Finite Group Databasebeta

← #6479823

Subgroups — #6479823

Order 512.

1694 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3#33148C212yesmaximal
4#33108C212yesmaximal
5#29908C212yesmaximal
6#29641C212yesmaximal
7#29512C214yesmaximal, complemented
8#29390C214yesmaximal, complemented
9#29347C212yesmaximal, complemented
10#29313C212yesmaximal, complemented
11#29273C212yesmaximal, complemented
12#29255C212yesmaximal, complemented
13#22078C214yesmaximal
14#20777C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4×Q8—21no—
17C2×#483C2214yes—
18#2864—21no—
19#2863—21no—
20#2863C2212yes—
21#2862C2211yescharacteristic
22#2859C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2855—21no—
26#2855C2212yes—
27#2851—21no—
28C4×C4⋊Q8—21no—
29C4×C4⋊Q8—21no—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8C2211yescharacteristic
33C4×C4⋊Q8C2212yes—
34C4×C4⋊Q8C2212yes—
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2211yescharacteristic
38C4×#125—21no—
39C4×C22⋊Q8—21no—
40C4×C22⋊Q8C2212yes—
41#2295—21no—
42#2295C2212yes—
43C2×#391—21no—
44C2×#389—21no—
45C2×#389C2212yes—
46C2×#384C2211yescharacteristic
47#2283—21no—
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4C2212yes—
50C2×C43C2211yescharacteristic
51#2099C2214yes—
52#2096C2214yes—
53#2084C2214yes—
54#2072C2214yes—
55#2058C2214yes—
56#2058C2214yes—
57#2024C2214yes—
58#2024C2214yes—
59#1987C2214yes—
60#1981C2214yes—
61#1948C2214yes—
62#1948C2214yes—
63#1941C2218yes—
64#1933C2218yes—
65#1927C2218yes—
66#1920C2218yes—
67#1919C2218yes—
68#1872C2214yes—
69#1865C2218yes—
70#1853C2214yes—
71#1839C2212yes—
72#1831C2212yes—
73#1823C2214yes—
74#1817C2212yes—
75#1797C2212yes—
76C4×C4⋊C8C2212yes—
77C4×C22⋊C8C2212yes—
78#1744C2212yes—
79C23×Q8—21no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×#125—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8C2312yes—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8—21no—
95C2×C4×Q8—48no—
96C2×C4×Q8C2312yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C42—22no—
104C22×C42D411yescharacteristic
105C22×C42D411yescharacteristic
106C22×C42D411yescharacteristic
107C22×C42D411yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111#483—216no—
112C2×#106C2318yes—
113C2×#106C2318yes—
114C2×C4⋊C8C2314yes—
115C2×#103C2318yes—
116C2×#103C2318yes—
117C2×C22⋊C8C2314yes—
118C2×C4×C8C2314yes—
119#391—22no—
120#391—22no—
121#391—22no—
122#391—22no—
123#391—21no—
124#391—21no—
125#391—21no—
126#391—21no—
127#391C2312yes—
128#391C2312yes—
129#391C2312yes—
130#391C2312yes—
131#389—21no—
132#389—21no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389—21no—
143#389—21no—
144#389—21no—
145#389C2314yes—
146#389C2314yes—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#385—22no—
160#385—22no—
161#384—21no—
162#384—21no—
163#384C2312yes—
164#384C2312yes—
165#384C2311yescharacteristic
166#384C2311yescharacteristic
167#383—22no—
168#382—21no—
169#382—21no—
170#382—21no—
171#381—22no—
172#380—21no—
173#380—21no—
174#380—21no—
175#380C2312yes—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4C2314yes—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—22no—
191#376—22no—
192C2×#95—22no—
193C2×#95—22no—
194C43D411yescharacteristic
195C43D411yescharacteristic
196C43D411yescharacteristic
197C43D411yescharacteristic
198C43C2311yescharacteristic
199C43C2311yescharacteristic
200C43C2311yescharacteristic
201C43C2311yescharacteristic
202#340C2318yes—
203#340C2318yes—
204#340C2318yes—
205#340C2318yes—
206#336C2314yes—
207#336C2314yes—
208#336C2314yes—
