Finite Group Databasebeta

← #7459857

Subgroups — #7459857

Order 512.

1429 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46152C211yesmaximal, characteristic, complemented
2#39489C211yesmaximal, characteristic
3#32695C212yesmaximal, complemented
4#32268C212yesmaximal, complemented
5#31687C214yesmaximal, complemented
6#31078C214yesmaximal, complemented
7#30304C212yesmaximal, complemented
8#29574C212yesmaximal, complemented
9#28865C214yesmaximal, complemented
10#26077C214yesmaximal, complemented
11#25850C212yesmaximal, complemented
12#24105C212yesmaximal
13#23272C211yesmaximal, characteristic, complemented
14C2×#555C2211yescharacteristic
15C2×#554C2211yescharacteristic
16C2×#501C2212yes—
17C2×#480C2212yes—
18C2×#474C2212yes—
19C2×#428C2212yes—
20#2868C2211yescharacteristic
21#2859C2212yes—
22#2844—21no—
23#2809—21no—
24#2720—24no—
25#2683—21no—
26#2674—24no—
27#2620—21no—
28#2567C2211yescharacteristic
29#2567C2212yes—
30#2563C2211yescharacteristic
31#2501—21no—
32#2499—21no—
33#2497C2214yes—
34#2487C2212yes—
35#2441C2214yes—
36#2440C2214yes—
37#2422C2214yes—
38#2379—21no—
39#2317—21no—
40C4×#124C2211yescharacteristic
41#2293C2211yescharacteristic
42#2100C2212yes—
43#2094C2212yes—
44#2086C2214yes—
45#2084C2212yes—
46#2083C2214yes—
47#2083C2214yes—
48#2077C2214yes—
49#2075C2214yes—
50#2071C2212yes—
51#2059C2214yes—
52#2057C2214yes—
53#2057C2214yes—
54#2057C2214yes—
55#2033C2214yes—
56#1989C2212yes—
57#1985C2212yes—
58#1944C2214yes—
59#1941C2214yes—
60#1934C2214yes—
61#1923C2214yes—
62#1920C2214yes—
63#1920C2214yes—
64#1895C2212yes—
65#1870C2212yes—
66#1864C2214yes—
67#1862C2214yes—
68#1854C2214yes—
69#1852C2214yes—
70#1845C2214yes—
71#1834C2214yes—
72#1825C2212yes—
73#1809C2214yes—
74#1762C2212yes—
75#1753C2214yes—
76#1744C2212yes—
77C22×M16C2312yes—
78#555—42no—
79#555—42no—
80#554—42no—
81#554—42no—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×#125—22no—
87C2×#125C2312yes—
88C2×#124—22no—
89C2×#124D411yescharacteristic
90C2×#123—24no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—24no—
94C2×#121—21no—
95C2×#121C2312yes—
96C2×C4×Q8C2311yescharacteristic
97C2×C4×D4C2311yescharacteristic
98C2×C4×D4C2312yes—
99C2×C4×D4C2311yescharacteristic
100C2×#117—21no—
101C2×#117—21no—
102C2×#117C2312yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2312yes—
105#501—44no—
106#480—44no—
107#474—44no—
108#428—44no—
109C2×#107C2314yes—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×#106C2314yes—
115C2×C4⋊C8C2312yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×C22⋊C8C2314yes—
119C2×C22⋊C8C2312yes—
120C2×#97C2312yes—
121C2×C4×C8C2314yes—
122#400—22no—
123#400—24no—
124#399—24no—
125#399—22no—
126#398—24no—
127#397—24no—
128#396—24no—
129#396—24no—
130#395—22no—
131#395—22no—
132#395—22no—
133#395—24no—
134#394—24no—
135#393—22no—
136#391—21no—
137#391—22no—
138#391D411yescharacteristic
139#390—21no—
140#390D411yescharacteristic
141#390C2312yes—
142#389—22no—
143#389—22no—
144#389—24no—
145#389C2314yes—
146#389C2314yes—
147#389C2314yes—
148#389C2314yes—
149#389C2312yes—
150#388—24no—
151#388—22no—
152#388D414yes—
153#387—24no—
154#387—24no—
155#386—22no—
156#385—24no—
157#385—22no—
158#385—24no—
159#384—21no—
160#384—22no—
161#384—21no—
162#384—21no—
163#384C2312yes—
164#384C2314yes—
165#384C2312yes—
166#382—21no—
167#382—22no—
168#382D411yescharacteristic
169#380C2314yes—
170#380C2314yes—
171#380C2312yes—
172C4×C4⋊C4C2311yescharacteristic
173C4×C4⋊C4C2312yes—
174C4×C22⋊C4—21no—
175C4×C22⋊C4C2314yes—
176C43C2311yescharacteristic
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#336C2314yes—
186#336C2314yes—
187#336C2312yes—
188#336C2312yes—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—22no—
192C22×Q8C2×D411yescharacteristic
193C22×D4—22no—
194C22×D4C2×D411yescharacteristic
195C22×D4C2411yescharacteristic
196C23×C4—21no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4C2412yes—
200C2×M16—28no—
