Finite Group Databasebeta

← #6852643

Subgroups — #6852643

Order 512.

1811 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#127C211yesmaximal, characteristic, complemented
2#34941C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic, complemented
4#32364C214yesmaximal, complemented
5#31538C214yesmaximal, complemented
6#29981C212yesmaximal, complemented
7#29256C212yesmaximal, complemented
8#29213C212yesmaximal, complemented
9#28687C211yesmaximal, characteristic, complemented
10#28541C212yesmaximal, complemented
11#28280C211yesmaximal, characteristic, complemented
12#27822C212yesmaximal, complemented
13#25834C212yesmaximal, complemented
14#24326C212yesmaximal, complemented
15#22825C211yesmaximal, characteristic, complemented
16#22151C212yesmaximal
17C4×#459C211yesmaximal, characteristic
18C2×D4×Q8—21no—
19C2×D4×Q8—21no—
20C2×D4×Q8C2212yes—
21C2×#498C2212yes—
22C2×#476C2212yes—
23C2×#474C2212yes—
24C2×#459C2211yescharacteristic
25C2×C4×D8C2212yes—
26#2868C2211yescharacteristic
27#2868C2212yes—
28#2574—22no—
29#2567C2211yescharacteristic
30#2563C2211yescharacteristic
31#2552C2211yescharacteristic
32#2542—22no—
33#2542—22no—
34#2542—21no—
35#2542—21no—
36#2542C2214yes—
37#2542C2212yes—
38#2538C2211yescharacteristic
39#2508C2212yes—
40#2497—21no—
41#2454C2212yes—
42#2423—21no—
43#2396—22no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2212yes—
46C4×#127C2211yescharacteristic
47C4×#125C2212yes—
48C4×C22⋊Q8—21no—
49#2293C2212yes—
50#2285—21no—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2100C2214yes—
54#2094C2214yes—
55#2092C2214yes—
56#2084C2214yes—
57#2065C2214yes—
58#2058C2214yes—
59#2058C2214yes—
60#2057C2214yes—
61#2025C2214yes—
62#2023C2214yes—
63#1992C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1982C2212yes—
69#1972C2214yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1947C2211yescharacteristic
74#1947C2212yes—
75#1947C2212yes—
76#1936C2214yes—
77#1925C2212yes—
78#1917C2214yes—
79#1879C2214yes—
80#1879C2212yes—
81#1864C2214yes—
82#1864C2214yes—
83#1862C2214yes—
84#1847C2214yes—
85#1842C2211yescharacteristic
86#1831C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C23×Q8—22no—
96C22×D8C2312yes—
97D4×Q8—44no—
98D4×Q8—44no—
99D4×Q8—48no—
100D4×Q8—48no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×#127C2311yescharacteristic
109C2×#125C2312yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—24no—
117C2×C4×Q8—21no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2314yes—
130C2×#117—22no—
131C22×C42—21no—
132#498—44no—
133#476—44no—
134#474—44no—
135#459Q818yescomplemented
136C4×D8—44no—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2311yescharacteristic
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#400—22no—
157#399—22no—
158#399—24no—
159#399—24no—
160#399—24no—
161#398—22no—
162#398—22no—
163#396—22no—
164#396—22no—
165#395—22no—
166#393—22no—
167#391—21no—
168#391—21no—
169#390—22no—
170#390—21no—
171#390—21no—
172#390—22no—
173#390C2312yes—
174#390C2314yes—
175#390C2312yes—
176#390C2312yes—
177#389—21no—
178#389—21no—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384—22no—
193#384—21no—
194#384—21no—
195#384—22no—
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2314yes—
201#382—22no—
202#382C2312yes—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C43C2312yes—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236#336C2314yes—
237C22×Q8—24no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—48no—
241C22×Q8—48no—
242C22×Q8—48no—
243C22×Q8C2×D411yescharacteristic
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2412yes—
247C22×D4C2411yescharacteristic
248C23×C4—21no—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C2×D8—42no—
254C2×D8—42no—
255C2×D8C2×Q818yescomplemented
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—48no—
266C4⋊Q8—44no—
267C4⋊Q8—48no—
268C4⋊Q8—44no—
269#127—22no—
270#127C2×Q814yes—
271#125—24no—
272#125—42no—
273C22⋊Q8—48no—
274C22⋊Q8—416no—
275C22⋊Q8—48no—
276C22⋊Q8—48no—
277C22⋊Q8—416no—
278C22⋊Q8—416no—
279C22⋊Q8—44no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—82no—
285C4×D4—22no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—41no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—48no—
