Finite Group Databasebeta

← #7623440

Subgroups — #7623440

Order 512.

1228 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60713C211yesmaximal, characteristic
2#60667C211yesmaximal, characteristic, complemented
3#33810C214yesmaximal, complemented
4#33809C214yesmaximal, complemented
5#33806C212yesmaximal, complemented
6#33806C212yesmaximal, complemented
7#33805C212yesmaximal, complemented
8#33804C212yesmaximal, complemented
9#33731C211yesmaximal, characteristic, complemented
10#33728C211yesmaximal, characteristic, complemented
11#33706C211yesmaximal, characteristic, complemented
12C2×#2243C211yesmaximal, characteristic, complemented
13#33651C212yesmaximal, complemented
14#33650C212yesmaximal
15#33632C212yesmaximal, complemented
16#33631C212yesmaximal, complemented
17#33421C211yesmaximal, characteristic, complemented
18#3510—21no—
19#3503—21no—
20#3183—22no—
21#3175C2212yes—
22#3175C2212yes—
23#3173—22no—
24#3168C2212yes—
25#3168C2212yes—
26#3165—21no—
27#3165—22no—
28#3164—21no—
29#3164—22no—
30#3164C2212yes—
31#3164C2214yescomplemented
32#3160C2211yescharacteristic
33#3160C2211yescharacteristic
34#3160C2212yescomplemented
35C2×#500C2211yescharacteristic
36C2×#500C2212yescomplemented
37C2×#495C2211yescharacteristic
38#3140—21no—
39#3059C2211yescharacteristic
40#3058C2211yescharacteristic
41#3057—21no—
42#2950C2211yescharacteristic
43#2949—21no—
44#2948C2211yescharacteristic
45#2913—21no—
46#2913C2212yes—
47C4×#131C2211yescharacteristic
48#2255C2211yescharacteristic, complemented
49#2255C2212yescomplemented
50#2255C2211yescharacteristic, complemented
51#2255C2212yescomplemented
52#2255C2211yescharacteristic, complemented
53#2255C2211yescharacteristic, complemented
54#2255C2212yescomplemented
55#2255C2212yescomplemented
56#2254C2212yescomplemented
57#2252C2212yescomplemented
58#2243C2212yes—
59#2243C2212yes—
60#2243C2212yes—
61#2243C2212yes—
62#2243C2214yes—
63#2242C2214yes—
64#2237C2214yescomplemented
65#2237C2214yescomplemented
66#2236C2214yescomplemented
67#2235C2214yescomplemented
68#2234C2214yes—
69#2234C2214yes—
70#2233C2214yes—
71#2232C2214yes—
72#2229C2214yescomplemented
73#2229C2214yescomplemented
74#2228C2214yes—
75#2228C2214yes—
76#2227C2214yescomplemented
77#2226C2214yes—
78#2225C2214yescomplemented
79#2224C2214yes—
80#2164C2214yescomplemented
81#2163C2212yescomplemented
82#2163C2212yescomplemented
83#2159C2212yes—
84#2159C2212yes—
85#2158C2212yes—
86#2158C2212yes—
87#2151C2212yes—
88#2141C2212yes—
89#2140C2212yes—
90C2×#358C2212yes—
91C2×#357C2212yes—
92#2113C2212yes—
93C4×M32C2211yescharacteristic
94C2×#346C2211yescharacteristic
95C22×Q16C2311yescharacteristic
96C22×SD16—21no—
97C22×D8C2311yescharacteristic
98C2×#131D411yescharacteristic
99C2×#131C2311yescharacteristic
100#559—41no—
101#559—41no—
102Q82—41no—
103#557—41no—
104#556—41no—
105#556—41no—
106#554—42no—
107#554—41no—
108#551—42no—
109#551—42no—
110D4×Q8—42no—
111D4×Q8—41no—
112#544—22no—
113#544—24no—
114#542—22no—
115#542—22no—
116#536—21no—
117#535—21no—
118#533—21no—
119#533—21no—
120#533—22no—
121C2×C4⋊Q8—21no—
122C2×C4⋊Q8—22no—
123#529—22no—
124#529—22no—
