Finite Group Databasebeta

← #6492320

Subgroups — #6492320

Order 512.

1960 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3255C211yesmaximal, characteristic, complemented
2C2×#3216C211yesmaximal, characteristic
3#35298C211yesmaximal, characteristic
4#34145C211yesmaximal, characteristic, complemented
5#33165C211yesmaximal, characteristic, complemented
6#33163C211yesmaximal, characteristic, complemented
7#33033C211yesmaximal, characteristic, complemented
8#33025C211yesmaximal, characteristic, complemented
9#29831C211yesmaximal, characteristic, complemented
10#29826C211yesmaximal, characteristic, complemented
11#29799C211yesmaximal, characteristic, complemented
12#29797C211yesmaximal, characteristic, complemented
13#25379C214yesmaximal, complemented
14#25287C214yesmaximal, complemented
15#25100C214yesmaximal, complemented
16#25098C214yesmaximal, complemented
17#21978C211yesmaximal, characteristic, complemented
18#21394C211yesmaximal, characteristic, complemented
19#21258C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#532C2211yescharacteristic
23#3255—28no—
24#3216—28no—
25C2×#497C2212yes—
26C2×#489C2212yes—
27C2×#475C2211yescharacteristic
28C2×#475C2211yescharacteristic
29C2×#475C2212yes—
30C2×#470C2212yes—
31C2×#470C2212yes—
32C2×#464C2214yes—
33C2×#462C2211yescharacteristic
34C2×#462C2212yes—
35C2×#462C2211yescharacteristic
36C2×#462C2211yescharacteristic
37C2×#462C2211yescharacteristic
38C2×#462C2214yes—
39C2×#460C2211yescharacteristic
40C2×#460C2211yescharacteristic
41C2×#452C2212yes—
42C2×#452C2212yes—
43C2×#432C2211yescharacteristic
44C2×#423C2211yescharacteristic
45C2×#417C2214yes—
46#2868C2211yescharacteristic
47#2862C2211yescharacteristic
48#2858C2211yescharacteristic
49#2858C2211yescharacteristic
50#2620—21no—
51#2620—21no—
52#2566—22no—
53#2545C2214yes—
54#2500—22no—
55#2432C2214yes—
56#2370—22no—
57#2369C2212yes—
58#2368—22no—
59#2365C2212yes—
60#2360C2212yes—
61#2360C2212yes—
62#2326—21no—
63#2326—21no—
64#2296—21no—
65#2295—21no—
66#2295—21no—
67#2295C2211yescharacteristic
68#2295C2211yescharacteristic
69C42×Q8C2211yescharacteristic
70C42×Q8C2211yescharacteristic
71C4×#117—21no—
72C4×#117C2211yescharacteristic
73C2×C4×C4⋊C4C2211yescharacteristic
74#2066C2214yes—
75#2065C2214yes—
76#2056C2214yes—
77#2055C2214yes—
78#2043C2214yes—
79#2042C2214yes—
80#2016C2214yes—
81#2015C2214yes—
82#1988C2211yescharacteristic
83#1988C2211yescharacteristic
84#1983C2214yes—
85#1983C2214yes—
86#1982C2211yescharacteristic
87#1982C2211yescharacteristic
88#1981C2211yescharacteristic
89#1981C2211yescharacteristic
90#1981C2214yes—
91#1981C2211yescharacteristic
92#1981C2211yescharacteristic
93#1940C2214yes—
94#1927C2214yes—
95#1926C2214yes—
96#1880C2214yes—
97#1878C2214yes—
98#1872C2214yes—
99#1851C2214yes—
100#1842C2211yescharacteristic
101#1842C2211yescharacteristic
102#1831C2211yescharacteristic
103#1831C2211yescharacteristic
104#1763C2214yes—
105C22×#41C2312yes—
106C22×Q16C2312yes—
107C22×Q16C2312yes—
108C22×Q16C2312yes—
109C22×Q16C2312yes—
110C22×SD16C2312yes—
111C22×SD16C2312yes—
112C22×M16C2312yes—
113#533—24no—
114#533—24no—
115#533—24no—
116#533D418yes—
117#532—24no—
118#532—24no—
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2314yes—
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2311yescharacteristic
125C2×#127C2311yescharacteristic
126C2×#125—21no—
127C2×#125—21no—
128C2×#124—21no—
