Finite Group Databasebeta

← #7624363

Subgroups — #7624363

Order 512.

1277 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62252C211yesmaximal, characteristic, complemented
2C2×#3341C211yesmaximal, characteristic, complemented
3#33921C212yesmaximal, complemented
4#33915C214yesmaximal, complemented
5#33907C212yesmaximal, complemented
6#33896C212yesmaximal, complemented
7#33888C214yesmaximal, complemented
8#33870C212yesmaximal, complemented
9#33804C211yesmaximal, characteristic, complemented
10#33795C211yesmaximal, characteristic, complemented
11#33760C212yesmaximal, complemented
12#33753C212yesmaximal, complemented
13#33670C211yesmaximal, characteristic, complemented
14#33665C211yesmaximal, characteristic, complemented
15#33649C212yesmaximal, complemented
16#33645C212yesmaximal, complemented
17#33488C211yesmaximal, characteristic, complemented
18#3520—21no—
19C2×#555—21no—
20#3342C2211yescharacteristic, complemented
21#3341—21no—
22#3341—22no—
23#3341—21no—
24#3341C2212yes—
25#3341C2214yes—
26#3341C2212yes—
27#3336C2212yescomplemented
28#3334C2211yescharacteristic, complemented
29#3302—21no—
30#3300—22no—
31#3298—21no—
32D4×Q16C2211yescharacteristic, complemented
33#3285C2212yescomplemented
34#3283C2211yescharacteristic, complemented
35#3248—21no—
36#3190—22no—
37#3170C2211yescharacteristic
38#3168C2211yescharacteristic
39C2×#500C2211yescharacteristic
40C2×#482—22no—
41C2×#476—21no—
42#3050C2212yes—
43C2×#467C2212yes—
44C2×#466C2212yes—
45C2×C4×D8C2211yescharacteristic
46C2×C8×D4C2211yescharacteristic
47C2×#427C2212yes—
48#2255C2212yescomplemented
49#2254C2212yescomplemented
50#2243C2212yescomplemented
51#2241C2212yescomplemented
52#2237C2214yescomplemented
53#2237C2214yescomplemented
54#2237C2214yescomplemented
55#2236C2212yes—
56#2236C2211yescharacteristic
57#2235C2212yes—
58#2235C2212yes—
59#2235C2212yes—
60#2234C2214yescomplemented
61#2234C2214yescomplemented
62#2233C2211yescharacteristic
63#2233C2214yescomplemented
64#2232C2212yes—
65#2228C2212yescomplemented
66#2227C2212yescomplemented
67#2226C2212yescomplemented
68#2225C2214yescomplemented
69#2224C2212yescomplemented
70#2220C2214yescomplemented
71#2218C2214yescomplemented
72#2217C2214yescomplemented
73#2217C2214yescomplemented
74#2210C2212yescomplemented
75#2208C2212yescomplemented
76#2187C2211yescharacteristic
77#2187C2212yes—
78#2186C2214yescomplemented
79#2185C2214yescomplemented
80#2184C2211yescharacteristic
81#2175C2214yescomplemented
82C4×SD32C2212yescomplemented
83C4×D16C2212yescomplemented
84#2169C2212yescomplemented
85#2168C2214yescomplemented
86D4×C16C2212yescomplemented
87#2163C2211yescharacteristic
88#2162C2211yescharacteristic
89#2159C2214yescomplemented
90#2158C2214yescomplemented
91#2141C2214yescomplemented
92#2138C2214yescomplemented
93#2112C2212yes—
94C2×C22⋊C16C2212yes—
95C2×#135C2312yes—
96C22×Q16C2312yes—
97C22×D8C2311yescharacteristic
98C23×C8C2312yes—
99#556—21no—
100#555—22no—
101#555—22no—
102#555—24no—
103#555—41no—
104#555—42no—
105#555—41no—
106#554—41no—
107#552—22no—
108#551—42no—
109D4×Q8—21no—
110#548—22no—
111#547—21no—
112#547—21no—
113#546—41no—
114#542—41no—
115#539—42no—
116#533—21no—
117C2×C4⋊Q8—21no—
118#529—21no—
119C2×#124—42no—
120#525—22no—
121C2×C22⋊Q8—22no—
122C2×#121—22no—
123C2×#121—41no—
124#520—41no—
