Finite Group Databasebeta

← #6471597

Subgroups — #6471597

Order 512.

1850 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#33005C212yesmaximal, complemented
4#32957C211yesmaximal, characteristic, complemented
5#32903C211yesmaximal, characteristic
6#29969C211yesmaximal, characteristic, complemented
7#29906C212yesmaximal, complemented
8#29755C211yesmaximal, characteristic
9#27567C214yesmaximal, complemented
10#27311C214yesmaximal, complemented
11#26397C214yesmaximal, complemented
12#26179C214yesmaximal, complemented
13#22082C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21454C212yesmaximal, complemented
16C2×#539—21no—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#481C2212yes—
21C2×#477C2212yes—
22C2×#460C2212yes—
23C2×#448C2212yes—
24C2×#440C2214yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854C2212yes—
28#2639—21no—
29#2618—21no—
30#2560—22no—
31#2558—22no—
32#2512C2212yes—
33#2506C2212yes—
34#2452C2212yes—
35#2445C2212yes—
36Q8×C4⋊C4—22no—
37Q8×C4⋊C4—22no—
38Q8×C4⋊C4C2214yes—
39Q8×C4⋊C4C2214yes—
40#2325—21no—
41C4×#121C2212yes—
42C2×#389—21no—
43C2×#389—21no—
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#384—21no—
47C2×#384C2211yescharacteristic
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2100C2212yes—
51#2099C2214yes—
52#2093C2214yes—
53#2087C2214yes—
54#2084C2212yes—
55#2083C2214yes—
56#2071C2214yes—
57#2065C2214yes—
58#2057C2212yes—
59#2041C2212yes—
60#1944C2214yes—
61#1936C2214yes—
62#1919C2214yes—
63#1917C2214yes—
64#1917C2214yes—
65#1880C2214yes—
66#1880C2214yes—
67#1880C2214yes—
68#1880C2214yes—
69#1874C2212yes—
70#1874C2212yes—
71#1865C2214yes—
72#1865C2214yes—
73#1863C2214yes—
74#1863C2214yes—
75#1862C2214yes—
76#1852C2212yes—
77#1852C2212yes—
78#1824C2212yes—
79#1817C2212yes—
80#1803C2212yes—
81#1803C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1787C2212yes—
85#1763C2214yes—
86#1749C2212yes—
87C23×Q8C2311yescharacteristic
88C22×SD16C2314yes—
89#539—48no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#126—22no—
93C2×#125—22no—
94C2×#123—22no—
95C2×#123—22no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—22no—
98C2×#121—21no—
99C2×#121—21no—
100C2×#121C2312yes—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—44no—
108C2×C4×Q8—44no—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2314yes—
113C2×C4×D4—22no—
114C2×C4×D4C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C42—21no—
128C22×C42—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#481—28no—
134#477—44no—
135#460—44no—
136#448—28no—
137#440—48no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×#97C2314yes—
152#400—22no—
153#398—22no—
154#397—22no—
155#396—22no—
156#396—22no—
157#394—22no—
158#394—22no—
159#393—22no—
160#391—21no—
161#390—21no—
162#390—21no—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#389—21no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—24no—
172#389—24no—
173#389—24no—
174#389—24no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2314yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#387—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#385—22no—
191#384—24no—
192#384—24no—
193#384—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2314yes—
197#380—21no—
198#380C2312yes—
199C4×C4⋊C4—24no—
200C4×C4⋊C4—24no—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220#336C2314yes—
221#336C2314yes—
222C22×Q8—24no—
223C22×Q8—24no—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8#4114yes—
228C22×Q8C2412yes—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×SD16—28no—
246C2×SD16—28no—
247C2×SD16—28no—
248C22×C8C2414yes—
249C22×C8C2414yes—
250C4⋊Q8—42no—
251#127—42no—
252#126—48no—
253#125—48no—
254#123—48no—
255#123—48no—
256C22⋊Q8—48no—
257C22⋊Q8—44no—
258#121—24no—
259#121—24no—
260#121—44no—
261#121—44no—
262C4×Q8—216no—
263C4×Q8—28no—
264C4×Q8—28no—
265C4×Q8—28no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—44no—
269C4×Q8—42no—
270C4×Q8—42no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—48no—
274C4×D4—44no—
275C4×D4—48no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4#4114yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C42—28no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42—44no—
342C2×C42—44no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351#107—44no—
352#107—44no—
353#106—28no—
354C4⋊C8—42no—
355C4⋊C8—42no—
356#103—28no—
357#103—28no—
358#103—28no—
359#102—28no—
360#102—44no—
361#102—44no—
362C22⋊C8—24no—
363C22⋊C8—24no—
364#97—44no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—22no—
379C24C22×D411yescharacteristic
380C2×Q8—24no—
381C2×Q8—216no—
382C2×Q8—24no—
383C2×Q8—216no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—42no—
389C2×Q8—81no—
390C2×Q8—81no—
391C2×Q8C2×#4114yes—
392C2×Q8H(5,𝔽2)14yes—
393C2×Q82−1+414yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—81no—
404C2×D4—81no—
405C2×D4—81no—
406C2×D4—81no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4AGL(1,ℤ/8)14yes—
448C22×C4#13714yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2511yescharacteristic
461SD16—416no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C4⋊C4—216no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—81no—
501C4⋊C4C2×#4114yes—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C42—216no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—41no—
521C42—41no—
522C42—42no—
523C42—44no—
524C42—44no—
525C42—48no—
526C42—44no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23#52512yes—
539C23C2×AGL(1,ℤ/8)12yes—
540C23C2×#13712yes—
541C23C23×D411yescharacteristic
542Q8—216no—
543Q8—24no—
544Q8—24no—
545Q8—24no—
546Q8—162no—
547Q8#53918yescomplemented
548D4—44no—
549D4—44no—
550D4—88no—
551D4—162no—
552D4—162no—
553C2×C4—216no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—48no—
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—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—164no—
610C2×C4#52512yes—
611C2×C4C2×AGL(1,ℤ/8)14yes—
612C2×C4C2×#13712yes—
613C2×C4C2×#13712yes—
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)12yes—
617C2×C4C2×2−1+412yes—
618C8—44no—
619C8—44no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22#311214yes—
634C22C2×#52511yescharacteristic
635C22#348411yescharacteristic
636C22#348611yescharacteristic
637C22C22×AGL(1,ℤ/8)11yescharacteristic
638C22C22×#13711yescharacteristic
639C22#358511yescharacteristic
640C22#358611yescharacteristic
641C4—24no—
642C4—22no—
643C4—22no—
644C4—28no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—41no—
649C4—41no—
650C4—42no—
651C4—44no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4C2×#53912yes—
658C4#349212yes—
659C2—24no—
660C2—162no—
661C2#3498911yescharacteristic
662C2C2×#311211yescharacteristic
663C2#6056511yescharacteristic
664C2#6157411yescharacteristic
665C2#6158711yescharacteristic
666C2#6182811yescharacteristic
667C2#6183011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
2. C2×Q8×C4⋊C4C2semidirect
3. #33005C2semidirect
4. #32957C2semidirect
6. #29969C2semidirect
7. #29906C2semidirect
9. #27567C2semidirect
10. #27311C2semidirect
11. #26397C2semidirect
12. #26179C2semidirect
13. #22082C2semidirect
14. #21916C2semidirect
15. #21454C2semidirect
547. Q8#539semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×Q8×C4⋊C42
#330052
#329572
#329032
#299692
#299062
#297552
#275672
#273112
#263972
#261792
#220822
#219162
#214542