋊Q8—44no—
275C4⋊Q8—41no—
276C4⋊Q8—42no—
277C4⋊Q8—42no—
278#127—22no—
279#126—42no—
280#126—42no—
281#125—24no—
282#125—24no—
283#125—22no—
284#125—42no—
285#125—42no—
286#125—44no—
287#125—41no—
288#125—42no—
289#125—42no—
290#124—22no—
291#124—42no—
292#124—42no—
293#123—41no—
294#123—42no—
295#123—42no—
296#123—42no—
297#123—42no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—42no—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305C22⋊Q8—42no—
306#121—22no—
307#121—22no—
308#121—21no—
309#121—21no—
310#121—41no—
311#121—41no—
312#121—42no—
313#121—41no—
314#121—41no—
315#121—42no—
316#121—42no—
317#120—41no—
318#120—42no—
319C4×Q8—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—41no—
324C4×D4—41no—
325C4×D4—42no—
326C4×D4—41no—
327C4×D4—41no—
328C4×D4—42no—
329C4×D4—42no—
330#117—22no—
331#117—22no—
332#117—21no—
333#117—41no—
334#117—41no—
335#117—41no—
336#117—42no—
337#117D812yes—
338#117D811yescharacteristic
339#117D811yescharacteristic
340#117C2×D411yescharacteristic
341#117C2×D411yescharacteristic
342#117C2×D412yes—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4—41no—
351C2×C4⋊C4—41no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—41no—
356C2×C4⋊C4D811yescharacteristic
357C2×C4⋊C4D811yescharacteristic
358C2×C22⋊C4—41no—
359C2×C22⋊C4—41no—
360C2×C22⋊C4—42no—
361C2×C22⋊C4D812yes—
362C2×C42—41no—
363C2×C42C2×D412yes—
364M32—24no—
365C2×C16—24no—
366C2×C16C2412yes—
367C2×C16C2414yes—
368C2×C16C2412yes—
369#107—22no—
370#107—22no—
371#107—22no—
372#107—24no—
373#107—24no—
374#107—21no—
375#107—21no—
376#107—21no—
377#107—22no—
378#107C2412yes—
379#107C2414yes—
380#107C2412yes—
381#107C2411yescharacteristic
382#107C2412yes—
383#107C2411yescharacteristic
384#106—22no—
385#106—22no—
386#106—24no—
387#106—24no—
388#106—21no—
389#106—22no—
390#103—21no—
391#103—21no—
392#103—22no—
393#102—22no—
394#102—22no—
395#102—21no—
396#102—21no—
397#102—22no—
398#102—22no—
399#102—22no—
400#97C2×D412yes—
401#97C2×D412yes—
402#97C2×D412yes—
403#97C2×D412yes—
404C4×C8C2×D412yes—
405C4×C8C2×D412yes—
406C4×C8C2412yes—
407C4×C8C2412yes—
408C24—41no—
409C24C2×D811yescharacteristic
410#41—42no—
411#41—42no—
412#41—164no—
413C2×Q8—21no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—164no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—42no—
423C2×D4—41no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—41no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—162no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—162no—
444C22×C4C2×D811yescharacteristic
445C22×C4C2×D812yes—
446C22×C4C2×D811yescharacteristic
447C22×C4C2×D811yescharacteristic
448C22×C4C2×D812yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452Q16—22no—
453SD16—42no—
454SD16—44no—
455D8—24no—
456D8—22no—
457D8—42no—
458D8—44no—
459M16—22no—
460M16—24no—
461M16#13714yes—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8C22×D411yescharacteristic
469C2×C8C22×D412yes—
470C2×C8C22×D411yescharacteristic
471C2×C8C22×D411yescharacteristic
472C2×C8C22×D412yes—
473C2×C82−1+414yes—
474C2×C8C2511yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—41no—
499C4⋊C4C2×D811yescharacteristic
500C4⋊C4C2×D811yescharacteristic
501C4⋊C4C2×D811yescharacteristic
502C4⋊C4C2×D811yescharacteristic
503C4⋊C4C2×D812yes—
504C4⋊C4C2×D812yes—
505C22⋊C4—41no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—41no—
