Finite Group Databasebeta

← #6470685

Subgroups — #6470685

Order 512.

2526 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C212yesmaximal
2#35011C211yesmaximal, characteristic, complemented
3C2×#2361C211yesmaximal, characteristic
4#33108C212yesmaximal, complemented
5#32905C212yesmaximal, complemented
6#29756C212yesmaximal, complemented
7#29714C212yesmaximal, complemented
8#25202C218yesmaximal, complemented
9#25078C218yesmaximal, complemented
10#21930C211yesmaximal, characteristic, complemented
11C2×#1872C212yesmaximal, complemented
12C22×C22⋊Q8C2212yes—
13C22×#121C2211yescharacteristic
14C22×C4×D4—21no—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#3187—216no—
18C2×#481C2214yes—
19C2×#483C2214yes—
20C2×#465C2218yes—
21C2×#461C2214yescomplemented
22C2×#448C2214yes—
23C2×#450C2214yes—
24C22×#103C2214yescomplemented
25C22×C22⋊C8C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2659—21no—
29#2659—21no—
30#2583—24no—
31#2502C2214yescomplemented
32#2434C2214yescomplemented
33#2361—24no—
34#2361C2218yes—
35#2315—21no—
36#2315—21no—
37C2×#390C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#388—21no—
40C2×#386—21no—
41C2×#385—21no—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44#2084C2214yescomplemented
45#2071C2214yescomplemented
46#2066C2218yescomplemented
47#2055C2218yescomplemented
48#2043C2214yescomplemented
49#2013C2214yescomplemented
50#1981C2214yescomplemented
51#1948C2214yescomplemented
52#1927C2218yescomplemented
53#1877C2218yescomplemented
54#1872C2214yes—
55#1872C2214yes—
56#1872C2218yescomplemented
57#1850C2218yescomplemented
58#1802C2214yescomplemented
59#1797C2212yes—
60#1797C2212yes—
61C2×#340C2214yescomplemented
62C23×Q8C2312yes—
63C23×D4—21no—
64C23×D4C2311yescharacteristic
65C24×C4—21no—
66C24×C4—21no—
67C24×C4C2311yescharacteristic
68C22×SD16C2318yes—
69C23×C8C2314yes—
70C2×C4⋊Q8C2312yes—
71C2×C4⋊Q8C2312yes—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8C2314yes—
77C2×C22⋊Q8C2314yes—
78C2×#121—22no—
79C2×#121—24no—
80C2×#121D418yes—
81C2×#121C2314yes—
82C2×C4×D4—28no—
83C2×C4×D4—22no—
84C2×C4×D4—44no—
85C2×C4×D4—48no—
86C2×C4×D4C2314yes—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4—21no—
98C22×C22⋊C4—22no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4D411yescharacteristic
103C22×C22⋊C4D411yescharacteristic
104C22×C42—21no—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#483—216no—
108#481—216no—
109#465—232no—
110#461—48no—
111#450—216no—
112#448—216no—
113C2×#106C2318yes—
114C2×C4⋊C8C2314yes—
115C2×C4⋊C8C2314yes—
116C2×#103—48no—
117C2×#103C2318yes—
118C2×#103C2318yes—
119C2×#103C2318yes—
120C2×#102C2318yes—
121C2×C22⋊C8—28no—
122C2×C22⋊C8C2314yes—
123C2×C22⋊C8C2314yes—
124#398—24no—
125#398—24no—
126#395—24no—
127#395—24no—
128#391C2314yes—
129#391C2314yes—
130#390—22no—
131#390C2314yes—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389C2314yes—
138#388—28no—
139#386—28no—
140#385—24no—
141#385—24no—
142#385—24no—
143#385—24no—
144#385—28no—
145#384C2312yes—
146#384C2312yes—
147#380—22no—
148#380—22no—
149#380—22no—
150#380—22no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4C2314yes—
153C4×C22⋊C4D414yes—
154C4×C22⋊C4C2314yes—
155C2×#95—21no—
156C2×#95—21no—
157C2×#95D411yescharacteristic
158C2×#95D411yescharacteristic
159#340C2318yes—
160#340C2318yes—
161#336C2314yes—
162#336C2314yes—
163C25—21no—
164C25C2×D411yescharacteristic
165C22×Q8—22no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—44no—
169C22×Q8—44no—
170C22×Q8C2414yes—
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—21no—
174C22×D4—22no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4C2×D414yes—
180C22×D4C2412yes—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—82no—
195C23×C4—82no—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D414yes—
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4#4114yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C2×SD16—216no—
207C2×SD16—48no—
208C2×SD16—48no—
209C22×C8—28no—
210C22×C8—44no—
211C22×C8—44no—
212C22×C8C2418yes—
213C4⋊Q8—44no—
214C4⋊Q8—44no—
215C22⋊Q8—28no—
216C22⋊Q8—28no—
217C22⋊Q8—28no—
218C22⋊Q8—28no—
219C22⋊Q8—48no—
220C22⋊Q8—48no—
221#121—216no—
222#121—28no—
223#121C2×D4116yes—
224C4×D4—28no—
225C4×D4—44no—
226C4×D4—48no—
227C4×D4—432no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
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—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
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—44no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—82no—
272C2×C22⋊C4SD1618yescomplemented
