Finite Group Databasebeta

← #6644702

Subgroups — #6644702

Order 512.

1792 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3#33002C212yesmaximal, complemented
4#32457C214yesmaximal, complemented
5#32371C214yesmaximal, complemented
6#31665C214yesmaximal, complemented
7#31547C214yesmaximal, complemented
8#29639C212yesmaximal, complemented
9#29109C211yesmaximal, characteristic, complemented
10#28310C211yesmaximal, characteristic
11#28117C212yesmaximal, complemented
12#27969C211yesmaximal, characteristic, complemented
13#27918C211yesmaximal, characteristic, complemented
14#27651C212yesmaximal, complemented
15#21916C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#481C2212yes—
20C2×#481C2212yes—
21C2×#480C2212yes—
22C2×#447C2212yes—
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2854C2212yes—
26#2639—21no—
27#2618—21no—
28#2563—22no—
29#2563—22no—
30#2563C2214yes—
31#2563C2214yes—
32#2560—22no—
33#2558—22no—
34#2512C2212yescomplemented
35#2506C2212yescomplemented
36#2452C2212yescomplemented
37#2445C2212yescomplemented
38C2×#393—21no—
39C2×#393—21no—
40#2325—21no—
41C4×#121C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380—21no—
47C2×#380—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49#2101C2214yescomplemented
50#2094C2212yes—
51#2092C2214yescomplemented
52#2085C2214yescomplemented
53#2073C2214yescomplemented
54#2071C2212yes—
55#2065C2214yes—
56#2058C2214yescomplemented
57#2058C2214yescomplemented
58#2054C2214yes—
59#2023C2212yes—
60#2023C2214yescomplemented
61#2023C2214yescomplemented
62#2011C2212yes—
63#1993C2212yescomplemented
64#1972C2214yes—
65#1961C2212yescomplemented
66#1949C2214yescomplemented
67#1936C2214yes—
68#1926C2214yes—
69#1917C2214yescomplemented
70#1917C2214yescomplemented
71#1879C2214yes—
72#1879C2214yes—
73#1879C2214yes—
74#1862C2214yes—
75#1862C2214yes—
76#1847C2212yescomplemented
77#1836C2212yescomplemented
78#1824C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1817C2212yes—
82#1804C2214yescomplemented
83#1797C2212yes—
84#1786C2212yes—
85#1749C2211yescharacteristic
86#1749C2211yescharacteristic
87C23×Q8—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×D8C2314yes—
91#539—48no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8C2314yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#126—22no—
98C2×#125—22no—
99C2×#123—22no—
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—22no—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—44no—
109C2×C4×D4C2314yes—
110C2×C4×D4C2312yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#481—44no—
126#481—44no—
127#480—44no—
128#447—44no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×#97C2312yes—
142C2×#97C2312yes—
143#400—22no—
144#398—22no—
145#397—22no—
146#396—22no—
147#396—22no—
148#394—22no—
149#394—22no—
150#393—22no—
151#393—24no—
152#393—24no—
153#393—24no—
154#393—24no—
155#391—21no—
156#390—21no—
157#390—22no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390C2314yes—
162#390C2312yes—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#389—21no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#387—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#385—22no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#382—22no—
185#382—21no—
186#382—21no—
187#382C2314yes—
188#382C2312yes—
189#382C2312yes—
190#380—21no—
191#380—24no—
192#380—24no—
193#380—24no—
194#380—24no—
195#380C2312yes—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C22⋊C4—22no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95D411yescharacteristic
207C2×#95D411yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219C25—21no—
220C22×Q8—21no—
221C22×Q8—24no—
222C22×Q8—44no—
223C22×Q8—82no—
224C22×Q8—82no—
225C22×D4—24no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—24no—
231C22×D4—42no—
232C22×D4C2412yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—24no—
238C23×C4—22no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×D8—28no—
250C2×D8—44no—
251C2×D8—44no—
252C22×C8C2414yes—
253C22×C8C2414yes—
254C4⋊Q8—48no—
255C4⋊Q8—42no—
256C4⋊Q8—48no—
257#127—42no—
258#126—48no—
