Finite Group Databasebeta

← #7624358

Subgroups — #7624358

Order 512.

1299 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62251C211yesmaximal, characteristic
2C2×#3338C211yesmaximal, characteristic, complemented
3#33924C212yesmaximal, complemented
4#33913C214yesmaximal, complemented
5#33905C212yesmaximal, complemented
6#33900C212yesmaximal, complemented
7#33881C214yesmaximal, complemented
8#33862C212yesmaximal, complemented
9#33804C211yesmaximal, characteristic, complemented
10#33792C211yesmaximal, characteristic, complemented
11#33759C212yesmaximal, complemented
12#33756C212yesmaximal, complemented
13#33672C211yesmaximal, characteristic, complemented
14#33668C211yesmaximal, characteristic, complemented
15#33646C212yesmaximal, complemented
16C2×#2232C212yesmaximal, complemented
17#33488C211yesmaximal, characteristic, complemented
18#3519—21no—
19C2×#554—21no—
20#3341C2211yescharacteristic
21#3339C2212yes—
22#3338—21no—
23#3338—22no—
24#3338—21no—
25#3338C2211yescharacteristic
26#3338C2212yes—
27#3338C2214yescomplemented
28#3338C2212yescomplemented
29#3305—21no—
30#3297—22no—
31#3295—21no—
32#3287C2211yescharacteristic
33#3280C2212yes—
34D4×D8C2211yescharacteristic
35#3247—21no—
36#3188—22no—
37#3169C2211yescharacteristic
38#3168C2211yescharacteristic
39C2×#500C2211yescharacteristic, complemented
40C2×#484—22no—
41C2×#477—21no—
42#3050C2212yes—
43C2×#467C2212yescomplemented
44C2×#466C2212yes—
45C2×C4×Q16C2211yescharacteristic, complemented
46C2×C8×D4C2211yescharacteristic
47C22×#107C2212yescomplemented
48#2255C2212yescomplemented
49#2252C2212yescomplemented
50#2243C2212yescomplemented
51#2240C2212yescomplemented
52#2237C2214yescomplemented
53#2236C2214yescomplemented
54#2236C2214yescomplemented
55#2235C2212yescomplemented
56#2235C2211yescharacteristic, complemented
57#2233C2214yescomplemented
58#2232C2212yes—
59#2232C2211yescharacteristic
60#2232C2212yescomplemented
61#2232C2212yescomplemented
62#2232C2214yescomplemented
63#2232C2214yescomplemented
64#2232C2212yescomplemented
65#2229C2212yescomplemented
66#2227C2212yescomplemented
67#2227C2212yescomplemented
68#2226C2212yescomplemented
69#2224C2214yescomplemented
70#2220C2214yescomplemented
71#2220C2214yescomplemented
72#2217C2214yescomplemented
73#2216C2214yescomplemented
74#2211C2212yescomplemented
75#2207C2212yescomplemented
76#2189C2214yescomplemented
77#2188C2211yescharacteristic, complemented
78#2187C2214yescomplemented
79#2185C2211yescharacteristic
80#2185C2212yescomplemented
81#2175C2214yescomplemented
82C4×Q32C2212yescomplemented
83C4×SD32C2212yescomplemented
84#2169C2212yescomplemented
85#2168C2214yescomplemented
86D4×C16C2212yes—
87#2163C2211yescharacteristic, complemented
88#2162C2211yescharacteristic, complemented
89#2159C2214yescomplemented
90C2×#366C2214yescomplemented
91#2139C2214yescomplemented
92C2×#357C2214yescomplemented
93#2112C2212yescomplemented
94C2×C22⋊C16C2212yescomplemented
95C2×#135C2312yes—
96C22×Q16C2311yescharacteristic
97C22×D8C2312yes—
98C23×C8C2312yes—
99#555—21no—
100#554—22no—
101#554—24no—
102#554—22no—
103#554—21no—
104#554—41no—
105#554—42no—
106#554—41no—
107#553—22no—
108#553—41no—
109#550—21no—
110#546—22no—
111#546—42no—
112D42—21no—
113D42—41no—
114#540—41no—
115#538—42no—
116#532—21no—
