Finite Group Databasebeta

← #6508727

Subgroups — #6508727

Order 512.

2250 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3192C211yesmaximal, characteristic
2C22×#480C211yesmaximal, characteristic, complemented
3C2×#2504C211yesmaximal, characteristic, complemented
4#34246C211yesmaximal, characteristic, complemented
5#28114C214yesmaximal, complemented
6#27651C214yesmaximal, complemented
7#25227C218yesmaximal, complemented
8#24583C214yesmaximal, complemented
9#24531C214yesmaximal, complemented
10#21932C211yesmaximal, characteristic, complemented
11#21761C211yesmaximal, characteristic, complemented
12#20700C211yesmaximal, characteristic, complemented
13C22×#121C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16#3192—28no—
17C2×#481C2212yes—
18C2×#481C2212yes—
19C2×#480—28no—
20C2×#480C2214yes—
21C2×#480C2214yescomplemented
22C2×#469C2214yes—
23C2×#447C2212yes—
24C2×#447C2212yes—
25C22×#107C2212yescomplemented
26C22×#102C2212yescomplemented
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29#2504—21no—
30#2504—22no—
31#2504—21no—
32#2504C2214yes—
33#2504C2214yescomplemented
34C2×#397—22no—
35#2364—22no—
36#2363C2214yescomplemented
37#2345C2212yescomplemented
38D4×C22⋊C4—22no—
39D4×C22⋊C4C2212yescomplemented
40#2333—22no—
41#2312—21no—
42C4×#121C2214yes—
43C2×#390C2212yescomplemented
44C2×#386—22no—
45C2×#382C2212yescomplemented
46#2280C2211yescharacteristic
47#2273—21no—
48C2×C4×C22⋊C4C2211yescharacteristic
49#2085C2218yescomplemented
50#2073C2218yescomplemented
51#2067C2218yescomplemented
52#2054C2218yescomplemented
53#1949C2218yescomplemented
54#1925C2218yescomplemented
55#1893C2214yes—
56#1879C2218yescomplemented
57#1875C2214yescomplemented
58#1851C2214yescomplemented
59#1850C2214yescomplemented
60#1824C2212yescomplemented
61#1824C2214yescomplemented
62#1823C2214yescomplemented
63#1804C2218yescomplemented
64#1797C2214yescomplemented
65#1786C2212yescomplemented
66#1736C2214yescomplemented
67C23×D4—21no—
68C23×D4—21no—
69C23×D4C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C24×C4—21no—
72C24×C4C2311yescharacteristic
73C22×D8C2314yes—
74C22×M16C2312yes—
75C23×C8C2312yes—
76C2×#121—21no—
77C2×#121—21no—
78C2×#121—24no—
79C2×#121—21no—
80C2×#121—21no—
81C2×#121—24no—
82C2×#121—24no—
83C2×#121D418yes—
84C2×#121C2312yes—
85C2×#121C2312yes—
86C2×#121C2312yes—
87C2×#121C2312yes—
88C2×C4×D4—24no—
89C2×C4×D4C2314yes—
90C2×#117C2312yes—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—22no—
105C22×C22⋊C4D412yes—
106C22×C42C2312yes—
107#481—28no—
108#481—28no—
109#480—216no—
110#480—216no—
111#469—216no—
112#447—28no—
113#447—28no—
114C2×#107—28no—
115C2×#107C2318yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2318yes—
118C2×#102—28no—
119C2×#102C2318yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2318yes—
123#407—24no—
124C2×C22⋊C8—24no—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2312yes—
129#397—28no—
130#397—24no—
131#397—24no—
132#390—24no—
133#390—22no—
134#390—22no—
135#390C2314yes—
136#390C2318yes—
137#389C2314yes—
138#388—24no—
139#388—24no—
140#387—24no—
141#386—24no—
142#386—24no—
143#386—24no—
144#386—24no—
145#386—28no—
146#386—24no—
147#382—22no—
148#382—22no—
149#382C2318yes—
150#381—21no—
151#380—22no—
152#380C2312yes—
153#380C2312yes—
154#379—21no—
155#379—22no—
156C4×C4⋊C4C2314yes—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—22no—
160C4×C22⋊C4—24no—
161C4×C22⋊C4—24no—
162C4×C22⋊C4D411yescharacteristic
163C4×C22⋊C4D412yes—
164C4×C22⋊C4D411yescharacteristic
165C4×C22⋊C4C2312yes—
166C4×C22⋊C4C2312yes—
167C2×#95—24no—
168C2×#95D412yes—
169#340C2318yes—
170#340C2318yes—
171#340C2318yes—
172#336C2318yes—
173C25—21no—
174C25—21no—
175C25C2×D411yescharacteristic
176C22×D4—22no—
177C22×D4—28no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—21no—
182C22×D4—22no—
183C22×D4—21no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—48no—
189C22×D4—44no—
190C22×D4—84no—
191C22×D4C2×D414yes—
192C22×D4C2412yes—
193C22×D4C2412yes—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—21no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—84no—
203C23×C4C2×D414yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D414yes—
208C23×C4#4114yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C2×D8—28no—
213C2×D8—48no—
214C2×M16—24no—
215C2×M16—44no—
216C22×C8—24no—
217C22×C8—44no—
218C22×C8C2414yes—
219C22×C8C2414yes—
220#121—28no—
221#121—28no—
222#121—24no—
223#121—24no—
224#121—24no—
225#121—24no—
226#121—24no—
227#121—24no—
228#121—44no—
229#121—44no—
230#121C2×D418yes—
231#121C2×D418yes—
232C4×D4—48no—
233C4×D4—88no—
234#117—44no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2412yes—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—88no—
