Finite Group Databasebeta

← #6475347

Subgroups — #6475347

Order 512.

1828 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic, complemented
2#34269C211yesmaximal, characteristic
3#33156C212yesmaximal, complemented
4#32966C212yesmaximal, complemented
5#29971C212yesmaximal, complemented
6#29950C212yesmaximal, complemented
7#29246C214yesmaximal, complemented
8#29163C212yesmaximal, complemented
9#29157C212yesmaximal, complemented
10#27677C212yesmaximal, complemented
11#24848C214yesmaximal, complemented
12#24304C212yesmaximal, complemented
13#21947C212yesmaximal, complemented
14#21770C212yesmaximal
15#20630C211yesmaximal, characteristic, complemented
16C2×#535C2211yescharacteristic
17C2×#533C2211yescharacteristic, complemented
18C2×#518C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#478C2214yes—
21C2×#424C2212yescomplemented
22C2×#408C2212yes—
23#2857C2212yes—
24#2857C2212yescomplemented
25#2629C2212yes—
26#2629C2212yescomplemented
27#2582—22no—
28#2538—22no—
29#2417—22no—
30#2417—22no—
31#2367—22no—
32#2366—22no—
33#2358—22no—
34#2358—22no—
35#2324C2212yes—
36#2324C2212yes—
37#2316C2212yescomplemented
38#2311C2212yes—
39#2311C2211yescharacteristic
40C4×#121C2212yes—
41C4×#121C2212yes—
42#2296C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#384C2211yescharacteristic, complemented
45#2274C2211yescharacteristic
46#2099C2212yescomplemented
47#2099C2212yescomplemented
48#2093C2212yes—
49#2093C2212yes—
50#2088C2214yescomplemented
51#2087C2214yescomplemented
52#2067C2214yes—
53#2067C2214yes—
54#2058C2212yescomplemented
55#2057C2212yescomplemented
56#2023C2212yes—
57#2023C2212yes—
58#1989C2214yescomplemented
59#1947C2214yescomplemented
60#1921C2214yescomplemented
61#1918C2214yescomplemented
62#1895C2214yes—
63#1893C2214yes—
64#1875C2214yes—
65#1874C2214yes—
66#1864C2214yes—
67#1862C2214yes—
68#1854C2214yes—
69#1854C2214yescomplemented
70#1852C2214yes—
71#1852C2214yescomplemented
72#1825C2214yes—
73#1809C2214yescomplemented
74#1808C2214yescomplemented
75#1802C2214yes—
76#1798C2212yes—
77#1798C2212yescomplemented
78#1797C2212yes—
79#1797C2212yescomplemented
80#1786C2214yes—
81#1728C2211yescharacteristic, complemented
82C2×#336C2211yescharacteristic, complemented
83C22×#41C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C22×M16C2312yes—
86C23×C8C2312yes—
87#535—24no—
88#535Q818yescomplemented
89#533—44no—
90C2×C4⋊Q8C2312yes—
91C2×#127—21no—
92C2×#125C2312yes—
93C2×#124—21no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×#121C2312yes—
98C2×#121C2312yes—
99C2×#120—21no—
100C2×#120—21no—
101#518—44no—
102C4×#41—44no—
103C2×C4×Q8—22no—
104C2×C4×Q8D412yes—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4D412yes—
109C2×C4×D4D412yes—
110C2×C4×D4D412yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117C2311yescharacteristic
114C2×#117C2312yes—
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—22no—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#478—48no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132#424—44no—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139#408—28no—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142#398—24no—
143#398—24no—
144#396—24no—
145#396—24no—
146#391—22no—
147#391—21no—
148#391—22no—
149#391—22no—
150#390—21no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390C2312yes—
156#390C2312yes—
157#389—24no—
158#389—22no—
159#389—22no—
160#389C2312yes—
161#389C2314yes—
162#389C2314yes—
163#389C2312yes—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#386—21no—
169#386—21no—
170#386—22no—
171#386—22no—
172#386—22no—
173#386—22no—
174#386—22no—
175#386—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#384—22no—
180#384C2312yes—
181#384C2312yes—
182#380—22no—
183#380—22no—
184#380—24no—
185#380C2312yes—
186#380C2314yes—
187#380C2312yes—
188#380C2314yes—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2312yes—
205#336C2312yes—
206#336C2312yes—
207#336C2312yes—
208C25—21no—
209C2×#41—42no—
210C2×#41—42no—
211C2×#41C2×Q814yes—
212C2×#41#4114yes—
213C22×Q8—22no—
214C22×Q8C2×D411yescharacteristic
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—21no—
218C22×D4—22no—
219C22×D4—42no—
220C22×D4—42no—
221C22×D4C2×D411yescharacteristic
222C22×D4C2×D411yescharacteristic
223C22×D4C2×D411yescharacteristic
224C22×D4C2×Q814yes—
225C22×D4C2412yes—
226C23×C4—21no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4C2×D412yes—
231C23×C4C2×D412yes—
232C23×C4C2×D412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C2×M16—24no—
237C2×M16—42no—
238C2×M16—42no—
239C22×C8—24no—
240C22×C8—42no—
241C22×C8—42no—