117C2×C4⋊Q8—21no—
118#529—21no—
119C2×#125—41no—
120#525—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—42no—
123C2×#121—22no—
124#518—41no—
125C2×C4×Q8—21no—
126C2×C4×D4—41no—
127C2×C4×D4D411yescharacteristic
128C22×C4⋊C4—22no—
129C2×Q32C2312yes—
130C2×SD32C2312yes—
131C2×SD32C2314yes—
132C2×M32C2314yes—
133C22×C16C2314yes—
134#500—21no—
135#500—21no—
136#500C2312yes—
137#500C2311yescharacteristic
138#500C2312yes—
139#499C2311yescharacteristic
140#495C2311yescharacteristic
141#493C2311yescharacteristic
142#488—21no—
143#486—21no—
144#484—24no—
145#484—24no—
146#484—24no—
147#484—24no—
148#484—22no—
149#482—22no—
150#482—22no—
151#481—22no—
152#478—22no—
153#477—22no—
154#477—22no—
155#477—24no—
156#477—21no—
157#476—21no—
158#467—22no—
159#467—22no—
160#467C2314yes—
161#467C2312yes—
162#467C2314yes—
163#467C2311yescharacteristic
164#466—21no—
165#466—22no—
166#466—22no—
167#466C2312yes—
168#466C2312yes—
169#466C2312yes—
170#466C2311yescharacteristic
171#466C2312yes—
172#465—21no—
173#465—22no—
174#465—22no—
175#463—22no—
176#460—21no—
177#459—21no—
178#450—22no—
179#449—22no—
180#449—22no—
181#447—22no—
182C4×Q16—21no—
183C4×Q16—21no—
184C4×Q16C2312yes—
185C4×Q16C2311yescharacteristic
186C4×Q16C2312yes—
187C4×SD16—21no—
188C4×D8—21no—
189C4×D8C2312yes—
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×#107C2314yes—
203C2×#107C2312yes—
204C2×#107C2314yes—
205C2×#107C2312yes—
206C2×#107C2311yescharacteristic
207C2×C4⋊C8D411yescharacteristic
208#416—22no—
209C2×#103—22no—
210C2×#103—21no—
211C2×#102—22no—
212C2×C22⋊C8D412yes—
213C2×C4×C8C2311yescharacteristic
214#367C2312yes—
215#367C2314yes—
216#366C2312yes—
217#366C2314yes—
218#366C2314yes—
219#366C2314yes—
220#366C2314yes—
221C4⋊C16C2312yes—
222C4⋊C16C2312yes—
223#358C2312yes—
224#358C2314yes—
225#358C2312yes—
226#358C2312yes—
227#358C2314yes—
228#357C2312yes—
229#357C2314yes—
230#357C2314yes—
231#357C2314yes—
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—41no—
244C22×D4—22no—
245C22×D4—42no—
246C22×D4—41no—
247C22×D4D811yescharacteristic
248C23×C4—42no—
249C23×C4C2×D412yes—
250#135—44no—
251C2×Q16—21no—
252C2×Q16—42no—
253C2×Q16C2412yes—
254C2×Q16C2411yescharacteristic
255C2×SD16—22no—
256C2×SD16—21no—
257C2×D8—22no—
258C2×D8—21no—
259C2×D8—44no—
260C2×D8C2411yescharacteristic
261C2×D8C2412yes—
262C22×C8—44no—
263C22×C8C2×D412yes—
264C22×C8C2×D412yes—
265C22×C8C2×D412yes—
266C22×C8C2×D411yescharacteristic
267C22×C8C2411yescharacteristic
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C4⋊Q8—21no—
273C4⋊Q8—41no—
274C4⋊Q8—41no—
275#127—21no—
276#127—41no—
277#126—42no—
278#126—41no—
279#125—21no—
280#125—42no—
281#125—42no—
282#125—44no—
283#125—41no—
284#124—21no—
285#124—42no—
286#124—41no—
287#123—42no—
288#123—41no—
289#123—42no—
290#123—42no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—21no—
294C22⋊Q8—22no—
295C22⋊Q8—42no—
296C22⋊Q8—42no—
297C22⋊Q8—44no—
298C22⋊Q8—44no—
299C22⋊Q8—44no—
300C22⋊Q8—44no—
301C22⋊Q8—42no—
302#121—21no—
303#121—22no—
304#121—22no—
305#121—22no—
306#121—22no—
307#121—42no—
308#121—41no—
309#121—42no—
310#121—42no—
311#121—41no—
312#121—41no—
313#121—42no—
314#121—42no—
315#120—42no—
316#120—42no—
317#120—42no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—21no—
321C4×Q8—41no—