125C2×#124—21no—
126#520—22no—
127#519—21no—
128#518—21no—
129C4×#41D411yescharacteristic
130C2×M32C2312yes—
131C22×C16C2312yes—
132#501—24no—
133#501—24no—
134#501—24no—
135#500—21no—
136#500—21no—
137#500—22no—
138#500—22no—
139#500C2312yes—
140#500C2312yes—
141#500C2311yescharacteristic
142#500C2314yes—
143#500C2311yescharacteristic
144#500C2311yescharacteristic
145#500C2311yescharacteristic
146#499—21no—
147#499—21no—
148#499—22no—
149#499C2312yes—
150#499C2311yescharacteristic
151#499C2311yescharacteristic
152#499C2314yes—
153#498—21no—
154#498—21no—
155#498—22no—
156#495—21no—
157#495—21no—
158#495C2312yes—
159#495C2312yes—
160#489—22no—
161#489—22no—
162#488—22no—
163#488—22no—
164#470—22no—
165#469—22no—
166#468—22no—
167#468—22no—
168#467C2312yes—
169#467C2312yes—
170#466C2312yes—
171#466C2312yes—
172#465—22no—
173#465—22no—
174#442—22no—
175#441—22no—
176#440—22no—
177#440—22no—
178C4×Q16C2312yes—
179C4×Q16C2312yes—
180C4×SD16—22no—
181C4×SD16—22no—
182C4×D8C2312yes—
183C4×D8C2312yes—
184#428—22no—
185#428—24no—
186#428—22no—
187#428—24no—
188#427—21no—
189#427—21no—
190#427—22no—
191#427C2312yes—
192#427C2311yescharacteristic
193#427C2311yescharacteristic
194#427C2314yes—
195C2×#107—21no—
196C2×#107—22no—
197C2×#107C2312yes—
198C2×#107C2314yes—
199#419—21no—
200#419—21no—
201#418—21no—
202#418—21no—
203#405D412yes—
204#405D411yescharacteristic
205#405C2311yescharacteristic
206#405C2312yes—
207C4×M16D411yescharacteristic
208C4×M16D412yes—
209C2×C4×C8C2311yescharacteristic
210C2×C4×C8C2312yes—
211#367C2312yes—
212#367C2312yes—
213#367C2312yes—
214#367C2312yes—
215#367C2314yes—
216#367C2314yes—
217#366C2312yes—
218#366C2312yes—
219#366C2312yes—
220#366C2312yes—
221#366C2314yes—
222#366C2314yes—
223C4⋊C16C2314yes—
224C4⋊C16C2314yes—
225#358C2314yes—
226#358C2314yes—
227#358C2314yes—
228#358C2314yes—
229#357C2314yes—
230#357C2314yes—
231#357C2314yes—
232#357C2314yes—
233C22⋊C16C2314yes—
234C22⋊C16C2314yes—
235#346C2311yescharacteristic
236#346C2311yescharacteristic
237#346C2312yes—
238#346C2312yes—
239C4×C16C2312yes—
240C2×#41C2×D411yescharacteristic
241C22×Q8—21no—
242C22×Q8—41no—
243C22×Q8—42no—
244C22×D4—21no—
245C2×Q16—21no—
246C2×Q16—24no—
247C2×Q16C2412yes—
248C2×SD16—22no—
249C2×SD16—22no—
250C2×SD16—44no—
251C2×D8—21no—
252C2×D8—24no—
253C2×D8C2412yes—
254#131—24no—
255#131—24no—
256C2×M16C2×D411yescharacteristic
257C2×M16C2×D412yes—
258C2×M16C2×D411yescharacteristic
259C2×M16C2×D412yes—
260C22×C8C2×D412yes—
261C22×C8C2×D411yescharacteristic
262C22×C8C2412yes—
263C22×C8C2411yescharacteristic
264C4⋊Q8—22no—
265C4⋊Q8—21no—
266C4⋊Q8—24no—
267C4⋊Q8—22no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8—41no—
272C4⋊Q8—42no—
273C4⋊Q8—44no—
274C4⋊Q8—42no—
275C4⋊Q8—41no—
276C4⋊Q8—42no—
277C4⋊Q8—41no—
278#127—41no—
279#126—42no—
280#126—42no—
281#126—42no—
282#126—42no—
283#125—21no—
284#125—21no—
285#125—24no—
286#125—22no—
287#125—42no—
288#125—44no—
289#124—22no—
290#124—22no—
291#124—41no—
292#124—41no—
293#124—42no—
294#124—42no—
295#124—41no—
296#124—41no—
297#123—42no—
298#123—42no—
299#123—42no—
300#123—42no—