125C2×C4×D4—21no—
126C2×C4×D4D411yescharacteristic
127C2×#117—22no—
128C2×#117—41no—
129C2×SD32C2312yes—
130C2×SD32C2314yes—
131C2×D16C2312yes—
132C2×M32C2314yes—
133C22×C16C2314yes—
134#500—21no—
135#500—21no—
136#500C2311yescharacteristic
137#500C2312yes—
138#500C2312yes—
139#499C2311yescharacteristic
140#495C2311yescharacteristic
141#494C2311yescharacteristic
142#488—21no—
143#487—21no—
144#485—22no—
145#485—22no—
146#483—22no—
147#482—24no—
148#482—24no—
149#482—24no—
150#482—24no—
151#480—22no—
152#479—22no—
153#476—22no—
154#476—22no—
155#476—24no—
156#475—21no—
157#474—21no—
158#467—21no—
159#467—22no—
160#467—22no—
161#467C2312yes—
162#467C2312yes—
163#467C2312yes—
164#467C2311yescharacteristic
165#467C2312yes—
166#466—22no—
167#466—22no—
168#466C2312yes—
169#466C2314yes—
170#466C2314yes—
171#466C2311yescharacteristic
172#465—21no—
173#465—22no—
174#465—22no—
175#464—22no—
176#462—21no—
177#461—21no—
178#452—22no—
179#452—22no—
180#451—22no—
181#448—22no—
182C4×Q16—21no—
183C4×Q16C2312yes—
184C4×Q16C2311yescharacteristic
185C4×SD16—21no—
186C4×D8—21no—
187C4×D8—21no—
188C4×D8C2311yescharacteristic
189C4×D8C2312yes—
190C4×D8C2312yes—
191C8×D4D411yescharacteristic
192C8×D4D412yes—
193C8×D4D411yescharacteristic
194C8×D4C2311yescharacteristic
195C8×D4C2312yes—
196C8×D4C2311yescharacteristic
197#427—22no—
198#427—22no—
199#427C2312yes—
200#427C2314yes—
201#427C2314yes—
202C2×#107—21no—
203C2×#107C2312yes—
204C2×#107C2312yes—
205C2×#107C2311yescharacteristic
206C2×#106—22no—
207C2×C4⋊C8D411yescharacteristic
208#416—22no—
209C2×#103—22no—
210C2×#102—21no—
211C2×#102—22no—
212C2×C22⋊C8D412yes—
213C2×C4×C8C2311yescharacteristic
214#367C2312yes—
215#367C2314yes—
216#367C2314yes—
217#367C2314yes—
218#366C2314yes—
219#366C2314yes—
220#366C2312yes—
221C4⋊C16C2312yes—
222C4⋊C16C2312yes—
223#358C2314yes—
224#358C2314yes—
225#358C2312yes—
226#358C2314yes—
227#358C2314yes—
228#357C2312yes—
229#357C2314yes—
230#357C2312yes—
231#357C2312yes—
232#357C2314yes—
233C22⋊C16C2311yescharacteristic
234C22⋊C16C2312yes—
235C22⋊C16C2314yes—
236C22⋊C16C2311yescharacteristic
237C22⋊C16C2314yes—
238#346C2312yes—
239C4×C16C2312yes—
240C2×#41—22no—
241C2×#41—41no—
242C22×Q8—22no—
243C22×Q8—41no—
244C22×D4—21no—
245C22×D4—42no—
246C22×D4D811yescharacteristic
247C23×C4—41no—
248C23×C4C2×D412yes—
249#135—44no—
250C2×Q16—22no—
251C2×Q16—21no—
252C2×Q16—44no—
253C2×Q16C2411yescharacteristic
254C2×Q16C2412yes—
255C2×SD16—22no—
256C2×SD16—21no—
257C2×D8—21no—
258C2×D8—42no—
259C2×D8C2412yes—
260C2×D8C2411yescharacteristic
261C22×C8—44no—
262C22×C8C2×D412yes—
263C22×C8C2×D412yes—
264C22×C8C2×D412yes—
265C22×C8C2×D411yescharacteristic
266C22×C8C2411yescharacteristic
267C22×C8C2412yes—
268C22×C8C2412yes—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—21no—
272C4⋊Q8—21no—
273C4⋊Q8—41no—
274C4⋊Q8—41no—
275C4⋊Q8—41no—
276#126—42no—
277#126—42no—
278#126—41no—
279#125—21no—
280#125—42no—
281#125—41no—
282#124—21no—
283#124—41no—
284#124—44no—
285#124—44no—
286#124—44no—
287#124—44no—
288#124—42no—
289#123—42no—
290#123—42no—
291#123—41no—
292#123—42no—
293#123—42no—
294C22⋊Q8—21no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—42no—
300C22⋊Q8—42no—
301C22⋊Q8—41no—
302C22⋊Q8—42no—
303C22⋊Q8—41no—
304C22⋊Q8—42no—
305#121—24no—
306#121—24no—