259#125—48no—
260#123—48no—
261#123—48no—
262C22⋊Q8—48no—
263C22⋊Q8—44no—
264#121—24no—
265#121—24no—
266#121—44no—
267#121—44no—
268C4×D4—48no—
269C4×D4—48no—
270C4×D4—44no—
271C4×D4—48no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
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×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—82no—
321C2×C22⋊C4—82no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4—82no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2412yes—
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#107—44no—
337#106—44no—
338#106—44no—
339C4⋊C8—44no—
340#102—28no—
341#102—44no—
342#102—44no—
343#102—44no—
344#102—44no—
345#102—44no—
346C22⋊C8—24no—
347C22⋊C8—24no—
348#97—42no—
349#97—42no—
350#95—24no—
351#95—22no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24—24no—
367C24—22no—
368C24—22no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24C22×D411yescharacteristic
376C2×Q8—48no—
377C2×Q8—42no—
378C2×Q8—84no—
379C2×Q8—84no—
380C2×Q8—81no—
381C2×Q8—81no—
382C2×Q8—816no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—416no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—81no—
400C2×D4—81no—
401C2×D4—81no—
402C2×D4—81no—
403C2×D42−1+414yes—
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—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—48no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—816no—
445C22×C4AGL(1,ℤ/8)14yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C4C2511yescharacteristic
457D8—88no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—84no—
500C22⋊C4—82no—
501C22⋊C4—84no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—42no—
512C42—41no—
513C42—41no—
514C42—44no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—24no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23#53512yes—
541C23C2×AGL(1,ℤ/8)12yes—
542C23#57512yes—
543C23C23×D411yescharacteristic
544Q8—816no—
545Q8—162no—
546D4—28no—
547D4—48no—
548D4—44no—
549D4—44no—
550D4—44no—
551D4—88no—
552D4—162no—
553D4—162no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—816no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—164no—
609C2×C4#53512yes—
610C2×C4C2×AGL(1,ℤ/8)14yes—
611C2×C4C2×H(5,𝔽2)12yes—
612C2×C4C2×2−1+412yes—
613C2×C4#58512yes—
614C8—44no—
615C8—44no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—816no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22#321314yes—
638C22C2×#53511yescharacteristic
639C22#348511yescharacteristic
640C22#348611yescharacteristic
641C22C22×AGL(1,ℤ/8)11yescharacteristic
642C22C2×#57511yescharacteristic
643C22#358511yescharacteristic
644C22#358511yescharacteristic
645C4—24no—
646C4—22no—
647C4—42no—
648C4—41no—
649C4—41no—
650C4—42no—
651C4—41no—
652C4—41no—
653C4—42no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4#353514yes—
661C2—24no—
662C2—84no—
663C2—162no—
664C2#3711811yescharacteristic
665C2C2×#321311yescharacteristic
666C2#6080611yescharacteristic
667C2#6171011yescharacteristic
668C2#6172011yescharacteristic
669C2#6182511yescharacteristic
670C2#6183111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28310C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35006C2semidirect
2. C2×#2563C2semidirect
3. #33002C2semidirect
4. #32457C2semidirect
5. #32371C2semidirect
6. #31665C2semidirect
7. #31547C2semidirect
8. #29639C2semidirect
9. #29109C2semidirect
11. #28117C2semidirect
12. #27969C2semidirect
13. #27918C2semidirect
14. #27651C2semidirect
15. #21916C2semidirect
34. #2512C22semidirect
35. #2506C22semidirect
36. #2452C22semidirect
37. #2445C22semidirect
49. #2101C22semidirect
51. #2092C22semidirect
52. #2085C22semidirect
53. #2073C22semidirect
56. #2058C22semidirect
57. #2058C22semidirect
60. #2023C22semidirect
61. #2023C22semidirect
63. #1993C22semidirect
65. #1961C22semidirect
66. #1949C22semidirect
69. #1917C22semidirect
70. #1917C22semidirect
76. #1847C22semidirect
77. #1836C22semidirect
82. #1804C22semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#25632
#330022
#324572
#323712
#316652
#315472
#296392
#291092
#283102
#281172
#279692
#279182
#276512
#219162