Finite Group Databasebeta

← #6633319

Subgroups — #6633319

Order 512.

1727 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37641C211yesmaximal, characteristic, complemented
2#35541C211yesmaximal, characteristic
3#32914C212yesmaximal, complemented
4#31661C214yesmaximal, complemented
5#31649C214yesmaximal, complemented
6#30757C214yesmaximal, complemented
7#30741C214yesmaximal, complemented
8#29726C212yesmaximal, complemented
9#29341C212yesmaximal, complemented
10#25732C212yesmaximal
11#23935C212yesmaximal, complemented
12#23933C212yesmaximal, complemented
13#21918C211yesmaximal, characteristic, complemented
14C2×#535C2211yescharacteristic
15C2×#533C2212yes—
16C2×#518C2211yescharacteristic
17C2×#497C2212yes—
18C2×#492C2212yes—
19C2×#461C2214yes—
20C2×#452C2214yes—
21C2×#450C2214yes—
22#2822—22no—
23#2625—24no—
24#2621C2212yes—
25#2621C2212yes—
26#2582—22no—
27#2568C2214yescomplemented
28#2546C2214yescomplemented
29#2447C2214yes—
30#2445C2214yes—
31#2422—22no—
32#2420—22no—
33#2412—24no—
34#2316C2211yescharacteristic
35#2311C2211yescharacteristic
36#2297C2211yescharacteristic
37C2×#390C2211yescharacteristic
38#2285C2211yescharacteristic
39#2101C2214yescomplemented
40#2074C2212yescomplemented
41#2073C2212yescomplemented
42#2056C2214yescomplemented
43#2055C2214yescomplemented
44#2043C2214yescomplemented
45#2024C2212yescomplemented
46#2023C2214yes—
47#2023C2212yescomplemented
48#2016C2212yescomplemented
49#2016C2212yescomplemented
50#2015C2214yes—
51#2014C2214yes—
52#2013C2214yes—
53#2013C2212yescomplemented
54#2013C2212yescomplemented
55#1961C2212yes—
56#1958C2212yes—
57#1935C2214yescomplemented
58#1933C2214yescomplemented
59#1917C2214yescomplemented
60#1916C2214yescomplemented
61#1880C2214yes—
62#1880C2214yes—
63#1880C2214yescomplemented
64#1878C2214yescomplemented
65#1877C2214yes—
66#1877C2214yes—
67#1873C2214yes—
68#1872C2214yes—
69#1817C2212yes—
70#1804C2214yes—
71#1798C2212yes—
72#1750C2212yes—
73C22×#41C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C22×Q16C2314yes—
76C22×SD16C2314yes—
77C22×SD16C2314yes—
78C22×SD16C2314yes—
79#535—44no—
80#533—48no—
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×#127C2312yes—
86C2×#125—22no—
87C2×#124C2312yes—
88C2×#123—24no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8C2314yes—
91C2×#121—22no—
92C2×#120—22no—
93#518—44no—
94C2×C4×Q8—22no—
95C2×C4×Q8C2314yes—
96C2×C4×D4—22no—
97C2×C4×D4C2314yes—
98C2×#117—22no—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—22no—
104C22×C22⋊C4—22no—
105C22×C42C2311yescharacteristic
106#497—44no—
107#492—44no—
108#461—48no—
109#452—48no—
110#450—48no—
111C2×C4⋊C8C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×C22⋊C8C2314yes—
121C2×#97C2312yes—
122C2×C4×C8C2312yes—
123#400—24no—
124#400—24no—
125#398—24no—
126#398—24no—
127#398—24no—
128#398—24no—
129#397—24no—
130#397—24no—
131#397—24no—
132#397—24no—
133#394—24no—
134#394—24no—
135#391—24no—
136#391—24no—
137#391C2314yes—
138#391C2312yes—
139#391C2314yes—
140#391C2312yes—
141#390—21no—
142#390—21no—
143#390C2312yes—
144#390C2314yes—
145#390C2312yes—
146#389—22no—
147#389—24no—
148#389—24no—
149#389C2314yes—
150#386—22no—
151#386—22no—
152#386—24no—
153#385—24no—
154#384D412yes—
155#382D412yes—
156#382C2312yes—
157#382C2312yes—
158#381—24no—
159#380—22no—
160#380—22no—
161#380D414yes—
162C4×C4⋊C4C2314yes—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165#340C2314yes—
166#340C2314yes—
167#340C2314yes—
168#340C2314yes—
169#336C2312yes—
170#336C2312yes—
171#336C2314yes—
172#336C2312yes—
173#336C2312yes—
174#336C2314yes—
175C25—21no—
176C2×#41—24no—
177C2×#41—42no—
178C2×#41—42no—
179C22×Q8—22no—
180C22×Q8C2414yes—
181C22×Q8C2411yescharacteristic
182C22×D4—24no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—42no—
188C22×D4C2411yescharacteristic
189C22×D4C2412yes—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—24no—
193C23×C4C2×D412yes—
194C23×C4C2411yescharacteristic
195C2×Q16—44no—
196C2×Q16—44no—
197C2×Q16—44no—
198C2×SD16—28no—
199C2×SD16—28no—
200C2×SD16—44no—
201C2×SD16—44no—
202C2×SD16—44no—
203C2×SD16—44no—
204C2×SD16—44no—
205C2×SD16—44no—
206C2×SD16—44no—
207C22×C8C2414yes—
208C22×C8C2414yes—
209C4⋊Q8—44no—
210C4⋊Q8—44no—
211C4⋊Q8—44no—
212C4⋊Q8—42no—
213C4⋊Q8—42no—
214#127—24no—
215#127—42no—
216#125—48no—
217#124—24no—
218#124—42no—
