Finite Group Databasebeta

← #7624268

Subgroups — #7624268

Order 512.

1272 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62247C211yesmaximal, characteristic
2C2×#3338C211yesmaximal, characteristic, complemented
3#33913C214yesmaximal, complemented
4#33911C214yesmaximal, complemented
5#33883C214yesmaximal, complemented
6#33881C214yesmaximal, complemented
7#33805C212yesmaximal, complemented
8#33762C212yesmaximal, complemented
9#33755C211yesmaximal, characteristic, complemented
10#33753C211yesmaximal, characteristic, complemented
11#33677C212yesmaximal, complemented
12#33646C212yesmaximal, complemented
13C2×#2236C211yesmaximal, characteristic, complemented
14C2×#2233C211yesmaximal, characteristic, complemented
15#33487C211yesmaximal, characteristic, complemented
16#3505—21no—
17C2×#554—21no—
18#3339C2212yes—
19#3338—22no—
20#3338—22no—
21#3338C2214yes—
22#3338C2214yescomplemented
23#3336C2212yes—
24#3301—22no—
25#3296—22no—
26#3285C2212yes—
27#3280C2212yes—
28#3250—21no—
29#3191—21no—
30#3187—21no—
31#3168C2212yes—
32C2×#500C2211yescharacteristic, complemented
33C2×#484—21no—
34C2×#484—21no—
35C2×#477—21no—
36#3050C2212yes—
37C2×#467C2211yescharacteristic
38C2×#467C2211yescharacteristic
39C2×#467C2211yescharacteristic, complemented
40C2×#466C2211yescharacteristic
41C2×C4×Q16C2211yescharacteristic, complemented
42C2×C8×D4C2211yescharacteristic
43C22×#107C2211yescharacteristic
44C22×#107C2211yescharacteristic, complemented
45#2255C2214yescomplemented
46#2243C2214yescomplemented
47#2237C2214yescomplemented
48#2236C2212yescomplemented
49#2236C2212yescomplemented
50#2236C2214yescomplemented
51#2236C2212yescomplemented
52#2236C2212yescomplemented
53#2235C2214yescomplemented
54#2235C2212yescomplemented
55#2233C2212yes—
56#2233C2212yes—
57#2233C2212yes—
58#2233C2214yescomplemented
59#2232C2214yescomplemented
60#2229C2214yescomplemented
61#2225C2214yescomplemented
62#2224C2214yescomplemented
63#2220C2214yescomplemented
64#2219C2214yescomplemented
65#2217C2214yescomplemented
66#2216C2214yescomplemented
67#2211C2214yescomplemented
68#2189C2214yescomplemented
69#2188C2212yescomplemented
70#2188C2212yescomplemented
71#2187C2212yescomplemented
72#2185C2212yes—
73#2176C2214yescomplemented
74#2175C2214yescomplemented
75#2168C2214yes—
76#2168C2214yescomplemented
77#2163C2212yescomplemented
78#2159C2214yescomplemented
79C2×#367C2212yescomplemented
80C2×#366C2212yescomplemented
81#2139C2214yescomplemented
82C2×#358C2212yescomplemented
83C2×#357C2212yescomplemented
84#2112C2212yescomplemented
85C2×C22⋊C16C2211yescharacteristic
86C2×C22⋊C16C2211yescharacteristic
87C2×#135C2312yes—
88C22×Q16C2311yescharacteristic
89C22×Q16C2311yescharacteristic
90C22×D8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93#554—24no—
94#554—24no—
95#554—42no—
96#554—42no—
97#553—22no—
98#553—42no—
99#552—22no—
100#551—42no—
101#548—22no—
102#546—22no—
103#539—41no—
104#538—41no—
105#534—41no—
106C2×C4⋊Q8—21no—
107#529—22no—
108C2×#125—41no—
109#525—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—41no—
114C2×C22⋊Q8—41no—
115C2×#121—21no—
116#518—41no—
