Finite Group Databasebeta

← #6478032

Subgroups — #6478032

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#33146C211yesmaximal, characteristic
4#33108C211yesmaximal, characteristic
5#33099C212yesmaximal
6#29966C211yesmaximal, characteristic
7#29756C211yesmaximal, characteristic
8#29641C212yesmaximal
9#27145C214yesmaximal, complemented
10#27017C214yesmaximal, complemented
11#26875C214yesmaximal, complemented
12#26707C214yesmaximal, complemented
13#22084C212yesmaximal
14#21916C211yesmaximal, characteristic
15#21579C212yesmaximal
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8—21no—
18C2×#483C2212yes—
19C2×#483C2212yes—
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#2855C2212yes—
23#2646—21no—
24#2627—21no—
25#2590—22no—
26#2588—22no—
27#2511C2212yes—
28#2507C2212yes—
29#2451C2212yes—
30#2446C2212yes—
31#2365—22no—
32#2365—22no—
33#2365C2214yes—
34#2365C2214yes—
35#2329—21no—
36#2313—21no—
37C4×C22⋊Q8C2212yes—
38C2×#391—21no—
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#387—21no—
42C2×#387—21no—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47#2099C2212yes—
48#2086C2214yes—
49#2084C2212yes—
50#2066C2214yes—
51#2058C2214yes—
52#2058C2214yes—
53#2057C2212yes—
54#2043C2212yes—
55#2024C2214yes—
56#2024C2214yes—
57#1988C2212yes—
58#1981C2212yes—
59#1948C2212yes—
60#1947C2212yes—
61#1944C2214yes—
62#1935C2214yes—
63#1928C2214yes—
64#1928C2214yes—
65#1928C2214yes—
66#1927C2214yes—
67#1921C2214yes—
68#1921C2214yes—
69#1920C2214yes—
70#1919C2214yes—
71#1918C2214yes—
72#1917C2214yes—
73#1917C2214yes—
74#1872C2212yes—
75#1872C2212yes—
76#1863C2214yes—
77#1852C2212yes—
78#1852C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1817C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1776C2214yes—
85#1763C2214yes—
86#1749C2212yes—
87C23×Q8—21no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2314yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8C2312yes—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—48no—
99C2×C4×Q8C2312yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C42—21no—
110C22×C42—21no—
111C22×C42—21no—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#483—28no—
116#483—28no—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2312yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2312yes—
129C2×#97C2314yes—
130#400—22no—
131#400—22no—
132#400—22no—
133#398—22no—
134#398—22no—
135#398—22no—
136#398—22no—
137#395—22no—
138#391—24no—
139#391—24no—
140#391—21no—
141#391—22no—
142#391C2312yes—
143#391C2312yes—
144#391C2312yes—
145#391C2312yes—
146#389—21no—
147#389—21no—
148#389—21no—
149#389—22no—
150#389—21no—
151#389—21no—
152#389—22no—
153#389C2312yes—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2314yes—
159#389C2312yes—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—24no—
170#387—24no—
171#387—24no—
172#387—24no—
173#385—22no—
174#385—22no—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#381—22no—
178#380—21no—
179#380—21no—
180#380—21no—
181#380C2312yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4D412yes—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2314yes—
191C4×C4⋊C4C2314yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95D411yescharacteristic
203C2×#95D411yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215#336C2314yes—
216#336C2314yes—
217C22×Q8—28no—
218C22×Q8—24no—
219C22×Q8—21no—
220C22×Q8—44no—
221C22×Q8—44no—
222C22×Q8C2412yes—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2411yescharacteristic
234C22×C8C2414yes—
235C22×C8C2414yes—
236C4⋊Q8—216no—
237C4⋊Q8—28no—
238C4⋊Q8—28no—
239C4⋊Q8—42no—
240C4⋊Q8—42no—
241C22⋊Q8—24no—
242C22⋊Q8—24no—
243C22⋊Q8—44no—
244C4×Q8—44no—
245C4×Q8—416no—
246C4×Q8—416no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4#4114yes—
282C2×C4⋊C4#4114yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2412yes—
310C2×C42C2412yes—
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313#107—28no—
314#107—44no—
315#106—28no—
316#106—28no—
317C4⋊C8—42no—
318C4⋊C8—42no—
319#103—28no—
320#103—28no—
321#103—28no—
322#103—44no—
323C22⋊C8—24no—
324C22⋊C8—24no—
325#97—44no—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95C2×D412yes—
348#95C2×D412yes—
349#95C2×D412yes—
350#95C2×D412yes—
351C24C22×D411yescharacteristic
352C2×Q8—216no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—24no—
356C2×Q8—24no—
357C2×Q8—48no—
358C2×Q8—432no—
359C2×Q8—42no—
360C2×Q8—48no—
361C2×Q8—81no—
362C2×Q8—81no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4AGL(1,ℤ/8)14yes—
402C22×C4#13714yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D412yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4C2511yescharacteristic
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C4⋊C4—24no—
422C4⋊C4—216no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—42no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4C2×#4114yes—
455C4⋊C4C2×#4114yes—
456C4⋊C42−1+414yes—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C42—24no—
460C42—22no—
461C42—22no—
462C42—24no—
463C42—22no—
464C42—22no—
465C42—41no—
466C42—41no—
467C42—42no—
468C42—48no—
469C42—48no—
470C42—44no—
471C42—44no—
472C42—82no—
473C42—82no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23#52512yes—
479C23C2×AGL(1,ℤ/8)12yes—
480C23C2×#13712yes—
481C23C23×D411yescharacteristic
482Q8—432no—
483Q8—44no—
484Q8—44no—
485Q8—162no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—48no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—48no—
521C2×C4—48no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—84no—
545C2×C4#52512yes—
546C2×C4C2×AGL(1,ℤ/8)12yes—
547C2×C4C2×AGL(1,ℤ/8)12yes—
548C2×C4C2×#13714yes—
549C2×C4C22×#4111yescharacteristic
550C2×C4C22×#4111yescharacteristic
551C2×C4C2×2−1+412yes—
552C2×C4#58512yes—
553C8—44no—
554C8—44no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22#311014yes—
562C22C2×#52511yescharacteristic
563C22#348511yescharacteristic
564C22#348611yescharacteristic
565C22C22×AGL(1,ℤ/8)11yescharacteristic
566C22C22×#13711yescharacteristic
567C22#358611yescharacteristic
568C22#358611yescharacteristic
569C4—22no—
570C4—22no—
571C4—22no—
572C4—24no—
573C4—42no—
574C4—42no—
575C4—41no—
576C4—41no—
577C4—44no—
578C4—44no—
579C4—44no—
580C4—42no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4#346112yes—
587C4#350712yes—
588C2—24no—
589C2#3509211yescharacteristic
590C2C2×#311011yescharacteristic
591C2#6056611yescharacteristic
592C2#6170611yescharacteristic
593C2#6172311yescharacteristic
594C2#6182711yescharacteristic
595C2#6182911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #27145C2semidirect
10. #27017C2semidirect
11. #26875C2semidirect
12. #26707C2semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#23652
#331462
#331082
#330992
#299662
#297562
#296412
#271452
#270172
#268752
#267072
#220842
#219162
#215792