201C2×M16—42no—
202C22×C8C2414yes—
203C22×C8C2414yes—
204C4⋊Q8—41no—
205C4⋊Q8—41no—
206C4⋊Q8—41no—
207C4⋊Q8—41no—
208C4⋊Q8—42no—
209C4⋊Q8—42no—
210C4⋊Q8—84no—
211#125—24no—
212#125—42no—
213#125—84no—
214#124—24no—
215#124—48no—
216#123—88no—
217C22⋊Q8—48no—
218C22⋊Q8—44no—
219C22⋊Q8—88no—
220#121—24no—
221#121—42no—
222#121—44no—
223C4×Q8—41no—
224C4×Q8—41no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—24no—
228C4×D4—42no—
229C4×D4—41no—
230C4×D4—41no—
231#117—24no—
232#117—42no—
233#117—44no—
234#117—44no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2411yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4C2×D414yes—
274C2×C22⋊C4C2×D414yes—
275C2×C22⋊C4C2×D412yes—
276C2×C42—22no—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42C2×D412yes—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2411yescharacteristic
283C2×C42C2412yes—
284C2×C42C2412yes—
285#107—44no—
286#107—44no—
287#107—44no—
288#107—44no—
289#106—28no—
290#106—28no—
291C4⋊C8—42no—
292#102—28no—
293#102—28no—
294C22⋊C8—28no—
295C22⋊C8—42no—
296#97—42no—
297C4×C8—28no—
298#95—22no—
299#95—21no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95C2×D414yes—
309#95C2×D414yes—
310C24—21no—
311C24C22×D412yes—
312C2×Q8—22no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—84no—
318C2×Q8—82no—
319C2×Q8—82no—
320C2×Q8—82no—
321C2×D4—22no—
322C2×D4—21no—
323C2×D4—24no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—44no—
327C2×D4—84no—
328C2×D4H(5,𝔽2)12yes—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—84no—
344C22×C4—82no—
345C22×C4—84no—
346C22×C4C22×D412yes—
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D412yes—
352C22×C4C22×D412yes—
353C22×C4C2511yescharacteristic
354M16—48no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C82−1+418yes—
361C4⋊C4—21no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—84no—
389C4⋊C4—82no—
390C4⋊C4—84no—
391C4⋊C4H(5,𝔽2)12yes—
392C22⋊C4—24no—
393C22⋊C4—28no—
394C22⋊C4—28no—
395C22⋊C4—42no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—82no—
402C22⋊C4—84no—
403C22⋊C4—84no—
404C42—24no—
405C42—22no—
406C42—22no—
407C42—21no—
408C42—21no—
409C42—24no—
410C42—41no—
411C42—44no—
412C42—42no—
413C42—82no—
414C42#13714yes—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—22no—
420C23—24no—
421C23—41no—
422C23—42no—
423C23—41no—
424C23—42no—
425C23C23×D411yescharacteristic
426Q8—82no—
427Q8—162no—
428Q8—162no—
429Q8—164no—
430D4—24no—
431D4—42no—
432D4—82no—
433D4—164no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—48no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—164no—
470C2×C4—164no—
471C2×C4C2×#13712yes—
472C2×C4C2×#13714yes—
473C2×C4#57614yes—
474C2×C4C2×H(5,𝔽2)11yescharacteristic
475C2×C4C2×H(5,𝔽2)11yescharacteristic
476C2×C4C2×2−1+414yes—
477C8—28no—
478C8—44no—
479C22—44no—
480C22—44no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—82no—
488C22—82no—
489C22#348411yescharacteristic
490C22#349912yes—
491C22C22×#13711yescharacteristic
492C22C2×#57612yes—
493C22#358611yescharacteristic
494C4—24no—
495C4—24no—
496C4—22no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—41no—
502C4—82no—
503C4—84no—
504C4—82no—
505C4—84no—
506C4—82no—
507C4#353812yes—
508C4#354412yes—
509C2—82no—
510C2—82no—
511C2#5154211yescharacteristic
512C2#6158611yescharacteristic
513C2#6166111yescharacteristic
514C2#6248012yes—
515C2#6251512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
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. #46152C2semidirect
3. #32695C2semidirect
4. #32268C2semidirect
5. #31687C2semidirect
6. #31078C2semidirect
7. #30304C2semidirect
8. #29574C2semidirect
9. #28865C2semidirect
10. #26077C2semidirect
11. #25850C2semidirect
13. #23272C2semidirect

Maximal subgroups

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

subgroupindex
#461522
#394892
#326952
#322682
#316872
#310782
#303042
#295742
#288652
#260772
#258502
#241052
#232722