Finite Group Databasebeta

← #6616253

Subgroups — #6616253

Order 512.

1813 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36116C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33108C211yesmaximal, characteristic, complemented
4#33099C212yesmaximal, complemented
5#27016C214yesmaximal, complemented
6#26704C214yesmaximal, complemented
7#26447C212yesmaximal, complemented
8#25902C211yesmaximal, characteristic, complemented
9#25275C212yesmaximal, complemented
10#24609C212yesmaximal, complemented
11#24298C211yesmaximal, characteristic
12#24066C212yesmaximal
13#23222C212yesmaximal, complemented
14#22968C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16C4×#480C211yesmaximal, characteristic
17#21391C212yesmaximal
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2212yes—
20C22×C4×Q8—21no—
21C2×#487C2212yes—
22C2×#483C2212yes—
23C2×#480C2211yescharacteristic
24C2×#474C2212yes—
25C2×#432C2212yes—
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2552C2211yescharacteristic
29#2547C2211yescharacteristic
30#2496—21no—
31#2486—21no—
32#2478—22no—
33#2453C2212yes—
34#2447C2212yes—
35#2419—22no—
36#2365—21no—
37#2365—22no—
38#2365—21no—
39#2365C2212yes—
40#2365C2214yes—
41#2365C2212yes—
42#2365C2211yescharacteristic
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2211yescharacteristic
45#2344—21no—
46C4×#121C2211yescharacteristic
47C2×#391—21no—
48C2×#389C2212yes—
49C2×#387—22no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2088C2214yes—
54#2068C2214yes—
55#2059C2214yes—
56#2057C2212yes—
57#2042C2214yes—
58#2024C2214yes—
59#2023C2212yes—
60#1990C2212yes—
61#1989C2214yes—
62#1988C2212yes—
63#1983C2214yes—
64#1981C2212yes—
65#1948C2212yes—
66#1943C2214yes—
67#1939C2212yes—
68#1935C2214yes—
69#1928C2214yes—
70#1928C2214yes—
71#1927C2214yes—
72#1925C2212yes—
73#1921C2214yes—
74#1921C2214yes—
75#1918C2212yes—
76#1917C2214yes—
77#1874C2212yes—
78#1874C2211yescharacteristic
79#1872C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1864C2214yes—
83#1863C2214yes—
84#1852C2212yes—
85#1852C2211yescharacteristic
86#1823C2211yescharacteristic
87#1817C2212yes—
88#1797C2211yescharacteristic
89C4×#107C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92#1762C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102#529—28no—
103#529—44no—
104C2×#126—22no—
105C2×#125C2312yes—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×C22⋊Q8C2312yes—
109C2×#121C2311yescharacteristic
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—24no—
113C2×C4×Q8—48no—
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2312yes—
119C2×#117—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C42—22no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#487—28no—
133#483—44no—
134#480—24no—
135#474—28no—
136#432—44no—
137C2×#107C2314yes—
138C2×#107C2312yes—
139C2×#107C2312yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2311yescharacteristic
153C2×#97C2314yes—
154C2×C4×C8C2312yes—
155#400—22no—
156#398—22no—
157#398—22no—
158#397—22no—
159#397—22no—
160#397—22no—
161#391—22no—
162#391—22no—
163#391—24no—
164#391C2312yes—
165#391C2312yes—
166#391C2312yes—
167#390C2311yescharacteristic
168#389—21no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389C2311yescharacteristic
176#389C2314yes—
177#389C2312yes—
178#389C2312yes—
179#389C2314yes—
180#388—21no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—24no—
185#387—24no—
186#387—24no—
187#387—24no—
188#387—22no—
189#386—21no—
190#385—22no—
191#385—22no—
192#385—22no—
193#384—21no—
194#384C2311yescharacteristic
195#381—22no—
196#381—22no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380C2311yescharacteristic
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C2×#95—21no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—28no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8C2412yes—
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2411yescharacteristic
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—28no—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8#4118yes—
264#126—48no—
265#125—24no—
266#125—42no—
267#124—24no—
268#124—42no—
269#123—48no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272#121—22no—
273#121#4114yes—
274C4×Q8—24no—
275C4×Q8—42no—
276C4×Q8—416no—
277C4×Q8—48no—
278C4×Q8—48no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—42no—
282C4×D4—48no—
283C4×D4—44no—
284#117—48no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
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—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—48no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—28no—
354#107—24no—
355#107—24no—
356#106—44no—
357C4⋊C8—24no—
358C4⋊C8—44no—
359#103—28no—
360#103—28no—
361#103—28no—
362#103—44no—
363#102—24no—
364#102—28no—
365#102—24no—
366C22⋊C8—22no—
367C22⋊C8—22no—
368C22⋊C8—24no—
369#97—44no—
370C4×C8—24no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—21no—
393C24C22×D411yescharacteristic
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—44no—
401C2×Q8—432no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4C2×#4112yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—82no—
448C22×C4AGL(1,ℤ/8)14yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4112yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C4C2511yescharacteristic
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4C2×#4118yes—
504C4⋊C4C2×#4112yes—
505C4⋊C42−1+414yes—
506C22⋊C4—24no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—81no—
515C42—22no—
516C42—22no—
517C42—24no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—44no—
525C42—48no—
526C42—48no—
527C42—44no—
528C42—44no—
529C42—42no—
530C42C2×#4114yes—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23#54612yes—
543C23C2×AGL(1,ℤ/8)12yes—
544C23#57812yes—
545C23C23×D411yescharacteristic
546Q8—416no—
547Q8—416no—
548Q8—44no—
549Q8—44no—
550D4—24no—
551D4—48no—
552D4—42no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—416no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—84no—
606C2×C4—88no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4#52512yes—
610C2×C4#52512yes—
611C2×C4#54612yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×AGL(1,ℤ/8)12yes—
614C2×C4#57512yes—
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4112yes—
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×2−1+412yes—
619C8—42no—
620C8—42no—
621C8—44no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22#333014yes—
636C22C2×#52511yescharacteristic
637C22C2×#54611yescharacteristic
638C22#348611yescharacteristic
639C22C22×AGL(1,ℤ/8)11yescharacteristic
640C22C2×#57511yescharacteristic
641C22C2×#57811yescharacteristic
642C22#358611yescharacteristic
643C4—22no—
644C4—22no—
645C4—22no—
646C4—48no—
647C4—44no—
648C4—41no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—82no—
655C4—81no—
656C4—81no—
657C4—82no—
658C4—82no—
659C4#311414yes—
660C4#348212yes—
661C4#352212yes—
662C2—24no—
663C2—82no—
664C2#3642211yescharacteristic
665C2C2×#311411yescharacteristic
666C2#6058311yescharacteristic
667C2C2×#333011yescharacteristic
668C2#6116511yescharacteristic
669C2#6182711yescharacteristic
670C2#6184211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#2365C2semidirect
3. #33108C2semidirect
4. #33099C2semidirect
5. #27016C2semidirect
6. #26704C2semidirect
7. #26447C2semidirect
8. #25902C2semidirect
9. #25275C2semidirect
10. #24609C2semidirect
13. #23222C2semidirect
14. #22968C2semidirect
15. #22603C2semidirect

Maximal subgroups

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

subgroupindex
#361162
C2×#23652
#331082
#330992
#270162
#267042
#264472
#259022
#252752
#246092
#242982
#240662
#232222
#229682
#226032
C4×#4802
#213912