Finite Group Databasebeta

← #6490587

Subgroups — #6490587

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 613 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#33147C212yesmaximal
4#33109C212yesmaximal
5#30152C214yesmaximal, complemented
6#29355C212yesmaximal, complemented
7#29320C212yesmaximal, complemented
8#29286C212yesmaximal, complemented
9#28283C212yesmaximal, complemented
10#27648C212yesmaximal
11#24606C214yesmaximal
12#24301C212yesmaximal
13#22118C214yesmaximal, complemented
14#20810C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4×Q8—21no—
17C2×#484C2214yes—
18C2×#475C2214yes—
19C2×#432C2212yes—
20#2864—21no—
21#2863—21no—
22#2863C2212yes—
23#2862C2211yescharacteristic
24#2859C2211yescharacteristic
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2855—21no—
28#2855C2212yes—
29#2851—21no—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2211yescharacteristic
40C4×#125—21no—
41C4×C22⋊Q8—21no—
42C4×C22⋊Q8C2212yes—
43#2295—21no—
44#2295C2212yes—
45C2×#391—21no—
46C2×#389—21no—
47C2×#389C2212yes—
48C2×#384C2211yescharacteristic
49#2283—21no—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2212yes—
52C2×C43C2211yescharacteristic
53#2086C2218yes—
54#2083C2214yes—
55#2072C2214yes—
56#2066C2218yes—
57#2058C2214yes—
58#2024C2214yes—
59#1988C2214yes—
60#1988C2214yes—
61#1981C2214yes—
62#1948C2214yes—
63#1947C2214yes—
64#1947C2214yes—
65#1944C2218yes—
66#1935C2218yes—
67#1928C2218yes—
68#1919C2218yes—
69#1872C2214yes—
70#1853C2214yes—
71#1842C2212yes—
72#1831C2212yes—
73#1824C2214yes—
74#1803C2214yes—
75#1797C2212yes—
76#1797C2212yes—
77#1767C2212yes—
78#1767C2212yes—
79C23×Q8—21no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
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#484—48no—
112#475—216no—
113#432—44no—
114C2×#107C2318yes—
115C2×#106C2318yes—
116C2×C4⋊C8C2314yes—
117C2×C4⋊C8C2314yes—
118C2×C4⋊C8C2314yes—
119C2×#103C2318yes—
120C2×#103C2318yes—
121C2×C22⋊C8C2314yes—
122C2×#97C2314yes—
123#391—22no—
124#391—22no—
125#391—22no—
126#391—22no—
127#391—21no—
128#391—21no—
129#391—21no—
130#391—21no—
131#391C2312yes—
132#391C2312yes—
133#391C2312yes—
134#391C2312yes—
135#389—21no—
136#389—21no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—21no—
144#389—21no—
145#389—21no—
146#389—21no—
147#389—21no—
148#389—21no—
149#389C2314yes—
150#389C2314yes—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#389C2312yes—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#385—22no—
164#385—22no—
165#384—21no—
166#384—21no—
167#384C2312yes—
168#384C2312yes—
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#383—22no—
172#382—21no—
173#382—21no—
174#382—21no—
175#381—22no—
176#380—21no—
177#380—21no—
178#380—21no—
179#380C2312yes—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4C2314yes—
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2312yes—
194C4×C22⋊C4—22no—
195#376—22no—
196C2×#95—22no—
197C2×#95—22no—
198C43D411yescharacteristic
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43C2311yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206#340C2318yes—
207#340C2318yes—
208#340C2318yes—
209#340C2318yes—
210#336C2314yes—
211#336C2314yes—
212C22×Q8—28no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—21no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8C2412yes—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2411yescharacteristic
228C22×C8C2418yes—
229C4⋊Q8—28no—
230C4⋊Q8—28no—
231C4⋊Q8—28no—
232C4⋊Q8—24no—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8#4118yes—
236#125—44no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8—44no—
240C4×Q8—28no—
241C4×Q8—48no—
242C4×Q8—48no—
243C4×Q8—48no—
244C4×Q8—44no—
245C4×Q8—48no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2414yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—42no—
295C2×C42—48no—
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×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324#107—48no—
325#106—216no—
326C4⋊C8—28no—
327C4⋊C8—28no—
328C4⋊C8—44no—
329#103—216no—
330#103—48no—
331C22⋊C8—28no—
332#97—44no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351C24C22×D411yescharacteristic
352C2×Q8—28no—
353C2×Q8—28no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—24no—
358C2×Q8—44no—
359C2×Q8—432no—
360C2×Q8—42no—
361C2×Q8—44no—
362C2×Q8—42no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—42no—
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—28no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4#13714yes—
397C22×C4#13714yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4H(5,𝔽2)12yes—
412C22×C42−1+412yes—
413C22×C4C2511yescharacteristic
414C2×C8—28no—
415C2×C8—28no—
416C2×C8—28no—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—28no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4C2×#4118yes—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—22no—
473C42—21no—
474C42—21no—
475C42—41no—
476C42—41no—
477C42—48no—
478C42—44no—
479C42—48no—
480C42—44no—
481C42AGL(1,ℤ/8)14yes—
482C42C2×#4114yes—
483C422−1+414yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#53712yes—
489C23C2×#13712yes—
490C23C2×#13712yes—
491C23C23×D411yescharacteristic
492Q8—416no—
493Q8—416no—
494Q8—44no—
495Q8—44no—
496Q8—84no—
497C2×C4—28no—
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—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
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—44no—
530C2×C4—44no—
531C2×C4—416no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—88no—
551C2×C4#52512yes—
552C2×C4#52512yes—
553C2×C4#53612yes—
554C2×C4#53612yes—
555C2×C4#53712yes—
556C2×C4C2×AGL(1,ℤ/8)12yes—
557C2×C4C2×AGL(1,ℤ/8)12yes—
558C2×C4C2×AGL(1,ℤ/8)12yes—
559C2×C4C2×#13712yes—
560C2×C4C2×#13712yes—
561C2×C4C2×#13712yes—
562C2×C4C2×#13712yes—
563C2×C4#57512yes—
564C2×C4C22×#4112yes—
565C2×C4C2×H(5,𝔽2)11yescharacteristic
566C2×C4C2×2−1+412yes—
567C2×C4C2×2−1+411yescharacteristic
568C8—48no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22#325514yes—
576C22C2×#52511yescharacteristic
577C22C2×#53611yescharacteristic
578C22C2×#53711yescharacteristic
579C22C22×AGL(1,ℤ/8)11yescharacteristic
580C22C22×#13711yescharacteristic
581C22C22×#13711yescharacteristic
582C22C2×#57511yescharacteristic
583C4—22no—
584C4—22no—
585C4—22no—
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—48no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—84no—
600C4#311014yes—
601C4#323914yes—
602C4#352212yes—
603C4#352312yes—
604C2—24no—
605C2#3524711yescharacteristic
606C2C2×#311011yescharacteristic
607C2#6056011yescharacteristic
608C2C2×#323911yescharacteristic
609C2#6086511yescharacteristic
610C2C2×#325511yescharacteristic
611C2#6091611yescharacteristic
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
5. #30152C2semidirect
6. #29355C2semidirect
7. #29320C2semidirect
8. #29286C2semidirect
9. #28283C2semidirect
13. #22118C2semidirect

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
#331472
#331092
#301522
#293552
#293202
#292862
#282832
#276482
#246062
#243012
#221182
#208102