Finite Group Databasebeta

← #6468220

Subgroups — #6468220

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic, complemented
2#34208C211yesmaximal, characteristic
3#29148C212yesmaximal, complemented
4#29138C212yesmaximal, complemented
5#29136C212yesmaximal, complemented
6#29120C214yesmaximal, complemented
7#28727C212yesmaximal, complemented
8#27969C212yesmaximal, complemented
9#25863C212yesmaximal
10#23829C212yesmaximal, complemented
11#23828C212yesmaximal, complemented
12#21655C212yesmaximal, complemented
13C4×#489C212yesmaximal, complemented
14#20746C214yesmaximal, complemented
15#20465C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#520C2212yes—
18C2×#489C2212yes—
19C2×#489C2212yes—
20C2×#433C2212yes—
21#2868C2211yescharacteristic
22#2862C2211yescharacteristic
23#2857C2212yes—
24#2647C2212yes—
25#2628C2212yes—
26#2511—22no—
27#2492—22no—
28#2451—22no—
29#2423—22no—
30#2364—22no—
31#2364—22no—
32#2352—22no—
33#2352—22no—
34#2331C2212yes—
35#2328C2211yescharacteristic
36#2328C2211yescharacteristic
37#2327C2211yescharacteristic
38#2327C2211yescharacteristic
39#2314C2212yes—
40#2311C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×#124C2212yes—
43C4×#124C2212yes—
44C2×#382C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2101C2214yes—
47#2101C2214yes—
48#2074C2214yes—
49#2073C2214yes—
50#2043C2214yes—
51#2041C2214yes—
52#2024C2214yes—
53#2023C2214yes—
54#1993C2212yes—
55#1993C2212yes—
56#1959C2212yes—
57#1959C2212yes—
58#1936C2214yes—
59#1935C2214yes—
60#1934C2214yes—
61#1933C2214yes—
62#1836C2212yes—
63#1836C2212yes—
64#1824C2214yes—
65#1823C2214yes—
66#1805C2214yes—
67#1804C2214yes—
68#1803C2214yes—
69#1802C2214yes—
70#1788C2214yes—
71#1788C2214yes—
72#1768C2212yes—
73#1767C2212yes—
74#1763C2214yes—
75#1762C2214yes—
76C4×#103C2214yes—
77C4×#102C2214yes—
78#1750C2212yes—
79#1749C2212yes—
80#1746C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83C22×#41C2312yes—
84#532—28no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#127C2311yescharacteristic
90C2×#124C2312yes—
91C2×C22⋊Q8—22no—
92C2×#121—22no—
93#520—216no—
94C2×C4×Q8—22no—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2312yes—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×#117—22no—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C4⋊C4—22no—
106C22×C4⋊C4C2312yes—
107C22×C42C2311yescharacteristic
108#489—28no—
109#489—28no—
110#433—28no—
111C2×C4⋊C8C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×C22⋊C8C2314yes—
121C2×#97C2312yes—
122C2×#97C2312yes—
123C2×#97C2312yes—
124C2×#97C2312yes—
125C2×C4×C8C2312yes—
126C2×C4×C8C2312yes—
127#400—22no—
128#400—24no—
129#399—22no—
130#398—24no—
131#395—22no—
132#393—22no—
133#391—22no—
134#391—22no—
135#391—22no—
136#391—22no—
137#391—24no—
138#391C2312yes—
139#391C2312yes—
140#390C2312yes—
141#390C2312yes—
142#389—22no—
143#389—22no—
144#389C2314yes—
145#389C2314yes—
146#388—24no—
147#388—22no—
148#388—22no—
149#388—22no—
150#388—22no—
151#387—24no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—24no—
157#386—22no—
158#386—22no—
159#386—22no—
160#386—22no—
161#386—22no—
162#386—22no—
163#384—22no—
164#384C2312yes—
165#384C2312yes—
166#382—22no—
167#382C2312yes—
168#382C2311yescharacteristic
169#382C2311yescharacteristic
170#382C2312yes—
171#382C2312yes—
172#380—22no—
173#380—22no—
174#380—22no—
175#380C2314yes—
176C4×C4⋊C4—22no—
177C4×C4⋊C4D412yes—
178C4×C4⋊C4D412yes—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4D412yes—
185C4×C22⋊C4D412yes—
186#376C2311yescharacteristic
187#376C2311yescharacteristic
188C2×#95—22no—
189C43C2311yescharacteristic
190C43C2311yescharacteristic
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C2×#41—24no—
202C2×#41—24no—
203C2×#41—28no—
204C2×#41—44no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8C2412yes—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4C2412yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4C2×D412yes—
