Finite Group Databasebeta

← #6499472

Subgroups — #6499472

Order 512.

1422 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#35343C211yesmaximal, characteristic
2#35286C211yesmaximal, characteristic, complemented
3#33282C212yesmaximal, complemented
4#33282C212yesmaximal, complemented
5#30235C212yesmaximal, complemented
6#30227C212yesmaximal, complemented
7#29345C214yesmaximal, complemented
8#29341C212yesmaximal, complemented
9#29340C212yesmaximal
10#29138C212yesmaximal, complemented
11#29136C212yesmaximal, complemented
12#29134C214yesmaximal, complemented
13#21868C212yesmaximal, complemented
14#21867C212yesmaximal, complemented
15#20467C211yesmaximal, characteristic, complemented
16C2×#529C2212yes—
17C2×#432C2212yes—
18#2861C2212yes—
19#2859C2212yes—
20#2859C2212yes—
21#2859C2211yescharacteristic
22#2858C2211yescharacteristic
23#2647C2212yes—
24#2647C2212yes—
25#2621C2211yescharacteristic
26#2620C2212yes—
27#2619C2211yescharacteristic
28#2602—22no—
29#2602—22no—
30#2601—22no—
31#2601—22no—
32#2571—22no—
33#2552—22no—
34#2442—22no—
35#2442—22no—
36#2328C2211yescharacteristic
37#2327C2211yescharacteristic
38#2326C2212yes—
39#2317C2212yes—
40#2316C2212yes—
41#2285C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C4×#117C2211yescharacteristic
44#2101C2214yes—
45#2101C2214yes—
46#2100C2214yes—
47#2099C2214yes—
48#2043C2214yes—
49#2042C2214yes—
50#2042C2214yes—
51#2041C2214yes—
52#1992C2212yes—
53#1992C2212yes—
54#1992C2212yes—
55#1992C2212yes—
56#1959C2212yes—
57#1959C2212yes—
58#1959C2212yes—
59#1959C2212yes—
60#1936C2214yes—
61#1936C2214yes—
62#1935C2214yes—
63#1935C2214yes—
64#1934C2214yes—
65#1934C2214yes—
66#1933C2214yes—
67#1933C2214yes—
68#1875C2214yes—
69#1874C2214yes—
70#1873C2214yes—
71#1872C2214yes—
72#1805C2214yes—
73#1804C2214yes—
74#1803C2214yes—
75#1802C2214yes—
76#1768C2212yes—
77#1768C2212yes—
78#1750C2212yes—
79#1750C2212yes—
80#1745C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×#127C2311yescharacteristic
85#529—216no—
86C2×#126—22no—
87C2×#125—22no—
88C2×#125—22no—
89C2×#125C2312yes—
90C2×#124C2312yes—
91C2×#124C2312yes—
92C2×#123—22no—
93C2×C22⋊Q8C2312yes—
94C2×#121C2312yes—
95C2×C4×Q8C2312yes—
96C2×C4×D4—22no—
97C2×C4×D4C2312yes—
98C2×#117—22no—
99C2×#117C2312yes—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—22no—
103C22×C4⋊C4C2312yes—
104C22×C42C2311yescharacteristic
105#432—28no—
106C2×C4⋊C8C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#102C2314yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×C22⋊C8C2314yes—
116C2×#97C2312yes—
117C2×#97C2312yes—
118C2×#97C2312yes—
119C2×#97C2312yes—
120C2×C4×C8C2312yes—
121C2×C4×C8C2312yes—
122#400—24no—
123#400—24no—
124#398—22no—
125#398—24no—
126#398—24no—
127#397—22no—
128#396—22no—
129#396—22no—
130#394—22no—
131#394—22no—
132#394—22no—
133#392—22no—
134#391C2312yes—
135#391C2312yes—
136#390—22no—
137#390—22no—
138#390C2312yes—
139#390C2312yes—
140#389—24no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389C2314yes—
148#389C2314yes—
149#389C2314yes—
150#389C2314yes—
151#388—22no—
152#388—22no—
153#388—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#387—22no—
158#387—22no—
159#386—22no—
160#386—22no—
161#385—22no—
162#385—22no—
163#385—22no—
164#385—22no—
165#384—22no—
166#384—22no—
167#384C2312yes—
168#384C2312yes—
169#384C2312yes—
170#384C2312yes—
171#382—22no—
172#382—22no—
173#382—22no—
174#382C2312yes—
175#382C2311yescharacteristic
176#382C2311yescharacteristic
177#380—22no—
178#380—24no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D412yes—
181C4×C4⋊C4D412yes—
182C4×C22⋊C4D412yes—
183C4×C22⋊C4D412yes—
184#376C2311yescharacteristic
185#376C2311yescharacteristic
