Finite Group Databasebeta

← #6816611

Subgroups — #6816611

Order 512.

2004 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic, complemented
3#36127C211yesmaximal, characteristic
4#33310C211yesmaximal, characteristic, complemented
5#31092C218yesmaximal, complemented
6#29877C212yesmaximal, complemented
7#28716C211yesmaximal, characteristic, complemented
8#28689C211yesmaximal, characteristic, complemented
9#26045C214yesmaximal, complemented
10#25812C212yesmaximal, complemented
11#25671C212yesmaximal, complemented
12#25013C214yesmaximal, complemented
13C4×#486C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic, complemented
15C2×#547—21no—
16C2×#533C2212yes—
17C2×#532C2212yes—
18#3228—216no—
19C2×#500C2212yes—
20C2×#498C2212yes—
21C2×#498C2212yes—
22C2×#486C2212yes—
23C2×#486C2211yescharacteristic
24C2×#482C2218yes—
25C2×#478C2218yes—
26C2×#431C2212yes—
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2857C2212yes—
30#2857C2212yes—
31#2552C2212yes—
32#2547—21no—
33#2542—21no—
34#2538—21no—
35#2537—21no—
36#2537C2212yes—
37#2441C2214yes—
38#2432—22no—
39#2429—22no—
40#2423—22no—
41#2422—22no—
42#2422C2214yes—
43#2365—21no—
44#2354C2212yes—
45Q8×C4⋊C4—21no—
46D4×C4⋊C4—21no—
47D4×C4⋊C4C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2088C2218yes—
52#2083C2214yes—
53#2077C2218yes—
54#2071C2214yes—
55#2059C2218yes—
56#2034C2218yes—
57#1990C2212yes—
58#1990C2212yes—
59#1990C2211yescharacteristic
60#1957C2211yescharacteristic
61#1954C2212yes—
62#1947C2214yes—
63#1939C2212yes—
64#1934C2214yes—
65#1923C2214yes—
66#1864C2218yes—
67#1854C2218yes—
68#1844C2214yes—
69#1844C2214yes—
70#1834C2211yescharacteristic
71#1802C2214yes—
72C4×C4⋊C8C2211yescharacteristic
73C4×#102C2214yes—
74C4×C22⋊C8C2212yes—
75C42×C8C2211yescharacteristic
76C22×#41—22no—
77#547—48no—
78#533—28no—
79#533—44no—
80#532—28no—
81#532—44no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×#127C2311yescharacteristic
90C2×#125C2314yes—
91C2×#124—22no—
92C2×#123—24no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×#121—22no—
98C2×#121—22no—
99C2×#121C2314yes—
100C2×C4×Q8—21no—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4—22no—
106C2×C4×D4D411yescharacteristic
107C2×C4×D4D411yescharacteristic
108C2×C4×D4C2312yes—
109C2×C4×D4C2314yes—
110C2×C4×D4C2311yescharacteristic
111C2×#117C2314yes—
112C22×C4⋊C4—22no—
113C22×C42C2312yes—
114#500—28no—
115#498—28no—
116#498—44no—
117#486—28no—
118#486D418yescomplemented
119#482—232no—
120#478—232no—
121#431—28no—
122C2×#107C2318yes—
123C2×#106C2318yes—
124C2×#106C2318yes—
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2312yes—
127C2×#102C2318yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132C2×C4×C8C2312yes—
133C2×C4×C8C2312yes—
134C2×C4×C8C2312yes—
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137#399—24no—
138#398—24no—
139#397—24no—
140#396—24no—
141#394—24no—
142#394—24no—
143#391—21no—
144#391—22no—
145#391—22no—
146#390—22no—
147#390—21no—
148#390—21no—
149#390C2312yes—
150#389—22no—
151#389—21no—
152#389—22no—
153#389—22no—
154#389—24no—
155#389C2312yes—
156#389C2314yes—
157#389C2314yes—
158#388—24no—
159#387—24no—
160#385—24no—
161#385—24no—
162#384—22no—
163#384—21no—
164#384—21no—
165#384—21no—
166#384—21no—
167#384C2311yescharacteristic
168#384C2311yescharacteristic
169#384C2312yes—
170#384C2312yes—
171#384C2312yes—
172#384C2312yes—
173#380—22no—
174#380—22no—
175#380—22no—
176#380—22no—
177#380C2314yes—
178#380C2314yes—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4D411yescharacteristic
182C4×C4⋊C4D411yescharacteristic
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2311yescharacteristic
185C4×C22⋊C4D412yes—
186C4×C22⋊C4D412yes—
187C4×C22⋊C4C2312yes—
188C43C2311yescharacteristic
189#340C2318yes—
190#340C2314yes—
191#340C2314yes—
192#340C2318yes—
193#336C2314yes—
194#336C2312yes—
195C2×#41—48no—
196C2×#41—48no—
197C2×#41—84no—
198C22×Q8—21no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×D4—24no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4C2×D411yescharacteristic
205C22×D4C2412yes—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4C2×D412yes—
209C23×C4C2×D412yes—
210C23×C4C2412yes—
211C22×C8C2414yes—
212C22×C8C2412yes—
213C22×C8C2412yes—
214C4⋊Q8—24no—
215C4⋊Q8—24no—
216C4⋊Q8—22no—
217C4⋊Q8—24no—
