Finite Group Databasebeta

← #6442521

Subgroups — #6442521

Order 512.

4704 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3449C211yesmaximal, characteristic
2C2×#3037C212yesmaximal, complemented
3C2×#3007C212yesmaximal, complemented
4C2×#2323C211yesmaximal, characteristic, complemented
5#24221C218yesmaximal, complemented
6#24220C218yesmaximal, complemented
7#23508C212yesmaximal, complemented
8#23429C212yesmaximal, complemented
9#21258C212yesmaximal, complemented
10#20627C211yesmaximal, characteristic, complemented
11#20601C212yesmaximal, complemented
12C23×#41C2212yes—
13C22×#137C2214yescomplemented
14#3449—216no—
15C2×#537—21no—
16C2×#537—21no—
17C2×#536C2214yes—
18C2×#535C2214yes—
19C2×#534—21no—
20C2×#534—21no—
21C2×#532C2212yes—
22C2×#532C2212yes—
23C2×#525—24no—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic, complemented
26C22×#117C2211yescharacteristic
27#3037—216no—
28C2×#464C2218yescomplemented
29C2×#462C2214yescomplemented
30C2×#460C2214yescomplemented
31#3007—216no—
32C2×#452C2218yes—
33C2×#451C2214yescomplemented
34C2×#450C2214yescomplemented
35C2×#423C2212yescomplemented
36C2×#417C2214yescomplemented
37C2×#417C2214yescomplemented
38C2×#417C2214yescomplemented
39C2×#407C2212yes—
40#2323—24no—
41#2323C2218yescomplemented
42C2×#391C2211yescharacteristic, complemented
43C2×#389—21no—
44C2×#389C2211yescharacteristic, complemented
45C2×#388—21no—
46C2×#388—21no—
47C2×#387—21no—
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic, complemented
50#2043C2214yescomplemented
51#2042C2218yescomplemented
52#2041C2214yescomplemented
53#2016C2214yescomplemented
54#2015C2218yescomplemented
55#2014C2214yescomplemented
56#1983C2218yescomplemented
57#1982C2214yescomplemented
58#1981C2214yescomplemented
59#1798C2218yescomplemented
60#1788C2218yes—
61#1787C2214yescomplemented
62#1786C2214yescomplemented
63#1728C2212yescomplemented
64C22×#41—216no—
65C22×#41—22no—
66C22×#41—22no—
67C22×#41—28no—
68C22×#41—28no—
69C22×#41—48no—
70C22×#41C2314yes—
71C22×#41C2314yes—
72C23×Q8C2311yescharacteristic
73C23×Q8C2312yes—
74C23×D4—22no—
75C23×D4C2312yes—
76C24×C4—22no—
77C24×C4C2311yescharacteristic
78C2×#137—216no—
79C2×#137—216no—
80C2×#137—48no—
81C22×Q16C2318yes—
82C22×SD16C2318yes—
83C22×M16C2314yes—
84#537—216no—
85#537—216no—
86#536—216no—
87#536—216no—
88#535—216no—
89#535—216no—
90#534—216no—
91#534—216no—
92#532—216no—
93#532—28no—
94#532D4116yes—
95C2×C4⋊Q8C2312yes—
96C2×C4⋊Q8C2312yes—
97C2×#127C2312yes—
98C2×#127C2312yes—
99C2×#124C2314yes—
100C2×#124C2314yes—
101#525—264no—
102C2×#123—24no—
103C2×#123—24no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8D418yescomplemented
109C2×C22⋊Q8C2314yes—
110C2×#121—22no—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121—24no—
115C2×#121—24no—
116C2×#120—24no—
117C2×#120—24no—
118C2×C4×Q8—22no—
119C2×C4×Q8—28no—
120C2×C4×Q8C2314yes—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×#117—28no—
126C2×#117D414yes—
127C2×#117C2314yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—21no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4D411yescharacteristic
139C22×C22⋊C4D411yescharacteristic
140C22×C42—21no—
141C22×C42—21no—
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#464—232no—
145#462—216no—
146#460—216no—
147#452—232no—
148#451—216no—
149#450—216no—
150#423—28no—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153#417—216no—
154#417—216no—
155#417—48no—
156C2×#103C2318yes—
157C2×#103C2318yes—
158C2×#103C2318yes—
159C2×#102C2318yes—
160C2×#102C2318yes—
161C2×#102C2318yes—
162#407—28no—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2314yes—
165#391—22no—
166#391C2314yes—
167#389—28no—
168#389—22no—
169#389C2314yes—
170#388—28no—
171#388—28no—
172#387—28no—
173#380—22no—
174#380C2314yes—
175C4×C4⋊C4—22no—
176C4×C4⋊C4C2314yes—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182#336C2314yes—
