Finite Group Databasebeta

← #6477621

Subgroups — #6477621

Order 512.

2028 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3095C211yesmaximal, characteristic, complemented
2C2×#3094C211yesmaximal, characteristic
3#35019C211yesmaximal, characteristic
4C2×#2346C211yesmaximal, characteristic, complemented
5#33111C214yesmaximal, complemented
6#29761C214yesmaximal, complemented
7#24599C214yesmaximal, complemented
8#24595C214yesmaximal, complemented
9#24499C214yesmaximal, complemented
10#24495C214yesmaximal, complemented
11#21934C211yesmaximal, characteristic, complemented
12#21765C211yesmaximal, characteristic, complemented
13#21764C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#3095—28no—
18#3094—28no—
19C2×#484C2212yescomplemented
20C2×#481C2212yes—
21C2×#485C2214yes—
22C2×#482C2214yes—
23C2×#483C2212yescomplemented
24C2×#480C2212yes—
25C2×#428C2212yes—
26C2×#428C2212yescomplemented
27C2×#419C2212yescomplemented
28C2×#418C2212yes—
29C2×#407C2211yescharacteristic
30C2×#407C2211yescharacteristic
31#2850C2214yes—
32#2639—21no—
33#2639—21no—
34#2591—22no—
35#2561—22no—
36#2504C2212yes—
37#2503C2212yes—
38#2436C2212yes—
39#2435C2212yes—
40#2346—22no—
41#2346—22no—
42#2346C2214yes—
43#2346C2214yescomplemented
44#2308—22no—
45C2×#389C2212yescomplemented
46C2×#388—22no—
47C2×#382—21no—
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2212yescomplemented
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×#376—21no—
52#2085C2218yescomplemented
53#2067C2218yescomplemented
54#2066C2218yescomplemented
55#2042C2218yescomplemented
56#1983C2218yescomplemented
57#1949C2218yescomplemented
58#1928C2218yescomplemented
59#1926C2218yescomplemented
60#1895C2214yes—
61#1894C2214yes—
62#1874C2214yes—
63#1872C2214yescomplemented
64#1851C2214yes—
65#1851C2214yescomplemented
66#1824C2212yes—
67#1823C2212yescomplemented
68#1797C2214yescomplemented
69#1787C2212yescomplemented
70#1786C2212yes—
71C23×Q8C2311yescharacteristic
72C23×D4—21no—
73C23×D4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C22×M16C2312yes—
76C22×M16C2312yes—
77C2×#124C2314yes—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8C2312yes—
83C2×C22⋊Q8C2312yes—
84C2×#121—21no—
85C2×#121—21no—
86C2×#121—24no—
87C2×#121—24no—
88C2×#121C2312yes—
89C2×#121C2312yes—
90C2×#117—21no—
91C2×#117—21no—
92C2×#117—24no—
93C2×#117D418yes—
94C2×#117C2312yes—
95C2×#117C2312yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4D412yes—
108C22×C42—21no—
109C22×C42—22no—
110C22×C42C2312yes—
111#485—216no—
112#484—28no—
113#483—28no—
114#482—216no—
115#481—28no—
116#480—28no—
117#428—28no—
118#428—28no—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8C2318yes—
124#419—28no—
125#418—28no—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2318yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2318yes—
132#407—24no—
133#407—24no—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138#400—22no—
139#400—22no—
140#397—22no—
141#397—22no—
142#394—22no—
143#394—22no—
144#393—22no—
145#393—22no—
146#391C2314yes—
147#390C2314yes—
148#389—24no—
149#389C2318yes—
150#388—28no—
151#388—28no—
152#388—24no—
153#388—24no—
154#386—24no—
155#386—24no—
156#386—24no—
157#386—24no—
158#382—24no—
159#382—24no—
160#382C2314yes—
161#381—24no—
162#380—21no—
163#380—21no—
164#380C2312yes—
165#380C2312yes—
166#379—24no—
167C4×C4⋊C4—24no—
168C4×C4⋊C4C2318yes—
169C4×C22⋊C4D412yes—
170C4×C22⋊C4D412yes—
171C4×C22⋊C4C2312yes—
172C4×C22⋊C4C2312yes—
173#376—24no—
174#376—24no—
175C2×#95—21no—
176C2×#95—22no—
177C2×#95—22no—
178C2×#95D411yescharacteristic
179C2×#95D411yescharacteristic
180#340C2318yes—
181#340C2318yes—
182#336C2318yes—
183C25—21no—
184C25C2×D411yescharacteristic
185C22×Q8—21no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—44no—
189C22×Q8C2412yes—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4—22no—
195C22×D4—44no—
196C22×D4—48no—
197C22×D4—44no—
198C22×D4C2412yes—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—22no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—44no—
206C23×C4C2×D412yes—
207C23×C4C2×D412yes—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D414yes—
212C23×C4#4114yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C2×M16—24no—
217C2×M16—24no—
218C2×M16—44no—
219C2×M16—44no—
220C22×C8C2414yes—
221C22×C8C2414yes—
222#124—48no—
223C22⋊Q8—24no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227C22⋊Q8—44no—
228C22⋊Q8—44no—
229#121—24no—
230#121—24no—
231#121—24no—
232#121—24no—
233#121—44no—
234#121—44no—
235#117—28no—