275C2×C22⋊C4C2×D412yes—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D414yes—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—44no—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2414yes—
284#107—216no—
285#106—28no—
286C4⋊C8—48no—
287#102—216no—
288#102—28no—
289#102—28no—
290#102—48no—
291C22⋊C8—24no—
292C22⋊C8—24no—
293C22⋊C8—24no—
294C22⋊C8—24no—
295#95—28no—
296#95—28no—
297#95—22no—
298#95—24no—
299#95—22no—
300#95—24no—
301#95C2×D412yes—
302#95C2×D414yes—
303#95C2×D412yes—
304C24—24no—
305C24—24no—
306C24—22no—
307C24—24no—
308C24—22no—
309C24—42no—
310C24—44no—
311C24—44no—
312C24—44no—
313C24—42no—
314C24—44no—
315C24—82no—
316C24—82no—
317C24#12114yes—
318C24C22×D411yescharacteristic
319C24C22×D411yescharacteristic
320C24C22×D411yescharacteristic
321C2×D4—28no—
322C2×D4—28no—
323C2×D4—24no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—48no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—42no—
337C2×D4—84no—
338C2×D4—84no—
339C2×D4—84no—
340C2×D4—82no—
341C2×D4—84no—
342C2×D4—816no—
343C2×D4—816no—
344C2×D4C22×D414yes—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—48no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—48no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—82no—
373C22×C4—816no—
374C22×C4#12114yes—
375C22×C4#12114yes—
376C22×C4#12114yes—
377C22×C4C22×D414yes—
378C22×C4C22×D414yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4C2×#4112yes—
388C22×C4C2511yescharacteristic
389D8—88no—
390M16—84no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—28no—
394C2×C8—28no—
395C2×C8—48no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—28no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—84no—
408C4⋊C4C22×D414yes—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—28no—
412C22⋊C4—28no—
413C22⋊C4—44no—
414C22⋊C4—48no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—84no—
422C22⋊C4—84no—
423C22⋊C4—82no—
424C42—28no—
425C42—42no—
426C42—42no—
427C42—82no—
428C42—84no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—28no—
435C23—28no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—44no—
440C23—416no—
441C23—44no—
442C23—42no—
443C23—42no—
444C23—44no—
445C23—48no—
446C23—42no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—84no—
451C23—84no—
452C23—84no—
453C23—84no—
454C23—816no—
455C23—84no—
456C23—168no—
457C23C2×#12112yes—
458C23C2×#12112yes—
459C23C2×#12112yes—
460C23#57514yes—
461C23#57614yes—
462C23C23×D411yescharacteristic
463D4—28no—
464D4—416no—
465D4—44no—
466D4—44no—
467D4—44no—
468D4—84no—
469D4—84no—
470D4—88no—
471D4—88no—
472D4—88no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—48no—
500C2×C4—88no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4—88no—
504C2×C4—88no—
505C2×C4—88no—
506C2×C4—168no—
507C2×C4C2×#12112yes—
508C2×C4C2×#12112yes—
509C2×C4C2×#12112yes—
510C2×C4C23×D411yescharacteristic
511C2×C4C22×#4111yescharacteristic
512C2×C4#58514yes—
513C8—44no—
514C8—44no—
515C22—22no—
516C22—28no—
517C22—28no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—416no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—84no—
530C22—84no—
531C22—82no—
532C22—88no—
533C22—82no—
534C22—82no—
535C22—168no—
536C22—168no—
537C22#304014yes—
538C22#304114yes—
539C22C22×#12111yescharacteristic
540C22#348512yes—
541C22C2×#57512yes—
542C22C2×#57612yes—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—44no—
547C4—44no—
548C4—44no—
549C4—84no—
550C4—84no—
551C4—82no—
552C4#348512yes—
553C4#349112yes—
554C2—24no—
555C2—24no—
556C2—42no—
557C2—84no—
558C2—162no—
559C2#3544711yescharacteristic
560C2C2×#304011yescharacteristic
561C2C2×#304111yescharacteristic
562C2#6172714yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3192C2
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×#480C2semidirect
3. C2×#2504C2semidirect
4. #34246C2semidirect
5. #28114C2semidirect
6. #27651C2semidirect
7. #25227C2semidirect
8. #24583C2semidirect
9. #24531C2semidirect
10. #21932C2semidirect
11. #21761C2semidirect
12. #20700C2semidirect
21. C2×#480C22semidirect
25. C22×#107C22semidirect
26. C22×#102C22semidirect
33. #2504C22semidirect
36. #2363C22semidirect
37. #2345C22semidirect
39. D4×C22⋊C4C22semidirect
43. C2×#390C22semidirect
45. C2×#382C22semidirect
49. #2085C22semidirect
50. #2073C22semidirect
51. #2067C22semidirect
52. #2054C22semidirect
53. #1949C22semidirect
54. #1925C22semidirect
56. #1879C22semidirect
57. #1875C22semidirect
58. #1851C22semidirect
59. #1850C22semidirect
60. #1824C22semidirect
61. #1824C22semidirect
62. #1823C22semidirect
63. #1804C22semidirect
64. #1797C22semidirect
65. #1786C22semidirect
66. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31922
C22×#4802
C2×#25042
#342462
#281142
#276512
#252272
#245832
#245312
#219322
#217612
#207002