Finite Group Databasebeta

← #6446781

Subgroups — #6446781

Order 512.

1758 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34948C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#26121C214yesmaximal, complemented
4#26120C214yesmaximal, complemented
5#26113C212yesmaximal, complemented
6#26112C212yesmaximal, complemented
7#26109C212yesmaximal, complemented
8#26108C212yesmaximal, complemented
9#23747C212yesmaximal
10#23670C212yesmaximal
11#21171C211yesmaximal, characteristic
12#21146C211yesmaximal, characteristic
13#20874C212yesmaximal
14#20764C211yesmaximal, characteristic
15#20647C211yesmaximal, characteristic
16#20647C212yesmaximal
17C4×#419C211yesmaximal, characteristic
18C22×C4×Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C2×#428C2212yes—
21C2×#423C2211yescharacteristic
22C2×#423C2212yes—
23C2×#419C2211yescharacteristic
24C2×#419C2212yes—
25C2×#419C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27#2571C2211yescharacteristic
28#2571C2211yescharacteristic
29#2570C2212yes—
30#2570C2212yes—
31#2569—22no—
32#2552C2211yescharacteristic
33#2552C2211yescharacteristic
34#2442—21no—
35#2442—21no—
36#2402—22no—
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4—22no—
40Q8×C4⋊C4C2212yes—
41Q8×C4⋊C4C2212yes—
42Q8×C4⋊C4C2214yes—
43C4×#125—21no—
44C4×#125—21no—
45#2296C2212yes—
46C2×#389—22no—
47C2×#389C2212yes—
48#2285C2211yescharacteristic
49#2285C2212yes—
50C2×#384—21no—
51C2×#384C2211yescharacteristic
52C4×#117C2211yescharacteristic
53C2×C4×C4⋊C4—21no—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2101C2214yes—
56#2101C2214yes—
57#2100C2214yes—
58#2099C2214yes—
59#2088C2214yes—
60#2088C2214yes—
61#2087C2214yes—
62#2086C2214yes—
63#1948C2212yes—
64#1948C2212yes—
65#1947C2212yes—
66#1947C2212yes—
67#1940C2214yes—
68#1939C2212yes—
69#1939C2212yes—
70#1865C2214yes—
71#1865C2214yes—
72#1863C2214yes—
73#1863C2214yes—
74#1836C2214yes—
75#1834C2212yes—
76#1834C2212yes—
77#1805C2214yes—
78#1805C2214yes—
79#1805C2214yes—
80#1803C2212yes—
81#1803C2214yes—
82#1803C2214yes—
83#1803C2212yes—
84#1763C2212yes—
85#1763C2212yes—
86C4×#103C2212yes—
87C4×#103C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1736C2211yescharacteristic
91#1736C2212yes—
92#1736C2212yes—
93#1736C2211yescharacteristic
94#1728C2212yes—
95C23×Q8C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C22×M16C2311yescharacteristic
98C22×M16C2312yes—
99C2×#125—21no—
100C2×#125—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—24no—
106C2×C4×Q8—28no—
107C2×C4×Q8—44no—
108C2×C4×Q8Q818yes—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C42—21no—
126C22×C42—22no—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#428—44no—
130C2×#107C2314yes—
131C2×#106C2314yes—
132#423—24no—
133#423—44no—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138#419—24no—
139#419—28no—
140#419—24no—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C4×M16—24no—
148C2×#97C2312yes—
149C2×C4×C8C2312yes—
150#400—22no—
151#400—22no—
152#400—22no—
153#400—22no—
154#398—22no—
155#398—22no—
156#398—22no—
157#398—22no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—24no—
165#389—24no—
166#389—24no—
167#389—24no—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2314yes—
173#389C2314yes—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#385—22no—
179#385—22no—
180#384—22no—
181#384—22no—
182#384—24no—
183#384—21no—
184#384—21no—
185#384C2312yes—
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—21no—
192#382—21no—
193#382C2312yes—
194#382C2312yes—
195#382C2312yes—
196#380—21no—
197#380—22no—
198#380—22no—
199#380—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—24no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221#376—21no—
222C2×#95—22no—
223C2×#95D412yes—
224C43—21no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#336C2314yes—
238#336C2314yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8C2×Q814yes—
244C22×Q8#4114yes—
245C22×Q8#4114yes—
246C22×Q8C2412yes—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×M16—22no—
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—24no—
260C2×M16—24no—
261C2×M16—22no—
262C2×M16—24no—
