Finite Group Databasebeta

← #6798699

Subgroups — #6798699

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36196C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#33232C211yesmaximal, characteristic, complemented
4#31291C214yesmaximal, complemented
5#31287C214yesmaximal, complemented
6#29852C212yesmaximal, complemented
7#28755C211yesmaximal, characteristic, complemented
8#28516C211yesmaximal, characteristic, complemented
9#27896C212yesmaximal, complemented
10#25892C214yesmaximal, complemented
11#25862C212yesmaximal, complemented
12#25289C214yesmaximal, complemented
13#21812C212yesmaximal, complemented
14#21656C211yesmaximal, characteristic, complemented
15#21271C211yesmaximal, characteristic, complemented
16C2×#547C2212yes—
17C2×#532C2212yes—
18C2×#491C2212yes—
19C2×#491C2214yes—
20C2×#490C2211yescharacteristic
21C2×#478C2214yes—
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2857C2212yes—
26#2851C2212yes—
27#2573—22no—
28#2562C2212yes—
29#2547C2212yes—
30#2537—22no—
31#2533—22no—
32#2497C2212yes—
33#2496C2212yes—
34#2466—22no—
35#2432C2214yes—
36#2422—22no—
37#2422—22no—
38#2373—22no—
39#2370C2212yes—
40D4×C4⋊C4—22no—
41C4×#127C2211yescharacteristic
42C4×#125C2211yescharacteristic
43#2283C2212yes—
44C42×D4C2211yescharacteristic
45#2094C2212yes—
46#2093C2212yes—
47#2068C2214yes—
48#2066C2214yes—
49#2043C2214yes—
50#2042C2214yes—
51#2042C2214yes—
52#2036C2214yes—
53#2036C2214yes—
54#2035C2214yes—
55#2035C2214yes—
56#1992C2211yescharacteristic
57#1983C2214yes—
58#1983C2214yes—
59#1981C2214yes—
60#1961C2212yes—
61#1961C2211yescharacteristic
62#1960C2212yes—
63#1960C2212yes—
64#1960C2212yes—
65#1956C2212yes—
66#1956C2212yes—
67#1955C2212yes—
68#1940C2212yes—
69#1934C2214yes—
70#1928C2214yes—
71#1927C2214yes—
72#1923C2214yes—
73#1847C2212yes—
74#1836C2211yescharacteristic
75#1788C2214yes—
76#1788C2214yes—
77#1788C2214yes—
78#1788C2214yes—
79#1767C2211yescharacteristic
80#1762C2214yes—
81#1749C2212yes—
82C4×#97C2211yescharacteristic
83C22×#41C2314yes—
84#547—48no—
85#532—28no—
86#532—44no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#127C2311yescharacteristic
91C2×#126—22no—
92C2×#125C2312yes—
93C2×#125C2311yescharacteristic
94C2×#124C2314yes—
95C2×#123—24no—
96C2×C22⋊Q8C2314yes—
97C2×#121—22no—
98C2×#121—22no—
99C2×#121—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4D412yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2311yescharacteristic
106C22×C4⋊C4—22no—
107C22×C42C2312yes—
108#491—44no—
109#491—48no—
110#490—24no—
111#478—48no—
112#432—44no—
113C2×#107C2314yes—
114C2×#106C2314yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2314yes—
131C2×#97C2311yescharacteristic
132C2×#97C2312yes—
133C2×#97C2312yes—
134C2×#97C2311yescharacteristic
135C2×C4×C8C2312yes—
136#400—22no—
137#400—22no—
138#400—22no—
139#397—22no—
140#397—22no—
141#397—22no—
142#396—22no—
143#394—22no—
144#394—24no—
145#394—24no—
146#391C2314yes—
147#390—22no—
148#390C2312yes—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#388—22no—
158#388—22no—
159#388—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#385—22no—
168#385—22no—
169#385—24no—
170#385—24no—
171#384—22no—
172#384—22no—
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#383—22no—
177#382C2311yescharacteristic
178#382C2312yes—
179#381—22no—
180#380—22no—
181#380—22no—
182#380—24no—
183#380C2312yes—
184#380C2312yes—
185#380C2312yes—
186#380C2312yes—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4D412yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2311yescharacteristic
194C4×C22⋊C4—22no—
195C4×C22⋊C4D412yes—
196C4×C22⋊C4D412yes—
197C4×C22⋊C4C2312yes—
198#376—22no—
199C43C2311yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#336C2312yes—
203#336C2314yes—
204C2×#41—28no—
205C2×#41—44no—
206C2×#41—44no—
207C2×#41—44no—
208C2×#41—44no—
209C2×#41—44no—
210C22×Q8C2414yes—
211C22×D4—24no—
