Finite Group Databasebeta

← #6470949

Subgroups — #6470949

Order 512.

2254 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic, complemented
2C2×#2361C211yesmaximal, characteristic
3#33148C212yesmaximal, complemented
4#32956C212yesmaximal, complemented
5#29968C212yesmaximal, complemented
6#29952C212yesmaximal, complemented
7#26410C218yesmaximal, complemented
8#26338C218yesmaximal, complemented
9#21930C211yesmaximal, characteristic, complemented
10#21758C212yesmaximal
11C2×#1853C212yesmaximal, complemented
12C2×#543—21no—
13C22×#121C2211yescharacteristic
14C22×C4×D4—21no—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#476C2214yescomplemented
18C22×#106C2214yescomplemented
19C22×C22⋊C8C2212yes—
20#2857C2212yes—
21#2857C2212yes—
22#2659—21no—
23#2659—21no—
24#2549—24no—
25#2440C2214yescomplemented
26#2440C2214yescomplemented
27#2361—24no—
28#2361C2218yes—
29#2315—21no—
30C2×#390C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#388—21no—
33C2×#386—21no—
34C2×#385—21no—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#2100C2214yescomplemented
39#2094C2214yescomplemented
40#2088C2218yescomplemented
41#2065C2218yescomplemented
42#2058C2214yescomplemented
43#2023C2214yescomplemented
44#1987C2214yescomplemented
45#1948C2214yescomplemented
46#1920C2218yescomplemented
47#1893C2218yes—
48#1874C2214yes—
49#1874C2214yes—
50#1864C2218yescomplemented
51#1853C2214yes—
52#1853C2214yes—
53#1853C2218yescomplemented
54#1850C2218yescomplemented
55#1823C2214yes—
56#1802C2214yescomplemented
57#1797C2212yes—
58#1797C2212yes—
59#1786C2214yes—
60C2×#340C2214yescomplemented
61C23×D4—21no—
62C23×D4C2311yescharacteristic
63C24×C4—21no—
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C23×C8C2314yes—
67#543—48no—
68C2×C4⋊Q8C2312yes—
69C2×C4⋊Q8C2312yes—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8—24no—
74C2×#121—22no—
75C2×#121—24no—
76C2×#121Q818yescomplemented
77C2×#121C2314yes—
78C2×C4×D4—28no—
79C2×C4×D4—22no—
80C2×C4×D4—44no—
81C2×C4×D4—48no—
82C2×C4×D4C2314yes—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4D411yescharacteristic
99C22×C22⋊C4D411yescharacteristic
100C22×C42—21no—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103#476—48no—
104C2×#106—48no—
105C2×#106C2318yes—
106C2×#106C2318yes—
107C2×C4⋊C8C2314yes—
108C2×C4⋊C8C2314yes—
109C2×#102C2318yes—
110C2×#102C2318yes—
111C2×C22⋊C8—28no—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2314yes—
114#398—24no—
115#398—24no—
116#395—24no—
117#395—24no—
118#391—22no—
119#391—22no—
120#390—22no—
121#390C2314yes—
122#389—22no—
123#389—22no—
124#389—22no—
125#389C2314yes—
126#389C2314yes—
127#389C2314yes—
128#388—28no—
129#386—28no—
130#385—24no—
131#385—24no—
132#385—28no—
133#384C2312yes—
134#384C2312yes—
135#380—22no—
136#380—22no—
137#380—22no—
138#380—22no—
139#380C2314yes—
140#380C2314yes—
141C4×C4⋊C4—22no—
142C4×C4⋊C4C2314yes—
143C4×C22⋊C4D414yes—
144C4×C22⋊C4C2314yes—
145C2×#95—21no—
146C2×#95—22no—
147C2×#95—21no—
148C2×#95D411yescharacteristic
149C2×#95D411yescharacteristic
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#336C2314yes—
155#336C2314yes—
156C25—21no—
157C25C2×D411yescharacteristic
158C22×Q8—22no—
159C22×Q8—22no—
160C22×D4—28no—
161C22×D4—24no—
162C22×D4—21no—
163C22×D4—22no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4C2×Q814yes—
169C22×D4C2412yes—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—82no—
184C23×C4—82no—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×Q814yes—
192C23×C4#4114yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C22×C8—28no—
196C22×C8—44no—
197C22×C8—44no—
198C22×C8C2418yes—
199C4⋊Q8—44no—
200C4⋊Q8—44no—
201C22⋊Q8—416no—
202C22⋊Q8—416no—
203C22⋊Q8—48no—
204C22⋊Q8—48no—
205#121—216no—
206#121—28no—
207#121C2×Q8116yes—
208C4×D4—28no—
209C4×D4—44no—
210C4×D4—48no—
211C4×D4—432no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4C2×Q814yes—
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2414yes—
240C2×C4⋊C4C2414yes—
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2412yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—82no—
255C2×C22⋊C4SD1618yes—
256C2×C22⋊C4C2×D414yes—
257C2×C22⋊C4C2×D414yes—
258C2×C42—24no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—44no—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2412yes—
265C2×C42C2412yes—
266#106—216no—
267#106—48no—
268C4⋊C8—44no—
269C4⋊C8—44no—
270#102—48no—
271#102—48no—
272C22⋊C8—28no—
