Finite Group Databasebeta

← #6541367

Subgroups — #6541367

Order 512.

1843 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34321C211yesmaximal, characteristic
3#29980C211yesmaximal, characteristic, complemented
4#29971C212yesmaximal, complemented
5#29273C211yesmaximal, characteristic, complemented
6#29206C211yesmaximal, characteristic, complemented
7#29163C212yesmaximal, complemented
8#26446C212yesmaximal, complemented
9#26359C214yesmaximal, complemented
10#26044C212yesmaximal, complemented
11#26007C214yesmaximal, complemented
12#24299C212yesmaximal, complemented
13#23774C211yesmaximal, characteristic, complemented
14#23552C212yesmaximal, complemented
15#22700C211yesmaximal, characteristic, complemented
16#22663C212yesmaximal, complemented
17#22503C212yesmaximal
18C2×#555C2211yescharacteristic
19C2×D4×Q8—21no—
20C22×C4⋊Q8C2212yes—
21C2×#476C2214yes—
22C2×#436C2212yes—
23C2×#428C2214yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2551—21no—
27#2551C2212yes—
28#2546—21no—
29#2440—22no—
30#2440C2214yes—
31#2434—22no—
32#2369C2211yescharacteristic
33#2367—22no—
34#2363C2212yes—
35#2361—21no—
36#2354C2211yescharacteristic
37D4×C4⋊C4—21no—
38D4×C4⋊C4C2211yescharacteristic
39D4×C4⋊C4C2212yes—
40#2347—21no—
41#2347C2212yes—
42#2337—22no—
43#2324—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#121C2212yes—
47#2296C2212yes—
48#2293C2211yescharacteristic
49#2274—21no—
50C42×D4C2211yescharacteristic
51#2099C2211yescharacteristic
52#2099C2212yes—
53#2099C2212yes—
54#2093C2212yes—
55#2093C2211yescharacteristic
56#2088C2214yes—
57#2087C2214yes—
58#2086C2212yes—
59#2067C2212yes—
60#2058C2212yes—
61#2058C2212yes—
62#2058C2214yes—
63#2058C2212yes—
64#2057C2211yescharacteristic
65#2057C2212yes—
66#2057C2214yes—
67#2023C2212yes—
68#2023C2211yescharacteristic
69#1990C2212yes—
70#1987C2212yes—
71#1987C2214yes—
72#1948C2214yes—
73#1948C2212yes—
74#1947C2212yes—
75#1939C2212yes—
76#1921C2214yes—
77#1919C2214yes—
78#1895C2214yes—
79#1874C2214yes—
80#1864C2214yes—
81#1854C2214yes—
82#1853C2214yes—
83#1851C2214yes—
84#1844C2212yes—
85#1831C2211yescharacteristic
86#1825C2214yes—
87#1809C2214yes—
88#1802C2212yes—
89#1797C2212yes—
90C4×#107C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93#1736C2214yes—
94#1728C2212yes—
95C23×Q8—22no—
96C22×M16C2314yes—
97#555—24no—
98#555Q818yescomplemented
99D4×Q8—48no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—48no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×#124—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—24no—
113C2×#121—21no—
114C2×#121C2312yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×#117—21no—
125C2×#117—21no—
126C2×#117C2312yes—
127C2×#117C2312yes—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2314yes—
131C22×C42C2312yes—
132#476—48no—
133#436—28no—
134#428—48no—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2314yes—
148C2×C4×C8C2312yes—
149#398—24no—
150#395—24no—
151#395—24no—
152#391—21no—
153#391—21no—
154#391—22no—
155#391—22no—
156#390—21no—
157#390—22no—
158#390C2311yescharacteristic
159#390C2311yescharacteristic
160#389—21no—
161#389—22no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389C2312yes—
167#389C2312yes—
168#389C2311yescharacteristic
169#389C2312yes—
170#389C2312yes—
171#389C2311yescharacteristic
172#389C2314yes—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—21no—
177#388—22no—
178#388—21no—
179#388—22no—
180#386—22no—
181#386—22no—
182#385—22no—
183#385—22no—
184#385—24no—
185#385—24no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#383—22no—
194#380—22no—
195#380—21no—
196#380C2312yes—
197#380C2314yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—22no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—24no—
239C22×Q8—48no—
240C22×Q8—48no—
241C22×Q8—84no—
242C22×D4—22no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—24no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C2×M16—28no—
252C2×M16—28no—
253C2×M16—44no—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—28no—
259C4⋊Q8—22no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C4⋊Q8—48no—
263C4⋊Q8—48no—
264C4⋊Q8—48no—
265C4⋊Q8—44no—
266C4⋊Q8—82no—
267C4⋊Q8C2×Q814yescomplemented
268#124—216no—
269C22⋊Q8—416no—
