Finite Group Databasebeta

← #7231157

Subgroups — #7231157

Order 512.

1571 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46147C211yesmaximal, characteristic, complemented
2#36406C211yesmaximal, characteristic
3#33072C212yesmaximal, complemented
4#32821C212yesmaximal, complemented
5#32693C212yesmaximal, complemented
6#32027C212yesmaximal, complemented
7#31494C214yesmaximal, complemented
8#30653C214yesmaximal, complemented
9#27051C214yesmaximal, complemented
10#26342C214yesmaximal, complemented
11#25661C212yesmaximal, complemented
12#24088C212yesmaximal
13#23352C211yesmaximal, characteristic, complemented
14C2×#563—21no—
15C2×#556C2211yescharacteristic
16C2×#547C2211yescharacteristic
17C2×#489C2212yes—
18C2×#480C2212yes—
19C2×#477C2212yes—
20C22×#107C2212yes—
21#2862C2211yescharacteristic
22#2859C2212yes—
23#2857—21no—
24#2822—21no—
25#2734—21no—
26#2724—21no—
27#2698—24no—
28#2553—24no—
29#2546C2211yescharacteristic
30#2546C2212yes—
31#2542C2211yescharacteristic
32#2499—21no—
33#2496C2214yes—
34#2486C2212yes—
35#2440C2214yes—
36#2432—21no—
37#2427C2214yes—
38#2360C2214yes—
39C4×#124C2211yescharacteristic
40#2297—21no—
41C42×D4C2211yescharacteristic
42#2095C2214yes—
43#2087C2214yes—
44#2072C2214yes—
45#2068C2214yes—
46#2068C2214yes—
47#2057C2214yes—
48#2057C2214yes—
49#2043C2214yes—
50#2042C2214yes—
51#2035C2214yes—
52#2035C2214yes—
53#2033C2214yes—
54#1985C2212yes—
55#1983C2212yes—
56#1948C2212yes—
57#1947C2212yes—
58#1944C2214yes—
59#1924C2214yes—
60#1921C2214yes—
61#1919C2214yes—
62#1919C2214yes—
63#1893C2212yes—
64#1892C2214yes—
65#1872C2214yes—
66#1869C2212yes—
67#1864C2214yes—
68#1863C2214yes—
69#1862C2214yes—
70#1852C2214yes—
71#1850C2214yes—
72#1831C2212yes—
73#1823C2212yes—
74#1802C2212yes—
75#1802C2212yes—
76#1763C2212yes—
77C22×#41—21no—
78C23×C8C2312yes—
79#563—48no—
80#556—42no—
81#556—42no—
82#547—42no—
83#547—42no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×#126—22no—
90C2×#126—24no—
91C2×#125—22no—
92C2×#124D411yescharacteristic
93C2×#124C2312yes—
94C2×#123—22no—
95C2×#123—22no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8C2314yes—
100C2×#121C2312yes—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2311yescharacteristic
103C2×C4×D4C2311yescharacteristic
104C2×C4×D4C2312yes—
105C2×C4×D4C2311yescharacteristic
106C2×#117—21no—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C42—21no—
110#489—44no—
111#480—44no—
112#477—44no—
113C2×#107—44no—
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2312yes—
130#400—24no—
131#399—22no—
132#398—22no—
133#398—22no—
134#398—22no—
135#398—22no—
136#398—24no—
137#397—22no—
138#397—24no—
139#397—24no—
140#395—24no—
141#394—24no—
142#391—22no—
143#391—21no—
144#391—21no—
145#391—21no—
146#391D411yescharacteristic
147#391C2314yes—
148#390—21no—
149#390D411yescharacteristic
150#389—22no—
151#389—22no—
152#389—24no—
153#389C2314yes—
154#389C2314yes—
155#389C2314yes—
156#389C2314yes—
157#388D414yes—
158#387—24no—
159#387—24no—
160#386—24no—
161#385—24no—
162#385—24no—
163#384—21no—
164#384C2312yes—
165#384C2312yes—
166#382—22no—
167#382—22no—
168#382D411yescharacteristic
169#381—22no—
170#380—24no—
171#380C2312yes—
172#380C2314yes—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2311yescharacteristic
175C4×C4⋊C4C2312yes—
176C4×C22⋊C4—24no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—22no—
179C4×C22⋊C4C2314yes—
180C43C2311yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2314yes—
189#336C2314yes—
190C2×#41—44no—
191C2×#41—44no—
192C2×#41—82no—
193C22×Q8—21no—
194C22×Q8—22no—
195C22×Q8C2×D411yescharacteristic
196C22×Q8C2412yes—
197C22×D4—22no—
198C22×D4C2×D411yescharacteristic
199C22×D4C2411yescharacteristic
200C23×C4—21no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4C2412yes—
205C22×C8—44no—
206C22×C8—42no—
207C22×C8C2414yes—
208C22×C8C2414yes—
209C4⋊Q8—41no—
210C4⋊Q8—41no—
211C4⋊Q8—42no—
212C4⋊Q8—42no—
213C4⋊Q8—41no—
214C4⋊Q8—41no—
