Finite Group Databasebeta

← #6446232

Subgroups — #6446232

Order 512.

1776 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34953C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#26290C214yesmaximal, complemented
4#26288C214yesmaximal, complemented
5#26282C212yesmaximal, complemented
6#26280C212yesmaximal, complemented
7#26278C212yesmaximal, complemented
8#26276C212yesmaximal, complemented
9#23745C212yesmaximal
10#23658C212yesmaximal
11#21322C211yesmaximal, characteristic
12#21312C211yesmaximal, characteristic
13#21303C211yesmaximal, characteristic
14#21171C211yesmaximal, characteristic
15#21033C212yesmaximal
16#21033C212yesmaximal
17C4×#428C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4×Q8—21no—
20C22×C4×Q8C2211yescharacteristic
21C2×#428C2212yes—
22C2×#428C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26#2584C2212yes—
27#2584C2212yes—
28#2583—22no—
29#2567C2211yescharacteristic
30#2567C2211yescharacteristic
31#2546C2211yescharacteristic
32#2546C2211yescharacteristic
33#2502—21no—
34#2434—21no—
35#2399—22no—
36Q8×C4⋊C4—21no—
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4—22no—
39Q8×C4⋊C4C2212yes—
40Q8×C4⋊C4C2212yes—
41Q8×C4⋊C4C2214yes—
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8—21no—
44#2296C2212yes—
45#2296C2212yes—
46C2×#389—22no—
47C2×#389—21no—
48C2×#384C2211yescharacteristic
49C2×#384C2212yes—
50C4×#117C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2100C2214yes—
54#2100C2214yes—
55#2099C2214yes—
56#2099C2214yes—
57#2084C2214yes—
58#2084C2214yes—
59#2083C2214yes—
60#2083C2214yes—
61#1992C2214yes—
62#1990C2212yes—
63#1990C2212yes—
64#1989C2214yes—
65#1988C2212yes—
66#1987C2212yes—
67#1985C2214yes—
68#1984C2212yes—
69#1984C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1921C2214yes—
75#1921C2214yes—
76#1920C2214yes—
77#1920C2214yes—
78#1919C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1918C2214yes—
82#1776C2212yes—
83#1776C2212yes—
84C4×#107C2212yes—
85C4×#106C2212yes—
86#1736C2212yes—
87#1736C2211yescharacteristic
88#1736C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92#1736C2211yescharacteristic
93#1728C2212yes—
94#1728C2212yes—
95C23×Q8D411yescharacteristic
96C22×M16C2311yescharacteristic
97C22×M16C2312yes—
98C22×M16C2311yescharacteristic
99#533—24no—
100#533—24no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125C2312yes—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C4×Q8—24no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—24no—
114C2×C4×Q8—24no—
115C2×C4×Q8—48no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D412yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D412yes—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C42—22no—
135C22×C42—21no—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138#428—24no—
139#428—28no—
140#428Q818yescomplemented
141C2×#107C2312yes—
142C2×#107C2314yes—
143C2×#107C2312yes—
144C2×#106C2312yes—
145C2×#106C2314yes—
146C2×#106C2312yes—
147#423—24no—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C4×M16—24no—
151C2×#97C2312yes—
152C2×C4×C8C2312yes—
153#398—22no—
154#398—22no—
155#398—22no—
156#398—22no—
157#395—22no—
158#395—22no—
159#395—22no—
160#395—22no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—24no—
172#389—24no—
173#389—24no—
174#389—24no—
175#389—24no—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#385—22no—
185#385—22no—
186#385—22no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384—22no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384C2314yes—
195#384C2314yes—
196#384C2312yes—
197#384C2312yes—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216#376—21no—
217C2×#95—22no—
218C2×#95—21no—
219C43—21no—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238C22×Q8—24no—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8C2×D412yes—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2×D411yescharacteristic
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×M16—22no—
255C2×M16—22no—
256C2×M16—22no—
257C2×M16—24no—
258C2×M16—24no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—24no—
262C2×M16C2×Q814yes—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272#125—24no—
