Finite Group Databasebeta

← #6464062

Subgroups — #6464062

Order 512.

1924 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3135C211yesmaximal, characteristic
2C2×#2961C211yesmaximal, characteristic, complemented
3#35180C211yesmaximal, characteristic
4#35173C211yesmaximal, characteristic, complemented
5#28652C214yesmaximal, complemented
6#28651C214yesmaximal, complemented
7#28650C214yesmaximal, complemented
8#28649C214yesmaximal, complemented
9#23611C212yesmaximal, complemented
10#23610C212yesmaximal, complemented
11#21312C211yesmaximal, characteristic, complemented
12#21258C211yesmaximal, characteristic
13#21092C212yesmaximal, complemented
14#21092C212yesmaximal, complemented
15C2×#1776C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×#532C2211yescharacteristic
19#3135—28no—
20C2×#491C2212yes—
21C2×#490C2212yes—
22C2×#489C2212yes—
23C2×#489C2212yes—
24#2961—28no—
25C2×#446C2214yes—
26C2×#446C2214yes—
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C2×#424C2212yes—
30C2×#423C2211yescharacteristic
31C2×#417C2212yes—
32C2×#417C2212yes—
33C2×#416C2212yes—
34C2×#416C2212yes—
35C2×#405C2212yes—
36C22×#97C2211yescharacteristic
37#2857—22no—
38#2850—22no—
39#2659—22no—
40#2639—22no—
41#2592C2214yes—
42#2584—22no—
43#2584C2214yes—
44#2568C2212yes—
45#2567—21no—
46#2567C2212yes—
47#2563—21no—
48#2562—22no—
49#2548C2212yes—
50#2547—21no—
51#2546—21no—
52#2546C2212yes—
53#2316C2212yes—
54#2316C2212yes—
55#2311C2212yes—
56#2311C2212yes—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#2068C2214yes—
60#2068C2214yes—
61#2067C2214yes—
62#2067C2214yes—
63#2066C2214yes—
64#2066C2214yes—
65#2065C2214yes—
66#2065C2214yes—
67#1989C2214yes—
68#1988C2212yes—
69#1987C2212yes—
70#1983C2214yes—
71#1982C2212yes—
72#1981C2212yes—
73#1936C2214yes—
74#1936C2214yes—
75#1935C2214yes—
76#1935C2214yes—
77#1934C2214yes—
78#1934C2214yes—
79#1933C2214yes—
80#1933C2214yes—
81#1776C2214yes—
82#1776C2214yes—
83#1736C2212yes—
84#1736C2212yes—
85C2×#340C2211yescharacteristic
86C2×#340C2211yescharacteristic
87C22×#41C2312yes—
88C22×M16C2312yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91#533—24no—
92#533—24no—
93#533—24no—
94#533—24no—
95#532—24no—
96#532—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8D411yescharacteristic
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127C2311yescharacteristic
104C2×#125D412yes—
105C2×#125C2312yes—
106C2×#124—22no—
107C2×#124C2312yes—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8D412yes—
111C2×C22⋊Q8D412yes—
112C2×#121—21no—
113C2×#121—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×#117C2312yes—
121C2×#117C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C42—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129#491—28no—
130#490—28no—
131#489—28no—
132#489—28no—
133#446—216no—
134#446—216no—
135C2×#107—24no—
136C2×#107C2314yes—
137C2×#106—24no—
138C2×#106C2314yes—
139#424—28no—
140#423—24no—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145#417—28no—
146#417—28no—
147#416—28no—
148#416—28no—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161#405—28no—
162C2×#97D418yes—
163C2×#97C2314yes—
164C2×#97C2312yes—
165C2×#97C2312yes—
166C2×C4×C8C2314yes—
167#400—24no—
168#398—24no—
169#398—24no—
170#397—24no—
171#395—24no—
172#395—24no—
173#394—24no—
174#393—24no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391C2314yes—
179#391C2314yes—
180#390—22no—
181#390C2314yes—
182#390C2314yes—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—24no—
188#389—24no—
189#389C2314yes—
190#389C2314yes—
191#389C2314yes—
192#389C2314yes—
193#388—24no—
194#388—24no—
195#386—24no—
196#386—24no—
197#384—22no—
198#382—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—24no—
204#380—24no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C2×#95—21no—
214C2×#95—21no—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221C2×#41—24no—
222C2×#41—24no—
223C2×#41—24no—
224C2×#41—24no—
225C2×#41—24no—
226C2×#41—24no—
227C22×Q8—21no—
228C22×Q8—21no—
229C22×Q8C2×D412yes—
230C22×Q8C2×D412yes—
