Finite Group Databasebeta

← #6626287

Subgroups — #6626287

Order 512.

2161 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#32959C212yesmaximal, complemented
4#32956C211yesmaximal, characteristic, complemented
5#29303C212yesmaximal, complemented
6#29235C212yesmaximal, complemented
7#27948C211yesmaximal, characteristic, complemented
8#27677C211yesmaximal, characteristic, complemented
9#27312C214yesmaximal, complemented
10#26390C214yesmaximal, complemented
11#26045C212yesmaximal, complemented
12#24812C212yesmaximal, complemented
13#22969C212yesmaximal, complemented
14#22670C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21452C212yesmaximal
17C4×#447C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4—21no—
21C2×#478C2214yes—
22C2×#476C2212yes—
23C2×#447C2211yescharacteristic
24C2×#442C2214yes—
25C2×C4×D8C2212yes—
26#2857C2211yescharacteristic
27#2851C2212yes—
28#2589—22no—
29#2563C2211yescharacteristic, complemented
30#2552C2211yescharacteristic, complemented
31#2509C2212yescomplemented
32#2497—21no—
33#2487—21no—
34#2454C2212yescomplemented
35#2450—22no—
36#2369C2211yescharacteristic, complemented
37D4×C4⋊C4—22no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—21no—
41D4×C4⋊C4C2214yes—
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2212yes—
44D4×C4⋊C4C2211yescharacteristic, complemented
45#2345—21no—
46C4×#121C2211yescharacteristic
47C4×#121C2212yes—
48C2×#389C2212yes—
49C2×#385—22no—
50C2×#384C2211yescharacteristic
51C2×#380—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2099C2212yescomplemented
54#2093C2212yescomplemented
55#2088C2214yescomplemented
56#2085C2214yescomplemented
57#2084C2214yescomplemented
58#2065C2214yescomplemented
59#2059C2214yescomplemented
60#1990C2212yescomplemented
61#1947C2212yescomplemented
62#1944C2214yescomplemented
63#1943C2214yescomplemented
64#1939C2212yescomplemented
65#1920C2214yescomplemented
66#1917C2214yescomplemented
67#1879C2212yes—
68#1879C2214yes—
69#1874C2212yes—
70#1874C2211yescharacteristic
71#1864C2214yes—
72#1864C2214yes—
73#1864C2214yes—
74#1864C2214yescomplemented
75#1862C2214yes—
76#1862C2212yes—
77#1853C2212yes—
78#1853C2212yes—
79#1853C2212yes—
80#1852C2212yes—
81#1852C2211yescharacteristic
82#1844C2212yescomplemented
83#1804C2214yes—
84#1802C2212yes—
85#1802C2212yescomplemented
86#1797C2211yescharacteristic
87#1797C2212yes—
88#1786C2212yes—
89#1786C2211yescharacteristic
90#1770C2212yescomplemented
91#1762C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C24×C4—22no—
98C22×D8C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#125C2312yes—
101C2×C22⋊Q8—21no—
102C2×#121C2311yescharacteristic
103C2×C4×D4—28no—
104C2×C4×D4—21no—
105C2×C4×D4—22no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—48no—
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2312yes—
113C2×C4×D4C2314yes—
114C2×C4×D4C2314yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—24no—
124C22×C4⋊C4—48no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—22no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#478—48no—
134#476—28no—
135#447Q818yescomplemented
136#442—48no—
137C4×D8—28no—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2311yescharacteristic
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#400—22no—
158#400—22no—
159#399—22no—
160#398—22no—
161#396—22no—
162#393—22no—
163#391—21no—
164#391—22no—
165#390C2312yes—
166#390C2311yescharacteristic
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2314yes—
179#389C2311yescharacteristic
180#388—21no—
181#388—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#386—21no—
186#386—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—24no—
192#385—24no—
193#385—24no—
194#385—24no—
195#384—21no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#382C2312yes—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—22no—
206#380—22no—
207#380—24no—
208#380—21no—
209#380—22no—
210#380C2311yescharacteristic
211#380C2312yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C25—21no—
237C22×Q8—21no—
238C22×D4—24no—
239C22×D4—24no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4C2×Q814yes—
244C22×D4C2411yescharacteristic
245C22×D4C2412yes—
246C23×C4—28no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—24no—
253C23×C4—24no—
254C23×C4—22no—
255C23×C4—44no—
256C23×C4—48no—
257C23×C4—44no—
258C23×C4—48no—
259C23×C4—84no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2411yescharacteristic
265C2×D8—24no—
266C2×D8—24no—
267C2×D8C2×Q818yes—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C4⋊Q8—24no—
272#125—44no—
273C22⋊Q8—82no—
274#121—22no—
275#121C2×Q814yes—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—28no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—48no—
289C4×D4—44no—
290C4×D4—84no—
291C4×D4#4118yes—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
