Finite Group Databasebeta

← #6353777

Subgroups — #6353777

Order 512.

2080 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60941C211yesmaximal, characteristic, complemented
2C2×#3208C211yesmaximal, characteristic, complemented
3C2×#3151C211yesmaximal, characteristic
4#60192C211yesmaximal, characteristic, complemented
5#19238C218yesmaximal, complemented
6#19203C214yesmaximal, complemented
7#19202C214yesmaximal, complemented
8#19170C218yesmaximal, complemented
9#17787C211yesmaximal, characteristic, complemented
10#17704C211yesmaximal, characteristic, complemented
11C2×#1545C211yesmaximal, characteristic, complemented
12C22×AGL(1,ℤ/8)—21no—
13C22×AGL(1,ℤ/8)—21no—
14C2×#533C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C22×#127C2211yescharacteristic
17#3246C2214yescomplemented
18#3229C2212yescomplemented
19#3227—21no—
20#3208—22no—
21#3208—22no—
22#3208C2214yes—
23#3208C2214yescomplemented
24#3208C2212yescomplemented
25#3206—21no—
26#3187—22no—
27#3167—22no—
28#3151—22no—
29#3151—22no—
30#3151C2214yes—
31#3151C2214yes—
32C2×#496—22no—
33C2×#493C2212yes—
34C2×#492—21no—
35C2×#492C2212yes—
36C2×#486—21no—
37#3103C2214yescomplemented
38#3101—22no—
39C2×#460C2214yescomplemented
40C2×#450—22no—
41Q8×M16—22no—
42Q8×M16C2214yescomplemented
43#2962—22no—
44#2962C2214yescomplemented
45#2961—22no—
46#2960C2214yescomplemented
47C2×#431C2211yescharacteristic
48C2×#431C2211yescharacteristic
49#2905—21no—
50C2×#423C2211yescharacteristic
51#2873—21no—
52C2×C4×M16C2211yescharacteristic
53#1688C22116yescomplemented
54#1686C22116yescomplemented
55#1668C22116yescomplemented
56#1666C22116yescomplemented
57#1565C2214yescomplemented
58#1565C2214yescomplemented
59#1564C2214yescomplemented
60#1547C2214yescomplemented
61#1546C2212yescomplemented
62#1545C2214yes—
63#1545C2214yescomplemented
64#1545C2212yescomplemented
65#1540C2214yescomplemented
66C2×#331C2214yescomplemented
67C22×#41—21no—
68C23×D4—21no—
69C23×D4—21no—
70C23×D4—21no—
71C2×AGL(1,ℤ/8)—24no—
72C2×AGL(1,ℤ/8)—24no—
73C2×AGL(1,ℤ/8)—24no—
74C2×AGL(1,ℤ/8)—24no—
75C2×AGL(1,ℤ/8)—48no—
76C2×AGL(1,ℤ/8)—48no—
77C22×SD16—24no—
78C22×SD16—22no—
79C22×SD16—22no—
80C22×D8—22no—
81C22×D8—22no—
82C22×M16—21no—
83C22×M16D411yescharacteristic
84C22×M16D411yescharacteristic
85#533—21no—
86#533—21no—
87#533—21no—
88#533—21no—
89#533D412yes—
90#533D412yes—
91#533C2312yes—
92#533C2312yes—
93#532—21no—
94#532—21no—
95#532—21no—
96#532—21no—
97#532—21no—
98#532—21no—
99#532C2312yes—
100#532C2312yes—
101C2×C4⋊Q8D411yescharacteristic
102C2×C4⋊Q8D411yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127—21no—
105C2×#127—21no—
106C2×#127—21no—
107C2×#127—21no—
108C2×#127—21no—
109C2×#127—21no—
110C2×#127—21no—
111C2×#127—21no—
112C2×#127—21no—
113C2×#127C2312yes—
114C2×#127C2312yes—
115C2×#127C2311yescharacteristic
116C2×#127C2312yes—
117C2×#125—21no—
118C2×#124—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8D412yes—
121C2×#121—21no—
122C2×#121—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8C2312yes—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×#117—21no—
128C22×C42C2311yescharacteristic
129#498—22no—
130#496—28no—
131#496—28no—
132#493—24no—
133#493C2318yes—
134#492—24no—
135#492—24no—
136#492—22no—
137#492—24no—
138#492C2318yes—
139#487—22no—
140#486—24no—