273#125—24no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—28no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—416no—
286C4×Q8—48no—
287C4×Q8—48no—
288C4×Q8—44no—
289C4×Q8—44no—
290#117—22no—
291#117—22no—
292#117—24no—
293#117—24no—
294#117C2×Q814yes—
295#117#4114yes—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
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—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—48no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4—84no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×Q814yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—48no—
364C2×C42—44no—
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42#4114yes—
370C2×C42#4114yes—
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—24no—
376#107—28no—
377#107C2×Q818yescomplemented
378#106—24no—
379#106—28no—
380#106C2×Q818yescomplemented
381C4⋊C8—24no—
382C4⋊C8—24no—
383#97—24no—
384C4×C8—24no—
385#95—24no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401C24C22×D411yescharacteristic
402C2×Q8—232no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—81no—
415C2×Q8—81no—
416C2×Q8AGL(1,ℤ/8)14yes—
417C2×Q8#13714yes—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4#12314yes—
447C22×C4#12314yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×Q812yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C42−1+412yes—
462C22×C42−1+412yes—
463C22×C4C2511yescharacteristic
464M16—28no—
465M16—44no—
466M16—48no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8C22×Q814yes—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4C22×Q812yes—
520C4⋊C4C22×Q814yes—
521C4⋊C4C22×Q812yes—
522C4⋊C4C2×#4112yes—
523C4⋊C4C2×#4112yes—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—44no—
527C42—24no—
528C42—24no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—44no—
536C42—48no—
537C42—44no—
538C42—48no—
539C42—44no—
540C42—42no—
541C42—42no—
542C42—81no—
543C42—81no—
544C42#12314yes—
545C42#12314yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C42C2×#4112yes—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23C2×#12312yes—
555C23#54812yes—
556C23D4×Q812yes—
557C23C23×D411yescharacteristic
558Q8—216no—
559Q8—28no—
560Q8—28no—
561Q8—84no—
562Q8—84no—
563C2×C4—216no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—88no—
612C2×C4—84no—
613C2×C4C2×#12312yes—
614C2×C4C2×#12312yes—
615C2×C4C2×#12312yes—
616C2×C4C2×#12312yes—
617C2×C4C2×#12312yes—
618C2×C4#53314yes—
619C2×C4#54114yes—
620C2×C4#54812yes—
621C2×C4D4×Q812yes—
622C2×C4#55614yes—
623C2×C4#55714yes—
624C2×C4C2×AGL(1,ℤ/8)14yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×#13714yes—
627C2×C4C2×#13712yes—
628C2×C4C23×Q811yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×2−1+411yescharacteristic
633C2×C4C2×2−1+411yescharacteristic
634C8—24no—
635C8—42no—
636C8—44no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22#244614yes—
644C22C22×#12311yescharacteristic
645C22C2×#54811yescharacteristic
646C22C2×D4×Q811yescharacteristic
647C22C22×AGL(1,ℤ/8)11yescharacteristic
648C22C22×#13711yescharacteristic
649C22#358511yescharacteristic
650C22#358611yescharacteristic
651C4—22no—
652C4—24no—
653C4—24no—
654C4—22no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—82no—
664C4—82no—
665C4—84no—
666C4—82no—
667C4#244614yes—
668C4#309214yes—
669C4C2×#53311yescharacteristic
670C4C2×#54111yescharacteristic
671C4C2×#55611yescharacteristic
672C4C2×#55711yescharacteristic
673C2—24no—
674C2C2×#244611yescharacteristic
675C2C2×#309211yescharacteristic
676C2#6051711yescharacteristic
677C2#6132111yescharacteristic
678C2#6132211yescharacteristic
679C2Q8×AGL(1,ℤ/8)11yescharacteristic
680C2Q8×#13711yescharacteristic
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
3. #26290C2semidirect
4. #26288C2semidirect
5. #26282C2semidirect
6. #26280C2semidirect
7. #26278C2semidirect
8. #26276C2semidirect
140. #428Q8semidirect
377. #107C2×Q8semidirect
380. #106C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#349532
C2×Q8×C4⋊C42
#262902
#262882
#262822
#262802
#262782
#262762
#237452
#236582
#213222
#213122
#213032
#211712
#210332
C4×#4282