Finite Group Databasebeta

← #6475473

Subgroups — #6475473

Order 512.

2112 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic
3#33049C214yesmaximal, complemented
4#30199C214yesmaximal, complemented
5#29335C214yesmaximal, complemented
6#29111C214yesmaximal, complemented
7#27972C212yesmaximal, complemented
8#27960C212yesmaximal, complemented
9#25902C212yesmaximal, complemented
10#25856C212yesmaximal, complemented
11#21765C214yesmaximal
12C4×#407C211yesmaximal, characteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8—21no—
16C22×C22⋊Q8C2212yes—
17C23×C42C2211yescharacteristic
18C2×#487C2214yescomplemented
19C2×#407C2212yes—
20C2×C4×M16C2212yes—
21#2857—21no—
22#2857—21no—
23#2857—21no—
24#2857—21no—
25#2857—21no—
26#2857—21no—
27#2857—21no—
28#2857—21no—
29#2857C2212yes—
30#2857C2212yes—
31#2857C2212yes—
32#2857C2212yes—
33#2857C2212yes—
34#2857C2212yes—
35#2857C2212yes—
36#2857C2212yes—
37C2×#391—21no—
38C2×#391C2212yes—
39C2×#389—21no—
40C2×#389—21no—
41C2×#389—21no—
42C2×#389C2212yes—
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#380—21no—
46C2×#380—21no—
47C2×#380—21no—
48C2×#380C2212yes—
49#2099C2218yescomplemented
50#2072C2218yescomplemented
51#2043C2218yescomplemented
52#2024C2218yescomplemented
53#1990C2214yescomplemented
54#1956C2214yescomplemented
55#1935C22116yescomplemented
56#1894C22116yes—
57#1872C2218yes—
58#1872C2218yes—
59#1834C2214yescomplemented
60#1823C2218yes—
61#1787C2218yes—
62#1749C2212yes—
63#1749C2212yes—
64C4×C22⋊C8C2212yes—
65C4×C22⋊C8C2212yes—
66#1728C2212yes—
67C23×Q8—21no—
68C23×Q8C2312yes—
69C24×C4D411yescharacteristic
70C24×C4D411yescharacteristic
71C24×C4C2311yescharacteristic
72C22×M16C2314yes—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8—44no—
78C2×C4⋊Q8—44no—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2312yes—
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2312yes—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—216no—
90C2×C22⋊Q8—48no—
91C2×C22⋊Q8C2314yes—
92C2×C22⋊Q8C2314yes—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—22no—
107C22×C42—44no—
108C22×C42D411yescharacteristic
109C22×C42D411yescharacteristic
110C22×C42D411yescharacteristic
111C22×C42D411yescharacteristic
112C22×C42D411yescharacteristic
113C22×C42D411yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#487—48no—
121C2×#103C23116yes—
122C2×#103C23116yes—
123#407—28no—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2314yes—
126C4×M16—44no—
127C2×#97C2314yes—
128C2×C4×C8C2314yes—
129#391—22no—
130#391—22no—
131#391—22no—
132#391—22no—
133#391—22no—
134#391—22no—
135#391C2314yes—
136#391C2314yes—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389C2314yes—
152#389C2314yes—
153#384—21no—
154#384—21no—
155#384—21no—
156#384—21no—
157#384C2312yes—
158#384C2312yes—
159#384C2312yes—
160#384C2312yes—
161#380—22no—
162#380—22no—
163#380—22no—
164#380—22no—
165#380—22no—
166#380—22no—
167#380—22no—
168#380—22no—
169#380—22no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380—22no—
174#380—22no—
175#380C2314yes—
176#380C2314yes—
177C2×#95—22no—
178C2×#95—22no—
179C2×#95—22no—
180C2×#95—22no—
181#340C23116yes—
182#340C23116yes—
183#336C2314yes—
184#336C2314yes—
185C25C2×D411yescharacteristic
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—28no—
190C22×Q8—44no—
191C22×Q8—44no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8—44no—
195C22×Q8C2414yes—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4#4118yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C2×M16—28no—