141#486—24no—
142#486—22no—
143#483—24no—
144#481—24no—
145#476—24no—
146#475C2318yescomplemented
147#465—28no—
148#461—24no—
149#460—28no—
150#460C23116yes—
151#460C2318yescomplemented
152#450—24no—
153#450—28no—
154#450—28no—
155#448—24no—
156#446—28no—
157#446—28no—
158#446—24no—
159#446C2318yescomplemented
160C8×Q8—24no—
161C8×Q8C2318yescomplemented
162C4×SD16—28no—
163#431—21no—
164#431—21no—
165#431—21no—
166#431—21no—
167#431C2312yes—
168#431C2312yes—
169#431C2312yes—
170#431C2312yes—
171#424—24no—
172#424—24no—
173#423—24no—
174#423—21no—
175#423—21no—
176#423C2312yes—
177#423C2312yes—
178C2×C4⋊C8—21no—
179C2×C4⋊C8C2312yes—
180C2×C4⋊C8C2312yes—
181C2×C4⋊C8C2312yes—
182C2×#102—24no—
183C2×C22⋊C8—21no—
184C2×C22⋊C8—21no—
185#405—24no—
186#405—24no—
187C4×M16—24no—
188C4×M16D412yes—
189C4×M16D412yes—
190C4×M16C2312yes—
191C4×M16C2312yes—
192C2×#97D412yes—
193C2×C4×C8—21no—
194C2×C4×C8—21no—
195C2×C4×C8C2312yes—
196#335C2318yescomplemented
197#334C2318yescomplemented
198#332C2318yescomplemented
199#331C23116yes—
200#331C2318yescomplemented
201C25—41no—
202C2×#41—22no—
203C2×#41—22no—
204C2×#41—41no—
205C2×#41—41no—
206C2×#41—44no—
207C2×#41—44no—
208C22×Q8—21no—
209C22×Q8—21no—
210C22×Q8C2×D412yes—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—21no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—41no—
224C22×D4—41no—
225C22×D4—41no—
226C22×D4—41no—
227C22×D4—41no—
228C22×D4—41no—
229C22×D4—41no—
230C22×D4—41no—
231C22×D4—41no—
232C22×D4—41no—
233C22×D4—44no—
234C22×D4—44no—
235C22×D4—84no—
236C22×D4—84no—
237C23×C4—41no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4#4111yescharacteristic
241AGL(1,ℤ/8)—416no—
242AGL(1,ℤ/8)—416no—
243AGL(1,ℤ/8)—416no—
244AGL(1,ℤ/8)—416no—
245C2×SD16—216no—
246C2×SD16—28no—
247C2×SD16—28no—
248C2×SD16—28no—
249C2×SD16—48no—
250C2×SD16—48no—
251C2×SD16—88no—
252C2×D8—28no—
253C2×D8—28no—
254C2×D8—48no—
255C2×D8—48no—
256C2×M16—24no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—44no—
260C2×M16SD1618yescomplemented
261C2×M16C2×D412yes—
262C2×M16C2×D412yes—
263C2×M16C2×D412yes—
264C2×M16C2×D412yes—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8C2×D412yes—
269C22×C8C2×D412yes—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8—21no—
273C4⋊Q8—21no—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8C2×D412yes—
278C4⋊Q8C2×D412yes—
279C4⋊Q8C2×D412yes—
280C4⋊Q8C2×D412yes—
281C4⋊Q8C2412yes—
282#127—22no—
283#127—22no—
284#127—22no—
285#127—22no—
286#127—22no—
287#127—22no—
288#127—21no—
289#127—21no—
290#127—22no—
291#127—22no—
292#127—22no—
293#127—21no—
294#127—41no—
295#127—41no—
296#127—41no—
297#127—41no—
298#127—41no—
299#127—41no—
300#127C2414yes—
301#127C2412yes—
302#125—22no—
303#125—22no—
304#125—22no—
305#125—22no—
306#124—22no—
307#124—22no—
308#124—41no—
309#124—41no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8C2×D414yes—
317C22⋊Q8C2×D414yes—
318#121—22no—
319#121—22no—
320#121—22no—
321#121—22no—
322#121—41no—
323#121—41no—
324#121—41no—
325#121—41no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—22no—
332C4×Q8C2×D414yes—
333C4×Q8C2414yes—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—41no—
339C4×D4—41no—
340C4×D4—41no—
341C4×D4—41no—
342#117—22no—
343#117—22no—
344#117—22no—