273C22⋊C8—28no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95C2×D414yes—
281#95C2×D414yes—
282C24—24no—
283C24—22no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24C22⋊Q814yes—
295C24C2×SD1614yes—
296C24AGL(1,ℤ/8)14yes—
297C24C22×D411yescharacteristic
298C24C22×D412yes—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—82no—
302C2×Q8—82no—
303C2×Q8—82no—
304C2×Q8—82no—
305C2×D4—216no—
306C2×D4—24no—
307C2×D4—22no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—48no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—432no—
317C2×D4C22×Q814yes—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—48no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—48no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4—816no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—816no—
366C22×C4C22⋊Q814yes—
367C22×C4C22⋊Q818yes—
368C22×C4C2×SD1614yes—
369C22×C4C2×SD1614yes—
370C22×C4C22×D412yes—
371C22×C4C22×D412yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C22×Q812yes—
377C22×C4C22×Q814yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4114yes—
380C22×C4C2511yescharacteristic
381C2×C8—28no—
382C2×C8—28no—
383C2×C8—28no—
384C2×C8—416no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—28no—
390C4⋊C4—24no—
391C4⋊C4—28no—
392C4⋊C4—48no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4C22×Q814yes—
406C4⋊C4H(5,𝔽2)14yes—
407C22⋊C4—28no—
408C22⋊C4—216no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—84no—
414C22⋊C4C2×SD16116yes—
415C42—24no—
416C42—24no—
417C42—42no—
418C42—42no—
419C42—48no—
420C23—21no—
421C23—22no—
422C23—21no—
423C23—22no—
424C23—21no—
425C23—22no—
426C23—21no—
427C23—22no—
428C23—24no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—48no—
436C23—48no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—48no—
444C23—84no—
445C23—84no—
446C23—84no—
447C23—84no—
448C23#476116yescomplemented
449C23C2×C22⋊Q812yes—
450C23C2×C22⋊Q814yes—
451C23C22×SD1614yes—
452C23C22×SD1612yes—
453C23C2×AGL(1,ℤ/8)14yes—
454C23C2×AGL(1,ℤ/8)12yes—
455C23C23×D411yescharacteristic
456Q8—164no—
457Q8—164no—
458D4—28no—
459D4—48no—
460D4—432no—
461D4—44no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—48no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—816no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—816no—
505C2×C4—84no—
506C2×C4—164no—
507C2×C4—164no—
508C2×C4C2×C22⋊Q814yes—
509C2×C4C2×C22⋊Q812yes—
510C2×C4C22×SD1614yes—
511C2×C4C22×SD1614yes—
512C2×C4C23×Q811yescharacteristic
513C2×C4C22×#4111yescharacteristic
514C2×C4C2×H(5,𝔽2)12yes—
515C2×C4#58512yes—
516C8—48no—
517C22—22no—
518C22—24no—
519C22—22no—
520C22—24no—
521C22—22no—
522C22—24no—
523C22—22no—
524C22—24no—
525C22—22no—
526C22—24no—
527C22—22no—
528C22—24no—
529C22—416no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—816no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22C2×#47614yes—
539C22C2×#47618yes—
540C22C22×C22⋊Q811yescharacteristic
541C22#348411yescharacteristic
542C22#348511yescharacteristic
543C22C23×SD1611yescharacteristic
544C22C22×AGL(1,ℤ/8)11yescharacteristic
545C22#358511yescharacteristic
546C22#358611yescharacteristic
547C4—24no—
548C4—24no—
549C4—22no—
550C4—24no—
551C4—42no—
552C4—44no—
553C4—44no—
554C4—42no—
555C4—84no—
556C4—82no—
557C4—84no—
558C4#345912yes—
559C4#350112yes—
560C2—24no—
561C2—28no—
562C2—84no—
563C2#3498511yescharacteristic
564C2C22×#47611yescharacteristic
565C2#6049211yescharacteristic
566C2#6156011yescharacteristic
567C2#6157211yescharacteristic
568C2#6169511yescharacteristic
569C2#6170711yescharacteristic
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
1. #35002C2semidirect
3. #33148C2semidirect
4. #32956C2semidirect
5. #29968C2semidirect
6. #29952C2semidirect
7. #26410C2semidirect
8. #26338C2semidirect
9. #21930C2semidirect
11. C2×#1853C2semidirect
17. C2×#476C22semidirect
18. C22×#106C22semidirect
25. #2440C22semidirect
26. #2440C22semidirect
38. #2100C22semidirect
39. #2094C22semidirect
40. #2088C22semidirect
41. #2065C22semidirect
42. #2058C22semidirect
43. #2023C22semidirect
44. #1987C22semidirect
45. #1948C22semidirect
46. #1920C22semidirect
50. #1864C22semidirect
53. #1853C22semidirect
54. #1850C22semidirect
56. #1802C22semidirect
60. C2×#340C22semidirect
76. C2×#121Q8semidirect
448. C23#476semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#23612
#331482
#329562
#299682
#299522
#264102
#263382
#219302
#217582
C2×#18532