Finite Group Databasebeta

← #6560008

Subgroups — #6560008

Order 512.

1698 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36406C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#30205C211yesmaximal, characteristic, complemented
4#30187C212yesmaximal, complemented
5#29355C211yesmaximal, characteristic, complemented
6#29148C211yesmaximal, characteristic, complemented
7#29113C212yesmaximal, complemented
8#27822C212yesmaximal, complemented
9#27777C212yesmaximal, complemented
10#27716C214yesmaximal, complemented
11#27267C214yesmaximal, complemented
12#25910C212yesmaximal, complemented
13#23817C211yesmaximal, characteristic, complemented
14#23451C212yesmaximal, complemented
15#22490C212yesmaximal, complemented
16#22417C212yesmaximal
17#20558C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×#547C2211yescharacteristic
20C2×#533C2212yes—
21C2×#491C2214yes—
22C2×C8×D4C2212yes—
23C2×#416C2214yes—
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2857C2212yes—
27#2857C2212yes—
28#2553—22no—
29#2553—22no—
30#2552C2212yes—
31#2546C2211yescharacteristic
32#2542—21no—
33#2542C2211yescharacteristic
34#2538—21no—
35#2499C2212yes—
36#2441C2214yes—
37#2432C2212yes—
38#2429—22no—
39#2427—21no—
40#2427—21no—
41#2423—22no—
42#2365—21no—
43#2360—21no—
44#2360—21no—
45#2354C2212yes—
46Q8×C4⋊C4—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#124C2211yescharacteristic
49#2297C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2212yes—
52#2100C2214yes—
53#2072C2211yescharacteristic
54#2072C2214yes—
55#2072C2212yes—
56#2071C2211yescharacteristic
57#2043C2214yes—
58#2042C2212yes—
59#2026C2214yes—
60#2025C2214yes—
61#2024C2211yescharacteristic
62#2024C2212yes—
63#2024C2212yes—
64#2024C2212yes—
65#2024C2212yes—
66#2023C2211yescharacteristic
67#2023C2212yes—
68#1992C2214yes—
69#1990C2212yes—
70#1988C2212yes—
71#1981C2212yes—
72#1959C2212yes—
73#1956C2212yes—
74#1936C2214yes—
75#1935C2214yes—
76#1935C2214yes—
77#1892C2214yes—
78#1892C2214yes—
79#1873C2214yes—
80#1873C2214yes—
81#1836C2212yes—
82#1825C2214yes—
83#1803C2212yes—
84#1802C2212yes—
85#1788C2214yes—
86#1770C2212yes—
87#1767C2211yescharacteristic
88C4×#103C2212yes—
89C4×#102C2212yes—
90#1752C2214yes—
91#1749C2212yes—
92#1744C2211yescharacteristic
93#1735C2214yes—
94C2×#336C2212yes—
95C22×#41C2312yes—
96C23×C8C2314yes—
97#556—28no—
98#547—28no—
99#533—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#126—22no—
108C2×#125C2314yes—
109C2×#124—21no—
110C2×#124C2311yescharacteristic
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×#121—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—22no—
123C2×C4×Q8—22no—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×#117C2312yes—
130C2×#117C2314yes—
131C22×C4⋊C4—21no—
132C22×C42C2312yes—
133#491—48no—
134C8×D4—28no—
135C2×C4⋊C8C2312yes—
136#416—48no—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2314yes—
148C2×#97C2312yes—
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151#399—24no—
152#399—22no—
153#398—24no—
154#398—22no—
155#397—22no—
156#397—22no—
157#396—24no—
158#394—22no—
159#394—22no—
160#391—21no—
161#391—22no—
162#391—21no—
163#391—21no—
164#391C2312yes—
165#391C2311yescharacteristic
166#391C2312yes—
167#391C2312yes—
168#390—22no—
169#390C2311yescharacteristic
170#390C2312yes—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—24no—
179#389C2312yes—
180#389C2314yes—
181#388—24no—
182#388—21no—
183#388—21no—
184#387—22no—
185#387—22no—
186#387—24no—
187#386—22no—
188#386—22no—
189#385—22no—
190#385—22no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#384C2312yes—
198#382—21no—
199#382—21no—
200#382C2311yescharacteristic
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2314yes—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2314yes—
234C2×#41—24no—
235C2×#41—28no—
236C2×#41—42no—