209#336C2314yes—
210C22×Q8—28no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—21no—
214C22×Q8—44no—
215C22×Q8—44no—
216C22×Q8C2412yes—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2411yescharacteristic
226C22×C8C2418yes—
227C4⋊Q8—28no—
228C4⋊Q8—28no—
229C4⋊Q8—28no—
230C4⋊Q8—28no—
231C4⋊Q8—24no—
232C4⋊Q8—24no—
233#125—44no—
234C22⋊Q8—28no—
235C22⋊Q8—44no—
236C4×Q8—44no—
237C4×Q8—44no—
238C4×Q8—48no—
239C4×Q8—48no—
240C4×Q8—48no—
241C4×Q8—48no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4#4118yes—
272C2×C4⋊C4C2414yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2414yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—48no—
290C2×C42SD1618yes—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
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
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319#106—216no—
320#106—48no—
321C4⋊C8—28no—
322#103—216no—
323#103—48no—
324C22⋊C8—28no—
325C4×C8—44no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344C24C22×D411yescharacteristic
345C2×Q8—28no—
346C2×Q8—28no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—432no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—44no—
357C2×Q8—44no—
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—28no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4C2×SD1614yes—
388C22×C4C2×SD1614yes—
389C22×C4C2×SD1614yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4114yes—
404C22×C4C2511yescharacteristic
405C2×C8—28no—
406C2×C8—28no—
407C2×C8—28no—
408C4⋊C4—28no—
409C4⋊C4—28no—
410C4⋊C4—28no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4C2×#4118yes—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—44no—
463C42—44no—
464C42—48no—
465C42—48no—
466C42C2×SD1614yes—
467C42C2×SD1614yes—
468C42C2×SD1614yes—
469C42C2×SD1614yes—
470C42#13714yes—
471C422−1+414yes—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23#52512yes—
477C23C22×SD1612yes—
478C23#57512yes—
479C23C23×D411yescharacteristic
480Q8—416no—
481Q8—416no—
482Q8—48no—
483Q8—84no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—416no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—88no—
537C2×C4#52512yes—
538C2×C4#53612yes—
539C2×C4#53612yes—
540C2×C4C22×SD1612yes—
541C2×C4C22×SD1612yes—
542C2×C4C22×SD1612yes—
543C2×C4C22×SD1612yes—
544C2×C4C22×SD1612yes—
545C2×C4C22×SD1612yes—
546C2×C4C2×#13712yes—
547C2×C4C2×#13712yes—
548C2×C4C2×#13712yes—
549C2×C4#57512yes—
550C2×C4C22×#4112yes—
551C2×C4C2×2−1+412yes—
552C2×C4#58512yes—
553C8—48no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22#310114yes—
561C22C2×#52511yescharacteristic
562C22C2×#53611yescharacteristic
563C22#344911yescharacteristic
564C22C23×SD1611yescharacteristic
565C22C22×#13711yescharacteristic
566C22C2×#57511yescharacteristic
567C22C2×#57511yescharacteristic
568C4—22no—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—41no—
579C4—48no—
580C4—41no—
581C4—84no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4#322914yes—
587C4#350714yes—
588C2—24no—
589C2#3511111yescharacteristic
590C2C2×#310111yescharacteristic
591C2#6056011yescharacteristic
592C2C2×#322911yescharacteristic
593C2#6086611yescharacteristic
594C2#6094111yescharacteristic
595C2#6096311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
7. #29512C2semidirect
8. #29390C2semidirect
9. #29347C2semidirect
10. #29313C2semidirect
11. #29273C2semidirect
12. #29255C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×C4×C4⋊Q82
#331482
#331082
#299082
#296412
#295122
#293902
#293472
#293132
#292732
#292552
#220782
#207772