Finite Group Databasebeta

← #6436856

Subgroups — #6436856

Order 512.

1918 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3#29995C214yesmaximal, complemented
4#29981C212yesmaximal, complemented
5#29980C212yesmaximal, complemented
6#29969C212yesmaximal, complemented
7#29967C212yesmaximal, complemented
8#29952C212yesmaximal
9#29950C212yesmaximal
10#23781C214yesmaximal, complemented
11#23780C214yesmaximal, complemented
12#21365C212yesmaximal, complemented
13#20810C211yesmaximal, characteristic, complemented
14#20738C212yesmaximal
15C2×#533—21no—
16C22×C4×D4C2211yescharacteristic
17C22×#117—21no—
18C2×#432C2212yes—
19#2868C2211yescharacteristic, complemented
20#2868C2211yescharacteristic, complemented
21#2862—21no—
22#2859—21no—
23#2859C2212yescomplemented
24#2857—21no—
25#2857—21no—
26#2857C2212yes—
27#2857C2212yes—
28C4×C4⋊Q8—21no—
29C4×C4⋊Q8—21no—
30C4×C4⋊Q8C2212yescomplemented
31C4×C4⋊Q8C2212yescomplemented
32#2293—21no—
33#2293—21no—
34#2293—21no—
35#2293—21no—
36#2293C2212yes—
37#2293C2212yes—
38#2293C2212yes—
39#2293C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#388—21no—
46#2285—21no—
47#2285C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4—21no—
50C2×C43C2211yescharacteristic
51#2100C2214yescomplemented
52#2099C2214yescomplemented
53#2094C2214yes—
54#2093C2214yes—
55#2088C2218yescomplemented
56#2087C2218yescomplemented
57#2067C2218yes—
58#2065C2218yes—
59#2058C2214yescomplemented
60#2057C2214yescomplemented
61#2023C2214yes—
62#2023C2214yes—
63#1948C2214yescomplemented
64#1947C2214yescomplemented
65#1853C2214yes—
66#1852C2214yes—
67#1842C2212yescomplemented
68#1831C2212yescomplemented
69#1824C2214yes—
70#1802C2214yes—
71#1802C2214yes—
72#1797C2212yes—
73#1797C2212yes—
74#1786C2214yes—
75#1776C2218yescomplemented
76#1767C2212yescomplemented
77#1767C2212yescomplemented
78#1762C2218yes—
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81#533—48no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#125—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—24no—
96C2×C4×D4Q818yescomplemented
97C2×C4×D4C2312yes—
98C2×C4×D4C2312yes—
99C2×#117—22no—
100C2×#117—22no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—22no—
104C2×#117—48no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C42—21no—
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#432—28no—
124C2×#107C2318yes—
125C2×#106C2318yes—
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2314yes—
129C2×#102C2318yes—
130C2×#102C2318yes—
131C2×C22⋊C8C2314yes—
132C2×#97C2314yes—
133#391—22no—
134#391—22no—
135#391—22no—
136#391—22no—
137#390—21no—
138#390—21no—
139#390—21no—
140#390—21no—
141#390C2312yes—
142#390C2312yes—
143#390C2312yes—
144#390C2312yes—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389C2314yes—
156#389C2314yes—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#384—21no—
170#384—21no—
171#384—21no—
172#384—21no—
173#384C2312yes—
174#384C2312yes—
175#384C2312yes—
176#384C2312yes—
177#382—22no—
178#382—22no—
179#380—22no—
180#380—22no—
181#380—22no—
182#380C2314yes—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C43D411yescharacteristic
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43C2311yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206#340C2318yes—
207#340C2318yes—
208#340C2318yes—
209#340C2318yes—
210#336C2314yes—
211#336C2314yes—
212C25—21no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×D4—21no—
216C22×D4—22no—
217C22×D4—42no—
218C22×D4—42no—
219C22×D4C2×Q814yes—
220C22×D4C2412yes—
221C23×C4—21no—
222C23×C4—28no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2411yescharacteristic
236C22×C8C2418yes—
237C4⋊Q8—44no—
238C4⋊Q8—44no—
239C4⋊Q8—44no—
240C4⋊Q8—44no—
241C4⋊Q8—44no—
242C4⋊Q8—44no—
243#125—48no—
244C22⋊Q8—48no—
245C22⋊Q8—48no—
246C4×Q8—48no—
247C4×Q8—48no—
248C4×D4—28no—
249C4×D4—28no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4C2×Q818yes—