345#117—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4C2×D412yes—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—41no—
356C2×C22⋊C4—41no—
357C2×C42—22no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42#4111yescharacteristic
365C2×C42#4111yescharacteristic
366C2×C42#4111yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#106—28no—
371C4⋊C8—28no—
372C4⋊C8—28no—
373C4⋊C8—22no—
374C4⋊C8—22no—
375C4⋊C8—22no—
376C4⋊C8—22no—
377C4⋊C8—22no—
378C4⋊C8C2414yes—
379C4⋊C8C2414yes—
380C4⋊C8C2414yes—
381#103—28no—
382#102—216no—
383#102—28no—
384C22⋊C8—22no—
385C22⋊C8—22no—
386C22⋊C8—22no—
387C22⋊C8—22no—
388#97—28no—
389#97C2×D414yes—
390#97C2×D414yes—
391C4×C8—28no—
392C4×C8—22no—
393C4×C8—22no—
394C4×C8—22no—
395C4×C8—22no—
396C4×C8C2×D414yes—
397C4×C8C2414yes—
398C24—41no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—84no—
403C24—84no—
404C24—84no—
405C24#12111yescharacteristic
406#41—44no—
407#41—44no—
408#41—44no—
409#41—44no—
410#41—84no—
411#41—84no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—44no—
417C2×Q8—41no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—82no—
421C2×Q8C22×D414yes—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—24no—
428C2×D4—48no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—41no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—82no—
452C2×D4—84no—
453C2×D4—84no—
454C2×D4—84no—
455C2×D4—84no—
456C2×D4—82no—
457C2×D4—84no—
458C2×D4—84no—
459C2×D4—84no—
460C2×D4—84no—
461C2×D4—84no—
462C2×D4—84no—
463C2×D4—84no—
464C2×D4—84no—
465C2×D4—84no—
466C2×D4—84no—
467C2×D4—84no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—41no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—84no—
476C22×C4#12111yescharacteristic
477C22×C4#12111yescharacteristic
478C22×C4#12111yescharacteristic
479C22×C4#12111yescharacteristic
480C22×C4#12111yescharacteristic
481C22×C4#12111yescharacteristic
482C22×C4#12111yescharacteristic
483C22×C4#12111yescharacteristic
484C22×C4#12111yescharacteristic
485C22×C4C2×SD1614yes—
486C22×C4AGL(1,ℤ/8)14yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C2×#4111yescharacteristic
494C22×C4C2×#4111yescharacteristic
495C22×C4C2×#4111yescharacteristic
496SD16—416no—
497SD16—416no—
498SD16—88no—
499SD16—816no—
500D8—416no—
501D8—416no—
502M16—24no—
503M16—24no—
504M16—44no—
505M16—44no—
506M16C2×SD1618yes—
507M16C2×SD1618yescomplemented
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—44no—
513C2×C8—44no—
514C2×C8—82no—
515C2×C8C2×SD1618yescomplemented
516C2×C8C22×D414yes—
517C2×C8C22×D414yes—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4C22×D414yes—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C42—24no—
541C42—24no—
542C42#12111yescharacteristic
543C42#12111yescharacteristic
544C42#12111yescharacteristic
545C42#12111yescharacteristic
546C42#12111yescharacteristic
547C42#12111yescharacteristic
548C42C22×D411yescharacteristic
549C42C22×D411yescharacteristic
550C42C22×D411yescharacteristic
551C42C22×D411yescharacteristic
552C42C22×D411yescharacteristic
553C42C22×D411yescharacteristic
554C42C2×#4111yescharacteristic
555C42C2×#4111yescharacteristic
556C42C2×#4111yescharacteristic
557C42C2511yescharacteristic
558C23—41no—
559C23—41no—
560C23—42no—
561C23—44no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—82no—
