Finite Group Databasebeta

← #6355360

Subgroups — #6355360

Order 512.

1818 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic, complemented
2C2×#3079C211yesmaximal, characteristic, complemented
3#60415C211yesmaximal, characteristic
4#60277C211yesmaximal, characteristic, complemented
5#19450C218yesmaximal, complemented
6#19439C214yesmaximal, complemented
7#19427C214yesmaximal, complemented
8#19299C214yesmaximal, complemented
9#19295C214yesmaximal, complemented
10#17814C211yesmaximal, characteristic, complemented
11C2×#1548C211yesmaximal, characteristic, complemented
12#17704C211yesmaximal, characteristic, complemented
13C2×#575—21no—
14C2×#532C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C2×C4×#41C2211yescharacteristic
17#3227—22no—
18#3227—22no—
19#3227C2214yes—
20#3227C2214yescomplemented
21C2×#498C2212yescomplemented
22C2×#486C2212yescomplemented
23C2×#481—22no—
24#3079—22no—
25#3079—22no—
26#3079C2214yes—
27#3079C2214yescomplemented
28C2×#477C2212yescomplemented
29C2×#475—21no—
30C2×#476—22no—
31C2×#476C2212yescomplemented
32C2×#474—21no—
33#3040—22no—
34#3039C2212yes—
35#3038C2212yes—
36#3037—21no—
37#3037C2212yes—
38#3036—21no—
39#3036C2212yes—
40#3035—22no—
41#3035C2214yes—
42#2989—22no—
43#2989C2214yescomplemented
44#2983C2212yescomplemented
45#2975C2212yescomplemented
46#2973—22no—
47#2965—22no—
48#2965C2214yes—
49#2920—22no—
50C2×#431C2211yescharacteristic
51C2×#431C2211yescharacteristic
52C2×#428—21no—
53C2×#423C2211yescharacteristic
54C2×#423C2211yescharacteristic
55#2896—21no—
56C2×#417—21no—
57#1698C22116yescomplemented
58#1697C22116yescomplemented
59#1692C22116yescomplemented
60#1691C22116yescomplemented
61#1571C2214yescomplemented
62#1571C2214yescomplemented
63#1567C2214yescomplemented
64#1548C2214yescomplemented
65#1548C2214yescomplemented
66#1547C2214yescomplemented
67#1546C2212yescomplemented
68#1545C2212yescomplemented
69C2×#333C2214yescomplemented
70#1540C2214yes—
71C22×#41—21no—
72C22×#41D411yescharacteristic
73C23×Q8—21no—
74#575—48no—
75C2×#137—22no—
76C2×#137—22no—
77C2×AGL(1,ℤ/8)—22no—
78C2×AGL(1,ℤ/8)—22no—
79C2×#135—24no—
80C22×M16—21no—
81C22×M16—21no—
82#532—21no—
83#532—21no—
84#532—21no—
85#532—21no—
86#532—21no—
87#532—21no—
88#532C2312yes—
89#532C2312yes—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127C2311yescharacteristic
104C2×#124—21no—
105C2×#121—21no—
106C2×#121—21no—
107C4×#41—21no—
108C4×#41—21no—
109C4×#41—21no—
110C4×#41—21no—
111C4×#41D412yes—
112C4×#41D412yes—
113C4×#41C2312yes—
114C4×#41C2312yes—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C2×#117D411yescharacteristic
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129#498—24no—
130#498C2318yes—
131#486—24no—
132#486C2318yes—
133#481—28no—
134#481—28no—
135#477—24no—
136#477C2318yes—
137#476—24no—
138#476—28no—
139#476—28no—
140#476C2318yes—
141#475—24no—
142#475—24no—
143#474—24no—
144#474—24no—
145#464—24no—
146#464—24no—
147#463—24no—
148#463—24no—
149#462—22no—
150#462C2314yes—
151#461—22no—
152#461C2314yes—
153#460—22no—
154#460C2314yes—
155#459—22no—
156#459C2314yes—
157#446—24no—
158#446C2318yes—
159#436—24no—
160#436C2318yes—
161#435—28no—
162#435—28no—
163#432—24no—
164#432—24no—
165#431—21no—
166#431—21no—
167#431—24no—
168#431D412yes—
169#431D412yes—
170#431C2312yes—
171#431C2312yes—
172#431C2312yes—
173#431C2312yes—
174#428—24no—
175#428—24no—
176C2×#107—22no—
177C2×#106—24no—
178C2×#106—22no—
179#423—21no—
180#423—21no—
181#423—21no—
182#423—21no—
183#423—24no—
184#423C2312yes—
185#423C2312yes—
186#423C2312yes—
187#423C2312yes—
188C2×C4⋊C8—21no—
189C2×C4⋊C8—21no—
190C2×C4⋊C8D412yes—
191C2×C4⋊C8C2312yes—
192C2×C4⋊C8C2312yes—
193C2×C4⋊C8C2312yes—
194#419—22no—
195#418—22no—
196#417—24no—
197#417—24no—
198#417—22no—
199#417—22no—
200#416—24no—
201C2×#103—22no—
202C2×#102—22no—
203C2×#102—24no—
204#407—24no—
205C2×C22⋊C8—21no—
206C2×C22⋊C8D412yes—
207C4×M16—24no—
208C2×C4×C8—21no—
209C2×C4×C8C2312yes—
