Finite Group Databasebeta

← #6793565

Subgroups — #6793565

Order 512.

1671 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34943C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#31450C214yesmaximal, complemented
5#31374C214yesmaximal, complemented
6#29942C212yesmaximal, complemented
7#29290C211yesmaximal, characteristic, complemented
8#29218C211yesmaximal, characteristic, complemented
9#28799C212yesmaximal, complemented
10#27821C212yesmaximal, complemented
11#26300C212yesmaximal, complemented
12#24965C212yesmaximal
13#24708C212yesmaximal, complemented
14#24299C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic, complemented
16#21840C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#476C2211yescharacteristic
21C2×#476C2212yes—
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2572—22no—
26#2571C2211yescharacteristic
27#2563C2211yescharacteristic
28#2563C2211yescharacteristic
29#2551—21no—
30#2551C2212yes—
31#2546—21no—
32#2537C2211yescharacteristic
33#2495—21no—
34#2495C2212yes—
35#2440—22no—
36#2440C2214yes—
37#2434—22no—
38#2420—21no—
39#2420C2212yes—
40#2398—22no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#121—21no—
46C4×#121C2212yes—
47#2283—21no—
48#2283C2212yes—
49C42×D4C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2100C2214yes—
53#2087C2214yes—
54#2085C2214yes—
55#2077C2214yes—
56#2077C2214yes—
57#2075C2214yes—
58#2058C2214yes—
59#2033C2214yes—
60#2023C2214yes—
61#1993C2214yes—
62#1993C2214yes—
63#1990C2212yes—
64#1987C2212yes—
65#1987C2212yes—
66#1987C2211yescharacteristic
67#1987C2212yes—
68#1987C2212yes—
69#1987C2212yes—
70#1954C2212yes—
71#1948C2211yescharacteristic
72#1948C2212yes—
73#1948C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1943C2214yes—
77#1936C2214yes—
78#1924C2214yes—
79#1920C2212yes—
80#1874C2214yes—
81#1864C2212yes—
82#1853C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1823C2214yes—
86#1823C2214yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1802C2212yes—
90#1797C2212yes—
91C4×#106C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C23×Q8—22no—
96D4×Q8—48no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—24no—
110C2×#121—21no—
111C2×#121C2312yes—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C22×C4⋊C4—22no—
123C22×C4⋊C4C2314yes—
124C22×C42—21no—
125C22×C42C2312yes—
126#476—24no—
127#476—44no—
128#432—44no—
129C2×#107C2314yes—
130C2×#106C2312yes—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×#97C2314yes—
144C2×C4×C8C2312yes—
145#400—22no—
146#400—22no—
147#398—24no—
148#397—22no—
149#397—22no—
150#395—24no—
151#395—24no—
152#394—22no—
153#394—22no—
154#393—22no—
155#393—22no—
156#391—21no—
157#391—21no—
158#391—22no—
159#390—21no—
160#390C2312yes—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#388—22no—
172#388—22no—
173#386—22no—
174#386—22no—
175#385—24no—
176#385—24no—
177#384—21no—
178#384—21no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#383—22no—
184#383—22no—
185#382—21no—
186#382—21no—
187#382C2312yes—
188#382C2312yes—
189#381—22no—
190#381—22no—
191#380—22no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380C2312yes—
196#380C2312yes—
197#380C2312yes—
198#380C2314yes—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4—22no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217#376—22no—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—48no—
239C22×Q8—48no—
240C22×Q8—84no—
241C22×D4—22no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2411yescharacteristic
244C23×C4—24no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C4⋊Q8—28no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—44no—
261C4⋊Q8—48no—
262C4⋊Q8—48no—
263C4⋊Q8—44no—
264C4⋊Q8—48no—
265C4⋊Q8—82no—
266C4⋊Q8#4114yes—
267C22⋊Q8—416no—
268C22⋊Q8—48no—
269#121—24no—
270#121—42no—
271#121—44no—
272C4×Q8—44no—
273C4×Q8—82no—
274C4×D4—24no—
275C4×D4—22no—
276C4×D4—24no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—41no—
284C4×D4—48no—
285C4×D4—84no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—44no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—44no—
349#106—24no—
350#106—42no—
351#106—44no—
352C4⋊C8—22no—
353C4⋊C8—22no—
354C4⋊C8—24no—
355C4⋊C8—24no—
356C4⋊C8—42no—
357#102—24no—
358#102—24no—
359#102—44no—
360C22⋊C8—44no—
361C22⋊C8—44no—
362#97—44no—
363C4×C8—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—21no—
385C24C22×D412yes—
386C2×Q8—24no—
387C2×Q8—48no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—48no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—81no—
396C2×Q8—81no—
397C2×Q8—84no—
398C2×Q8—832no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—82no—
409C2×D4AGL(1,ℤ/8)14yes—
410C2×D42−1+412yes—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—42no—
433C22×C4—48no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—84no—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2511yescharacteristic
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—24no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—82no—
493C4⋊C4—81no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4C2×#4114yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—21no—
519C42—41no—
520C42—44no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42#13714yes—
526C42C2×#4112yes—
527C42H(5,𝔽2)12yes—
528C422−1+412yes—
529C422−1+412yes—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23C2×AGL(1,ℤ/8)14yes—
544C23C23×D411yescharacteristic
545Q8—48no—
546Q8—88no—
547Q8—84no—
548Q8—816no—
549D4—28no—
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554D4—88no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—84no—
604C2×C4—816no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4#53612yes—
608C2×C4#53612yes—
609C2×C4#54612yes—
610C2×C4#54612yes—
611C2×C4C2×AGL(1,ℤ/8)12yes—
612C2×C4C2×#13712yes—
613C2×C4C2×#13712yes—
614C2×C4C2×#13712yes—
615C2×C4#57712yes—
616C2×C4C22×#4111yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C2×C4C2×2−1+411yescharacteristic
621C2×C4#58511yescharacteristic
622C8—42no—
623C8—42no—
624C8—44no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22C2×#53611yescharacteristic
640C22C2×#54611yescharacteristic
641C22#348911yescharacteristic
642C22C22×AGL(1,ℤ/8)11yescharacteristic
643C22C22×#13711yescharacteristic
644C22C2×#57711yescharacteristic
645C22#358511yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—41no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—42no—
654C4—81no—
655C4—82no—
656C4—82no—
657C4—81no—
658C4—82no—
659C4—81no—
660C4—84no—
661C4#323814yes—
662C4#332914yes—
663C4#351811yescharacteristic
664C4#352311yescharacteristic
665C4#355211yescharacteristic
666C4#355411yescharacteristic
667C2—44no—
668C2—82no—
669C2#3891111yescharacteristic
670C2C2×#323811yescharacteristic
671C2#6089211yescharacteristic
672C2C2×#332911yescharacteristic
673C2#6115511yescharacteristic
674C2#6210811yescharacteristic
675C2#6212811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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. #33197C2semidirect
4. #31450C2semidirect
5. #31374C2semidirect
6. #29942C2semidirect
7. #29290C2semidirect
8. #29218C2semidirect
9. #28799C2semidirect
10. #27821C2semidirect
11. #26300C2semidirect
13. #24708C2semidirect
14. #24299C2semidirect
15. #22799C2semidirect
16. #21840C2semidirect
17. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#349432
#331972
#314502
#313742
#299422
#292902
#292182
#287992
#278212
#263002
#249652
#247082
#242992
#227992
#218402
C4×#4762