Finite Group Databasebeta

← #6508713

Subgroups — #6508713

Order 512.

2318 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2C22×#483C211yesmaximal, characteristic, complemented
3C2×#2501C211yesmaximal, characteristic
4#34212C211yesmaximal, characteristic, complemented
5#28120C214yesmaximal, complemented
6#27647C214yesmaximal, complemented
7#25204C218yesmaximal, complemented
8#24578C214yesmaximal, complemented
9#24528C214yesmaximal, complemented
10#21930C211yesmaximal, characteristic, complemented
11#21758C211yesmaximal, characteristic
12C2×#1823C211yesmaximal, characteristic, complemented
13C22×#123—21no—
14C22×#123—21no—
15C22×C22⋊Q8C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#3187—28no—
19C2×#481C2212yes—
20C2×#483—28no—
21C2×#483C2212yes—
22C2×#483C2214yes—
23C2×#483C2214yescomplemented
24C2×#465C2214yes—
25C2×#448C2212yes—
26C2×#450C2212yes—
27C22×#106C2212yescomplemented
28C22×#103C2212yescomplemented
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2501—22no—
32#2501—22no—
33#2501C2214yes—
34#2501C2214yes—
35C2×#399—22no—
36#2368—22no—
37#2360C2214yescomplemented
38#2345—22no—
39#2345C2212yescomplemented
40Q8×C22⋊C4C2212yescomplemented
41#2329—22no—
42#2315—21no—
43C4×C22⋊Q8C2214yes—
44C2×#390C2212yes—
45C2×#384C2212yes—
46C2×#380C2211yescharacteristic
47#2273—21no—
48C2×C4×C22⋊C4C2211yescharacteristic
49#2084C2218yescomplemented
50#2072C2218yescomplemented
51#2065C2218yescomplemented
52#2055C2218yescomplemented
53#1948C2218yescomplemented
54#1926C2218yescomplemented
55#1893C2214yes—
56#1880C2218yescomplemented
57#1874C2214yes—
58#1850C2214yescomplemented
59#1850C2214yescomplemented
60#1823C2214yes—
61#1823C2212yes—
62#1823C2214yescomplemented
63#1803C2218yescomplemented
64#1797C2214yes—
65#1786C2212yes—
66C2×#340C2214yescomplemented
67C23×Q8C2311yescharacteristic
68C23×D4—21no—
69C23×D4C2311yescharacteristic
70C24×C4—21no—
71C24×C4C2311yescharacteristic
72C22×SD16C2314yes—
73C23×C8C2312yes—
74C23×C8C2312yes—
75C2×#123—24no—
76C2×#123—24no—
77C2×#123—24no—
78C2×#123—24no—
79C2×#123—48no—
80C2×#123—48no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8D418yes—
85C2×C22⋊Q8C2312yes—
86C2×C22⋊Q8C2312yes—
87C2×#121—21no—
88C2×#121—21no—
89C2×#121—24no—
90C2×#121—24no—
91C2×#121C2312yes—
92C2×#121C2312yes—
93C2×C4×Q8C2314yes—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—22no—
112C22×C22⋊C4—22no—
113C22×C22⋊C4D411yescharacteristic
114C22×C22⋊C4D411yescharacteristic
115C22×C42C2312yes—
116#483—216no—
117#483—216no—
118#483—28no—
119#481—28no—
120#465—216no—
121#450—28no—
122#448—28no—
123C2×#106—28no—
124C2×#106C2318yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2318yes—
127C2×#103—28no—
128C2×#103C2318yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2318yes—
132C2×C22⋊C8—24no—
133C2×C22⋊C8—24no—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138#399—28no—
139#399—28no—
140#391—24no—
141#391C2314yes—
142#390—24no—
143#390C2318yes—
144#389—24no—
145#389C2314yes—
146#387—24no—
147#387—24no—
148#387—24no—
149#385—24no—
150#385—24no—
151#384—24no—
152#384C2318yes—
153#380—21no—
154#380—21no—
155#380—22no—
156#380—22no—
157#380C2312yes—
158#380C2312yes—
159C4×C4⋊C4—24no—
160C4×C4⋊C4C2314yes—
161C4×C22⋊C4—24no—
162C4×C22⋊C4—22no—
163C4×C22⋊C4—22no—
164C4×C22⋊C4—24no—
165C4×C22⋊C4D412yes—
166C4×C22⋊C4D412yes—
167C4×C22⋊C4C2312yes—
168C4×C22⋊C4C2312yes—
169C2×#95—21no—
170C2×#95D412yes—
171#340C2318yes—
172#340C2318yes—
173#340C2318yes—
174#336C2318yes—
175C25—21no—
176C25C2×D411yescharacteristic
177C22×Q8—21no—
178C22×Q8—22no—
179C22×Q8—44no—
180C22×Q8C2×D414yes—
181C22×Q8C2412yes—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—21no—
186C22×D4—22no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4C2412yes—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—22no—
195C23×C4—21no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×D412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D414yes—
208C23×C4C2×D412yes—
209C23×C4#4114yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C2×SD16—28no—
214C2×SD16—48no—
215C22×C8—24no—
216C22×C8—24no—
217C22×C8—44no—
218C22×C8—44no—
219C22×C8C2414yes—
220C22×C8C2414yes—
221#123—416no—
222#123—416no—
223#123—416no—
224#123—416no—
225C22⋊Q8—28no—
226C22⋊Q8—28no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8C2×D418yes—
230C22⋊Q8C2×D418yes—
231#121—24no—
232#121—24no—
233#121—24no—
