Finite Group Databasebeta

← #6643293

Subgroups — #6643293

Order 512.

1530 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#35015C211yesmaximal, characteristic, complemented
2C2×#2512C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic, complemented
4#33108C211yesmaximal, characteristic, complemented
5#32525C214yesmaximal, complemented
6#32474C214yesmaximal, complemented
7#31900C214yesmaximal, complemented
8#31804C214yesmaximal, complemented
9#29952C211yesmaximal, characteristic
10#29714C211yesmaximal, characteristic
11#28120C212yesmaximal
12#27680C212yesmaximal, complemented
13#24601C212yesmaximal, complemented
14#24501C212yesmaximal
15#21951C211yesmaximal, characteristic, complemented
16C22×#126—21no—
17C2×#483C2212yes—
18C2×#450C2212yes—
19C2×#431C2212yes—
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#2855C2212yes—
23#2646—21no—
24#2627—21no—
25#2590—22no—
26#2588—22no—
27#2512—22no—
28#2512—22no—
29#2512C2214yes—
30#2512C2214yes—
31#2511C2212yescomplemented
32#2507C2212yescomplemented
33#2451C2212yescomplemented
34#2446C2212yescomplemented
35C2×#397—21no—
36C2×#396—21no—
37#2329—21no—
38#2313—21no—
39C4×C22⋊Q8C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#384C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48#2099C2214yescomplemented
49#2096C2214yescomplemented
50#2094C2212yes—
51#2088C2214yescomplemented
52#2071C2212yes—
53#2066C2214yescomplemented
54#2065C2214yes—
55#2065C2214yes—
56#2065C2214yes—
57#2055C2214yes—
58#2055C2214yes—
59#2055C2214yes—
60#2023C2212yes—
61#2013C2212yes—
62#1987C2212yescomplemented
63#1981C2212yescomplemented
64#1968C2214yes—
65#1948C2212yescomplemented
66#1948C2214yescomplemented
67#1948C2212yescomplemented
68#1941C2214yescomplemented
69#1939C2214yescomplemented
70#1939C2214yescomplemented
71#1934C2214yes—
72#1933C2214yescomplemented
73#1927C2214yescomplemented
74#1926C2214yes—
75#1920C2214yescomplemented
76#1880C2214yes—
77#1872C2212yes—
78#1872C2212yes—
79#1853C2212yes—
80#1853C2212yes—
81#1823C2212yes—
82#1803C2214yescomplemented
83#1797C2211yescharacteristic
84#1797C2211yescharacteristic
85#1797C2212yes—
86#1786C2212yes—
87C23×D4C2311yescharacteristic
88C22×SD16C2314yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#126—24no—
92C2×#126—24no—
93C2×#126—48no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8C2312yes—
96C2×C4×Q8C2312yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4D411yescharacteristic
101C22×C4⋊C4D411yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C42—21no—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#483—44no—
116#450—44no—
117#431—44no—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133C2×C4×C8C2314yes—
134#400—22no—
135#400—22no—
136#400—22no—
137#398—22no—
138#398—22no—
139#398—22no—
140#398—22no—
141#397—24no—
142#397—24no—
143#396—24no—
144#396—24no—
145#395—22no—
146#391—21no—
147#391—22no—
148#391C2312yes—
149#391C2312yes—
150#391C2312yes—
151#391C2312yes—
152#390—22no—
153#390—22no—
154#390—21no—
155#390—21no—
156#390C2312yes—
157#390C2314yes—
158#390C2314yes—
159#390C2312yes—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#389C2312yes—
170#388—24no—
171#388—24no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#385—22no—
182#385—22no—
183#384—22no—
184#384C2314yes—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#381—22no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380C2312yes—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4D412yes—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95D411yescharacteristic
206C2×#95D411yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2312yes—
214#336C2312yes—
215#336C2314yes—
216C25—21no—
217C22×Q8—21no—
218C22×Q8C2412yes—
219C22×D4—21no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—42no—
223C22×D4—42no—
224C22×D4C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2411yescharacteristic
235C2×SD16—44no—
236C2×SD16—44no—
237C2×SD16—44no—
238C22×C8C2414yes—
239C22×C8C2414yes—
240C4⋊Q8—42no—
241C4⋊Q8—42no—