307#121—21no—
308#121—22no—
309#121—42no—
310#121—42no—
311#121—44no—
312#121—42no—
313#121—41no—
314#121—42no—
315#121—42no—
316#120—42no—
317C4×Q8—21no—
318C4×Q8—22no—
319C4×Q8—41no—
320C4×Q8—41no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—21no—
324C4×D4—41no—
325C4×D4—42no—
326C4×D4—42no—
327C4×D4—41no—
328C4×D4—42no—
329C4×D4—41no—
330C4×D4—41no—
331C4×D4D811yescharacteristic
332C4×D4D812yes—
333C4×D4D811yescharacteristic
334C4×D4C2×D411yescharacteristic
335C4×D4C2×D411yescharacteristic
336C4×D4C2×D412yes—
337#117—24no—
338#117—24no—
339#117—22no—
340#117—42no—
341#117—42no—
342#117—41no—
343#117—42no—
344#117—44no—
345#117—42no—
346#117—42no—
347#117—41no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—41no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—41no—
354C2×C4⋊C4D811yescharacteristic
355C2×C22⋊C4—42no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4—41no—
359C2×C22⋊C4D812yes—
360C2×C42—42no—
361C2×C42C2×D411yescharacteristic
362SD32—42no—
363SD32—44no—
364D16—42no—
365M32—44no—
366C2×C16—44no—
367C2×C16C2412yes—
368C2×C16C2412yes—
369C2×C16C2414yes—
370#107—22no—
371#107—22no—
372#107—21no—
373#107—22no—
374#107—22no—
375#107—21no—
376#107C2411yescharacteristic
377#107C2411yescharacteristic
378#107C2412yes—
379#107C2412yes—
380#107C2414yes—
381#106—24no—
382#106—24no—
383#106—22no—
384#106—21no—
385C4⋊C8C2×D411yescharacteristic
386C4⋊C8C2×D411yescharacteristic
387C4⋊C8C2×D412yes—
388#103—21no—
389#103—22no—
390#103—22no—
391#103—22no—
392#103—22no—
393#103—22no—
394#103—21no—
395#102—22no—
396#102—22no—
397#102—24no—
398#102—24no—
399#102—21no—
400#102—22no—
401#102—21no—
402C22⋊C8C2×D412yes—
403C22⋊C8C2×D412yes—
404C22⋊C8C2×D412yes—
405C22⋊C8C2×D412yes—
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D411yescharacteristic
408C4×C8C2411yescharacteristic
409C4×C8C2411yescharacteristic
410C24—41no—
411C24C2×D812yes—
412#41—82no—
413#41—82no—
414#41—84no—
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—41no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—82no—
422C2×Q8—82no—
423C2×Q8—162no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—41no—
435C2×D4—164no—
436C2×D4C2×D811yescharacteristic
437C2×D4C2×D811yescharacteristic
438C2×D4C2×D812yes—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—84no—
450C22×C4C2×D811yescharacteristic
451C22×C4C2×D811yescharacteristic
452C22×C4C2×D812yes—
453C22×C4C2×D812yes—
454C22×C4C2×D812yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458Q16—22no—
459Q16—24no—
460Q16—81no—
461Q16—82no—
462SD16—81no—
463SD16—82no—
464D8—22no—
465D8—22no—
466D8—22no—
467D8—42no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—41no—
475C2×C8C22×D411yescharacteristic
476C2×C8C22×D411yescharacteristic
477C2×C8C22×D411yescharacteristic
478C2×C8C22×D412yes—
479C2×C8C22×D412yes—
480C2×C8C2511yescharacteristic
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—41no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—42no—
500C4⋊C4—41no—
501C4⋊C4—42no—
502C4⋊C4—41no—
503C4⋊C4C2×D811yescharacteristic
504C4⋊C4C2×D812yes—
505C4⋊C4C2×D811yescharacteristic
506C22⋊C4—41no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—42no—
522C22⋊C4—41no—
523C22⋊C4C2×D812yes—
524C22⋊C4C2×D812yes—
525C22⋊C4C2×D812yes—
526C22⋊C4C2×D812yes—