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198C22×Q8C2412yes—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4C2412yes—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4C2×D412yes—
205C23×C4C2411yescharacteristic
206C22×C8C2414yes—
207C22×C8C2414yes—
208C4⋊Q8—42no—
209#127—42no—
210#126—48no—
211#125—28no—
212#125—84no—
213#125—84no—
214#124—28no—
215#124—44no—
216#123—48no—
217C22⋊Q8—28no—
218#121—28no—
219C4×Q8—28no—
220C4×D4—28no—
221C4×D4—48no—
222#117—42no—
223#117—42no—
224#117—42no—
225#117—42no—
226#117—48no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—84no—
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2414yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4C2×D412yes—
266C2×C22⋊C4C2×D411yescharacteristic
267C2×C22⋊C4C2×D411yescharacteristic
268C2×C42—22no—
269C2×C42—22no—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42#4118yes—
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280C2×C42C2411yescharacteristic
281C2×C42C2411yescharacteristic
282C4⋊C8—28no—
283#103—44no—
284#103—44no—
285#103—44no—
286#103—44no—
287#102—44no—
288#102—44no—
289#102—44no—
290#102—44no—
291C22⋊C8—28no—
292#97—24no—
293#97—24no—
294#97—42no—
295#97—42no—
296C4×C8—42no—
297C4×C8—42no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95C2×D412yes—
310#95C2×D412yes—
311#95C2×D412yes—
312#95C2×D412yes—
313C24—22no—
314C24C22×D411yescharacteristic
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—82no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4—82no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—84no—
341C22×C4#12418yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D412yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C2×#4114yes—
352C22×C4C2511yescharacteristic
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4—84no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—42no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—44no—
409C42—44no—
410C42#12418yes—
411C42C2×#4114yes—
412C42C2×#4114yes—
413C23—22no—
414C23—22no—
415C23—21no—
416C23—21no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23C2×#12412yes—
425C23C23×D411yescharacteristic
426Q8—48no—
427D4—48no—
428D4—88no—
429D4—164no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—28no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—48no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—88no—
468C2×C4—164no—
469C2×C4C2×#12414yes—
470C2×C4C2×#12412yes—
471C2×C4C2×#12414yes—
472C2×C4C22×#4112yes—
473C2×C4#58512yes—
474C2×C4#58512yes—
475C8—44no—
476C8—44no—
477C22—22no—
478C22—24no—
479C22—42no—
480C22—41no—
481C22—41no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22C22×#12411yescharacteristic
490C22#344911yescharacteristic
491C22#344911yescharacteristic
492C22#358512yes—
493C22#358612yes—
494C4—22no—
495C4—24no—
496C4—24no—
497C4—42no—
498C4—42no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—82no—
504C4—84no—
505C4—84no—
506C4—82no—
507C4—82no—
508C4#351214yes—
509C2—42no—
510C2—162no—
511C2#3534111yescharacteristic
512C2#6062112yes—
513C2#6101612yes—
514C2#6101911yescharacteristic
515C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
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
2. #35286C2semidirect
3. #33282C2semidirect
4. #33282C2semidirect
5. #30235C2semidirect
6. #30227C2semidirect
7. #29345C2semidirect
8. #29341C2semidirect
10. #29138C2semidirect
11. #29136C2semidirect
12. #29134C2semidirect
13. #21868C2semidirect
14. #21867C2semidirect
15. #20467C2semidirect

Maximal subgroups

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

subgroupindex
#353432
#352862
#332822
#302352
#302272
#293452
#293412
#293402
#291382
#291362
#291342
#218682
#218672
#204672