Finite Group Databasebeta

← #6433883

Subgroups — #6433883

Order 512.

1931 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34635C211yesmaximal, characteristic
2C2×#2621C211yesmaximal, characteristic, complemented
3#30333C218yesmaximal, complemented
4#30331C218yesmaximal, complemented
5#23659C211yesmaximal, characteristic, complemented
6#23631C211yesmaximal, characteristic, complemented
7#23507C211yesmaximal, characteristic, complemented
8#23481C212yesmaximal, complemented
9#23432C211yesmaximal, characteristic, complemented
10#23392C212yesmaximal
11#21237C212yesmaximal, complemented
12#21094C212yesmaximal, complemented
13#20377C211yesmaximal, characteristic, complemented
14C22×#137C2212yes—
15C22×C4⋊Q8C2211yescharacteristic
16C2×#525C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×#117C2211yescharacteristic
19C2×#428C2212yes—
20C2×#417C2212yes—
21C2×#419C2212yes—
22C2×#419C2212yes—
23C2×#419C2212yes—
24C2×#407C2212yes—
25C2×C4×M16C2212yes—
26#2625C2214yes—
27#2624C2214yes—
28#2623—22no—
29#2622—22no—
30#2621—24no—
31#2621C2218yes—
32#2395—24no—
33C2×#398—22no—
34C2×#397—22no—
35#2297C2211yescharacteristic
36#2297C2211yescharacteristic
37#2296C2211yescharacteristic
38#2296C2211yescharacteristic
39C2×#391C2212yes—
40C2×#382C2211yescharacteristic
41#2280C2211yescharacteristic
42#2085C2218yescomplemented
43#2074C2214yescomplemented
44#2073C2214yescomplemented
45#2043C2218yescomplemented
46#2036C2218yes—
47#2035C2218yes—
48#2016C2214yescomplemented
49#2013C2214yescomplemented
50#2001C2218yes—
51#2000C2218yes—
52#1973C2214yescomplemented
53#1971C2214yescomplemented
54#1969C2214yescomplemented
55#1968C2214yescomplemented
56#1935C2218yescomplemented
57#1933C2218yescomplemented
58#1750C2218yes—
59#1736C2212yes—
60#1736C2212yes—
61#1728C2211yescharacteristic
62#1728C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C23×Q8C2312yes—
65C24×C4C2311yescharacteristic
66C2×#137—28no—
67C2×#137—44no—
68C2×#137—44no—
69C22×Q16C2314yes—
70C22×SD16C2314yes—
71C22×M16C2312yes—
72C22×M16C2312yes—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—44no—
75C2×C4⋊Q8C2314yes—
76#525—44no—
77C2×#123—21no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—44no—
80C2×C22⋊Q8—44no—
81C2×C22⋊Q8C2318yes—
82C2×C22⋊Q8C2312yes—
83C2×#121C2312yes—
84C2×#120—21no—
85C2×C4×D4—22no—
86C2×#117—44no—
87C2×#117D414yes—
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C2×#117C2314yes—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4D412yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4—22no—
97C22×C22⋊C4D412yes—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#428—44no—
101C2×#107C2314yes—
102C2×#106C2314yes—
103#419—28no—
104#419—44no—
105#419—44no—
106#417—44no—
107C2×#103C2318yes—
108C2×#103C2314yes—
109C2×#103C2318yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#102C2314yes—
113#407—28no—
114C2×C22⋊C8C2318yes—
115C4×M16—44no—
116C2×#97C2314yes—
117C2×C4×C8C2314yes—
118#400—24no—
119#399—24no—
120#398—28no—
121#398—28no—
122#398—24no—
123#397—24no—
124#397—28no—
125#397—28no—
126#397—24no—
127#397—24no—
128#394—24no—
129#394—24no—
130#391—24no—
131#391C2312yes—
132#391C2314yes—
133#391C2314yes—
134#391C2312yes—
135#390C2312yes—
136#390C2312yes—
137#389C2314yes—
138#389C2314yes—
139#386—22no—
140#386—22no—
141#385—22no—
142#385—22no—
143#382—22no—
144#382C2314yes—
145#381—21no—
146#380C2314yes—
147#379—21no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4—22no—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—22no—
152C4×C22⋊C4—22no—
153C2×#95—22no—
154C2×#95—24no—
155#340C2318yes—
156#340C2318yes—
157#336C2314yes—
158#336C2314yes—
159C25C2×D411yescharacteristic
160C2×#41—24no—
161C2×#41—24no—
162C2×#41—22no—
163C2×#41—22no—
164C22×Q8—22no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8C2414yes—
172C22×Q8C2411yescharacteristic
173C22×D4—21no—
174C22×D4—22no—
175C22×D4C2411yescharacteristic
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189#137—432no—
190C2×Q16—28no—
191C2×Q16—44no—
192C2×Q16—44no—
193C2×SD16—28no—
194C2×SD16—44no—
195C2×SD16—44no—
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—24no—
201C2×M16#4118yes—
202C22×C8C2414yes—
203C22×C8C2414yes—
204C4⋊Q8—48no—
205C4⋊Q8—48no—
206#123—44no—
207C22⋊Q8—48no—