301#123—42no—
302C22⋊Q8—22no—
303C22⋊Q8—22no—
304C22⋊Q8—41no—
305C22⋊Q8—42no—
306C22⋊Q8—41no—
307C22⋊Q8—41no—
308C22⋊Q8—41no—
309C22⋊Q8—42no—
310C22⋊Q8—42no—
311C22⋊Q8—42no—
312C22⋊Q8—44no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315#121—22no—
316#121—22no—
317#120—42no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—42no—
321C4×Q8—41no—
322C4×Q8—41no—
323C4×Q8—42no—
324C4×Q8—41no—
325C4×Q8—42no—
326C4×Q8—41no—
327C4×Q8—41no—
328C4×Q8—41no—
329C4×Q8D811yescharacteristic
330C4×Q8D811yescharacteristic
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—44no—
334C4×D4—42no—
335C4×D4—42no—
336C4×D4—42no—
337C4×D4—41no—
338C4×D4—41no—
339C4×D4D812yes—
340C4×D4D811yescharacteristic
341C4×D4D812yes—
342C4×D4D811yescharacteristic
343#117—21no—
344#117—21no—
345#117—24no—
346#117—22no—
347#117—42no—
348#117—42no—
349#117—42no—
350#117C2×D411yescharacteristic
351#117C2×D412yes—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—41no—
355C2×C4⋊C4—42no—
356C2×C22⋊C4—41no—
357C2×C22⋊C4—41no—
358C2×C22⋊C4—42no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361M32—24no—
362C2×C16—24no—
363C2×C16C2414yes—
364C2×C16C2412yes—
365C2×C16C2412yes—
366#107—22no—
367#107—21no—
368#107—24no—
369#107—22no—
370#107—24no—
371#107—22no—
372#107—21no—
373#107—21no—
374#107—21no—
375#107C2414yes—
376#107C2412yes—
377#107C2412yes—
378#107C2411yescharacteristic
379#107C2411yescharacteristic
380#107C2411yescharacteristic
381#107C2411yescharacteristic
382#106—24no—
383#106—22no—
384#106—21no—
385#106—21no—
386#106—24no—
387#106—22no—
388#106—21no—
389#106—21no—
390#103—22no—
391#103—22no—
392#103—22no—
393#103—22no—
394#102—22no—
395#102—22no—
396#102—22no—
397#102—22no—
398#97C2×D412yes—
399#97C2×D412yes—
400#97C2×D412yes—
401C4×C8C2×D412yes—
402C4×C8C2×D411yescharacteristic
403C4×C8C2×D412yes—
404C4×C8C2×D411yescharacteristic
405C4×C8C2411yescharacteristic
406C4×C8C2411yescharacteristic
407C4×C8C2412yes—
408C24—41no—
409#41—22no—
410#41—22no—
411C2×Q8—22no—
412C2×Q8—41no—
413C2×Q8—41no—
414C2×Q8—42no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×Q8—41no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—164no—
423C2×Q8—162no—
424C2×Q8C2×D811yescharacteristic
425C2×D4—22no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4C2×D812yes—
432C2×D4C2×D811yescharacteristic
433C22×C4—41no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4C2×D811yescharacteristic
439C22×C4C2×D811yescharacteristic
440C22×C4C2×D812yes—
441C22×C4C2×D812yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444Q16—24no—
445Q16—42no—
446SD16—44no—
447SD16—44no—
448D8—24no—
449D8—42no—
450M16—22no—
451M16—22no—
452M16—24no—
453M16—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8#13714yes—
457C2×C8C22×D412yes—
458C2×C8C22×D412yes—
459C2×C8C22×D411yescharacteristic
460C2×C8C22×D411yescharacteristic
461C2×C8C22×D411yescharacteristic
462C2×C82−1+414yes—
463C2×C8C2511yescharacteristic
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—41no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4C2×D811yescharacteristic
493C4⋊C4C2×D811yescharacteristic
