Finite Group Databasebeta

← #6529061

Subgroups — #6529061

Order 512.

3202 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3509C211yesmaximal, characteristic, complemented
2C2×#3101C211yesmaximal, characteristic, complemented
3C22×#474C211yesmaximal, characteristic, complemented
4#34935C211yesmaximal, characteristic
5#24328C214yesmaximal, complemented
6#24300C214yesmaximal, complemented
7#24153C214yesmaximal, complemented
8#24119C214yesmaximal, complemented
9#24094C214yesmaximal, complemented
10#23966C214yesmaximal, complemented
11#21885C211yesmaximal, characteristic, complemented
12#21847C211yesmaximal, characteristic, complemented
13#21833C211yesmaximal, characteristic, complemented
14#3509—28no—
15#3509—28no—
16C2×#557—21no—
17C2×#557C2212yescomplemented
18C2×#556—21no—
19C2×#556C2212yescomplemented
20C2×#552—22no—
21C2×#552C2214yes—
22C2×#541—22no—
23C2×#536—21no—
24C2×#533C2211yescharacteristic
25C2×#533C2212yescomplemented
26C2×#533C2212yescomplemented
27C22×C4⋊Q8C2211yescharacteristic
28C2×#525—21no—
29C2×#519—21no—
30C2×C4×#41C2211yescharacteristic
31C22×C4×D4C2211yescharacteristic
32#3101—28no—
33C2×#483C2212yes—
34C2×#476C2212yes—
35C2×#474—28no—
36C2×#474C2214yes—
37C2×#474C2214yes—
38C2×#461C2212yes—
39C2×#450C2212yes—
40C2×C4×SD16C2214yes—
41C2×#431C2211yescharacteristic
42C22×#107C2212yes—
43C22×C4⋊C8C2211yescharacteristic
44C22×#102C2212yes—
45#2538C2212yes—
46#2537C2212yes—
47#2536—21no—
48#2535—21no—
49#2534—22no—
50#2486C2212yes—
51#2408C2212yes—
52#2392—22no—
53C4×#121C2214yes—
54#2285C2211yescharacteristic
55#2280—21no—
56#2059C2218yescomplemented
57#2036C2218yescomplemented
58#2025C2218yes—
59#2000C2218yes—
60#1990C2214yescomplemented
61#1988C2214yescomplemented
62#1987C2212yescomplemented
63#1984C2212yescomplemented
64#1981C2212yescomplemented
65#1954C2212yescomplemented
66#1892C2214yes—
67#1880C2218yes—
68#1875C2218yes—
69#1869C2214yes—
70#1863C2218yescomplemented
71#1854C2218yescomplemented
72#1823C2214yes—
73#1803C2214yescomplemented
74#1797C2214yescomplemented
75C22×#41D411yescharacteristic
76C23×Q8—21no—
77C23×D4C2311yescharacteristic
78C24×C4—21no—
79C22×SD16C2314yes—
80C23×C8C2312yes—
81#557—216no—
82#557—28no—
83#557—28no—
84#556—216no—
85#556—28no—
86#556—28no—
87#552—232no—
88#552—216no—
89#552—216no—
90#541—232no—
91#536—216no—
92#533—28no—
93#533—28no—
94#533—28no—
95#533—28no—
96#533—24no—
97#533D418yescomplemented
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8D418yescomplemented
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#126—24no—
108C2×#126—24no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—21no—
112C2×#125—22no—
113C2×#125C2314yes—
114C2×#124—22no—
115#525—48no—
116C2×#123—24no—
117C2×#123—22no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8C2314yes—
124C2×C22⋊Q8C2312yes—
125C2×#121—22no—
126C2×#121C2314yes—
127C2×#120—22no—
128#519—216no—
129C4×#41—24no—
130C4×#41—24no—
131C2×C4×Q8—22no—
132C2×C4×Q8—22no—
133C2×C4×Q8—22no—
134C2×C4×Q8—22no—
135C2×C4×Q8—21no—
136C2×C4×Q8—22no—
137C2×C4×Q8D411yescharacteristic
138C2×C4×Q8C2314yes—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2311yescharacteristic
141C2×C4×D4—24no—
142C2×C4×D4—21no—
143C2×C4×D4—21no—
144C2×C4×D4—24no—
145C2×C4×D4—24no—
146C2×C4×D4D412yes—
147C2×C4×D4D411yescharacteristic
148C2×C4×D4C2312yes—
149C2×C4×D4C2312yes—
150C2×C4×D4C2312yes—
151C2×C4×D4C2311yescharacteristic
152C2×#117—21no—
153C2×#117—22no—
154C2×#117—21no—
155C2×#117—24no—
156C2×#117D412yes—
157C2×#117D411yescharacteristic
158C2×#117C2314yes—
159C22×C4⋊C4—21no—
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2312yes—
