Finite Group Databasebeta

← #6500755

Subgroups — #6500755

Order 512.

2444 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#27589C214yesmaximal, complemented
4#24884C218yesmaximal, complemented
5#24880C218yesmaximal, complemented
6#24006C212yesmaximal, complemented
7#24003C212yesmaximal, complemented
8#21944C212yesmaximal, complemented
9#21742C211yesmaximal, characteristic
10#20599C212yesmaximal, complemented
11C24×Q8C2211yescharacteristic
12C22×C22⋊Q8C2212yes—
13C22×C4×D4C2211yescharacteristic
14C2×#484C2214yescomplemented
15C2×#483C2214yescomplemented
16C2×#419C2218yescomplemented
17C2×#407C2212yes—
18#2409—22no—
19#2409—22no—
20#2409C2214yes—
21#2409C2214yes—
22C2×#395—21no—
23C2×#393—21no—
24#2353—22no—
25#2353—22no—
26#2352C2214yes—
27#2352C2214yes—
28#2313—21no—
29#2312—21no—
30#2308—21no—
31C2×#391C2212yes—
32C2×#386—21no—
33C2×#385—21no—
34#2283C2212yes—
35#2280C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39#2273—21no—
40#2097C2218yescomplemented
41#2074C2218yescomplemented
42#2036C2218yescomplemented
43#2035C2218yescomplemented
44#1928C2218yescomplemented
45#1927C2218yescomplemented
46#1872C2214yescomplemented
47#1872C2214yescomplemented
48#1870C2214yes—
49#1870C2214yes—
50#1854C2218yescomplemented
51#1853C2214yes—
52#1852C2214yes—
53#1817C2212yes—
54#1809C2218yescomplemented
55#1787C2214yes—
56#1787C2214yes—
57#1787C2218yescomplemented
58#1779C2212yes—
59C23×Q8—21no—
60C23×Q8—22no—
61C23×Q8—28no—
62C23×Q8—48no—
63C23×Q8C2314yes—
64C23×Q8C2312yes—
65C23×D4D411yescharacteristic
66C24×C4—21no—
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C22×M16C2314yes—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—48no—
73C2×C22⋊Q8C2314yes—
74C2×C22⋊Q8C2314yes—
75C2×C4×D4—28no—
76C2×C4×D4D412yes—
77C2×C4×D4D412yes—
78C2×C4×D4C2312yes—
79C2×C4×D4C2312yes—
80C2×#117C2314yes—
81C22×C4⋊C4—22no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4D411yescharacteristic
91C22×C22⋊C4D411yescharacteristic
92C22×C42C2311yescharacteristic
93#484—48no—
94#483—48no—
95C2×#107C2318yes—
96C2×#106C2318yes—
97#419—416no—
98C2×#103C2318yes—
99C2×#103C2318yes—
100C2×#103C2318yes—
101C2×#103C2318yes—
102#407—28no—
103C2×C22⋊C8C2314yes—
104C2×C22⋊C8C2314yes—
105C2×C22⋊C8C2314yes—
106C2×C22⋊C8C2314yes—
107#395—24no—
108#395—24no—
109#393—24no—
110#393—24no—
111#391—24no—
112#391C2318yes—
113#388—24no—
114#388—24no—
115#387—24no—
116#387—24no—
117#386—24no—
118#386—24no—
119#386—24no—
120#386—24no—
121#385—24no—
122#385—24no—
123#383—22no—
124#382C2314yes—
125#381—22no—
126#381—22no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—22no—
132#380C2314yes—
133#380C2314yes—
134#380C2312yes—
135#380C2312yes—
136#380C2312yes—
137#380C2312yes—
138#379—22no—
139C4×C4⋊C4C2314yes—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—24no—
145#376—22no—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95—21no—
149C2×#95—22no—
150C2×#95—22no—
151#340C2318yes—
152#340C2318yes—
153#336C2318yes—
154C25C2×D411yescharacteristic
155C25C2×D411yescharacteristic
156C22×Q8—28no—
157C22×Q8—22no—
158C22×Q8—24no—
159C22×Q8—216no—
160C22×Q8—28no—
161C22×Q8—28no—
162C22×Q8—28no—
163C22×Q8—44no—
164C22×Q8—48no—
165C22×Q8—48no—
166C22×Q8—464no—
167C22×Q8—416no—
168C22×Q8—44no—
169C22×Q8—48no—
170C22×Q8C2414yes—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—42no—
174C22×D4C2×D412yes—
175C22×D4C2×D412yes—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—48no—
185C23×C4—44no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4#4118yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C2×M16—28no—
199C2×M16—48no—
200C22×C8C2418yes—
201C22⋊Q8—28no—
202C22⋊Q8—28no—
203C22⋊Q8—44no—
204C22⋊Q8—44no—
205C22⋊Q8—48no—
206C22⋊Q8—48no—
207C4×D4—28no—
208C4×D4—28no—
209C4×D4—24no—
210C4×D4—24no—
211C4×D4—24no—
212C4×D4—24no—
213#117—48no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2414yes—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2414yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—48no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4C2×D412yes—
245C2×C22⋊C4C2×D412yes—
246C2×C22⋊C4C2×D412yes—
