Finite Group Databasebeta

← #6459924

Subgroups — #6459924

Order 512.

1694 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35673C211yesmaximal, characteristic
2#35557C211yesmaximal, characteristic
3#28867C214yesmaximal, complemented
4#28864C214yesmaximal, complemented
5#28852C214yesmaximal, complemented
6#28847C214yesmaximal, complemented
7C2×#2096C211yesmaximal, characteristic, complemented
8#23656C211yesmaximal, characteristic, complemented
9#23640C211yesmaximal, characteristic
10#23487C211yesmaximal, characteristic
11#21157C211yesmaximal, characteristic, complemented
12#21094C211yesmaximal, characteristic, complemented
13#21038C211yesmaximal, characteristic, complemented
14#20974C211yesmaximal, characteristic
15#20880C211yesmaximal, characteristic, complemented
16#20764C211yesmaximal, characteristic, complemented
17#20662C211yesmaximal, characteristic
18#20417C211yesmaximal, characteristic, complemented
19#20387C211yesmaximal, characteristic
20C22×#131C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#519C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C2×#408C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C2×C4×M16C2211yescharacteristic
34#2512—22no—
35#2511—22no—
36#2498—24no—
37#2497C2214yes—
38#2496C2214yes—
39#2481—24no—
40#2448—22no—
41#2446—22no—
42#2423C2214yes—
43#2421C2214yes—
44#2316C2211yescharacteristic
45#2316C2211yescharacteristic
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48#2311C2211yescharacteristic
49#2297C2211yescharacteristic
50#2296C2211yescharacteristic
51C2×#391C2211yescharacteristic
52#2285C2211yescharacteristic
53C2×#384C2211yescharacteristic
54#2274C2211yescharacteristic
55#2096C2212yes—
56#2096C2212yes—
57#2096C2212yes—
58#2096C2212yes—
59#2085C2212yes—
60#2085C2212yes—
61#2084C2212yes—
62#2083C2212yes—
63#2077C2214yes—
64#2077C2214yes—
65#2036C2214yes—
66#2035C2214yes—
67#1944C2214yes—
68#1941C2214yes—
69#1935C2214yes—
70#1933C2214yes—
71#1920C2214yes—
72#1919C2214yes—
73#1894C2214yes—
74#1892C2214yes—
75#1865C2214yes—
76#1863C2214yes—
77#1836C2214yes—
78#1834C2214yes—
79#1809C2214yes—
80#1807C2214yes—
81#1763C2214yes—
82C4×#103C2214yes—
83#1753C2212yes—
84#1753C2212yes—
85#1752C2212yes—
86C8×C22⋊C4C2212yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C2×#131—24no—
100C2×#131—24no—
101C2×#131—42no—
102C22×M16C2311yescharacteristic
103C22×M16C2312yes—
104C22×M16C2311yescharacteristic
105C23×C8C2312yes—
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#536—24no—
109#536—42no—
110C2×C4⋊Q8—21no—
111C2×#126—24no—
112C2×#124—21no—
113C2×#123—22no—
114C2×C22⋊Q8C2314yes—
115#519—24no—
116#519—42no—
117C2×C4×Q8—22no—
118C2×C4×Q8C2314yes—
119C2×#117—22no—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—22no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#427—42no—
131#427—42no—
132C2×#107C2312yes—
133C2×#107C2312yes—
134C2×#106—24no—
135C2×#106—24no—
136C2×#106C2312yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142#419—24no—
143#419—24no—
144C2×#103—24no—
145C2×#103—24no—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152#408—24no—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155#405—42no—
156C4×M16—24no—
157C2×#97C2312yes—
158C2×#97C2312yes—
159C2×C4×C8C2312yes—
160C2×C4×C8C2312yes—
161#400—24no—
162#400—24no—
163#400—24no—
164#399—24no—
165#398—24no—
166#397—24no—
167#397—24no—
168#396—24no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—22no—
173#391D412yes—
174#391C2312yes—
175#391C2312yes—
176#390—21no—
177#390—21no—
178#390—21no—
179#390D412yes—
180#389—22no—
181#389—22no—
182#389C2314yes—
183#389C2314yes—
184#389C2312yes—
185#389C2312yes—
186#388—24no—
187#388—24no—
188#388—24no—
189#387—24no—
190#387—24no—
191#387—24no—
192#387—24no—
193#386—22no—
194#386—22no—
195#386—22no—
196#386D414yes—
197#385—22no—
198#384—22no—
199#384C2312yes—
200#384C2312yes—
201#384C2312yes—
202#382C2312yes—
203#380—22no—
204#380—22no—
205#380—22no—
206#380C2314yes—
207#380C2314yes—
208#380C2314yes—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4C2314yes—
213C4×C22⋊C4—22no—
214C4×C22⋊C4C2312yes—
215C4×C22⋊C4C2312yes—
216C2×#95—22no—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8#4114yes—
247C22×Q8C2412yes—
248C22×D4C2×D412yes—
249C22×D4C2×D411yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2412yes—