117C2×C4×Q8—21no—
118C2×C4×D4—41no—
119C2×C4×D4D411yescharacteristic
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C2×Q32C2314yes—
123C2×SD32C2314yes—
124C2×M32C2314yes—
125C22×C16C2312yes—
126C22×C16C2312yes—
127#500—22no—
128#500C2314yes—
129#499C2312yes—
130#495C2312yes—
131#488—22no—
132#485—22no—
133#484—24no—
134#484—24no—
135#484—24no—
136#484—24no—
137#483—22no—
138#482—22no—
139#481—22no—
140#479—22no—
141#478—22no—
142#477—24no—
143#477—24no—
144#467—21no—
145#467—21no—
146#467—22no—
147#467—21no—
148#467—21no—
149#467C2312yes—
150#467C2312yes—
151#467C2312yes—
152#467C2314yes—
153#467C2312yes—
154#467C2312yes—
155#466—21no—
156#466—21no—
157#466C2312yes—
158#466C2312yes—
159#466C2312yes—
160#465—22no—
161#465—22no—
162#465—22no—
163#464—22no—
164#463—22no—
165#452—22no—
166#450—22no—
167#449—22no—
168#448—22no—
169C4×Q16—22no—
170C4×Q16C2312yes—
171C4×Q16C2314yes—
172C4×SD16—22no—
173C4×D8C2312yes—
174C8×D4D412yes—
175C8×D4D412yes—
176C8×D4C2312yes—
177C8×D4C2312yes—
178#427C2312yes—
179C2×#107—21no—
180C2×#107—21no—
181C2×#107—21no—
182C2×#107—21no—
183C2×#107—22no—
184C2×#107—21no—
185C2×#107C2311yescharacteristic
186C2×#107C2312yes—
187C2×#107C2314yes—
188C2×#107C2312yes—
189C2×#107C2311yescharacteristic
190C2×#107C2311yescharacteristic
191C2×#107C2311yescharacteristic
192C2×#107C2311yescharacteristic
193C2×C4⋊C8D411yescharacteristic
194#416—22no—
195C2×#103—21no—
196C2×#103—21no—
197C2×#103—21no—
198C2×#103—21no—
199C2×#102—21no—
200C2×C22⋊C8D411yescharacteristic
201C2×C22⋊C8D411yescharacteristic
202C2×C4×C8C2311yescharacteristic
203#367C2314yes—
204#367C2314yes—
205#367C2314yes—
206#366C2314yes—
207#366C2314yes—
208#366C2314yes—
209C4⋊C16C2314yes—
210#358C2314yes—
211#358C2314yes—
212#358C2314yes—
213#358C2314yes—
214#358C2314yes—
215#357C2314yes—
216#357C2314yes—
217#357C2314yes—
218C22⋊C16C2312yes—
219C22⋊C16C2312yes—
220C22⋊C16C2312yes—
221C22⋊C16C2312yes—
222C22⋊C16C2314yes—
223#346C2314yes—
224C2×#41—22no—
225C22×Q8—21no—
226C22×Q8—21no—
227C22×Q8—41no—
228C22×D4—21no—
229C22×D4—41no—
230C22×D4D811yescharacteristic
231C23×C4—41no—
232C23×C4—41no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235#135—44no—
236C2×Q16—21no—
237C2×Q16—21no—
238C2×Q16—42no—
239C2×Q16—42no—
240C2×Q16C2412yes—
241C2×Q16C2412yes—
242C2×SD16—22no—
243C2×SD16—22no—
244C2×D8—21no—
245C2×D8—42no—
246C2×D8C2412yes—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8C2×D411yescharacteristic
250C22×C8C2×D411yescharacteristic
251C22×C8C2×D411yescharacteristic
252C22×C8C2×D411yescharacteristic
253C22×C8C2×D412yes—
254C22×C8C2×D411yescharacteristic
255C22×C8C2411yescharacteristic
256C22×C8C2411yescharacteristic
257C22×C8C2411yescharacteristic
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—42no—
261#126—42no—
262#126—42no—
263#126—42no—
264#125—22no—
265#125—44no—
266#125—44no—
267#125—42no—
268#124—22no—
269#124—42no—
270#123—42no—
271#123—42no—
272#123—42no—
273#123—42no—
274#123—42no—
275C22⋊Q8—22no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279C22⋊Q8—24no—
280C22⋊Q8—22no—
281C22⋊Q8—41no—
282C22⋊Q8—41no—