162C22×C22⋊C4—21no—
163C22×C42C2312yes—
164C22×C42C2311yescharacteristic
165#483—28no—
166#476—28no—
167#474—216no—
168#474—216no—
169#461—28no—
170#450—28no—
171C4×SD16—216no—
172#431—24no—
173C2×#107—28no—
174C2×#107C2318yes—
175C2×#106C2314yes—
176C2×C4⋊C8—24no—
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2312yes—
179C2×C4⋊C8C2312yes—
180C2×#103C2314yes—
181C2×#103C2318yes—
182C2×#102—28no—
183C2×#102C2318yes—
184C2×#102C2314yes—
185C2×C22⋊C8C2312yes—
186C2×C22⋊C8C2318yes—
187C2×C4×C8C2312yes—
188#398—22no—
189#397—22no—
190#396—24no—
191#394—22no—
192#394—24no—
193#392—22no—
194#390C2314yes—
195#389C2314yes—
196#388—24no—
197#386—24no—
198#384C2312yes—
199#382—21no—
200#381—22no—
201#380—21no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2314yes—
206#379—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—24no—
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213#336C2318yes—
214C25—21no—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41—24no—
218C22×Q8—28no—
219C22×Q8—21no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—44no—
223C22×Q8—44no—
224C22×Q8C2×D411yescharacteristic
225C22×Q8C2412yes—
226C22×D4—24no—
227C22×D4—24no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—42no—
232C22×D4C2×D412yes—
233C22×D4C2×D411yescharacteristic
234C22×D4C2412yes—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C2×SD16—28no—
249C2×SD16—28no—
250C2×SD16—44no—
251C22×C8—24no—
252C22×C8—44no—
253C22×C8C2414yes—
254C22×C8C2414yes—
255C4⋊Q8—216no—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8C2×D414yes—
263C4⋊Q8C2×D418yes—
264C4⋊Q8C2×D418yes—
265C4⋊Q8C2×D414yes—
266#126—232no—
267#126—232no—
268#125—216no—
269#125—232no—
270#125—232no—
271#125—28no—
272#125—28no—
273#125—28no—
274#124—216no—
275#123—232no—
276#123—48no—
277C22⋊Q8—216no—
278C22⋊Q8—216no—
279C22⋊Q8—216no—
280C22⋊Q8—28no—
281C22⋊Q8—28no—
282C22⋊Q8—24no—
283C22⋊Q8—28no—
284C22⋊Q8—48no—
285C22⋊Q8C2×D418yes—
286#121—28no—
287#121—44no—
288#121—48no—
289#120—48no—
290C4×Q8—216no—
291C4×Q8—216no—
292C4×Q8—216no—
293C4×Q8—216no—
294C4×Q8—216no—
295C4×Q8—28no—
296C4×Q8—28no—
297C4×Q8—28no—
298C4×Q8—24no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8—24no—
302C4×Q8C2×D418yes—
303C4×D4—28no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—416no—
314C4×D4—44no—
315C4×D4—44no—
316#117—232no—
317#117—216no—
318#117—28no—
319#117—24no—
320#117—28no—
321#117—28no—
322#117—28no—
323#117—44no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D418yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2414yes—
351C2×C4⋊C4C2414yes—
352C2×C4⋊C4C2412yes—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—44no—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C42—24no—
368C2×C42—24no—
369C2×C42—22no—
370C2×C42—24no—
371C2×C42—24no—
372C2×C42—22no—
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D414yes—
378C2×C42C2×D414yes—
379C2×C42C2412yes—
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382#107—216no—
383#106—28no—
384C4⋊C8—24no—
385C4⋊C8—24no—
386C4⋊C8—24no—
387#103—28no—
388#103—48no—
389#102—216no—
390#102—28no—
391C22⋊C8—24no—
392C22⋊C8—48no—
393C4×C8—24no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398C24—22no—
399C24—44no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—82no—
404C24C22×D412yes—
405C24C22×D411yescharacteristic
406#41—416no—
407C2×Q8—24no—
408C2×Q8—28no—
409C2×Q8—28no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—432no—
422C2×Q8C22×D414yes—
423C2×D4—28no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—48no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—48no—