129C2×#124—21no—
130C2×#124C2312yes—
131C2×#124C2314yes—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8—21no—
134C2×C22⋊Q8C2312yes—
135C2×C22⋊Q8C2311yescharacteristic
136C2×C22⋊Q8C2311yescharacteristic
137C2×#121—21no—
138C2×#121—21no—
139C2×C4×Q8—21no—
140C2×C4×Q8—21no—
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2312yes—
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×D4—21no—
147C2×C4×D4—21no—
148C2×#117—21no—
149C2×#117—21no—
150C2×#117D411yescharacteristic
151C2×#117D411yescharacteristic
152C22×C4⋊C4C2312yes—
153C22×C42—21no—
154C22×C42C2311yescharacteristic
155#497—28no—
156#489—28no—
157#475—28no—
158#475—24no—
159#475—24no—
160#470—28no—
161#470—28no—
162#464—48no—
163#462—28no—
164#462—24no—
165#462—24no—
166#462—24no—
167#462—24no—
168#462—48no—
169#460—24no—
170#460—24no—
171#452—28no—
172#452—28no—
173#432—24no—
174C2×#106C2312yes—
175C2×#106C2312yes—
176#423—24no—
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2311yescharacteristic
179C2×C4⋊C8C2312yes—
180C2×C4⋊C8C2311yescharacteristic
181C2×C4⋊C8C2311yescharacteristic
182C2×C4⋊C8C2314yes—
183C2×C4⋊C8C2314yes—
184#417—48no—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2312yes—
188C2×#103C2312yes—
189C2×#103C2312yes—
190C2×#103C2312yes—
191C2×#103C2314yes—
192C2×#103C2314yes—
193C2×#103C2314yes—
194C2×#103C2314yes—
195C2×#102C2312yes—
196C2×#102C2312yes—
197C2×#102C2314yes—
198C2×#102C2314yes—
199C2×C22⋊C8C2312yes—
200C2×#97C2312yes—
201#396—24no—
202#396—24no—
203#394—24no—
204#394—24no—
205#391—21no—
206#391—21no—
207#391—22no—
208#391—22no—
209#391C2312yes—
210#391C2312yes—
211#391C2314yes—
212#390—22no—
213#390—22no—
214#390—21no—
215#390—21no—
216#390C2314yes—
217#389—22no—
218#389—22no—
219#389—22no—
220#389—22no—
221#389C2312yes—
222#389C2312yes—
223#388—24no—
224#388—24no—
225#387—22no—
226#387—22no—
227#387—22no—
228#387—22no—
229#387—22no—
230#387—22no—
231#385—22no—
232#385—22no—
233#385—24no—
234#385—24no—
235#384—22no—
236#384—21no—
237#384—21no—
238#382—22no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4D412yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2311yescharacteristic
254C4×C4⋊C4C2311yescharacteristic
255C4×C4⋊C4C2312yes—
256C4×C22⋊C4—22no—
257C4×C22⋊C4—22no—
258C4×C22⋊C4D412yes—
259C4×C22⋊C4C2312yes—
260#376—21no—
261#376—21no—
262#376D411yescharacteristic
263#376D411yescharacteristic
264C2×#95—21no—
265C43—21no—
266C43—21no—
267C43C2311yescharacteristic
268C43C2311yescharacteristic
269#340C2314yes—
270#340C2314yes—
271#336C2314yes—
272#336C2314yes—
273C2×#41—24no—
274C2×#41—24no—
275C2×#41—44no—
276C2×#41—44no—
277C22×Q8—21no—
278C22×Q8—21no—
279C22×Q8C2412yes—
280C22×Q8C2412yes—
281C22×Q8C2411yescharacteristic
282C22×Q8C2411yescharacteristic
283C22×D4—22no—
284C22×D4—21no—
285C22×D4—21no—
286C22×D4C2412yes—
287C23×C4—22no—
288C23×C4—22no—
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2411yescharacteristic
292C2×Q16—24no—
293C2×Q16—24no—
294C2×Q16—24no—
295C2×Q16—24no—
296C2×Q16—44no—
297C2×Q16—44no—
298C2×Q16—44no—
299C2×Q16—44no—
300C2×SD16—24no—
301C2×SD16—24no—
302C2×SD16—44no—
303C2×SD16—44no—
304C2×M16—24no—
305C2×M16—44no—
306C22×C8C2412yes—
307C22×C8C2412yes—
308C22×C8C2412yes—
309C22×C8C2412yes—
310C4⋊Q8—22no—
311C4⋊Q8—22no—
312C4⋊Q8—22no—
313C4⋊Q8—22no—
314C4⋊Q8—22no—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C4⋊Q8—22no—
318C4⋊Q8—44no—
319C4⋊Q8—44no—