494C4⋊C4C2×D812yes—
495C4⋊C4C2×D812yes—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—42no—
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⋊C4C2×D812yes—
511C22⋊C4C2×D812yes—
512C22⋊C4C2×D812yes—
513C42—41no—
514C42—44no—
515C42—42no—
516C42—41no—
517C42—42no—
518C42—42no—
519C42C2×D812yes—
520C42C2×D812yes—
521C42C2×D811yescharacteristic
522C42C2×D811yescharacteristic
523C42C22×D411yescharacteristic
524C42C22×D412yes—
525C42C22×D411yescharacteristic
526C16—22no—
527C16—22no—
528C16—24no—
529C23—42no—
530C23—82no—
531C23—161no—
532C23—161no—
533C23C22×D812yes—
534C23C22×D811yescharacteristic
535Q8—22no—
536Q8—42no—
537Q8—42no—
538Q8—82no—
539Q8—84no—
540Q8—82no—
541Q8—82no—
542Q8—161no—
543Q8—161no—
544Q8—164no—
545Q8—162no—
546D4—24no—
547D4—22no—
548D4—42no—
549D4—42no—
550D4—82no—
551D4—162no—
552D4—162no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—82no—
568C2×C4—162no—
569C2×C4—162no—
570C2×C4—162no—
571C2×C4—164no—
572C2×C4#53612yes—
573C2×C4C22×D812yes—
574C2×C4C22×D812yes—
575C2×C4C22×D812yes—
576C2×C4C22×D811yescharacteristic
577C2×C4C22×D811yescharacteristic
578C2×C4C22×D811yescharacteristic
579C2×C4C22×D811yescharacteristic
580C2×C4C22×D811yescharacteristic
581C2×C4C2×#13712yes—
582C2×C4C23×D411yescharacteristic
583C8—22no—
584C8—22no—
585C8#53612yes—
586C8C2×#13712yes—
587C8C2×#13712yes—
588C8C2×2−1+412yes—
589C22—21no—
590C22—21no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—322no—
598C22#322612yes—
599C22C23×D811yescharacteristic
600C4—22no—
601C4—21no—
602C4—22no—
603C4—21no—
604C4—21no—
605C4—22no—
606C4—21no—
607C4—82no—
608C4—84no—
609C4—82no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—82no—
615C4#322612yes—
616C4C2×#53611yescharacteristic
617C4C22×#13711yescharacteristic
618C2—22no—
619C2—42no—
620C2—162no—
621C2C2×#322611yescharacteristic
622C2#6263611yescharacteristic
623C2#6263711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#3058C22
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
2. #60667C2semidirect
3. #33810C2semidirect
4. #33809C2semidirect
5. #33806C2semidirect
6. #33806C2semidirect
7. #33805C2semidirect
8. #33804C2semidirect
9. #33731C2semidirect
10. #33728C2semidirect
11. #33706C2semidirect
12. C2×#2243C2semidirect
13. #33651C2semidirect
15. #33632C2semidirect
16. #33631C2semidirect
17. #33421C2semidirect
31. #3164C22semidirect
34. #3160C22semidirect
36. C2×#500C22semidirect
48. #2255C22semidirect
49. #2255C22semidirect
50. #2255C22semidirect
51. #2255C22semidirect
52. #2255C22semidirect
53. #2255C22semidirect
54. #2255C22semidirect
55. #2255C22semidirect
56. #2254C22semidirect
57. #2252C22semidirect
64. #2237C22semidirect
65. #2237C22semidirect
66. #2236C22semidirect
67. #2235C22semidirect
72. #2229C22semidirect
73. #2229C22semidirect
76. #2227C22semidirect
78. #2225C22semidirect
80. #2164C22semidirect
81. #2163C22semidirect
82. #2163C22semidirect

Maximal subgroups

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

subgroupindex
#607132
#606672
#338102
#338092
#338062
#338052
#338042
#337312
#337282
#337062
C2×#22432
#336512
#336502
#336322
#336312
#334212