439C2×D4—48no—
440C2×D4—42no—
441C2×D4—84no—
442C2×D42−1+414yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—28no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—816no—
473C22×C4#12014yes—
474C22×C4#12014yes—
475C22×C4#13514yes—
476C22×C4AGL(1,ℤ/8)14yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C2511yescharacteristic
488SD16—416no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—28no—
494C2×C8—48no—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—216no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—82no—
521C4⋊C4C22×D414yes—
522C4⋊C4C22×D418yes—
523C22⋊C4—216no—
524C22⋊C4—216no—
525C22⋊C4—28no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—84no—
536C42—216no—
537C42—216no—
538C42—24no—
539C42—24no—
540C42—28no—
541C42—22no—
542C42—22no—
543C42—44no—
544C42#12014yes—
545C42#12014yes—
546C42C22×D412yes—
547C42C22×D412yes—
548C42C22×D412yes—
549C42C22×D412yes—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—22no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—48no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—88no—
566C23—88no—
567C23—84no—
568C23—84no—
569C23C2×#12012yes—
570C23C2×#13512yes—
571C23C2×AGL(1,ℤ/8)12yes—
572C23C23×D411yescharacteristic
573Q8—28no—
574Q8—48no—
575Q8—42no—
576Q8—42no—
577Q8—44no—
578Q8—416no—
579Q8—416no—
580D4—24no—
581D4—24no—
582D4—48no—
583D4—48no—
584D4—42no—
585D4—42no—
586D4—44no—
587D4—816no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—416no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—84no—
622C2×C4—88no—
623C2×C4—88no—
624C2×C4#449116yes—
625C2×C4C2×#12012yes—
626C2×C4C2×#12012yes—
627C2×C4C2×#12012yes—
628C2×C4C2×#12012yes—
629C2×C4C2×#12012yes—
630C2×C4C2×#13512yes—
631C2×C4C2×#13512yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C23×D411yescharacteristic
636C2×C4C2×2−1+412yes—
637C2×C4#58512yes—
638C8—44no—
639C8—44no—
640C22—22no—
641C22—24no—
642C22—24no—
643C22—22no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—42no—
651C22—42no—
652C22—816no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22C2×#44914yes—
658C22C22×#12011yescharacteristic
659C22#348912yes—
660C22C22×#13511yescharacteristic
661C22C22×AGL(1,ℤ/8)11yescharacteristic
662C22#358612yes—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—48no—
670C4—42no—
671C4—44no—
672C4—84no—
673C4—82no—
674C4C2×#44914yes—
675C4C2×#44914yes—
676C4#348611yescharacteristic
677C4#348711yescharacteristic
678C4#348912yes—
679C2—24no—
680C2—82no—
681C2—84no—
682C2#3565911yescharacteristic
683C2C22×#44911yescharacteristic
684C2#6036811yescharacteristic
685C2#6209712yes—
686C2#6210912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34935C2
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. C2×#3509C2semidirect
2. C2×#3101C2semidirect
3. C22×#474C2semidirect
5. #24328C2semidirect
6. #24300C2semidirect
7. #24153C2semidirect
8. #24119C2semidirect
9. #24094C2semidirect
10. #23966C2semidirect
11. #21885C2semidirect
12. #21847C2semidirect
13. #21833C2semidirect
17. C2×#557C22semidirect
19. C2×#556C22semidirect
25. C2×#533C22semidirect
26. C2×#533C22semidirect
56. #2059C22semidirect
57. #2036C22semidirect
60. #1990C22semidirect
61. #1988C22semidirect
62. #1987C22semidirect
63. #1984C22semidirect
64. #1981C22semidirect
65. #1954C22semidirect
70. #1863C22semidirect
71. #1854C22semidirect
73. #1803C22semidirect
74. #1797C22semidirect
97. #533D4semidirect
102. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#35092
C2×#31012
C22×#4742
#349352
#243282
#243002
#241532
#241192
#240942
#239662
#218852
#218472
#218332