Finite Group Databasebeta

← #6499939

Subgroups — #6499939

Order 512.

3144 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
1C2×#3538C211yesmaximal, characteristic
2C2×#3202C212yesmaximal, complemented
3#35633C211yesmaximal, characteristic
4#25562C214yesmaximal, complemented
5#25546C214yesmaximal, complemented
6#24453C214yesmaximal, complemented
7#24437C214yesmaximal, complemented
8#24072C212yesmaximal, complemented
9#24010C212yesmaximal, complemented
10#23943C212yesmaximal, complemented
11#23921C212yesmaximal, complemented
12#21744C211yesmaximal, characteristic
13#20913C212yesmaximal, complemented
14#3538—216no—
15C2×#563—22no—
16C2×#552C2214yes—
17C2×#551—22no—
18C2×D4×Q8C2214yes—
19C2×#544—22no—
20C2×#541C2212yes—
21C2×#539—22no—
22C2×#536C2211yescharacteristic
23C2×#536C2212yes—
24C2×#525C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×C22⋊Q8C2211yescharacteristic
27C2×#520C2211yescharacteristic
28#3202—216no—
29C2×#485C2214yes—
30C2×#485C2214yescomplemented
31C2×#470C2214yescomplemented
32C2×#467C2214yescomplemented
33C2×#451C2214yes—
34C2×#451C2214yes—
35C2×#451C2214yescomplemented
36C2×#419C2212yescomplemented
37C22×#103C2212yescomplemented
38C2×#408C2212yes—
39#2534—22no—
40#2486—22no—
41#2429—22no—
42#2427—22no—
43#2417C2214yes—
44#2415C2214yes—
45#2316C2211yescharacteristic
46#2310C2211yescharacteristic
47#2280C2211yescharacteristic
48#2059C2214yescomplemented
49#2043C2214yescomplemented
50#2036C2214yes—
51#2035C2214yes—
52#2026C2212yescomplemented
53#2026C2212yescomplemented
54#2016C2212yescomplemented
55#2016C2212yescomplemented
56#2001C2214yes—
57#2001C2214yes—
58#1878C2214yescomplemented
59#1878C2214yescomplemented
60#1873C2214yes—
61#1873C2214yescomplemented
62#1872C2214yes—
63#1872C2214yes—
64#1872C2214yescomplemented
65#1870C2214yes—
66#1869C2214yes—
67#1865C2214yescomplemented
68#1863C2214yescomplemented
69#1853C2212yes—
70#1852C2212yes—
71#1818C2212yes—
72#1787C2212yes—
73#1787C2212yes—
74#1779C2212yes—
75#1736C2212yescomplemented
76#1735C2212yescomplemented
77C22×#41C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×Q8C2312yes—
80C23×Q8C2312yes—
81C24×C4C2311yescharacteristic
82C22×Q16C2314yes—
83C22×Q16C2314yes—
84C22×M16C2312yes—
85C23×C8C2312yes—
86#563—232no—
87#552—216no—
88#552—216no—
89#551—232no—
90D4×Q8—216no—
91D4×Q8—216no—
92#544—232no—
93#541—216no—
94#539—232no—
95#536—28no—
96#536—28no—
97#536D4116yes—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8C2312yes—
102C2×#126—24no—
103C2×#126—24no—
104C2×#125—24no—
105C2×#125—24no—
106C2×#125C2312yes—
107C2×#124—22no—
108C2×#124—21no—
109#525—28no—
110C2×#123—22no—
111C2×#123—22no—
112C2×#123—24no—
113C2×#123—22no—
114C2×#123D412yes—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—216no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—44no—
122C2×C22⋊Q8D412yes—
123C2×C22⋊Q8C2314yes—
124C2×C22⋊Q8C2314yes—
125C2×C22⋊Q8C2314yes—
126C2×C22⋊Q8C2314yes—
127C2×C22⋊Q8C2314yes—
128C2×#121D412yes—
129C2×#120D412yes—
130#520—44no—
131C2×C4×Q8—21no—
132C2×C4×Q8C2314yes—
133C2×C4×D4—24no—
134C2×C4×D4—21no—
135C2×C4×D4—22no—
136C2×C4×D4C2314yes—
137C2×#117—24no—
138C2×#117C2311yescharacteristic
139C2×#117C2312yes—
140C2×#117C2312yes—
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C22×C4⋊C4—22no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2312yes—
148C22×C22⋊C4—22no—
149C22×C22⋊C4—21no—
