Finite Group Databasebeta

← #6506502

Subgroups — #6506502

Order 512.

2202 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37425C211yesmaximal, characteristic, complemented
2#35666C211yesmaximal, characteristic
3#29117C212yesmaximal, complemented
4#29114C212yesmaximal, complemented
5#27283C214yesmaximal, complemented
6#27272C214yesmaximal, complemented
7#27268C214yesmaximal, complemented
8#27267C214yesmaximal, complemented
9#25868C212yesmaximal, complemented
10#25861C212yesmaximal, complemented
11#21771C212yesmaximal, complemented
12#20603C212yesmaximal
13#20532C211yesmaximal, characteristic, complemented
14C22×2−1+4C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#536C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×#491C2214yes—
19C2×#491C2212yes—
20C2×#488C2212yes—
21C2×#417C2212yes—
22C2×#419C2212yes—
23C2×#416C2212yes—
24C22×#103C2212yes—
25C2×#408C2212yes—
26#2851C2212yes—
27#2809—22no—
28#2780—24no—
29#2748—22no—
30#2629C2212yes—
31#2442C2214yes—
32#2441C2214yes—
33#2434—22no—
34#2434C2214yes—
35#2432—22no—
36#2432C2214yes—
37#2406—24no—
38#2311C2212yes—
39#2310C2211yescharacteristic
40#2297C2211yescharacteristic
41#2297C2211yescharacteristic
42#2074C2212yes—
43#2073C2212yes—
44#2072C2214yes—
45#2043C2214yes—
46#2043C2214yes—
47#2043C2214yes—
48#2042C2214yes—
49#2026C2214yes—
50#2026C2212yes—
51#2025C2212yes—
52#1960C2214yes—
53#1959C2212yes—
54#1956C2212yes—
55#1936C2214yes—
56#1935C2214yes—
57#1935C2214yes—
58#1933C2214yes—
59#1894C2214yes—
60#1892C2214yes—
61#1873C2214yes—
62#1872C2214yes—
63#1825C2214yes—
64#1788C2214yes—
65#1788C2214yes—
66#1787C2214yes—
67#1787C2214yes—
68#1787C2214yes—
69#1753C2214yes—
70#1752C2214yes—
71#1750C2212yes—
72#1749C2212yes—
73C2×2−1+4—28no—
74C2×2−1+4—44no—
75C2×2−1+4—42no—
76C2×2−1+4—42no—
77C22×#41—22no—
78C22×#41—21no—
79C22×#41C2312yes—
80C22×#41C2311yescharacteristic
81C22×#41C2311yescharacteristic
82C23×Q8—21no—
83C23×Q8C2311yescharacteristic
84C23×Q8C2312yes—
85C23×Q8C2311yescharacteristic
86C22×M16C2312yes—
87C23×C8C2312yes—
88#536—28no—
89#536—42no—
90#536—42no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8C2314yes—
95C2×C4⋊Q8C2312yes—
96C2×#125C2312yes—
97C2×#125C2312yes—
98C2×#124—21no—
99C2×#124C2312yes—
100C2×#123—22no—
101C2×#123—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8C2314yes—
104C2×#117C2311yescharacteristic
105C2×#117C2312yes—
106C2×#117C2311yescharacteristic
107C22×C4⋊C4—22no—
108C22×C4⋊C4C2312yes—
109C22×C42C2311yescharacteristic
110#491—44no—
111#491—48no—
112#488—44no—
113#419—44no—
114#417—44no—
115#416—44no—
116C2×#103—44no—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129#408—28no—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132C2×#97C2312yes—
133C2×#97C2314yes—
134C2×C4×C8C2312yes—
135#400—24no—
136#399—24no—
137#398—24no—
138#398—24no—
139#396—24no—
140#396—24no—
141#395—24no—
142#395—24no—
143#394—24no—
144#394—24no—
145#391—22no—
146#391—22no—
147#391D412yes—
148#391C2314yes—
149#391C2312yes—
150#390—22no—
151#390D412yes—
152#390C2312yes—
153#389—24no—
154#389C2314yes—
155#389C2314yes—
156#387—24no—
157#387—24no—
158#386—22no—
159#386—22no—
160#386—22no—
161#386D414yes—
162#385—24no—
163#385—24no—
164#385—24no—
165#385—24no—
166#385—24no—
167#385—24no—
168#384C2312yes—
169#382C2312yes—
170#381—22no—
171#380—24no—
172#380C2314yes—
173#380C2314yes—
174C4×C22⋊C4—22no—
175C4×C22⋊C4C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#336C2314yes—
181#336C2312yes—
182#336C2312yes—
1832−1+4—432no—
1842−1+4—432no—
185C2×#41—22no—
186C2×#41—22no—
187C2×#41—24no—
188C2×#41—24no—
189C2×#41—24no—
190C2×#41—24no—
191C2×#41—24no—
192C2×#41—42no—
193C2×#41—41no—
194C2×#41—41no—
195C2×#41—42no—
196C2×#41—42no—
197C2×#41—42no—
198C2×#41—48no—
199C2×#41—44no—
200C2×#41—48no—
201C2×#41—44no—
202C2×#41—48no—
203C2×#41—44no—
204C2×#41#4118yes—
205C22×Q8—24no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8—22no—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—24no—
213C22×Q8—24no—
214C22×Q8—24no—
215C22×Q8—42no—
216C22×Q8—42no—
217C22×Q8—41no—
218C22×Q8—41no—
219C22×Q8—42no—
220C22×Q8—41no—
221C22×Q8—41no—
222C22×Q8—44no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8C2×D411yescharacteristic
226C22×Q8C2412yes—
227C22×Q8C2412yes—
228C22×Q8C2411yescharacteristic
229C22×D4—21no—
230C22×D4—22no—
231C22×D4C2×D412yes—
232C22×D4C2×D411yescharacteristic
233C22×D4C2411yescharacteristic
