Finite Group Databasebeta

← #6470700

Subgroups — #6470700

Order 512.

2548 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3193C212yesmaximal, complemented
2#35001C211yesmaximal, characteristic
3C2×#2362C211yesmaximal, characteristic, complemented
4#33106C212yesmaximal, complemented
5#32903C212yesmaximal, complemented
6#29754C212yesmaximal, complemented
7#29713C212yesmaximal, complemented
8#25217C218yesmaximal, complemented
9#25093C218yesmaximal, complemented
10#21934C211yesmaximal, characteristic, complemented
11#20912C212yesmaximal, complemented
12C2×#534—21no—
13C2×#534—21no—
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2212yes—
16C22×C4×Q8C2211yescharacteristic, complemented
17C22×#117C2211yescharacteristic
18#3193—216no—
19C2×#484C2214yescomplemented
20C2×#480C2214yescomplemented
21C2×#468C2218yes—
22C2×#460C2214yescomplemented
23C2×#448C2214yescomplemented
24C2×#450C2214yescomplemented
25C2×#418C2214yescomplemented
26C2×#407C2212yes—
27#2856C2212yes—
28#2856C2212yes—
29#2656—21no—
30#2656—21no—
31#2539—24no—
32#2501C2214yes—
33#2433C2214yes—
34#2362—24no—
35#2362C2218yescomplemented
36C2×#391C2211yescharacteristic, complemented
37C2×#389C2211yescharacteristic, complemented
38C2×#388—21no—
39C2×#388—21no—
40C2×#387—21no—
41C2×C4×C4⋊C4—21no—
42C2×C4×C4⋊C4C2211yescharacteristic, complemented
43C2×C4×C22⋊C4C2211yescharacteristic
44#2083C2214yescomplemented
45#2072C2214yescomplemented
46#2067C2218yescomplemented
47#2055C2218yescomplemented
48#2041C2214yescomplemented
49#2014C2214yescomplemented
50#1982C2214yescomplemented
51#1947C2214yescomplemented
52#1925C2218yescomplemented
53#1880C2218yescomplemented
54#1875C2218yes—
55#1874C2214yescomplemented
56#1874C2214yescomplemented
57#1851C2218yescomplemented
58#1803C2214yescomplemented
59#1797C2212yescomplemented
60#1797C2212yescomplemented
61#1736C2214yescomplemented
62C23×Q8C2311yescharacteristic
63C23×D4—22no—
64C23×D4C2312yes—
65C24×C4C2311yescharacteristic
66C22×SD16C2318yes—
67C22×M16C2314yes—
68#534—48no—
69#534—48no—
70C2×#127C2312yes—
71C2×#127C2312yes—
72C2×#123—24no—
73C2×#123—24no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8D418yescomplemented
77C2×C22⋊Q8C2314yes—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121—22no—
81C2×#121—22no—
82C2×#121—22no—
83C2×#121—22no—
84C2×#121—28no—
85C2×#121—28no—
86C2×#121C2314yes—
87C2×#121C2314yes—
88C2×#120—24no—
89C2×#120—24no—
90C2×C4×Q8—22no—
91C2×C4×Q8—44no—
92C2×C4×Q8C2314yes—
93C2×#117—22no—
94C2×#117—44no—
95C2×#117C2314yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4D411yescharacteristic
107C22×C22⋊C4D411yescharacteristic
108C22×C42—21no—
109C22×C42—21no—
110C22×C42—21no—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113#484—216no—
114#480—216no—
115#468—232no—
116#460—48no—
117#450—216no—
118#448—216no—
119C2×#107C2318yes—
120C2×C4⋊C8C2314yes—
121C2×C4⋊C8C2314yes—
122#418—48no—
123C2×#103C2318yes—
124C2×#102C2318yes—
125C2×#102C2318yes—
126C2×#102C2318yes—
127#407—28no—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130#399—24no—
131#399—24no—
132#392—24no—
133#392—24no—
134#391—22no—
135#391C2314yes—
136#390—22no—
137#390—22no—
138#390—22no—
139#390—22no—
140#390C2314yes—
141#390C2314yes—
142#389—22no—
143#389C2314yes—
144#388—28no—
145#388—28no—
146#387—28no—
147#384C2312yes—
148#384C2312yes—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4C2314yes—
152C4×C22⋊C4D414yes—
153C4×C22⋊C4C2314yes—
154C2×#95—21no—
155C2×#95—21no—
156C2×#95—21no—
157C2×#95—21no—
158C2×#95D411yescharacteristic
159C2×#95D411yescharacteristic
160#340C2318yes—
161#340C2318yes—
162#336C2314yes—
163#336C2314yes—
164C25—22no—
165C25C2×D411yescharacteristic
166C22×Q8—21no—
167C22×Q8—22no—
168C22×Q8—42no—
169C22×Q8—42no—
170C22×Q8C2×D414yes—
171C22×Q8C2412yes—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—48no—
184C22×D4—48no—
185C22×D4—48no—
186C22×D4C2414yes—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—22no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D412yes—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D414yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4#4114yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C2×SD16—216no—
205C2×SD16—48no—
206C2×SD16—48no—
207C2×M16—28no—
208C2×M16—44no—
209C2×M16—44no—
210C22×C8C2418yes—
211#127—44no—
212#127—44no—
213#123—416no—
214#123—416no—
215C22⋊Q8—216no—
216C22⋊Q8—28no—
217C22⋊Q8C2×D4116yes—
218#121—28no—
219#121—28no—
220#121—28no—
221#121—28no—
222#121—48no—
223#121—48no—
224#121—48no—
225#121—48no—
226#121—48no—
227#121—48no—
228#120—416no—
229#120—416no—
230C4×Q8—28no—