242#126—416no—
243#126—416no—
244C22⋊Q8—24no—
245C22⋊Q8—24no—
246C22⋊Q8—44no—
247C4×Q8—44no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4SD1618yescomplemented
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—82no—
301C2×C22⋊C4—82no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—44no—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2412yes—
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2412yes—
319#106—44no—
320#106—44no—
321#106—44no—
322C4⋊C8—44no—
323C4⋊C8—42no—
324C4⋊C8—42no—
325C4⋊C8—44no—
326#103—44no—
327#103—44no—
328#103—44no—
329#102—44no—
330#102—44no—
331#102—44no—
332C22⋊C8—24no—
333C22⋊C8—24no—
334C4×C8—44no—
335#95—22no—
336#95—22no—
337#95—24no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95C2×D412yes—
353#95C2×D412yes—
354#95C2×D412yes—
355#95C2×D412yes—
356C24—22no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24C22×D411yescharacteristic
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—42no—
370C2×Q8—81no—
371C2×Q8—81no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—84no—
380C2×D4—84no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4C2×SD1614yes—
415C22×C4C2×SD1614yes—
416C22×C4C2×SD1614yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4H(5,𝔽2)12yes—
427C22×C42−1+412yes—
428C22×C4C2511yescharacteristic
429SD16—88no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—84no—
462C4⋊C4—82no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4C2×SD1618yes—
467C4⋊C4C2×SD1618yes—
468C22⋊C4—48no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—84no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C42—24no—
479C42—22no—
480C42—22no—
481C42—24no—
482C42—22no—
483C42—22no—
484C42—42no—
485C42—41no—
486C42—41no—
487C42—44no—
488C42—44no—
489C42—82no—
490C42—82no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—84no—
503C23—84no—
504C23—84no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23#53712yes—
510C23C22×SD1612yes—
511C23#57512yes—
512C23C23×D411yescharacteristic
513Q8—44no—
514Q8—44no—
515Q8—162no—
516D4—44no—
517D4—44no—
518D4—84no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4#53712yes—
574C2×C4C22×SD1612yes—
575C2×C4C22×SD1612yes—
576C2×C4C22×SD1614yes—
577C2×C4C22×SD1614yes—
578C2×C4C2×H(5,𝔽2)11yescharacteristic
579C2×C4C2×2−1+411yescharacteristic
580C2×C4#58512yes—
581C2×C4#58512yes—
582C8—44no—
583C8—44no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—168no—
598C22#324614yes—
599C22C2×#53711yescharacteristic
600C22#348511yescharacteristic
601C22#348711yescharacteristic
602C22C23×SD1611yescharacteristic
603C22C2×#57511yescharacteristic
604C22#358511yescharacteristic
605C22#358611yescharacteristic
606C4—22no—
607C4—24no—
608C4—24no—
609C4—22no—
610C4—22no—
611C4—42no—
612C4—41no—
613C4—41no—
614C4—42no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4#356012yes—
623C4#356412yes—
624C2—24no—
625C2—162no—
626C2#3697111yescharacteristic
627C2C2×#324611yescharacteristic
628C2#6091911yescharacteristic
629C2#6169511yescharacteristic
630C2#6171811yescharacteristic
631C2#6188111yescharacteristic
632C2#6189411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35015C2semidirect
3. #33148C2semidirect
4. #33108C2semidirect
5. #32525C2semidirect
6. #32474C2semidirect
7. #31900C2semidirect
8. #31804C2semidirect
12. #27680C2semidirect
13. #24601C2semidirect
15. #21951C2semidirect
31. #2511C22semidirect
32. #2507C22semidirect
33. #2451C22semidirect
34. #2446C22semidirect
48. #2099C22semidirect
49. #2096C22semidirect
51. #2088C22semidirect
53. #2066C22semidirect
62. #1987C22semidirect
63. #1981C22semidirect
65. #1948C22semidirect
66. #1948C22semidirect
67. #1948C22semidirect
68. #1941C22semidirect
69. #1939C22semidirect
70. #1939C22semidirect
72. #1933C22semidirect
73. #1927C22semidirect
75. #1920C22semidirect
82. #1803C22semidirect
275. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#25122
#331482
#331082
#325252
#324742
#319002
#318042
#299522
#297142
#281202
#276802
#246012
#245012
#219512