Finite Group Databasebeta

← #6632011

Subgroups — #6632011

Order 512.

2650 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35165C211yesmaximal, characteristic
2C2×#2409C211yesmaximal, characteristic, complemented
3#32879C212yesmaximal, complemented
4#31262C214yesmaximal, complemented
5#30506C214yesmaximal, complemented
6#30399C218yesmaximal, complemented
7#29892C212yesmaximal, complemented
8#27583C212yesmaximal, complemented
9#24381C214yesmaximal, complemented
10#23931C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic, complemented
12C24×Q8C2211yescharacteristic
13C2×#535—21no—
14C2×#532C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C4×D4C2211yescharacteristic
17C2×#465C2218yescomplemented
18C2×#448C2214yes—
19C2×#448C2218yescomplemented
20C22×C22⋊C8C2212yes—
21#2847C2212yes—
22#2847C2212yes—
23#2617—21no—
24#2617—21no—
25#2547C2212yes—
26#2544—22no—
27#2541C2212yes—
28#2537C2214yes—
29#2535—22no—
30#2409—22no—
31#2409—22no—
32#2409C2214yes—
33#2409C2214yes—
34C2×#395—21no—
35C2×#393—21no—
36#2311—21no—
37C2×#391C2212yes—
38C2×#386—21no—
39C2×#385—21no—
40#2285C2211yescharacteristic
41C2×#380C2211yescharacteristic
42#2096C2214yescomplemented
43#2077C2218yescomplemented
44#2072C2214yescomplemented
45#2035C2218yescomplemented
46#2034C2218yescomplemented
47#2024C2214yescomplemented
48#2014C2214yescomplemented
49#2000C2218yescomplemented
50#1927C2218yescomplemented
51#1892C2218yescomplemented
52#1877C2218yescomplemented
53#1874C2214yes—
54#1869C2214yes—
55#1869C2214yes—
56#1853C2214yes—
57#1817C2212yes—
58#1787C2214yes—
59#1787C2218yescomplemented
60#1779C2212yes—
61C22×#41—21no—
62C23×Q8—21no—
63C23×Q8—22no—
64C23×Q8—44no—
65C23×Q8—48no—
66C23×Q8C2312yes—
67C23×Q8C2314yes—
68C23×D4D411yescharacteristic
69C24×C4—21no—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C22×SD16C2318yes—
73C23×C8C2314yes—
74#535—48no—
75#532—44no—
76C2×C4⋊Q8C2311yescharacteristic
77C2×#127—22no—
78C2×#127C2311yescharacteristic
79C2×#124—22no—
80C2×#124—21no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—48no—
83C2×C22⋊Q8C2318yes—
84C2×#121—24no—
85C2×#121—21no—
86C2×#121C2312yes—
87C2×#121C2314yes—
88C2×#121C2314yes—
89C2×#120—22no—
90C2×#120—22no—
91C2×C4×D4—21no—
92C2×C4×D4—44no—
93C2×C4×D4D412yes—
94C2×C4×D4D412yes—
95C2×C4×D4C2312yes—
96C2×C4×D4C2312yes—
97C2×C4×D4C2312yes—
98C2×#117—21no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4D411yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4—22no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4D411yescharacteristic
105C22×C22⋊C4D411yescharacteristic
106C22×C42C2311yescharacteristic
107#465—416no—
108#448—48no—
109#448—416no—
110C2×#106C2318yes—
111C2×#103C2318yes—
112C2×#103C2318yes—
113C2×#103C2318yes—
114C2×#102C2318yes—
115C2×C22⋊C8—44no—
116C2×C22⋊C8C2314yes—
117C2×C22⋊C8C2318yes—
118C2×C22⋊C8C2314yes—
119#397—22no—
120#397—22no—
121#395—24no—
122#395—24no—
123#394—24no—
124#394—24no—
125#393—24no—
126#393—24no—
127#392—22no—
128#392—22no—
129#391—22no—
130#391—22no—
131#391—22no—
132#391C2314yes—
133#391C2314yes—
134#390—22no—
135#386—24no—
136#386—24no—
137#386—22no—
138#386—22no—
139#386—24no—
140#386—24no—
141#385—24no—
142#385—24no—
143#384C2312yes—
144#382—21no—
145#380—21no—
146#380—22no—
147#380—24no—
148#380C2314yes—
149#380C2314yes—
150#380C2312yes—
151#380C2314yes—
152C4×C4⋊C4—21no—
153C4×C4⋊C4C2312yes—
154C2×#95—21no—
155C2×#95—22no—
156C2×#95—22no—
157#340C2318yes—
158#340C2318yes—
159#336C2314yes—
160#336C2314yes—
161C25C2×D411yescharacteristic
162C25C2×D411yescharacteristic
163C2×#41—44no—
164C2×#41—44no—
165C2×#41—44no—
166C22×Q8—28no—
167C22×Q8—22no—
168C22×Q8—21no—
169C22×Q8—24no—
170C22×Q8—216no—
171C22×Q8—28no—
172C22×Q8—44no—
173C22×Q8—48no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×Q8—48no—
177C22×Q8—44no—
178C22×Q8—48no—
179C22×Q8—416no—
180C22×Q8—464no—
181C22×Q8C2414yes—
182C22×D4—22no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4SD1618yescomplemented
190C22×D4C2×D412yes—
191C22×D4C2×D412yes—
192C22×D4C2412yes—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—48no—
202C23×C4—44no—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C2×SD16—216no—
213C2×SD16—48no—
214C2×SD16—48no—
215C22×C8—44no—
216C22×C8—48no—
217C22×C8C2418yes—
218C4⋊Q8—42no—
219#127—42no—
220#127—48no—
221#124—44no—
222#124—48no—
223C22⋊Q8—216no—
224C22⋊Q8—48no—
225C22⋊Q8—416no—
226#121—28no—
227#121—44no—
228#121—416no—
229#121—48no—
230#121—48no—
231#120—48no—
232#120—48no—
233C4×D4—44no—
234C4×D4—44no—
235C4×D4—44no—
236C4×D4—42no—
237C4×D4—42no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—44no—
