Finite Group Databasebeta

← #7624277

Subgroups — #7624277

Order 512.

1284 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62247C211yesmaximal, characteristic, complemented
2C2×#3339C211yesmaximal, characteristic, complemented
3#33922C214yesmaximal, complemented
4#33918C214yesmaximal, complemented
5#33893C214yesmaximal, complemented
6#33889C214yesmaximal, complemented
7#33798C212yesmaximal, complemented
8#33760C212yesmaximal, complemented
9#33759C211yesmaximal, characteristic, complemented
10#33758C211yesmaximal, characteristic, complemented
11#33665C212yesmaximal, complemented
12#33649C211yesmaximal, characteristic, complemented
13#33648C211yesmaximal, characteristic, complemented
14#33646C212yesmaximal, complemented
15#33489C211yesmaximal, characteristic, complemented
16#3505—21no—
17C2×#553—21no—
18#3339—22no—
19#3339—22no—
20#3339C2214yes—
21#3339C2212yescomplemented
22#3339C2214yescomplemented
23#3336C2212yescomplemented
24#3301—22no—
25#3296—22no—
26#3285C2212yescomplemented
27#3280C2212yescomplemented
28#3250—21no—
29#3191—21no—
30#3187—21no—
31#3168C2212yes—
32C2×#499C2211yescharacteristic, complemented
33C2×#481—21no—
34C2×#482—21no—
35C2×#478—21no—
36#3050C2212yes—
37C2×#467C2211yescharacteristic
38C2×#466C2211yescharacteristic
39C2×#466C2211yescharacteristic
40C2×#466C2211yescharacteristic, complemented
41C2×C4×D8C2211yescharacteristic, complemented
42C2×C8×D4C2211yescharacteristic
43C2×#427C2211yescharacteristic
44C22×#107C2211yescharacteristic, complemented
45#2253C2214yescomplemented
46#2242C2214yescomplemented
47#2237C2214yescomplemented
48#2236C2212yescomplemented
49#2236C2212yescomplemented
50#2235C2212yescomplemented
51#2235C2214yescomplemented
52#2234C2214yescomplemented
53#2234C2214yescomplemented
54#2233C2212yes—
55#2233C2212yes—
56#2233C2212yescomplemented
57#2232C2212yes—
58#2232C2214yescomplemented
59#2232C2212yescomplemented
60#2229C2214yescomplemented
61#2228C2214yescomplemented
62#2228C2214yescomplemented
63#2220C2214yescomplemented
64#2220C2214yescomplemented
65#2218C2214yescomplemented
66#2218C2214yescomplemented
67#2210C2214yescomplemented
68#2187C2212yescomplemented
69#2186C2214yescomplemented
70#2185C2212yescomplemented
71#2184C2212yes—
72#2184C2212yescomplemented
73C4×SD32C2214yescomplemented
74C4×D16C2214yescomplemented
75#2169C2214yescomplemented
76#2169C2214yescomplemented
77#2162C2212yescomplemented
78#2159C2214yescomplemented
79#2159C2212yescomplemented
80#2159C2212yescomplemented
81#2141C2212yescomplemented
82#2140C2212yescomplemented
83#2139C2214yescomplemented
84#2112C2211yescharacteristic
85#2112C2211yescharacteristic
86#2112C2212yescomplemented
87C2×#135C2312yes—
88C22×Q16C2311yescharacteristic
89C22×D8C2311yescharacteristic
90C22×D8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93#553—24no—
94#553—24no—
95#553—22no—
96#553—42no—
97#553—42no—
98#553—42no—
99#552—22no—
100#551—42no—
101#548—22no—
102#546—22no—
103#539—41no—
104#538—41no—
105#534—41no—
106#529—22no—
107C2×#126—41no—
108C2×#125—21no—
109#525—22no—
110C2×#123—41no—
111C2×C22⋊Q8—21no—
112C2×#121—21no—
113C2×#121—21no—
114C2×#121—21no—
115C2×#121—41no—
116#518—41no—
117C2×C4×D4—21no—
118C2×C4×D4—41no—
119C2×C4×D4D411yescharacteristic
120C2×#117—21no—
121C22×C4⋊C4—21no—
122C2×SD32C2314yes—
123C2×D16C2314yes—
124C2×M32C2312yes—
125C2×M32C2312yes—
126C2×M32C2314yes—
127#499—22no—
128#499C2312yes—
129#499C2314yes—
130#495C2312yes—
131#488—22no—
132#485—22no—
133#483—22no—
134#482—24no—
135#482—24no—
136#482—22no—
137#481—24no—
138#481—24no—
139#481—22no—
140#479—22no—
141#478—24no—
142#478—24no—
143#478—22no—
144#467—21no—
145#467—21no—
146#467C2312yes—
147#467C2312yes—
148#467C2312yes—
149#466—21no—
150#466—21no—
151#466—22no—
152#466—21no—
153#466—21no—
154#466C2312yes—
155#466C2314yes—
156#466C2312yes—
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×Q16C2312yes—
170C4×SD16—22no—
171C4×D8—22no—
172C4×D8C2314yes—
