Finite Group Databasebeta

← #6500796

Subgroups — #6500796

Order 512.

2134 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2417C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#27614C212yesmaximal, complemented
4#27603C212yesmaximal, complemented
5#27586C212yesmaximal, complemented
6#26204C218yesmaximal, complemented
7#26132C218yesmaximal, complemented
8#24006C212yesmaximal, complemented
9#21759C212yesmaximal, complemented
10#21738C211yesmaximal, characteristic
11C2×#1852C212yesmaximal, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C2×#518—21no—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#484C2214yescomplemented
18C22×#107C2214yescomplemented
19C22×C22⋊C8C2212yes—
20#2417—24no—
21#2417C2218yes—
22C2×#398—21no—
23C2×#396—21no—
24#2349—24no—
25D4×C4⋊C4C2214yes—
26D4×C4⋊C4C2214yes—
27#2316—21no—
28#2315—21no—
29#2315—21no—
30C2×#391C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#386—21no—
33#2285C2212yes—
34C2×#380—21no—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38#2100C2214yescomplemented
39#2095C2214yescomplemented
40#2083C2214yescomplemented
41#2076C2218yescomplemented
42#2072C2214yescomplemented
43#2036C2218yescomplemented
44#1928C2218yescomplemented
45#1921C2218yescomplemented
46#1892C2218yescomplemented
47#1872C2214yescomplemented
48#1872C2214yescomplemented
49#1872C2214yescomplemented
50#1869C2218yes—
51#1852C2214yes—
52#1852C2214yes—
53#1852C2214yescomplemented
54#1852C2218yescomplemented
55#1824C2214yes—
56#1824C2214yescomplemented
57#1817C2212yes—
58#1787C2214yes—
59#1779C2212yes—
60C2×#340C2214yescomplemented
61C23×Q8C2311yescharacteristic
62C23×D4D411yescharacteristic
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C23×C8C2314yes—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8—44no—
71C2×C22⋊Q8C2314yes—
72C2×C22⋊Q8C2314yes—
73#518—48no—
74C2×C4×D4—28no—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4D414yes—
78C2×C4×D4C2314yes—
79C2×#117—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—44no—
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4—22no—
96C22×C22⋊C4—22no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4D411yescharacteristic
101C22×C22⋊C4D411yescharacteristic
102C22×C42C2311yescharacteristic
103#484—48no—
104C2×#107—48no—
105C2×#107C2318yes—
106C2×#107C2318yes—
107C2×#103C2318yes—
108C2×#103C2318yes—
109C2×C22⋊C8—28no—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2314yes—
114#398—28no—
115#396—28no—
116#391—22no—
117#391C2314yes—
118#389—24no—
119#389—24no—
120#389—22no—
121#389C2314yes—
122#386—28no—
123#385—24no—
124#385—24no—
125#385—24no—
126#385—24no—
127#384C2312yes—
128#384C2312yes—
129#382—22no—
130#380—22no—
131#380—22no—
132#380—22no—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—28no—
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×D4Q1618yes—
165C22×D4C2×D414yes—
166C23×C4—21no—
167C23×C4—24no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—82no—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D412yes—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4#4118yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C22×C8—28no—
194C22×C8—44no—
195C22×C8—44no—
196C22×C8C2418yes—
197C22⋊Q8—28no—
198C22⋊Q8—28no—
199C22⋊Q8—28no—
200C22⋊Q8—44no—
201C22⋊Q8—48no—
202C22⋊Q8—48no—
203C4×D4—216no—
204C4×D4—28no—
205C4×D4—28no—
206C4×D4—416no—
207C4×D4—416no—
208#117—48no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2414yes—
242C2×C4⋊C4C2414yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2412yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—48no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4C2×D414yes—
259C2×C22⋊C4C2×D414yes—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42C2×D412yes—
263C2×C42C2412yes—
264#107—216no—
265#107—48no—
266#103—48no—
267#103—48no—
268C22⋊C8—28no—
269C22⋊C8—28no—
270C22⋊C8—44no—
271C22⋊C8—44no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280C24—22no—
281C24—22no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24#12314yes—
291C24C2×Q1614yes—