150#485—216no—
151#485—216no—
152#470—216no—
153#467—216no—
154#451—216no—
155#451—216no—
156#451—48no—
157C2×#107C2314yes—
158C2×#106C2314yes—
159#419—44no—
160C2×#103—44no—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169#408—28no—
170C2×C22⋊C8C2314yes—
171C2×C22⋊C8C2314yes—
172C2×C22⋊C8C2312yes—
173C2×C22⋊C8C2312yes—
174C2×C22⋊C8C2314yes—
175#398—24no—
176#397—24no—
177#396—24no—
178#394—24no—
179#391—22no—
180#391C2312yes—
181#391C2312yes—
182#389—22no—
183#389—24no—
184#389C2314yes—
185#388—24no—
186#386—24no—
187#381—21no—
188#381—22no—
189#380—22no—
190#380—24no—
191#380C2312yes—
192#380C2312yes—
193#379—21no—
194C4×C22⋊C4—22no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2312yes—
201#336C2312yes—
202C25C2×D411yescharacteristic
203C2×#41—42no—
204C2×#41—41no—
205C2×#41—41no—
206C2×#41C2×D418yes—
207C22×Q8—28no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—21no—
212C22×Q8—24no—
213C22×Q8—42no—
214C22×Q8—41no—
215C22×Q8—41no—
216C22×Q8—42no—
217C22×Q8—42no—
218C22×Q8—42no—
219C22×Q8—42no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8—44no—
223C22×Q8C2×D418yes—
224C22×Q8C2×D411yescharacteristic
225C22×Q8C2412yes—
226C22×Q8C2414yes—
227C22×D4C2×D414yes—
228C22×D4C2×D412yes—
229C22×D4C2×D411yescharacteristic
230C23×C4—22no—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4—42no—
238C23×C4—41no—
239C23×C4—41no—
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C2×Q16—28no—
246C2×Q16—28no—
247C2×Q16—44no—
248C2×Q16—44no—
249C2×Q16—44no—
250C2×Q16—44no—
251C2×M16—24no—
252C2×M16—42no—
253C2×M16—42no—
254C22×C8—24no—
255C22×C8—42no—
256C22×C8—42no—
257C22×C8C2414yes—
258C22×C8C2414yes—
259C4⋊Q8—232no—
260C4⋊Q8—216no—
261C4⋊Q8—28no—
262C4⋊Q8—44no—
263#126—232no—
264#126—232no—
265#125—232no—
266#125—232no—
267#125—28no—
268#124—216no—
269#124—44no—
270#123—232no—
271#123—216no—
272#123—216no—
273#123—28no—
274#123—48no—
275C22⋊Q8—216no—
276C22⋊Q8—216no—
277C22⋊Q8—232no—
278C22⋊Q8—232no—
279C22⋊Q8—28no—
280C22⋊Q8—28no—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—28no—
286C22⋊Q8—28no—
287C22⋊Q8—44no—
288C22⋊Q8—48no—
289C22⋊Q8C2×D4116yes—
290C22⋊Q8C2×D4116yes—
291#121—28no—
292#120—28no—
293C4×Q8—28no—
294C4×Q8—28no—
295C4×Q8—44no—
296C4×D4—232no—
297C4×D4—28no—
298C4×D4—28no—
299C4×D4—44no—
300C4×D4—48no—
301#117—232no—
302#117—24no—
303#117—24no—
304#117—24no—
305#117—24no—
306#117—42no—
307#117—42no—
308#117C2×D418yes—
309C2×C4⋊C4—216no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—216no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4C2×D414yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D414yes—
350C2×C42—22no—
351C2×C42—24no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—21no—
355C2×C42C2412yes—
356#107—28no—
357#106—28no—
358#103—28no—
359#103—28no—
360#103—28no—
361#103—28no—
362#103—28no—
363#103—28no—
364#103—44no—
365#103—44no—
366C22⋊C8—28no—
367C22⋊C8—28no—
368C22⋊C8—42no—
369C22⋊C8—42no—
370C22⋊C8—44no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—24no—
376C24—22no—
377C24—24no—
378C24—41no—
379C24—42no—
380C24—41no—
381C24C22×D411yescharacteristic
382C24C22×D412yes—
383C24C22×D412yes—
384#41—48no—
385#41—48no—
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—42no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—84no—
409C2×Q8—84no—
410C2×Q8—84no—
411C2×Q8—84no—
412C2×Q8—88no—
413C2×Q8—88no—