231C4×Q8—44no—
232C4×Q8—48no—
233#117—28no—
234#117—44no—
235#117—48no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4C2×D414yes—
274C2×C22⋊C4C2×D414yes—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2412yes—
286C2×C42C2412yes—
287#107—216no—
288C4⋊C8—44no—
289C4⋊C8—44no—
290#103—216no—
291#102—216no—
292#102—216no—
293#102—48no—
294C22⋊C8—28no—
295C22⋊C8—28no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95C2×D414yes—
301#95C2×D414yes—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—44no—
310C24—44no—
311C24—44no—
312C24—82no—
313C24—82no—
314C24—82no—
315C24—82no—
316C24#12114yes—
317C24C22×D412yes—
318C24C22×D411yescharacteristic
319C2×Q8—24no—
320C2×Q8—22no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—48no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—48no—
327C2×Q8C22×D414yes—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—416no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—88no—
345C2×D4—82no—
346C2×D4—82no—
347C2×D4—82no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4—82no—
354C2×D4—82no—
355C2×D4—832no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4#12118yes—
397C22×C4#12114yes—
398C22×C4AGL(1,ℤ/8)14yes—
399C22×C4AGL(1,ℤ/8)14yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C22×D414yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4114yes—
410C22×C4C2511yescharacteristic
411SD16—816no—
412M16—88no—
413C2×C8—28no—
414C2×C8—28no—
415C2×C8—28no—
416C4⋊C4—24no—
417C4⋊C4—216no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—48no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4C22×D414yes—
433C4⋊C4H(5,𝔽2)14yes—
434C22⋊C4—216no—
435C22⋊C4—28no—
436C22⋊C4—28no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C42—24no—
445C42—24no—
446C42—42no—
447C42—42no—
448C42—48no—
449C42—48no—
450C42—48no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—28no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—88no—
476C23—88no—
477C23—88no—
478C23—164no—
479C23—164no—
480C23—164no—
481C23—164no—
482C23C2×#12114yes—
483C23C2×#12112yes—
484C23C2×AGL(1,ℤ/8)12yes—
485C23C2×AGL(1,ℤ/8)12yes—
486C23C23×D411yescharacteristic
487Q8—28no—
488Q8—48no—
489Q8—44no—
490D4—48no—
491D4—48no—
492D4—88no—
493D4—816no—
494D4—816no—
495D4—164no—
496D4—164no—
497D4—164no—
498D4—164no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—48no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—84no—
542C2×C4—88no—
543C2×C4C2×#12112yes—
544C2×C4C2×#12114yes—
545C2×C4C2×AGL(1,ℤ/8)14yes—
546C2×C4C2×AGL(1,ℤ/8)14yes—
547C2×C4C23×D411yescharacteristic
548C2×C4C22×#4111yescharacteristic
549C2×C4C2×H(5,𝔽2)12yes—
550C2×C4#58512yes—
551C8—48no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—88no—
566C22—84no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22—1616no—
574C22#303814yes—
575C22C22×#12111yescharacteristic
576C22#348411yescharacteristic
577C22#348511yescharacteristic
578C22C22×AGL(1,ℤ/8)11yescharacteristic
579C22C22×AGL(1,ℤ/8)11yescharacteristic
580C22#358611yescharacteristic
581C22#358611yescharacteristic
582C4—24no—
583C4—24no—
584C4—24no—
585C4—22no—
586C4—42no—
587C4—42no—
588C4—44no—
589C4—44no—
590C4—44no—
591C4—82no—
592C4—84no—
593C4#345912yes—
594C4#349612yes—
595C2—24no—
596C2—44no—
597C2—164no—
598C2#3498411yescharacteristic
599C2C2×#303811yescharacteristic
600C2#6042411yescharacteristic
601C2#6157211yescharacteristic
602C2#6158511yescharacteristic
603C2#6170411yescharacteristic
604C2#6172211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35001C2
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×#3193C2semidirect
3. C2×#2362C2semidirect
4. #33106C2semidirect
5. #32903C2semidirect
6. #29754C2semidirect
7. #29713C2semidirect
8. #25217C2semidirect
9. #25093C2semidirect
10. #21934C2semidirect
11. #20912C2semidirect
16. C22×C4×Q8C22semidirect
19. C2×#484C22semidirect
20. C2×#480C22semidirect
22. C2×#460C22semidirect
23. C2×#448C22semidirect
24. C2×#450C22semidirect
25. C2×#418C22semidirect
35. #2362C22semidirect
36. C2×#391C22semidirect
37. C2×#389C22semidirect
42. C2×C4×C4⋊C4C22semidirect
44. #2083C22semidirect
45. #2072C22semidirect
46. #2067C22semidirect
47. #2055C22semidirect
48. #2041C22semidirect
49. #2014C22semidirect
50. #1982C22semidirect
51. #1947C22semidirect
52. #1925C22semidirect
53. #1880C22semidirect
55. #1874C22semidirect
56. #1874C22semidirect
57. #1851C22semidirect
58. #1803C22semidirect
59. #1797C22semidirect
60. #1797C22semidirect
61. #1736C22semidirect
76. C2×C22⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
#350012
C2×#23622
#331062
#329032
#297542
#297132
#252172
#250932
#219342
#209122