295C4×D4—44no—
296C4×D4—48no—
297C4×D4—44no—
298C4×D4C2×Q814yescomplemented
299#117—84no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—44no—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—42no—
365#107—44no—
366#106—44no—
367#106—44no—
368#106—44no—
369C4⋊C8—24no—
370C4⋊C8—22no—
371C4⋊C8—24no—
372C4⋊C8—42no—
373C4⋊C8C2×Q814yescomplemented
374#102—24no—
375#102—44no—
376#102—42no—
377#102—42no—
378#102—44no—
379#102—44no—
380#102C2×Q818yescomplemented
381#97—44no—
382C4×C8—24no—
383#95—21no—
384#95—21no—
385#95—21no—
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—
397C24—21no—
398C24—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—48no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—81no—
409C2×Q8—81no—
410C2×Q8—832no—
411C2×Q8AGL(1,ℤ/8)14yes—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4C22×Q814yes—
429C2×D4C22×Q812yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—42no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—81no—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2511yescharacteristic
463D8—48no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8C22×Q814yes—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—81no—
510C4⋊C4—82no—
511C4⋊C4C22×Q812yes—
512C4⋊C42−1+412yes—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C42—24no—
527C42—22no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—22no—
535C42—22no—
536C42—21no—
537C42—41no—
538C42—42no—
539C42—42no—
540C42—42no—
541C42—42no—
542C42—44no—
543C42AGL(1,ℤ/8)14yes—
544C42C22×Q812yes—
545C42H(5,𝔽2)12yes—
546C42H(5,𝔽2)12yes—
547C422−1+412yes—
548C23—22no—
549C23—24no—
550C23—41no—
551C23—42no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23C23×D411yescharacteristic
563Q8—28no—
564Q8—816no—
565Q8—84no—
566Q8—88no—
567D4—22no—
568D4—24no—
569D4—22no—
570D4—24no—
571D4—48no—
572D4—48no—
573D4—41no—
574D4—42no—
575D4—41no—
576D4—42no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
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—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—84no—
620C2×C4—816no—
621C2×C4—84no—
622C2×C4—88no—
623C2×C4#53512yes—
624C2×C4#53512yes—
625C2×C4D4×Q812yes—
626C2×C4D4×Q812yes—
627C2×C4C2×AGL(1,ℤ/8)14yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4C2×AGL(1,ℤ/8)12yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4C2×AGL(1,ℤ/8)12yes—
632C2×C4#57712yes—
633C2×C4C23×Q811yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C2×C4#58511yescharacteristic
639C8—24no—
640C8—42no—
641C8—44no—
642C22—42no—
643C22—44no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—42no—
648C22—82no—
649C22—81no—
650C22—81no—
651C22—82no—
652C22—81no—
653C22—82no—
654C22—81no—
655C22—82no—
656C22C2×#53511yescharacteristic
657C22C2×D4×Q811yescharacteristic
658C22#348911yescharacteristic
659C22C22×AGL(1,ℤ/8)11yescharacteristic
660C22C22×AGL(1,ℤ/8)11yescharacteristic
661C22C2×#57711yescharacteristic
662C22#358611yescharacteristic
663C4—24no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—41no—
672C4—42no—
673C4—81no—
674C4—84no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4#321714yes—
679C4#338414yes—
680C4#350111yescharacteristic
681C4#350311yescharacteristic
682C4#355011yescharacteristic
683C4#355411yescharacteristic
684C2—82no—
685C2—84no—
686C2#3959611yescharacteristic
687C2C2×#321711yescharacteristic
688C2#6083611yescharacteristic
689C2C2×#338411yescharacteristic
690C2Q8×AGL(1,ℤ/8)11yescharacteristic
691C2#6210711yescharacteristic
692C2#6215011yescharacteristic
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. Q8×#127C2semidirect
2. #34941C2semidirect
3. #33195C2semidirect
4. #32364C2semidirect
5. #31538C2semidirect
6. #29981C2semidirect
7. #29256C2semidirect
8. #29213C2semidirect
9. #28687C2semidirect
10. #28541C2semidirect
11. #28280C2semidirect
12. #27822C2semidirect
13. #25834C2semidirect
14. #24326C2semidirect
15. #22825C2semidirect
135. #459Q8semidirect
255. C2×D8C2×Q8semidirect
298. C4×D4C2×Q8semidirect
373. C4⋊C8C2×Q8semidirect
380. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×#1272
#349412
#331952
#323642
#315382
#299812
#292562
#292132
#286872
#285412
#282802
#278222
#258342
#243262
#228252
#221512
C4×#4592