569C23—82no—
570C23—82no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—1616no—
575C23#46014yes—
576C23C2×#12111yescharacteristic
577C23C2×#12111yescharacteristic
578C23C2×#12111yescharacteristic
579Q8—28no—
580Q8—44no—
581Q8—42no—
582Q8—42no—
583Q8—82no—
584Q8—82no—
585Q8—82no—
586D4—48no—
587D4—44no—
588D4—42no—
589D4—44no—
590D4—42no—
591D4—44no—
592D4—82no—
593D4—84no—
594D4—84no—
595D4—82no—
596D4—84no—
597D4—84no—
598D4—84no—
599D4—84no—
600D4—84no—
601D4—84no—
602D4—84no—
603D4—84no—
604D4—84no—
605D4—82no—
606D4—82no—
607D4—84no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4#46014yes—
629C2×C4#46014yes—
630C2×C4#46014yes—
631C2×C4C2×#12111yescharacteristic
632C2×C4C2×#12111yescharacteristic
633C2×C4C2×#12111yescharacteristic
634C2×C4C2×#12111yescharacteristic
635C2×C4C2×#12111yescharacteristic
636C2×C4C2×#12111yescharacteristic
637C2×C4C2×#12111yescharacteristic
638C2×C4C2×#12111yescharacteristic
639C2×C4C2×#12111yescharacteristic
640C2×C4C22×SD1612yes—
641C2×C4C22×SD1612yes—
642C2×C4C22×SD1612yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4C2×AGL(1,ℤ/8)12yes—
646C2×C4C23×D411yescharacteristic
647C2×C4C23×D411yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C8—24no—
650C8—44no—
651C8—44no—
652C8—44no—
653C8C22×SD1618yes—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—44no—
661C22—84no—
662C22—82no—
663C22—84no—
664C22—82no—
665C22—82no—
666C22—84no—
667C22—164no—
668C22—168no—
669C22—168no—
670C22—168no—
671C22C2×#46012yes—
672C22C2×#46012yes—
673C22C2×#46012yes—
674C22C22×#12111yescharacteristic
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—21no—
681C4—44no—
682C4—44no—
683C4—82no—
684C4C2×#46012yes—
685C4C2×#46012yes—
686C4C2×#46012yes—
687C4#306812yes—
688C4C23×SD1611yescharacteristic
689C4C22×AGL(1,ℤ/8)11yescharacteristic
690C4#358512yes—
691C2—22no—
692C2—22no—
693C2—22no—
694C2—162no—
695C2—164no—
696C2C22×#46011yescharacteristic
697C2#6045411yescharacteristic
698C2C2×#306811yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#3151C2
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. #60941C2semidirect
2. C2×#3208C2semidirect
4. #60192C2semidirect
5. #19238C2semidirect
6. #19203C2semidirect
7. #19202C2semidirect
8. #19170C2semidirect
9. #17787C2semidirect
10. #17704C2semidirect
11. C2×#1545C2semidirect
17. #3246C22semidirect
18. #3229C22semidirect
23. #3208C22semidirect
24. #3208C22semidirect
37. #3103C22semidirect
39. C2×#460C22semidirect
42. Q8×M16C22semidirect
44. #2962C22semidirect
46. #2960C22semidirect
53. #1688C22semidirect
54. #1686C22semidirect
55. #1668C22semidirect
56. #1666C22semidirect
57. #1565C22semidirect
58. #1565C22semidirect
59. #1564C22semidirect
60. #1547C22semidirect
61. #1546C22semidirect
63. #1545C22semidirect
64. #1545C22semidirect
65. #1540C22semidirect
66. C2×#331C22semidirect
146. #475C23semidirect
151. #460C23semidirect
159. #446C23semidirect
161. C8×Q8C23semidirect
196. #335C23semidirect
197. #334C23semidirect
198. #332C23semidirect
200. #331C23semidirect
260. C2×M16SD16semidirect
507. M16C2×SD16semidirect
515. C2×C8C2×SD16semidirect

Maximal subgroups

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

subgroupindex
#609412
C2×#32082
C2×#31512
#601922
#192382
#192032
#192022
#191702
#177872
#177042
C2×#15452