Finite Group Databasebeta

← #6537282

Subgroups — #6537282

Order 512.

1703 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34316C211yesmaximal, characteristic
3#29981C211yesmaximal, characteristic, complemented
4#29955C212yesmaximal, complemented
5#29255C211yesmaximal, characteristic, complemented
6#29159C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#26461C212yesmaximal, complemented
9#26417C214yesmaximal, complemented
10#26021C212yesmaximal, complemented
11#25953C214yesmaximal, complemented
12#24306C212yesmaximal, complemented
13#23774C211yesmaximal, characteristic, complemented
14#23451C212yesmaximal, complemented
15#22708C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22417C212yesmaximal
18C2×#547C2211yescharacteristic
19C2×#533C2212yes—
20C2×#479C2214yes—
21C2×C8×D4C2212yes—
22C2×#416C2214yes—
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2552C2212yes—
26#2542—21no—
27#2538—21no—
28#2441C2214yes—
29#2429—22no—
30#2423—22no—
31#2374—22no—
32#2365—21no—
33#2365C2211yescharacteristic
34#2363—21no—
35#2362C2212yes—
36#2354C2211yescharacteristic
37#2354C2211yescharacteristic
38#2354C2212yes—
39Q8×C4⋊C4—21no—
40#2345—21no—
41#2341C2212yes—
42#2335—22no—
43#2324—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47#2297C2212yes—
48#2293C2211yescharacteristic
49#2274—21no—
50C42×D4C2211yescharacteristic
51#2100C2212yes—
52#2099C2211yescharacteristic
53#2096C2212yes—
54#2094C2212yes—
55#2093C2211yescharacteristic
56#2086C2212yes—
57#2068C2214yes—
58#2067C2212yes—
59#2065C2214yes—
60#2058C2212yes—
61#2057C2211yescharacteristic
62#2026C2214yes—
63#2025C2214yes—
64#2024C2212yes—
65#2024C2212yes—
66#2024C2212yes—
67#2023C2212yes—
68#2023C2211yescharacteristic
69#1989C2214yes—
70#1981C2212yes—
71#1956C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1939C2212yes—
75#1919C2214yes—
76#1892C2214yes—
77#1873C2214yes—
78#1865C2214yes—
79#1865C2214yes—
80#1854C2214yes—
81#1854C2214yes—
82#1850C2214yes—
83#1834C2212yes—
84#1831C2211yescharacteristic
85#1807C2214yes—
86#1803C2214yes—
87#1802C2212yes—
88#1797C2212yes—
89#1788C2214yes—
90C4×#107C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93#1735C2214yes—
94C2×#336C2212yes—
95C22×#41C2312yes—
96C23×C8C2314yes—
97#547—24no—
98#547Q818yescomplemented
99#533—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125C2314yes—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×#121—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—21no—
122C2×#117—21no—
123C2×#117C2314yes—
124C2×#117C2312yes—
125C22×C4⋊C4—21no—
126C22×C42C2312yes—
127#479—48no—
128C8×D4—28no—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2312yes—
136#416—48no—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C4×C8C2312yes—
146#399—24no—
147#398—24no—
148#396—24no—
149#391—21no—
150#391—22no—
151#391C2312yes—
152#391C2312yes—
153#390—21no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390C2312yes—
158#390C2311yescharacteristic
159#390C2311yescharacteristic
160#389—21no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—24no—
167#389C2312yes—
168#389C2311yescharacteristic
169#389C2314yes—
170#389C2311yescharacteristic
171#388—22no—
172#388—22no—
173#388—22no—
174#388—24no—
175#388—21no—
176#388—21no—
177#388—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—24no—
182#386—22no—
183#386—22no—
184#385—22no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#380—22no—
193#380—21no—
194#380—22no—
195#380C2312yes—
196#380C2314yes—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219C43C2311yescharacteristic
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C2×#41—24no—
235C2×#41—44no—
236C2×#41—44no—
237C2×#41C2×Q818yes—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8C2412yes—
241C22×D4—22no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C22×C8—28no—
250C22×C8—44no—
251C22×C8—44no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260C4⋊Q8C2×Q814yes—
261#125—28no—
262#125—44no—
