Finite Group Databasebeta

← #6500818

Subgroups — #6500818

Order 512.

2036 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2415C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#27613C212yesmaximal, complemented
4#27600C212yesmaximal, complemented
5#27586C212yesmaximal, complemented
6#26216C218yesmaximal, complemented
7#26144C218yesmaximal, complemented
8#24003C212yesmaximal, complemented
9#21765C212yesmaximal, complemented
10#21742C211yesmaximal, characteristic
11#20846C212yesmaximal, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C2×#518—21no—
15C22×C4×D4C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C2×#483C2214yescomplemented
18C2×#428C2214yescomplemented
19C2×#407C2212yes—
20#2415—24no—
21#2415C2218yes—
22C2×#397—21no—
23C2×#394—21no—
24#2349—24no—
25D4×C4⋊C4C2214yes—
26D4×C4⋊C4C2214yes—
27#2316—21no—
28#2315—21no—
29#2315—21no—
30C2×#391C2211yescharacteristic
31C2×#389—21no—
32C2×#389C2211yescharacteristic
33C2×#388—21no—
34#2285C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38#2100C2214yescomplemented
39#2095C2214yescomplemented
40#2084C2214yescomplemented
41#2075C2218yescomplemented
42#2072C2214yescomplemented
43#2035C2218yescomplemented
44#1928C2218yescomplemented
45#1919C2218yescomplemented
46#1894C2218yescomplemented
47#1872C2214yescomplemented
48#1872C2214yescomplemented
49#1872C2214yescomplemented
50#1870C2218yes—
51#1854C2218yescomplemented
52#1853C2214yes—
53#1852C2214yes—
54#1852C2214yescomplemented
55#1823C2214yes—
56#1823C2214yescomplemented
57#1817C2212yes—
58#1787C2214yes—
59#1779C2212yes—
60#1736C2214yescomplemented
61C23×Q8C2311yescharacteristic
62C23×D4D411yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C22×M16C2314yes—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—44no—
70C2×C22⋊Q8C2314yes—
71C2×C22⋊Q8C2314yes—
72#518—48no—
73C2×C4×D4—28no—
74C2×C4×D4—24no—
75C2×C4×D4—24no—
76C2×C4×D4D414yes—
77C2×C4×D4C2314yes—
78C2×#117—22no—
79C2×#117—22no—
80C2×#117—44no—
81C2×#117C2314yes—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4D411yescharacteristic
100C22×C22⋊C4D411yescharacteristic
101C22×C42—21no—
102C22×C42C2311yescharacteristic
103#483—48no—
104#428—48no—
105C2×#107C2318yes—
106C2×#106C2318yes—
107C2×#103C2318yes—
108C2×#103C2318yes—
109#407—28no—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2314yes—
114#397—28no—
115#394—28no—
116#391—22no—
117#391C2314yes—
118#389—24no—
119#389—24no—
120#389—28no—
121#389—22no—
122#389C2314yes—
123#388—28no—
124#385—24no—
125#385—24no—
126#385—24no—
127#385—24no—
128#384C2312yes—
129#384C2312yes—
130#382—22no—
131#380—22no—
132#380—22no—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380C2314yes—
138#380C2314yes—
139#380C2314yes—
140#380C2314yes—
141C4×C4⋊C4—22no—
142C4×C4⋊C4C2314yes—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—21no—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95—21no—
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#336C2314yes—
154#336C2314yes—
155C25C2×D411yescharacteristic
156C25C2×D411yescharacteristic
157C22×Q8—21no—
158C22×Q8—24no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8C2412yes—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4C2×D414yes—
166C23×C4—21no—
167C23×C4—24no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4#4118yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C2×M16—28no—
190C2×M16—44no—
191C2×M16—44no—
192C22×C8C2418yes—
193C22⋊Q8—28no—
194C22⋊Q8—28no—
195C22⋊Q8—28no—
196C22⋊Q8—44no—
197C22⋊Q8—48no—
198C22⋊Q8—48no—
199C4×D4—216no—
200C4×D4—28no—
201C4×D4—28no—
202C4×D4—416no—
203C4×D4—416no—
204#117—28no—
205#117—44no—
206#117—48no—
207#117—48no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2414yes—
