Finite Group Databasebeta

← #6495695

Subgroups — #6495695

Order 512.

1758 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35277C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic, complemented
3#33121C212yesmaximal, complemented
4#32917C212yesmaximal, complemented
5#29981C212yesmaximal, complemented
6#29800C212yesmaximal, complemented
7#29570C214yesmaximal, complemented
8#29368C212yesmaximal, complemented
9#29365C212yesmaximal, complemented
10#29338C211yesmaximal, characteristic, complemented
11#29331C211yesmaximal, characteristic, complemented
12#28310C211yesmaximal, characteristic
13#28302C211yesmaximal, characteristic, complemented
14#24074C212yesmaximal, complemented
15#23945C212yesmaximal, complemented
16#22161C214yesmaximal, complemented
17#20784C211yesmaximal, characteristic, complemented
18C2×#542C2212yes—
19C2×#518C2211yescharacteristic
20C2×#481C2212yes—
21C2×#483C2212yes—
22C2×#463C2214yes—
23C2×#450C2212yes—
24C2×#447C2212yes—
25#2868C2211yescharacteristic, complemented
26#2858C2211yescharacteristic, complemented
27#2857C2211yescharacteristic
28#2856C2211yescharacteristic
29#2640C2212yescomplemented
30#2620C2212yes—
31#2512—22no—
32#2491—22no—
33#2452—22no—
34#2422—22no—
35#2357—22no—
36#2354—22no—
37#2342—22no—
38#2342—22no—
39#2326C2212yescomplemented
40#2319C2212yes—
41#2319C2212yes—
42#2317C2211yescharacteristic
43#2309C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic, complemented
45C4×#127C2211yescharacteristic, complemented
46#2296C2211yescharacteristic
47#2293C2212yes—
48#2293C2212yes—
49C2×#390C2211yescharacteristic
50#2274C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2212yescomplemented
53#2100C2212yescomplemented
54#2094C2212yes—
55#2094C2212yes—
56#2058C2214yescomplemented
57#2043C2212yescomplemented
58#2042C2214yescomplemented
59#2041C2212yescomplemented
60#2023C2214yes—
61#2013C2212yes—
62#2012C2214yes—
63#2011C2212yes—
64#1983C2214yescomplemented
65#1972C2214yes—
66#1968C2214yes—
67#1948C2214yescomplemented
68#1935C2214yescomplemented
69#1934C2214yescomplemented
70#1927C2214yescomplemented
71#1926C2214yescomplemented
72#1880C2214yes—
73#1879C2214yes—
74#1879C2214yes—
75#1877C2214yes—
76#1875C2212yes—
77#1875C2212yes—
78#1873C2212yes—
79#1873C2212yes—
80#1865C2214yescomplemented
81#1864C2214yescomplemented
82#1839C2212yescomplemented
83#1831C2212yescomplemented
84#1818C2212yes—
85#1802C2214yes—
86#1798C2212yes—
87#1768C2212yescomplemented
88#1749C2211yescharacteristic
89C4×C22⋊C8C2211yescharacteristic
90#1746C2212yescomplemented
91C23×D4C2311yescharacteristic
92C22×SD16C2314yes—
93C22×D8C2314yes—
94#542—44no—
95#542—44no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#127C2311yescharacteristic
100C2×#126—22no—
101C2×#124—22no—
102C2×#124C2312yes—
103C2×#124C2312yes—
104C2×#123—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2312yes—
107C2×#121—22no—
108C2×#121C2312yes—
109#518—28no—
110C2×C4×Q8C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×#117—22no—
117C2×#117—22no—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—22no—
124C22×C22⋊C4—22no—
125C22×C42C2311yescharacteristic
126#483—44no—
127#481—44no—
128#463—216no—
129#450—44no—
130#447—44no—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×#97C2312yes—
145C2×C4×C8C2312yes—
146#397—24no—
147#396—24no—
148#394—24no—
149#393—24no—
150#391C2312yes—
151#391C2312yes—
152#390—21no—
153#390—21no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390—22no—
158#390—22no—
159#390C2312yes—
160#390C2312yes—
161#390C2312yes—
162#390C2312yes—
163#390C2312yes—
164#390C2312yes—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—24no—
180#385—22no—
181#385—21no—
182#385—21no—
183#385—24no—
184#384—21no—
185#384—22no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#382—21no—
190#381—22no—
191#381—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#379—21no—
196#379—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4D412yes—
208C4×C22⋊C4D412yes—
209#376C2311yescharacteristic
210#376C2311yescharacteristic
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219#336C2314yes—
220#336C2312yes—
221#336C2312yes—
222C25—21no—
223C22×Q8—22no—
224C22×Q8C2412yes—
225C22×D4—24no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4—24no—
230C22×D4—42no—
231C22×D4C2412yes—
232C22×D4C2412yes—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4C2×D412yes—
238C23×C4C2411yescharacteristic
239C2×SD16—28no—
240C2×SD16—44no—
241C2×SD16—44no—
242C2×D8—28no—
243C2×D8—44no—
244C2×D8—44no—
245C22×C8C2414yes—
246C22×C8C2414yes—
247C4⋊Q8—42no—
248C4⋊Q8—42no—
249C4⋊Q8—42no—
