Finite Group Databasebeta

← #7624367

Subgroups — #7624367

Order 512.

1267 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62252C211yesmaximal, characteristic
2C2×#3338C211yesmaximal, characteristic, complemented
3#33923C212yesmaximal, complemented
4#33911C214yesmaximal, complemented
5#33902C212yesmaximal, complemented
6#33898C212yesmaximal, complemented
7#33883C214yesmaximal, complemented
8#33866C212yesmaximal, complemented
9#33804C211yesmaximal, characteristic, complemented
10#33795C211yesmaximal, characteristic, complemented
11#33761C212yesmaximal, complemented
12#33756C212yesmaximal, complemented
13#33673C211yesmaximal, characteristic, complemented
14#33668C211yesmaximal, characteristic, complemented
15#33646C212yesmaximal, complemented
16C2×#2235C212yesmaximal, complemented
17#33488C211yesmaximal, characteristic, complemented
18#3520—21no—
19C2×#554—21no—
20#3342C2211yescharacteristic
21#3338—21no—
22#3338—22no—
23#3338—21no—
24#3338C2212yes—
25#3338C2214yes—
26#3338C2212yes—
27#3336C2212yes—
28#3334C2211yescharacteristic
29#3302—21no—
30#3300—22no—
31#3298—21no—
32D4×Q16C2211yescharacteristic
33#3285C2212yes—
34#3283C2211yescharacteristic
35#3248—21no—
36#3190—22no—
37#3170C2211yescharacteristic
38#3168C2211yescharacteristic
39C2×#500C2211yescharacteristic
40C2×#484—22no—
41C2×#477—21no—
42#3050C2212yes—
43C2×#467C2212yes—
44C2×#467C2212yes—
45C2×C4×Q16C2211yescharacteristic
46C2×C8×D4C2211yescharacteristic
47C22×#107C2212yes—
48#2255C2212yescomplemented
49#2254C2212yescomplemented
50#2243C2212yescomplemented
51#2241C2212yescomplemented
52#2237C2214yescomplemented
53#2236C2214yescomplemented
54#2235C2212yes—
55#2235C2212yes—
56#2235C2212yes—
57#2235C2211yescharacteristic
58#2235C2212yes—
59#2235C2214yescomplemented
60#2235C2214yescomplemented
61#2235C2214yescomplemented
62#2233C2214yescomplemented
63#2232C2212yes—
64#2232C2211yescharacteristic
65#2229C2212yescomplemented
66#2227C2212yescomplemented
67#2226C2212yescomplemented
68#2225C2214yescomplemented
69#2225C2212yescomplemented
70#2219C2214yescomplemented
71#2219C2214yescomplemented
72#2217C2214yescomplemented
73#2216C2214yescomplemented
74#2211C2212yescomplemented
75#2209C2212yescomplemented
76#2189C2214yescomplemented
77#2188C2211yescharacteristic
78#2188C2212yes—
79#2188C2214yescomplemented
80#2185C2211yescharacteristic
81#2176C2214yescomplemented
82C4×Q32C2212yescomplemented
83C4×SD32C2212yescomplemented
84#2169C2212yes—
85#2168C2214yes—
86D4×C16C2212yes—
87#2163C2211yescharacteristic
88#2162C2211yescharacteristic
89#2159C2214yescomplemented
90C2×#367C2214yescomplemented
91#2139C2214yescomplemented
92C2×#358C2214yescomplemented
93#2112C2212yes—
94C2×C22⋊C16C2212yes—
95C2×#135C2312yes—
96C22×Q16C2312yes—
97C22×Q16C2311yescharacteristic
98C23×C8C2312yes—
99#556—21no—
100#554—22no—
101#554—22no—
102#554—24no—
103#554—41no—
104#554—42no—
105#554—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×#125—41no—
120#525—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—42no—
124#520—41no—
125C2×C4×Q8—21no—
126C2×C4×D4—41no—
127C2×C4×D4D411yescharacteristic
128C22×C4⋊C4—22no—
129C2×Q32C2312yes—
130C2×Q32C2314yes—
131C2×SD32C2312yes—
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#484—24no—
147#484—24no—
148#484—24no—
149#484—24no—
150#483—22no—
151#480—22no—
152#479—22no—
153#477—22no—
154#477—22no—
155#477—24no—
156#475—21no—
157#474—21no—
158#467—22no—
159#467—22no—
160#467—21no—
161#467—22no—
162#467—22no—
163#467C2312yes—
164#467C2312yes—
165#467C2314yes—
166#467C2314yes—
167#467C2312yes—
168#467C2311yescharacteristic
169#467C2312yes—
170#466C2312yes—
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×Q16—21no—
184C4×Q16—21no—
185C4×Q16C2312yes—
186C4×Q16C2312yes—
187C4×Q16C2312yes—
188C4×Q16C2311yescharacteristic
189C4×SD16—21no—
190C4×D8C2311yescharacteristic
191C8×D4D411yescharacteristic
192C8×D4D412yes—
193C8×D4D411yescharacteristic
194C8×D4C2311yescharacteristic