247C2×C22⋊C4C2×D412yes—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42C2×D412yes—
252C2×C42C2412yes—
253#107—48no—
254#106—48no—
255#103—216no—
256#103—216no—
257#103—48no—
258#103—48no—
259C22⋊C8—28no—
260C22⋊C8—28no—
261C22⋊C8—44no—
262C22⋊C8—44no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—24no—
268#95—28no—
269#95—28no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24#12314yes—
283C24#13714yes—
284C24C22×D412yes—
285C24C22×D411yescharacteristic
286C24C22×D411yescharacteristic
287C24C22×D411yescharacteristic
288C2×Q8—28no—
289C2×Q8—24no—
290C2×Q8—232no—
291C2×Q8—28no—
292C2×Q8—28no—
293C2×Q8—416no—
294C2×Q8—432no—
295C2×Q8—432no—
296C2×Q8—44no—
297C2×Q8—416no—
298C2×Q8—432no—
299C2×Q8—464no—
300C2×Q8—44no—
301C2×Q8—48no—
302C2×Q8H(5,𝔽2)18yes—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—416no—
310C22×C4—24no—
311C22×C4—28no—
312C22×C4—28no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—416no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—416no—
336C22×C4—44no—
337C22×C4—432no—
338C22×C4—44no—
339C22×C4—48no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—88no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4#12314yes—
351C22×C4#12318yes—
352C22×C4C22×D412yes—
353C22×C4C22×D412yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D412yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4114yes—
359C22×C4C2×#4114yes—
360C22×C4C2×#4114yes—
361C22×C4C2511yescharacteristic
362M16—88no—
363C2×C8—28no—
364C2×C8—28no—
365C2×C8—28no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—48no—
373C4⋊C4—48no—
374C4⋊C4—44no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—44no—
384C22⋊C4—48no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—84no—
389C22⋊C4—84no—
390C42—24no—
391C42—22no—
392C42—22no—
393C42—84no—
394C42—84no—
395C23—22no—
396C23—22no—
397C23—24no—
398C23—22no—
399C23—22no—
400C23—28no—
401C23—21no—
402C23—21no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—22no—
407C23—22no—
408C23—21no—
409C23—21no—
410C23—28no—
411C23—44no—
412C23—44no—
413C23—48no—
414C23—48no—
415C23—48no—
416C23—82no—
417C23—82no—
418C23—84no—
419C23C2×#12314yes—
420C23C2×#12312yes—
421C23C2×#13712yes—
422C23C2×#13712yes—
423C23C2×#13712yes—
424C23#57512yes—
425C23C23×D411yescharacteristic
426Q8—216no—
427Q8—416no—
428Q8—432no—
429Q8—48no—
430D4—48no—
431D4—48no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—28no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—28no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—48no—
448C2×C4—44no—
449C2×C4—416no—
450C2×C4—432no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—88no—
470C2×C4—164no—
471C2×C4—164no—
472C2×C4C2×#12312yes—
473C2×C4C2×#12314yes—
474C2×C4C22×#4112yes—
475C2×C4C2×H(5,𝔽2)14yes—
476C8—48no—
477C22—22no—
478C22—22no—
479C22—24no—
480C22—24no—
481C22—24no—
482C22—24no—
483C22—22no—
484C22—22no—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—416no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—44no—
494C22—42no—
495C22—44no—
496C22—42no—
497C22—88no—
498C22#309514yes—
499C22C22×#12311yescharacteristic
500C22#348412yes—
501C22C22×#13711yescharacteristic
502C22C2×#57511yescharacteristic
503C22#358612yes—
504C4—22no—
505C4—24no—
506C4—41no—
507C4—41no—
508C4—48no—
509C4—42no—
510C4—42no—
511C4—84no—
512C4—84no—
513C4—84no—
514C4#349014yes—
515C2—24no—
516C2—44no—
517C2—42no—
518C2—42no—
519C2#3535011yescharacteristic
520C2C2×#309511yescharacteristic
521C2#6051911yescharacteristic
522C2#6157412yes—
523C2#6158312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27589C2semidirect
4. #24884C2semidirect
5. #24880C2semidirect
6. #24006C2semidirect
7. #24003C2semidirect
8. #21944C2semidirect
10. #20599C2semidirect
14. C2×#484C22semidirect
15. C2×#483C22semidirect
16. C2×#419C22semidirect
40. #2097C22semidirect
41. #2074C22semidirect
42. #2036C22semidirect
43. #2035C22semidirect
44. #1928C22semidirect
45. #1927C22semidirect
46. #1872C22semidirect
47. #1872C22semidirect
50. #1854C22semidirect
54. #1809C22semidirect
57. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
#341852
#275892
#248842
#248802
#240062
#240032
#219442
#217422
#205992