320C4⋊Q8C2×D414yes—
321C4⋊Q8C2×D414yes—
322#127—22no—
323#127—22no—
324#125—28no—
325#125—44no—
326#124—24no—
327#124—24no—
328#124—44no—
329#124—44no—
330#124—44no—
331#124—44no—
332C22⋊Q8—28no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—24no—
336C22⋊Q8—24no—
337C22⋊Q8—44no—
338C22⋊Q8C2×D414yes—
339C22⋊Q8C2×D414yes—
340#121—44no—
341#121—44no—
342C4×Q8—28no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—24no—
346C4×Q8—24no—
347C4×Q8—24no—
348C4×Q8—24no—
349C4×Q8—42no—
350C4×Q8—42no—
351C4×Q8—44no—
352C4×Q8C2×D414yes—
353C4×Q8C2×D414yes—
354C4×D4—44no—
355C4×D4—44no—
356#117—28no—
357#117—24no—
358#117—42no—
359#117—44no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—42no—
383C2×C4⋊C4C2×D412yes—
384C2×C4⋊C4C2×D412yes—
385C2×C4⋊C4C2412yes—
386C2×C4⋊C4C2412yes—
387C2×C4⋊C4C2411yescharacteristic
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2412yes—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4C2×D412yes—
400C2×C22⋊C4C2×D412yes—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42—22no—
411C2×C42—44no—
412C2×C42C2×D411yescharacteristic
413C2×C42C2×D411yescharacteristic
414C2×C42C2×D411yescharacteristic
415C2×C42C2×D412yes—
416C2×C42C2×D412yes—
417C2×C42C2×D411yescharacteristic
418C2×C42C2×D411yescharacteristic
419C2×C42C2×D414yes—
420C2×C42C2×D411yescharacteristic
421C2×C42#4114yes—
422C2×C42C2412yes—
423C2×C42C2412yes—
424C2×C42C2411yescharacteristic
425C2×C42C2411yescharacteristic
426#106—24no—
427#106—24no—
428C4⋊C8—24no—
429C4⋊C8—22no—
430C4⋊C8—22no—
431C4⋊C8—22no—
432C4⋊C8—22no—
433C4⋊C8—44no—
434C4⋊C8—44no—
435#103—24no—
436#103—24no—
437#103—24no—
438#103—24no—
439#103—24no—
440#103—24no—
441#103—28no—
442#103—28no—
443#103—44no—
444#103—44no—
445#102—24no—
446#102—24no—
447#102—28no—
448#102—44no—
449C22⋊C8—24no—
450#97—24no—
451#95—21no—
452#95—21no—
453#95—22no—
454#95—22no—
455#95—22no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—22no—
461#95—22no—
462#95—24no—
463#95—24no—
464#95C2×D412yes—
465#95C2×D412yes—
466#95C2×D412yes—
467#95C2×D412yes—
468C24—22no—
469C24C22×D411yescharacteristic
470#41—416no—
471#41—84no—
472#41—84no—
473C2×Q8—24no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—22no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8—22no—
482C2×Q8—24no—
483C2×Q8—24no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—24no—
487C2×Q8—24no—
488C2×Q8—42no—
489C2×Q8—42no—
490C2×Q8—42no—
491C2×Q8—82no—
492C2×Q8C22×D412yes—
493C2×Q8C22×D412yes—
494C2×D4—22no—
495C2×D4—22no—
496C2×D4—24no—
497C2×D4—24no—
498C2×D4—44no—
499C2×D4—44no—
500C2×D4—42no—
501C2×D4—42no—
502C2×D4—42no—
503C2×D4—42no—
504C2×D4—84no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—42no—
519C22×C4—44no—
520C22×C4—44no—
521C22×C4—44no—
522C22×C4—84no—
523C22×C4—82no—
524C22×C4#12114yes—
525C22×C4#12114yes—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D411yescharacteristic
532C22×C4C22×D411yescharacteristic
533C22×C4C22×D412yes—
534C22×C4C22×D412yes—
535C22×C4C22×D411yescharacteristic
536C22×C4C2×#4112yes—
537C22×C4C2511yescharacteristic
538Q16—84no—
539Q16—84no—
540Q16—84no—
541Q16—84no—
542SD16—84no—
543SD16—84no—
544M16—84no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—24no—
550C2×C8—24no—
551C2×C8—24no—
552C2×C8—24no—
553C4⋊C4—24no—
554C4⋊C4—24no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—24no—
558C4⋊C4—24no—
559C4⋊C4—24no—
