Finite Group Databasebeta

← #6585012

Subgroups — #6585012

Order 512.

1948 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3181C212yesmaximal, complemented
2#36151C211yesmaximal, characteristic, complemented
3#36136C211yesmaximal, characteristic
4#33233C211yesmaximal, characteristic, complemented
5#33051C212yesmaximal, complemented
6#28695C211yesmaximal, characteristic, complemented
7#28447C211yesmaximal, characteristic, complemented
8#26981C214yesmaximal, complemented
9#26978C214yesmaximal, complemented
10#25679C212yesmaximal, complemented
11#25603C212yesmaximal, complemented
12#25389C214yesmaximal, complemented
13#23006C214yesmaximal, complemented
14C4×#491C211yesmaximal, characteristic, complemented
15D4×#97C211yesmaximal, characteristic, complemented
16C2×#533C2212yes—
17C2×#529C2212yes—
18#3181—216no—
19C2×#501C2212yes—
20C2×#497C2212yes—
21C2×#491C2211yescharacteristic
22C2×#489C2212yes—
23C2×#489C2212yes—
24C2×#467C2214yes—
25C2×#465C2214yes—
26C2×#464C2214yes—
27C2×#441C2214yes—
28C2×#440C2214yes—
29C2×C8×D4C2212yes—
30C22×#97C2212yes—
31#2862C2211yescharacteristic
32#2858C2211yescharacteristic
33#2857C2212yes—
34#2850C2212yes—
35#2554—22no—
36#2540C2212yes—
37#2536C2212yes—
38#2489—22no—
39#2479—22no—
40#2456—22no—
41#2453—22no—
42#2429C2212yes—
43#2428C2212yes—
44#2421C2214yes—
45#2415—22no—
46#2367C2212yes—
47#2363—22no—
48#2362—22no—
49#2351C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#125C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2085C2214yes—
54#2035C2214yes—
55#2033C2214yes—
56#2016C2212yes—
57#2014C2212yes—
58#2001C2214yes—
59#2000C2214yes—
60#1993C2212yes—
61#1992C2211yescharacteristic
62#1961C2212yes—
63#1960C2211yescharacteristic
64#1960C2212yes—
65#1959C2212yes—
66#1958C2214yes—
67#1949C2214yes—
68#1936C2214yes—
69#1935C2214yes—
70#1935C2214yes—
71#1893C2214yes—
72#1892C2214yes—
73#1880C2214yes—
74#1878C2214yes—
75#1864C2214yes—
76#1863C2214yes—
77#1836C2211yescharacteristic
78#1770C2212yes—
79#1767C2211yescharacteristic
80C4×#103C2214yes—
81#1752C2214yes—
82#1749C2212yes—
83C4×#97C2211yescharacteristic
84C22×Q16C2314yes—
85C22×SD16C2314yes—
86C23×C8C2314yes—
87#533—28no—
88#533—44no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2312yes—
92C2×#127C2311yescharacteristic
93#529—28no—
94#529—44no—
95C2×#126—22no—
96C2×#125—22no—
97C2×#125C2311yescharacteristic
98C2×#124C2312yes—
99C2×#124C2312yes—
100C2×#123—22no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2314yes—
105C2×C22⋊Q8C2312yes—
106C2×#121—22no—
107C2×#121C2312yes—
108C2×#121C2312yes—
109C2×C4×Q8—22no—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2312yes—
113C2×C4×D4—22no—
114C2×C4×D4D412yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2311yescharacteristic
117C2×#117—22no—
118C2×#117—22no—
119C22×C42C2312yes—
120#501—28no—
121#497—28no—
122#491D418yescomplemented
123#489—28no—
124#489—44no—
125#467—216no—
126#465—216no—
127#464—216no—
128#441—216no—
129#440—216no—
130C8×D4—28no—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2314yes—
143C2×#97—28no—
144C2×#97C2312yes—
145C2×#97C2314yes—
146C2×#97C2311yescharacteristic
147C2×#97C2312yes—
148C2×#97C2311yescharacteristic
149C2×C4×C8C2312yes—
150#400—24no—
151#399—22no—
152#398—22no—
153#398—22no—
154#397—24no—
155#396—22no—
156#394—22no—
157#394—22no—
158#392—22no—
159#392—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391C2312yes—
164#391C2312yes—
165#390—22no—
166#390—22no—
167#390—22no—
168#390—22no—
169#390C2312yes—
170#389—24no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2314yes—
176#388—22no—
177#388—24no—
178#388—24no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—24no—
185#386—22no—
186#386—22no—
187#386—22no—
188#385—22no—
189#384C2311yescharacteristic
190#382—22no—
191#382C2311yescharacteristic
192#382C2312yes—
193#382C2312yes—
194#380—22no—
195#380C2314yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2311yescharacteristic
202C4×C22⋊C4—22no—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4D412yes—
205C4×C22⋊C4C2312yes—
206C43C2311yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#336C2312yes—
212#336C2314yes—
213C22×Q8—22no—
214C22×Q8C2412yes—