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—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—416no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—48no—
349C2×C42—24no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—24no—
362#107—44no—
363#106—28no—
364#106—44no—
365#106—44no—
366C4⋊C8—24no—
367C4⋊C8—44no—
368#102—24no—
369#102—28no—
370#102—28no—
371#102—24no—
372#102—24no—
373#102—44no—
374#102C2×Q818yes—
375C22⋊C8—24no—
376C22⋊C8—22no—
377C22⋊C8C2×Q814yes—
378#97—44no—
379C4×C8—24no—
380#95—24no—
381#95—24no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—24no—
398C24—24no—
399C24—24no—
400C24—22no—
401C24—21no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24C22×D411yescharacteristic
407C2×Q8—42no—
408C2×Q8—81no—
409C2×Q8—81no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—216no—
413C2×D4—216no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4C22×Q812yes—
425C2×D4C22×Q814yes—
426C2×D4C2×#4114yes—
427C2×D4H(5,𝔽2)14yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—28no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—42no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—432no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4—832no—
474C22×C4AGL(1,ℤ/8)14yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×Q812yes—
485C22×C42−1+412yes—
486C22×C4C2511yescharacteristic
487D8—216no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8C22×Q814yes—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—81no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4C22×Q812yes—
530C4⋊C4C2×#4114yes—
531C22⋊C4—28no—
532C22⋊C4—28no—
533C22⋊C4—24no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—24no—
548C42—24no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—82no—
553C42C2×#4114yes—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—216no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—22no—
563C23—48no—
564C23—48no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—42no—
569C23—48no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23D4×Q812yes—
580C23C2×AGL(1,ℤ/8)12yes—
581C23#57712yes—
582C23C23×D411yescharacteristic
583Q8—162no—
584D4—216no—
585D4—24no—
586D4—24no—
587D4—24no—
588D4—24no—
589D4—48no—
590D4—42no—
591D4#55418yes—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—44no—
619C2×C4—416no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—416no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—84no—
643C2×C4—816no—
644C2×C4—816no—
645C2×C4—88no—
646C2×C4—84no—
647C2×C4—164no—
648C2×C4#52512yes—
649C2×C4#52512yes—
650C2×C4D4×Q812yes—
651C2×C4C2×AGL(1,ℤ/8)12yes—
652C2×C4C2×AGL(1,ℤ/8)12yes—
653C2×C4#57512yes—
654C2×C4C23×Q811yescharacteristic
655C2×C4C22×#4112yes—
656C2×C4C2×H(5,𝔽2)12yes—
657C2×C4C2×2−1+411yescharacteristic
658C8—24no—
659C8—42no—
660C8—44no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—28no—
668C22—432no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22#338414yes—
683C22C2×#52511yescharacteristic
684C22C2×D4×Q811yescharacteristic
685C22#348411yescharacteristic
686C22C22×AGL(1,ℤ/8)11yescharacteristic
687C22C2×#57511yescharacteristic
688C22C2×#57711yescharacteristic
689C22#358511yescharacteristic
690C4—22no—
691C4—22no—
692C4—22no—
693C4—44no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—41no—
699C4—42no—
700C4—81no—
701C4—81no—
702C4—84no—
703C4—82no—
704C4—82no—
705C4#311414yes—
706C4C2×#55412yes—
707C4#349212yes—
708C2—24no—
709C2—48no—
710C2—82no—
711C2#3654611yescharacteristic
712C2C2×#311411yescharacteristic
713C2#6058811yescharacteristic
714C2C2×#338411yescharacteristic
715C2#6136711yescharacteristic
716C2#6158211yescharacteristic
717C2#6160411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
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
1. #36551C2semidirect
2. C2×D4×C4⋊C4C2semidirect
3. #32959C2semidirect
4. #32956C2semidirect
5. #29303C2semidirect
6. #29235C2semidirect
7. #27948C2semidirect
8. #27677C2semidirect
9. #27312C2semidirect
10. #26390C2semidirect
11. #26045C2semidirect
12. #24812C2semidirect
13. #22969C2semidirect
14. #22670C2semidirect
15. #22629C2semidirect
29. #2563C22semidirect
30. #2552C22semidirect
31. #2509C22semidirect
34. #2454C22semidirect
36. #2369C22semidirect
44. D4×C4⋊C4C22semidirect
53. #2099C22semidirect
54. #2093C22semidirect
55. #2088C22semidirect
56. #2085C22semidirect
57. #2084C22semidirect
58. #2065C22semidirect
59. #2059C22semidirect
60. #1990C22semidirect
61. #1947C22semidirect
62. #1944C22semidirect
63. #1943C22semidirect
64. #1939C22semidirect
65. #1920C22semidirect
66. #1917C22semidirect
74. #1864C22semidirect
82. #1844C22semidirect
85. #1802C22semidirect
90. #1770C22semidirect
135. #447Q8semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×D4×C4⋊C42
#329592
#329562
#293032
#292352
#279482
#276772
#273122
#263902
#260452
#248122
#229692
#226702
#226292
#214522
C4×#4472