Finite Group Databasebeta

← #6475730

Subgroups — #6475730

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C212yesmaximal
2#33210C211yesmaximal, characteristic, complemented
3#33176C211yesmaximal, characteristic, complemented
4#33049C212yesmaximal, complemented
5#29743C214yesmaximal
6#29479C218yesmaximal, complemented
7#27892C212yesmaximal, complemented
8#27876C212yesmaximal, complemented
9#25672C212yesmaximal, complemented
10#25600C212yesmaximal, complemented
11#22189C214yesmaximal, complemented
12C4×#407C211yesmaximal, characteristic
13C2×#533C2212yes—
14C23×C42C2211yescharacteristic
15C2×#498C2211yescharacteristic, complemented
16C2×#494C2211yescharacteristic, complemented
17C2×#487C2212yescomplemented
18C2×#470C2214yes—
19C2×#407C2212yes—
20C2×C4×M16C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2857C2212yes—
23#2857C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26#2855—22no—
27#2855C2214yes—
28#2851—22no—
29#2851—22no—
30#2848—22no—
31#2848C2214yes—
32C4×C22⋊Q8—22no—
33C4×C22⋊Q8—22no—
34C4×C22⋊Q8C2214yes—
35C4×C22⋊Q8C2214yes—
36#2296—22no—
37#2285C2212yes—
38#2280—22no—
39#2077C2218yes—
40#2036C2218yes—
41#2035C2218yes—
42#2001C2218yes—
43#1990C2212yescomplemented
44#1990C2212yescomplemented
45#1971C2214yescomplemented
46#1956C2212yescomplemented
47#1956C2212yescomplemented
48#1943C2214yescomplemented
49#1935C2218yescomplemented
50#1928C2218yescomplemented
51#1919C2218yescomplemented
52#1894C2218yes—
53#1878C2218yescomplemented
54#1865C2218yescomplemented
55#1845C2211yescharacteristic, complemented
56#1844C2211yescharacteristic, complemented
57#1834C2212yescomplemented
58#1809C2214yes—
59#1749C2214yes—
60C4×C22⋊C8C2212yes—
61C4×C22⋊C8C2212yes—
62#1728C2212yes—
63C24×C4D412yes—
64C24×C4C2311yescharacteristic
65C22×Q16C2314yes—
66C22×M16C2312yes—
67C22×M16C2312yes—
68#533—44no—
69#533—44no—
70C2×C4⋊Q8C2312yes—
71C2×C4⋊Q8C2312yes—
72C2×#125—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8C2314yes—
77C2×C22⋊Q8C2314yes—
78C2×C22⋊Q8C2314yes—
79C2×C4×Q8—22no—
80C2×C4×Q8C2314yes—
81C2×#117—22no—
82C2×#117—22no—
83C22×C42—42no—
84C22×C42—42no—
85C22×C42D412yes—
86C22×C42D412yes—
87C22×C42D412yes—
88C22×C42C2311yescharacteristic
89C22×C42C2312yes—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92#498—42no—
93#494—42no—
94#487—44no—
95#470—216no—
96C2×#107C2314yes—
97C2×#106C2314yes—
98C2×#106C2314yes—
99C2×#103C2318yes—
100C2×#103C2318yes—
101C2×#103C2318yes—
102#407—28no—
103C2×C22⋊C8C2314yes—
104C2×C22⋊C8C2314yes—
105C4×M16—42no—
106C4×M16—42no—
107C2×#97C2312yes—
108C2×#97C2312yes—
109C2×C4×C8C2312yes—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#391—22no—
113#391—22no—
114#391C2314yes—
115#391C2314yes—
116#389—22no—
117#389—22no—
118#389—22no—
119#389—22no—
120#389—22no—
121#389—22no—
122#389C2314yes—
123#389C2314yes—
124#387—24no—
125#387—24no—
126#387—24no—
127#387—24no—
128#385—24no—
129#385—24no—
130#385—24no—
131#385—24no—
132#385—24no—
133#385—24no—
134#384C2312yes—
135#384C2312yes—
136#382—22no—
137#381—24no—
138#380—22no—
139#380—22no—
140#380—22no—
141#380—22no—
142#380—22no—
143#380—22no—
144#380—22no—
145#380C2314yes—
146#380C2314yes—
147#380C2314yes—
148#379—24no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4C2314yes—
153C4×C22⋊C4—24no—
154C4×C22⋊C4—24no—
155#340C2318yes—
156#340C2318yes—
157#340C2318yes—
158#336C2314yes—
159#336C2314yes—
160C25C2×D411yescharacteristic
161C22×Q8—22no—
162C22×Q8C2414yes—
163C23×C4—28no—
164C23×C4—24no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—42no—
168C23×C4—41no—
169C23×C4—41no—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4C2412yes—
179C23×C4C2411yescharacteristic
180C2×Q16—28no—
181C2×Q16—44no—
182C2×Q16—44no—
183C2×M16—24no—
184C2×M16—42no—
185C2×M16—42no—
186C2×M16—42no—
187C2×M16—42no—
188C2×M16#4118yes—
189C22×C8C2414yes—
190C22×C8C2412yes—
191C22×C8C2412yes—
192C4⋊Q8—24no—
193C4⋊Q8—24no—
194C4⋊Q8—24no—
195C4⋊Q8—24no—
196#125—48no—
197C22⋊Q8—216no—
198C22⋊Q8—48no—
199C22⋊Q8—44no—
200C22⋊Q8—44no—
201C22⋊Q8—48no—
202C22⋊Q8—48no—
203C22⋊Q8—48no—
204C4×Q8—44no—