322C4×Q8—41no—
323C4×D4—21no—
324C4×D4—22no—
325C4×D4—44no—
326C4×D4—42no—
327C4×D4—42no—
328C4×D4—41no—
329C4×D4—42no—
330C4×D4—42no—
331C4×D4—41no—
332C4×D4—41no—
333C4×D4—42no—
334C4×D4—41no—
335C4×D4D811yescharacteristic
336C4×D4D812yes—
337C4×D4D811yescharacteristic
338C4×D4C2×D411yescharacteristic
339C4×D4C2×D412yes—
340C4×D4C2×D411yescharacteristic
341#117—22no—
342#117—41no—
343#117—41no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—41no—
353C2×C4⋊C4—41no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4D811yescharacteristic
356C2×C22⋊C4—42no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4—42no—
359C2×C22⋊C4D812yes—
360C2×C42—41no—
361C2×C42C2×D411yescharacteristic
362Q32—42no—
363SD32—44no—
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#107C2412yes—
382#107C2411yescharacteristic
383#106—22no—
384#106—21no—
385C4⋊C8C2×D411yescharacteristic
386C4⋊C8C2×D411yescharacteristic
387C4⋊C8C2×D412yes—
388#103—22no—
389#103—22no—
390#103—24no—
391#103—24no—
392#103—21no—
393#103—22no—
394#103—21no—
395#102—21no—
396#102—22no—
397#102—22no—
398#102—22no—
399#102—22no—
400#102—21no—
401#102—22no—
402C22⋊C8C2×D412yes—
403C22⋊C8C2×D412yes—
404C22⋊C8C2×D412yes—
405C22⋊C8C2×D412yes—
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D411yescharacteristic
408C4×C8C2411yescharacteristic
409C4×C8C2411yescharacteristic
410C24—42no—
411C24C2×D812yes—
412#41—82no—
413#41—82no—
414#41—162no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—41no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—41no—
423C2×Q8—162no—
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—42no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—84no—
439C2×D4—164no—
440C2×D4C2×D812yes—
441C2×D4C2×D811yescharacteristic
442C2×D4C2×D811yescharacteristic
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—164no—
455C22×C4C2×D811yescharacteristic
456C22×C4C2×D811yescharacteristic
457C22×C4C2×D812yes—
458C22×C4C2×D812yes—
459C22×C4C2×D812yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463Q16—22no—
464Q16—22no—
465Q16—22no—
466Q16—42no—
467SD16—81no—
468SD16—82no—
469D8—22no—
470D8—24no—
471D8—81no—
472D8—82no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—41no—
480C2×C8C22×D411yescharacteristic
481C2×C8C22×D411yescharacteristic
482C2×C8C22×D411yescharacteristic
483C2×C8C22×D412yes—
484C2×C8C22×D412yes—
485C2×C8C2511yescharacteristic
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—41no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—41no—
508C4⋊C4C2×D811yescharacteristic
509C4⋊C4C2×D811yescharacteristic
510C4⋊C4C2×D812yes—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—42no—
518C22⋊C4—41no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—41no—
522C22⋊C4—42no—
523C22⋊C4—41no—
524C22⋊C4—42no—
525C22⋊C4C2×D812yes—
526C22⋊C4C2×D812yes—
527C22⋊C4C2×D812yes—
528C22⋊C4C2×D812yes—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—41no—
533C42C2×D811yescharacteristic
534C42C2×D811yescharacteristic
535C42C22×D411yescharacteristic
536C42C22×D411yescharacteristic
537C16—22no—
538C16—24no—
539C16—22no—
540C23—24no—
541C23—24no—
542C23—42no—
543C23—41no—
544C23—42no—
545C23—84no—
546C23—162no—
547C23—162no—
548C23C22×D812yes—
549C23C22×D812yes—
550C23C22×D811yescharacteristic
