Finite Group Databasebeta

← #6470764

Subgroups — #6470764

Order 512.

2016 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3202C212yesmaximal
2#37386C211yesmaximal, characteristic
3#34266C211yesmaximal, characteristic
4#33117C212yesmaximal, complemented
5#32906C212yesmaximal, complemented
6#29759C212yesmaximal, complemented
7#29724C212yesmaximal, complemented
8#25220C214yesmaximal, complemented
9#25201C214yesmaximal, complemented
10#25095C214yesmaximal, complemented
11#25082C214yesmaximal, complemented
12#21937C211yesmaximal, characteristic
13#20913C212yesmaximal
14C2×#536C2212yes—
15C2×#536C2211yescharacteristic
16C2×#520C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18#3202—216no—
19C2×#485C2214yes—
20C2×#485C2214yes—
21C2×#470C2214yes—
22C2×#467C2214yes—
23C2×#462C2214yes—
24C2×#451C2214yes—
25C2×#451C2214yes—
26C2×#419C2212yes—
27C22×#103C2212yes—
28C2×#408C2212yes—
29#2857C2212yes—
30#2734—22no—
31#2724—22no—
32#2717—22no—
33#2717—22no—
34#2621C2212yes—
35#2499C2214yes—
36#2434C2214yes—
37#2367—22no—
38#2365—22no—
39#2362C2214yes—
40#2360C2214yes—
41#2322—22no—
42#2310C2211yescharacteristic
43C4×#123—22no—
44#2297C2211yescharacteristic
45#2296C2211yescharacteristic
46#2274C2211yescharacteristic
47#2084C2212yes—
48#2083C2212yes—
49#2074C2212yes—
50#2074C2212yes—
51#2068C2214yes—
52#2066C2214yes—
53#2056C2214yes—
54#2056C2214yes—
55#2043C2212yes—
56#2043C2212yes—
57#2016C2212yes—
58#2016C2212yes—
59#1981C2214yes—
60#1949C2214yes—
61#1928C2214yes—
62#1927C2214yes—
63#1878C2214yes—
64#1878C2214yes—
65#1873C2214yes—
66#1873C2214yes—
67#1872C2214yes—
68#1872C2214yes—
69#1851C2214yes—
70#1850C2214yes—
71#1803C2214yes—
72#1798C2212yes—
73#1797C2212yes—
74#1736C2212yes—
75#1735C2212yes—
76C22×#41C2311yescharacteristic
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C22×Q16C2314yes—
80C22×Q16C2314yes—
81C22×M16C2312yes—
82C23×C8C2312yes—
83#536—24no—
84#536—28no—
85#536—28no—
86#536D418yes—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#124—21no—
92C2×#124—22no—
93C2×#123—22no—
94C2×#123—22no—
95C2×#123—22no—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2314yes—
99#520—44no—
100C2×C4×Q8—21no—
101C2×C4×Q8—22no—
102C2×C4×Q8—44no—
103C2×C4×Q8C2314yes—
104C2×C4×D4—22no—
105C2×C4×D4—21no—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116#485—216no—
117#485—216no—
118#470—216no—
119#467—216no—
120#462—48no—
121#451—216no—
122#451—216no—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2314yes—
127#419—44no—
128C2×#103—44no—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137#408—28no—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140#398—24no—
141#397—24no—
142#397—24no—
143#397—24no—
144#395—24no—
145#395—24no—
146#391—22no—
147#391D412yes—
148#391C2312yes—
149#391C2312yes—
150#391C2314yes—
151#391C2314yes—
152#390D412yes—
153#389—22no—
154#389—22no—
155#389—24no—
156#389C2312yes—
157#389C2312yes—
158#388—24no—
159#388—24no—
160#387—24no—
161#387—24no—
162#387—24no—
163#387—24no—
164#386D412yes—
165#386D412yes—
166#385—21no—
167#385—21no—
168#385—24no—
169#385—24no—
170#385—24no—
171#384C2312yes—
172#382C2312yes—
173#381—22no—
174#380—22no—
175#380—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—24no—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—24no—
182C4×C22⋊C4C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2312yes—
188#336C2312yes—
189#336C2312yes—
190#336C2312yes—
191C2×#41—42no—
192C2×#41—42no—
193C2×#41C2×D414yes—
194C2×#41#4114yes—
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8C2×D414yes—
207C22×Q8C2×D411yescharacteristic
208C22×Q8C2414yes—
209C22×Q8C2412yes—
210C22×D4C2×D412yes—
211C22×D4C2×D411yescharacteristic
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C2×Q16—28no—
221C2×Q16—28no—
222C2×Q16—44no—
223C2×Q16—44no—
224C2×Q16—44no—
225C2×Q16—44no—
226C2×M16—24no—
227C2×M16—42no—
228C2×M16—42no—
229C22×C8—24no—
230C22×C8—42no—
231C22×C8—42no—
