Finite Group Databasebeta

← #6432503

Subgroups — #6432503

Order 512.

1786 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34268C211yesmaximal, characteristic
2#34189C211yesmaximal, characteristic
3#23773C214yesmaximal, complemented
4#23769C214yesmaximal, complemented
5#23765C214yesmaximal, complemented
6#23759C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8C2×#2097C211yesmaximal, characteristic, complemented
9#23656C211yesmaximal, characteristic, complemented
10#23640C211yesmaximal, characteristic
11#23639C211yesmaximal, characteristic
12#23629C211yesmaximal, characteristic, complemented
13#23487C211yesmaximal, characteristic
14#23487C211yesmaximal, characteristic
15#20455C211yesmaximal, characteristic, complemented
16#20454C211yesmaximal, characteristic, complemented
17#20417C211yesmaximal, characteristic, complemented
18#20417C211yesmaximal, characteristic, complemented
19#20378C211yesmaximal, characteristic
20C2×#536C2211yescharacteristic
21C2×#519C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C2×#408C2211yescharacteristic
32C2×#408C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C2×#405C2211yescharacteristic
35#2367—22no—
36#2365—22no—
37#2359—24no—
38#2357—22no—
39#2356—22no—
40#2353—24no—
41#2323C2214yes—
42#2314C2214yes—
43#2311C2211yescharacteristic
44#2311C2211yescharacteristic
45#2311C2211yescharacteristic
46C4×C22⋊Q8C2214yes—
47#2296C2211yescharacteristic
48C2×#391C2211yescharacteristic
49#2285C2211yescharacteristic
50#2283C2214yes—
51C2×#382C2211yescharacteristic
52#2274C2211yescharacteristic
53#2274C2211yescharacteristic
54#2274C2211yescharacteristic
55#2101C2214yes—
56#2101C2214yes—
57#2097C2212yes—
58#2097C2212yes—
59#2097C2212yes—
60#2097C2212yes—
61#2085C2212yes—
62#2085C2212yes—
63#2084C2212yes—
64#2083C2212yes—
65#2077C2214yes—
66#2077C2214yes—
67#2076C2214yes—
68#2075C2214yes—
69#2074C2214yes—
70#2072C2214yes—
71#2036C2214yes—
72#2036C2214yes—
73#2035C2214yes—
74#2035C2214yes—
75#1776C2214yes—
76#1776C2214yes—
77C4×#107C2214yes—
78C4×#106C2214yes—
79#1763C2214yes—
80#1763C2214yes—
81C4×#103C2214yes—
82C4×#103C2214yes—
83#1750C2212yes—
84#1750C2212yes—
85#1749C2212yes—
86C4×C22⋊C8C2212yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#536—24no—
104#536—24no—
105C2×C4⋊Q8—21no—
106C2×#124—21no—
107C2×#123—22no—
108C2×C22⋊Q8C2314yes—
109#519—24no—
110#519—24no—
111C4×#41—24no—
112C4×#41—24no—
113C2×C4×Q8—22no—
114C2×C4×Q8D412yes—
115C2×C4×Q8C2314yes—
116C2×C4×D4D414yes—
117C2×C4×D4D412yes—
118C2×#117—22no—
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C2×#117C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#427—24no—
131#427—24no—
132C2×#107—24no—
133C2×#107C2312yes—
134C2×#107C2312yes—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#106—24no—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142#419—24no—
143#419—24no—
144C2×#103—24no—
145C2×#103—24no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152#408—24no—
153#408—24no—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158#405—24no—
159#405—24no—
160C2×#97C2312yes—
161C2×#97C2312yes—
162C2×C4×C8C2312yes—
163C2×C4×C8C2312yes—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—22no—
168#391C2312yes—
169#391C2312yes—
170#390—21no—
171#390—21no—
172#390—21no—
173#389—24no—
174#389C2312yes—
175#389C2312yes—
176#388—24no—
177#388—24no—
178#388—24no—
179#388—24no—
180#387—24no—
181#387—24no—
182#387—24no—
183#387—24no—
184#386—22no—
185#386—22no—
186#386—22no—
187#385—22no—
188#385—24no—
189#384C2312yes—
190#383—24no—
191#382—22no—
192#382C2312yes—
193#382C2312yes—
194#382C2312yes—
195#381—24no—
196#380—22no—
197#380C2314yes—
198#380C2314yes—
199#380C2314yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—24no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2314yes—
209C4×C22⋊C4—24no—
210C4×C22⋊C4—24no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214#376—24no—
215C2×#95—22no—
216C2×#95—22no—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2311yescharacteristic
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×C4C2×D414yes—
256C23×C4C2412yes—
257C23×C4C2411yescharacteristic
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8#4114yes—