195C8×D4C2312yes—
196C8×D4C2311yescharacteristic
197#427C2312yes—
198C2×#107—22no—
199C2×#107—22no—
200C2×#107—22no—
201C2×#107—21no—
202C2×#107C2312yes—
203C2×#107C2314yes—
204C2×#107C2314yes—
205C2×#107C2312yes—
206C2×#107C2311yescharacteristic
207C2×C4⋊C8D411yescharacteristic
208#416—22no—
209C2×#103—22no—
210C2×#103—21no—
211C2×#103—22no—
212C2×C22⋊C8D412yes—
213C2×C4×C8C2311yescharacteristic
214#367C2312yes—
215#367C2314yes—
216#367C2314yes—
217#367C2314yes—
218#367C2314yes—
219#366C2312yes—
220#366C2314yes—
221C4⋊C16C2312yes—
222C4⋊C16C2312yes—
223#358C2312yes—
224#358C2314yes—
225#358C2314yes—
226#358C2314yes—
227#358C2314yes—
228#358C2312yes—
229#358C2312yes—
230#358C2314yes—
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—21no—
243C22×Q8—22no—
244C22×Q8—41no—
245C22×D4D811yescharacteristic
246C23×C4—42no—
247C23×C4C2×D412yes—
248#135—44no—
249C2×Q16—21no—
250C2×Q16—22no—
251C2×Q16—21no—
252C2×Q16—42no—
253C2×Q16—44no—
254C2×Q16C2411yescharacteristic
255C2×Q16C2412yes—
256C2×Q16C2412yes—
257C2×SD16—22no—
258C2×SD16—21no—
259C2×D8C2411yescharacteristic
260C22×C8—44no—
261C22×C8C2×D412yes—
262C22×C8C2×D412yes—
263C22×C8C2×D412yes—
264C22×C8C2×D411yescharacteristic
265C22×C8C2411yescharacteristic
266C22×C8C2412yes—
267C22×C8C2412yes—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8—41no—
273C4⋊Q8—41no—
274C4⋊Q8—41no—
275#126—42no—
276#126—42no—
277#126—41no—
278#125—21no—
279#125—42no—
280#125—44no—
281#125—42no—
282#125—42no—
283#125—41no—
284#124—21no—
285#124—41no—
286#124—42no—
287#123—42no—
288#123—42no—
289#123—41no—
290#123—42no—
291#123—42no—
292C22⋊Q8—21no—
293C22⋊Q8—22no—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—22no—
299C22⋊Q8—42no—
300C22⋊Q8—42no—
301C22⋊Q8—42no—
302C22⋊Q8—42no—
303C22⋊Q8—41no—
304C22⋊Q8—42no—
305C22⋊Q8—41no—
306C22⋊Q8—44no—
307C22⋊Q8—44no—
308C22⋊Q8—44no—
309C22⋊Q8—44no—
310C22⋊Q8—42no—
311#121—21no—
312#121—22no—
313#121—42no—
314#121—41no—
315#120—42no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—21no—
319C4×Q8—22no—
320C4×Q8—41no—
321C4×Q8—41no—
322C4×Q8—41no—
323C4×Q8—41no—
324C4×D4—21no—
325C4×D4—44no—
326C4×D4—42no—
327C4×D4—42no—
328C4×D4—41no—
329C4×D4—42no—
330C4×D4—42no—
331C4×D4—41no—
332C4×D4—42no—
333C4×D4D811yescharacteristic
334C4×D4D812yes—
335C4×D4D811yescharacteristic
336C4×D4C2×D411yescharacteristic
337C4×D4C2×D411yescharacteristic
338C4×D4C2×D412yes—
339#117—22no—
340#117—41no—
341#117—42no—
342#117—41no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—41no—
352C2×C4⋊C4—41no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—41no—
356C2×C4⋊C4D811yescharacteristic
357C2×C22⋊C4—42no—
358C2×C22⋊C4—42no—
359C2×C22⋊C4D812yes—
360C2×C42—41no—
361C2×C42C2×D411yescharacteristic
362Q32—44no—
363Q32—42no—
364SD32—42no—
365M32—44no—
366C2×C16—44no—
367C2×C16C2412yes—
368C2×C16C2412yes—
369C2×C16C2414yes—
370#107—21no—
371#107—22no—
372#107—24no—
373#107—24no—
374#107—22no—
375#107—22no—
376#107—22no—
377#107—21no—
378#107C2412yes—
379#107C2414yes—
380#107C2411yescharacteristic
381#107C2411yescharacteristic
382#107C2412yes—
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—24no—
395#103—24no—
396#103—22no—
397#103—22no—
398#103—21no—
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
410C24C2×D812yes—
411#41—82no—
412#41—82no—
413#41—84no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—41no—
420C2×Q8—41no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—82no—
425C2×Q8—82no—
426C2×Q8—162no—
427C2×D4—24no—
428C2×D4—21no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4C2×D812yes—
433C2×D4C2×D811yescharacteristic
434C2×D4C2×D811yescharacteristic
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—164no—
447C22×C4C2×D811yescharacteristic
448C22×C4C2×D811yescharacteristic
449C22×C4C2×D812yes—
450C22×C4C2×D812yes—
451C22×C4C2×D812yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455Q16—22no—
456Q16—22no—
457Q16—22no—
458Q16—24no—
459Q16—42no—
