Finite Group Databasebeta

← #6852098

Subgroups — #6852098

Order 512.

1701 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊Q8C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic
4#32683C214yesmaximal, complemented
5#32566C214yesmaximal, complemented
6#29968C212yesmaximal
7#29297C212yesmaximal
8#29240C212yesmaximal
9#28677C212yesmaximal, complemented
10#26278C212yesmaximal, complemented
11#25837C212yesmaximal, complemented
12#25811C211yesmaximal, characteristic
13#25472C212yesmaximal, complemented
14#24301C211yesmaximal, characteristic
15Q8×C22⋊C8C211yesmaximal, characteristic
16#22078C212yesmaximal
17C4×#483C211yesmaximal, characteristic
18C2×Q82C2211yescharacteristic
19C22×C4⋊Q8—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C2×#498C2212yes—
23C2×#483C2211yescharacteristic
24C2×#475C2212yes—
25C2×C8×Q8C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2582C2212yes—
29#2582C2211yescharacteristic
30#2567C2211yescharacteristic
31#2567C2212yes—
32#2546—21no—
33#2546—21no—
34#2546—22no—
35#2546C2212yes—
36#2546C2214yes—
37#2546C2212yes—
38#2451—21no—
39#2451—22no—
40#2440—22no—
41#2440—21no—
42C2×#398—22no—
43Q8×C4⋊C4—21no—
44Q8×C4⋊C4C2211yescharacteristic
45Q8×C22⋊C4—21no—
46C4×C22⋊Q8C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48C2×#391—21no—
49C2×#389C2212yes—
50C2×#389C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2214yes—
53#2088C2214yes—
54#2088C2214yes—
55#2088C2214yes—
56#2084C2214yes—
57#2066C2214yes—
58#2058C2214yes—
59#2058C2214yes—
60#2058C2214yes—
61#2058C2212yes—
62#2024C2212yes—
63#1990C2212yes—
64#1988C2212yes—
65#1987C2212yes—
66#1984C2214yes—
67#1984C2212yes—
68#1948C2212yes—
69#1941C2214yes—
70#1941C2214yes—
71#1941C2214yes—
72#1933C2214yes—
73#1927C2212yes—
74#1920C2214yes—
75#1920C2212yes—
76#1919C2214yes—
77#1919C2214yes—
78#1872C2211yescharacteristic
79#1872C2212yes—
80#1865C2214yes—
81#1865C2214yes—
82#1853C2212yes—
83#1853C2212yes—
84#1853C2212yes—
85#1853C2212yes—
86#1853C2211yescharacteristic
87#1823C2211yescharacteristic
88#1823C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8D411yescharacteristic
97Q82—44no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8—48no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2311yescharacteristic
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—44no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D412yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4D412yes—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#498—44no—
137#483Q818yescomplemented
138#475—44no—
139C8×Q8—44no—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C4×C8C2312yes—
155C2×C4×C8C2314yes—
156#398—22no—
157#398—22no—
158#398—22no—
159#398—24no—
160#398—24no—
161#398—24no—
162#398—24no—
163#395—22no—
164#395—22no—
165#395—22no—
166#391—21no—
167#391—22no—
168#391—21no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391—24no—
173#391—21no—
174#391—22no—
175#391C2312yes—
176#391C2311yescharacteristic
177#389—21no—
178#389—22no—
179#389—21no—
180#389—22no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389C2311yescharacteristic
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2314yes—
193#389C2314yes—
194#389C2314yes—
195#389C2314yes—
196#387—21no—
197#387—22no—
198#385—21no—
199#385—22no—
200#384—21no—
201#384—22no—
202#384C2312yes—
203#384C2311yescharacteristic
204#380—21no—
205#380—22no—
206#380C2311yescharacteristic
207#380C2312yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C2×#95—21no—
219C2×#95—22no—
220C2×#95—22no—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8SD1618yes—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2×D412yes—
249C22×Q8C2411yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269C4⋊Q8—44no—
