Finite Group Databasebeta

← #6529836

Subgroups — #6529836

Order 512.

2016 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3103C211yesmaximal, characteristic
2C2×#2983C211yesmaximal, characteristic, complemented
3#35178C211yesmaximal, characteristic
4C2×#2582C211yesmaximal, characteristic, complemented
5#28272C214yesmaximal, complemented
6#28195C214yesmaximal, complemented
7#25479C214yesmaximal, complemented
8#25303C214yesmaximal, complemented
9#24607C214yesmaximal, complemented
10#24509C214yesmaximal, complemented
11#21982C211yesmaximal, characteristic, complemented
12#21859C211yesmaximal, characteristic
13#20734C211yesmaximal, characteristic, complemented
14C2×#536—21no—
15C2×#532C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18#3103—28no—
19C2×#481C2212yes—
20C2×#475C2212yes—
21C2×#460C2212yes—
22C2×#448C2212yes—
23#2983—28no—
24C2×#446C2214yes—
25C2×#446C2214yes—
26C2×C4×SD16C2214yes—
27C2×#432C2214yes—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30#2854C2214yes—
31#2618—22no—
32#2582—22no—
33#2582—22no—
34#2582C2214yes—
35#2582C2214yes—
36#2548C2212yes—
37#2546C2212yes—
38#2545—22no—
39#2538C2212yes—
40#2538C2212yes—
41#2536—22no—
42C2×#398—22no—
43#2311—21no—
44C2×#391—21no—
45C2×#391C2212yes—
46C2×#389—21no—
47C2×#389C2212yes—
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50#2088C2218yes—
51#2068C2218yes—
52#2067C2218yes—
53#2055C2218yes—
54#1990C2212yes—
55#1973C2218yes—
56#1957C2212yes—
57#1947C2214yes—
58#1943C2218yes—
59#1936C2218yes—
60#1935C2218yes—
61#1874C2214yes—
62#1852C2214yes—
63#1845C2212yes—
64#1834C2212yes—
65#1824C2212yes—
66#1803C2212yes—
67#1803C2212yes—
68#1802C2214yes—
69#1787C2212yes—
70C4×#103C2214yes—
71C22×#41—21no—
72C23×Q8D411yescharacteristic
73C23×Q8C2311yescharacteristic
74C22×SD16C2314yes—
75#536—48no—
76#532—24no—
77#532—24no—
78C2×C4⋊Q8—22no—
79C2×C4⋊Q8C2311yescharacteristic
80C2×#127C2311yescharacteristic
81C2×#124—22no—
82C2×#124—21no—
83C2×#123—24no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—22no—
86C2×#121—21no—
87C2×#121C2312yes—
88C2×C4×Q8—24no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8D412yes—
95C2×C4×Q8D412yes—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2312yes—
99C2×C4×Q8C2312yes—
100C2×C4×D4—21no—
101C2×C4×D4C2312yes—
102C2×#117—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D412yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C42—21no—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#481—28no—
115#475—28no—
116#460—28no—
117#448—28no—
118#446—216no—
119#446—216no—
120C4×SD16—216no—
121#432—216no—
122#431—24no—
123#431D418yes—
124C2×#107C2318yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2318yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#102C2314yes—
133C2×#102C2318yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×#97C2318yes—
137C2×C4×C8C2312yes—
138C2×C4×C8C2312yes—
139#398—28no—
140#398—28no—
141#398—24no—
142#396—24no—
143#396—24no—
144#394—24no—
145#391—24no—
146#391—24no—
147#391—22no—
148#391—24no—
149#391C2318yes—
150#390—22no—
151#390C2314yes—
152#390C2314yes—
153#389—24no—
154#389—24no—
155#389—24no—
156#389C2318yes—
157#389C2314yes—
158#389C2314yes—
159#388—24no—
160#388—24no—
161#386—24no—
162#384—21no—
163#384—21no—
164#384C2312yes—
165#384C2311yescharacteristic
166#384C2311yescharacteristic
167#384C2312yes—
168#382—21no—
169#380—21no—
170#380—22no—
171#380—22no—
172#380C2312yes—
173C4×C4⋊C4—21no—
174C4×C4⋊C4C2312yes—
175C2×#95—22no—
176C2×#95—22no—
177C2×#95—22no—
178#340C2318yes—
179#340C2318yes—
180#340C2314yes—
181#340C2314yes—
182C2×#41—44no—
183C2×#41—44no—
184C2×#41—44no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—42no—
194C22×Q8C2×D412yes—
195C22×Q8C2×D412yes—
196C22×Q8C2412yes—
197C22×D4—22no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4C2412yes—
201C23×C4—21no—
202C23×C4—22no—
203C23×C4—21no—
204C23×C4—22no—
205C23×C4C2×D412yes—
206C23×C4C2×D412yes—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2411yescharacteristic
210C2×SD16—28no—
211C2×SD16—28no—
212C2×SD16—44no—
