Finite Group Databasebeta

← #6855855

Subgroups — #6855855

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36596C211yesmaximal, characteristic
2C2×#2551C211yesmaximal, characteristic, complemented
3#33107C211yesmaximal, characteristic, complemented
4#32230C214yesmaximal, complemented
5#30813C214yesmaximal, complemented
6#29755C212yesmaximal, complemented
7#28791C212yesmaximal, complemented
8#27973C211yesmaximal, characteristic, complemented
9#27767C212yesmaximal, complemented
10#27678C211yesmaximal, characteristic, complemented
11#26464C212yesmaximal, complemented
12#26024C212yesmaximal, complemented
13#25903C212yesmaximal, complemented
14#24299C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16#22076C212yesmaximal, complemented
17C4×#481C211yesmaximal, characteristic, complemented
18C2×#553—21no—
19C22×C4⋊Q8—21no—
20C22×C4⋊Q8C2212yes—
21C2×#486C2214yes—
22C2×#481C2211yescharacteristic
23C2×#481C2212yes—
24C2×#476C2214yes—
25#2856C2212yes—
26#2856C2211yescharacteristic
27#2554—22no—
28#2552C2211yescharacteristic
29#2551—21no—
30#2551—21no—
31#2551—22no—
32#2551C2214yes—
33#2551C2212yes—
34#2551C2212yes—
35#2543—22no—
36#2541C2211yescharacteristic
37#2512C2212yes—
38#2490—21no—
39#2445C2212yes—
40#2416—21no—
41C2×#395—22no—
42#2366C2211yescharacteristic
43#2363—21no—
44#2354C2211yescharacteristic
45C4×#121C2211yescharacteristic
46C4×#121C2212yes—
47C2×#389—21no—
48C2×#389C2212yes—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2212yes—
53#2094C2212yes—
54#2084C2214yes—
55#2071C2214yes—
56#2065C2214yes—
57#2065C2214yes—
58#2058C2214yes—
59#2041C2214yes—
60#2023C2214yes—
61#2023C2214yes—
62#2023C2214yes—
63#1990C2214yes—
64#1987C2214yes—
65#1982C2212yes—
66#1957C2212yes—
67#1948C2212yes—
68#1948C2212yes—
69#1941C2214yes—
70#1939C2212yes—
71#1934C2214yes—
72#1926C2212yes—
73#1926C2214yes—
74#1920C2214yes—
75#1919C2212yes—
76#1916C2214yes—
77#1874C2212yes—
78#1874C2211yescharacteristic
79#1874C2212yes—
80#1874C2212yes—
81#1874C2212yes—
82#1864C2214yes—
83#1853C2212yes—
84#1853C2211yescharacteristic
85#1834C2212yes—
86#1824C2211yescharacteristic
87#1802C2212yes—
88#1797C2211yescharacteristic
89#1797C2212yes—
90C4×#106C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—22no—
96#553—48no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—48no—
103C2×C4⋊Q8—48no—
104C2×C4⋊Q8C2314yes—
105C2×C4⋊Q8C2314yes—
106C2×#127C2311yescharacteristic
107C2×#127C2312yes—
108C2×#126—22no—
109C2×#125—21no—
110C2×#123—22no—
111C2×#121—21no—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121—21no—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121C2311yescharacteristic
118C2×#121C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×#117—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#486—48no—
138#481—24no—
139#481—28no—
140#476—48no—
141C2×#106C2312yes—
142C2×#106C2312yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×C22⋊C8C2311yescharacteristic
157C2×C4×C8C2312yes—
158C2×C4×C8C2314yes—
159#398—22no—
160#397—22no—
161#396—22no—
162#395—24no—
163#395—24no—
164#395—24no—
165#395—24no—
166#394—22no—
167#394—22no—
168#392—22no—
169#390—21no—
170#390—21no—
171#390—22no—
172#390—22no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#390C2311yescharacteristic
178#390C2312yes—
179#389—21no—
180#389—22no—
181#389—22no—
182#389—24no—
183#389—22no—
184#389—22no—
185#389C2312yes—
186#389C2314yes—
187#389C2314yes—
188#389C2311yescharacteristic
189#388—22no—
190#388—21no—
191#388—22no—
192#388—22no—
193#386—22no—
194#386—21no—
195#386—22no—
196#386—22no—
197#384—21no—
198#384—21no—
199#384C2312yes—
200#384C2312yes—
201#384C2312yes—
202#384C2311yescharacteristic
203#380—21no—
204#380C2311yescharacteristic
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C2×#95—21no—
220C2×#95—22no—
221C2×#95D412yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#336C2314yes—
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—48no—
238C22×Q8—48no—
239C22×Q8—84no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4C2412yes—
247C22×D4C2411yescharacteristic
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—48no—
266C4⋊Q8—48no—
267C4⋊Q8—416no—
268C4⋊Q8—48no—
269C4⋊Q8—48no—
270#127—24no—
271#127—44no—
272#126—48no—
273#125—44no—
274#123—48no—
275#121—22no—
276#121—24no—
277#121—22no—
278#121—24no—
279#121—48no—
280#121—44no—
