Finite Group Databasebeta

← #6464007

Subgroups — #6464007

Order 512.

2224 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic, complemented
2C2×#3141C211yesmaximal, characteristic
3#36639C211yesmaximal, characteristic
4C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
5#28659C214yesmaximal, complemented
6#28658C214yesmaximal, complemented
7#28656C212yesmaximal, complemented
8#28655C212yesmaximal, complemented
9#28654C212yesmaximal, complemented
10#28653C212yesmaximal, complemented
11#23617C214yesmaximal, complemented
12#21322C211yesmaximal, characteristic, complemented
13#21200C211yesmaximal, characteristic
14#21059C214yesmaximal, complemented
15C4×#423C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8—21no—
20#3165—28no—
21C2×#501C2214yes—
22C2×#500C2212yes—
23C2×#498C2212yes—
24#3141—28no—
25C2×#489C2214yes—
26C2×#487C2212yes—
27C2×#486C2212yes—
28C2×#428C2214yes—
29C2×#423C2212yes—
30C2×#417C2214yes—
31C2×#417C2214yes—
32C2×C4×M16C2211yescharacteristic
33C2×C4×M16C2211yescharacteristic
34#2862C2211yescharacteristic
35#2862C2211yescharacteristic
36#2861C2212yes—
37#2861C2212yes—
38#2858C2211yescharacteristic
39#2858C2211yescharacteristic
40#2850—21no—
41#2850—21no—
42#2850—21no—
43#2850—21no—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8—21no—
47C4×C4⋊Q8—21no—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2212yes—
50C4×C4⋊Q8C2212yes—
51C4×C4⋊Q8C2212yes—
52C4×#124—21no—
53C4×#124—21no—
54C4×#124—21no—
55C4×#124—21no—
56#2297C2212yes—
57#2297C2212yes—
58C2×#391—21no—
59C2×#389—21no—
60C2×#389C2212yes—
61C2×#384C2211yescharacteristic
62C2×C4×C4⋊C4—21no—
63C2×C4×C4⋊C4C2212yes—
64C2×C43C2211yescharacteristic
65#2068C2218yes—
66#2067C2218yes—
67#2066C2218yes—
68#2065C2218yes—
69#1992C2214yes—
70#1990C2212yes—
71#1990C2212yes—
72#1959C2214yes—
73#1957C2212yes—
74#1956C2212yes—
75#1928C2218yes—
76#1927C2218yes—
77#1926C2218yes—
78#1925C2218yes—
79#1767C2212yes—
80#1767C2212yes—
81C4×C4⋊C8C2212yes—
82C4×C4⋊C8C2212yes—
83#1736C2214yes—
84#1728C2212yes—
85C22×#41C2312yes—
86C23×Q8—21no—
87C22×M16C2312yes—
88C22×M16C2312yes—
89#533—24no—
90#533—24no—
91#532—24no—
92#532—24no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8D418yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#125C2312yes—
104C2×#124—21no—
105C2×#124—21no—
106C2×#124—21no—
107C2×#124—21no—
108C2×#124C2312yes—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×#121—21no—
112C2×#121—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—48no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×#117C2312yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C42—22no—
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#501—216no—
138#500—28no—
139#498—28no—
140#489—216no—
141#487—28no—
142#486—28no—
143#428—216no—
144C2×#107C2318yes—
145C2×#106C2318yes—
146#423—28no—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149#417—216no—
150#417—216no—
151C2×#103C2318yes—
152C2×#103C2318yes—
153C2×#102C2318yes—
154C2×#102C2318yes—
155C4×M16—24no—
156C4×M16—24no—
157C2×#97C2312yes—
158C2×#97C2312yes—
159C2×C4×C8C2312yes—
160C2×C4×C8C2312yes—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391C2312yes—
170#391C2312yes—
171#391C2312yes—
172#391C2312yes—
173#390—21no—
174#390—21no—
175#390—21no—
176#390—21no—
177#390C2312yes—
178#390C2312yes—
179#390C2312yes—
180#390C2312yes—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389C2314yes—
188#389C2314yes—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#386—22no—
198#386—22no—
199#386—22no—
200#386—22no—
201#384—21no—
202#384—21no—
203#384C2312yes—
204#384C2312yes—
205#382—21no—
206#382—21no—
207#382—21no—
208#382—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2314yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C2×#95—22no—
222C2×#95—22no—
223C43D411yescharacteristic
224C43D411yescharacteristic
225C43D411yescharacteristic
226C43D411yescharacteristic
227C43C2311yescharacteristic
228C43C2311yescharacteristic
229C43C2311yescharacteristic
230C43C2311yescharacteristic
231#340C2318yes—
232#340C2318yes—
233#336C2314yes—
234#336C2314yes—
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—28no—
238C2×#41—28no—
239C22×Q8—28no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8C2412yes—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2412yes—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×M16—24no—