231C22×Q8C2412yes—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×M16—24no—
245C2×M16—24no—
246C2×M16—24no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8C2×D414yes—
252C22×C8C2×D414yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—84no—
265#127—22no—
266#127—22no—
267#125—28no—
268#125—24no—
269#125—24no—
270#124—24no—
271#124—24no—
272#124—84no—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277#121—44no—
278#121—44no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×D4—44no—
286C4×D4—44no—
287#117—24no—
288#117—24no—
289#117—24no—
290#117—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4#4114yes—
318C2×C4⋊C4#4114yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4C2×D414yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—82no—
339C2×C42—82no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D414yes—
343C2×C42#4114yes—
344C2×C42C2414yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—28no—
348#106—28no—
349C4⋊C8—28no—
350C4⋊C8—28no—
351C4⋊C8—24no—
352C4⋊C8—24no—
353#103—24no—
354#103—24no—
355#103—24no—
356#103—24no—
357#103—28no—
358#103—28no—
359#102—24no—
360#102—24no—
361#102—24no—
362#102—24no—
363#102—28no—
364#102—28no—
365#97—28no—
366#97C2×D418yes—
367#97C2×D418yes—
368C4×C8—28no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381C24—22no—
382C24C22×D411yescharacteristic
383#41—48no—
384#41—48no—
385#41—48no—
386#41—48no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4—81no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4#12114yes—
447C22×C4#12114yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C42−1+412yes—
460C22×C42−1+412yes—
461C22×C4C2511yescharacteristic
462M16—48no—
463C2×C8—28no—
464C2×C8—28no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8C22×D414yes—
473C2×C8C22×D414yes—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4C2×#4114yes—
505C4⋊C4C2×#4114yes—
506C22⋊C4—28no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C42—24no—
515C42—24no—
516C42—44no—
517C42—44no—
518C42—44no—
519C42—82no—
520C42—82no—
521C42#12114yes—
522C42#12114yes—
523C42C22×D412yes—
524C42C22×D412yes—
525C42C2×#4112yes—
526C42C2×#4112yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23C2×#12112yes—
539C23#54812yes—
540C23#54812yes—
541C23C23×D411yescharacteristic
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—84no—
547Q8—84no—
548Q8—84no—
549Q8—84no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—84no—
555D4—84no—
556D4—88no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—88no—
599C2×C4C2×#12112yes—
600C2×C4C2×#12112yes—
601C2×C4C2×#12112yes—
602C2×C4C2×#12112yes—
603C2×C4C2×#12112yes—
604C2×C4#53214yes—
605C2×C4#53614yes—
606C2×C4#54812yes—
607C2×C4#54812yes—
608C2×C4#55614yes—
609C2×C4#55614yes—
610C2×C4C23×D411yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C2×2−1+411yescharacteristic
615C2×C4C2×2−1+411yescharacteristic
616C8—24no—
617C8—24no—
618C8—44no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22#254514yes—
633C22C22×#12111yescharacteristic
634C22C2×#54811yescharacteristic
635C22C2×#54811yescharacteristic
636C22#358511yescharacteristic
637C22#358511yescharacteristic
638C22#358611yescharacteristic
639C22#358611yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4#254514yes—
654C4C2×#53211yescharacteristic
655C4C2×#53611yescharacteristic
656C4C2×#55611yescharacteristic
657C4C2×#55611yescharacteristic
658C2—24no—
659C2—162no—
660C2C2×#254511yescharacteristic
661C2#6045611yescharacteristic
662C2#6045611yescharacteristic
663C2#6133211yescharacteristic
664C2#6133311yescharacteristic
665C2#6133411yescharacteristic
666C2#6133511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2961C2semidirect
4. #35173C2semidirect
5. #28652C2semidirect
6. #28651C2semidirect
7. #28650C2semidirect
8. #28649C2semidirect
9. #23611C2semidirect
10. #23610C2semidirect
11. #21312C2semidirect
13. #21092C2semidirect
14. #21092C2semidirect
15. C2×#1776C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31352
C2×#29612
#351802
#351732
#286522
#286512
#286502
#286492
#236112
#236102
#213122
#212582
#210922
C2×#17762