460Q16—81no—
461Q16—82no—
462SD16—81no—
463SD16—82no—
464D8—22no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—41no—
472C2×C8C22×D411yescharacteristic
473C2×C8C22×D411yescharacteristic
474C2×C8C22×D411yescharacteristic
475C2×C8C22×D412yes—
476C2×C8C22×D412yes—
477C2×C8C2511yescharacteristic
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—41no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—41no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—41no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—42no—
503C4⋊C4—41no—
504C4⋊C4—42no—
505C4⋊C4—41no—
506C4⋊C4C2×D811yescharacteristic
507C4⋊C4C2×D812yes—
508C4⋊C4C2×D811yescharacteristic
509C22⋊C4—41no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—42no—
521C22⋊C4—41no—
522C22⋊C4C2×D812yes—
523C22⋊C4C2×D812yes—
524C22⋊C4C2×D812yes—
525C22⋊C4C2×D812yes—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—41no—
531C42C2×D811yescharacteristic
532C42C2×D811yescharacteristic
533C42C22×D411yescharacteristic
534C42C22×D411yescharacteristic
535C16—22no—
536C16—24no—
537C16—22no—
538C23—24no—
539C23—24no—
540C23—42no—
541C23—41no—
542C23C22×D812yes—
543C23C22×D812yes—
544C23C22×D811yescharacteristic
545Q8—41no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—41no—
551Q8—82no—
552Q8—82no—
553Q8—162no—
554Q8—162no—
555Q8—162no—
556D4—24no—
557D4—22no—
558D4—41no—
559D4—41no—
560D4—84no—
561D4—82no—
562D4—162no—
563D4#51014yes—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—82no—
584C2×C4—84no—
585C2×C4—82no—
586C2×C4—84no—
587C2×C4—82no—
588C2×C4—162no—
589C2×C4—164no—
590C2×C4—164no—
591C2×C4—162no—
592C2×C4C22×D811yescharacteristic
593C2×C4C22×D811yescharacteristic
594C2×C4C22×D812yes—
595C2×C4C22×D811yescharacteristic
596C2×C4C22×D811yescharacteristic
597C2×C4C22×D812yes—
598C2×C4C22×D811yescharacteristic
599C2×C4C23×D411yescharacteristic
600C8—21no—
601C8—21no—
602C8—21no—
603C8—22no—
604C8—22no—
605C8#58512yes—
606C22—21no—
607C22—22no—
608C22—21no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—48no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#51014yes—
618C22#341612yes—
619C22C23×D811yescharacteristic
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—22no—
625C4—22no—
626C4—81no—
627C4—82no—
628C4—82no—
629C4—84no—
630C4—81no—
631C4—82no—
632C4—82no—
633C4—81no—
634C4C2×#51012yes—
635C4#348511yescharacteristic
636C4#358611yescharacteristic
637C2—22no—
638C2—24no—
639C2—42no—
640C2—161no—
641C2#6169411yescharacteristic
642C2C22×#51011yescharacteristic
643C2C2×#341611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#62252C2
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
2. C2×#3338C2semidirect
3. #33923C2semidirect
4. #33911C2semidirect
5. #33902C2semidirect
6. #33898C2semidirect
7. #33883C2semidirect
8. #33866C2semidirect
9. #33804C2semidirect
10. #33795C2semidirect
11. #33761C2semidirect
12. #33756C2semidirect
13. #33673C2semidirect
14. #33668C2semidirect
15. #33646C2semidirect
16. C2×#2235C2semidirect
17. #33488C2semidirect
48. #2255C22semidirect
49. #2254C22semidirect
50. #2243C22semidirect
51. #2241C22semidirect
52. #2237C22semidirect
53. #2236C22semidirect
59. #2235C22semidirect
60. #2235C22semidirect
61. #2235C22semidirect
62. #2233C22semidirect
65. #2229C22semidirect
66. #2227C22semidirect
67. #2226C22semidirect
68. #2225C22semidirect
69. #2225C22semidirect
70. #2219C22semidirect
71. #2219C22semidirect
72. #2217C22semidirect
73. #2216C22semidirect
74. #2211C22semidirect
75. #2209C22semidirect
76. #2189C22semidirect
79. #2188C22semidirect
81. #2176C22semidirect
82. C4×Q32C22semidirect
83. C4×SD32C22semidirect
89. #2159C22semidirect
90. C2×#367C22semidirect
91. #2139C22semidirect
92. C2×#358C22semidirect

Maximal subgroups

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

subgroupindex
#622522
C2×#33382
#339232
#339112
#339022
#338982
#338832
#338662
#338042
#337952
#337612
#337562
#336732
#336682
#336462
C2×#22352
#334882