Finite Group Databasebeta

← #6532181

Subgroups — #6532181

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2584C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#27617C212yesmaximal
4#27616C212yesmaximal
5#27614C211yesmaximal, characteristic
6#27613C211yesmaximal, characteristic
7#27611C211yesmaximal, characteristic
8#27610C211yesmaximal, characteristic
9#26287C214yesmaximal, complemented
10#26286C214yesmaximal, complemented
11#26285C214yesmaximal, complemented
12#26284C214yesmaximal, complemented
13#21836C212yesmaximal
14#21818C211yesmaximal, characteristic
15#21361C212yesmaximal
16C22×#125C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#431C2212yes—
19#2584—22no—
20#2584—22no—
21#2584C2214yes—
22#2584C2214yes—
23C2×#398—21no—
24C2×#395—21no—
25#2356—22no—
26#2356—22no—
27#2355C2212yes—
28#2355C2212yes—
29#2355C2212yes—
30#2355C2212yes—
31#2313—21no—
32#2313—21no—
33#2310—21no—
34C2×#389—21no—
35C2×#389—21no—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40#2285C2211yescharacteristic
41#2285C2211yescharacteristic
42#2283C2212yes—
43#2283C2212yes—
44#2274—21no—
45C2×C4×C4⋊C4C2211yescharacteristic
46#2101C2214yes—
47#2101C2214yes—
48#2100C2212yes—
49#2100C2212yes—
50#2099C2212yes—
51#2099C2212yes—
52#2085C2214yes—
53#2085C2214yes—
54#2084C2212yes—
55#2084C2212yes—
56#2083C2212yes—
57#2083C2212yes—
58#1991C2214yes—
59#1991C2214yes—
60#1989C2214yes—
61#1989C2214yes—
62#1984C2214yes—
63#1984C2214yes—
64#1949C2214yes—
65#1949C2214yes—
66#1921C2214yes—
67#1921C2214yes—
68#1920C2214yes—
69#1920C2214yes—
70#1919C2214yes—
71#1919C2214yes—
72#1918C2214yes—
73#1918C2214yes—
74#1853C2212yes—
75#1853C2212yes—
76#1852C2212yes—
77#1852C2212yes—
78#1824C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1817C2211yescharacteristic
83#1817C2211yescharacteristic
84#1817C2212yes—
85C4×#107C2214yes—
86C4×#106C2214yes—
87C23×Q8D411yescharacteristic
88C2×#125—22no—
89C2×#125—28no—
90C2×#125C2314yes—
91C2×C4×Q8—28no—
92C2×C4×Q8D412yes—
93C2×C4×Q8D412yes—
94C2×C4×Q8C2312yes—
95C2×C4×Q8C2312yes—
96C2×#117—21no—
97C2×#117—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C42—21no—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#431—28no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2314yes—
121C2×C22⋊C8C2312yes—
122C2×C22⋊C8C2312yes—
123C2×C4×C8C2314yes—
124#398—24no—
125#398—24no—
126#395—24no—
127#395—24no—
128#391—22no—
129#391—22no—
130#391—22no—
131#391—22no—
132#389—24no—
133#389—24no—
134#389—24no—
135#389—24no—
136#389—22no—
137#389—21no—
138#389—21no—
139#389—22no—
140#389—21no—
141#389—21no—
142#389C2312yes—
143#389C2312yes—
144#389C2312yes—
145#389C2312yes—
146#389C2314yes—
147#389C2314yes—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#384C2311yescharacteristic
157#384C2311yescharacteristic
158#384C2311yescharacteristic
159#384C2311yescharacteristic
160#383—22no—
161#383—22no—
162#382—21no—
163#382—21no—
164#382C2312yes—
165#382C2312yes—
166#382C2312yes—
167#382C2312yes—
168#381—22no—
169#381—22no—
170#380—21no—
171#380—21no—
172#380C2312yes—
173#380C2312yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190#376—22no—
191#376—22no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2312yes—
210#336C2312yes—
211#336C2312yes—
212#336C2312yes—
213#336C2314yes—
214#336C2314yes—
215C22×Q8—24no—
216C22×Q8—24no—
217C22×Q8—42no—
218C22×Q8C2×D412yes—
219C22×Q8C2×D412yes—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2411yescharacteristic
231C22×C8C2414yes—
232C22×C8C2414yes—
233#125—28no—
234#125—28no—
235#125—48no—
236C4×Q8—28no—
237C4×Q8—28no—
238C4×Q8—24no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—24no—
242#117—44no—
243#117—44no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4—82no—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—82no—
303C2×C42C2×D412yes—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D412yes—
307C2×C42#4118yes—
308C2×C42C2412yes—
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2412yes—
312#107—28no—
313#107—28no—
314#106—28no—
315#106—28no—
316C4⋊C8—28no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319C4×C8—28no—
320#95—24no—
321#95—24no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
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—
343C24C22×D411yescharacteristic
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—416no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4#12314yes—
389C22×C4#12314yes—
390C22×C4#13514yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C2×#4114yes—
401C22×C4C2511yescharacteristic
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—48no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C4⋊C4—84no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C42−1+414yes—
443C4⋊C42−1+414yes—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C42—22no—
447C42—22no—
448C42—24no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—44no—
454C42—44no—
455C42—82no—
456C42—82no—
457C42—82no—
458C42—82no—
459C42—84no—
460C42#12318yes—
461C42C2×#4114yes—
462C42C2×#4114yes—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23C2×#12312yes—
468C23C2×#13512yes—
469C23#57512yes—
470C23C23×D411yescharacteristic
471Q8—48no—
472Q8—48no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—44no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4C2×#12312yes—
531C2×C4C2×#12314yes—
532C2×C4C2×#12314yes—
533C2×C4C2×#13512yes—
534C2×C4C2×#13512yes—
535C2×C4C22×#4112yes—
536C2×C4C2×2−1+412yes—
537C2×C4C2×2−1+412yes—
538C8—44no—
539C8—44no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22#309114yes—
547C22C22×#12311yescharacteristic
548C22#348611yescharacteristic
549C22#348611yescharacteristic
550C22C22×#13511yescharacteristic
551C22C2×#57511yescharacteristic
552C22#358511yescharacteristic
553C22#358611yescharacteristic
554C4—22no—
555C4—24no—
556C4—22no—
557C4—22no—
558C4—24no—
559C4—42no—
560C4—42no—
561C4—41no—
562C4—41no—
563C4—44no—
564C4—44no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4#350914yes—
572C2—24no—
573C2#3567411yescharacteristic
574C2C2×#309111yescharacteristic
575C2#6052011yescharacteristic
576C2#6181911yescharacteristic
577C2#6182211yescharacteristic
578C2#6183111yescharacteristic
579C2#6183211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
9. #26287C2semidirect
10. #26286C2semidirect
11. #26285C2semidirect
12. #26284C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25842
#341872
#276172
#276162
#276142
#276132
#276112
#276102
#262872
#262862
#262852
#262842
#218362
#218182
#213612