Finite Group Databasebeta

← #6508728

Subgroups — #6508728

Order 512.

2302 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3193C211yesmaximal, characteristic
2C22×#484C211yesmaximal, characteristic, complemented
3C2×#2501C211yesmaximal, characteristic
4#34245C211yesmaximal, characteristic, complemented
5#28123C214yesmaximal, complemented
6#27648C214yesmaximal, complemented
7#25217C218yesmaximal, complemented
8#24576C214yesmaximal, complemented
9#24528C214yesmaximal, complemented
10#21932C211yesmaximal, characteristic, complemented
11#21761C211yesmaximal, characteristic
12#20700C211yesmaximal, characteristic, complemented
13C22×#123—22no—
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#3193—28no—
18C2×#484—28no—
19C2×#484C2212yes—
20C2×#484C2214yes—
21C2×#484C2214yescomplemented
22C2×#480C2212yes—
23C2×#468C2214yes—
24C2×#448C2212yes—
25C2×#450C2212yes—
26C22×#107C2212yescomplemented
27C22×#103C2212yescomplemented
28C2×#407C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30#2501—21no—
31#2501—22no—
32#2501—21no—
33#2501C2214yes—
34#2501C2214yes—
35C2×#399—22no—
36#2368—22no—
37#2362C2214yescomplemented
38#2344C2212yescomplemented
39#2342—22no—
40Q8×C22⋊C4C2212yescomplemented
41#2332—22no—
42#2313—21no—
43C4×C22⋊Q8C2214yes—
44C2×#390C2212yes—
45C2×#384C2212yes—
46#2280C2211yescharacteristic
47#2274—21no—
48C2×C4×C22⋊C4C2211yescharacteristic
49#2083C2218yescomplemented
50#2072C2218yescomplemented
51#2067C2218yescomplemented
52#2055C2218yescomplemented
53#1947C2218yescomplemented
54#1925C2218yescomplemented
55#1893C2214yes—
56#1880C2218yescomplemented
57#1875C2214yes—
58#1851C2214yescomplemented
59#1850C2214yescomplemented
60#1824C2212yes—
61#1824C2214yes—
62#1823C2214yescomplemented
63#1803C2218yescomplemented
64#1797C2214yes—
65#1786C2212yes—
66#1736C2214yescomplemented
67C23×Q8C2311yescharacteristic
68C23×D4—21no—
69C23×D4C2311yescharacteristic
70C24×C4—21no—
71C24×C4C2311yescharacteristic
72C22×SD16C2314yes—
73C22×M16C2312yes—
74C23×C8C2312yes—
75C2×#123—24no—
76C2×#123—28no—
77C2×#123—24no—
78C2×#123—416no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8D418yes—
83C2×C22⋊Q8C2312yes—
84C2×C22⋊Q8C2312yes—
85C2×#121—21no—
86C2×#121—21no—
87C2×#121—24no—
88C2×#121—24no—
89C2×#121C2312yes—
90C2×#121C2312yes—
91C2×C4×Q8C2314yes—
92C2×#117—22no—
93C2×#117C2312yes—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4D412yes—
110C22×C42C2312yes—
111#484—216no—
112#484—216no—
113#484—28no—
114#480—28no—
115#468—216no—
116#450—28no—
117#448—28no—
118C2×#107—28no—
119C2×#107C2314yes—
120C2×#107C2318yes—
121C2×C4⋊C8C2318yes—
122C2×#103—28no—
123C2×#103C2318yes—
124C2×#103C2314yes—
125C2×#102C2314yes—
126C2×#102C2318yes—
127#407—24no—
128C2×C22⋊C8—24no—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133#399—24no—
134#399—24no—
135#399—28no—
136#391—24no—
137#391C2314yes—
138#390—22no—
139#390—22no—
140#390C2318yes—
141#389C2314yes—
142#388—24no—
143#388—24no—
144#387—24no—
145#387—24no—
146#387—24no—
147#385—24no—
148#384—22no—
149#384—22no—
150#384C2318yes—
151#381—21no—
152#381—22no—
153#380—22no—
154#380C2312yes—
155#380C2312yes—
156#379—21no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4C2314yes—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—24no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—22no—
164C4×C22⋊C4D411yescharacteristic
165C4×C22⋊C4D412yes—
166C4×C22⋊C4D411yescharacteristic
167C4×C22⋊C4C2312yes—
168C4×C22⋊C4C2312yes—
169C2×#95D412yes—
170#340C2318yes—
171#340C2318yes—
172#340C2318yes—
173#336C2318yes—
174C25—21no—
175C25C2×D411yescharacteristic
176C22×Q8—21no—
177C22×Q8—22no—
178C22×Q8—44no—
179C22×Q8C2×D414yes—
180C22×Q8C2412yes—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—21no—
185C22×D4—22no—
186C22×D4—44no—
187C22×D4—48no—
188C22×D4—44no—
189C22×D4C2412yes—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—22no—
193C23×C4—21no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—44no—
197C23×C4—48no—
198C23×C4—44no—
199C23×C4C2×D414yes—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D414yes—
204C23×C4#4114yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C2×SD16—28no—
209C2×SD16—48no—
210C2×M16—24no—
211C2×M16—44no—
212C22×C8—24no—
213C22×C8—44no—
214C22×C8C2414yes—
215C22×C8C2414yes—
216#123—416no—
217#123—432no—
218#123—416no—
219C22⋊Q8—28no—
220C22⋊Q8—28no—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C22⋊Q8C2×D418yes—
224C22⋊Q8C2×D418yes—
225#121—24no—
226#121—24no—
227#121—24no—
228#121—24no—
229#121—44no—
230#121—44no—
