Finite Group Databasebeta

← #6650287

Subgroups — #6650287

Order 512.

2298 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35000C211yesmaximal, characteristic
2C2×#2542C211yesmaximal, characteristic, complemented
3#33095C212yesmaximal, complemented
4#32027C214yesmaximal, complemented
5#30705C218yesmaximal, complemented
6#30500C214yesmaximal, complemented
7#29624C212yesmaximal, complemented
8#24381C214yesmaximal, complemented
9#24003C212yesmaximal, complemented
10#23919C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic, complemented
12C2×#534—21no—
13C22×C4⋊Q8C2211yescharacteristic, complemented
14C22×C22⋊Q8C2212yes—
15C22×C22⋊Q8C2212yescomplemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#483C2214yescomplemented
19C2×#465C2218yescomplemented
20C2×#448C2214yes—
21C2×#450C2214yescomplemented
22C22×C22⋊C8C2212yescomplemented
23#2854C2212yes—
24#2847C2212yes—
25#2618—21no—
26#2617—21no—
27#2543—22no—
28#2542—22no—
29#2542—22no—
30#2542C2214yes—
31#2542C2214yescomplemented
32#2534—22no—
33#2486C2212yes—
34#2416C2214yes—
35#2408C2212yes—
36C2×#399—21no—
37C2×#399—21no—
38#2308—21no—
39C2×#390—21no—
40C2×#384—21no—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic, complemented
43#2043C2218yescomplemented
44#2035C2218yescomplemented
45#2035C2218yescomplemented
46#2013C2218yescomplemented
47#2000C2218yescomplemented
48#2000C2218yescomplemented
49#1926C2218yescomplemented
50#1919C2218yescomplemented
51#1892C2218yescomplemented
52#1874C2214yes—
53#1872C2214yescomplemented
54#1872C2214yescomplemented
55#1869C2214yes—
56#1869C2214yescomplemented
57#1853C2214yescomplemented
58#1817C2212yescomplemented
59#1787C2214yescomplemented
60#1779C2212yes—
61C23×Q8C2312yes—
62C23×D4—21no—
63C23×D4D411yescharacteristic
64C23×D4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C22×SD16C2318yes—
68C23×C8C2314yes—
69#534—48no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—44no—
72C2×C4⋊Q8C2314yes—
73C2×#123—22no—
74C2×#123—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—48no—
79C2×C22⋊Q8—48no—
80C2×C22⋊Q8C2314yes—
81C2×C22⋊Q8C2314yes—
82C2×C22⋊Q8C2318yes—
83C2×#121—24no—
84C2×#121—21no—
85C2×#121—21no—
86C2×#121—44no—
87C2×#121C2312yes—
88C2×#121C2314yes—
89C2×#121C2312yes—
90C2×#120—22no—
91C2×#120—22no—
92C2×C4×D4—44no—
93C2×C4×D4D412yes—
94C2×C4×D4D412yes—
95C2×C4×D4C2312yes—
96C2×C4×D4C2312yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4D411yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—21no—
106C22×C22⋊C4—22no—
107C22×C22⋊C4—22no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4D411yescharacteristic
110C22×C22⋊C4D411yescharacteristic
111C22×C42—21no—
112C22×C42C2311yescharacteristic
113#483—48no—
114#465—416no—
115#450—48no—
116#448—48no—
117C2×#106C2318yes—
118C2×#103C2318yes—
119C2×#103C2318yes—
120C2×#103C2318yes—
121C2×#102C2318yes—
122C2×C22⋊C8—44no—
123C2×C22⋊C8C2314yes—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2314yes—
127#399—24no—
128#399—24no—
129#399—24no—
130#399—24no—
131#398—22no—
132#397—22no—
133#396—24no—
134#394—22no—
135#394—24no—
136#392—22no—
137#390—24no—
138#390—24no—
139#390C2314yes—
140#389C2314yes—
141#388—24no—
142#386—22no—
143#386—22no—
144#386—24no—
145#386—24no—
146#384—24no—
147#384—24no—
148#381—22no—
149#380—21no—
150#380—21no—
151#380C2312yes—
152#380C2314yes—
153#380C2312yes—
154#379—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4C2314yes—
157C2×#95—21no—
158C2×#95—21no—
159#340C2318yes—
160#340C2318yes—
161#336C2318yes—
162C25—21no—
163C25C2×D411yescharacteristic
164C25C2×D411yescharacteristic
165C22×Q8—22no—
166C22×Q8—28no—
167C22×Q8—44no—
168C22×Q8—44no—
169C22×Q8C2414yes—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—24no—
173C22×D4—21no—
174C22×D4—42no—
175C22×D4—44no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4SD1618yescomplemented
182C22×D4C2×D412yes—
183C22×D4C2×D412yes—
184C22×D4C2412yes—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—21no—
190C23×C4—22no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C2×SD16—48no—
205C2×SD16—48no—
206C2×SD16—48no—
207C22×C8—44no—
208C22×C8—48no—
209C22×C8C2418yes—
210C4⋊Q8—44no—
211C4⋊Q8—44no—
212C4⋊Q8—48no—
213#123—48no—
214#123—48no—
215C22⋊Q8—28no—
216C22⋊Q8—216no—
217C22⋊Q8—28no—
218C22⋊Q8—44no—
219C22⋊Q8—44no—
220C22⋊Q8—48no—
221C22⋊Q8—48no—
222C22⋊Q8—48no—
223C22⋊Q8—416no—
224#121—28no—
225#121—44no—
226#121—44no—
227#121—44no—
228#121—416no—
229#121—48no—
230#120—48no—
231#120—48no—
232C4×D4—44no—
233C4×D4—44no—
234C4×D4—42no—
235C4×D4—42no—
236C4×D4—42no—
237C4×D4—42no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2414yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D412yes—