205C4×Q8—44no—
206C4×Q8—48no—
207#117—48no—
208#117—48no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4C2414yes—
223C2×C4⋊C4C2414yes—
224C2×C4⋊C4C2414yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—41no—
243C2×C42—41no—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D411yescharacteristic
251C2×C42C2412yes—
252C2×C42C2411yescharacteristic
253C2×C42C2411yescharacteristic
254#107—44no—
255#106—28no—
256#106—44no—
257#103—216no—
258#103—48no—
259#103—48no—
260C22⋊C8—28no—
261C22⋊C8—28no—
262#97—42no—
263#97—42no—
264C4×C8—24no—
265C4×C8—22no—
266C4×C8—22no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277C24—22no—
278C24—24no—
279C24—41no—
280C24—41no—
281C24—42no—
282C24C22×D411yescharacteristic
283C24C22×D412yes—
284C2×Q8—28no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—28no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—44no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—432no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4#13718yes—
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C22×D412yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D412yes—
334C22×C4C22×D412yes—
335C22×C4C2×#4114yes—
336C22×C4C2511yescharacteristic
337Q16—88no—
338M16—48no—
339M16—48no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8C2×#4114yes—
348C2×C8C2×#4114yes—
349C4⋊C4—28no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—28no—
353C4⋊C4—48no—
354C4⋊C4—48no—
355C4⋊C4—48no—
356C4⋊C4—48no—
357C4⋊C4—48no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—48no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C22⋊C4—48no—
367C22⋊C4—48no—
368C22⋊C4—48no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—48no—
372C42—24no—
373C42—24no—
374C42—24no—
375C42—21no—
376C42—21no—
377C42—22no—
378C42—22no—
379C42—41no—
380C42—42no—
381C42—41no—
382C42—42no—
383C42—48no—
384C42—48no—
385C42H(5,𝔽2)12yes—
386C42H(5,𝔽2)12yes—
387C23—22no—
388C23—22no—
389C23—21no—
390C23—22no—
391C23—21no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—48no—
398C23—44no—
399C23—48no—
400C23—44no—
401C23—44no—
402C23#53212yes—
403C23C2×#13714yes—
404C23C23×D411yescharacteristic
405Q8—48no—
406Q8—48no—
407Q8—88no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—216no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—48no—
428C2×C4—44no—
429C2×C4—48no—
430C2×C4—48no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4#53212yes—
446C2×C4#53414yes—
447C2×C4C2×#13714yes—
448C2×C4C2×#13714yes—
449C2×C4#57514yes—
450C2×C4C22×#4112yes—
451C2×C4C2×H(5,𝔽2)11yescharacteristic
452C2×C4C2×H(5,𝔽2)11yescharacteristic
453C2×C4#58512yes—
454C8—24no—
455C8—24no—
456C8—44no—
457C22—24no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22—22no—
462C22—416no—
463C22—42no—
464C22—41no—
465C22—42no—
466C22—41no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22C2×#53211yescharacteristic
471C22C2×#53411yescharacteristic
472C22#344911yescharacteristic
473C22C22×#13712yes—
474C22C2×#57512yes—
475C4—24no—
476C4—22no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—84no—
481C4—84no—
482C4—84no—
483C4—84no—
484C4#319618yes—
485C4#350512yes—
486C4#350612yes—
487C2—42no—
488C2—44no—
489C2#3503211yescharacteristic
490C2#6070112yes—
491C2C2×#319612yes—
492C2#6097111yescharacteristic
493C2#6097211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33210C2semidirect
3. #33176C2semidirect
4. #33049C2semidirect
6. #29479C2semidirect
7. #27892C2semidirect
8. #27876C2semidirect
9. #25672C2semidirect
10. #25600C2semidirect
11. #22189C2semidirect
15. C2×#498C22semidirect
16. C2×#494C22semidirect
17. C2×#487C22semidirect
43. #1990C22semidirect
44. #1990C22semidirect
45. #1971C22semidirect
46. #1956C22semidirect
47. #1956C22semidirect
48. #1943C22semidirect
49. #1935C22semidirect
50. #1928C22semidirect
51. #1919C22semidirect
53. #1878C22semidirect
54. #1865C22semidirect
55. #1845C22semidirect
56. #1844C22semidirect
57. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#366252
#332102
#331762
#330492
#297432
#294792
#278922
#278762
#256722
#256002
#221892
C4×#4072