231C4×Q8—48no—
232#117—48no—
233#117—44no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2412yes—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—416no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D414yes—
279C2×C22⋊C4C2×D412yes—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—44no—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2414yes—
287#107—216no—
288#107—28no—
289C4⋊C8—48no—
290#103—216no—
291#103—28no—
292#102—28no—
293#102—48no—
294C22⋊C8—24no—
295C22⋊C8—24no—
296C22⋊C8—24no—
297C22⋊C8—24no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95C2×D412yes—
305#95C2×D412yes—
306#95C2×D414yes—
307C24—24no—
308C24—24no—
309C24—22no—
310C24—42no—
311C24—44no—
312C24—44no—
313C24—42no—
314C24—44no—
315C24#12114yes—
316C24C22×D411yescharacteristic
317C24C22×D411yescharacteristic
318C24C22×D411yescharacteristic
319C2×Q8—24no—
320C2×Q8—22no—
321C2×Q8—28no—
322C2×Q8—48no—
323C2×Q8—82no—
324C2×Q8—84no—
325C2×Q8C22×D414yes—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—28no—
329C2×D4—44no—
330C2×D4—48no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—48no—
334C2×D4—42no—
335C2×D4—84no—
336C2×D4—84no—
337C2×D4—816no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—48no—
358C22×C4—44no—
359C22×C4—48no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—48no—
365C22×C4—82no—
366C22×C4—816no—
367C22×C4#12114yes—
368C22×C4#12114yes—
369C22×C4#12114yes—
370C22×C4C22×D414yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C22×D412yes—
377C22×C4C22×D414yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4C2×#4112yes—
381C22×C4C2511yescharacteristic
382SD16—88no—
383M16—84no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—28no—
387C2×C8—28no—
388C2×C8—48no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—28no—
398C4⋊C4—416no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—416no—
404C4⋊C4—48no—
405C4⋊C4—84no—
406C4⋊C4C22×D414yes—
407C22⋊C4—28no—
408C22⋊C4—28no—
409C22⋊C4—24no—
410C22⋊C4—28no—
411C22⋊C4—416no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—416no—
417C22⋊C4—82no—
418C22⋊C4AGL(1,ℤ/8)18yes—
419C42—28no—
420C42—42no—
421C42—42no—
422C42—48no—
423C42—82no—
424C42—84no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—28no—
431C23—28no—
432C23—22no—
433C23—22no—
434C23—44no—
435C23—416no—
436C23—42no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—84no—
441C23—84no—
442C23—816no—
443C23—84no—
444C23C2×#12112yes—
445C23C2×#12112yes—
446C23C2×#12112yes—
447C23C2×AGL(1,ℤ/8)14yes—
448C23#57814yes—
449C23C23×D411yescharacteristic
450Q8—28no—
451Q8—44no—
452Q8—84no—
453D4—44no—
454D4—44no—
455D4—84no—
456D4—88no—
457D4—88no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—416no—
478C2×C4—44no—
479C2×C4—48no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—84no—
490C2×C4—88no—
491C2×C4—88no—
492C2×C4—84no—
493C2×C4—88no—
494C2×C4C2×#12112yes—
495C2×C4C2×#12112yes—
496C2×C4C2×#12112yes—
497C2×C4C2×AGL(1,ℤ/8)18yes—
498C2×C4C23×D411yescharacteristic
499C2×C4C22×#4111yescharacteristic
500C2×C4#58514yes—
501C8—44no—
502C8—44no—
503C22—22no—
504C22—28no—
505C22—28no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—416no—
513C22—44no—
514C22—82no—
515C22—88no—
516C22—82no—
517C22—82no—
518C22—168no—
519C22#303814yes—
520C22#304414yes—
521C22C22×#12111yescharacteristic
522C22#348512yes—
523C22C22×AGL(1,ℤ/8)12yes—
524C22C2×#57812yes—
525C4—28no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—44no—
530C4—44no—
531C4—44no—
532C4—84no—
533C4—84no—
534C4—82no—
535C4#348512yes—
536C4#349112yes—
537C2—24no—
538C2—24no—
539C2—42no—
540C2—162no—
541C2#3544711yescharacteristic
542C2C2×#303811yescharacteristic
543C2C2×#304411yescharacteristic
544C2#6173714yes—
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×#484C2semidirect
4. #34245C2semidirect
5. #28123C2semidirect
6. #27648C2semidirect
7. #25217C2semidirect
8. #24576C2semidirect
9. #24528C2semidirect
10. #21932C2semidirect
12. #20700C2semidirect
21. C2×#484C22semidirect
26. C22×#107C22semidirect
27. C22×#103C22semidirect
37. #2362C22semidirect
38. #2344C22semidirect
40. Q8×C22⋊C4C22semidirect
49. #2083C22semidirect
50. #2072C22semidirect
51. #2067C22semidirect
52. #2055C22semidirect
53. #1947C22semidirect
54. #1925C22semidirect
56. #1880C22semidirect
58. #1851C22semidirect
59. #1850C22semidirect
62. #1823C22semidirect
63. #1803C22semidirect
66. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
C22×#4842
C2×#25012
#342452
#281232
#276482
#252172
#245762
#245282
#219322
#217612
#207002