560C4⋊C4—24no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—42no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—44no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4—42no—
575C4⋊C4—42no—
576C4⋊C4—44no—
577C4⋊C4—44no—
578C4⋊C4—42no—
579C4⋊C4—82no—
580C4⋊C4—82no—
581C4⋊C4—84no—
582C4⋊C4—84no—
583C4⋊C4C22×D412yes—
584C4⋊C4C22×D412yes—
585C4⋊C4H(5,𝔽2)14yes—
586C22⋊C4—24no—
587C22⋊C4—24no—
588C22⋊C4—24no—
589C22⋊C4—42no—
590C22⋊C4—48no—
591C22⋊C4—44no—
592C22⋊C4—44no—
593C22⋊C4—44no—
594C22⋊C4—48no—
595C22⋊C4—84no—
596C22⋊C4—84no—
597C42—24no—
598C42—24no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—22no—
605C42—22no—
606C42—42no—
607C42—42no—
608C42—44no—
609C42—44no—
610C42—44no—
611C42—44no—
612C42—42no—
613C42—42no—
614C42—82no—
615C42—82no—
616C42#12114yes—
617C42#12114yes—
618C42#13714yes—
619C42C22×D412yes—
620C42C22×D412yes—
621C42C2×#4112yes—
622C42C2×#4112yes—
623C23—22no—
624C23—22no—
625C23—21no—
626C23—21no—
627C23—44no—
628C23—42no—
629C23—44no—
630C23—42no—
631C23—42no—
632C23C2×#12112yes—
633C23C23×D411yescharacteristic
634Q8—22no—
635Q8—22no—
636Q8—22no—
637Q8—22no—
638Q8—42no—
639Q8—42no—
640Q8—44no—
641Q8—44no—
642Q8—42no—
643Q8—42no—
644Q8—42no—
645Q8—42no—
646Q8—84no—
647Q8—84no—
648D4—44no—
649D4—42no—
650D4—42no—
651D4—84no—
652D4—88no—
653D4—84no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—24no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—28no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—22no—
674C2×C4—22no—
675C2×C4—24no—
676C2×C4—22no—
677C2×C4—22no—
678C2×C4—22no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—44no—
691C2×C4—42no—
692C2×C4—44no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—88no—
697C2×C4—88no—
698C2×C4C2×#12112yes—
699C2×C4C2×#12112yes—
700C2×C4C2×#12112yes—
701C2×C4C2×#12112yes—
702C2×C4C2×#12112yes—
703C2×C4C2×#13714yes—
704C2×C4C2×#13712yes—
705C2×C4#57512yes—
706C2×C4C23×D411yescharacteristic
707C2×C4C22×#4111yescharacteristic
708C2×C4C2×H(5,𝔽2)12yes—
709C2×C4#58512yes—
710C8—42no—
711C8—42no—
712C8—42no—
713C8—42no—
714C22—22no—
715C22—22no—
716C22—22no—
717C22—42no—
718C22—42no—
719C22—82no—
720C22—84no—
721C22—84no—
722C22—82no—
723C22—82no—
724C22C22×#12111yescharacteristic
725C22#348812yes—
726C22C22×#13711yescharacteristic
727C22C2×#57511yescharacteristic
728C22#358612yes—
729C4—22no—
730C4—22no—
731C4—24no—
732C4—22no—
733C4—22no—
734C4—42no—
735C4—42no—
736C4—42no—
737C4—42no—
738C4—42no—
739C4—44no—
740C4—42no—
741C4—42no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4#303714yes—
747C4#346012yes—
748C4#349811yescharacteristic
749C4#350011yescharacteristic
750C2—42no—
751C2—162no—
752C2#3525611yescharacteristic
753C2C2×#303711yescharacteristic
754C2#6042711yescharacteristic
755C2#6196612yes—
756C2#6196712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3255C2semidirect
4. #34145C2semidirect
5. #33165C2semidirect
6. #33163C2semidirect
7. #33033C2semidirect
8. #33025C2semidirect
9. #29831C2semidirect
10. #29826C2semidirect
11. #29799C2semidirect
12. #29797C2semidirect
13. #25379C2semidirect
14. #25287C2semidirect
15. #25100C2semidirect
16. #25098C2semidirect
17. #21978C2semidirect
18. #21394C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32552
C2×#32162
#352982
#341452
#331652
#331632
#330332
#330252
#298312
#298262
#297992
#297972
#253792
#252872
#251002
#250982
#219782
#213942
#212582