256C23×C4C2411yescharacteristic
257#131—416no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—42no—
266C2×M16#4114yes—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8—42no—
274C22×C8#4114yes—
275C22×C8#4114yes—
276C22×C8C2412yes—
277C22×C8C2411yescharacteristic
278C22×C8C2411yescharacteristic
279C22×C8C2412yes—
280C22×C8C2411yescharacteristic
281C22×C8C2411yescharacteristic
282C4⋊Q8—44no—
283#126—88no—
284#124—44no—
285#123—48no—
286C22⋊Q8—28no—
287C22⋊Q8—44no—
288C4×Q8—28no—
289C4×Q8—44no—
290C4×Q8—48no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—48no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D414yes—
340C2×C22⋊C4C2×D414yes—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—24no—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—42no—
354#107—42no—
355#106—24no—
356#106—24no—
357#106—24no—
358#106—24no—
359#106—42no—
360#106—42no—
361#103—28no—
362#103—24no—
363#103—24no—
364#103—24no—
365#103—24no—
366#103—28no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369#97—24no—
370#97—42no—
371C4×C8—24no—
372C4×C8—42no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95C2×D414yes—
386#95C2×D414yes—
387C24C22×D412yes—
388C24C22×D411yescharacteristic
389#41—44no—
390#41—44no—
391#41—44no—
392#41—44no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8C2×#4114yes—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C42−1+412yes—
452C22×C4C2511yescharacteristic
453M16—28no—
454M16—48no—
455M16—44no—
456C2×C8—28no—
457C2×C8—28no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—22no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—48no—
472C2×C8C2×#4114yes—
473C2×C8C2×#4112yes—
474C2×C8C2×#4112yes—
475C2×C8C2×#4112yes—
476C2×C8C2×#4112yes—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
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—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—82no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C22⋊C4—24no—
506C22⋊C4—28no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C42—24no—
517C42—21no—
518C42—21no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—21no—
523C42—48no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—84no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23#53212yes—
537C23#54612yes—
538C23#54712yes—
539C23C23×D411yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—42no—
543Q8—42no—
544Q8—48no—
545Q8—44no—
546D4—48no—
547D4—44no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—48no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—162no—
587C2×C4—162no—
588C2×C4—162no—
589C2×C4#52914yes—
590C2×C4#53212yes—
591C2×C4#53914yes—
592C2×C4#54612yes—
593C2×C4#54712yes—
594C2×C4#55114yes—
595C2×C4#55214yes—
596C2×C4#57714yes—
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4C22×#4111yescharacteristic
600C2×C4C22×#4111yescharacteristic
601C2×C4C2×H(5,𝔽2)11yescharacteristic
602C2×C4C2×2−1+411yescharacteristic
603C8—22no—
604C8—22no—
605C8—22no—
606C8—22no—
607C8—24no—
608C8—42no—
609C22—24no—
610C22—22no—
611C22—22no—
612C22—24no—
613C22—42no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22#255314yes—
619C22C2×#53211yescharacteristic
620C22C2×#54611yescharacteristic
621C22C2×#54711yescharacteristic
622C22C2×#57712yes—
623C22#358612yes—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—42no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4#255314yes—
637C4C2×#52911yescharacteristic
638C4C2×#53911yescharacteristic
639C4C2×#55111yescharacteristic
640C4C2×#55211yescharacteristic
641C2—24no—
642C2—82no—
643C2C2×#255311yescharacteristic
644C2#6071211yescharacteristic
645C2#6072011yescharacteristic
646C2#6120212yes—
647C2#6127112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #28867C2semidirect
4. #28864C2semidirect
5. #28852C2semidirect
6. #28847C2semidirect
7. C2×#2096C2semidirect
8. #23656C2semidirect
11. #21157C2semidirect
12. #21094C2semidirect
13. #21038C2semidirect
15. #20880C2semidirect
16. #20764C2semidirect
18. #20417C2semidirect

Maximal subgroups

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

subgroupindex
#356732
#355572
#288672
#288642
#288522
#288472
C2×#20962
#236562
#236402
#234872
#211572
#210942
#210382
#209742
#208802
#207642
#206622
#204172
#203872