212C22×D4—22no—
213C22×D4C2×D411yescharacteristic
214C22×D4C2412yes—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2412yes—
220C22×C8C2412yes—
221C22×C8C2414yes—
222C22×C8C2412yes—
223C4⋊Q8—24no—
224C4⋊Q8—24no—
225C4⋊Q8—28no—
226C4⋊Q8—42no—
227C4⋊Q8—42no—
228#127—22no—
229#127#4114yes—
230#126—48no—
231#125—22no—
232#125—28no—
233#125—22no—
234#124—28no—
235#124—44no—
236#123—416no—
237C22⋊Q8—44no—
238C22⋊Q8—44no—
239#121—48no—
240#121—48no—
241#121—48no—
242C4×D4—24no—
243C4×D4—28no—
244C4×D4—44no—
245C4×D4—48no—
246C4×D4—48no—
247C4×D4—48no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
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—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D412yes—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2412yes—
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296#107—44no—
297#106—44no—
298C4⋊C8—24no—
299C4⋊C8—44no—
300#103—44no—
301#103—44no—
302#103—44no—
303#103—44no—
304#103—44no—
305#103—44no—
306#103—44no—
307#103—44no—
308#102—28no—
309#102—44no—
310#102—44no—
311#102—44no—
312C22⋊C8—44no—
313C22⋊C8—44no—
314#97—22no—
315#97—22no—
316#97—24no—
317#97—24no—
318C4×C8—42no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95C2×D411yescharacteristic
335#95C2×D411yescharacteristic
336#95C2×D412yes—
337#95C2×D412yes—
338C24—22no—
339C24C22×D412yes—
340#41—88no—
341#41—88no—
342#41—88no—
343#41—88no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×D4—22no—
351C2×D4—28no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—24no—
355C2×D4—42no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—84no—
361C2×D4C2×#4114yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—44no—
382C22×C4—84no—
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C2511yescharacteristic
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—42no—
408C4⋊C4—48no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—84no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—48no—
433C22⋊C4—82no—
434C22⋊C4—82no—
435C42—24no—
436C42—24no—
437C42—24no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—42no—
442C42—44no—
443C42—82no—
444C42—82no—
445C42C2×#4112yes—
446C42H(5,𝔽2)14yes—
447C42H(5,𝔽2)12yes—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23C23×D411yescharacteristic
461Q8—44no—
462Q8—44no—
463Q8—44no—
464Q8—44no—
465D4—28no—
466D4—44no—
467D4—416no—
468D4—44no—
469D4—816no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4—88no—
507C2×C4—816no—
508C2×C4#53512yes—
509C2×C4#53512yes—
510C2×C4#54614yes—
511C2×C4#57514yes—
512C2×C4C22×#4111yescharacteristic
513C2×C4C2×H(5,𝔽2)12yes—
514C2×C4C2×H(5,𝔽2)11yescharacteristic
515C2×C4#58512yes—
516C8—42no—
517C8—42no—
518C8—44no—
519C22—42no—
520C22—44no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—48no—
528C22—84no—
529C22—82no—
530C22—82no—
531C22C2×#53511yescharacteristic
532C22C2×#54611yescharacteristic
533C22#350011yescharacteristic
534C22C2×#57512yes—
535C22#358612yes—
536C4—24no—
537C4—22no—
538C4—22no—
539C4—24no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—84no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4#351412yes—
550C4#354612yes—
551C2—82no—
552C2—84no—
553C2#3895711yescharacteristic
554C2#6080111yescharacteristic
555C2#6083111yescharacteristic
556C2#6115912yes—
557C2#6257912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36196C2semidirect
3. #33232C2semidirect
4. #31291C2semidirect
5. #31287C2semidirect
6. #29852C2semidirect
7. #28755C2semidirect
8. #28516C2semidirect
9. #27896C2semidirect
10. #25892C2semidirect
11. #25862C2semidirect
12. #25289C2semidirect
13. #21812C2semidirect
14. #21656C2semidirect
15. #21271C2semidirect

Maximal subgroups

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

subgroupindex
#361962
#361272
#332322
#312912
#312872
#298522
#287552
#285162
#278962
#258922
#258622
#252892
#218122
#216562
#212712