236#117—28no—
237#117—24no—
238#117—24no—
239#117C2×D418yes—
240#117C2×D418yes—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—84no—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—82no—
282C2×C22⋊C4C2×D414yes—
283C2×C22⋊C4C2×D414yes—
284C2×C42—28no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—28no—
288C2×C42—44no—
289C2×C42—48no—
290C2×C42C2×D414yes—
291C2×C42C2414yes—
292#107—28no—
293#107—28no—
294#106—28no—
295#106—28no—
296C4⋊C8—48no—
297#103—28no—
298#103—28no—
299#103—48no—
300#102—28no—
301#102—28no—
302#102—48no—
303C22⋊C8—24no—
304C22⋊C8—24no—
305C22⋊C8—24no—
306C22⋊C8—24no—
307#95—22no—
308#95—22no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95C2×D412yes—
316#95C2×D412yes—
317#95C2×D412yes—
318#95C2×D412yes—
319C24—24no—
320C24—22no—
321C24—41no—
322C24—42no—
323C24—41no—
324C24—44no—
325C24—42no—
326C24—82no—
327C24—81no—
328C24—81no—
329C24#12114yes—
330C24C22×D411yescharacteristic
331C24C22×D411yescharacteristic
332C24C22×D411yescharacteristic
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—28no—
336C2×Q8—48no—
337C2×Q8—42no—
338C2×Q8—82no—
339C2×Q8—84no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—28no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—48no—
348C2×D4—42no—
349C2×D4—84no—
350C2×D4—82no—
351C2×D4—816no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—82no—
387C22×C4#12114yes—
388C22×C4#12114yes—
389C22×C4#12114yes—
390C22×C4C22×D414yes—
391C22×C4C22×D414yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C2×#4112yes—
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2511yescharacteristic
402M16—48no—
403M16—48no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—28no—
407C2×C8—28no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—28no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—88no—
423C4⋊C4C22×D414yes—
424C4⋊C4C22×D414yes—
425C22⋊C4—28no—
426C22⋊C4—28no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—82no—
438C22⋊C4—82no—
439C42—28no—
440C42—28no—
441C42—28no—
442C42—44no—
443C42—416no—
444C23—21no—
445C23—21no—
446C23—22no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—28no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—84no—
464C23—84no—
465C23—84no—
466C23—84no—
467C23—84no—
468C23—164no—
469C23—164no—
470C23C2×#12112yes—
471C23C2×#12112yes—
472C23C2×#12112yes—
473C23#57514yes—
474C23C23×D411yescharacteristic
475Q8—44no—
476Q8—44no—
477Q8—84no—
478D4—44no—
479D4—44no—
480D4—84no—
481D4—88no—
482D4—88no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
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—42no—
506C2×C4—42no—
507C2×C4—48no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—84no—
520C2×C4—88no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4C2×#12112yes—
525C2×C4C2×#12112yes—
526C2×C4C2×#12112yes—
527C2×C4C23×D411yescharacteristic
528C2×C4C22×#4111yescharacteristic
529C2×C4#58514yes—
530C8—44no—
531C8—44no—
532C22—22no—
533C22—24no—
534C22—22no—
535C22—24no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—88no—
547C22—82no—
548C22—84no—
549C22—82no—
550C22—84no—
551C22—82no—
552C22—168no—
553C22#304014yes—
554C22C22×#12111yescharacteristic
555C22#348512yes—
556C22C2×#57512yes—
557C22#358511yescharacteristic
558C22#358611yescharacteristic
559C4—22no—
560C4—22no—
561C4—22no—
562C4—44no—
563C4—44no—
564C4—44no—
565C4—44no—
566C4—82no—
567C4—84no—
568C4—82no—
569C4—82no—
570C4#344912yes—
571C4#349112yes—
572C2—24no—
573C2—42no—
574C2—42no—
575C2—162no—
576C2#3508911yescharacteristic
577C2C2×#304011yescharacteristic
578C2#6042511yescharacteristic
579C2#6171012yes—
580C2#6171612yes—
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
1. C2×#3095C2semidirect
4. C2×#2346C2semidirect
5. #33111C2semidirect
6. #29761C2semidirect
7. #24599C2semidirect
8. #24595C2semidirect
9. #24499C2semidirect
10. #24495C2semidirect
11. #21934C2semidirect
12. #21765C2semidirect
19. C2×#484C22semidirect
23. C2×#483C22semidirect
26. C2×#428C22semidirect
27. C2×#419C22semidirect
43. #2346C22semidirect
45. C2×#389C22semidirect
49. C2×C4×C4⋊C4C22semidirect
52. #2085C22semidirect
53. #2067C22semidirect
54. #2066C22semidirect
55. #2042C22semidirect
56. #1983C22semidirect
57. #1949C22semidirect
58. #1928C22semidirect
59. #1926C22semidirect
63. #1872C22semidirect
65. #1851C22semidirect
67. #1823C22semidirect
68. #1797C22semidirect
69. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30952
C2×#30942
#350192
C2×#23462
#331112
#297612
#245992
#245952
#244992
#244952
#219342
#217652
#217642