218C4⋊Q8—24no—
219C4⋊Q8—44no—
220C4⋊Q8—44no—
221C4⋊Q8—82no—
222C4⋊Q8—82no—
223C4⋊Q8C2×D414yes—
224#127—22no—
225#127C2×D414yes—
226#125—28no—
227#125—44no—
228#124—48no—
229#123—416no—
230C22⋊Q8—48no—
231C22⋊Q8—48no—
232C22⋊Q8—48no—
233C22⋊Q8—84no—
234#121—28no—
235#121—44no—
236#121—48no—
237#121—48no—
238C4×Q8—48no—
239C4×Q8—48no—
240C4×Q8—44no—
241C4×D4—28no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—28no—
246C4×D4—44no—
247C4×D4—48no—
248C4×D4—44no—
249C4×D4—44no—
250#117—28no—
251#117—44no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4C2×D411yescharacteristic
274C2×C4⋊C4C2×D411yescharacteristic
275C2×C4⋊C4C2×D411yescharacteristic
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C22⋊C4C2×D412yes—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2412yes—
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306#107—216no—
307#106—216no—
308#106—48no—
309C4⋊C8—28no—
310C4⋊C8—24no—
311#102—216no—
312#102—28no—
313#102C2×D4116yes—
314C22⋊C8—28no—
315C22⋊C8—28no—
316C4×C8—24no—
317C4×C8—24no—
318C4×C8—22no—
319C4×C8—24no—
320C4×C8C2×D414yes—
321#95—24no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95C2×D412yes—
328#95C2×D412yes—
329#95C2×D412yes—
330C24—22no—
331C24C22×D412yes—
332#41—832no—
333C2×Q8—24no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—42no—
338C2×Q8—44no—
339C2×Q8—42no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—84no—
357C2×D4C22×D414yes—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—84no—
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2511yescharacteristic
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8C22×D414yes—
397C2×C82−1+414yes—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—84no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4C22×D414yes—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—84no—
440C42—24no—
441C42—24no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—48no—
448C42—44no—
449C42#13514yes—
450C42AGL(1,ℤ/8)14yes—
451C42C22×D412yes—
452C42H(5,𝔽2)12yes—
453C422−1+414yes—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23C23×D411yescharacteristic
469Q8—48no—
470Q8—88no—
471Q8—88no—
472D4—28no—
473D4—44no—
474D4—48no—
475D4—48no—
476D4—44no—
477D4—816no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—48no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—816no—
513C2×C4—88no—
514C2×C4#53612yes—
515C2×C4#53612yes—
516C2×C4D4212yes—
517C2×C4D4212yes—
518C2×C4C2×#13512yes—
519C2×C4C2×#13512yes—
520C2×C4C2×#13512yes—
521C2×C4C2×AGL(1,ℤ/8)12yes—
522C2×C4C2×AGL(1,ℤ/8)12yes—
523C2×C4C2×AGL(1,ℤ/8)12yes—
524C2×C4#57712yes—
525C2×C4#57712yes—
526C2×C4C23×D411yescharacteristic
527C2×C4C2×H(5,𝔽2)11yescharacteristic
528C2×C4C2×2−1+412yes—
529C2×C4C2×2−1+412yes—
530C8—24no—
531C8—24no—
532C8—44no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—44no—
537C22—42no—
538C22—44no—
539C22—42no—
540C22—42no—
541C22—44no—
542C22—42no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22C2×#53611yescharacteristic
548C22C2×D4211yescharacteristic
549C22#349911yescharacteristic
550C22C22×#13511yescharacteristic
551C22C22×AGL(1,ℤ/8)11yescharacteristic
552C22C2×#57711yescharacteristic
553C22C2×#57711yescharacteristic
554C4—22no—
555C4—22no—
556C4—22no—
557C4—42no—
558C4—44no—
559C4—42no—
560C4—42no—
561C4—82no—
562C4—84no—
563C4—84no—
564C4—82no—
565C4#322814yescomplemented
566C4#327214yes—
567C4#328414yes—
568C4#349912yes—
569C4#355212yes—
570C2—82no—
571C2—84no—
572C2#3936011yescharacteristic
573C2C2×#322811yescharacteristic
574C2#6088111yescharacteristic
575C2C2×#327211yescharacteristic
576C2C2×#328411yescharacteristic
577C2#6247011yescharacteristic
578C2#6247711yescharacteristic
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. C2×#3228C2semidirect
2. #36134C2semidirect
4. #33310C2semidirect
5. #31092C2semidirect
6. #29877C2semidirect
7. #28716C2semidirect
8. #28689C2semidirect
9. #26045C2semidirect
10. #25812C2semidirect
11. #25671C2semidirect
12. #25013C2semidirect
13. C4×#486C2semidirect
14. #21264C2semidirect
118. #486D4semidirect
565. C4#3228semidirect

Maximal subgroups

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

subgroupindex
C2×#32282
#361342
#361272
#333102
#310922
#298772
#287162
#286892
#260452
#258122
#256712
#250132
C4×#4862
#212642