283C22⋊Q8—41no—
284C22⋊Q8—41no—
285C22⋊Q8—42no—
286C22⋊Q8—42no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C22⋊Q8—44no—
291#121—22no—
292#121—22no—
293#121—22no—
294#121—42no—
295#121—42no—
296#121—41no—
297#121—41no—
298#121—42no—
299#120—42no—
300#120—42no—
301C4×Q8—24no—
302C4×Q8—22no—
303C4×Q8—42no—
304C4×D4—22no—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4—42no—
308C4×D4—42no—
309C4×D4—42no—
310C4×D4—42no—
311C4×D4—42no—
312C4×D4D812yes—
313C4×D4D812yes—
314C4×D4C2×D412yes—
315C4×D4C2×D412yes—
316#117—22no—
317#117—42no—
318#117—42no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—41no—
328C2×C4⋊C4—41no—
329C2×C4⋊C4—41no—
330C2×C4⋊C4—41no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4—41no—
334C2×C4⋊C4—41no—
335C2×C4⋊C4—41no—
336C2×C4⋊C4—41no—
337C2×C4⋊C4D811yescharacteristic
338C2×C22⋊C4—41no—
339C2×C22⋊C4—41no—
340C2×C22⋊C4—41no—
341C2×C22⋊C4—41no—
342C2×C22⋊C4—42no—
343C2×C22⋊C4D811yescharacteristic
344C2×C22⋊C4D811yescharacteristic
345C2×C42—41no—
346C2×C42C2×D411yescharacteristic
347Q32—44no—
348SD32—44no—
349M32—44no—
350C2×C16—42no—
351C2×C16—42no—
352C2×C16C2414yes—
353C2×C16C2414yes—
354#107—21no—
355#107—21no—
356#107—21no—
357#107—22no—
358#107—24no—
359#107—22no—
360#107—24no—
361#107—21no—
362#107—21no—
363#107C2412yes—
364#107C2412yes—
365#107C2412yes—
366#107C2412yes—
367#107C2412yes—
368#106—22no—
369#106—22no—
370C4⋊C8C2×D412yes—
371C4⋊C8C2×D412yes—
372#103—22no—
373#103—24no—
374#103—22no—
375#103—22no—
376#103—24no—
377#103—22no—
378#103—22no—
379#103—22no—
380#102—22no—
381#102—22no—
382#102—22no—
383#102—22no—
384C22⋊C8C2×D412yes—
385C22⋊C8C2×D412yes—
386C22⋊C8C2×D412yes—
387C22⋊C8C2×D412yes—
388C4×C8C2×D412yes—
389C4×C8C2412yes—
390C24—41no—
391C24C2×D811yescharacteristic
392C24C2×D811yescharacteristic
393#41—82no—
394#41—82no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8—44no—
402C2×Q8—81no—
403C2×Q8—81no—
404C2×Q8—162no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—41no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4—84no—
415C2×D4C2×D812yes—
416C2×D4C2×D812yes—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—162no—
434C22×C4—162no—
435C22×C4C2×D811yescharacteristic
436C22×C4C2×D811yescharacteristic
437C22×C4C2×D811yescharacteristic
438C22×C4C2×D811yescharacteristic
439C22×C4C2×D811yescharacteristic
440C22×C4C2×D811yescharacteristic
441C22×C4C2×D812yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446Q16—24no—
447Q16—24no—
448Q16—42no—
449Q16—81no—
450SD16—82no—
451SD16—82no—
452D8—24no—
453D8—81no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—41no—
459C2×C8—42no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8C22×D411yescharacteristic
465C2×C8C22×D412yes—
466C2×C8C22×D411yescharacteristic
467C2×C8C22×D412yes—
468C2×C8C22×D411yescharacteristic
469C2×C8C2511yescharacteristic
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—41no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4C2×D812yes—
495C4⋊C4C2×D812yes—
496C22⋊C4—42no—
497C22⋊C4—42no—
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⋊C4C2×D812yes—
510C22⋊C4C2×D812yes—