263#123—216no—
264C22⋊Q8—24no—
265C22⋊Q8—48no—
266C22⋊Q8—48no—
267C22⋊Q8—84no—
268C22⋊Q8C2×Q818yes—
269#121—28no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—48no—
273C4×Q8—48no—
274C4×Q8—44no—
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—42no—
278C4×D4C2×Q814yes—
279C4×D4C2×Q814yes—
280C4×D4#4114yes—
281#117—28no—
282#117—44no—
283#117—44no—
284#117—44no—
285#117—44no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—44no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—24no—
352#107—42no—
353#106—44no—
354#106—44no—
355C4⋊C8—24no—
356C4⋊C8—42no—
357C4⋊C8—44no—
358#103—28no—
359#103—44no—
360#103—44no—
361#103—44no—
362#102—24no—
363#102—42no—
364#102—44no—
365C22⋊C8—28no—
366C4×C8—24no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—24no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—21no—
384C24C22×D412yes—
385#41—416no—
386#41—88no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8C22×Q814yes—
397C2×D4—28no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4C22⋊Q814yes—
406C2×D4C22×Q812yes—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×Q814yes—
439C22×C4C2×#4114yes—
440C22×C4C2511yescharacteristic
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—88no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—28no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—81no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4C22⋊Q814yes—
484C4⋊C4C22×Q812yes—
485C4⋊C4H(5,𝔽2)14yes—
486C22⋊C4—28no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—42no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—81no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—84no—
500C22⋊C4C22⋊Q818yes—
501C42—22no—
502C42—22no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—42no—
507C42—42no—
508C42—41no—
509C42—48no—
510C42—44no—
511C42—42no—
512C42—44no—
513C42—82no—
514C42—82no—
515C42—82no—
516C42#13514yes—
517C42C22×Q812yes—
518C42C2×#4112yes—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×C22⋊Q814yes—
533C23C23×D411yescharacteristic
534Q8—28no—
535Q8—48no—
536Q8—44no—
537Q8—88no—
538D4—28no—
539D4—24no—
540D4—42no—
541D4—88no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—88no—
591C2×C4C2×C22⋊Q812yes—
592C2×C4C2×C22⋊Q814yes—
593C2×C4C2×C22⋊Q812yes—
594C2×C4C2×#13512yes—
595C2×C4C2×#13512yes—
596C2×C4C2×#13512yes—
597C2×C4#57512yes—
598C2×C4#57712yes—
599C2×C4C23×Q811yescharacteristic
600C2×C4C22×#4111yescharacteristic
601C2×C4C2×H(5,𝔽2)12yes—
602C2×C4#58512yes—
603C8—42no—
604C8—44no—
605C8—42no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22C22×C22⋊Q811yescharacteristic
621C22#348411yescharacteristic
622C22#350011yescharacteristic
623C22C22×#13511yescharacteristic
624C22C2×#57511yescharacteristic
625C22C2×#57711yescharacteristic
626C22#358611yescharacteristic
627C4—22no—
628C4—22no—
629C4—24no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—44no—
634C4—41no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—81no—
639C4—82no—
640C4—82no—
641C4—81no—
642C4—84no—
643C4#307614yes—
644C4#350112yes—
645C4#351412yes—
646C2—44no—
647C2—82no—
648C2#3570211yescharacteristic
649C2C2×#307611yescharacteristic
650C2#6049711yescharacteristic
651C2#6156511yescharacteristic
652C2#6159611yescharacteristic
653C2#6253211yescharacteristic
654C2#6258711yescharacteristic
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
1. #36134C2semidirect
3. #29981C2semidirect
4. #29955C2semidirect
5. #29255C2semidirect
6. #29159C2semidirect
7. #29141C2semidirect
8. #26461C2semidirect
9. #26417C2semidirect
10. #26021C2semidirect
11. #25953C2semidirect
12. #24306C2semidirect
13. #23774C2semidirect
14. #23451C2semidirect
15. #22708C2semidirect
16. #22696C2semidirect
98. #547Q8semidirect

Maximal subgroups

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

subgroupindex
#361342
#343162
#299812
#299552
#292552
#291592
#291412
#264612
#264172
#260212
#259532
#243062
#237742
#234512
#227082
#226962
#224172