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4C2×D412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C2×M16—24no—
244C2×M16—42no—
245C2×M16—42no—
246C22×C8—24no—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—28no—
252C4⋊Q8—24no—
253C4⋊Q8—44no—
254C4⋊Q8—42no—
255C4⋊Q8—48no—
256C4⋊Q8—44no—
257#125—42no—
258#125—44no—
259#125—42no—
260#124—24no—
261#124—42no—
262#124—44no—
263#123—48no—
264#123—48no—
265C22⋊Q8—28no—
266C22⋊Q8—28no—
267C22⋊Q8—48no—
268#117—24no—
269#117—41no—
270#117—41no—
271#117—42no—
272#117—42no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2414yes—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×D414yes—
299C2×C22⋊C4C2×D414yes—
300C2×C42—21no—
301C2×C42—22no—
302C2×C42—42no—
303C2×C42C2×D412yes—
304C2×C42C2412yes—
305C2×C42C2412yes—
306#103—28no—
307#103—28no—
308#103—28no—
309#103—44no—
310#103—44no—
311#103—44no—
312#103—44no—
313#103—44no—
314#103—44no—
315#103—44no—
316#102—28no—
317#102—44no—
318C22⋊C8—28no—
319C22⋊C8—28no—
320#97—44no—
321#97—42no—
322C4×C8—42no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95C2×D414yes—
332#95C2×D414yes—
333C24—21no—
334C24C22×D412yes—
335C24C22×D411yescharacteristic
336#41—216no—
337#41—48no—
338#41—48no—
339#41—48no—
340#41—48no—
341#41—48no—
342#41—48no—
343#41—48no—
344#41—432no—
345#41—464no—
346C2×Q8—24no—
347C2×Q8—22no—
348C2×Q8—21no—
349C2×Q8—216no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—24no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—42no—
369C2×Q8—48no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—432no—
373C2×Q8—42no—
374C2×Q8—44no—
375C2×Q8—42no—
376C2×Q8—84no—
377C2×Q8—84no—
378C2×Q8#12414yes—
379C2×Q8H(5,𝔽2)12yes—
380C2×Q8H(5,𝔽2)14yes—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—41no—
386C2×D4—42no—
387C2×D4—41no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—44no—
394C2×D4#12418yes—
395C2×D4#12414yes—
396C2×D4H(5,𝔽2)12yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—82no—
428C22×C4—84no—
429C22×C4C22×D414yes—
430C22×C4C22×D414yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4114yes—
436C22×C4C2511yescharacteristic
437M16—84no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—84no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—42no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—82no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—42no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—42no—
467C22⋊C4—48no—
468C22⋊C4—84no—
469C22⋊C4—84no—
470C22⋊C4—82no—
471C42—22no—
472C42—24no—
473C42—22no—
474C42—42no—
475C42—42no—
476C42—82no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—21no—
481C23—22no—
482C23—21no—
483C23—22no—
484C23—22no—
485C23—44no—
486C23—41no—
487C23—41no—
488C23C2×#12414yes—
489C23C2×#12412yes—
490C23C23×D411yescharacteristic
491Q8—24no—
492Q8—24no—
493Q8—24no—
494Q8—44no—
495Q8—416no—
496Q8—48no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—44no—
501Q8—44no—
502Q8—84no—
503D4—24no—
504D4—48no—
505D4—44no—
506D4—48no—
507D4—416no—
508D4—42no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—48no—
528C2×C4—416no—
529C2×C4—48no—
530C2×C4—416no—
531C2×C4—48no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—84no—
548C2×C4—164no—
549C2×C4C2×#12412yes—
550C2×C4C2×#12414yes—
551C2×C4C22×#4112yes—
552C2×C4C2×H(5,𝔽2)12yes—
553C2×C4C2×H(5,𝔽2)11yescharacteristic
554C2×C4C2×H(5,𝔽2)11yescharacteristic
555C8—44no—
556C8—44no—
557C22—22no—
558C22—24no—
559C22—28no—
560C22—24no—
561C22—42no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—44no—
568C22—82no—
569C22—82no—
570C22C22×#12411yescharacteristic
571C22#348411yescharacteristic
572C22#348411yescharacteristic
573C22#358614yes—
574C4—24no—
575C4—22no—
576C4—42no—
577C4—44no—
578C4—44no—
579C4—44no—
580C4—82no—
581C4—84no—
582C4—82no—
583C4—84no—
584C4#349012yes—
585C4#349012yes—
586C2—42no—
587C2—44no—
588C2—82no—
589C2#3536711yescharacteristic
590C2#6062712yes—
591C2#6164512yes—
592C2#6167612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×2−1+4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37425C2semidirect
3. #29117C2semidirect
4. #29114C2semidirect
5. #27283C2semidirect
6. #27272C2semidirect
7. #27268C2semidirect
8. #27267C2semidirect
9. #25868C2semidirect
10. #25861C2semidirect
11. #21771C2semidirect
13. #20532C2semidirect

Maximal subgroups

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

subgroupindex
#374252
#356662
#291172
#291142
#272832
#272722
#272682
#272672
#258682
#258612
#217712
#206032
#205322