213C22×C8C2414yes—
214C22×C8C2414yes—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—48no—
218#127—22no—
219#127—22no—
220#124—48no—
221#124—44no—
222#123—416no—
223C22⋊Q8—48no—
224C22⋊Q8—48no—
225#121—24no—
226#121—24no—
227#121—44no—
228C4×Q8—28no—
229C4×Q8—28no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—44no—
239C4×Q8—44no—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—44no—
243#117—44no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—82no—
268C2×C4⋊C4—84no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—44no—
291C2×C42C2×D414yes—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D414yes—
295C2×C42#4114yes—
296C2×C42C2414yes—
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299#107—48no—
300#106—28no—
301C4⋊C8—216no—
302C4⋊C8—24no—
303C4⋊C8C2×D418yes—
304#103—28no—
305#103—28no—
306#103—44no—
307#102—28no—
308#102—48no—
309C22⋊C8—24no—
310C22⋊C8C2×D418yes—
311#97—216no—
312C4×C8—24no—
313C4×C8C2×D418yes—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—24no—
321C24—22no—
322C24C22×D411yescharacteristic
323#41—816no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—416no—
344C2×Q82−1+414yes—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—81no—
357C2×D4—81no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4#12114yes—
387C22×C4#12114yes—
388C22×C4AGL(1,ℤ/8)14yes—
389C22×C4#13714yes—
390C22×C4C22×D414yes—
391C22×C4C22×D414yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4112yes—
399C22×C4C2511yescharacteristic
400SD16—416no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8C22×D418yes—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—28no—
410C4⋊C4—28no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—28no—
417C4⋊C4—24no—
418C4⋊C4—44no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—88no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C42—28no—
442C42—24no—
443C42—24no—
444C42—44no—
445C42—44no—
446C42—44no—
447C42#12114yes—
448C42#12114yes—
449C42C22×D412yes—
450C42C22×D412yes—
451C42C2×#4112yes—
452C42C2×#4112yes—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C2×#12112yes—
465C23C2×AGL(1,ℤ/8)12yes—
466C23C2×#13712yes—
467C23C23×D411yescharacteristic
468Q8—24no—
469Q8—24no—
470Q8—48no—
471Q8—42no—
472Q8—42no—
473Q8—48no—
474Q8—48no—
475Q8—84no—
476D4—44no—
477D4—44no—
478D4—84no—
479D4—88no—
480D4—84no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—44no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—48no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—88no—
519C2×C4—88no—
520C2×C4#468116yes—
521C2×C4C2×#12112yes—
522C2×C4C2×#12112yes—
523C2×C4C2×#12112yes—
524C2×C4C2×#12112yes—
525C2×C4C2×#12112yes—
526C2×C4C2×AGL(1,ℤ/8)12yes—
527C2×C4C2×AGL(1,ℤ/8)12yes—
528C2×C4C2×#13712yes—
529C2×C4C2×#13712yes—
530C2×C4C23×D411yescharacteristic
531C2×C4C22×#4111yescharacteristic
532C2×C4C2×2−1+412yes—
533C2×C4#58512yes—
534C8—28no—
535C8—44no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22C2×#46814yes—
550C22C22×#12111yescharacteristic
551C22#348611yescharacteristic
552C22#348711yescharacteristic
553C22C22×AGL(1,ℤ/8)11yescharacteristic
554C22C22×#13711yescharacteristic
555C22#358511yescharacteristic
556C22#358611yescharacteristic
557C4—22no—
558C4—22no—
559C4—22no—
560C4—44no—
561C4—44no—
562C4—44no—
563C4—82no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4—84no—
568C4C2×#46814yes—
569C4C2×#46814yes—
570C4#348912yes—
571C4#349512yes—
572C2—24no—
573C2—162no—
574C2#3566111yescharacteristic
575C2C22×#46811yescharacteristic
576C2#6045611yescharacteristic
577C2#6182811yescharacteristic
578C2#6183011yescharacteristic
579C2#6189111yescharacteristic
580C2#6189311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2983C2semidirect
4. C2×#2582C2semidirect
5. #28272C2semidirect
6. #28195C2semidirect
7. #25479C2semidirect
8. #25303C2semidirect
9. #24607C2semidirect
10. #24509C2semidirect
11. #21982C2semidirect
13. #20734C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31032
C2×#29832
#351782
C2×#25822
#282722
#281952
#254792
#253032
#246072
#245092
#219822
#218592
#207342