183#336C2314yes—
184C25—22no—
185C25C2×D411yescharacteristic
186C2×#41—216no—
187C2×#41—216no—
188C2×#41—28no—
189C2×#41—28no—
190C2×#41—28no—
191C2×#41—28no—
192C2×#41—28no—
193C2×#41—28no—
194C2×#41—28no—
195C2×#41—28no—
196C2×#41—28no—
197C2×#41—44no—
198C2×#41—44no—
199C2×#41—464no—
200C2×#41—464no—
201C2×#41—48no—
202C2×#41—48no—
203C2×#41—48no—
204C2×#41—48no—
205C2×#41C2×D4116yes—
206C22×Q8—28no—
207C22×Q8—21no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—24no—
213C22×Q8—24no—
214C22×Q8—44no—
215C22×Q8C2×D414yes—
216C22×Q8C2412yes—
217C22×Q8C2414yes—
218C22×D4—216no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—28no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—24no—
228C22×D4—24no—
229C22×D4—44no—
230C22×D4—48no—
231C22×D4—48no—
232C22×D4C2414yes—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—216no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—24no—
240C23×C4—24no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—48no—
244C23×C4—48no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D414yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4#4114yes—
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255#137—464no—
256C2×Q16—216no—
257C2×Q16—216no—
258C2×Q16—48no—
259C2×SD16—216no—
260C2×SD16—216no—
261C2×SD16—48no—
262C2×M16—28no—
263C2×M16—28no—
264C2×M16—28no—
265C22×C8C2418yes—
266C4⋊Q8—28no—
267C4⋊Q8—24no—
268C4⋊Q8C2×D418yes—
269#127—28no—
270#127—24no—
271#127C2×D418yes—
272#124—28no—
273#124—28no—
274#124—28no—
275#124C2×D4116yes—
276#123—232no—
277#123—232no—
278C22⋊Q8—232no—
279C22⋊Q8—232no—
280C22⋊Q8—216no—
281C22⋊Q8—28no—
282C22⋊Q8C2×D4116yes—
283#121—232no—
284#121—232no—
285#121—216no—
286#121—216no—
287#121—216no—
288#121—216no—
289#120—232no—
290#120—232no—
291C4×Q8—28no—
292C4×Q8—28no—
293C4×Q8—48no—
294C4×D4—216no—
295C4×D4—216no—
296C4×D4—216no—
297C4×D4—216no—
298#117—216no—
299#117—28no—
300#117—28no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4C2×D414yes—
336C2×C22⋊C4C2×D414yes—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—44no—
341C2×C42—82no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D418yes—
345C2×C42C2412yes—
346C2×C42C2412yes—
347C4⋊C8—28no—
348C4⋊C8—28no—
349#103—216no—
350#103—216no—
351#103—48no—
352#102—216no—
353#102—216no—
354#102—48no—
355C22⋊C8—28no—
356C22⋊C8—28no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362C24—22no—
363C24—22no—
364C24—28no—
365C24—24no—
366C24—41no—
367C24—41no—
368C24—41no—
369C24—41no—
370C24—44no—
371C24—44no—
372C24—82no—
373C24—82no—
374C24—82no—
375C24—82no—
376C24#12114yes—
377C24C22×D412yes—
378C24C22×D411yescharacteristic
379#41—232no—
380#41—232no—
381#41—416no—
382#41—416no—
383#41—416no—
384#41—416no—
385#41—464no—
386#41—464no—
387#41—416no—
388#41—416no—
389C2×Q8—28no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—216no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—416no—
401C2×Q8—416no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—48no—
407C2×Q8C22×D418yes—
408C2×Q8C22×D414yes—
409C2×D4—216no—
410C2×D4—28no—
411C2×D4—216no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—416no—
422C2×D4—416no—
423C2×D4—44no—
424C2×D4—464no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—44no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4C22×D418yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—216no—
444C22×C4—216no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—216no—
448C22×C4—28no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—216no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—48no—
468C22×C4—464no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4#12014yes—
485C22×C4#12014yes—
486C22×C4#12114yes—
487C22×C4#12118yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D414yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C2×#4114yes—