273C2×C22⋊C4C2×D414yes—
274C2×C22⋊C4C2×D414yes—
275C2×C42—24no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2412yes—
282C2×C42C2412yes—
283#106—216no—
284C4⋊C8—44no—
285C4⋊C8—44no—
286#103—216no—
287#103—216no—
288#103—48no—
289#102—216no—
290C22⋊C8—28no—
291C22⋊C8—28no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95C2×D414yes—
299#95C2×D414yes—
300C24—24no—
301C24—22no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24#12114yes—
313C24C2×SD1614yes—
314C24#13714yes—
315C24C22×D411yescharacteristic
316C24C22×D412yes—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—28no—
320C2×Q8—28no—
321C2×Q8—416no—
322C2×Q8—44no—
323C2×Q8—82no—
324C2×Q8—82no—
325C2×Q8—88no—
326C2×D4—216no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—48no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—432no—
338C2×D4C22×D414yes—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—84no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—84no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—816no—
386C22×C4—816no—
387C22×C4#12114yes—
388C22×C4#12118yes—
389C22×C4C2×SD1614yes—
390C22×C4C2×SD1614yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D414yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C2×#4114yes—
400C22×C4C2×#4112yes—
401C22×C4C2511yescharacteristic
402SD16—816no—
403C2×C8—28no—
404C2×C8—28no—
405C2×C8—28no—
406C2×C8—88no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—48no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4C22×D414yes—
429C4⋊C4H(5,𝔽2)14yes—
430C22⋊C4—28no—
431C22⋊C4—216no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—84no—
437C22⋊C4C2×SD16116yes—
438C42—24no—
439C42—24no—
440C42—42no—
441C42—42no—
442C42—48no—
443C23—21no—
444C23—22no—
445C23—21no—
446C23—22no—
447C23—21no—
448C23—22no—
449C23—21no—
450C23—22no—
451C23—24no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—48no—
459C23—48no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—48no—
465C23—42no—
466C23—42no—
467C23—84no—
468C23—84no—
469C23—84no—
470C23—84no—
471C23#461116yescomplemented
472C23C2×#12112yes—
473C23C2×#12114yes—
474C23C22×SD1614yes—
475C23C22×SD1612yes—
476C23C2×#13714yes—
477C23C2×#13712yes—
478C23C23×D411yescharacteristic
479Q8—48no—
480Q8—48no—
481Q8—88no—
482D4—28no—
483D4—48no—
484D4—432no—
485D4—44no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—48no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—816no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—816no—
529C2×C4—84no—
530C2×C4—88no—
531C2×C4—88no—
532C2×C4C2×#12114yes—
533C2×C4C2×#12112yes—
534C2×C4C22×SD1614yes—
535C2×C4C22×SD1614yes—
536C2×C4C23×D411yescharacteristic
537C2×C4C22×#4111yescharacteristic
538C2×C4C2×H(5,𝔽2)12yes—
539C2×C4#58512yes—
540C8—48no—
541C22—24no—
542C22—22no—
543C22—22no—
544C22—24no—
545C22—22no—
546C22—24no—
547C22—22no—
548C22—24no—
549C22—24no—
550C22—22no—
551C22—22no—
552C22—24no—
553C22—416no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—816no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22C2×#46114yes—
563C22C2×#46118yes—
564C22C22×#12111yescharacteristic
565C22#348411yescharacteristic
566C22#348511yescharacteristic
567C22C23×SD1611yescharacteristic
568C22C22×#13711yescharacteristic
569C22#358511yescharacteristic
570C22#358611yescharacteristic
571C4—24no—
572C4—22no—
573C4—24no—
574C4—24no—
575C4—42no—
576C4—44no—
577C4—42no—
578C4—44no—
579C4—84no—
580C4—82no—
581C4—84no—
582C4#345912yes—
583C4#349612yes—
584C2—24no—
585C2—28no—
586C2—84no—
587C2#3498411yescharacteristic
588C2C22×#46111yescharacteristic
589C2#6042511yescharacteristic
590C2#6156011yescharacteristic
591C2#6157311yescharacteristic
592C2#6169511yescharacteristic
593C2#6170811yescharacteristic
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. #35011C2semidirect
4. #33108C2semidirect
5. #32905C2semidirect
6. #29756C2semidirect
7. #29714C2semidirect
8. #25202C2semidirect
9. #25078C2semidirect
10. #21930C2semidirect
11. C2×#1872C2semidirect
21. C2×#461C22semidirect
24. C22×#103C22semidirect
31. #2502C22semidirect
32. #2434C22semidirect
44. #2084C22semidirect
45. #2071C22semidirect
46. #2066C22semidirect
47. #2055C22semidirect
48. #2043C22semidirect
49. #2013C22semidirect
50. #1981C22semidirect
51. #1948C22semidirect
52. #1927C22semidirect
53. #1877C22semidirect
56. #1872C22semidirect
57. #1850C22semidirect
58. #1802C22semidirect
61. C2×#340C22semidirect
272. C2×C22⋊C4SD16semidirect
471. C23#461semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
#350112
C2×#23612
#331082
#329052
#297562
#297142
#252022
#250782
#219302
C2×#18722