Finite Group Databasebeta

← #6436294

Subgroups — #6436294

Order 512.

1459 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#35294C211yesmaximal, characteristic, complemented
2#34280C211yesmaximal, characteristic
3#30179C214yesmaximal, complemented
4#30098C212yesmaximal, complemented
5#30076C212yesmaximal, complemented
6#29981C212yesmaximal, complemented
7#29949C212yesmaximal, complemented
8#29870C212yesmaximal, complemented
9#29700C212yesmaximal
10#23845C212yesmaximal, complemented
11#23828C212yesmaximal, complemented
12#23760C212yesmaximal, complemented
13#23755C212yesmaximal, complemented
14#21619C214yesmaximal, complemented
15#20784C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C2×#478C2214yes—
18#2868C2211yescharacteristic
19#2859C2212yes—
20#2859C2211yescharacteristic
21#2629C2212yes—
22#2628C2212yes—
23#2570—22no—
24#2550—22no—
25#2441—22no—
26#2439—22no—
27#2374—22no—
28#2364—22no—
29#2363—22no—
30#2359—22no—
31#2331C2212yes—
32#2330C2211yescharacteristic
33#2330C2212yes—
34#2328C2211yescharacteristic
35#2324C2212yes—
36#2314C2211yescharacteristic
37C4×#127C2211yescharacteristic
38C4×#125C2211yescharacteristic
39C4×#121C2212yes—
40#2293C2212yes—
41#2285C2212yes—
42#2275C2212yes—
43C4×#117C2211yescharacteristic
44#2101C2214yes—
45#2100C2212yes—
46#2100C2212yes—
47#2094C2212yes—
48#2094C2212yes—
49#2085C2214yes—
50#2084C2214yes—
51#2084C2212yes—
52#2083C2212yes—
53#2077C2214yes—
54#2077C2214yes—
55#2073C2214yes—
56#2071C2212yes—
57#2071C2212yes—
58#2059C2214yes—
59#2058C2214yes—
60#2041C2214yes—
61#2034C2214yes—
62#2033C2214yes—
63#2023C2214yes—
64#1989C2214yes—
65#1948C2214yes—
66#1919C2214yes—
67#1918C2214yes—
68#1864C2214yes—
69#1864C2214yes—
70#1839C2212yes—
71#1831C2212yes—
72#1818C2212yes—
73#1802C2214yes—
74#1798C2212yes—
75#1776C2214yes—
76C4×#106C2214yes—
77#1768C2212yes—
78#1762C2214yes—
79C4×#102C2214yes—
80#1749C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82#1746C2212yes—
83#540—28no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2312yes—
86C2×#127C2311yescharacteristic
87C2×#125—22no—
88C2×#125C2312yes—
89C2×#125C2311yescharacteristic
90C2×#123—22no—
91C2×#121—22no—
92C2×#121C2312yes—
93C2×C4×D4—22no—
94C2×C4×D4—22no—
95C2×C4×D4C2312yes—
96C2×C4×D4C2312yes—
97C2×C4×D4C2312yes—
98C2×#117—22no—
99C2×#117—22no—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C42C2311yescharacteristic
103#478—216no—
104C2×#107C2314yes—
105C2×#106C2314yes—
106C2×#106C2314yes—
107C2×#106C2314yes—
108C2×C4⋊C8C2314yes—
109C2×#102C2314yes—
110C2×#102C2314yes—
111C2×#102C2314yes—
112C2×#102C2314yes—
113C2×C22⋊C8C2312yes—
114C2×C22⋊C8C2312yes—
115C2×#97C2312yes—
116C2×C4×C8C2312yes—
117#400—22no—
118#399—22no—
119#398—24no—
120#396—24no—
121#395—22no—
122#393—22no—
123#391—24no—
124#390—22no—
125#390—22no—
126#390C2312yes—
127#390C2312yes—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—24no—
134#389C2312yes—
135#389C2312yes—
136#389C2314yes—
137#389C2314yes—
138#388—22no—
139#388—22no—
140#388—22no—
141#388—22no—
142#388—22no—
143#387—22no—
144#387—22no—
145#387—24no—
146#387—24no—
147#387—22no—
148#386—22no—
149#386—22no—
150#386—22no—
151#386—22no—
152#385—22no—
153#384—22no—
154#384C2311yescharacteristic
155#384C2312yes—
156#384C2312yes—
157#384C2312yes—
158#382—22no—
159#382—22no—
160#382C2311yescharacteristic
161#382C2312yes—
162#380—22no—
163#380C2312yes—
164#380C2314yes—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4D412yes—
168C4×C4⋊C4D412yes—
169C4×C4⋊C4C2312yes—
170C4×C4⋊C4C2312yes—
171C4×C4⋊C4C2314yes—
172C4×C4⋊C4C2312yes—
173C4×C22⋊C4—22no—
174C4×C22⋊C4—22no—
175C4×C22⋊C4—22no—
176C4×C22⋊C4—22no—
177C4×C22⋊C4—22no—
178C4×C22⋊C4D412yes—
179C4×C22⋊C4D412yes—
180#376C2311yescharacteristic
181#376C2311yescharacteristic
182C2×#95—22no—
183C43C2311yescharacteristic
184C43C2311yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195#336C2312yes—
196#336C2312yes—