511C22⋊C4C2×D812yes—
512C22⋊C4C2×D812yes—
513C42—44no—
514C42—42no—
515C42—42no—
516C42C2×D812yes—
517C42C22×D412yes—
518C16—24no—
519C16—24no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—82no—
527C23—161no—
528C23—161no—
529C23#51014yes—
530C23C22×D812yes—
531C23C22×D811yescharacteristic
532C23C22×D811yescharacteristic
533C23C22×D811yescharacteristic
534Q8—42no—
535Q8—42no—
536Q8—42no—
537Q8—42no—
538Q8—82no—
539Q8—161no—
540Q8—164no—
541D4—24no—
542D4—24no—
543D4—42no—
544D4—42no—
545D4—84no—
546D4—84no—
547D4—161no—
548D4—162no—
549D4—162no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—162no—
578C2×C4—162no—
579C2×C4—161no—
580C2×C4—162no—
581C2×C4—162no—
582C2×C4—164no—
583C2×C4—161no—
584C2×C4C22×D812yes—
585C2×C4C22×D812yes—
586C2×C4C22×D812yes—
587C2×C4C22×D811yescharacteristic
588C2×C4C22×D811yescharacteristic
589C2×C4C22×D811yescharacteristic
590C2×C4C23×D411yescharacteristic
591C8—22no—
592C8—21no—
593C8—21no—
594C8—22no—
595C8—21no—
596C8#58512yes—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—44no—
607C22—44no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—322no—
612C22C2×#51012yes—
613C22C2×#51012yes—
614C22C2×#51012yes—
615C22C23×D811yescharacteristic
616C4—22no—
617C4—22no—
618C4—21no—
619C4—22no—
620C4—21no—
621C4—21no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4#348511yescharacteristic
630C4#358611yescharacteristic
631C2—22no—
632C2—22no—
633C2—22no—
634C2—42no—
635C2—162no—
636C2#6169411yescharacteristic
637C2C22×#51011yescharacteristic
638C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#62247C2
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. #33913C2semidirect
4. #33911C2semidirect
5. #33883C2semidirect
6. #33881C2semidirect
7. #33805C2semidirect
8. #33762C2semidirect
9. #33755C2semidirect
10. #33753C2semidirect
11. #33677C2semidirect
12. #33646C2semidirect
13. C2×#2236C2semidirect
14. C2×#2233C2semidirect
15. #33487C2semidirect
22. #3338C22semidirect
32. C2×#500C22semidirect
39. C2×#467C22semidirect
41. C2×C4×Q16C22semidirect
44. C22×#107C22semidirect
45. #2255C22semidirect
46. #2243C22semidirect
47. #2237C22semidirect
48. #2236C22semidirect
49. #2236C22semidirect
50. #2236C22semidirect
51. #2236C22semidirect
52. #2236C22semidirect
53. #2235C22semidirect
54. #2235C22semidirect
58. #2233C22semidirect
59. #2232C22semidirect
60. #2229C22semidirect
61. #2225C22semidirect
62. #2224C22semidirect
63. #2220C22semidirect
64. #2219C22semidirect
65. #2217C22semidirect
66. #2216C22semidirect
67. #2211C22semidirect
68. #2189C22semidirect
69. #2188C22semidirect
70. #2188C22semidirect
71. #2187C22semidirect
73. #2176C22semidirect
74. #2175C22semidirect
76. #2168C22semidirect
77. #2163C22semidirect
78. #2159C22semidirect
79. C2×#367C22semidirect
80. C2×#366C22semidirect
81. #2139C22semidirect
82. C2×#358C22semidirect
83. C2×#357C22semidirect
84. #2112C22semidirect

Maximal subgroups

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

subgroupindex
#622472
C2×#33382
#339132
#339112
#338832
#338812
#338052
#337622
#337552
#337532
#336772
#336462
C2×#22362
C2×#22332
#334872