292C24C2×Q1614yes—
293C24AGL(1,ℤ/8)14yes—
294C24C22×D412yes—
295C24C22×D412yes—
296C24C22×D411yescharacteristic
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—88no—
304C2×D4—28no—
305C2×D4—28no—
306C2×D4—232no—
307C2×D4C2×Q16116yes—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—416no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—48no—
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—42no—
348C22×C4—42no—
349C22×C4—48no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—88no—
356C22×C4—84no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—816no—
366C22×C4#12314yes—
367C22×C4#12318yes—
368C22×C4C2×Q1614yes—
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C2×#4114yes—
375C22×C4C2×#4114yes—
376C22×C4C2×#4114yes—
377C22×C4C2511yescharacteristic
378C2×C8—28no—
379C2×C8—28no—
380C2×C8—28no—
381C2×C8—416no—
382C4⋊C4—24no—
383C4⋊C4—28no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—28no—
387C4⋊C4—24no—
388C4⋊C4—28no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—48no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—84no—
403C4⋊C4H(5,𝔽2)14yes—
404C22⋊C4—28no—
405C22⋊C4—28no—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—84no—
412C42—24no—
413C42—24no—
414C42—48no—
415C23—21no—
416C23—22no—
417C23—21no—
418C23—22no—
419C23—21no—
420C23—22no—
421C23—21no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—28no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23#484116yescomplemented
443C23C2×#12314yes—
444C23C2×#12312yes—
445C23C22×Q1614yes—
446C23C22×Q1612yes—
447C23C22×Q1614yes—
448C23C2×AGL(1,ℤ/8)12yes—
449C23C2×AGL(1,ℤ/8)14yes—
450C23C23×D411yescharacteristic
451Q8—48no—
452Q8—84no—
453D4—232no—
454C2×C4—28no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—48no—
472C2×C4—44no—
473C2×C4—48no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—48no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—84no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—816no—
496C2×C4—84no—
497C2×C4—84no—
498C2×C4—84no—
499C2×C4—164no—
500C2×C4—164no—
501C2×C4—164no—
502C2×C4—164no—
503C2×C4C2×#12312yes—
504C2×C4C2×#12314yes—
505C2×C4C22×Q1614yes—
506C2×C4C22×#4112yes—
507C2×C4C2×H(5,𝔽2)12yes—
508C2×C4#58512yes—
509C8—48no—
510C22—22no—
511C22—24no—
512C22—24no—
513C22—22no—
514C22—22no—
515C22—24no—
516C22—22no—
517C22—24no—
518C22—22no—
519C22—24no—
520C22—22no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—416no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—88no—
531C22C2×#48414yes—
532C22C2×#48418yes—
533C22C22×#12311yescharacteristic
534C22#348411yescharacteristic
535C22#348511yescharacteristic
536C22C23×Q1611yescharacteristic
537C22C22×AGL(1,ℤ/8)11yescharacteristic
538C22#358511yescharacteristic
539C22#358611yescharacteristic
540C4—22no—
541C4—24no—
542C4—24no—
543C4—42no—
544C4—42no—
545C4—44no—
546C4—84no—
547C4—82no—
548C4—82no—
549C4—82no—
550C4—84no—
551C4#350514yes—
552C2—24no—
553C2—28no—
554C2—44no—
555C2#3535111yescharacteristic
556C2C22×#48411yescharacteristic
557C2#6052011yescharacteristic
558C2#6156311yescharacteristic
559C2#6156911yescharacteristic
560C2#6169811yescharacteristic
561C2#6170411yescharacteristic
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. #27614C2semidirect
4. #27603C2semidirect
5. #27586C2semidirect
6. #26204C2semidirect
7. #26132C2semidirect
8. #24006C2semidirect
9. #21759C2semidirect
11. C2×#1852C2semidirect
17. C2×#484C22semidirect
18. C22×#107C22semidirect
38. #2100C22semidirect
39. #2095C22semidirect
40. #2083C22semidirect
41. #2076C22semidirect
42. #2072C22semidirect
43. #2036C22semidirect
44. #1928C22semidirect
45. #1921C22semidirect
46. #1892C22semidirect
47. #1872C22semidirect
48. #1872C22semidirect
49. #1872C22semidirect
53. #1852C22semidirect
54. #1852C22semidirect
56. #1824C22semidirect
60. C2×#340C22semidirect
442. C23#484semidirect

Maximal subgroups

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

subgroupindex
C2×#24172
#341812
#276142
#276032
#275862
#262042
#261322
#240062
#217592
#217382
C2×#18522