235C2×C4⋊C4C2414yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2412yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—48no—
254C2×C22⋊C4C2×D414yes—
255C2×C22⋊C4C2×D414yes—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—44no—
260C2×C42C2×D412yes—
261C2×C42C2412yes—
262#107—48no—
263#106—216no—
264#103—48no—
265#103—48no—
266C22⋊C8—28no—
267C22⋊C8—28no—
268C22⋊C8—44no—
269C22⋊C8—44no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24#12314yes—
291C24AGL(1,ℤ/8)14yes—
292C24C22×D412yes—
293C24C22×D412yes—
294C24C22×D411yescharacteristic
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—88no—
302C2×D4—232no—
303C2×D4—28no—
304C2×D4—28no—
305C2×D4—28no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—416no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—88no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4#12314yes—
361C22×C4#12318yes—
362C22×C4#13714yes—
363C22×C4C22×D412yes—
364C22×C4C22×D412yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C2×#4114yes—
369C22×C4C2×#4114yes—
370C22×C4C2×#4114yes—
371C22×C4C2511yescharacteristic
372M16—416no—
373C2×C8—28no—
374C2×C8—28no—
375C2×C8—28no—
376C4⋊C4—24no—
377C4⋊C4—28no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—28no—
381C4⋊C4—24no—
382C4⋊C4—28no—
383C4⋊C4—48no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4H(5,𝔽2)14yes—
397C22⋊C4—28no—
398C22⋊C4—28no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—48no—
406C42—24no—
407C42—24no—
408C42—48no—
409C42—48no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—28no—
427C23—28no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23C2×#12314yes—
441C23C2×#12312yes—
442C23C2×AGL(1,ℤ/8)14yes—
443C23C2×AGL(1,ℤ/8)12yes—
444C23C2×#13712yes—
445C23C23×D411yescharacteristic
446Q8—48no—
447Q8—84no—
448D4—232no—
449C2×C4—28no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—48no—
467C2×C4—44no—
468C2×C4—48no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—48no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—84no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4—164no—
494C2×C4—164no—
495C2×C4—164no—
496C2×C4—164no—
497C2×C4C2×#12312yes—
498C2×C4C2×#12314yes—
499C2×C4C2×#13714yes—
500C2×C4C22×#4112yes—
501C2×C4C2×H(5,𝔽2)12yes—
502C2×C4#58512yes—
503C8—48no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—88no—
525C22—88no—
526C22#309214yes—
527C22C22×#12311yescharacteristic
528C22#348411yescharacteristic
529C22#348511yescharacteristic
530C22C22×AGL(1,ℤ/8)11yescharacteristic
531C22C22×#13711yescharacteristic
532C22#358611yescharacteristic
533C22#358611yescharacteristic
534C4—22no—
535C4—24no—
536C4—24no—
537C4—42no—
538C4—42no—
539C4—44no—
540C4—84no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—84no—
545C4#350514yes—
546C2—24no—
547C2—44no—
548C2—44no—
549C2#3535111yescharacteristic
550C2C2×#309211yescharacteristic
551C2#6051911yescharacteristic
552C2#6157211yescharacteristic
553C2#6158611yescharacteristic
554C2#6170411yescharacteristic
555C2#6172311yescharacteristic
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. #27613C2semidirect
4. #27600C2semidirect
5. #27586C2semidirect
6. #26216C2semidirect
7. #26144C2semidirect
8. #24003C2semidirect
9. #21765C2semidirect
11. #20846C2semidirect
17. C2×#483C22semidirect
18. C2×#428C22semidirect
38. #2100C22semidirect
39. #2095C22semidirect
40. #2084C22semidirect
41. #2075C22semidirect
42. #2072C22semidirect
43. #2035C22semidirect
44. #1928C22semidirect
45. #1919C22semidirect
46. #1894C22semidirect
47. #1872C22semidirect
48. #1872C22semidirect
49. #1872C22semidirect
51. #1854C22semidirect
54. #1852C22semidirect
56. #1823C22semidirect
60. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24152
#341812
#276132
#276002
#275862
#262162
#261442
#240032
#217652
#217422
#208462