269C22×C8#4114yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2411yescharacteristic
273C22×C8C2411yescharacteristic
274C22×C8C2411yescharacteristic
275C22×C8C2411yescharacteristic
276C4⋊Q8—28no—
277#124—28no—
278#123—216no—
279C22⋊Q8—28no—
280C22⋊Q8—28no—
281C4×Q8—28no—
282C4×Q8—28no—
283C4×Q8—28no—
284C4×Q8—48no—
285C4×D4—28no—
286C4×D4—216no—
287#117—22no—
288#117—24no—
289#117—24no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—48no—
297#117#4114yes—
298#117#4114yes—
299#117#4114yes—
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—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4C2×D414yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42C2×D414yes—
347C2×C42C2×D412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351#107—24no—
352#107—24no—
353#107—24no—
354#107—24no—
355#106—24no—
356#106—24no—
357#106—24no—
358#106—24no—
359#103—28no—
360#103—28no—
361#103—24no—
362#103—24no—
363#103—24no—
364#103—24no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369#97—24no—
370#97—24no—
371C4×C8—24no—
372C4×C8—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—22no—
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—44no—
407C2×Q8—42no—
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—22no—
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—28no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2511yescharacteristic
456M16—44no—
457M16—44no—
458C2×C8—28no—
459C2×C8—28no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
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—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8C2×#4112yes—
475C2×C8C2×#4112yes—
476C2×C8C2×#4112yes—
477C2×C8C2×#4112yes—
478C4⋊C4—216no—
479C4⋊C4—28no—
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—22no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4#12318yes—
502C4⋊C4C2×#4114yes—
503C22⋊C4—216no—
504C22⋊C4—24no—
505C22⋊C4—28no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4#12318yes—
513C42—24no—
514C42—28no—
515C42—24no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—42no—
521C42—48no—
522C42—42no—
523C42—42no—
524C42—84no—
525C42—84no—
526C42—84no—
527C42C2×#4114yes—
528C42C2×#4114yes—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23C2×#12312yes—
537C23#52512yes—
538C23#53212yes—
539C23C23×D411yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—48no—
543Q8—44no—
544Q8—42no—
545Q8—42no—
546D4—48no—
547D4—44no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
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—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
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—88no—
586C2×C4—84no—
587C2×C4—162no—
588C2×C4—162no—
589C2×C4—162no—
590C2×C4C2×#12318yes—
591C2×C4C2×#12312yes—
592C2×C4#52512yes—
593C2×C4#52914yes—
594C2×C4#52914yes—
595C2×C4#52914yes—
596C2×C4#52914yes—
597C2×C4#53212yes—
598C2×C4C22×#4111yescharacteristic
599C2×C4C22×#4111yescharacteristic
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C8—22no—
605C8—22no—
606C8—22no—
607C8—22no—
608C8—42no—
609C8—42no—
610C22—22no—
611C22—22no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22#239514yes—
620C22C22×#12311yescharacteristic
621C22C2×#52511yescharacteristic
622C22C2×#53211yescharacteristic
623C22#358412yes—
624C22#358612yes—
625C4—28no—
626C4—42no—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—44no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4#239514yes—
638C4C2×#52911yescharacteristic
639C4C2×#52911yescharacteristic
640C4C2×#52911yescharacteristic
641C4C2×#52911yescharacteristic
642C2—24no—
643C2—82no—
644C2C2×#239511yescharacteristic
645C2#6052412yes—
646C2#6059812yes—
647C2#6071311yescharacteristic
648C2#6072011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #23773C2semidirect
4. #23769C2semidirect
5. #23765C2semidirect
6. #23759C2semidirect
7. #23746C2semidirect
8. C2×#2097C2semidirect
9. #23656C2semidirect
12. #23629C2semidirect
15. #20455C2semidirect
16. #20454C2semidirect
17. #20417C2semidirect
18. #20417C2semidirect

Maximal subgroups

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

subgroupindex
#342682
#341892
#237732
#237692
#237652
#237592
#237462
C2×#20972
#236562
#236402
#236392
#236292
#234872
#204552
#204542
#204172
#203782