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16#4118yes—
267C22×C8C2414yes—
268C22×C8C2414yes—
269C4⋊Q8—28no—
270C4⋊Q8—28no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8C2×D418yes—
280C4⋊Q8C2×D418yes—
281#127—22no—
282#127—22no—
283#125—24no—
284#125—24no—
285#124—24no—
286#124—24no—
287#124—44no—
288#124—44no—
289#124—44no—
290#124—44no—
291C22⋊Q8—44no—
292C22⋊Q8—44no—
293#121—44no—
294#121—44no—
295C4×Q8—48no—
296C4×Q8—48no—
297C4×Q8—48no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×Q8—48no—
301C4×D4—44no—
302C4×D4—44no—
303#117—24no—
304#117—24no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—48no—
327C2×C4⋊C4C2×D418yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—42no—
351C2×C42—42no—
352C2×C42—48no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D414yes—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42#4114yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—216no—
384#106—216no—
385C4⋊C8—28no—
386C4⋊C8—28no—
387#103—216no—
388#103—216no—
389#102—216no—
390#102—216no—
391#97—24no—
392#97—24no—
393C4×C8—24no—
394C4×C8—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407C24—22no—
408C24C22×D411yescharacteristic
409#41—416no—
410#41—416no—
411C2×Q8—28no—
412C2×Q8—28no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—432no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—44no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—24no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4#12114yes—
472C22×C4#12114yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4114yes—
489C22×C42−1+414yes—
490C22×C4C2511yescharacteristic
491M16—216no—
492M16—48no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8C2×#4118yes—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—28no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4C22×D418yes—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—48no—
556C42—44no—
557C42—48no—
558C42#12114yes—
559C42#12114yes—
560C42AGL(1,ℤ/8)14yes—
561C42#13714yes—
562C42C22×D412yes—
563C42C22×D412yes—
564C42C2×#4112yes—
565C42C2×#4112yes—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23C2×#12112yes—
578C23#53212yes—
579C23#53612yes—
580C23C23×D411yescharacteristic
581Q8—416no—
582Q8—416no—
583Q8—44no—
584Q8—44no—
585Q8—84no—
586Q8—84no—
587D4—44no—
588D4—44no—
589D4—88no—
590D4—84no—
591D4—84no—
592C2×C4—28no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—416no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—88no—
639C2×C4—88no—
640C2×C4C2×#12112yes—
641C2×C4C2×#12112yes—
642C2×C4C2×#12112yes—
643C2×C4C2×#12112yes—
644C2×C4C2×#12112yes—
645C2×C4#53212yes—
646C2×C4#53612yes—
647C2×C4#54818yes—
648C2×C4#55618yes—
649C2×C4C2×AGL(1,ℤ/8)12yes—
650C2×C4C2×AGL(1,ℤ/8)12yes—
651C2×C4C2×AGL(1,ℤ/8)12yes—
652C2×C4C2×#13712yes—
653C2×C4C2×#13712yes—
654C2×C4C2×#13712yes—
655C2×C4#57512yes—
656C2×C4#57512yes—
657C2×C4C23×D411yescharacteristic
658C2×C4C22×#4112yes—
659C2×C4C22×#4111yescharacteristic
660C2×C4C2×2−1+412yes—
661C8—28no—
662C8—44no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22#254514yes—
677C22C22×#12111yescharacteristic
678C22C2×#53211yescharacteristic
679C22C2×#53611yescharacteristic
680C22C22×AGL(1,ℤ/8)11yescharacteristic
681C22C22×#13711yescharacteristic
682C22C2×#57511yescharacteristic
683C22C2×#57511yescharacteristic
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—48no—
695C4—84no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4#254514yes—
700C4#303614yes—
701C4#303714yes—
702C4C2×#54812yes—
703C4C2×#55612yes—
704C2—24no—
705C2—162no—
706C2C2×#254511yescharacteristic
707C2C2×#303611yescharacteristic
708C2C2×#303711yescharacteristic
709C2#6070011yescharacteristic
710C2#6070111yescharacteristic
711C2#6086011yescharacteristic
712C2#6086211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. C2×#3165C2semidirect
4. C2×C4×C4⋊Q8C2semidirect
5. #28659C2semidirect
6. #28658C2semidirect
7. #28656C2semidirect
8. #28655C2semidirect
9. #28654C2semidirect
10. #28653C2semidirect
11. #23617C2semidirect
12. #21322C2semidirect
14. #21059C2semidirect
15. C4×#423C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
C2×#31412
#366392
C2×C4×C4⋊Q82
#286592
#286582
#286562
#286552
#286542
#286532
#236172
#213222
#212002
#210592
C4×#4232