215C22×Q8C2412yes—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4C2×D411yescharacteristic
220C22×D4C2412yes—
221C23×C4—22no—
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4C2412yes—
225C2×Q16—28no—
226C2×Q16—28no—
227C2×Q16—44no—
228C2×SD16—28no—
229C2×SD16—28no—
230C2×SD16—44no—
231C22×C8—28no—
232C22×C8—28no—
233C22×C8—44no—
234C22×C8C2412yes—
235C22×C8C2412yes—
236C22×C8C2414yes—
237C4⋊Q8—24no—
238C4⋊Q8—22no—
239C4⋊Q8—24no—
240C4⋊Q8—42no—
241C4⋊Q8C2×D414yes—
242#127—42no—
243#126—48no—
244#125—22no—
245#125—48no—
246#125C2×D414yes—
247#124—24no—
248#124—28no—
249#124—24no—
250#123—48no—
251C22⋊Q8—28no—
252C22⋊Q8—24no—
253C22⋊Q8—42no—
254C22⋊Q8—44no—
255C22⋊Q8—48no—
256C22⋊Q8—48no—
257C22⋊Q8—44no—
258#121—24no—
259#121—48no—
260#121—42no—
261#121—44no—
262C4×Q8—28no—
263C4×Q8—24no—
264C4×Q8—28no—
265C4×Q8—42no—
266C4×Q8—44no—
267C4×Q8—48no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—42no—
271C4×D4—44no—
272C4×D4—48no—
273#117—48no—
274#117—48no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4C2×D412yes—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324#107—28no—
325#106—28no—
326C4⋊C8#4118yes—
327#103—28no—
328#103—28no—
329#103—44no—
330#103—44no—
331#103C2×D4116yes—
332#102—28no—
333#102—28no—
334#102—44no—
335C22⋊C8—28no—
336#97—28no—
337#97—24no—
338#97—22no—
339#97—42no—
340#97C2×D414yes—
341C4×C8—24no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95C2×D411yescharacteristic
354#95C2×D411yescharacteristic
355#95C2×D412yes—
356#95C2×D412yes—
357C24—22no—
358C24C22×D412yes—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8C22×D414yes—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—24no—
374C2×D4—22no—
375C2×D4—24no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—82no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—82no—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C2511yescharacteristic
412Q16—416no—
413SD16—416no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—416no—
422C2×C8C22×D414yes—
423C2×C8C2×#4118yes—
424C4⋊C4—22no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4C22×D414yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—42no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—48no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—48no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C42—24no—
462C42—24no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—42no—
467C42—42no—
468C42—44no—
469C42—48no—
470C42—44no—
471C42—82no—
472C42C22×D412yes—
473C42C2×#4114yes—
474C42H(5,𝔽2)12yes—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23#57514yes—
489C23C23×D411yescharacteristic
490Q8—28no—
491Q8—44no—
492Q8—44no—
493Q8—44no—
494Q8—88no—
495D4—28no—
496D4—44no—
497D4—44no—
498D4—88no—
499D4—164no—
500D4—164no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—84no—
538C2×C4—88no—
539C2×C4#53212yes—
540C2×C4#53212yes—
541C2×C4D4214yes—
542C2×C4C23×D411yescharacteristic
543C2×C4C22×#4112yes—
544C2×C4C2×H(5,𝔽2)11yescharacteristic
545C2×C4#58512yes—
546C8—24no—
547C8—28no—
548C8—42no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22C2×#53211yescharacteristic
564C22C2×D4211yescharacteristic
565C22#350011yescharacteristic
566C22C2×#57512yes—
567C22#358612yes—
568C4—22no—
569C4—22no—
570C4—24no—
571C4—42no—
572C4—44no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—44no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—84no—
581C4D4×#4112yes—
582C4#351712yes—
583C2—44no—
584C2—162no—
585C2#3612511yescharacteristic
586C2#6068811yescharacteristic
587C2#6072611yescharacteristic
588C2#6106812yes—
589C2#6258512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36136C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3181C2semidirect
2. #36151C2semidirect
4. #33233C2semidirect
5. #33051C2semidirect
6. #28695C2semidirect
7. #28447C2semidirect
8. #26981C2semidirect
9. #26978C2semidirect
10. #25679C2semidirect
11. #25603C2semidirect
12. #25389C2semidirect
13. #23006C2semidirect
14. C4×#491C2semidirect
15. D4×#97C2semidirect
122. #491D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31812
#361512
#361362
#332332
#330512
#286952
#284472
#269812
#269782
#256792
#256032
#253892
#230062
C4×#4912
D4×#972