414C2×Q8#12014yes—
415C2×Q8C22×D418yes—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—41no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—41no—
422C2×D4—42no—
423C2×D4#12018yes—
424C2×D4#12014yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—41no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—84no—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C2511yescharacteristic
475Q16—88no—
476Q16—88no—
477M16—84no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—84no—
485C4⋊C4—24no—
486C4⋊C4—216no—
487C4⋊C4—28no—
488C4⋊C4—216no—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—24no—
494C4⋊C4—28no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—84no—
504C4⋊C4C22×D418yes—
505C4⋊C4H(5,𝔽2)14yes—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—28no—
509C22⋊C4—216no—
510C22⋊C4—24no—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—82no—
520C42—216no—
521C42—28no—
522C42—24no—
523C42—44no—
524C42—42no—
525C42—42no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—21no—
533C23—22no—
534C23—21no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—44no—
541C23—48no—
542C23—82no—
543C23—84no—
544C23—82no—
545C23C2×#12012yes—
546C23C2×#12014yes—
547C23#57814yes—
548C23C23×D411yescharacteristic
549Q8—28no—
550Q8—28no—
551Q8—44no—
552Q8—44no—
553Q8—44no—
554Q8—44no—
555Q8—44no—
556Q8—88no—
557Q8—84no—
558Q8—88no—
559D4—44no—
560D4—48no—
561D4—416no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—416no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—84no—
600C2×C4—82no—
601C2×C4—88no—
602C2×C4—84no—
603C2×C4—88no—
604C2×C4—84no—
605C2×C4—164no—
606C2×C4C2×#12014yes—
607C2×C4C2×#12012yes—
608C2×C4C23×D412yes—
609C2×C4C2×H(5,𝔽2)12yes—
610C2×C4#58511yescharacteristic
611C2×C4#58511yescharacteristic
612C8—44no—
613C8—44no—
614C22—22no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—28no—
620C22—24no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—41no—
627C22—41no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—88no—
632C22C22×#12011yescharacteristic
633C22#348411yescharacteristic
634C22#348511yescharacteristic
635C22C2×#57812yes—
636C22#358612yes—
637C4—22no—
638C4—24no—
639C4—42no—
640C4—48no—
641C4—44no—
642C4—44no—
643C4—48no—
644C4—82no—
645C4—81no—
646C4—81no—
647C4—82no—
648C4#349612yes—
649C4#349612yes—
650C2—44no—
651C2—42no—
652C2—44no—
653C2#3534711yescharacteristic
654C2#6037512yes—
655C2#6163411yescharacteristic
656C2#6164911yescharacteristic
657C2#6175712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3202C2semidirect
4. #25562C2semidirect
5. #25546C2semidirect
6. #24453C2semidirect
7. #24437C2semidirect
8. #24072C2semidirect
9. #24010C2semidirect
10. #23943C2semidirect
11. #23921C2semidirect
13. #20913C2semidirect
30. C2×#485C22semidirect
31. C2×#470C22semidirect
32. C2×#467C22semidirect
35. C2×#451C22semidirect
36. C2×#419C22semidirect
37. C22×#103C22semidirect
48. #2059C22semidirect
49. #2043C22semidirect
52. #2026C22semidirect
53. #2026C22semidirect
54. #2016C22semidirect
55. #2016C22semidirect
58. #1878C22semidirect
59. #1878C22semidirect
61. #1873C22semidirect
64. #1872C22semidirect
67. #1865C22semidirect
68. #1863C22semidirect
75. #1736C22semidirect
76. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35382
C2×#32022
#356332
#255622
#255462
#244532
#244372
#240722
#240102
#239432
#239212
#217442
#209132