Finite Group Databasebeta

← #6628108

Subgroups — #6628108

Order 512.

1874 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36596C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33107C212yesmaximal, complemented
4#33107C211yesmaximal, characteristic, complemented
5#29334C212yesmaximal, complemented
6#29238C212yesmaximal, complemented
7#27143C214yesmaximal, complemented
8#26705C214yesmaximal, complemented
9#26510C212yesmaximal, complemented
10#25903C211yesmaximal, characteristic, complemented
11#25272C212yesmaximal, complemented
12#24298C211yesmaximal, characteristic, complemented
13#23220C212yesmaximal, complemented
14#23066C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21580C212yesmaximal, complemented
17C4×#481C211yesmaximal, characteristic, complemented
18C2×#553—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8—21no—
21C2×#486C2212yes—
22C2×#481C2211yescharacteristic
23C2×#481C2212yes—
24C2×#474C2212yes—
25#2856C2211yescharacteristic
26#2856C2212yes—
27#2554—22no—
28#2552C2211yescharacteristic
29#2543—22no—
30#2541C2211yescharacteristic
31#2512C2212yes—
32#2490—21no—
33#2445C2212yes—
34#2416—21no—
35#2366C2211yescharacteristic
36#2365—21no—
37#2365—21no—
38#2365—22no—
39#2365C2214yes—
40#2365C2212yes—
41#2365C2212yes—
42#2363—21no—
43#2354C2211yescharacteristic
44C4×#121C2212yes—
45C4×#121C2211yescharacteristic
46C2×#391—21no—
47C2×#389C2212yes—
48C2×#387—22no—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2214yes—
53#2088C2214yes—
54#2067C2214yes—
55#2057C2212yes—
56#2041C2214yes—
57#2023C2214yes—
58#2023C2212yes—
59#1990C2212yes—
60#1988C2212yes—
61#1982C2214yes—
62#1982C2212yes—
63#1948C2214yes—
64#1948C2212yes—
65#1939C2212yes—
66#1936C2214yes—
67#1936C2214yes—
68#1926C2214yes—
69#1926C2214yes—
70#1926C2214yes—
71#1926C2212yes—
72#1925C2214yes—
73#1921C2214yes—
74#1920C2214yes—
75#1919C2212yes—
76#1917C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1874C2211yescharacteristic
81#1874C2212yes—
82#1864C2214yes—
83#1853C2211yescharacteristic
84#1852C2212yes—
85#1824C2211yescharacteristic
86#1797C2212yes—
87#1797C2211yescharacteristic
88#1776C2214yes—
89C4×#106C2212yes—
90#1770C2212yes—
91#1762C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96#553—48no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#127C2312yes—
104C2×#126—22no—
105C2×#125—21no—
106C2×#123—22no—
107C2×#121—21no—
108C2×#121—22no—
109C2×#121—22no—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121—21no—
113C2×#121C2311yescharacteristic
114C2×#121C2312yes—
115C2×C4×Q8—22no—
116C2×C4×Q8—24no—
117C2×C4×Q8—22no—
118C2×C4×Q8—48no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×#117—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C42—22no—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#486—28no—
137#481—24no—
138#481—28no—
139#474—28no—
140C2×#107C2314yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×C22⋊C8C2311yescharacteristic
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158#398—22no—
159#397—22no—
160#396—22no—
161#394—22no—
162#394—22no—
163#392—22no—
164#391—22no—
165#391—22no—
166#391—24no—
167#390—21no—
168#390—21no—
169#390—22no—
170#390—22no—
171#390—21no—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#390C2311yescharacteristic
177#389—21no—
178#389—22no—
179#389—22no—
180#389C2311yescharacteristic
181#389C2314yes—
182#389C2312yes—
183#389C2314yes—
184#388—22no—
185#388—21no—
186#388—22no—
187#388—22no—
188#387—24no—
189#387—24no—
190#387—24no—
191#387—24no—
192#386—22no—
193#386—21no—
194#386—22no—
195#386—22no—
196#384C2311yescharacteristic
197#384C2312yes—
198#380—21no—
199#380C2311yescharacteristic
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C2×#95—21no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95D412yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—28no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—44no—
239C22×Q8—44no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4C2411yescharacteristic
247C22×D4C2412yes—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264C4⋊Q8—24no—
265C4⋊Q8#4118yes—
266#127—24no—
267#127—44no—
268#126—48no—
269#125—44no—
270#123—48no—
271#121—22no—
272#121—24no—
273#121—22no—
274#121—24no—
275#121—44no—
276#121—48no—
