Finite Group Databasebeta

← #6627610

Subgroups — #6627610

Order 512.

1837 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2C4×D4×Q8C211yesmaximal, characteristic, complemented
3#33197C211yesmaximal, characteristic, complemented
4#33030C212yesmaximal
5#29354C212yesmaximal, complemented
6#29277C212yesmaximal, complemented
7#29218C211yesmaximal, characteristic, complemented
8#28279C211yesmaximal, characteristic, complemented
9#27571C214yesmaximal, complemented
10#27143C214yesmaximal, complemented
11#24992C212yesmaximal, complemented
12#24326C212yesmaximal, complemented
13#23066C212yesmaximal, complemented
14#22825C211yesmaximal, characteristic, complemented
15#22758C212yesmaximal, complemented
16#21614C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556—21no—
19C2×D4×Q8C2211yescharacteristic
20C22×C4×Q8—21no—
21C2×#476C2212yes—
22C2×#474C2212yes—
23C2×#459C2211yescharacteristic
24C2×C4×D8C2212yes—
25#2868C2211yescharacteristic
26#2858C2212yes—
27#2547—21no—
28#2545—22no—
29#2538—21no—
30#2538C2212yes—
31#2512—22no—
32#2512C2214yes—
33#2365C2212yes—
34#2365C2211yescharacteristic
35#2360—22no—
36#2360—21no—
37#2354—21no—
38#2354C2212yes—
39Q8×C4⋊C4C2211yescharacteristic
40D4×C4⋊C4C2212yes—
41D4×C4⋊C4C2211yescharacteristic
42Q8×C22⋊C4—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×#127C2211yescharacteristic
47C4×C22⋊Q8—21no—
48C4×C22⋊Q8—22no—
49C42×Q8C2211yescharacteristic
50C42×D4C2212yes—
51C42×D4C2211yescharacteristic
52#2100C2214yes—
53#2100C2214yes—
54#2071C2214yes—
55#2067C2214yes—
56#2057C2214yes—
57#2023C2214yes—
58#1990C2212yes—
59#1987C2212yes—
60#1987C2212yes—
61#1987C2212yes—
62#1984C2212yes—
63#1982C2212yes—
64#1982C2211yescharacteristic
65#1948C2212yes—
66#1948C2212yes—
67#1947C2211yescharacteristic
68#1947C2212yes—
69#1936C2214yes—
70#1936C2214yes—
71#1926C2214yes—
72#1926C2214yes—
73#1925C2212yes—
74#1921C2214yes—
75#1921C2214yes—
76#1919C2214yes—
77#1917C2214yes—
78#1879C2214yes—
79#1879C2214yes—
80#1879C2212yes—
81#1862C2214yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1831C2212yes—
85#1802C2212yes—
86#1802C2211yescharacteristic
87#1802C2212yes—
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1762C2214yes—
94C4×#102C2212yes—
95C23×Q8—21no—
96C22×D8C2312yes—
97#556—48no—
98D4×Q8—24no—
99D4×Q8—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#126—24no—
107C2×#124—22no—
108C2×#124—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—48no—
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2312yes—
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×#117—22no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—22no—
133C22×C42—22no—
134C22×C42—21no—
135#476—28no—
136#474—28no—
137#459—24no—
138C4×D8—28no—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#397—24no—
157#396—24no—
158#396—24no—
159#391—21no—
160#391—22no—
161#391—21no—
162#391—22no—
163#391—22no—
164#390—21no—
165#390—21no—
166#390—22no—
167#390—22no—
168#390C2312yes—
169#390C2314yes—
170#390C2314yes—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—21no—
175#389—22no—
176#389C2314yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#388—24no—
181#388—24no—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#384C2312yes—
195#380—21no—
196#380—22no—
197#380—22no—
198#380—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C43C2311yescharacteristic
222C43C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—28no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8C2×D411yescharacteristic
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2412yes—
246C22×D4C2411yescharacteristic
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—21no—
253C2×D8—24no—
254C2×D8—24no—
255C2×D8—24no—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265C4⋊Q8—82no—
266C4⋊Q8#4114yes—
267#127—22no—
268#127—22no—
269#126—416no—
270#124—48no—
271#124—84no—
272C22⋊Q8—28no—
273C22⋊Q8—216no—
274C22⋊Q8—48no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—48no—
280C4×Q8—48no—
281C4×Q8—44no—