216C23×C4C2411yescharacteristic
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—24no—
220C4⋊Q8—24no—
221C4⋊Q8—24no—
222C4⋊Q8—44no—
223C4⋊Q8—84no—
224#127—24no—
225#124—24no—
226#124#4118yes—
227C22⋊Q8—84no—
228#121—216no—
229C4×Q8—28no—
230C4×Q8—28no—
231C4×Q8—84no—
232C4×D4—216no—
233C4×D4—28no—
234C4×D4—28no—
235C4×D4—48no—
236C4×D4—48no—
237#117—42no—
238#117—48no—
239#117#4118yes—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2414yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4C2×D411yescharacteristic
270C2×C22⋊C4C2×D411yescharacteristic
271C2×C22⋊C4C2×D412yes—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2411yescharacteristic
282C2×C42C2411yescharacteristic
283C2×C42C2411yescharacteristic
284C2×C42C2411yescharacteristic
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287C4⋊C8—28no—
288#103—28no—
289#103—28no—
290#103—44no—
291#103—44no—
292#102—28no—
293#102—28no—
294#102—44no—
295#102—44no—
296C22⋊C8—28no—
297#97—24no—
298#97—24no—
299#97—24no—
300#97—42no—
301C4×C8—24no—
302C4×C8—42no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—22no—
316#95—24no—
317#95—22no—
318#95C2×D412yes—
319#95C2×D412yes—
320#95C2×D412yes—
321#95C2×D412yes—
322C24—22no—
323C24C22×D411yescharacteristic
324#41—416no—
325#41—416no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—84no—
334C2×Q8C2×#4114yes—
335C2×D4—28no—
336C2×D4—28no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4C2×#4114yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D412yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C2×#4114yes—
377C22×C4C2511yescharacteristic
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C4⋊C4—28no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4#12418yes—
404C22⋊C4—28no—
405C22⋊C4—28no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—84no—
415C22⋊C4#12418yes—
416C42—24no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—44no—
425C42—44no—
426C42—84no—
427C42C2×#4114yes—
428C42C2×#4114yes—
429C42C2×#4114yes—
430C23—22no—
431C23—22no—
432C23—21no—
433C23—21no—
434C23—24no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23C2×#12412yes—
442C23C23×D411yescharacteristic
443Q8—28no—
444Q8—44no—
445Q8—88no—
446Q8—164no—
447D4—28no—
448D4—416no—
449D4—44no—
450D4—88no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—28no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—416no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—84no—
488C2×C4—164no—
489C2×C4—164no—
490C2×C4—164no—
491C2×C4#52514yes—
492C2×C4C2×#12414yes—
493C2×C4C2×#12414yes—
494C2×C4C2×#12412yes—
495C2×C4C22×#4112yes—
496C2×C4C22×#4112yes—
497C2×C4#58512yes—
498C8—44no—
499C8—44no—
500C22—22no—
501C22—24no—
502C22—41no—
503C22—41no—
504C22—42no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22C2×#52511yescharacteristic
513C22C22×#12411yescharacteristic
514C22#344911yescharacteristic
515C22#358512yes—
516C22#358612yes—
517C4—22no—
518C4—24no—
519C4—24no—
520C4—24no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—44no—
526C4—42no—
527C4—84no—
528C4—84no—
529C4—82no—
530C4—82no—
531C4#347514yes—
532C2—42no—
533C2—84no—
534C2#3494611yescharacteristic
535C2#6056712yes—
536C2#6062512yes—
537C2#6101911yescharacteristic
538C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35283C2semidirect
3. #29148C2semidirect
4. #29138C2semidirect
5. #29136C2semidirect
6. #29120C2semidirect
7. #28727C2semidirect
8. #27969C2semidirect
10. #23829C2semidirect
11. #23828C2semidirect
12. #21655C2semidirect
13. C4×#489C2semidirect
14. #20746C2semidirect
15. #20465C2semidirect

Maximal subgroups

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

subgroupindex
#352832
#342082
#291482
#291382
#291362
#291202
#287272
#279692
#258632
#238292
#238282
#216552
C4×#4892
#207462
#204652