Finite Group Databasebeta

← #6852324

Subgroups — #6852324

Order 512.

1931 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3#32956C211yesmaximal, characteristic, complemented
4#32362C214yesmaximal, complemented
5#31538C214yesmaximal, complemented
6#29956C212yesmaximal, complemented
7#28541C212yesmaximal, complemented
8#27953C212yesmaximal, complemented
9#27822C212yesmaximal, complemented
10#27678C212yesmaximal, complemented
11#25811C211yesmaximal, characteristic, complemented
12#25451C212yesmaximal, complemented
13#24811C212yesmaximal, complemented
14#24298C211yesmaximal, characteristic, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22046C212yesmaximal
17C4×#447C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8—21no—
21C22×C4×D4C2211yescharacteristic
22C2×#498C2212yescomplemented
23C2×#476C2212yes—
24C2×#474C2212yescomplemented
25C2×#447C2211yescharacteristic
26C2×C4×D8C2212yes—
27#2857C2211yescharacteristic
28#2851C2212yes—
29#2589—22no—
30#2563C2211yescharacteristic, complemented
31#2552C2211yescharacteristic, complemented
32#2542—22no—
33#2542—21no—
34#2542—21no—
35#2542C2214yes—
36#2542C2212yes—
37#2542C2212yes—
38#2509C2212yescomplemented
39#2497—21no—
40#2487—21no—
41#2454C2212yescomplemented
42#2450—22no—
43C2×#399—22no—
44#2369C2211yescharacteristic, complemented
45D4×C4⋊C4—21no—
46D4×C4⋊C4C2211yescharacteristic, complemented
47#2345—21no—
48C4×#121C2212yes—
49C4×#121C2211yescharacteristic
50C2×#390C2212yes—
51C2×#384C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2100C2214yescomplemented
54#2094C2214yescomplemented
55#2092C2214yes—
56#2084C2214yescomplemented
57#2083C2214yescomplemented
58#2067C2214yes—
59#2065C2214yes—
60#2058C2214yescomplemented
61#2057C2212yescomplemented
62#2025C2214yes—
63#2023C2212yescomplemented
64#1992C2214yescomplemented
65#1990C2212yescomplemented
66#1988C2212yescomplemented
67#1972C2214yes—
68#1948C2214yescomplemented
69#1939C2212yescomplemented
70#1936C2214yes—
71#1926C2214yes—
72#1917C2214yescomplemented
73#1879C2214yes—
74#1879C2212yes—
75#1874C2212yes—
76#1874C2211yescharacteristic
77#1864C2214yes—
78#1862C2212yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1852C2212yes—
83#1852C2211yescharacteristic
84#1847C2214yescomplemented
85#1802C2212yes—
86#1802C2214yescomplemented
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1786C2211yescharacteristic
90#1786C2212yes—
91#1770C2212yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×D8C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#125C2312yes—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—48no—
114C2×C22⋊Q8—48no—
115C2×#121C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—44no—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—22no—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#498—44no—
139#476—44no—
140#474—44no—
141#447Q818yescomplemented
142C4×D8—44no—
143C2×#107C2314yes—
144C2×#107C2312yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2311yescharacteristic
159C2×C22⋊C8C2311yescharacteristic
160C2×#97C2314yes—
161C2×C4×C8C2312yes—
162#400—22no—
163#400—22no—
164#399—22no—
165#399—24no—
166#399—24no—
167#399—24no—
168#399—24no—
169#398—22no—
170#396—22no—
171#393—22no—
172#391—21no—
173#391—22no—
174#390—22no—
175#390—22no—
176#390C2312yes—
177#390C2314yes—
178#390C2314yes—
179#390C2311yescharacteristic
180#389—22no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389C2311yescharacteristic
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#388—21no—
190#388—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#386—21no—
195#386—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384—21no—
201#384—22no—
202#384—22no—
203#384C2311yescharacteristic
204#384C2314yes—
205#384C2314yes—
206#382C2312yes—
207#380—21no—
208#380—21no—
209#380—21no—
210#380—21no—
211#380—22no—
212#380C2311yescharacteristic
213#380C2312yes—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225C2×#95D412yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236#336C2314yes—
237C25—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—42no—
248C22×D4—42no—
249C22×D4C2×Q814yes—
250C22×D4C2411yescharacteristic
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—44no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C2×D8—42no—
267C2×D8—42no—
268C2×D8C2×Q818yescomplemented
269C22×C8C2412yes—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#125—44no—
279C22⋊Q8—416no—
280C22⋊Q8—48no—
281C22⋊Q8—48no—
282C22⋊Q8—48no—
283C22⋊Q8—416no—
284C22⋊Q8—48no—
285C22⋊Q8—82no—
286#121—22no—
287#121C2×Q814yes—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—84no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—44no—
353C2×C22⋊C4—48no—
354C2×C22⋊C4—48no—
355C2×C42—24no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—44no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2412yes—
