Finite Group Databasebeta

← #6489124

Subgroups — #6489124

Order 512.

2008 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#32907C214yesmaximal, complemented
4#30042C212yesmaximal, complemented
5#30039C212yesmaximal, complemented
6#29148C212yesmaximal, complemented
7#28449C212yesmaximal, complemented
8#28161C212yesmaximal, complemented
9#28157C212yesmaximal, complemented
10#27980C212yesmaximal, complemented
11#25728C212yesmaximal
12#23935C214yesmaximal, complemented
13#22273C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic, complemented
15C2×#533C2212yes—
16C2×#532C2211yescharacteristic
17C2×C4×#41C2212yes—
18C2×#497C2214yes—
19C2×#462C2214yes—
20C2×#461C2214yes—
21C2×#452C2218yes—
22#2868C2211yescharacteristic
23#2868C2211yescharacteristic
24#2862C2212yes—
25#2859—21no—
26#2859—21no—
27#2857—21no—
28#2857—21no—
29#2857C2212yes—
30#2857C2212yes—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8C2212yes—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2211yescharacteristic
37C4×#127C2211yescharacteristic
38C4×#127C2211yescharacteristic
39C4×#124C2212yes—
40C4×#124C2212yes—
41C4×#121—21no—
42C4×#121—21no—
43C4×#121—21no—
44C4×#121—21no—
45#2297C2212yes—
46#2293—21no—
47#2293—21no—
48#2285—21no—
49#2285—21no—
50#2285C2211yescharacteristic
51C42×D4—21no—
52C42×D4—21no—
53C2×C43C2211yescharacteristic
54#2072C2214yes—
55#2072C2214yes—
56#2024C2214yes—
57#2023C2214yes—
58#2016C2214yes—
59#2016C2214yes—
60#2014C2214yes—
61#2013C2214yes—
62#1960C2214yes—
63#1959C2214yes—
64#1880C2218yes—
65#1878C2218yes—
66#1878C2218yes—
67#1877C2218yes—
68#1873C2218yes—
69#1839C2212yes—
70#1836C2214yes—
71#1831C2212yes—
72#1817C2212yes—
73#1803C2214yes—
74#1802C2214yes—
75#1797C2212yes—
76#1767C2212yes—
77#1749C2212yes—
78#1745C2212yes—
79C22×#41C2312yes—
80C22×Q16C2318yes—
81C22×SD16C2318yes—
82#533—48no—
83#532—28no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#127C2311yescharacteristic
92C2×#125—22no—
93C2×#124C2312yes—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8C2314yes—
96C2×#121—21no—
97C2×#121—21no—
98C4×#41—48no—
99C2×C4×Q8—22no—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2314yes—
102C2×C4×Q8C2312yes—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117—22no—
114C2×#117—22no—
115C2×#117C2312yes—
116C22×C42—22no—
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#497—216no—
125#462—48no—
126#461—48no—
127#452—416no—
128C2×C4⋊C8C2314yes—
129C2×#103C2318yes—
130C2×#103C2318yes—
131C2×#103C2318yes—
132C2×#102C2318yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2314yes—
135#391—22no—
136#391—22no—
137#391C2314yes—
138#391C2314yes—
139#391C2312yes—
140#391C2312yes—
141#390—21no—
142#390—21no—
143#390—21no—
144#390—21no—
145#390C2312yes—
146#390C2312yes—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#386—22no—
164#386—22no—
165#384—21no—
166#384—21no—
167#384—21no—
168#384—21no—
169#384—21no—
170#384—21no—
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#384C2312yes—
174#384C2312yes—
175#382—22no—
176#382—22no—
177#382C2312yes—
178#380—22no—
179#380—22no—
180#380—22no—
181#380—22no—
182#380—21no—
183#380—21no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208#340C2318yes—
209#340C2318yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214C2×#41—24no—
215C2×#41—24no—
216C2×#41—28no—
217C2×#41—44no—
218C22×Q8—22no—
219C22×Q8C2414yes—
220C22×Q8C2412yes—
221C22×D4—22no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C2412yes—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2411yescharacteristic
235C2×Q16—216no—
236C2×Q16—48no—
237C2×Q16—48no—
