Finite Group Databasebeta

← #6434309

Subgroups — #6434309

Order 512.

1699 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic, complemented
2#35267C211yesmaximal, characteristic
3#30355C214yesmaximal, complemented
4#30351C214yesmaximal, complemented
5#30345C212yesmaximal, complemented
6#30344C212yesmaximal, complemented
7#30339C212yesmaximal, complemented
8#30338C212yesmaximal, complemented
9#23615C212yesmaximal, complemented
10#23554C211yesmaximal, characteristic, complemented
11#23545C212yesmaximal
12C2×#2043C211yesmaximal, characteristic, complemented
13#23451C211yesmaximal, characteristic
14#23432C211yesmaximal, characteristic
15#21234C212yesmaximal
16#21090C212yesmaximal, complemented
17#20430C211yesmaximal, characteristic, complemented
18C22×#137C2212yes—
19C2×#533C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#428C2212yes—
22C2×#424C2212yes—
23C2×#419C2212yes—
24C2×#416C2212yes—
25C22×#103C2212yes—
26C22×#103C2212yes—
27C2×#405C2212yes—
28#2863C2212yes—
29#2862C2212yes—
30#2862C2212yes—
31#2860C2212yes—
32#2645C2212yes—
33#2644C2212yes—
34#2642C2214yes—
35#2568—21no—
36#2568—21no—
37#2568—21no—
38#2568—21no—
39#2510—22no—
40#2503—22no—
41#2503—22no—
42#2497—21no—
43#2496—21no—
44#2495—21no—
45#2494—21no—
46#2481—22no—
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2311C2211yescharacteristic
50#2311C2211yescharacteristic
51#2310C2212yes—
52#2310C2212yes—
53#2297C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C4×#117C2211yescharacteristic
56#2068C2214yes—
57#2068C2214yes—
58#2066C2214yes—
59#2066C2214yes—
60#2059C2214yes—
61#2058C2212yes—
62#2057C2212yes—
63#2056C2214yes—
64#2056C2214yes—
65#2055C2214yes—
66#2055C2214yes—
67#2043C2212yes—
68#2043C2212yes—
69#2043C2214yes—
70#2026C2212yes—
71#2025C2212yes—
72#2024C2211yescharacteristic
73#2024C2211yescharacteristic
74#2023C2211yescharacteristic
75#2023C2211yescharacteristic
76#2016C2212yes—
77#2016C2212yes—
78#2013C2212yes—
79#2013C2212yes—
80#1973C2214yes—
81#1971C2214yes—
82#1969C2214yes—
83#1968C2214yes—
84#1935C2214yes—
85#1935C2214yes—
86#1933C2214yes—
87#1933C2214yes—
88#1768C2214yes—
89#1767C2212yes—
90C4×C4⋊C8C2212yes—
91#1736C2212yes—
92#1735C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8C2312yes—
97C2×#137—28no—
98C2×#137—44no—
99C2×#137—44no—
100C22×Q16C2314yes—
101C22×SD16C2314yes—
102C22×M16C2312yes—
103C23×C8C2312yes—
104#533—44no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—44no—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#126—22no—
113C2×#125C2312yes—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×#117C2312yes—
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#428—44no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131#424—28no—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2314yes—
134#419—28no—
135#416—44no—
136C2×#103—44no—
137C2×#103—44no—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148#405—44no—
149C2×#97C2314yes—
150C2×C4×C8C2314yes—
151#401—22no—
152#400—24no—
153#400—24no—
154#400—24no—
155#400—24no—
156#400—22no—
157#400—22no—
158#400—22no—
159#397—22no—
160#397—22no—
161#397—22no—
162#396—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391C2314yes—
172#391C2314yes—
173#391C2314yes—
174#391C2314yes—
175#391C2312yes—
176#391C2312yes—
177#391C2311yescharacteristic
178#391C2311yescharacteristic
179#390C2312yes—
180#390C2312yes—
181#390C2311yescharacteristic
182#390C2311yescharacteristic
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#388—22no—
194#388—22no—
195#388—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#386—21no—
200#386—21no—
201#386—22no—
202#386—22no—
203#385—22no—
204#384—21no—
205#383—22no—
206#382—21no—
207#382—22no—
208#382—22no—
209#382—22no—
210#382—22no—
211#381—24no—
212#381—24no—
213#380—21no—
214#380—21no—
215#380—21no—
216#380—21no—
217#380—22no—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226#376C2312yes—
227C2×#95—21no—
228C43C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2311yescharacteristic
