Finite Group Databasebeta

← #6507902

Subgroups — #6507902

Order 512.

1994 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic, complemented
2C2×#2420C211yesmaximal, characteristic, complemented
3#29109C211yesmaximal, characteristic, complemented
4#28992C214yesmaximal, complemented
5#28862C214yesmaximal, complemented
6#28310C211yesmaximal, characteristic, complemented
7#27982C211yesmaximal, characteristic, complemented
8#27956C211yesmaximal, characteristic, complemented
9#26968C214yesmaximal, complemented
10#25038C214yesmaximal, complemented
11#24557C212yesmaximal, complemented
12#24384C212yesmaximal, complemented
13#22076C212yesmaximal, complemented
14C4×#447C212yesmaximal, complemented
15#20524C211yesmaximal, characteristic, complemented
16C2×#538—21no—
17C22×#121C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#481C2212yes—
20C2×#469C2214yes—
21C2×#447C2212yes—
22C2×C4×D8C2214yescomplemented
23C2×#407C2212yes—
24#2857C2211yescharacteristic
25#2856C2211yescharacteristic
26#2847C2212yes—
27#2639—21no—
28#2617—21no—
29#2559—22no—
30#2557—22no—
31#2496C2212yescomplemented
32#2491C2212yescomplemented
33#2428C2212yescomplemented
34#2422C2212yescomplemented
35#2420—22no—
36#2420—22no—
37#2420C2214yes—
38#2420C2214yes—
39C2×#397—21no—
40C2×#394—21no—
41#2325—21no—
42C4×#121C2212yes—
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#386—21no—
46C2×#382C2211yescharacteristic
47C2×#381—21no—
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2094C2212yescomplemented
51#2085C2214yescomplemented
52#2077C2214yescomplemented
53#2073C2214yescomplemented
54#2071C2212yescomplemented
55#2034C2214yes—
56#2033C2214yescomplemented
57#2023C2212yescomplemented
58#2011C2212yescomplemented
59#1999C2214yes—
60#1993C2212yescomplemented
61#1972C2214yescomplemented
62#1961C2212yescomplemented
63#1941C2214yescomplemented
64#1936C2214yescomplemented
65#1934C2214yescomplemented
66#1920C2214yescomplemented
67#1895C2214yes—
68#1895C2214yescomplemented
69#1879C2214yescomplemented
70#1874C2212yes—
71#1874C2212yes—
72#1864C2214yescomplemented
73#1862C2214yescomplemented
74#1853C2212yes—
75#1852C2212yes—
76#1847C2212yescomplemented
77#1836C2212yescomplemented
78#1824C2212yes—
79#1809C2214yescomplemented
80#1786C2212yes—
81C4×#102C2214yescomplemented
82#1753C2214yescomplemented
83#1753C2214yescomplemented
84#1749C2211yescharacteristic
85#1749C2211yescharacteristic
86C4×C22⋊C8C2212yes—
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×D8C2314yes—
92C22×M16C2314yes—
93#538—48no—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#126—22no—
97C2×#124—22no—
98C2×#123—22no—
99C2×C22⋊Q8—21no—
100C2×#121—21no—
101C2×#121—21no—
102C2×#121—22no—
103C2×#121—22no—
104C2×#121—44no—
105C2×#121C2312yes—
106C2×#121C2314yes—
107C2×#121C2312yes—
108C2×#121C2312yes—
109C2×#120—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—44no—
113C2×C4×D4C2314yes—
114C2×C4×D4C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#481—28no—
129#469—48no—
130#447—28no—
131C4×D8—48no—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141#407—28no—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×#97C2312yes—
146C2×#97C2312yes—
147C2×C4×C8C2314yes—
148#400—22no—
149#397—22no—
150#397—22no—
151#397—24no—
152#397—24no—
153#394—22no—
154#394—22no—
155#394—24no—
156#394—24no—
157#394—22no—
158#393—22no—
159#392—22no—
160#391—21no—
161#390—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#389—21no—
170#389—22no—
171#389C2312yes—
172#389C2312yes—
173#389C2314yes—
174#388—22no—
175#388—22no—
176#387—22no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#386—24no—
182#386—24no—
183#386—22no—
184#385—22no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#382—21no—
188#382—21no—
189#382C2312yes—
190#382C2312yes—
191#381—24no—
192#381—24no—
193#380—21no—
194#380—22no—
195#380C2312yes—
196#380C2314yes—
197#380C2312yes—
198#380C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4C2312yes—
204C4×C22⋊C4C2312yes—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C25—21no—
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×D4—24no—
224C22×D4—22no—
225C22×D4—24no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—24no—
232C22×D4—42no—
233C22×D4—44no—
234C22×D4—44no—
235C22×D4—82no—
236C22×D4C2412yes—
237C22×D4C2412yes—
238C23×C4—24no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2412yes—
256C23×C4C2411yescharacteristic
257C2×D8—28no—
258C2×D8—28no—
259C2×D8—44no—
260C2×M16—28no—
261C2×M16—44no—
262C2×M16—44no—
263C22×C8C2414yes—
264C22×C8C2414yes—
265C4⋊Q8—42no—
266#127—42no—
267#126—48no—
268#124—48no—
269#123—48no—
270C22⋊Q8—44no—
271#121—28no—
272#121—24no—
273#121—24no—
274#121—48no—
275#121—44no—
