Finite Group Databasebeta

← #6519410

Subgroups — #6519410

Order 512.

2332 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2540C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic, complemented
3#26244C218yesmaximal, complemented
4#26242C218yesmaximal, complemented
5#24006C212yesmaximal, complemented
6#24005C212yesmaximal, complemented
7#24003C212yesmaximal, complemented
8#24002C212yesmaximal, complemented
9#21808C212yesmaximal, complemented
10#21790C211yesmaximal, characteristic, complemented
11#21365C212yesmaximal, complemented
12C22×#124C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C2×#518—21no—
18C22×C4×D4C2211yescharacteristic
19C2×#484C2214yescomplemented
20C2×#481C2214yescomplemented
21C2×#483C2214yescomplemented
22C2×#480C2214yescomplemented
23C2×#432C2212yes—
24#2540—24no—
25#2540C2218yes—
26C2×#399—21no—
27C2×#392—21no—
28#2349—24no—
29D4×C4⋊C4C2214yescomplemented
30D4×C4⋊C4C2214yescomplemented
31#2316—21no—
32#2315—21no—
33#2315—21no—
34C2×#390—21no—
35C2×#390—21no—
36#2285C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40#2036C2218yescomplemented
41#2035C2218yescomplemented
42#2034C2218yescomplemented
43#2033C2218yescomplemented
44#1983C2218yescomplemented
45#1959C2214yescomplemented
46#1959C2214yescomplemented
47#1955C2218yescomplemented
48#1948C2214yescomplemented
49#1947C2214yescomplemented
50#1874C2214yescomplemented
51#1874C2214yescomplemented
52#1872C2214yescomplemented
53#1872C2214yescomplemented
54#1853C2214yes—
55#1852C2214yes—
56#1787C2214yes—
57#1786C2214yes—
58#1779C2212yes—
59#1779C2212yes—
60#1776C2218yescomplemented
61C23×Q8C2311yescharacteristic
62C23×D4—21no—
63C23×D4—21no—
64C23×D4D411yescharacteristic
65C23×D4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C2×#124—22no—
69C2×#124—28no—
70C2×#124C2314yes—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—44no—
74C2×C22⋊Q8—44no—
75C2×C22⋊Q8C2314yes—
76C2×C22⋊Q8C2314yes—
77C2×#121—22no—
78C2×#121—22no—
79C2×#121—44no—
80C2×#121—44no—
81C2×#121C2314yes—
82C2×#121C2314yes—
83#518—48no—
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C2×C4×D4—28no—
87C2×C4×D4D414yes—
88C2×C4×D4C2314yes—
89C2×#117—22no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4D411yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4—22no—
100C22×C22⋊C4—22no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4D411yescharacteristic
108C22×C22⋊C4D411yescharacteristic
109C22×C42—21no—
110C22×C42C2311yescharacteristic
111#484—48no—
112#483—48no—
113#481—48no—
114#480—48no—
115#432—28no—
116C2×#107C2318yes—
117C2×#106C2318yes—
118C2×C4⋊C8C2314yes—
119C2×#103C2318yes—
120C2×#103C2318yes—
121C2×#102C2318yes—
122C2×#102C2318yes—
123C2×C22⋊C8C2314yes—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2314yes—
128C2×#97C2314yes—
129#399—28no—
130#392—28no—
131#390—28no—
132#390—28no—
133#389—24no—
134#389—24no—
135#385—24no—
136#385—24no—
137#385—24no—
138#385—24no—
139#384C2312yes—
140#384C2312yes—
141#382—22no—
142#380—22no—
143#380—22no—
144#380—22no—
145#380—22no—
146#380—22no—
147#380C2314yes—
148#380C2314yes—
149#380C2314yes—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4C2314yes—
153C4×C4⋊C4C2314yes—
154C2×#95—22no—
155C2×#95—22no—
156C2×#95—21no—
157#340C2318yes—
158#340C2318yes—
159C25—21no—
160C25C2×D411yescharacteristic
161C25C2×D411yescharacteristic
162C22×Q8—21no—
163C22×Q8—24no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8C2412yes—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—21no—
173C22×D4—24no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—82no—
181C22×D4—82no—
182C22×D4C2×D414yes—
183C22×D4C2412yes—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—21no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2×D411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C22×C8C2418yes—
207#124—28no—
208#124—28no—
209#124—48no—
210C22⋊Q8—28no—
211C22⋊Q8—28no—
212C22⋊Q8—44no—
213C22⋊Q8—44no—
214C22⋊Q8—48no—
215C22⋊Q8—48no—
216#121—28no—
217#121—28no—
218#121—44no—
219#121—44no—
220#121—48no—
221#121—48no—
222C4×D4—216no—
223C4×D4—28no—
224C4×D4—28no—
225C4×D4—416no—
226C4×D4—416no—