241#117—44no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2414yes—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4C2×D412yes—
269C2×C22⋊C4C2×D412yes—
270C2×C22⋊C4C2×D412yes—
271C2×C22⋊C4C2×D412yes—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278#106—48no—
279#103—216no—
280#103—48no—
281#103—48no—
282#102—48no—
283C22⋊C8—28no—
284C22⋊C8—44no—
285C22⋊C8—48no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—22no—
292#95—22no—
293#95—24no—
294#95—24no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24C2×SD1614yes—
305C24C2×SD1614yes—
306C24C22×D412yes—
307C24C22×D412yes—
308C24C22×D411yescharacteristic
309#41—816no—
310C2×Q8—28no—
311C2×Q8—24no—
312C2×Q8—28no—
313C2×Q8—28no—
314C2×Q8—416no—
315C2×Q8—432no—
316C2×Q8—44no—
317C2×Q8—48no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—416no—
321C2×Q8—432no—
322C2×Q8—464no—
323C2×Q8—432no—
324C2×Q8—416no—
325C2×Q8—44no—
326C2×Q8—42no—
327C2×Q8H(5,𝔽2)18yes—
328C2×D4—28no—
329C2×D4—24no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—416no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—42no—
344C2×D4—82no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4C2×SD1618yes—
350C2×D4C2×SD1618yes—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—28no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—416no—
376C22×C4—416no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—48no—
381C22×C4—432no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—88no—
388C22×C4C2×SD1614yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4H(5,𝔽2)14yes—
397C22×C4C2511yescharacteristic
398SD16—816no—
399C2×C8—28no—
400C2×C8—28no—
401C2×C8—28no—
402C2×C8—88no—
403C4⋊C4—28no—
404C4⋊C4—28no—
405C4⋊C4—24no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—416no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—82no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—416no—
418C22⋊C4—44no—
419C22⋊C4—48no—
420C22⋊C4—44no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—44no—
424C22⋊C4—84no—
425C22⋊C4—84no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—44no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—28no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—22no—
441C23—22no—
442C23—28no—
443C23—48no—
444C23—48no—
445C23—44no—
446C23—44no—
447C23—48no—
448C23—44no—
449C23—42no—
450C23—44no—
451C23—42no—
452C23—42no—
453C23—82no—
454C23—82no—
455C23—84no—
456C23#53412yes—
457C23C22×SD1614yes—
458C23C22×SD1614yes—
459C23C22×SD1612yes—
460C23#57512yes—
461C23C23×D411yescharacteristic
462Q8—216no—
463Q8—416no—
464Q8—432no—
465Q8—48no—
466Q8—84no—
467D4—48no—
468D4—48no—
469D4—48no—
470D4—84no—
471D4—84no—
472D4—88no—
473D4—168no—
474D4—168no—
475C2×C4—28no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—28no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—416no—
494C2×C4—432no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—84no—
511C2×C4—88no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—88no—
515C2×C4#53412yes—
516C2×C4C22×SD1612yes—
517C2×C4C22×SD1612yes—
518C2×C4C2×H(5,𝔽2)12yes—
519C2×C4C2×H(5,𝔽2)14yes—
520C8—48no—
521C22—22no—
522C22—22no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—28no—
528C22—28no—
529C22—416no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—44no—
535C22—44no—
536C22—88no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—84no—
541C22—84no—
542C22#318714yes—
543C22C2×#53411yescharacteristic
544C22#348412yes—
545C22C23×SD1611yescharacteristic
546C22C2×#57511yescharacteristic
547C22#358612yes—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—41no—
552C4—41no—
553C4—42no—
554C4—42no—
555C4—48no—
556C4—82no—
557C4—84no—
558C4—84no—
559C4#353214yes—
560C2—24no—
561C2—44no—
562C2—44no—
563C2—162no—
564C2#3670211yescharacteristic
565C2C2×#318711yescharacteristic
566C2#6075411yescharacteristic
567C2#6156212yes—
568C2#6157812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35165C2
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. C2×#2409C2semidirect
3. #32879C2semidirect
4. #31262C2semidirect
5. #30506C2semidirect
6. #30399C2semidirect
7. #29892C2semidirect
8. #27583C2semidirect
9. #24381C2semidirect
10. #23931C2semidirect
11. #21738C2semidirect
17. C2×#465C22semidirect
19. C2×#448C22semidirect
42. #2096C22semidirect
43. #2077C22semidirect
44. #2072C22semidirect
45. #2035C22semidirect
46. #2034C22semidirect
47. #2024C22semidirect
48. #2014C22semidirect
49. #2000C22semidirect
50. #1927C22semidirect
51. #1892C22semidirect
52. #1877C22semidirect
59. #1787C22semidirect
189. C22×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#351652
C2×#24092
#328792
#312622
#305062
#303992
#298922
#275832
#243812
#239312
#217382