210#335C23116yes—
211#333C23116yes—
212#332C2318yes—
213#331C2318yes—
214C2×#41—21no—
215C2×#41—21no—
216C2×#41—24no—
217C2×#41—24no—
218C2×#41—22no—
219C2×#41—22no—
220C2×#41—41no—
221C2×#41—41no—
222C2×#41—44no—
223C2×#41—44no—
224C2×#41C2×D412yes—
225C2×#41C2×D412yes—
226C22×Q8—21no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—41no—
231C22×Q8—41no—
232C22×Q8—41no—
233C22×Q8—84no—
234C22×Q8C2×D411yescharacteristic
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—41no—
240C22×D4C2×D411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—41no—
245C23×C4—41no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4#4111yescharacteristic
249#137—48no—
250#137—48no—
251AGL(1,ℤ/8)—48no—
252AGL(1,ℤ/8)—48no—
253#135—416no—
254C2×Q16—24no—
255C2×SD16—24no—
256C2×SD16—24no—
257C2×D8—24no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—28no—
263C2×M16—44no—
264C2×M16—44no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8C2×D414yes—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8—41no—
282C4⋊Q8—41no—
283C4⋊Q8—41no—
284C4⋊Q8—41no—
285C4⋊Q8—41no—
286C4⋊Q8—41no—
287C4⋊Q8C2414yes—
288C4⋊Q8C2412yes—
289#127—21no—
290#127—21no—
291#127—21no—
292#127C2412yes—
293#124—22no—
294#124—22no—
295#124—41no—
296#124—41no—
297#121—22no—
298#121—22no—
299#121—22no—
300#121—22no—
301#121—41no—
302#121—41no—
303#121—41no—
304#121—41no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8C2×D412yes—
310C4×Q8C2×D412yes—
311C4×Q8C2×D412yes—
312C4×Q8C2412yes—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—21no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—41no—
330C4×D4—41no—
331C4×D4—41no—
332C4×D4—41no—
333C4×D4C2×D412yes—
334C4×D4C2×D412yes—
335C4×D4C2×D412yes—
336C4×D4C2412yes—
337#117—21no—
338#117—21no—
339#117—22no—
340#117—22no—
341#117—22no—
342#117—22no—
343#117C2×D412yes—
344#117C2×D412yes—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—41no—
356C2×C4⋊C4—41no—
357C2×C4⋊C4—41no—
358C2×C4⋊C4—41no—
359C2×C4⋊C4—41no—
360C2×C4⋊C4—41no—
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—41no—
367C2×C22⋊C4—41no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42#4111yescharacteristic
382C2×C42#4111yescharacteristic
383C2×C42#4111yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—28no—
388#106—216no—
389#106—28no—
390C4⋊C8—28no—
391C4⋊C8—22no—
392C4⋊C8—22no—
393C4⋊C8—22no—
394C4⋊C8—22no—
395C4⋊C8—22no—
396C4⋊C8—22no—
397C4⋊C8—22no—
398C4⋊C8C2×D414yes—
399C4⋊C8C2×D414yes—
400C4⋊C8C2414yes—
401C4⋊C8C2414yes—
402C4⋊C8C2414yes—
403#103—28no—
404#103—24no—
405#103—24no—
406#102—28no—
407#102—216no—
408#102—24no—
409#102—24no—
410C22⋊C8—22no—
411C22⋊C8—22no—
412C22⋊C8—28no—
413C22⋊C8C2×D414yes—
414C22⋊C8C2×D414yes—
415#97—28no—
416C4×C8—22no—
417C4×C8—22no—
418C4×C8C2×D414yes—
419C4×C8C2414yes—
420C24—21no—
421C24—41no—
422C24#12111yescharacteristic
423#41—24no—
424#41—24no—
425#41—24no—
426#41—24no—
427#41—44no—
428#41—44no—
429#41—44no—
430#41—44no—
431#41—44no—
432#41—44no—
433#41—84no—
434#41—84no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—41no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—84no—
447C2×Q8—84no—
448C2×Q8—84no—
449C2×Q8—84no—
450C2×Q8—84no—
451C2×Q8—84no—
452C2×Q8C22×D412yes—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—82no—
471C2×D4—84no—
472C2×D4C22×D412yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—41no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—84no—
494C22×C4—84no—
495C22×C4#12111yescharacteristic
496C22×C4#12111yescharacteristic
497C22×C4#12111yescharacteristic
498C22×C4#12111yescharacteristic
499C22×C4#12111yescharacteristic
500C22×C4#12111yescharacteristic
501C22×C4#12111yescharacteristic
502C22×C4#12111yescharacteristic
503C22×C4#12111yescharacteristic
504C22×C4AGL(1,ℤ/8)14yes—
505C22×C4#13714yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C2×#4111yescharacteristic
513C22×C4C2×#4111yescharacteristic
514C22×C4C2×#4111yescharacteristic
515Q16—48no—