173C4×D8C2312yes—
174C8×D4D412yes—
175C8×D4D412yes—
176C8×D4C2312yes—
177C8×D4C2312yes—
178#427—21no—
179#427—21no—
180#427C2312yes—
181#427C2312yes—
182#427C2312yes—
183C2×#107—21no—
184C2×#107—22no—
185C2×#107C2311yescharacteristic
186C2×#107C2314yes—
187C2×#107C2311yescharacteristic
188C2×#107C2311yescharacteristic
189C2×#107C2311yescharacteristic
190C2×#107C2311yescharacteristic
191C2×#106—21no—
192C2×#106—21no—
193C2×C4⋊C8D411yescharacteristic
194#416—22no—
195C2×#103—21no—
196C2×#102—21no—
197C2×#102—21no—
198C2×#102—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#357C2314yes—
214#357C2314yes—
215#357C2314yes—
216#357C2314yes—
217#357C2314yes—
218C22⋊C16C2312yes—
219C22⋊C16C2312yes—
220C22⋊C16C2312yes—
221C22⋊C16C2312yes—
222C22⋊C16C2314yes—
223C4×C16C2314yes—
224C2×#41—22no—
225C22×Q8—21no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4—41no—
229C22×D4—41no—
230C22×D4—41no—
231C22×D4D811yescharacteristic
232C23×C4—41no—
233C23×C4—41no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236#135—44no—
237C2×Q16—21no—
238C2×Q16—42no—
239C2×Q16C2412yes—
240C2×SD16—22no—
241C2×SD16—22no—
242C2×D8—21no—
243C2×D8—21no—
244C2×D8—42no—
245C2×D8—42no—
246C2×D8C2412yes—
247C2×D8C2412yes—
248C22×C8—42no—
249C22×C8—42no—
250C22×C8C2×D411yescharacteristic
251C22×C8C2×D411yescharacteristic
252C22×C8C2×D411yescharacteristic
253C22×C8C2×D411yescharacteristic
254C22×C8C2×D412yes—
255C22×C8C2×D411yescharacteristic
256C22×C8C2411yescharacteristic
257C22×C8C2411yescharacteristic
258C22×C8C2411yescharacteristic
259C22×C8C2412yes—
260#126—44no—
261#126—44no—
262#126—42no—
263#126—42no—
264#126—42no—
265#125—24no—
266#125—22no—
267#125—42no—
268#125—42no—
269#124—22no—
270#124—42no—
271#123—42no—
272#123—42no—
273#123—42no—
274#123—42no—
275#123—44no—
276#123—44no—
277#123—42no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281C22⋊Q8—41no—
282C22⋊Q8—41no—
283C22⋊Q8—42no—
284#121—22no—
285#121—22no—
286#121—24no—
287#121—22no—
288#121—22no—
289#121—22no—
290#121—44no—
291#121—44no—
292#121—42no—
293#121—42no—
294#121—41no—
295#121—41no—
296#121—41no—
297#121—41no—
298#121—42no—
299#121—42no—
300#120—42no—
301#120—42no—
302C4×Q8—22no—
303C4×D4—24no—
304C4×D4—22no—
305C4×D4—42no—
306C4×D4—42no—
307C4×D4—42no—
308C4×D4—44no—
309C4×D4—44no—
310C4×D4—42no—
311C4×D4—42no—
312C4×D4—42no—
313C4×D4D812yes—
314C4×D4D812yes—
315C4×D4C2×D412yes—
316C4×D4C2×D412yes—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—41no—
321#117—41no—
322#117—42no—
323#117—42no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—41no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4—41no—
334C2×C4⋊C4—41no—
335C2×C4⋊C4D811yescharacteristic
336C2×C22⋊C4—41no—
337C2×C22⋊C4—41no—
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
347SD32—44no—
348D16—44no—
349M32—42no—
350M32—44no—
351M32—42no—
352C2×C16C2414yes—
353C2×C16C2414yes—
354#107—21no—
355#107—21no—
356#107—24no—
357#107—21no—
358#107—21no—
359#107—21no—
360#107C2412yes—
361#107C2412yes—
362#107C2412yes—
363#107C2412yes—
364#107C2412yes—
365#106—22no—
366#106—22no—
367#106—24no—
368#106—22no—
369#106—22no—
370C4⋊C8C2×D412yes—
371C4⋊C8C2×D412yes—
372#103—22no—
373#103—22no—
374#103—22no—
375#103—22no—
376#102—22no—
377#102—24no—
378#102—22no—
379#102—22no—
380#102—22no—
381#102—24no—
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—
391C24—41no—
392C24C2×D811yescharacteristic
393C24C2×D811yescharacteristic
394#41—82no—
395#41—82no—
396C2×Q8—22no—
397C2×Q8—41no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4—84no—
417C2×D4—84no—
418C2×D4—162no—
419C2×D4C2×D812yes—
420C2×D4C2×D812yes—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—162no—
436C22×C4—162no—
437C22×C4C2×D811yescharacteristic
438C22×C4C2×D811yescharacteristic
439C22×C4C2×D811yescharacteristic
440C22×C4C2×D811yescharacteristic
441C22×C4C2×D811yescharacteristic
442C22×C4C2×D811yescharacteristic