250C4⋊Q8—84no—
251#127—42no—
252#126—48no—
253#124—28no—
254#124—42no—
255#124—42no—
256#124—48no—
257#123—48no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—48no—
261#121—24no—
262#121—24no—
263#121—84no—
264C4×Q8—42no—
265C4×Q8—42no—
266C4×D4—216no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—42no—
271C4×D4—42no—
272C4×D4—84no—
273C4×D4#4118yes—
274#117—24no—
275#117—24no—
276#117—24no—
277#117—42no—
278#117—84no—
279#117—84no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—216no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—84no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D411yescharacteristic
325C2×C22⋊C4C2×D411yescharacteristic
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—42no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#106—44no—
344#106—44no—
345C4⋊C8—28no—
346#103—28no—
347#103—44no—
348#103—44no—
349#102—28no—
350#102—44no—
351#102—44no—
352#102—44no—
353#102—44no—
354C22⋊C8—24no—
355C22⋊C8—24no—
356#97—42no—
357C4×C8—42no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—24no—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—42no—
374C24—42no—
375C24—82no—
376C24—82no—
377C24C22×D411yescharacteristic
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—44no—
383C2×Q8—84no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—416no—
393C2×D4—42no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—42no—
398C2×D4—44no—
399C2×D4—84no—
400C2×D4C2×#4114yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4—84no—
421C22×C4—82no—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4H(5,𝔽2)14yes—
432C22×C4C2511yescharacteristic
433SD16—88no—
434D8—88no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—84no—
463C4⋊C4C2×#4114yes—
464C22⋊C4—24no—
465C22⋊C4—28no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—24no—
469C22⋊C4—28no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—84no—
486C22⋊C4—84no—
487C42—24no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—44no—
500C42—82no—
501C42—82no—
502C42C2×#4114yes—
503C23—22no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—24no—
508C23—48no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—88no—
521C23—162no—
522C23—162no—
523C23—164no—
524C23#53512yes—
525C23C23×D411yescharacteristic
526Q8—44no—
527Q8—44no—
528Q8—164no—
529D4—28no—
530D4—48no—
531D4—44no—
532D4—44no—
533D4—44no—
534D4—88no—
535D4—164no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—84no—
574C2×C4—88no—
575C2×C4—88no—
576C2×C4—164no—
577C2×C4—162no—
578C2×C4#52514yes—
579C2×C4#53512yes—
580C2×C4#57714yes—
581C2×C4C22×#4112yes—
582C2×C4C2×H(5,𝔽2)12yes—
583C2×C4#58512yes—
584C8—44no—
585C8—44no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—42no—
590C22—42no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—84no—
599C22—82no—
600C22—82no—
601C22—84no—
602C22—82no—
603C22—168no—
604C22C2×#52511yescharacteristic
605C22C2×#53511yescharacteristic
606C22#344911yescharacteristic
607C22C2×#57712yes—
608C22#358512yes—
609C4—24no—
610C4—24no—
611C4—22no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—42no—
618C4—44no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4#349414yes—
625C2—42no—
626C2—84no—
627C2—162no—
628C2#3530111yescharacteristic
629C2#6059712yes—
630C2#6082011yescharacteristic
631C2#6082711yescharacteristic
632C2#6098012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28310C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35277C2semidirect
2. #34255C2semidirect
3. #33121C2semidirect
4. #32917C2semidirect
5. #29981C2semidirect
6. #29800C2semidirect
7. #29570C2semidirect
8. #29368C2semidirect
9. #29365C2semidirect
10. #29338C2semidirect
11. #29331C2semidirect
13. #28302C2semidirect
14. #24074C2semidirect
15. #23945C2semidirect
16. #22161C2semidirect
17. #20784C2semidirect
25. #2868C22semidirect
26. #2858C22semidirect
29. #2640C22semidirect
39. #2326C22semidirect
44. C4×C4⋊Q8C22semidirect
45. C4×#127C22semidirect
52. #2100C22semidirect
53. #2100C22semidirect
56. #2058C22semidirect
57. #2043C22semidirect
58. #2042C22semidirect
59. #2041C22semidirect
64. #1983C22semidirect
67. #1948C22semidirect
68. #1935C22semidirect
69. #1934C22semidirect
70. #1927C22semidirect
71. #1926C22semidirect
80. #1865C22semidirect
81. #1864C22semidirect
82. #1839C22semidirect
83. #1831C22semidirect
87. #1768C22semidirect
90. #1746C22semidirect

Maximal subgroups

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

subgroupindex
#352772
#342552
#331212
#329172
#299812
#298002
#295702
#293682
#293652
#293382
#293312
#283102
#283022
#240742
#239452
#221612
#207842