208C22⋊Q8—48no—
209C22⋊Q8—44no—
210C22⋊Q8—416no—
211#121—44no—
212#120—44no—
213C4×D4—48no—
214#117—24no—
215#117—24no—
216#117—22no—
217#117—22no—
218#117—48no—
219#117—44no—
220#117—44no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2414yes—
235C2×C4⋊C4C2412yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2412yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—48no—
249C2×C22⋊C4C2×D414yes—
250C2×C22⋊C4C2×D414yes—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2412yes—
259C2×C42C2412yes—
260#107—44no—
261#106—44no—
262#103—216no—
263#103—44no—
264#103—44no—
265#103—44no—
266#103—48no—
267#102—44no—
268C22⋊C8—216no—
269#97—44no—
270C4×C8—44no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—21no—
283C24—44no—
284C24C22×D412yes—
285C24C22×D411yescharacteristic
286#41—48no—
287#41—48no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—22no—
291C2×Q8—21no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—21no—
298C2×Q8—44no—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—48no—
302C2×Q8—48no—
303C2×Q8—48no—
304C2×Q8—48no—
305C2×D4—22no—
306C2×D4—21no—
307C2×D4—22no—
308C2×D4—21no—
309C2×D4—44no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—84no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—48no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4C22×D414yes—
345C22×C4C22×D414yes—
346C22×C4C22×D412yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C2×#4114yes—
349C22×C4H(5,𝔽2)14yes—
350C22×C4H(5,𝔽2)12yes—
351C22×C4H(5,𝔽2)12yes—
352C22×C4C2511yescharacteristic
353Q16—416no—
354SD16—416no—
355M16—216no—
356M16—48no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8C2×#4118yes—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—416no—
372C4⋊C4—416no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—416no—
381C22⋊C4—44no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—84no—
386C22⋊C4—84no—
387C22⋊C4—84no—
388C42—24no—
389C42—24no—
390C42—22no—
391C42—22no—
392C42—24no—
393C42—42no—
394C42—42no—
395C42—42no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—48no—
403C23—44no—
404C23—44no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—84no—
409C23—84no—
410C23#53212yes—
411C23#53412yes—
412C23#53512yes—
413C23C23×D411yescharacteristic
414Q8—44no—
415Q8—42no—
416Q8—44no—
417Q8—44no—
418Q8—44no—
419Q8—42no—
420Q8—88no—
421D4—42no—
422D4—42no—
423D4—88no—
424D4—84no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—48no—
435C2×C4—44no—
436C2×C4—48no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—48no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—84no—
454C2×C4—88no—
455C2×C4—88no—
456C2×C4—84no—
457C2×C4#53212yes—
458C2×C4#53412yes—
459C2×C4#53512yes—
460C2×C4#53818yes—
461C2×C4#54018yes—
462C2×C4C22×#4112yes—
463C2×C4C2×H(5,𝔽2)11yescharacteristic
464C2×C4C2×H(5,𝔽2)12yes—
465C2×C4C2×H(5,𝔽2)11yescharacteristic
466C8—28no—
467C8—44no—
468C22—24no—
469C22—24no—
470C22—24no—
471C22—42no—
472C22—44no—
473C22—42no—
474C22—42no—
475C22—44no—
476C22—88no—
477C22—81no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22#266114yes—
482C22C2×#53211yescharacteristic
483C22C2×#53411yescharacteristic
484C22C2×#53511yescharacteristic
485C22#358614yes—
486C4—42no—
487C4—44no—
488C4—44no—
489C4—81no—
490C4—84no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4—84no—
495C4#266114yes—
496C4C2×#53812yes—
497C4C2×#54012yes—
498C2—24no—
499C2—82no—
500C2—161no—
501C2C2×#266111yescharacteristic
502C2#6070512yes—
503C2#6075412yes—
504C2#6080712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2621C2semidirect
3. #30333C2semidirect
4. #30331C2semidirect
5. #23659C2semidirect
6. #23631C2semidirect
7. #23507C2semidirect
8. #23481C2semidirect
9. #23432C2semidirect
11. #21237C2semidirect
12. #21094C2semidirect
13. #20377C2semidirect
42. #2085C22semidirect
43. #2074C22semidirect
44. #2073C22semidirect
45. #2043C22semidirect
48. #2016C22semidirect
49. #2013C22semidirect
52. #1973C22semidirect
53. #1971C22semidirect
54. #1969C22semidirect
55. #1968C22semidirect
56. #1935C22semidirect
57. #1933C22semidirect

Maximal subgroups

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

subgroupindex
#346352
C2×#26212
#303332
#303312
#236592
#236312
#235072
#234812
#234322
#233922
#212372
#210942
#203772