443C22×C4C2×D812yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448Q16—24no—
449Q16—81no—
450SD16—82no—
451SD16—82no—
452D8—24no—
453D8—24no—
454D8—42no—
455D8—81no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—41no—
461C2×C8—42no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8C22×D411yescharacteristic
467C2×C8C22×D412yes—
468C2×C8C22×D411yescharacteristic
469C2×C8C22×D412yes—
470C2×C8C22×D411yescharacteristic
471C2×C8C2511yescharacteristic
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—44no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4C2×D812yes—
492C4⋊C4C2×D812yes—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4C2×D812yes—
510C22⋊C4C2×D812yes—
511C22⋊C4C2×D812yes—
512C22⋊C4C2×D812yes—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42C2×D812yes—
518C42C22×D412yes—
519C16—24no—
520C16—24no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23—161no—
534C23—161no—
535C23C22×D811yescharacteristic
536C23C22×D812yes—
537C23C22×D811yescharacteristic
538C23C22×D811yescharacteristic
539Q8—42no—
540Q8—42no—
541Q8—161no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—82no—
549D4—84no—
550D4—88no—
551D4—84no—
552D4—161no—
553D4—162no—
554D4—162no—
555D4—162no—
556D4—162no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—162no—
583C2×C4—164no—
584C2×C4—162no—
585C2×C4—161no—
586C2×C4—164no—
587C2×C4—161no—
588C2×C4C22×D811yescharacteristic
589C2×C4C22×D811yescharacteristic
590C2×C4C22×D812yes—
591C2×C4C22×D812yes—
592C2×C4C22×D811yescharacteristic
593C2×C4C22×D812yes—
594C2×C4C23×D411yescharacteristic
595C8—22no—
596C8—21no—
597C8—21no—
598C8—22no—
599C8—21no—
600C8#58512yes—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—22no—
605C22—21no—
606C22—22no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—21no—
612C22—44no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—164no—
621C22—322no—
622C22#341412yes—
623C22C23×D811yescharacteristic
624C4—22no—
625C4—22no—
626C4—21no—
627C4—22no—
628C4—21no—
629C4—21no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4#348511yescharacteristic
637C4#358611yescharacteristic
638C2—22no—
639C2—42no—
640C2—41no—
641C2—41no—
642C2—162no—
643C2—162no—
644C2#6169411yescharacteristic
645C2C2×#341411yescharacteristic
646C2#6263611yescharacteristic
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. #62247C2semidirect
2. C2×#3339C2semidirect
3. #33922C2semidirect
4. #33918C2semidirect
5. #33893C2semidirect
6. #33889C2semidirect
7. #33798C2semidirect
8. #33760C2semidirect
9. #33759C2semidirect
10. #33758C2semidirect
11. #33665C2semidirect
12. #33649C2semidirect
13. #33648C2semidirect
14. #33646C2semidirect
15. #33489C2semidirect
21. #3339C22semidirect
22. #3339C22semidirect
23. #3336C22semidirect
26. #3285C22semidirect
27. #3280C22semidirect
32. C2×#499C22semidirect
40. C2×#466C22semidirect
41. C2×C4×D8C22semidirect
44. C22×#107C22semidirect
45. #2253C22semidirect
46. #2242C22semidirect
47. #2237C22semidirect
48. #2236C22semidirect
49. #2236C22semidirect
50. #2235C22semidirect
51. #2235C22semidirect
52. #2234C22semidirect
53. #2234C22semidirect
56. #2233C22semidirect
58. #2232C22semidirect
59. #2232C22semidirect
60. #2229C22semidirect
61. #2228C22semidirect
62. #2228C22semidirect
63. #2220C22semidirect
64. #2220C22semidirect
65. #2218C22semidirect
66. #2218C22semidirect
67. #2210C22semidirect
68. #2187C22semidirect
69. #2186C22semidirect
70. #2185C22semidirect
72. #2184C22semidirect
73. C4×SD32C22semidirect
74. C4×D16C22semidirect
75. #2169C22semidirect
76. #2169C22semidirect
77. #2162C22semidirect
78. #2159C22semidirect
79. #2159C22semidirect
80. #2159C22semidirect
81. #2141C22semidirect
82. #2140C22semidirect
83. #2139C22semidirect
86. #2112C22semidirect

Maximal subgroups

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

subgroupindex
#622472
C2×#33392
#339222
#339182
#338932
#338892
#337982
#337602
#337592
#337582
#336652
#336492
#336482
#336462
#334892