237C2×#41—44no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8C2411yescharacteristic
242C22×Q8C2412yes—
243C22×D4—22no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2411yescharacteristic
246C23×C4—21no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C22×C8—28no—
252C22×C8—44no—
253C22×C8—44no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264#126—48no—
265#125—28no—
266#125—44no—
267#124—22no—
268#124—41no—
269#124—44no—
270#123—48no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C22⋊Q8—48no—
276C22⋊Q8—48no—
277C22⋊Q8—84no—
278#121—44no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—48no—
282C4×Q8—48no—
283C4×Q8—44no—
284C4×D4—24no—
285C4×D4—42no—
286C4×D4—42no—
287C4×D4#4118yes—
288#117—28no—
289#117—28no—
290#117—44no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C4⋊C8—24no—
354#103—28no—
355#103—24no—
356#103—44no—
357#103—42no—
358#103—44no—
359#103—44no—
360#103—44no—
361#102—24no—
362#102—24no—
363#102—44no—
364C22⋊C8—28no—
365#97—42no—
366#97—44no—
367C4×C8—24no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C24C22×D412yes—
387#41—416no—
388#41—88no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—82no—
414C2×D4#12414yes—
415C2×D42−1+412yes—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—82no—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4118yes—
449C22×C4C2511yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—88no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—84no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4#12414yes—
493C4⋊C4H(5,𝔽2)14yes—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—48no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—84no—
508C22⋊C4#12418yes—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—22no—
515C42—21no—
516C42—21no—
517C42—42no—
518C42—42no—
519C42—48no—
520C42—44no—
521C42—42no—
522C42—44no—
523C42C2×#4114yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C2×#12414yes—
538C23C23×D411yescharacteristic
539Q8—48no—
540Q8—42no—
541Q8—44no—
542Q8—42no—
543Q8—44no—
544Q8—88no—
545Q8—84no—
546D4—28no—
547D4—24no—
548D4—42no—
549D4—88no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—88no—
599C2×C4—84no—
600C2×C4C2×#12414yes—
601C2×C4C2×#12412yes—
602C2×C4C2×#12412yes—
603C2×C4#57512yes—
604C2×C4#57512yes—
605C2×C4#57712yes—
606C2×C4#57812yes—
607C2×C4C22×#4112yes—
608C2×C4C2×H(5,𝔽2)12yes—
609C2×C4C2×2−1+411yescharacteristic
610C2×C4#58511yescharacteristic
611C8—44no—
612C8—42no—
613C8—42no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22C22×#12411yescharacteristic
629C22#348411yescharacteristic
630C22#348911yescharacteristic
631C22C2×#57511yescharacteristic
632C22C2×#57511yescharacteristic
633C22C2×#57711yescharacteristic
634C22C2×#57811yescharacteristic
635C4—24no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—42no—
640C4—44no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—82no—
645C4—81no—
646C4—84no—
647C4—82no—
648C4—81no—
649C4—82no—
650C4—81no—
651C4#313514yes—
652C4#351412yes—
653C4#352012yes—
654C2—44no—
655C2—82no—
656C2#3593911yescharacteristic
657C2C2×#313511yescharacteristic
658C2#6062311yescharacteristic
659C2#6159511yescharacteristic
660C2#6159611yescharacteristic
661C2#6213311yescharacteristic
662C2#6213411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36134C2semidirect
3. #30205C2semidirect
4. #30187C2semidirect
5. #29355C2semidirect
6. #29148C2semidirect
7. #29113C2semidirect
8. #27822C2semidirect
9. #27777C2semidirect
10. #27716C2semidirect
11. #27267C2semidirect
12. #25910C2semidirect
13. #23817C2semidirect
14. #23451C2semidirect
15. #22490C2semidirect
17. #20558C2semidirect

Maximal subgroups

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

subgroupindex
#364062
#361342
#302052
#301872
#293552
#291482
#291132
#278222
#277772
#277162
#272672
#259102
#238172
#234512
#224902
#224172
#205582