516SD16—48no—
517SD16—48no—
518D8—48no—
519M16—44no—
520M16—44no—
521M16—44no—
522M16—44no—
523M16—88no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—44no—
528C2×C8—44no—
529C2×C8—44no—
530C2×C8—82no—
531C2×C8C22×D418yes—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4C22×D412yes—
551C4⋊C4C22×D412yes—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42#12111yescharacteristic
569C42#12111yescharacteristic
570C42#12111yescharacteristic
571C42#12111yescharacteristic
572C42#12111yescharacteristic
573C42#12111yescharacteristic
574C42C22×D411yescharacteristic
575C42C22×D411yescharacteristic
576C42C22×D411yescharacteristic
577C42C22×D411yescharacteristic
578C42C22×D411yescharacteristic
579C42C22×D411yescharacteristic
580C42C2×#4111yescharacteristic
581C42C2×#4111yescharacteristic
582C42C2×#4111yescharacteristic
583C42C2511yescharacteristic
584C23—22no—
585C23—41no—
586C23—41no—
587C23—42no—
588C23—42no—
589C23—81no—
590C23—81no—
591C23—82no—
592C23—82no—
593C23—82no—
594C23#46814yes—
595C23C2×#12111yescharacteristic
596C23C2×#12111yescharacteristic
597C23C2×#12111yescharacteristic
598Q8—22no—
599Q8—22no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—84no—
604Q8—84no—
605Q8—84no—
606Q8—82no—
607Q8—84no—
608D4—22no—
609D4—22no—
610D4—44no—
611D4—44no—
612D4—44no—
613D4—42no—
614D4—42no—
615D4—42no—
616D4—82no—
617D4—82no—
618D4—84no—
619D4—84no—
620D4—82no—
621D4—82no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—82no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4#46814yes—
655C2×C4#46814yes—
656C2×C4#46814yes—
657C2×C4C2×#12111yescharacteristic
658C2×C4C2×#12111yescharacteristic
659C2×C4C2×#12111yescharacteristic
660C2×C4C2×#12111yescharacteristic
661C2×C4C2×#12111yescharacteristic
662C2×C4C2×#12111yescharacteristic
663C2×C4C2×#12111yescharacteristic
664C2×C4C2×#12111yescharacteristic
665C2×C4C2×#12111yescharacteristic
666C2×C4C2×AGL(1,ℤ/8)12yes—
667C2×C4C2×AGL(1,ℤ/8)12yes—
668C2×C4C2×AGL(1,ℤ/8)12yes—
669C2×C4C2×#13712yes—
670C2×C4C2×#13712yes—
671C2×C4C2×#13712yes—
672C2×C4C23×D411yescharacteristic
673C2×C4C23×D411yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C8—28no—
676C8—44no—
677C8—44no—
678C8—44no—
679C22—21no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—44no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—84no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—164no—
694C22C2×#46812yes—
695C22C2×#46812yes—
696C22C2×#46812yes—
697C22C22×#12111yescharacteristic
698C4—21no—
699C4—21no—
700C4—21no—
701C4—21no—
702C4—21no—
703C4—21no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—82no—
708C4—84no—
709C4C2×#46812yes—
710C4C2×#46812yes—
711C4C2×#46812yes—
712C4#306812yes—
713C4C22×AGL(1,ℤ/8)11yescharacteristic
714C4C22×#13711yescharacteristic
715C4#358612yes—
716C2—22no—
717C2—22no—
718C2—22no—
719C2—82no—
720C2—162no—
721C2#6042711yescharacteristic
722C2C22×#46811yescharacteristic
723C2C2×#306811yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radical#60415C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3227C2semidirect
2. C2×#3079C2semidirect
4. #60277C2semidirect
5. #19450C2semidirect
6. #19439C2semidirect
7. #19427C2semidirect
8. #19299C2semidirect
9. #19295C2semidirect
10. #17814C2semidirect
11. C2×#1548C2semidirect
12. #17704C2semidirect
20. #3227C22semidirect
21. C2×#498C22semidirect
22. C2×#486C22semidirect
27. #3079C22semidirect
28. C2×#477C22semidirect
31. C2×#476C22semidirect
43. #2989C22semidirect
44. #2983C22semidirect
45. #2975C22semidirect
57. #1698C22semidirect
58. #1697C22semidirect
59. #1692C22semidirect
60. #1691C22semidirect
61. #1571C22semidirect
62. #1571C22semidirect
63. #1567C22semidirect
64. #1548C22semidirect
65. #1548C22semidirect
66. #1547C22semidirect
67. #1546C22semidirect
68. #1545C22semidirect
69. C2×#333C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
C2×#30792
#604152
#602772
#194502
#194392
#194272
#192992
#192952
#178142
C2×#15482
#177042