527C42—44no—
528C42—44no—
529C42—42no—
530C42—42no—
531C42—41no—
532C42C2×D811yescharacteristic
533C42C2×D811yescharacteristic
534C42C22×D411yescharacteristic
535C42C22×D411yescharacteristic
536C16—22no—
537C16—24no—
538C16—22no—
539C23—24no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—82no—
545C23—82no—
546C23—82no—
547C23C22×D812yes—
548C23C22×D812yes—
549C23C22×D811yescharacteristic
550Q8—41no—
551Q8—42no—
552Q8—42no—
553Q8—41no—
554Q8—82no—
555Q8—162no—
556Q8—162no—
557Q8—162no—
558D4—22no—
559D4—22no—
560D4—42no—
561D4—41no—
562D4—42no—
563D4—42no—
564D4—41no—
565D4—82no—
566D4—84no—
567D4—82no—
568D4—164no—
569D4—164no—
570D4—162no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—82no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—82no—
595C2×C4—162no—
596C2×C4—162no—
597C2×C4C22×D812yes—
598C2×C4C22×D811yescharacteristic
599C2×C4C22×D812yes—
600C2×C4C22×D811yescharacteristic
601C2×C4C22×D811yescharacteristic
602C2×C4C22×D811yescharacteristic
603C2×C4C22×D811yescharacteristic
604C2×C4C23×D411yescharacteristic
605C8—21no—
606C8—21no—
607C8—21no—
608C8—22no—
609C8—22no—
610C8#58512yes—
611C22—21no—
612C22—21no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—84no—
620C22—82no—
621C22—81no—
622C22—81no—
623C22—82no—
624C22—81no—
625C22—82no—
626C22—164no—
627C22#341612yes—
628C22C23×D811yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—22no—
634C4—22no—
635C4—82no—
636C4—81no—
637C4—82no—
638C4—84no—
639C4—81no—
640C4—82no—
641C4—81no—
642C4#341412yes—
643C4#348511yescharacteristic
644C4#358611yescharacteristic
645C2—22no—
646C2—42no—
647C2—42no—
648C2—161no—
649C2—162no—
650C2#6169411yescharacteristic
651C2C2×#341411yescharacteristic
652C2C2×#341611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62252C2semidirect
2. C2×#3341C2semidirect
3. #33921C2semidirect
4. #33915C2semidirect
5. #33907C2semidirect
6. #33896C2semidirect
7. #33888C2semidirect
8. #33870C2semidirect
9. #33804C2semidirect
10. #33795C2semidirect
11. #33760C2semidirect
12. #33753C2semidirect
13. #33670C2semidirect
14. #33665C2semidirect
15. #33649C2semidirect
16. #33645C2semidirect
17. #33488C2semidirect
20. #3342C22semidirect
27. #3336C22semidirect
28. #3334C22semidirect
32. D4×Q16C22semidirect
33. #3285C22semidirect
34. #3283C22semidirect
48. #2255C22semidirect
49. #2254C22semidirect
50. #2243C22semidirect
51. #2241C22semidirect
52. #2237C22semidirect
53. #2237C22semidirect
54. #2237C22semidirect
60. #2234C22semidirect
61. #2234C22semidirect
63. #2233C22semidirect
65. #2228C22semidirect
66. #2227C22semidirect
67. #2226C22semidirect
68. #2225C22semidirect
69. #2224C22semidirect
70. #2220C22semidirect
71. #2218C22semidirect
72. #2217C22semidirect
73. #2217C22semidirect
74. #2210C22semidirect
75. #2208C22semidirect
78. #2186C22semidirect
79. #2185C22semidirect
81. #2175C22semidirect
82. C4×SD32C22semidirect
83. C4×D16C22semidirect
84. #2169C22semidirect
85. #2168C22semidirect
86. D4×C16C22semidirect
89. #2159C22semidirect
90. #2158C22semidirect
91. #2141C22semidirect
92. #2138C22semidirect

Maximal subgroups

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

subgroupindex
#622522
C2×#33412
#339212
#339152
#339072
#338962
#338882
#338702
#338042
#337952
#337602
#337532
#336702
#336652
#336492
#336452
#334882