227#117—48no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2414yes—
251C2×C4⋊C4C2414yes—
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2412yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4C2×D414yes—
274C2×C22⋊C4C2×D414yes—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—82no—
278C2×C42C2×D412yes—
279C2×C42#4118yes—
280C2×C42C2412yes—
281#107—216no—
282#106—216no—
283C4⋊C8—28no—
284#103—48no—
285#103—48no—
286#102—48no—
287#102—48no—
288C22⋊C8—28no—
289C22⋊C8—44no—
290C22⋊C8—44no—
291C22⋊C8—44no—
292C22⋊C8—44no—
293#97—28no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24AGL(1,ℤ/8)14yes—
320C24#13714yes—
321C24C22×D412yes—
322C24C22×D412yes—
323C24C22×D411yescharacteristic
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—88no—
331C2×D4—232no—
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—28no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—42no—
345C2×D4—88no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—816no—
349C2×D4—816no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—88no—
393C22×C4—88no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4#12314yes—
403C22×C4#12314yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C2×#4114yes—
410C22×C4C2511yescharacteristic
411C2×C8—28no—
412C2×C8—28no—
413C2×C8—28no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—28no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4H(5,𝔽2)14yes—
432C4⋊C4H(5,𝔽2)14yes—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C42—48no—
445C42—84no—
446C42#12318yes—
447C42C2×#4114yes—
448C42C2×#4114yes—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—28no—
466C23—28no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—84no—
487C23—84no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23C2×#12312yes—
494C23C2×AGL(1,ℤ/8)14yes—
495C23C2×AGL(1,ℤ/8)12yes—
496C23C2×#13714yes—
497C23C2×#13712yes—
498C23C23×D411yescharacteristic
499Q8—48no—
500Q8—84no—
501D4—232no—
502D4—48no—
503D4—84no—
504D4—816no—
505D4—816no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—44no—
523C2×C4—48no—
524C2×C4—48no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—84no—
545C2×C4—164no—
546C2×C4—164no—
547C2×C4—164no—
548C2×C4—164no—
549C2×C4C2×#12312yes—
550C2×C4C2×#12314yes—
551C2×C4C2×#12314yes—
552C2×C4C22×#4112yes—
553C2×C4C2×H(5,𝔽2)12yes—
554C2×C4C2×H(5,𝔽2)12yes—
555C8—48no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—24no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—88no—
577C22—88no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—168no—
586C22#309214yes—
587C22C22×#12311yescharacteristic
588C22#348411yescharacteristic
589C22#348411yescharacteristic
590C22C22×AGL(1,ℤ/8)11yescharacteristic
591C22C22×#13711yescharacteristic
592C22#358511yescharacteristic
593C22#358611yescharacteristic
594C4—24no—
595C4—24no—
596C4—22no—
597C4—44no—
598C4—44no—
599C4—42no—
600C4—42no—
601C4—84no—
602C4—84no—
603C4—82no—
604C4#350814yes—
605C2—24no—
606C2—44no—
607C2—44no—
608C2—162no—
609C2#3555411yescharacteristic
610C2C2×#309211yescharacteristic
611C2#6052011yescharacteristic
612C2#6156911yescharacteristic
613C2#6157011yescharacteristic
614C2#6157211yescharacteristic
615C2#6157311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34181C2semidirect
3. #26244C2semidirect
4. #26242C2semidirect
5. #24006C2semidirect
6. #24005C2semidirect
7. #24003C2semidirect
8. #24002C2semidirect
9. #21808C2semidirect
10. #21790C2semidirect
11. #21365C2semidirect
19. C2×#484C22semidirect
20. C2×#481C22semidirect
21. C2×#483C22semidirect
22. C2×#480C22semidirect
29. D4×C4⋊C4C22semidirect
30. D4×C4⋊C4C22semidirect
40. #2036C22semidirect
41. #2035C22semidirect
42. #2034C22semidirect
43. #2033C22semidirect
44. #1983C22semidirect
45. #1959C22semidirect
46. #1959C22semidirect
47. #1955C22semidirect
48. #1948C22semidirect
49. #1947C22semidirect
50. #1874C22semidirect
51. #1874C22semidirect
52. #1872C22semidirect
53. #1872C22semidirect
60. #1776C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25402
#341812
#262442
#262422
#240062
#240052
#240032
#240022
#218082
#217902
#213652