270C4⋊Q8—48no—
271C4⋊Q8—48no—
272C4⋊Q8—416no—
273C22⋊Q8—22no—
274C22⋊Q8—82no—
275C22⋊Q8—84no—
276C22⋊Q8C2×Q814yes—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—42no—
283C4×Q8—42no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×Q8—44no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×Q814yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42—44no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354#106—24no—
355#106—28no—
356#106—44no—
357#106—44no—
358#106—44no—
359#106C2×Q818yescomplemented
360C4⋊C8—24no—
361C4⋊C8—44no—
362#103—24no—
363#103—44no—
364#103C2×Q818yescomplemented
365C22⋊C8—22no—
366C22⋊C8—24no—
367C22⋊C8C2×Q814yes—
368C4×C8—24no—
369C4×C8—44no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—21no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389C24C22×D411yescharacteristic
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—416no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—81no—
409C2×Q8—82no—
410C2×Q8—81no—
411C2×Q8—82no—
412C2×Q8—816no—
413C2×Q8C2×SD1614yes—
414C2×Q8C2×SD1614yes—
415C2×Q8C2×SD1618yes—
416C2×Q8C22×Q812yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4C2×SD1614yes—
448C22×C4C2×SD1618yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×Q812yes—
459C22×C42−1+412yes—
460C22×C4C2511yescharacteristic
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8C22×Q814yes—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—42no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4C22×Q814yes—
510C4⋊C4C22×Q812yes—
511C4⋊C42−1+414yes—
512C22⋊C4—24no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C42—22no—
516C42—22no—
517C42—24no—
518C42—22no—
519C42—24no—
520C42—42no—
521C42—42no—
522C42—44no—
523C42—44no—
524C42—42no—
525C42—44no—
526C42—44no—
527C42—42no—
528C42H(5,𝔽2)14yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23D4×Q812yes—
534C23C22×SD1612yes—
535C23#57712yes—
536C23C23×D411yescharacteristic
537Q8—28no—
538Q8—28no—
539Q8—24no—
540Q8—48no—
541Q8—42no—
542Q8—88no—
543Q8—88no—
544Q8—162no—
545Q8—164no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—48no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—88no—
601C2×C4#53412yes—
602C2×C4#53412yes—
603C2×C4D4×Q812yes—
604C2×C4C22×SD1612yes—
605C2×C4C22×SD1614yes—
606C2×C4C22×SD1614yes—
607C2×C4C22×SD1612yes—
608C2×C4#57512yes—
609C2×C4C23×Q811yescharacteristic
610C2×C4C2×H(5,𝔽2)12yes—
611C2×C4C2×2−1+412yes—
612C2×C4C2×2−1+411yescharacteristic
613C8—24no—
614C8—42no—
615C8—44no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22Q8×SD1614yes—
623C22C2×#53411yescharacteristic
624C22C2×D4×Q811yescharacteristic
625C22#348611yescharacteristic
626C22C23×SD1611yescharacteristic
627C22C2×#57511yescharacteristic
628C22C2×#57711yescharacteristic
629C22#358611yescharacteristic
630C4—22no—
631C4—24no—
632C4—24no—
633C4—22no—
634C4—22no—
635C4—41no—
636C4—42no—
637C4—42no—
638C4—44no—
639C4—44no—
640C4—42no—
641C4—42no—
642C4—84no—
643C4—82no—
644C4—81no—
645C4—82no—
646C4—81no—
647C4#318714yes—
648C4#350312yes—
649C4#355212yes—
650C2—24no—
651C2#3959311yescharacteristic
652C2C2×#318711yescharacteristic
653C2#6075811yescharacteristic
654C2C2×Q8×SD1611yescharacteristic
655C2#6136911yescharacteristic
656C2#6181811yescharacteristic
657C2#6183611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32683C2semidirect
5. #32566C2semidirect
9. #28677C2semidirect
10. #26278C2semidirect
11. #25837C2semidirect
13. #25472C2semidirect
137. #483Q8semidirect
359. #106C2×Q8semidirect
364. #103C2×Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊Q82
C2×#25462
#331482
#326832
#325662
#299682
#292972
#292402
#286772
#262782
#258372
#258112
#254722
#243012
Q8×C22⋊C82
#220782
C4×#4832