Finite Group Databasebeta

← #6461288

Subgroups — #6461288

Order 512.

1986 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic, complemented
2#34267C211yesmaximal, characteristic
3#29173C214yesmaximal, complemented
4#29168C214yesmaximal, complemented
5#29165C212yesmaximal, complemented
6#29163C212yesmaximal, complemented
7#29159C212yesmaximal, complemented
8#29157C212yesmaximal, complemented
9C2×#2099C211yesmaximal, characteristic, complemented
10#23734C211yesmaximal, characteristic, complemented
11#23554C211yesmaximal, characteristic, complemented
12#23451C211yesmaximal, characteristic, complemented
13#21036C212yesmaximal, complemented
14#20976C212yesmaximal
15#20878C212yesmaximal, complemented
16#20664C212yesmaximal, complemented
17#20336C211yesmaximal, characteristic, complemented
18C22×#131C2212yes—
19C2×#537C2211yescharacteristic
20C2×#533C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C2×#428C2212yes—
24C2×#428C2212yes—
25C2×#417C2212yes—
26C2×#416C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2629C2214yes—
30#2582—21no—
31#2582—21no—
32#2538—21no—
33#2538—21no—
34#2417—22no—
35#2417—22no—
36#2364—22no—
37#2363—21no—
38#2362—21no—
39#2361—22no—
40#2361—21no—
41#2360—21no—
42#2324C2212yes—
43#2323C2212yes—
44#2316C2212yes—
45#2311C2211yescharacteristic
46#2311C2212yes—
47C4×C22⋊Q8C2212yes—
48C4×#121C2212yes—
49#2297C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#384C2211yescharacteristic
52#2274C2211yescharacteristic
53#2099C2212yes—
54#2099C2212yes—
55#2099C2214yes—
56#2096C2212yes—
57#2095C2212yes—
58#2094C2212yes—
59#2093C2212yes—
60#2059C2214yes—
61#2058C2212yes—
62#2057C2212yes—
63#2026C2212yes—
64#2025C2212yes—
65#2024C2211yescharacteristic
66#2024C2211yescharacteristic
67#2023C2211yescharacteristic
68#2023C2211yescharacteristic
69#1921C2214yes—
70#1920C2214yes—
71#1919C2214yes—
72#1918C2214yes—
73#1895C2214yes—
74#1894C2214yes—
75#1893C2214yes—
76#1892C2214yes—
77#1865C2214yes—
78#1864C2214yes—
79#1863C2214yes—
80#1862C2214yes—
81#1809C2214yes—
82#1809C2214yes—
83#1808C2214yes—
84#1807C2214yes—
85#1736C2212yes—
86#1735C2212yes—
87C2×#340C2212yes—
88C2×#340C2212yes—
89#1729C2212yes—
90#1729C2214yes—
91#1728C2211yescharacteristic
92#1728C2212yes—
93C2×#336C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41C2311yescharacteristic
96C22×#41C2311yescharacteristic
97C2×#131—28no—
98C2×#131—28no—
99C2×#131—44no—
100C22×M16C2314yes—
101C22×M16C2312yes—
102C23×C8C2312yes—
103C23×C8C2314yes—
104#537—24no—
105#537Q818yescomplemented
106#533—44no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×#125C2312yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8C2312yes—
116C2×#121—21no—
117C2×#121—22no—
118C2×#121C2312yes—
119#520—44no—
120C4×#41—44no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8D411yescharacteristic
124C2×C4×Q8D411yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×D4—22no—
127C2×C4×D4—22no—
128C2×C4×D4D412yes—
129C2×C4×D4D411yescharacteristic
130C2×C4×D4D411yescharacteristic
131C2×C4×D4D412yes—
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C2×#117C2312yes—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2312yes—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4C2312yes—
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C42C2311yescharacteristic
146C22×C42C2312yes—
147#428—28no—
148#428—44no—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153#417—28no—
154#416—44no—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#102C2314yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161#398—24no—
162#398—24no—
163#396—24no—
164#396—24no—
165#391—22no—
166#391—21no—
167#391—22no—
168#391—22no—
169#391C2311yescharacteristic
170#391C2311yescharacteristic
171#390—21no—
172#390—22no—
173#390—22no—
174#390C2311yescharacteristic
175#390C2311yescharacteristic
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—21no—
182#389—21no—
183#389C2312yes—
184#389C2312yes—
185#389C2314yes—
186#389C2314yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—22no—
192#387—22no—
193#386—22no—
194#386—24no—
195#386—21no—
196#386—22no—
197#386—21no—
198#386—22no—
199#385—22no—
200#385—22no—
201#384—21no—
202#384—21no—
203#384C2312yes—
204#384C2312yes—
205#380—22no—
206#380—22no—
207#380—24no—
208#380C2312yes—
209#380C2314yes—
210#380C2312yes—
211#380C2314yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C2×#95—21no—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2311yescharacteristic
236#336C2311yescharacteristic
237#336C2312yes—
238#336C2314yes—
239#336C2314yes—
240C2×#41—24no—
241C2×#41—22no—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41C2×Q814yes—
246C2×#41C2×Q814yes—
247C2×#41#4114yes—
248C22×Q8—21no—
249C22×Q8—21no—
250C22×Q8C2×D411yescharacteristic
251C22×Q8C2411yescharacteristic
252C22×D4—21no—
253C22×D4—22no—
254C22×D4C2×D412yes—
255C22×D4C2×D411yescharacteristic
256C22×D4C2411yescharacteristic
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—22no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2412yes—
267C23×C4C2411yescharacteristic
268#131—432no—
269C2×M16—24no—