197#336C2314yes—
198C22×Q8—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4C2412yes—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4C2×D412yes—
206C23×C4C2411yescharacteristic
207C22×C8C2414yes—
208C22×C8C2414yes—
209C4⋊Q8—44no—
210C4⋊Q8—84no—
211#127—24no—
212#125—24no—
213#125—28no—
214#125—84no—
215#123—216no—
216#121—216no—
217#121—24no—
218#121—24no—
219C4×D4—24no—
220C4×D4—28no—
221C4×D4—44no—
222C4×D4—84no—
223C4×D4—84no—
224C4×D4#4118yes—
225#117—24no—
226#117—42no—
227#117—84no—
228#117—84no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2412yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4C2×D412yes—
261C2×C22⋊C4C2×D411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2411yescharacteristic
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279#107—28no—
280#106—28no—
281#106—44no—
282#106—44no—
283C4⋊C8—28no—
284#102—28no—
285#102—28no—
286#102—44no—
287#102—44no—
288C22⋊C8—42no—
289C22⋊C8—42no—
290#97—24no—
291C4×C8—24no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—22no—
305#95C2×D412yes—
306#95C2×D412yes—
307#95C2×D412yes—
308#95C2×D412yes—
309C24—22no—
310C24C22×D411yescharacteristic
311C2×Q8—44no—
312C2×Q8—84no—
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—22no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—82no—
319C2×D4—82no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4C2×#4114yes—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—28no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D412yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D412yes—
350C22×C4C22×D412yes—
351C22×C4C22×D412yes—
352C22×C4C2511yescharacteristic
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C4⋊C4—28no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—48no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C4⋊C4—84no—
382C4⋊C4C2×#4114yes—
383C22⋊C4—28no—
384C22⋊C4—28no—
385C22⋊C4—24no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—84no—
395C22⋊C4—82no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—42no—
404C42—42no—
405C42—82no—
406C42—84no—
407C42—82no—
408C42—82no—
409C42C2×#4114yes—
410C422−1+414yes—
411C23—22no—
412C23—21no—
413C23—24no—
414C23—21no—
415C23—21no—
416C23—22no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23C23×D411yescharacteristic
424Q8—164no—
425D4—28no—
426D4—44no—
427D4—88no—
428D4—88no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—24no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—48no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—88no—
467C2×C4—164no—
468C2×C4—164no—
469C2×C4#52514yes—
470C2×C4#53614yes—
471C2×C4#57714yes—
472C2×C4C22×#4112yes—
473C2×C4C2×2−1+412yes—
474C2×C4#58512yes—
475C8—44no—
476C8—44no—
477C22—41no—
478C22—44no—
479C22—41no—
480C22—41no—
481C22—44no—
482C22—42no—
483C22—44no—
484C22—42no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22C2×#52511yescharacteristic
490C22C2×#53611yescharacteristic
491C22#344911yescharacteristic
492C22C2×#57712yes—
493C22#358512yes—
494C4—24no—
495C4—24no—
496C4—24no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—42no—
503C4—44no—
504C4—82no—
505C4—84no—
506C4—84no—
507C4—82no—
508C4#346114yes—
509C2—81no—
510C2—84no—
511C2#3462711yescharacteristic
512C2#6059712yes—
513C2#6087312yes—
514C2#6101211yescharacteristic
515C2#6101911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
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. #35294C2semidirect
3. #30179C2semidirect
4. #30098C2semidirect
5. #30076C2semidirect
6. #29981C2semidirect
7. #29949C2semidirect
8. #29870C2semidirect
10. #23845C2semidirect
11. #23828C2semidirect
12. #23760C2semidirect
13. #23755C2semidirect
14. #21619C2semidirect
15. #20784C2semidirect

Maximal subgroups

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

subgroupindex
#352942
#342802
#301792
#300982
#300762
#299812
#299492
#298702
#297002
#238452
#238282
#237602
#237552
#216192
#207842