232C22×C8C2414yes—
233C22×C8C2414yes—
234C4⋊Q8—28no—
235C4⋊Q8—44no—
236C4⋊Q8—44no—
237C4⋊Q8—48no—
238#124—28no—
239#124—48no—
240#123—216no—
241#123—48no—
242#123—48no—
243C22⋊Q8—28no—
244C22⋊Q8—28no—
245C22⋊Q8—28no—
246C22⋊Q8—28no—
247C22⋊Q8—28no—
248C22⋊Q8C2×D4116yes—
249C4×Q8—28no—
250C4×Q8—44no—
251C4×Q8—48no—
252C4×Q8—44no—
253C4×D4—44no—
254C4×D4—48no—
255#117—22no—
256#117—24no—
257#117—24no—
258#117—24no—
259#117—42no—
260#117—42no—
261#117—42no—
262#117C2×D414yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2411yescharacteristic
282C2×C4⋊C4C2411yescharacteristic
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4C2×D414yes—
294C2×C22⋊C4C2×D412yes—
295C2×C22⋊C4C2×D414yes—
296C2×C42—21no—
297C2×C42—22no—
298C2×C42—21no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42C2×D412yes—
305C2×C42C2412yes—
306C2×C42C2412yes—
307#107—28no—
308#106—28no—
309C4⋊C8—44no—
310C4⋊C8—44no—
311#103—28no—
312#103—28no—
313#103—28no—
314#103—28no—
315#103—28no—
316#103—28no—
317#103—44no—
318#103—44no—
319C22⋊C8—28no—
320C22⋊C8—28no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95C2×D414yes—
332#95C2×D414yes—
333C24C22×D412yes—
334C24C22×D411yescharacteristic
335#41—48no—
336#41—48no—
337C2×Q8—24no—
338C2×Q8—22no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—48no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—44no—
356C2×Q8—82no—
357C2×Q8—82no—
358C2×Q8—84no—
359C2×Q8—84no—
360C2×Q8#12114yes—
361C2×Q8C22×D414yes—
362C2×D4—41no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4#12118yes—
367C2×D4#12114yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—84no—
395C22×C4C22×D414yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D414yes—
401C22×C4C22×D412yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4114yes—
404C22×C4C2511yescharacteristic
405Q16—88no—
406Q16—88no—
407M16—84no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—84no—
415C4⋊C4—24no—
416C4⋊C4—216no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4—84no—
435C4⋊C4C22×D414yes—
436C4⋊C4H(5,𝔽2)14yes—
437C22⋊C4—24no—
438C22⋊C4—216no—
439C22⋊C4—28no—
440C22⋊C4—44no—
441C22⋊C4—42no—
442C22⋊C4—44no—
443C22⋊C4—42no—
444C22⋊C4—44no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C42—24no—
448C42—24no—
449C42—24no—
450C42—42no—
451C42—48no—
452C42—44no—
453C42—42no—
454C42—44no—
455C42—84no—
456C23—24no—
457C23—22no—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—22no—
462C23—24no—
463C23—22no—
464C23C2×#12114yes—
465C23C2×#12112yes—
466C23C23×D411yescharacteristic
467Q8—28no—
468Q8—44no—
469Q8—48no—
470Q8—44no—
471Q8—44no—
472Q8—44no—
473Q8—44no—
474Q8—44no—
475Q8—88no—
476D4—44no—
477D4—48no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—84no—
517C2×C4—82no—
518C2×C4—88no—
519C2×C4—164no—
520C2×C4C2×#12112yes—
521C2×C4C2×#12114yes—
522C2×C4#57814yes—
523C2×C4C23×D411yescharacteristic
524C2×C4C22×#4111yescharacteristic
525C2×C4C2×H(5,𝔽2)12yes—
526C2×C4#58512yes—
527C8—44no—
528C8—44no—
529C22—28no—
530C22—24no—
531C22—22no—
532C22—24no—
533C22—44no—
534C22—42no—
535C22—41no—
536C22—42no—
537C22—41no—
538C22—42no—
539C22C22×#12111yescharacteristic
540C22#348411yescharacteristic
541C22#348511yescharacteristic
542C22C2×#57812yes—
543C22#358612yes—
544C4—22no—
545C4—24no—
546C4—24no—
547C4—44no—
548C4—44no—
549C4—44no—
550C4—42no—
551C4—44no—
552C4—84no—
553C4—84no—
554C4—81no—
555C4—81no—
556C4#345912yes—
557C4#349612yes—
558C2—42no—
559C2—44no—
560C2#3498411yescharacteristic
561C2#6043212yes—
562C2#6163411yescharacteristic
563C2#6164911yescharacteristic
564C2#6180612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
4. #33117C2semidirect
5. #32906C2semidirect
6. #29759C2semidirect
7. #29724C2semidirect
8. #25220C2semidirect
9. #25201C2semidirect
10. #25095C2semidirect
11. #25082C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32022
#373862
#342662
#331172
#329062
#297592
#297242
#252202
#252012
#250952
#250822
#219372
#209132