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)14yes—
500C22×C4C2511yescharacteristic
501Q16—432no—
502SD16—432no—
503M16—416no—
504C2×C8—28no—
505C2×C8—28no—
506C2×C8—28no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—216no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—42no—
519C4⋊C4—48no—
520C4⋊C4—42no—
521C4⋊C4—84no—
522C4⋊C4C22×D414yes—
523C22⋊C4—216no—
524C22⋊C4—216no—
525C22⋊C4—216no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—216no—
529C22⋊C4—216no—
530C22⋊C4—84no—
531C42—24no—
532C42—24no—
533C42—48no—
534C42—84no—
535C42#12018yes—
536C42C22×D414yes—
537C42C22×D414yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—28no—
549C23—28no—
550C23—44no—
551C23—44no—
552C23—416no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—82no—
560C23—82no—
561C23—82no—
562C23—82no—
563C23—88no—
564C23—88no—
565C23—88no—
566C23—88no—
567C23—88no—
568C23—88no—
569C23C2×#12012yes—
570C23C2×#12114yes—
571C23C2×#12112yes—
572C23#53512yes—
573C23C23×D411yescharacteristic
574Q8—28no—
575Q8—28no—
576Q8—28no—
577Q8—416no—
578Q8—48no—
579Q8—44no—
580Q8—44no—
581Q8—44no—
582D4—28no—
583D4—28no—
584D4—416no—
585D4—432no—
586D4—432no—
587D4—44no—
588D4—44no—
589D4—88no—
590D4—88no—
591D4—88no—
592D4—88no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—216no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—432no—
615C2×C4—48no—
616C2×C4—416no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—432no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4C2×#12014yes—
636C2×C4C2×#12014yes—
637C2×C4C2×#12012yes—
638C2×C4C2×#12114yes—
639C2×C4C2×#12112yes—
640C2×C4#53512yes—
641C2×C4D4218yes—
642C2×C4#54618yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C23×D412yes—
645C2×C4C22×#4111yescharacteristic
646C2×C4C2×H(5,𝔽2)12yes—
647C8—48no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—48no—
662C22—48no—
663C22—48no—
664C22—88no—
665C22—832no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22#240814yes—
671C22C22×#12011yescharacteristic
672C22C22×#12111yescharacteristic
673C22C2×#53511yescharacteristic
674C22#358511yescharacteristic
675C22#358611yescharacteristic
676C22#358611yescharacteristic
677C22#358611yescharacteristic
678C4—24no—
679C4—24no—
680C4—24no—
681C4—44no—
682C4—48no—
683C4—44no—
684C4—42no—
685C4—42no—
686C4—82no—
687C4—82no—
688C4#240814yes—
689C4C2×D4212yes—
690C4C2×#54612yes—
691C2—24no—
692C2—44no—
693C2—88no—
694C2C2×#240811yescharacteristic
695C2#6037011yescharacteristic
696C2#6037111yescharacteristic
697C2#6042411yescharacteristic
698C2#6042511yescharacteristic
699C2#6080911yescharacteristic
700C2#6081011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3449C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3037C2semidirect
3. C2×#3007C2semidirect
4. C2×#2323C2semidirect
5. #24221C2semidirect
6. #24220C2semidirect
7. #23508C2semidirect
8. #23429C2semidirect
9. #21258C2semidirect
10. #20627C2semidirect
11. #20601C2semidirect
13. C22×#137C22semidirect
25. C22×C4×Q8C22semidirect
28. C2×#464C22semidirect
29. C2×#462C22semidirect
30. C2×#460C22semidirect
33. C2×#451C22semidirect
34. C2×#450C22semidirect
35. C2×#423C22semidirect
36. C2×#417C22semidirect
37. C2×#417C22semidirect
38. C2×#417C22semidirect
41. #2323C22semidirect
42. C2×#391C22semidirect
44. C2×#389C22semidirect
49. C2×C4×C4⋊C4C22semidirect
50. #2043C22semidirect
51. #2042C22semidirect
52. #2041C22semidirect
53. #2016C22semidirect
54. #2015C22semidirect
55. #2014C22semidirect
56. #1983C22semidirect
57. #1982C22semidirect
58. #1981C22semidirect
59. #1798C22semidirect
61. #1787C22semidirect
62. #1786C22semidirect
63. #1728C22semidirect
108. C2×C22⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#34492
C2×#30372
C2×#30072
C2×#23232
#242212
#242202
#235082
#234292
#212582
#206272
#206012