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367#107—42no—
368#107—44no—
369#106—44no—
370#106—44no—
371#106—44no—
372C4⋊C8—24no—
373C4⋊C8—44no—
374#102—44no—
375#102—42no—
376#102—42no—
377#102—42no—
378#102—44no—
379#102—44no—
380#102C2×Q818yescomplemented
381C22⋊C8—24no—
382C22⋊C8—22no—
383C22⋊C8C2×Q814yes—
384#97—44no—
385C4×C8—24no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24—24no—
402C24—21no—
403C24—22no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24C22×D411yescharacteristic
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—81no—
419C2×Q8—81no—
420C2×Q8—816no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—48no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—48no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4C22×Q814yes—
438C2×D4C22×Q812yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—816no—
478C22×C4AGL(1,ℤ/8)14yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×Q812yes—
489C22×C42−1+412yes—
490C22×C4C2511yescharacteristic
491D8—48no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8C22×Q814yes—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—42no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—48no—
526C4⋊C4—42no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—81no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4AGL(1,ℤ/8)14yes—
534C4⋊C4C22×Q812yes—
535C22⋊C4—24no—
536C22⋊C4—44no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—42no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—42no—
544C22⋊C4—48no—
545C22⋊C4—42no—
546C22⋊C4—81no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—24no—
554C42—24no—
555C42—42no—
556C42—42no—
557C42—42no—
558C42—82no—
559C42H(5,𝔽2)14yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—24no—
566C23—48no—
567C23—48no—
568C23—41no—
569C23—42no—
570C23—48no—
571C23—42no—
572C23—41no—
573C23—41no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—42no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23—84no—
585C23D4×Q812yes—
586C23C2×AGL(1,ℤ/8)12yes—
587C23#57712yes—
588C23C23×D411yescharacteristic
589Q8—88no—
590Q8—88no—
591Q8—162no—
592D4—24no—
593D4—24no—
594D4—24no—
595D4—48no—
596D4—48no—
597D4—42no—
598D4—42no—
599D4—42no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—88no—
648C2×C4—84no—
649C2×C4—88no—
650C2×C4—84no—
651C2×C4—88no—
652C2×C4—164no—
653C2×C4#53512yes—
654C2×C4#53512yes—
655C2×C4D4×Q812yes—
656C2×C4C2×AGL(1,ℤ/8)14yes—
657C2×C4C2×AGL(1,ℤ/8)12yes—
658C2×C4C2×AGL(1,ℤ/8)12yes—
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4C23×Q811yescharacteristic
661C2×C4C2×H(5,𝔽2)12yes—
662C2×C4C2×2−1+411yescharacteristic
663C2×C4#58512yes—
664C8—24no—
665C8—42no—
666C8—44no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—44no—
674C22—42no—
675C22—42no—
676C22—44no—
677C22—42no—
678C22—44no—
679C22—816no—
680C22—81no—
681C22—82no—
682C22—81no—
683C22—82no—
684C22—82no—
685C22—81no—
686C22—81no—
687C22—82no—
688C22#338414yes—
689C22C2×#53511yescharacteristic
690C22C2×D4×Q811yescharacteristic
691C22#348511yescharacteristic
692C22C22×AGL(1,ℤ/8)11yescharacteristic
693C22C22×AGL(1,ℤ/8)11yescharacteristic
694C22C2×#57711yescharacteristic
695C22#358611yescharacteristic
696C4—24no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—41no—
704C4—42no—
705C4—81no—
706C4—82no—
707C4—81no—
708C4—82no—
709C4—84no—
710C4—82no—
711C4#321714yes—
712C4#350112yes—
713C4#354112yes—
714C2—24no—
715C2—82no—
716C2—84no—
717C2#3959411yescharacteristic
718C2C2×#321711yescharacteristic
719C2#6081311yescharacteristic
720C2C2×#338411yescharacteristic
721C2Q8×AGL(1,ℤ/8)11yescharacteristic
722C2#6171711yescharacteristic
723C2#6172811yescharacteristic
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×#2542C2semidirect
3. #32956C2semidirect
4. #32362C2semidirect
5. #31538C2semidirect
6. #29956C2semidirect
7. #28541C2semidirect
8. #27953C2semidirect
9. #27822C2semidirect
10. #27678C2semidirect
11. #25811C2semidirect
12. #25451C2semidirect
13. #24811C2semidirect
14. #24298C2semidirect
15. #22629C2semidirect
22. C2×#498C22semidirect
24. C2×#474C22semidirect
30. #2563C22semidirect
31. #2552C22semidirect
38. #2509C22semidirect
41. #2454C22semidirect
44. #2369C22semidirect
46. D4×C4⋊C4C22semidirect
53. #2100C22semidirect
54. #2094C22semidirect
56. #2084C22semidirect
57. #2083C22semidirect
60. #2058C22semidirect
61. #2057C22semidirect
63. #2023C22semidirect
64. #1992C22semidirect
65. #1990C22semidirect
66. #1988C22semidirect
68. #1948C22semidirect
69. #1939C22semidirect
72. #1917C22semidirect
84. #1847C22semidirect
86. #1802C22semidirect
91. #1770C22semidirect
141. #447Q8semidirect
268. C2×D8C2×Q8semidirect
380. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×#25422
#329562
#323622
#315382
#299562
#285412
#279532
#278222
#276782
#258112
#254512
#248112
#242982
#226292
#220462
C4×#4472