219#123—88no—
220C22⋊Q8—28no—
221C22⋊Q8—28no—
222C22⋊Q8—48no—
223#121—48no—
224#120—48no—
225C4×Q8—44no—
226C4×Q8—44no—
227C4×Q8—48no—
228C4×D4—28no—
229C4×D4—44no—
230C4×D4—48no—
231#117—42no—
232#117—42no—
233#117—42no—
234#117—48no—
235C2×C4⋊C4—21no—
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⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2414yes—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—84no—
263C2×C22⋊C4C2×D412yes—
264C2×C22⋊C4C2×D412yes—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—21no—
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2412yes—
271C2×C42C2412yes—
272C2×C42C2412yes—
273C4⋊C8—44no—
274#103—44no—
275#103—44no—
276#103—44no—
277#103—44no—
278#103—44no—
279#102—44no—
280#102—44no—
281#102—44no—
282C22⋊C8—28no—
283#97—42no—
284C4×C8—42no—
285#95—22no—
286#95—22no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95C2×D414yes—
296#95C2×D414yes—
297C24—24no—
298C24—22no—
299C24—21no—
300C24—42no—
301C24—42no—
302C24—82no—
303C24—82no—
304C24C22×D411yescharacteristic
305#41—88no—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—22no—
325C2×D4—416no—
326C2×D4—42no—
327C2×D4—48no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—84no—
333C2×D4—84no—
334C2×D4H(5,𝔽2)14yes—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—48no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—82no—
353C22×C4C22×D414yes—
354C22×C4C22×D414yes—
355C22×C4C22×D412yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D412yes—
358C22×C4H(5,𝔽2)14yes—
359C22×C4C2511yescharacteristic
360Q16—88no—
361SD16—88no—
362SD16—88no—
363SD16—88no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C4⋊C4—28no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—48no—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—48no—
390C22⋊C4—48no—
391C22⋊C4—42no—
392C22⋊C4—48no—
393C22⋊C4—48no—
394C22⋊C4—44no—
395C22⋊C4—82no—
396C22⋊C4—84no—
397C22⋊C4—84no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—24no—
403C42—24no—
404C42—42no—
405C42—48no—
406C23—22no—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—24no—
411C23—48no—
412C23—44no—
413C23—42no—
414C23—42no—
415C23—44no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23—88no—
422C23—164no—
423C23—162no—
424C23—162no—
425C23#53512yes—
426C23C23×D411yescharacteristic
427Q8—44no—
428Q8—44no—
429Q8—44no—
430Q8—44no—
431Q8—44no—
432Q8—88no—
433D4—28no—
434D4—48no—
435D4—44no—
436D4—44no—
437D4—84no—
438D4—168no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—28no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—84no—
469C2×C4—168no—
470C2×C4—162no—
471C2×C4#53512yes—
472C2×C4#57714yes—
473C2×C4C2×H(5,𝔽2)12yes—
474C2×C4C2×H(5,𝔽2)12yes—
475C2×C4#58512yes—
476C8—44no—
477C8—44no—
478C22—22no—
479C22—24no—
480C22—41no—
481C22—42no—
482C22—41no—
483C22—44no—
484C22—48no—
485C22—84no—
486C22—82no—
487C22—82no—
488C22—81no—
489C22—82no—
490C22—81no—
491C22—82no—
492C22—81no—
493C22—168no—
494C22C2×#53511yescharacteristic
495C22#348411yescharacteristic
496C22#348711yescharacteristic
497C22C2×#57712yes—
498C22#358612yes—
499C4—22no—
500C4—24no—
501C4—42no—
502C4—44no—
503C4—44no—
504C4—81no—
505C4—84no—
506C4—84no—
507C4—82no—
508C4—82no—
509C4#353414yes—
510C2—42no—
511C2—84no—
512C2—161no—
513C2#3671011yescharacteristic
514C2#6081312yes—
515C2#6168012yes—
516C2#6193311yescharacteristic
517C2#6194311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37641C2semidirect
3. #32914C2semidirect
4. #31661C2semidirect
5. #31649C2semidirect
6. #30757C2semidirect
7. #30741C2semidirect
8. #29726C2semidirect
9. #29341C2semidirect
11. #23935C2semidirect
12. #23933C2semidirect
13. #21918C2semidirect
27. #2568C22semidirect
28. #2546C22semidirect
39. #2101C22semidirect
40. #2074C22semidirect
41. #2073C22semidirect
42. #2056C22semidirect
43. #2055C22semidirect
44. #2043C22semidirect
45. #2024C22semidirect
47. #2023C22semidirect
48. #2016C22semidirect
49. #2016C22semidirect
53. #2013C22semidirect
54. #2013C22semidirect
57. #1935C22semidirect
58. #1933C22semidirect
59. #1917C22semidirect
60. #1916C22semidirect
63. #1880C22semidirect
64. #1878C22semidirect

Maximal subgroups

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

subgroupindex
#376412
#355412
#329142
#316612
#316492
#307572
#307412
#297262
#293412
#257322
#239352
#239332
#219182