517C22⋊C4—42no—
518C22⋊C4—41no—
519C22⋊C4C2×D812yes—
520C22⋊C4C2×D812yes—
521C22⋊C4C2×D812yes—
522C22⋊C4C2×D812yes—
523C42—42no—
524C42—42no—
525C42—41no—
526C42—42no—
527C42C2×D812yes—
528C42C2×D812yes—
529C42C22×D412yes—
530C42C22×D412yes—
531C16—22no—
532C16—22no—
533C16—24no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—42no—
538C23—41no—
539C23—82no—
540C23—161no—
541C23—161no—
542C23C22×D811yescharacteristic
543C23C22×D811yescharacteristic
544C23C22×D811yescharacteristic
545Q8—41no—
546Q8—41no—
547Q8—164no—
548Q8—164no—
549Q8—162no—
550D4—42no—
551D4—41no—
552D4—41no—
553D4—42no—
554D4—84no—
555D4—82no—
556D4—82no—
557D4—82no—
558D4—162no—
559D4—162no—
560D4—161no—
561D4—162no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—84no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—162no—
583C2×C4—162no—
584C2×C4—162no—
585C2×C4—162no—
586C2×C4—162no—
587C2×C4—161no—
588C2×C4#53612yes—
589C2×C4C22×D811yescharacteristic
590C2×C4C22×D812yes—
591C2×C4C22×D811yescharacteristic
592C2×C4C22×D812yes—
593C2×C4C22×D812yes—
594C2×C4C22×D812yes—
595C2×C4C22×D811yescharacteristic
596C2×C4C2×#13712yes—
597C2×C4C23×D411yescharacteristic
598C8—21no—
599C8—21no—
600C8—22no—
601C8#53612yes—
602C8C2×#13714yes—
603C8C2×2−1+412yes—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—44no—
612C22—81no—
613C22—82no—
614C22—81no—
615C22—82no—
616C22—82no—
617C22—81no—
618C22—322no—
619C22#322612yes—
620C22#341612yes—
621C22C23×D811yescharacteristic
622C4—21no—
623C4—21no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—81no—
629C4—82no—
630C4—82no—
631C4—81no—
632C4—84no—
633C4—82no—
634C4—81no—
635C4#322612yes—
636C4C2×#53611yescharacteristic
637C4C22×#13711yescharacteristic
638C2—22no—
639C2—22no—
640C2—41no—
641C2—161no—
642C2—162no—
643C2C2×#322611yescharacteristic
644C2C2×#341611yescharacteristic
645C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#60707C2
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
2. C2×#3164C2semidirect
3. #33810C2semidirect
4. #33807C2semidirect
5. #33805C2semidirect
6. #33800C2semidirect
7. #33798C2semidirect
8. #33793C2semidirect
9. #33730C2semidirect
10. #33702C2semidirect
11. #33648C2semidirect
12. #33646C2semidirect
13. #33645C2semidirect
14. C2×#2233C2semidirect
15. #33634C2semidirect
16. #33631C2semidirect
17. #33422C2semidirect
32. #3164C22semidirect
34. C2×#500C22semidirect
35. C2×#499C22semidirect
48. #2255C22semidirect
49. #2255C22semidirect
50. #2254C22semidirect
51. #2253C22semidirect
52. #2244C22semidirect
53. #2243C22semidirect
54. #2242C22semidirect
55. #2240C22semidirect
56. #2237C22semidirect
57. #2236C22semidirect
58. #2236C22semidirect
59. #2235C22semidirect
60. #2235C22semidirect
62. #2234C22semidirect
67. #2233C22semidirect
68. #2233C22semidirect
70. #2232C22semidirect
71. #2232C22semidirect
72. #2229C22semidirect
73. #2228C22semidirect
74. #2228C22semidirect
75. #2228C22semidirect
76. #2226C22semidirect
77. #2226C22semidirect
78. #2225C22semidirect
79. #2224C22semidirect
80. #2164C22semidirect
81. #2163C22semidirect
82. #2162C22semidirect

Maximal subgroups

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

subgroupindex
#607072
C2×#31642
#338102
#338072
#338052
#338002
#337982
#337932
#337302
#337022
#336482
#336462
#336452
C2×#22332
#336342
#336312
#334222