270C2×M16—24no—
271C2×M16—28no—
272C2×M16—28no—
273C2×M16—44no—
274C2×M16#4118yes—
275C22×C8—28no—
276C22×C8—24no—
277C22×C8—24no—
278C22×C8—24no—
279C22×C8—28no—
280C22×C8—44no—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2414yes—
284C4⋊Q8—42no—
285C4⋊Q8—42no—
286#125—42no—
287#125—42no—
288C22⋊Q8—28no—
289C22⋊Q8—24no—
290C22⋊Q8—44no—
291C22⋊Q8—44no—
292C22⋊Q8—84no—
293C22⋊Q8—84no—
294C22⋊Q8C2×Q818yes—
295#121—28no—
296#121—216no—
297#121—24no—
298#121C2×Q818yes—
299C4×Q8—42no—
300C4×Q8—42no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×D4—42no—
305C4×D4—42no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—44no—
309C4×D4—48no—
310C4×D4—84no—
311#117—24no—
312#117—24no—
313#117—24no—
314#117—24no—
315#117—44no—
316#117—44no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4—82no—
340C2×C4⋊C4—82no—
341C2×C4⋊C4—82no—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4C2×Q814yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2414yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4C2×D412yes—
368C2×C22⋊C4C2×D412yes—
369C2×C22⋊C4C2×D412yes—
370C2×C42—24no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—44no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D412yes—
380C2×C42C2×D412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#107—28no—
385#107—44no—
386#106—28no—
387#106—44no—
388#103—28no—
389#103—42no—
390#103—42no—
391#102—28no—
392#102—42no—
393#102—42no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—21no—
402#95—21no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408C24—21no—
409C24C22×D412yes—
410C24C22×D411yescharacteristic
411#41—216no—
412#41—216no—
413#41—216no—
414#41—48no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8C22⋊Q814yes—
427C2×Q8C22×Q812yes—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—28no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4C22⋊Q818yes—
442C2×D4C22⋊Q814yes—
443C2×D4C22×Q812yes—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—82no—
478C22×C4—81no—
479C22×C4—82no—
480C22×C4—81no—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×Q814yes—
490C22×C4C22×Q812yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4114yes—
494C22×C4H(5,𝔽2)14yes—
495C22×C4C2511yescharacteristic
496M16—216no—
497M16—416no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—416no—
507C2×C8—48no—
508C2×C8C2×#4118yes—
509C4⋊C4—24no—
510C4⋊C4—28no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—84no—
539C4⋊C4—84no—
540C4⋊C4C22×Q814yes—
541C22⋊C4—28no—
542C22⋊C4—24no—
543C22⋊C4—28no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4—42no—
548C22⋊C4—44no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—81no—
552C22⋊C4—81no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C22⋊C4—84no—
556C22⋊C4—84no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—24no—
562C42—42no—
563C42—42no—
564C42—44no—
565C42—42no—
566C42—44no—
567C42—82no—
568C42—82no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23C2×C22⋊Q814yes—
587C23C2×C22⋊Q812yes—
588C23#53212yes—
589C23#53412yes—
590C23C23×D411yescharacteristic
591Q8—28no—
592Q8—24no—
593Q8—42no—
594Q8—84no—
595Q8—84no—
596D4—216no—
597D4—28no—
598D4—24no—
599D4—48no—
600D4—42no—
601D4—88no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—48no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—164no—
653C2×C4—164no—
654C2×C4—162no—
655C2×C4C2×C22⋊Q814yes—
656C2×C4C2×C22⋊Q812yes—
657C2×C4#53212yes—
658C2×C4#53412yes—
659C2×C4#55118yes—
660C2×C4#55418yes—
661C2×C4#57612yes—
662C2×C4#57712yes—
663C2×C4C23×Q811yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C22×#4112yes—
666C2×C4C2×H(5,𝔽2)12yes—
667C8—28no—
668C8—42no—
669C8—42no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22#256414yes—
691C22C22×C22⋊Q811yescharacteristic
692C22C2×#53211yescharacteristic
693C22C2×#53411yescharacteristic
694C22C2×#57611yescharacteristic
695C22C2×#57711yescharacteristic
696C22#358611yescharacteristic
697C22#358611yescharacteristic
698C4—24no—
699C4—22no—
700C4—22no—
701C4—42no—
702C4—44no—
703C4—41no—
704C4—42no—
705C4—41no—
706C4—42no—
707C4—81no—
708C4—84no—
709C4—81no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4#256414yes—
714C4C2×#55112yes—
715C4C2×#55412yes—
716C2—24no—
717C2—44no—
718C2—82no—
719C2C2×#256411yescharacteristic
720C2#6049411yescharacteristic
721C2#6049711yescharacteristic
722C2#6072211yescharacteristic
723C2#6072611yescharacteristic
724C2#6076211yescharacteristic
725C2#6076311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35280C2semidirect
3. #29173C2semidirect
4. #29168C2semidirect
5. #29165C2semidirect
6. #29163C2semidirect
7. #29159C2semidirect
8. #29157C2semidirect
9. C2×#2099C2semidirect
10. #23734C2semidirect
11. #23554C2semidirect
12. #23451C2semidirect
13. #21036C2semidirect
15. #20878C2semidirect
16. #20664C2semidirect
17. #20336C2semidirect
105. #537Q8semidirect

Maximal subgroups

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

subgroupindex
#352802
#342672
#291732
#291682
#291652
#291632
#291592
#291572
C2×#20992
#237342
#235542
#234512
#210362
#209762
#208782
#206642
#203362