281#121—44no—
282#121—44no—
283#121—82no—
284#121—84no—
285C4×D4—24no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—44no—
289C4×D4—44no—
290#117—44no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
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—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4SD1618yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4#4114yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—44no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#106—24no—
362#106—28no—
363#106—24no—
364#106—44no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367#102—24no—
368#102—28no—
369#102—28no—
370#102—24no—
371#102—44no—
372#102—44no—
373#102—44no—
374C22⋊C8—24no—
375C22⋊C8—22no—
376C22⋊C8—22no—
377C4×C8—24no—
378C4×C8—44no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—21no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—21no—
393C24—22no—
394C24C22×D411yescharacteristic
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—84no—
400C2×Q8—84no—
401C2×Q8—832no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—24no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—81no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—82no—
427C2×D4—81no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4C2×SD1614yes—
463C22×C4C2×SD1618yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4112yes—
474C22×C42−1+412yes—
475C22×C4C2511yescharacteristic
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—81no—
513C4⋊C4—82no—
514C4⋊C4C2×SD1614yes—
515C4⋊C4C2×SD1614yes—
516C4⋊C4C2×SD1618yes—
517C4⋊C4AGL(1,ℤ/8)14yes—
518C4⋊C4C2×#4114yes—
519C4⋊C4H(5,𝔽2)14yes—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—81no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C42—22no—
536C42—22no—
537C42—24no—
538C42—24no—
539C42—22no—
540C42—42no—
541C42—42no—
542C42—44no—
543C42—44no—
544C42—42no—
545C42—44no—
546C422−1+414yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—42no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23—42no—
560C23—41no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23#54812yes—
566C23C22×SD1612yes—
567C23#57612yes—
568C23C23×D411yescharacteristic
569Q8—816no—
570Q8—816no—
571D4—44no—
572D4—42no—
573D4—44no—
574D4—42no—
575D4—88no—
576D4—84no—
577D4—84no—
578D4—164no—
579D4—164no—
580D4—162no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—88no—
629C2×C4—84no—
630C2×C4—164no—
631C2×C4#53612yes—
632C2×C4#53612yes—
633C2×C4#54812yes—
634C2×C4C22×SD1614yes—
635C2×C4C22×SD1612yes—
636C2×C4C22×SD1614yes—
637C2×C4C22×SD1612yes—
638C2×C4C2×AGL(1,ℤ/8)14yes—
639C2×C4C2×AGL(1,ℤ/8)12yes—
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×H(5,𝔽2)12yes—
642C2×C4C2×2−1+412yes—
643C2×C4C2×2−1+411yescharacteristic
644C8—44no—
645C8—42no—
646C8—42no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—81no—
654C22—82no—
655C22—82no—
656C22—81no—
657C22—82no—
658C22—81no—
659C22—81no—
660C22—82no—
661C22—81no—
662C22—82no—
663C22—81no—
664C22—82no—
665C22—82no—
666C22—81no—
667C22#336214yes—
668C22C2×#53611yescharacteristic
669C22C2×#54811yescharacteristic
670C22#348411yescharacteristic
671C22C23×SD1611yescharacteristic
672C22C22×AGL(1,ℤ/8)11yescharacteristic
673C22C2×#57611yescharacteristic
674C22#358611yescharacteristic
675C4—22no—
676C4—24no—
677C4—24no—
678C4—24no—
679C4—22no—
680C4—22no—
681C4—44no—
682C4—42no—
683C4—44no—
684C4—41no—
685C4—82no—
686C4—81no—
687C4—81no—
688C4—84no—
689C4—81no—
690C4#322714yes—
691C4#351112yes—
692C4#355212yes—
693C2—24no—
694C2—162no—
695C2—161no—
696C2#3972011yescharacteristic
697C2C2×#322711yescharacteristic
698C2#6087111yescharacteristic
699C2C2×#336211yescharacteristic
700C2#6131711yescharacteristic
701C2#6156211yescharacteristic
702C2#6159111yescharacteristic
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×#2551C2semidirect
3. #33107C2semidirect
4. #32230C2semidirect
5. #30813C2semidirect
6. #29755C2semidirect
7. #28791C2semidirect
8. #27973C2semidirect
9. #27767C2semidirect
10. #27678C2semidirect
11. #26464C2semidirect
12. #26024C2semidirect
13. #25903C2semidirect
14. #24299C2semidirect
15. #22625C2semidirect
16. #22076C2semidirect
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×#25512
#331072
#322302
#308132
#297552
#287912
#279732
#277672
#276782
#264642
#260242
#259032
#242992
#226252
#220762
C4×#4812