282C4×Q8—48no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—22no—
293C4×D4—42no—
294C4×D4—41no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4#4114yes—
298C4×D4#4114yes—
299#117—48no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—48no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—24no—
367#107—28no—
368#106—28no—
369C4⋊C8—22no—
370C4⋊C8—24no—
371C4⋊C8—24no—
372C4⋊C8—22no—
373C4⋊C8—42no—
374#102—24no—
375#102—28no—
376#102—24no—
377#102—24no—
378#102—28no—
379#102—42no—
380#102—44no—
381#97—44no—
382C4×C8—24no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—21no—
398C24—22no—
399C2×Q8—28no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—432no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8AGL(1,ℤ/8)14yes—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C2×#4112yes—
428C2×D4C2×#4112yes—
429C2×D42−1+412yes—
430C22×C4—28no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—81no—
454C22×C4—82no—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2511yescharacteristic
466D8—48no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C4⋊C4—28no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4C2×#4112yes—
511C4⋊C4C2×#4114yes—
512C4⋊C4C2×#4112yes—
513C22⋊C4—28no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—48no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C42—24no—
526C42—22no—
527C42—24no—
528C42—22no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—21no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—42no—
537C42—41no—
538C42—44no—
539C42—44no—
540C42—48no—
541C42—42no—
542C42—42no—
543C42—44no—
544C42—42no—
545C42AGL(1,ℤ/8)14yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C422−1+412yes—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23C23×D411yescharacteristic
565Q8—28no—
566Q8—416no—
567Q8—48no—
568Q8—88no—
569D4—22no—
570D4—22no—
571D4—22no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—84no—
577D4—84no—
578D4#55614yescomplemented
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—416no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—84no—
624C2×C4—88no—
625C2×C4—88no—
626C2×C4#52512yes—
627C2×C4#52512yes—
628C2×C4#54812yes—
629C2×C4#54812yes—
630C2×C4C2×AGL(1,ℤ/8)14yes—
631C2×C4C2×AGL(1,ℤ/8)12yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4#57812yes—
636C2×C4C22×#4111yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×2−1+411yescharacteristic
640C2×C4C2×2−1+411yescharacteristic
641C2×C4#58511yescharacteristic
642C8—42no—
643C8—44no—
644C8—42no—
645C22—24no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22C2×#52511yescharacteristic
660C22C2×#54811yescharacteristic
661C22#348911yescharacteristic
662C22C22×AGL(1,ℤ/8)11yescharacteristic
663C22C22×AGL(1,ℤ/8)11yescharacteristic
664C22C2×#57811yescharacteristic
665C22#358511yescharacteristic
666C4—24no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—42no—
671C4—44no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—41no—
677C4—82no—
678C4—84no—
679C4—81no—
680C4—82no—
681C4#310914yes—
682C4#336714yes—
683C4C2×#55611yescharacteristic
684C4#348011yescharacteristic
685C4#350711yescharacteristic
686C4#352311yescharacteristic
687C2—44no—
688C2—162no—
689C2#3657911yescharacteristic
690C2C2×#310911yescharacteristic
691C2#6060011yescharacteristic
692C2C2×#336711yescharacteristic
693C2#6132111yescharacteristic
694C2#6211011yescharacteristic
695C2#6214811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36594C2semidirect
2. C4×D4×Q8C2semidirect
3. #33197C2semidirect
5. #29354C2semidirect
6. #29277C2semidirect
7. #29218C2semidirect
8. #28279C2semidirect
9. #27571C2semidirect
10. #27143C2semidirect
11. #24992C2semidirect
12. #24326C2semidirect
13. #23066C2semidirect
14. #22825C2semidirect
15. #22758C2semidirect
16. #21614C2semidirect
578. D4#556semidirect

Maximal subgroups

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

subgroupindex
#365942
C4×D4×Q82
#331972
#330302
#293542
#292772
#292182
#282792
#275712
#271432
#249922
#243262
#230662
#228252
#227582
#216142
C4×#4592