Finite Group Databasebeta

← #6480210

Subgroups — #6480210

Order 512.

2194 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2858C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#33025C214yesmaximal, complemented
4#30062C212yesmaximal, complemented
5#30041C212yesmaximal, complemented
6#29555C214yesmaximal, complemented
7#29346C212yesmaximal, complemented
8#28374C212yesmaximal, complemented
9#27585C212yesmaximal, complemented
10#24301C212yesmaximal, complemented
11#24211C212yesmaximal
12#23933C212yesmaximal, complemented
13#22043C214yesmaximal, complemented
14#20777C211yesmaximal, characteristic, complemented
15C2×#533C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×#124—21no—
19C22×#124—21no—
20C2×#475C2214yescomplemented
21C2×#460C2214yes—
22C2×#450C2214yes—
23#2858—21no—
24#2858—21no—
25#2858—21no—
26#2858—21no—
27#2858C2212yes—
28#2858C2212yes—
29#2858C2212yes—
30#2858C2212yes—
31C4×C4⋊Q8C2211yescharacteristic, complemented
32C4×C4⋊Q8C2211yescharacteristic, complemented
33C4×C4⋊Q8C2211yescharacteristic, complemented
34C4×C4⋊Q8C2211yescharacteristic, complemented
35C4×#125—21no—
36C4×#125—21no—
37C4×C22⋊Q8—21no—
38C4×C22⋊Q8—21no—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41#2296—21no—
42#2295—21no—
43#2295C2212yescomplemented
44C2×#390—21no—
45C2×#390—21no—
46C2×#390C2212yes—
47C2×#390C2212yes—
48#2285C2211yescharacteristic
49C42×Q8—21no—
50C42×Q8C2212yescomplemented
51C4×#117—21no—
52C2×C43C2211yescharacteristic
53#2094C2214yescomplemented
54#2071C2214yescomplemented
55#2058C2214yescomplemented
56#2041C2214yes—
57#2024C2214yescomplemented
58#2023C2214yescomplemented
59#2014C2214yes—
60#2013C2214yescomplemented
61#1988C2214yescomplemented
62#1982C2214yes—
63#1968C2218yescomplemented
64#1934C2218yescomplemented
65#1926C2218yescomplemented
66#1926C2218yescomplemented
67#1880C2218yescomplemented
68#1880C2218yescomplemented
69#1872C2214yes—
70#1853C2214yes—
71#1839C2212yescomplemented
72#1831C2212yescomplemented
73#1817C2212yes—
74#1797C2212yes—
75#1786C2214yes—
76C4×C4⋊C8C2212yescomplemented
77C4×C22⋊C8C2212yes—
78#1744C2212yescomplemented
79C23×Q8—21no—
80C23×D4—21no—
81C23×D4—21no—
82C23×D4C2312yes—
83C22×SD16C2318yes—
84#533—28no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—44no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2312yes—
92C2×#127—21no—
93C2×#127—21no—
94C2×#127—44no—
95C2×#127C2312yes—
96C2×#127C2312yes—
97C2×#125—21no—
98C2×#124—22no—
99C2×#124—22no—
100C2×#124—22no—
101C2×#124—22no—
102C2×#124—22no—
103C2×#124—22no—
104C2×#124—22no—
105C2×#124—22no—
106C2×#124—48no—
107C2×#124—48no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×#117—21no—
117C2×#117—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4—22no—
121C22×C22⋊C4—22no—
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#475—48no—
132#460—48no—
133#450—216no—
134C2×#106C2318yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2318yes—
137C2×#102C2318yes—
138C2×#102C2318yes—
139C2×#102C2318yes—
140C2×C22⋊C8C2314yes—
141C2×C4×C8C2314yes—
142#391—21no—
143#391—21no—
144#391C2312yes—
145#391C2312yes—
146#390—22no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390—22no—
158#390C2314yes—
159#390C2314yes—
160#390C2314yes—
161#390C2314yes—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389C2312yes—
169#389C2312yes—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#385—22no—
175#385—22no—
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#382—21no—
179#380—21no—
180#380C2312yes—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199#376—22no—
200C43—22no—
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43D411yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208C43C2311yescharacteristic
209#340C2318yes—
210#340C2318yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2314yes—
215C25—22no—
216C22×Q8—21no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—44no—
220C22×Q8—44no—
221C22×Q8—44no—
222C22×Q8C2412yes—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—28no—
229C22×D4—44no—
230C22×D4—44no—
231C22×D4—44no—
232C22×D4—44no—
233C22×D4—44no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4C2414yes—
238C23×C4—22no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×SD16—216no—
247C2×SD16—48no—
248C2×SD16—48no—
249C22×C8C2418yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—28no—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256#127—28no—
257#127—44no—
258#127—44no—
259#127—44no—
260#125—28no—
261#124—48no—