238C2×SD16—216no—
239C2×SD16—48no—
240C2×SD16—48no—
241C22×C8C2418yes—
242C4⋊Q8—24no—
243C4⋊Q8—28no—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8—44no—
247C4⋊Q8—44no—
248C4⋊Q8—44no—
249C4⋊Q8—44no—
250#127—24no—
251#125—48no—
252#124—24no—
253#124#4118yes—
254C22⋊Q8—216no—
255C22⋊Q8—48no—
256#121—28no—
257#121—28no—
258C4×Q8—28no—
259C4×Q8—44no—
260C4×Q8—48no—
261C4×Q8—48no—
262C4×D4—28no—
263C4×D4—28no—
264C4×D4—28no—
265C4×D4—44no—
266C4×D4—44no—
267C4×D4—44no—
268C4×D4—48no—
269C4×D4—48no—
270C4×D4—44no—
271C4×D4—44no—
272#117—48no—
273#117—44no—
274#117—48no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2414yes—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—48no—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C4⋊C8—28no—
350#103—48no—
351#103—48no—
352#103—48no—
353#102—48no—
354C22⋊C8—28no—
355#97—28no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366C24—22no—
367C24C22×D411yescharacteristic
368#41—416no—
369#41—416no—
370C2×Q8—22no—
371C2×Q8—28no—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—28no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8C2×#4114yes—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—28no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4C2×#4114yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4H(5,𝔽2)14yes—
442C22×C4C2511yescharacteristic
443Q16—816no—
444SD16—816no—
445C2×C8—28no—
446C2×C8—28no—
447C2×C8—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—28no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—48no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—44no—
507C42—44no—
508C42—48no—
509C42—44no—
510C42#13714yes—
511C42#13714yes—
512C42C2×#4114yes—
513C42H(5,𝔽2)14yes—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—24no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23#53512yes—
528C23C23×D411yescharacteristic
529Q8—28no—
530Q8—44no—
531Q8—48no—
532Q8—48no—
533Q8—88no—
534D4—28no—
535D4—48no—
536D4—48no—
537D4—416no—
538D4—44no—
539C2×C4—28no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—416no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—88no—
589C2×C4—88no—
590C2×C4#52512yes—
591C2×C4#52512yes—
592C2×C4#53512yes—
593C2×C4#53512yes—
594C2×C4#53512yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13712yes—
599C2×C4C2×#13712yes—
600C2×C4C2×#13712yes—
601C2×C4#57512yes—
602C2×C4#57512yes—
603C2×C4C22×#4112yes—
604C2×C4C2×H(5,𝔽2)12yes—
605C2×C4C2×H(5,𝔽2)12yes—
606C8—48no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22C2×#52511yescharacteristic
622C22C2×#53511yescharacteristic
623C22C2×#53511yescharacteristic
624C22C22×#13711yescharacteristic
625C22C22×#13711yescharacteristic
626C22C2×#57511yescharacteristic
627C22C2×#57511yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—44no—
641C4—82no—
642C4—84no—
643C4—82no—
644C4—84no—
645C4#321614yes—
646C4#351914yes—
647C2—42no—
648C2—84no—
649C2#3523711yescharacteristic
650C2#6056211yescharacteristic
651C2#6056211yescharacteristic
652C2C2×#321611yescharacteristic
653C2#6079011yescharacteristic
654C2#6080011yescharacteristic
655C2#6080411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #32907C2semidirect
4. #30042C2semidirect
5. #30039C2semidirect
6. #29148C2semidirect
7. #28449C2semidirect
8. #28161C2semidirect
9. #28157C2semidirect
10. #27980C2semidirect
12. #23935C2semidirect
13. #22273C2semidirect
14. #20778C2semidirect

Maximal subgroups

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

subgroupindex
#366452
C4×#5322
#329072
#300422
#300392
#291482
#284492
#281612
#281572
#279802
#257282
#239352
#222732
#207782