263C2×M16—22no—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C22×C8C2412yes—
267#125—44no—
268#125—44no—
269C4×Q8—28no—
270C4×Q8—28no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—28no—
274C4×Q8—28no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—48no—
278C4×Q8—48no—
279C4×Q8C2×Q818yes—
280C4×Q8C2×Q818yes—
281#117—22no—
282#117—22no—
283#117—24no—
284#117—22no—
285#117—22no—
286#117—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4—84no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×Q814yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C42—28no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—48no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×Q814yes—
359C2×C42#4114yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—44no—
365#106—44no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—44no—
369C4⋊C8—44no—
370#103—28no—
371#103—28no—
372#103—24no—
373#103—24no—
374#103—24no—
375#103—24no—
376#97—24no—
377C4×C8—24no—
378#95—24no—
379#95—24no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24C22×D411yescharacteristic
399C2×Q8—24no—
400C2×Q8—28no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—28no—
404C2×Q8—28no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8#125116yescomplemented
409C2×Q8C22×Q814yes—
410C2×Q8C2×#4114yes—
411C2×Q8C2×#4114yes—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4C22⋊Q814yes—
441C22×C4C22⋊Q814yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×Q812yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C42−1+412yes—
457C22×C4C2511yescharacteristic
458M16—44no—
459M16—44no—
460M16—48no—
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×C8—22no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4#13714yes—
515C4⋊C4C22×Q814yes—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C42—28no—
520C42—28no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—48no—
528C42—48no—
529C42—42no—
530C42—42no—
531C42—44no—
532C42—42no—
533C42—42no—
534C42—81no—
535C42—81no—
536C42C22⋊Q814yes—
537C42C22⋊Q814yes—
538C42C22×Q812yes—
539C42C22×Q812yes—
540C42C2×#4112yes—
541C42C2×#4112yes—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23C2×C22⋊Q812yes—
547C23#54612yes—
548C23#54812yes—
549C23C23×D411yescharacteristic
550Q8—24no—
551Q8—216no—
552Q8—24no—
553Q8—24no—
554Q8C2×#12518yes—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—216no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—84no—
605C2×C4—88no—
606C2×C4C2×C22⋊Q812yes—
607C2×C4C2×C22⋊Q812yes—
608C2×C4C2×C22⋊Q812yes—
609C2×C4C2×C22⋊Q812yes—
610C2×C4C2×C22⋊Q812yes—
611C2×C4C2×#12514yes—
612C2×C4#54414yes—
613C2×C4#54612yes—
614C2×C4#54812yes—
615C2×C4#55214yes—
616C2×C4#55314yes—
617C2×C4C2×#13714yes—
618C2×C4C2×#13712yes—
619C2×C4#57512yes—
620C2×C4C23×Q811yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C8—42no—
627C8—42no—
628C8—44no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22#244214yes—
636C22C22×C22⋊Q811yescharacteristic
637C22C2×#54611yescharacteristic
638C22C2×#54811yescharacteristic
639C22C22×#13711yescharacteristic
640C22C2×#57511yescharacteristic
641C22#358511yescharacteristic
642C22#358611yescharacteristic
643C4—22no—
644C4—22no—
645C4—22no—
646C4—28no—
647C4—24no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—81no—
653C4—82no—
654C4—81no—
655C4—82no—
656C4—81no—
657C4—81no—
658C4—84no—
659C4#244214yes—
660C4#307814yes—
661C4C22×#12511yescharacteristic
662C4C2×#54411yescharacteristic
663C4C2×#55211yescharacteristic
664C4C2×#55311yescharacteristic
665C2—24no—
666C2C2×#244211yescharacteristic
667C2C2×#307811yescharacteristic
668C2#6049211yescharacteristic
669C2#6115711yescharacteristic
670C2#6116011yescharacteristic
671C2#6131911yescharacteristic
672C2#6132011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #26121C2semidirect
4. #26120C2semidirect
5. #26113C2semidirect
6. #26112C2semidirect
7. #26109C2semidirect
8. #26108C2semidirect
408. C2×Q8#125semidirect

Maximal subgroups

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

subgroupindex
#349482
C2×Q8×C4⋊C42
#261212
#261202
#261132
#261122
#261092
#261082
#237472
#236702
#211712
#211462
#208742
#207642
#206472
C4×#4192