262#124—48no—
263#124—48no—
264#124—48no—
265#124—48no—
266#124—48no—
267#124—48no—
268#124—48no—
269C22⋊Q8—28no—
270C22⋊Q8—24no—
271C22⋊Q8#4118yes—
272C4×Q8—28no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—28no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—44no—
279#117—28no—
280#117—44no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4C2414yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—48no—
314C2×C22⋊C4—48no—
315C2×C22⋊C4—48no—
316C2×C22⋊C4—48no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42SD1618yescomplemented
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#106—48no—
359C4⋊C8—28no—
360#103—48no—
361#102—216no—
362#102—48no—
363#102—48no—
364C22⋊C8—28no—
365C4×C8—44no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376C24—24no—
377C24—44no—
378C24—44no—
379C24—44no—
380C24—44no—
381C24—44no—
382C24—44no—
383C24—44no—
384C24C22×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—816no—
398C2×Q8C2×#4114yes—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—816no—
419C2×D4—816no—
420C2×D4—816no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4C2×SD1614yes—
447C22×C4C2×SD1614yes—
448C22×C4C2×SD1614yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2×#4114yes—
463C22×C4C2511yescharacteristic
464SD16—816no—
465C2×C8—28no—
466C2×C8—28no—
467C2×C8—28no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—28no—
471C4⋊C4—24no—
472C4⋊C4—28no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—28no—
476C4⋊C4—24no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4C2×#4114yes—
489C22⋊C4—28no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C42—28no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—44no—
523C42—44no—
524C42—44no—
525C42—44no—
526C42—44no—
527C42C2×SD1614yes—
528C42C2×SD1614yes—
529C42C2×SD1614yes—
530C42C2×SD1614yes—
531C42AGL(1,ℤ/8)14yes—
532C42H(5,𝔽2)14yes—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—88no—
545C23—88no—
546C23—88no—
547C23—88no—
548C23—88no—
549C23—88no—
550C23—88no—
551C23#52512yes—
552C23C22×SD1612yes—
553C23#57512yes—
554C23C23×D411yescharacteristic
555Q8—28no—
556Q8—48no—
557Q8—44no—
558Q8—88no—
559Q8—88no—
560D4—48no—
561D4—48no—
562D4—88no—
563D4—88no—
564D4—88no—
565D4—88no—
566D4—88no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—21no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—88no—
616C2×C4#52512yes—
617C2×C4#53512yes—
618C2×C4#53512yes—
619C2×C4C22×SD1612yes—
620C2×C4C22×SD1612yes—
621C2×C4C22×SD1612yes—
622C2×C4C22×SD1612yes—
623C2×C4C22×SD1612yes—
624C2×C4C22×SD1612yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4#57512yes—
629C2×C4C22×#4112yes—
630C2×C4C2×H(5,𝔽2)12yes—
631C2×C4#58512yes—
632C8—48no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—1616no—
647C22#310314yes—
648C22C2×#52511yescharacteristic
649C22C2×#53511yescharacteristic
650C22#344911yescharacteristic
651C22C23×SD1611yescharacteristic
652C22C22×AGL(1,ℤ/8)11yescharacteristic
653C22C2×#57511yescharacteristic
654C22C2×#57511yescharacteristic
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—41no—
666C4—44no—
667C4—41no—
668C4—44no—
669C4—82no—
670C4—84no—
671C4—82no—
672C4#320814yes—
673C4#351414yes—
674C2—24no—
675C2—164no—
676C2#3512911yescharacteristic
677C2C2×#310311yescharacteristic
678C2#6056111yescharacteristic
679C2C2×#320811yescharacteristic
680C2#6079911yescharacteristic
681C2#6094111yescharacteristic
682C2#6096211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#533C2semidirect
3. #33025C2semidirect
4. #30062C2semidirect
5. #30041C2semidirect
6. #29555C2semidirect
7. #29346C2semidirect
8. #28374C2semidirect
9. #27585C2semidirect
10. #24301C2semidirect
12. #23933C2semidirect
13. #22043C2semidirect
14. #20777C2semidirect
20. C2×#475C22semidirect
31. C4×C4⋊Q8C22semidirect
32. C4×C4⋊Q8C22semidirect
33. C4×C4⋊Q8C22semidirect
34. C4×C4⋊Q8C22semidirect
43. #2295C22semidirect
50. C42×Q8C22semidirect
53. #2094C22semidirect
54. #2071C22semidirect
55. #2058C22semidirect
57. #2024C22semidirect
58. #2023C22semidirect
60. #2013C22semidirect
61. #1988C22semidirect
63. #1968C22semidirect
64. #1934C22semidirect
65. #1926C22semidirect
66. #1926C22semidirect
67. #1880C22semidirect
68. #1880C22semidirect
71. #1839C22semidirect
72. #1831C22semidirect
76. C4×C4⋊C8C22semidirect
78. #1744C22semidirect
329. C2×C42SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#28582
C4×#5332
#330252
#300622
#300412
#295552
#293462
#283742
#275852
#243012
#242112
#239332
#220432
#207772