228C2×M16—44no—
229C2×M16—44no—
230C22×C8C2418yes—
231C4⋊Q8—28no—
232C4⋊Q8—28no—
233C4⋊Q8—44no—
234C4⋊Q8—44no—
235C4⋊Q8—44no—
236C4⋊Q8—44no—
237C4⋊Q8—44no—
238C4⋊Q8—44no—
239C22⋊Q8—216no—
240C22⋊Q8—216no—
241C22⋊Q8—48no—
242C22⋊Q8—48no—
243C22⋊Q8—48no—
244C22⋊Q8—48no—
245C22⋊Q8—48no—
246C22⋊Q8—48no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—48no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2414yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317#103—416no—
318#103—416no—
319C22⋊C8—28no—
320C22⋊C8—28no—
321#97—44no—
322C4×C8—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—41no—
336C24—41no—
337C24#12414yes—
338C24C22×D411yescharacteristic
339C24C22×D411yescharacteristic
340C24C22×D411yescharacteristic
341C2×Q8—28no—
342C2×Q8—28no—
343C2×Q8—28no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—816no—
354C2×Q8—816no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—416no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4#12414yes—
419C22×C4#12414yes—
420C22×C4#12414yes—
421C22×C4AGL(1,ℤ/8)14yes—
422C22×C4#13714yes—
423C22×C4#13714yes—
424C22×C4#13714yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2×#4114yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4114yes—
439C22×C4C2511yescharacteristic
440M16—416no—
441C2×C8—28no—
442C2×C8—28no—
443C2×C8—28no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42H(5,𝔽2)14yes—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—28no—
497C23—28no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23C2×#12412yes—
510C23C2×#12412yes—
511C23C2×#12412yes—
512C23C2×AGL(1,ℤ/8)12yes—
513C23C2×#13712yes—
514C23C2×#13712yes—
515C23C2×#13712yes—
516C23C23×D411yescharacteristic
517Q8—416no—
518Q8—88no—
519Q8—88no—
520Q8—88no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—88no—
572C2×C4—88no—
573C2×C4—88no—
574C2×C4—88no—
575C2×C4C2×#12412yes—
576C2×C4C2×#12412yes—
577C2×C4C2×#12412yes—
578C2×C4#53412yes—
579C2×C4#53412yes—
580C2×C4C2×AGL(1,ℤ/8)12yes—
581C2×C4C2×AGL(1,ℤ/8)12yes—
582C2×C4C2×#13712yes—
583C2×C4C2×#13712yes—
584C2×C4C2×#13712yes—
585C2×C4C2×#13712yes—
586C2×C4C2×#13712yes—
587C2×C4C2×#13712yes—
588C2×C4C22×#4112yes—
589C2×C4C2×H(5,𝔽2)12yes—
590C2×C4#58512yes—
591C8—48no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—416no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22#313914yes—
613C22#314114yes—
614C22C22×#12411yescharacteristic
615C22C2×#53411yescharacteristic
616C22#344911yescharacteristic
617C22C22×AGL(1,ℤ/8)11yescharacteristic
618C22C22×#13711yescharacteristic
619C22C22×#13711yescharacteristic
620C22C22×#13711yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—84no—
631C4—84no—
632C4—84no—
633C4—84no—
634C4#319314yes—
635C4#319514yes—
636C4#350514yes—
637C2—24no—
638C2—24no—
639C2—42no—
640C2#3501911yescharacteristic
641C2C2×#313911yescharacteristic
642C2C2×#314111yescharacteristic
643C2C2×#319311yescharacteristic
644C2C2×#319511yescharacteristic
645C2#6096911yescharacteristic
646C2#6097111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33049C2semidirect
4. #30199C2semidirect
5. #29335C2semidirect
6. #29111C2semidirect
7. #27972C2semidirect
8. #27960C2semidirect
9. #25902C2semidirect
10. #25856C2semidirect
18. C2×#487C22semidirect
49. #2099C22semidirect
50. #2072C22semidirect
51. #2043C22semidirect
52. #2024C22semidirect
53. #1990C22semidirect
54. #1956C22semidirect
55. #1935C22semidirect
59. #1834C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#330492
#301992
#293352
#291112
#279722
#279602
#259022
#258562
#217652
C4×#4072