Finite Group Databasebeta

← #6852157

Subgroups — #6852157

Order 512.

1763 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#121C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#33146C211yesmaximal, characteristic, complemented
4#32694C214yesmaximal, complemented
5#32593C214yesmaximal, complemented
6#29968C212yesmaximal, complemented
7#29301C212yesmaximal, complemented
8#29238C212yesmaximal
9#28684C212yesmaximal, complemented
10#26282C212yesmaximal, complemented
11#25834C212yesmaximal, complemented
12#25811C211yesmaximal, characteristic, complemented
13#25477C212yesmaximal, complemented
14#24299C211yesmaximal, characteristic
15#22510C211yesmaximal, characteristic, complemented
16#22081C212yesmaximal
17C4×#480C211yesmaximal, characteristic
18C2×D4×Q8—21no—
19C2×D4×Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C2×#498C2212yes—
23C2×#480C2211yescharacteristic
24C2×#476C2212yes—
25C2×#446C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2567—21no—
29#2567—21no—
30#2567—22no—
31#2567C2212yes—
32#2567C2214yes—
33#2567C2212yes—
34#2546C2211yescharacteristic
35#2546C2212yes—
36#2538C2212yes—
37#2538C2211yescharacteristic
38#2429—21no—
39#2429—22no—
40#2417—22no—
41#2417—21no—
42C2×#395—22no—
43Q8×C4⋊C4C2211yescharacteristic
44Q8×C22⋊C4—21no—
45C4×#121C2212yes—
46C4×#121C2211yescharacteristic
47C2×#391—21no—
48C2×#389C2212yes—
49C2×#389C2212yes—
50C2×#384—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2214yes—
53#2088C2214yes—
54#2088C2214yes—
55#2088C2214yes—
56#2083C2214yes—
57#2067C2214yes—
58#2058C2214yes—
59#2058C2214yes—
60#2058C2214yes—
61#2058C2212yes—
62#2023C2212yes—
63#1990C2212yes—
64#1988C2212yes—
65#1987C2212yes—
66#1985C2214yes—
67#1984C2212yes—
68#1947C2212yes—
69#1943C2214yes—
70#1943C2214yes—
71#1943C2214yes—
72#1936C2214yes—
73#1925C2212yes—
74#1921C2214yes—
75#1921C2214yes—
76#1918C2212yes—
77#1918C2214yes—
78#1874C2211yescharacteristic
79#1874C2212yes—
80#1864C2214yes—
81#1864C2214yes—
82#1853C2212yes—
83#1853C2212yes—
84#1853C2212yes—
85#1852C2212yes—
86#1852C2211yescharacteristic
87#1823C2211yescharacteristic
88#1823C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8—21no—
97C23×Q8D411yescharacteristic
98D4×Q8—44no—
99D4×Q8—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—22no—
115C2×#121C2311yescharacteristic
116C2×C4×Q8—21no—
117C2×C4×Q8—44no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8D412yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2312yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4D412yes—
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#498—44no—
140#480Q818yescomplemented
141#476—44no—
142#446—44no—
143C2×#107C2314yes—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2314yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×C22⋊C8C2311yescharacteristic
157C2×#97C2314yes—
158C2×C4×C8C2312yes—
159#398—22no—
160#398—22no—
161#398—22no—
162#396—22no—
163#396—22no—
164#396—22no—
165#395—24no—
166#395—24no—
167#395—24no—
168#395—24no—
169#391—22no—
170#391—24no—
171#391—22no—
172#391—21no—
173#391—22no—
174#391—21no—
175#391—22no—
176#390C2312yes—
177#390C2311yescharacteristic
178#389—21no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—21no—
185#389—22no—
186#389C2314yes—
187#389C2314yes—
188#389C2314yes—
189#389C2314yes—
190#389C2312yes—
191#389C2311yescharacteristic
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#388—21no—
196#388—22no—
197#386—21no—
198#386—22no—
199#384—22no—
200#384—22no—
201#384—24no—
202#384C2312yes—
203#384C2311yescharacteristic
204#380—21no—
205#380—22no—
206#380—21no—
207#380—22no—
208#380C2311yescharacteristic
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
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—24no—
237C22×Q8—24no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—42no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8—82no—
248C22×Q8—82no—
249C22×Q8C2×D411yescharacteristic
250C22×Q8C2×D412yes—
251C22×Q8C2×D411yescharacteristic
252C22×D4—21no—
253C22×D4C2411yescharacteristic
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264C22×C8C2412yes—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—48no—