270C22⋊Q8—48no—
271#121—28no—
272#121—24no—
273#121C2×Q818yescomplemented
274C4×Q8—44no—
275C4×Q8—82no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—22no—
279C4×D4—44no—
280C4×D4—48no—
281C4×D4—84no—
282C4×D4C2×Q814yes—
283C4×D4C2×Q814yescomplemented
284C4×D4#4114yes—
285#117—24no—
286#117—28no—
287#117—44no—
288#117—44no—
289#117C2×Q818yescomplemented
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—44no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—24no—
357#107—24no—
358#107—44no—
359#106—28no—
360#106—44no—
361#106—44no—
362C4⋊C8—24no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365#102—24no—
366#102—24no—
367#102—44no—
368C22⋊C8—28no—
369C4×C8—24no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—24no—
382#95—24no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—21no—
387C24C22×D412yes—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—48no—
391C2×Q8—42no—
392C2×Q8—48no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—81no—
398C2×Q8—81no—
399C2×Q8—84no—
400C2×Q8—832no—
401C2×D4—28no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4C22⋊Q814yes—
410C2×D4C22×Q812yes—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—48no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4—84no—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×Q814yes—
446C22×C4C2×#4114yes—
447C22×C4C2511yescharacteristic
448M16—416no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C82−1+414yes—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—82no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4C22⋊Q814yes—
490C4⋊C4#13714yes—
491C4⋊C4C22×Q812yes—
492C4⋊C4C22×Q814yes—
493C4⋊C4H(5,𝔽2)14yes—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—48no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4C22⋊Q818yes—
508C42—28no—
509C42—22no—
510C42—22no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—42no—
515C42—42no—
516C42—41no—
517C42—42no—
518C42—44no—
519C42—82no—
520C42—82no—
521C42—82no—
522C42AGL(1,ℤ/8)14yes—
523C42C22×Q812yes—
524C42C2×#4112yes—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23C2×C22⋊Q814yes—
539C23C23×D411yescharacteristic
540Q8—48no—
541Q8—84no—
542Q8—88no—
543Q8—816no—
544D4—28no—
545D4—24no—
546D4—42no—
547D4—88no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—48no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—816no—
596C2×C4—164no—
597C2×C4C2×C22⋊Q814yes—
598C2×C4C2×C22⋊Q812yes—
599C2×C4C2×C22⋊Q812yes—
600C2×C4C2×AGL(1,ℤ/8)12yes—
601C2×C4C2×AGL(1,ℤ/8)12yes—
602C2×C4C2×AGL(1,ℤ/8)12yes—
603C2×C4C2×#13714yes—
604C2×C4C2×#13712yes—
605C2×C4#57712yes—
606C2×C4C23×Q811yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C2×H(5,𝔽2)12yes—
609C2×C4C2×2−1+412yes—
610C8—24no—
611C8—42no—
612C8—44no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22C22×C22⋊Q811yescharacteristic
628C22#348411yescharacteristic
629C22#349911yescharacteristic
630C22C22×AGL(1,ℤ/8)11yescharacteristic
631C22C22×#13711yescharacteristic
632C22C2×#57711yescharacteristic
633C22#358511yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—24no—
639C4—24no—
640C4—41no—
641C4—42no—
642C4—44no—
643C4—42no—
644C4—44no—
645C4—82no—
646C4—81no—
647C4—81no—
648C4—84no—
649C4—82no—
650C4#307914yes—
651C4#350312yes—
652C4#352012yes—
653C2—44no—
654C2—82no—
655C2#3573911yescharacteristic
656C2C2×#307911yescharacteristic
657C2#6049511yescharacteristic
658C2#6158211yescharacteristic
659C2#6159311yescharacteristic
660C2#6247711yescharacteristic
661C2#6251011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
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. D4×C4⋊Q8C2semidirect
3. #29980C2semidirect
4. #29971C2semidirect
5. #29273C2semidirect
6. #29206C2semidirect
7. #29163C2semidirect
8. #26446C2semidirect
9. #26359C2semidirect
10. #26044C2semidirect
11. #26007C2semidirect
12. #24299C2semidirect
13. #23774C2semidirect
14. #23552C2semidirect
15. #22700C2semidirect
16. #22663C2semidirect
98. #555Q8semidirect
267. C4⋊Q8C2×Q8semidirect
273. #121C2×Q8semidirect
283. C4×D4C2×Q8semidirect
289. #117C2×Q8semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#343212
#299802
#299712
#292732
#292062
#291632
#264462
#263592
#260442
#260072
#242992
#237742
#235522
#227002
#226632
#225032