253C4×D4C2×Q818yes—
254#117—48no—
255#117—48no—
256#117—48no—
257#117—48no—
258#117—48no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×Q814yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42—44no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×Q814yes—
338C2×C42#4114yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—48no—
346#106—48no—
347C4⋊C8—28no—
348C4⋊C8—44no—
349C4⋊C8—44no—
350#102—48no—
351#102—48no—
352C22⋊C8—28no—
353#97—28no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364C24—24no—
365C24—22no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24C22×D411yescharacteristic
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×D4—24no—
380C2×D4—22no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—48no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—48no—
387C2×D4C22×Q814yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—432no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4C22⋊Q814yes—
426C22×C4C22⋊Q814yes—
427C22×C4AGL(1,ℤ/8)14yes—
428C22×C4#13714yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×Q812yes—
443C22×C4C2×#4112yes—
444C22×C4C2511yescharacteristic
445C2×C8—28no—
446C2×C8—28no—
447C2×C8—28no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4C22×Q814yes—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—48no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—44no—
508C42—48no—
509C42—44no—
510C42C22⋊Q814yes—
511C42C22⋊Q814yes—
512C42AGL(1,ℤ/8)14yes—
513C42C22×Q812yes—
514C42C22×Q812yes—
515C42C2×#4112yes—
516C42C2×#4112yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—24no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23C2×C22⋊Q812yes—
536C23C2×AGL(1,ℤ/8)12yes—
537C23C2×#13712yes—
538C23C23×D411yescharacteristic
539Q8—88no—
540Q8—88no—
541D4—28no—
542D4—48no—
543D4—44no—
544C2×C4—28no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—416no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—416no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4C2×C22⋊Q812yes—
599C2×C4C2×C22⋊Q812yes—
600C2×C4C2×C22⋊Q812yes—
601C2×C4C2×C22⋊Q812yes—
602C2×C4C2×C22⋊Q812yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4C2×#13712yes—
609C2×C4C2×#13712yes—
610C2×C4#57512yes—
611C2×C4C23×Q811yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4#58512yes—
614C2×C4#58512yes—
615C8—48no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—816no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22#307914yes—
631C22C22×C22⋊Q811yescharacteristic
632C22#344911yescharacteristic
633C22#344911yescharacteristic
634C22C22×AGL(1,ℤ/8)11yescharacteristic
635C22C22×AGL(1,ℤ/8)11yescharacteristic
636C22C22×#13711yescharacteristic
637C22C2×#57511yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—44no—
651C4—84no—
652C4—82no—
653C4—84no—
654C4—82no—
655C4#307714yes—
656C4#345512yes—
657C4#345812yes—
658C2—24no—
659C2—84no—
660C2#3463411yescharacteristic
661C2C2×#307711yescharacteristic
662C2C2×#307911yescharacteristic
663C2#6095811yescharacteristic
664C2#6096011yescharacteristic
665C2#6096811yescharacteristic
666C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #29995C2semidirect
4. #29981C2semidirect
5. #29980C2semidirect
6. #29969C2semidirect
7. #29967C2semidirect
10. #23781C2semidirect
11. #23780C2semidirect
12. #21365C2semidirect
13. #20810C2semidirect
19. #2868C22semidirect
20. #2868C22semidirect
23. #2859C22semidirect
30. C4×C4⋊Q8C22semidirect
31. C4×C4⋊Q8C22semidirect
51. #2100C22semidirect
52. #2099C22semidirect
55. #2088C22semidirect
56. #2087C22semidirect
59. #2058C22semidirect
60. #2057C22semidirect
63. #1948C22semidirect
64. #1947C22semidirect
67. #1842C22semidirect
68. #1831C22semidirect
75. #1776C22semidirect
76. #1767C22semidirect
77. #1767C22semidirect
96. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#366452
C2×#22932
#299952
#299812
#299802
#299692
#299672
#299522
#299502
#237812
#237802
#213652
#208102
#207382