275C4⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—48no—
278C22⋊Q8—48no—
279C22⋊Q8—44no—
280C22⋊Q8—48no—
281C22⋊Q8—44no—
282#121—22no—
283#121C2×Q814yes—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×Q8—42no—
289C4×Q8—42no—
290C4×Q8—44no—
291C4×D4—24no—
292C4×D4—44no—
293C4×D4—48no—
294C4×D4—44no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4—84no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×Q814yescomplemented
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—44no—
354C2×C42—44no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363#107—24no—
364#107—28no—
365#107C2×Q818yescomplemented
366#106—44no—
367#106—44no—
368#106—44no—
369C4⋊C8—24no—
370C4⋊C8—44no—
371#102—24no—
372#102—44no—
373#102C2×Q818yescomplemented
374C22⋊C8—24no—
375C22⋊C8—22no—
376C22⋊C8C2×Q814yescomplemented
377#97—44no—
378C4×C8—24no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—21no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394C24—21no—
395C24C22×D411yescharacteristic
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—42no—
404C2×Q8—416no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×Q8—816no—
418C2×Q8—816no—
419C2×Q8#13714yes—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4C22×Q812yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—42no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4AGL(1,ℤ/8)14yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×Q812yes—
474C22×C42−1+412yes—
475C22×C4C2511yescharacteristic
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8C22×Q814yes—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—81no—
516C4⋊C4—82no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4C22×Q814yes—
522C4⋊C4C22×Q812yes—
523C4⋊C42−1+414yes—
524C22⋊C4—24no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—82no—
532C22⋊C4—81no—
533C42—22no—
534C42—22no—
535C42—24no—
536C42—22no—
537C42—24no—
538C42—42no—
539C42—42no—
540C42—44no—
541C42—44no—
542C42—44no—
543C42—42no—
544C42H(5,𝔽2)14yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23D4×Q812yes—
557C23C2×AGL(1,ℤ/8)12yes—
558C23#57712yes—
559C23C23×D411yescharacteristic
560Q8—28no—
561Q8—48no—
562Q8—44no—
563Q8—88no—
564Q8—88no—
565Q8—88no—
566Q8—84no—
567D4—24no—
568D4—48no—
569D4—42no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—42no—
600C2×C4—48no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—88no—
622C2×C4—84no—
623C2×C4—88no—
624C2×C4—84no—
625C2×C4#53412yes—
626C2×C4#53412yes—
627C2×C4D4×Q812yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4C2×AGL(1,ℤ/8)12yes—
630C2×C4C2×#13714yes—
631C2×C4C2×#13712yes—
632C2×C4C23×Q811yescharacteristic
633C2×C4C2×H(5,𝔽2)12yes—
634C2×C4C2×2−1+412yes—
635C2×C4C2×2−1+411yescharacteristic
636C8—24no—
637C8—42no—
638C8—44no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22#338414yes—
653C22C2×#53411yescharacteristic
654C22C2×D4×Q811yescharacteristic
655C22#348611yescharacteristic
656C22C22×AGL(1,ℤ/8)11yescharacteristic
657C22C22×#13711yescharacteristic
658C22C2×#57711yescharacteristic
659C22#358511yescharacteristic
660C4—22no—
661C4—24no—
662C4—22no—
663C4—22no—
664C4—42no—
665C4—41no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—44no—
670C4—42no—
671C4—82no—
672C4—81no—
673C4—84no—
674C4—82no—
675C4—81no—
676C4#319314yes—
677C4#350312yes—
678C4#355212yes—
679C2—24no—
680C2—82no—
681C2#3959311yescharacteristic
682C2C2×#319311yescharacteristic
683C2#6075711yescharacteristic
684C2C2×#338411yescharacteristic
685C2Q8×#13711yescharacteristic
686C2#6182511yescharacteristic
687C2#6183811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#2567C2semidirect
3. #33146C2semidirect
4. #32694C2semidirect
5. #32593C2semidirect
6. #29968C2semidirect
7. #29301C2semidirect
9. #28684C2semidirect
10. #26282C2semidirect
11. #25834C2semidirect
12. #25811C2semidirect
13. #25477C2semidirect
15. #22510C2semidirect
140. #480Q8semidirect
330. C2×C4⋊C4C2×Q8semidirect
365. #107C2×Q8semidirect
373. #102C2×Q8semidirect
376. C22⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×#1212
C2×#25672
#331462
#326942
#325932
#299682
#293012
#292382
#286842
#262822
#258342
#258112
#254772
#242992
#225102
#220812
C4×#4802