280C2×C22⋊C4C2×D412yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—44no—
284C2×C42C2×D412yes—
285C2×C42C2412yes—
286#106—216no—
287#103—48no—
288#103—48no—
289#103—48no—
290#102—216no—
291C22⋊C8—28no—
292C22⋊C8—44no—
293C22⋊C8—44no—
294C22⋊C8—44no—
295#95—24no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—24no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—82no—
313C24—82no—
314C24C2×SD1614yes—
315C24C2×SD1614yes—
316C24C22×D411yescharacteristic
317C24C22×D412yes—
318C24C22×D411yescharacteristic
319C24C22×D411yescharacteristic
320C2×Q8—24no—
321C2×Q8—28no—
322C2×Q8—28no—
323C2×Q8—24no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—816no—
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—44no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—82no—
344C2×D4—88no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—816no—
348C2×D4C2×SD1618yes—
349C2×D4C2×SD1618yes—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4C2×SD1614yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4H(5,𝔽2)14yes—
398C22×C4C2511yescharacteristic
399SD16—816no—
400C2×C8—28no—
401C2×C8—28no—
402C2×C8—28no—
403C2×C8—88no—
404C4⋊C4—28no—
405C4⋊C4—24no—
406C4⋊C4—28no—
407C4⋊C4—24no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—416no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—42no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—82no—
418C4⋊C4H(5,𝔽2)18yes—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—48no—
424C22⋊C4—416no—
425C22⋊C4—48no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—48no—
430C22⋊C4—44no—
431C22⋊C4—48no—
432C22⋊C4—84no—
433C22⋊C4—84no—
434C42—24no—
435C42—22no—
436C42—22no—
437C42—48no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—22no—
443C23—21no—
444C23—21no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—22no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—28no—
453C23—28no—
454C23—48no—
455C23—48no—
456C23—44no—
457C23—42no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—84no—
466C23—84no—
467C23—84no—
468C23—84no—
469C23—84no—
470C23—164no—
471C23—164no—
472C23#53412yes—
473C23C22×SD1612yes—
474C23C22×SD1614yes—
475C23C22×SD1612yes—
476C23C22×SD1612yes—
477C23#57512yes—
478C23C23×D411yescharacteristic
479Q8—48no—
480Q8—48no—
481Q8—88no—
482D4—48no—
483D4—48no—
484D4—48no—
485D4—84no—
486D4—88no—
487D4—88no—
488D4—168no—
489C2×C4—28no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—28no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—44no—
504C2×C4—416no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—84no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—84no—
526C2×C4—88no—
527C2×C4—164no—
528C2×C4#53412yes—
529C2×C4C22×SD1614yes—
530C2×C4C2×H(5,𝔽2)12yes—
531C2×C4C2×H(5,𝔽2)14yes—
532C8—48no—
533C22—22no—
534C22—22no—
535C22—24no—
536C22—24no—
537C22—22no—
538C22—22no—
539C22—24no—
540C22—24no—
541C22—28no—
542C22—24no—
543C22—41no—
544C22—42no—
545C22—41no—
546C22—42no—
547C22—41no—
548C22—41no—
549C22—42no—
550C22—44no—
551C22—42no—
552C22—88no—
553C22—88no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—84no—
558C22—84no—
559C22—168no—
560C22#318714yes—
561C22C2×#53411yescharacteristic
562C22#348412yes—
563C22C23×SD1611yescharacteristic
564C22C2×#57511yescharacteristic
565C22#358612yes—
566C4—24no—
567C4—22no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—41no—
572C4—48no—
573C4—82no—
574C4—84no—
575C4—84no—
576C4#354314yes—
577C2—24no—
578C2—42no—
579C2—44no—
580C2—42no—
581C2—162no—
582C2#3723211yescharacteristic
583C2C2×#318711yescharacteristic
584C2#6075411yescharacteristic
585C2#6156212yes—
586C2#6157512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35000C2
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×#2542C2semidirect
3. #33095C2semidirect
4. #32027C2semidirect
5. #30705C2semidirect
6. #30500C2semidirect
7. #29624C2semidirect
8. #24381C2semidirect
9. #24003C2semidirect
10. #23919C2semidirect
11. #21738C2semidirect
13. C22×C4⋊Q8C22semidirect
15. C22×C22⋊Q8C22semidirect
18. C2×#483C22semidirect
19. C2×#465C22semidirect
21. C2×#450C22semidirect
22. C22×C22⋊C8C22semidirect
31. #2542C22semidirect
42. C2×C4×C4⋊C4C22semidirect
43. #2043C22semidirect
44. #2035C22semidirect
45. #2035C22semidirect
46. #2013C22semidirect
47. #2000C22semidirect
48. #2000C22semidirect
49. #1926C22semidirect
50. #1919C22semidirect
51. #1892C22semidirect
53. #1872C22semidirect
54. #1872C22semidirect
56. #1869C22semidirect
57. #1853C22semidirect
58. #1817C22semidirect
59. #1787C22semidirect
181. C22×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#350002
C2×#25422
#330952
#320272
#307052
#305002
#296242
#243812
#240032
#239192
#217382