551Q8—41no—
552Q8—42no—
553Q8—41no—
554Q8—42no—
555Q8—82no—
556Q8—162no—
557Q8—162no—
558Q8—161no—
559D4—24no—
560D4—22no—
561D4—42no—
562D4—41no—
563D4—41no—
564D4—42no—
565D4—84no—
566D4—82no—
567D4—84no—
568D4—82no—
569D4—162no—
570D4—161no—
571D4—162no—
572D4—162no—
573D4—162no—
574D4#51014yes—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—84no—
596C2×C4—82no—
597C2×C4—162no—
598C2×C4—162no—
599C2×C4—164no—
600C2×C4—164no—
601C2×C4—162no—
602C2×C4C22×D811yescharacteristic
603C2×C4C22×D812yes—
604C2×C4C22×D811yescharacteristic
605C2×C4C22×D811yescharacteristic
606C2×C4C22×D811yescharacteristic
607C2×C4C22×D812yes—
608C2×C4C22×D811yescharacteristic
609C2×C4C23×D411yescharacteristic
610C8—21no—
611C8—21no—
612C8—21no—
613C8—22no—
614C8—22no—
615C8#58512yes—
616C22—21no—
617C22—22no—
618C22—21no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—48no—
624C22—81no—
625C22—82no—
626C22—82no—
627C22—81no—
628C22—82no—
629C22—81no—
630C22—324no—
631C22C2×#51014yes—
632C22#341512yes—
633C22C23×D811yescharacteristic
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—22no—
639C4—22no—
640C4—81no—
641C4—82no—
642C4—84no—
643C4—81no—
644C4—82no—
645C4—82no—
646C4—81no—
647C4C2×#51012yes—
648C4#348511yescharacteristic
649C4#358611yescharacteristic
650C2—22no—
651C2—24no—
652C2—42no—
653C2—162no—
654C2—161no—
655C2#6169411yescharacteristic
656C2C22×#51011yescharacteristic
657C2C2×#341511yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#62251C2
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. #33924C2semidirect
4. #33913C2semidirect
5. #33905C2semidirect
6. #33900C2semidirect
7. #33881C2semidirect
8. #33862C2semidirect
9. #33804C2semidirect
10. #33792C2semidirect
11. #33759C2semidirect
12. #33756C2semidirect
13. #33672C2semidirect
14. #33668C2semidirect
15. #33646C2semidirect
16. C2×#2232C2semidirect
17. #33488C2semidirect
27. #3338C22semidirect
28. #3338C22semidirect
39. C2×#500C22semidirect
43. C2×#467C22semidirect
45. C2×C4×Q16C22semidirect
47. C22×#107C22semidirect
48. #2255C22semidirect
49. #2252C22semidirect
50. #2243C22semidirect
51. #2240C22semidirect
52. #2237C22semidirect
53. #2236C22semidirect
54. #2236C22semidirect
55. #2235C22semidirect
56. #2235C22semidirect
57. #2233C22semidirect
60. #2232C22semidirect
61. #2232C22semidirect
62. #2232C22semidirect
63. #2232C22semidirect
64. #2232C22semidirect
65. #2229C22semidirect
66. #2227C22semidirect
67. #2227C22semidirect
68. #2226C22semidirect
69. #2224C22semidirect
70. #2220C22semidirect
71. #2220C22semidirect
72. #2217C22semidirect
73. #2216C22semidirect
74. #2211C22semidirect
75. #2207C22semidirect
76. #2189C22semidirect
77. #2188C22semidirect
78. #2187C22semidirect
80. #2185C22semidirect
81. #2175C22semidirect
82. C4×Q32C22semidirect
83. C4×SD32C22semidirect
84. #2169C22semidirect
85. #2168C22semidirect
87. #2163C22semidirect
88. #2162C22semidirect
89. #2159C22semidirect
90. C2×#366C22semidirect
91. #2139C22semidirect
92. C2×#357C22semidirect
93. #2112C22semidirect
94. C2×C22⋊C16C22semidirect

Maximal subgroups

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

subgroupindex
#622512
C2×#33382
#339242
#339132
#339052
#339002
#338812
#338622
#338042
#337922
#337592
#337562
#336722
#336682
#336462
C2×#22322
#334882