242C22×C8C2414yes—
243C22×C8C2414yes—
244C4⋊Q8—44no—
245#127—28no—
246#125—28no—
247#124—28no—
248C22⋊Q8—48no—
249C22⋊Q8—84no—
250C22⋊Q8—84no—
251#121—24no—
252#121—24no—
253#121C2×Q818yes—
254#121C2×Q818yes—
255#120—28no—
256#120—28no—
257C4×Q8—44no—
258C4×Q8—48no—
259C4×D4—28no—
260C4×D4—44no—
261C4×D4—44no—
262C4×D4—44no—
263C4×D4—44no—
264C4×D4—48no—
265C4×D4—84no—
266C4×D4—84no—
267#117—22no—
268#117—24no—
269#117—24no—
270#117—42no—
271#117—42no—
272#117—42no—
273#117—42no—
274#117C2×Q814yes—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—84no—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—48no—
314C2×C22⋊C4—84no—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C42—21no—
319C2×C42—24no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333#107—44no—
334#107—44no—
335#107—44no—
336#106—44no—
337C4⋊C8—28no—
338C4⋊C8—44no—
339#102—44no—
340#102—44no—
341#102—44no—
342#102—44no—
343C22⋊C8—28no—
344C22⋊C8—28no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355C24—22no—
356C24—24no—
357C24—42no—
358C24—42no—
359C24—82no—
360C24—82no—
361C24C22×D411yescharacteristic
362C24C22×D411yescharacteristic
363C24C22×D411yescharacteristic
364#41—48no—
365#41—48no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8C22⋊Q814yes—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—48no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—48no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4—82no—
392C2×D4C22⋊Q814yes—
393C2×D4C22⋊Q814yes—
394C2×D4C22⋊Q814yes—
395C2×D4C22×Q814yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—84no—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×Q812yes—
434C22×C4C22×Q812yes—
435C22×C4C22×Q812yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2511yescharacteristic
440M16—84no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—84no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—82no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4C22×Q814yes—
470C22⋊C4—24no—
471C22⋊C4—28no—
472C22⋊C4—28no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—42no—
476C22⋊C4—44no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—42no—
494C42—42no—
495C42—42no—
496C42—44no—
497C42—82no—
498C42—82no—
499C42H(5,𝔽2)14yes—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—48no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—88no—
521C23—162no—
522C23—164no—
523C23—162no—
524C23C2×C22⋊Q812yes—
525C23C2×C22⋊Q812yes—
526C23C2×C22⋊Q812yes—
527C23C23×D411yescharacteristic
528Q8—44no—
529Q8—88no—
530D4—28no—
531D4—44no—
532D4—44no—
533D4—44no—
534D4—48no—
535D4—44no—
536D4—88no—
537D4—88no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—164no—
579C2×C4—164no—
580C2×C4—164no—
581C2×C4—162no—
582C2×C4C2×C22⋊Q812yes—
583C2×C4C2×C22⋊Q812yes—
584C2×C4C2×C22⋊Q812yes—
585C2×C4#53414yes—
586C2×C4#57614yes—
587C2×C4C23×Q811yescharacteristic
588C2×C4C22×#4111yescharacteristic
589C2×C4C2×H(5,𝔽2)12yes—
590C2×C4#58512yes—
591C8—44no—
592C8—44no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—168no—
616C22C22×C22⋊Q811yescharacteristic
617C22C2×#53411yescharacteristic
618C22#344911yescharacteristic
619C22C2×#57612yes—
620C22#358612yes—
621C4—24no—
622C4—24no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—82no—
630C4—84no—
631C4—82no—
632C4—84no—
633C4#350112yes—
634C4#350512yes—
635C2—42no—
636C2—42no—
637C2—42no—
638C2—84no—
639C2#3501011yescharacteristic
640C2#6049412yes—
641C2#6076212yes—
642C2#6101111yescharacteristic
643C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35280C2semidirect
3. #33156C2semidirect
4. #32966C2semidirect
5. #29971C2semidirect
6. #29950C2semidirect
7. #29246C2semidirect
8. #29163C2semidirect
9. #29157C2semidirect
10. #27677C2semidirect
11. #24848C2semidirect
12. #24304C2semidirect
13. #21947C2semidirect
15. #20630C2semidirect
17. C2×#533C22semidirect
21. C2×#424C22semidirect
24. #2857C22semidirect
26. #2629C22semidirect
37. #2316C22semidirect
44. C2×#384C22semidirect
46. #2099C22semidirect
47. #2099C22semidirect
50. #2088C22semidirect
51. #2087C22semidirect
54. #2058C22semidirect
55. #2057C22semidirect
58. #1989C22semidirect
59. #1947C22semidirect
60. #1921C22semidirect
61. #1918C22semidirect
69. #1854C22semidirect
71. #1852C22semidirect
73. #1809C22semidirect
74. #1808C22semidirect
77. #1798C22semidirect
79. #1797C22semidirect
81. #1728C22semidirect
82. C2×#336C22semidirect
88. #535Q8semidirect

Maximal subgroups

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

subgroupindex
#352802
#342692
#331562
#329662
#299712
#299502
#292462
#291632
#291572
#276772
#248482
#243042
#219472
#217702
#206302