276#121—44no—
277#121—44no—
278#121—48no—
279#121—44no—
280#121—44no—
281#120—48no—
282C4×D4—28no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—48no—
286C4×D4—48no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—82no—
331C2×C22⋊C4—82no—
332C2×C22⋊C4—82no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—22no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346#107—44no—
347#106—28no—
348#106—44no—
349#102—28no—
350#102—28no—
351#102—28no—
352#102—44no—
353#102—44no—
354#102—44no—
355C22⋊C8—24no—
356C22⋊C8—28no—
357C22⋊C8#4118yes—
358#97—42no—
359#97—42no—
360C4×C8—44no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—24no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—41no—
384C24—41no—
385C24—41no—
386C24—41no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24C22×D412yes—
394C24C22×D411yescharacteristic
395C2×Q8—42no—
396C2×Q8—81no—
397C2×Q8—81no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—416no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—84no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—82no—
427C2×D4—816no—
428C2×D4H(5,𝔽2)14yes—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—41no—
458C22×C4—48no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—88no—
471C22×C4—84no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—816no—
475C22×C4AGL(1,ℤ/8)14yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C2×#4114yes—
485C22×C4C2511yescharacteristic
486D8—416no—
487M16—88no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8C2×#4118yes—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—81no—
510C4⋊C4—84no—
511C4⋊C4—81no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—84no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—41no—
538C42—41no—
539C42—42no—
540C42—44no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—48no—
554C23—48no—
555C23—44no—
556C23—48no—
557C23—44no—
558C23—42no—
559C23—42no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23—82no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23#53212yes—
581C23C2×AGL(1,ℤ/8)12yes—
582C23#57512yes—
583C23C23×D411yescharacteristic
584Q8—162no—
585D4—28no—
586D4—48no—
587D4—44no—
588D4—44no—
589D4—44no—
590D4—816no—
591D4—88no—
592D4—162no—
593D4—162no—
594D4—164no—
595D4—164no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—28no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—84no—
641C2×C4—816no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—164no—
650C2×C4#53212yes—
651C2×C4C2×AGL(1,ℤ/8)14yes—
652C2×C4C22×#4112yes—
653C2×C4C2×H(5,𝔽2)12yes—
654C2×C4#58512yes—
655C8—28no—
656C8—44no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—44no—
669C22—44no—
670C22—42no—
671C22—42no—
672C22—44no—
673C22—88no—
674C22—816no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22#317114yes—
687C22C2×#53211yescharacteristic
688C22#348411yescharacteristic
689C22#348511yescharacteristic
690C22C22×AGL(1,ℤ/8)11yescharacteristic
691C22C2×#57511yescharacteristic
692C22#358511yescharacteristic
693C22#358511yescharacteristic
694C4—22no—
695C4—24no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—42no—
701C4—42no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4#351314yes—
709C2—24no—
710C2—82no—
711C2—84no—
712C2—162no—
713C2#3539411yescharacteristic
714C2C2×#317111yescharacteristic
715C2#6070411yescharacteristic
716C2#6156911yescharacteristic
717C2#6157611yescharacteristic
718C2#6171811yescharacteristic
719C2#6172011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #35005C2semidirect
2. C2×#2420C2semidirect
3. #29109C2semidirect
4. #28992C2semidirect
5. #28862C2semidirect
6. #28310C2semidirect
7. #27982C2semidirect
8. #27956C2semidirect
9. #26968C2semidirect
10. #25038C2semidirect
11. #24557C2semidirect
12. #24384C2semidirect
13. #22076C2semidirect
14. C4×#447C2semidirect
15. #20524C2semidirect
22. C2×C4×D8C22semidirect
31. #2496C22semidirect
32. #2491C22semidirect
33. #2428C22semidirect
34. #2422C22semidirect
50. #2094C22semidirect
51. #2085C22semidirect
52. #2077C22semidirect
53. #2073C22semidirect
54. #2071C22semidirect
56. #2033C22semidirect
57. #2023C22semidirect
58. #2011C22semidirect
60. #1993C22semidirect
61. #1972C22semidirect
62. #1961C22semidirect
63. #1941C22semidirect
64. #1936C22semidirect
65. #1934C22semidirect
66. #1920C22semidirect
68. #1895C22semidirect
69. #1879C22semidirect
72. #1864C22semidirect
73. #1862C22semidirect
76. #1847C22semidirect
77. #1836C22semidirect
79. #1809C22semidirect
81. C4×#102C22semidirect
82. #1753C22semidirect
83. #1753C22semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#24202
#291092
#289922
#288622
#283102
#279822
#279562
#269682
#250382
#245572
#243842
#220762
C4×#4472
#205242