277#121—44no—
278#121—44no—
279#121—84no—
280#121—82no—
281C4×Q8—48no—
282C4×Q8—416no—
283C4×Q8—48no—
284C4×D4—24no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—44no—
289#117—44no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4#4114yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42—44no—
348C2×C42—48no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—28no—
358#106—24no—
359#106—28no—
360#106—42no—
361C4⋊C8—24no—
362C4⋊C8—44no—
363#102—28no—
364#102—28no—
365#102—24no—
366#102—28no—
367#102—28no—
368#102—44no—
369#102—42no—
370C22⋊C8—24no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373#97—44no—
374C4×C8—24no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—21no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—21no—
391C24—22no—
392C24C22×D411yescharacteristic
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—44no—
396C2×Q8—432no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—82no—
421C2×D4—81no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—81no—
425C2×D4—82no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4AGL(1,ℤ/8)14yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C2×#4112yes—
472C22×C42−1+412yes—
473C22×C4C2511yescharacteristic
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C4⋊C4—28no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—42no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—82no—
511C4⋊C4AGL(1,ℤ/8)14yes—
512C4⋊C4C2×#4114yes—
513C4⋊C4C2×#4118yes—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—81no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—24no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42—48no—
538C42—44no—
539C42—48no—
540C42—44no—
541C42—42no—
542C42C2×#4114yes—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—41no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—41no—
555C23—42no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23#54812yes—
562C23C2×AGL(1,ℤ/8)12yes—
563C23#57712yes—
564C23C23×D411yescharacteristic
565Q8—416no—
566Q8—416no—
567D4—42no—
568D4—44no—
569D4—42no—
570D4—44no—
571D4—88no—
572D4—84no—
573D4—84no—
574D4—164no—
575D4—162no—
576D4—164no—
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—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—416no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—164no—
629C2×C4#52512yes—
630C2×C4#52512yes—
631C2×C4#54812yes—
632C2×C4C2×AGL(1,ℤ/8)14yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4C2×AGL(1,ℤ/8)12yes—
636C2×C4C22×#4111yescharacteristic
637C2×C4C22×#4112yes—
638C2×C4C2×2−1+411yescharacteristic
639C2×C4#58512yes—
640C8—42no—
641C8—44no—
642C8—42no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—82no—
650C22—81no—
651C22—81no—
652C22—82no—
653C22—82no—
654C22—81no—
655C22—81no—
656C22—82no—
657C22—81no—
658C22—82no—
659C22—82no—
660C22—81no—
661C22—81no—
662C22—82no—
663C22#336814yes—
664C22C2×#52511yescharacteristic
665C22C2×#54811yescharacteristic
666C22#348511yescharacteristic
667C22C22×AGL(1,ℤ/8)11yescharacteristic
668C22C22×AGL(1,ℤ/8)11yescharacteristic
669C22C2×#57711yescharacteristic
670C22#358611yescharacteristic
671C4—24no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—48no—
679C4—44no—
680C4—41no—
681C4—81no—
682C4—82no—
683C4—82no—
684C4—81no—
685C4—81no—
686C4#310914yes—
687C4#348012yes—
688C4#351612yes—
689C2—24no—
690C2—162no—
691C2—161no—
692C2#3658211yescharacteristic
693C2C2×#310911yescharacteristic
694C2#6058211yescharacteristic
695C2C2×#336811yescharacteristic
696C2#6131711yescharacteristic
697C2#6172211yescharacteristic
698C2#6172811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36596C2
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. #33107C2semidirect
4. #33107C2semidirect
5. #29334C2semidirect
6. #29238C2semidirect
7. #27143C2semidirect
8. #26705C2semidirect
9. #26510C2semidirect
10. #25903C2semidirect
11. #25272C2semidirect
12. #24298C2semidirect
13. #23220C2semidirect
14. #23066C2semidirect
15. #22629C2semidirect
16. #21580C2semidirect
17. C4×#481C2semidirect

Maximal subgroups

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

subgroupindex
#365962
C2×#23652
#331072
#293342
#292382
#271432
#267052
#265102
#259032
#252722
#242982
#232202
#230662
#226292
#215802
C4×#4812