238#336C2311yescharacteristic
239#336C2312yes—
240C2×#41—24no—
241C2×#41—22no—
242C2×#41—24no—
243C2×#41—22no—
244C22×Q8—21no—
245C22×Q8—21no—
246C22×Q8—22no—
247C22×Q8—24no—
248C22×Q8—24no—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8—24no—
252C22×Q8—24no—
253C22×Q8C2414yes—
254C22×Q8C2411yescharacteristic
255C22×D4—21no—
256C22×D4C2411yescharacteristic
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2411yescharacteristic
264#137—432no—
265C2×Q16—28no—
266C2×Q16—44no—
267C2×Q16—44no—
268C2×SD16—28no—
269C2×SD16—44no—
270C2×SD16—44no—
271C2×M16—24no—
272C2×M16—24no—
273C2×M16#4118yes—
274C22×C8—24no—
275C22×C8—24no—
276C22×C8—24no—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2414yes—
280C4⋊Q8—42no—
281C4⋊Q8—42no—
282C4⋊Q8—44no—
283C4⋊Q8—44no—
284C4⋊Q8—44no—
285C4⋊Q8—44no—
286#126—84no—
287#125—42no—
288#125—42no—
289C22⋊Q8—44no—
290C22⋊Q8—44no—
291C4×Q8—44no—
292C4×Q8—44no—
293#117—24no—
294#117—22no—
295#117—22no—
296#117—24no—
297#117—24no—
298#117—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—44no—
357#106—44no—
358C4⋊C8—28no—
359C4⋊C8—28no—
360#103—28no—
361#103—28no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#103—42no—
367#103—42no—
368#102—42no—
369#102—42no—
370#97—44no—
371C4×C8—44no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—21no—
399C24C22×D411yescharacteristic
400#41—48no—
401#41—48no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—21no—
410C2×Q8—24no—
411C2×Q8—21no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—48no—
424C2×Q8—48no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2×#4114yes—
468C22×C4H(5,𝔽2)14yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C42−1+412yes—
471C22×C4C2511yescharacteristic
472Q16—416no—
473SD16—416no—
474M16—216no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—48no—
484C2×C8C2×#4118yes—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—81no—
523C22⋊C4—81no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—42no—
536C42—42no—
537C42—42no—
538C42—44no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23#53212yes—
551C23#53512yes—
552C23#53712yes—
553C23C23×D411yescharacteristic
554Q8—44no—
555Q8—42no—
556Q8—44no—
557Q8—44no—
558Q8—44no—
559Q8—42no—
560Q8—84no—
561Q8—84no—
562Q8—88no—
563D4—42no—
564D4—42no—
565D4—84no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—84no—
615C2×C4—88no—
616C2×C4—84no—
617C2×C4—162no—
618C2×C4#53212yes—
619C2×C4#53512yes—
620C2×C4#53712yes—
621C2×C4#53818yes—
622C2×C4#54218yes—
623C2×C4#57712yes—
624C2×C4#57812yes—
625C2×C4C22×#4112yes—
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×H(5,𝔽2)12yes—
628C2×C4C2×2−1+411yescharacteristic
629C8—28no—
630C8—42no—
631C8—42no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22#266214yes—
646C22C2×#53211yescharacteristic
647C22C2×#53511yescharacteristic
648C22C2×#53711yescharacteristic
649C22C2×#57711yescharacteristic
650C22C2×#57811yescharacteristic
651C22#358611yescharacteristic
652C22#358611yescharacteristic
653C4—22no—
654C4—22no—
655C4—42no—
656C4—42no—
657C4—41no—
658C4—41no—
659C4—42no—
660C4—42no—
661C4—81no—
662C4—84no—
663C4—81no—
664C4—81no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4#266214yes—
670C4C2×#53812yes—
671C4C2×#54212yes—
672C2—24no—
673C2—161no—
674C2C2×#266211yescharacteristic
675C2#6072311yescharacteristic
676C2#6072611yescharacteristic
677C2#6081311yescharacteristic
678C2#6081411yescharacteristic
679C2#6092311yescharacteristic
680C2#6092511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35320C2semidirect
3. #30355C2semidirect
4. #30351C2semidirect
5. #30345C2semidirect
6. #30344C2semidirect
7. #30339C2semidirect
8. #30338C2semidirect
9. #23615C2semidirect
10. #23554C2semidirect
12. C2×#2043C2semidirect
16. #21090C2semidirect
17. #20430C2semidirect

Maximal subgroups

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

subgroupindex
#353202
#352672
#303552
#303512
#303452
#303442
#303392
#303382
#236152
#235542
#235452
C2×#20432
#234512
#234322
#212342
#210902
#204302