215C4⋊Q8—48no—
216C4⋊Q8—82no—
217#126—48no—
218#126—416no—
219#125—48no—
220#124—24no—
221#124—24no—
222#124—42no—
223#123—48no—
224#123—48no—
225C22⋊Q8—28no—
226C22⋊Q8—416no—
227C22⋊Q8—44no—
228C22⋊Q8—44no—
229C22⋊Q8—44no—
230#121—24no—
231#121—42no—
232C4×Q8—24no—
233C4×Q8—41no—
234C4×Q8—41no—
235C4×Q8—42no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—24no—
239C4×D4—42no—
240C4×D4—41no—
241C4×D4—41no—
242#117—44no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2411yescharacteristic
270C2×C4⋊C4C2412yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4C2×D414yes—
280C2×C22⋊C4C2×D414yes—
281C2×C22⋊C4C2×D412yes—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42—82no—
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D412yes—
289C2×C42C2×D411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2412yes—
292C2×C42C2412yes—
293#107—28no—
294#107—44no—
295#107—44no—
296#106—28no—
297C4⋊C8—42no—
298C4⋊C8—42no—
299C4⋊C8—42no—
300#103—44no—
301#103—44no—
302#103—44no—
303#103—44no—
304#102—28no—
305#102—44no—
306C22⋊C8—44no—
307C22⋊C8—42no—
308#97—42no—
309#95—22no—
310#95—21no—
311#95—24no—
312#95—24no—
313#95—22no—
314#95—24no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95C2×D414yes—
321#95C2×D414yes—
322C24—21no—
323C24C22×D412yes—
324#41—816no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—44no—
335C2×Q8—42no—
336C2×Q8—82no—
337C2×Q8—82no—
338C2×D4—21no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—82no—
346C2×D4AGL(1,ℤ/8)14yes—
347C2×D4H(5,𝔽2)12yes—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C2511yescharacteristic
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—48no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—48no—
407C4⋊C4—44no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—84no—
412C4⋊C4H(5,𝔽2)12yes—
413C4⋊C42−1+414yes—
414C22⋊C4—24no—
415C22⋊C4—28no—
416C22⋊C4—28no—
417C22⋊C4—42no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—48no—
422C22⋊C4—44no—
423C22⋊C4—48no—
424C22⋊C4—44no—
425C42—22no—
426C42—24no—
427C42—22no—
428C42—22no—
429C42—21no—
430C42—21no—
431C42—22no—
432C42—22no—
433C42—41no—
434C42—44no—
435C42—42no—
436C42—44no—
437C42—82no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—41no—
444C23—42no—
445C23—41no—
446C23—42no—
447C23C2×AGL(1,ℤ/8)14yes—
448C23C23×D411yescharacteristic
449Q8—42no—
450Q8—42no—
451Q8—44no—
452Q8—82no—
453Q8—84no—
454Q8—164no—
455D4—24no—
456D4—44no—
457D4—41no—
458D4—41no—
459D4—88no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—48no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—88no—
498C2×C4—88no—
499C2×C4—164no—
500C2×C4C2×AGL(1,ℤ/8)12yes—
501C2×C4#57814yes—
502C2×C4C2×H(5,𝔽2)11yescharacteristic
503C2×C4C2×H(5,𝔽2)11yescharacteristic
504C2×C4C2×2−1+412yes—
505C2×C4#58512yes—
506C8—44no—
507C8—44no—
508C22—24no—
509C22—28no—
510C22—42no—
511C22—44no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—82no—
517C22—82no—
518C22#348411yescharacteristic
519C22#348912yes—
520C22C22×AGL(1,ℤ/8)11yescharacteristic
521C22C2×#57812yes—
522C22#358511yescharacteristic
523C4—22no—
524C4—24no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—44no—
530C4—42no—
531C4—41no—
532C4—82no—
533C4—84no—
534C4—82no—
535C4—84no—
536C4#352911yescharacteristic
537C4#355011yescharacteristic
538C4#355912yes—
539C2—44no—
540C2—82no—
541C2#4622811yescharacteristic
542C2#6156911yescharacteristic
543C2#6166511yescharacteristic
544C2#6213212yes—
545C2#6216512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #46147C2semidirect
3. #33072C2semidirect
4. #32821C2semidirect
5. #32693C2semidirect
6. #32027C2semidirect
7. #31494C2semidirect
8. #30653C2semidirect
9. #27051C2semidirect
10. #26342C2semidirect
11. #25661C2semidirect
13. #23352C2semidirect

Maximal subgroups

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

subgroupindex
#461472
#364062
#330722
#328212
#326932
#320272
#314942
#306532
#270512
#263422
#256612
#240882
#233522