234#121—24no—
235#121—44no—
236#121—44no—
237C4×Q8—48no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4SD1618yescomplemented
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—44no—
293C2×C42C2×D414yes—
294C2×C42C2414yes—
295#106—216no—
296#106—28no—
297C4⋊C8—48no—
298#103—216no—
299#103—28no—
300#102—28no—
301#102—48no—
302C22⋊C8—24no—
303C22⋊C8—24no—
304C22⋊C8—24no—
305C22⋊C8—24no—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95C2×D414yes—
313#95C2×D414yes—
314C24—22no—
315C24—42no—
316C24—42no—
317C24—44no—
318C24—44no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24#12114yes—
323C24C2×SD1614yes—
324C24AGL(1,ℤ/8)14yes—
325C24C22×D411yescharacteristic
326C24C22×D411yescharacteristic
327C24C22×D411yescharacteristic
328C2×Q8—24no—
329C2×Q8—22no—
330C2×Q8—28no—
331C2×Q8—48no—
332C2×Q8—82no—
333C2×Q8—84no—
334C2×Q8C22×D414yes—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—28no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—42no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—816no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—816no—
387C22×C4#12114yes—
388C22×C4#12114yes—
389C22×C4#12114yes—
390C22×C4C2×SD1618yes—
391C22×C4C22×D414yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D414yes—
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4112yes—
402C22×C4C2511yescharacteristic
403SD16—88no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—28no—
407C2×C8—28no—
408C2×C8—48no—
409C2×C8—84no—
410C4⋊C4—28no—
411C4⋊C4—28no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—28no—
419C4⋊C4—416no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—416no—
425C4⋊C4—84no—
426C4⋊C4—82no—
427C4⋊C4—84no—
428C4⋊C4C22×D414yes—
429C22⋊C4—28no—
430C22⋊C4—28no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—416no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—416no—
439C22⋊C4C2×SD1618yes—
440C22⋊C4C2×SD1618yes—
441C42—28no—
442C42—44no—
443C42—84no—
444C42—84no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—44no—
452C23—44no—
453C23—48no—
454C23—48no—
455C23—44no—
456C23—42no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—84no—
461C23—88no—
462C23—84no—
463C23—88no—
464C23—84no—
465C23#460116yes—
466C23C2×#12112yes—
467C23C2×#12112yes—
468C23C2×#12112yes—
469C23C22×SD1612yes—
470C23C22×SD1612yes—
471C23C22×SD1612yes—
472C23C2×AGL(1,ℤ/8)12yes—
473C23C2×AGL(1,ℤ/8)12yes—
474C23C2×AGL(1,ℤ/8)12yes—
475C23C23×D411yescharacteristic
476Q8—28no—
477Q8—44no—
478Q8—84no—
479D4—44no—
480D4—44no—
481D4—84no—
482D4—88no—
483D4—88no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—88no—
519C2×C4—88no—
520C2×C4—88no—
521C2×C4—84no—
522C2×C4—164no—
523C2×C4C2×#12112yes—
524C2×C4C2×#12112yes—
525C2×C4C2×#12112yes—
526C2×C4C22×SD1618yes—
527C2×C4C23×D411yescharacteristic
528C2×C4C22×#4111yescharacteristic
529C2×C4#58514yes—
530C8—44no—
531C8—44no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—416no—
545C22—84no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—84no—
550C22—168no—
551C22C2×#46014yes—
552C22C2×#46014yes—
553C22C2×#46014yes—
554C22C22×#12111yescharacteristic
555C22#348512yes—
556C22C23×SD1611yescharacteristic
557C22C22×AGL(1,ℤ/8)11yescharacteristic
558C22#358612yes—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—28no—
563C4—44no—
564C4—44no—
565C4—44no—
566C4—84no—
567C4—84no—
568C4—82no—
569C4#348512yes—
570C4#349112yes—
571C2—24no—
572C2—24no—
573C2—24no—
574C2—162no—
575C2#3544711yescharacteristic
576C2C22×#46011yescharacteristic
577C2#6042711yescharacteristic
578C2#6169512yes—
579C2#6170612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#483C2semidirect
4. #34212C2semidirect
5. #28120C2semidirect
6. #27647C2semidirect
7. #25204C2semidirect
8. #24578C2semidirect
9. #24528C2semidirect
10. #21930C2semidirect
12. C2×#1823C2semidirect
23. C2×#483C22semidirect
27. C22×#106C22semidirect
28. C22×#103C22semidirect
37. #2360C22semidirect
39. #2345C22semidirect
40. Q8×C22⋊C4C22semidirect
49. #2084C22semidirect
50. #2072C22semidirect
51. #2065C22semidirect
52. #2055C22semidirect
53. #1948C22semidirect
54. #1926C22semidirect
56. #1880C22semidirect
58. #1850C22semidirect
59. #1850C22semidirect
62. #1823C22semidirect
63. #1803C22semidirect
66. C2×#340C22semidirect
285. C2×C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
C22×#4832
C2×#25012
#342122
#281202
#276472
#252042
#245782
#245282
#219302
#217582
C2×#18232