Finite Group Databasebeta

← #6470967

Subgroups — #6470967

Order 512.

2048 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic, complemented
2C2×#2363C211yesmaximal, characteristic
3#33149C212yesmaximal, complemented
4#32953C212yesmaximal, complemented
5#29967C212yesmaximal, complemented
6#29952C212yesmaximal, complemented
7#26427C218yesmaximal, complemented
8#26357C218yesmaximal, complemented
9#21934C211yesmaximal, characteristic, complemented
10#21764C212yesmaximal
11#20846C212yesmaximal, complemented
12C2×#543—21no—
13C22×#121C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#474C2214yescomplemented
17C2×#428C2214yescomplemented
18C2×#407C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2659—21no—
22#2659—21no—
23#2549—24no—
24#2440C2214yescomplemented
25#2440C2214yescomplemented
26#2363—24no—
27#2363C2218yes—
28#2315—21no—
29C2×#390—21no—
30C2×#390C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#388—21no—
33C2×#386—21no—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37C2×C4×C22⋊C4C2211yescharacteristic
38#2099C2214yescomplemented
39#2094C2214yescomplemented
40#2088C2218yescomplemented
41#2065C2218yescomplemented
42#2058C2214yescomplemented
43#2023C2214yescomplemented
44#1987C2214yescomplemented
45#1947C2214yescomplemented
46#1921C2218yescomplemented
47#1895C2218yes—
48#1874C2214yes—
49#1874C2214yes—
50#1862C2218yescomplemented
51#1854C2218yescomplemented
52#1853C2214yes—
53#1852C2214yes—
54#1851C2218yescomplemented
55#1824C2214yes—
56#1802C2214yescomplemented
57#1797C2212yes—
58#1797C2212yes—
59#1786C2214yes—
60#1736C2214yescomplemented
61C23×D4—21no—
62C23×D4C2311yescharacteristic
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C22×M16C2314yes—
66#543—48no—
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2312yes—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—24no—
72C2×C22⋊Q8—24no—
73C2×#121—22no—
74C2×#121—24no—
75C2×#121Q818yescomplemented
76C2×#121C2314yes—
77C2×C4×D4—22no—
78C2×C4×D4—44no—
79C2×C4×D4C2314yes—
80C2×#117—22no—
81C2×#117—44no—
82C2×#117C2314yes—
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4D411yescharacteristic
96C22×C22⋊C4D411yescharacteristic
97C22×C42—21no—
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101#474—48no—
102#428—48no—
103C2×#107C2318yes—
104C2×#106C2318yes—
105C2×C4⋊C8C2314yes—
106C2×C4⋊C8C2314yes—
107C2×#102C2318yes—
108C2×#102C2318yes—
109#407—28no—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112#398—24no—
113#398—24no—
114#395—24no—
115#395—24no—
116#391—22no—
117#391—22no—
118#390—28no—
119#390—22no—
120#390C2314yes—
121#389—22no—
122#389—22no—
123#389—22no—
124#389C2314yes—
125#389C2314yes—
126#389C2314yes—
127#388—28no—
128#386—28no—
129#385—24no—
130#385—24no—
131#384C2312yes—
132#384C2312yes—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380C2314yes—
138#380C2314yes—
139C4×C4⋊C4—22no—
140C4×C4⋊C4C2314yes—
141C4×C22⋊C4—28no—
142C4×C22⋊C4D414yes—
143C4×C22⋊C4C2314yes—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95—22no—
147C2×#95D411yescharacteristic
148C2×#95D411yescharacteristic
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#336C2314yes—
154#336C2314yes—
155C25—21no—
156C25C2×D411yescharacteristic
157C22×Q8—22no—
158C22×Q8—22no—
159C22×D4—28no—
160C22×D4—24no—
161C22×D4—21no—
162C22×D4—22no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—82no—
166C22×D4—82no—
167C22×D4C2×Q814yes—
168C22×D4C2412yes—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—82no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×Q814yes—
187C23×C4#4114yes—
188C23×C4C2411yescharacteristic
189C23×C4C2412yes—
190C2×M16—28no—
191C2×M16—44no—
192C2×M16—44no—
193C22×C8C2418yes—
194C4⋊Q8—44no—
195C4⋊Q8—44no—
196C22⋊Q8—416no—
197C22⋊Q8—416no—
198C22⋊Q8—48no—
199C22⋊Q8—48no—
200#121—216no—
201#121—28no—
202#121C2×Q8116yes—
203C4×D4—28no—
204C4×D4—44no—
205C4×D4—48no—
206#117—28no—
207#117—44no—
208#117—48no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4C2×Q814yes—
230C2×C4⋊C4C2412yes—
231C2×C4⋊C4C2414yes—
232C2×C4⋊C4C2414yes—
233C2×C4⋊C4C2412yes—
234C2×C4⋊C4C2412yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4C2×D414yes—
251C2×C22⋊C4C2×D414yes—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—44no—
257C2×C42—82no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2412yes—
261C2×C42C2412yes—
262#107—216no—
263#106—48no—
264C4⋊C8—44no—
265C4⋊C8—44no—
266#102—48no—
267#102—48no—
268C22⋊C8—28no—
269C22⋊C8—28no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95C2×D414yes—
277#95C2×D414yes—
278C24—22no—
279C24—22no—
280C24—24no—
281C24—22no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24C22⋊Q814yes—
293C24C22×D412yes—
294C24C22×D411yescharacteristic
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—82no—
298C2×Q8—82no—
299C2×Q8—82no—
300C2×Q8—82no—
301C2×D4—216no—
302C2×D4—24no—
303C2×D4—22no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—48no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—48no—
310C2×D4—84no—
311C2×D4—84no—
312C2×D4—816no—
313C2×D4C22×Q814yes—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—48no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—48no—
352C22×C4—84no—
353C22×C4—816no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4C22⋊Q814yes—
359C22×C4C22⋊Q818yes—
360C22×C4AGL(1,ℤ/8)14yes—
361C22×C4#13714yes—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×Q812yes—
369C22×C4C22×Q814yes—
370C22×C4C2×#4112yes—
371C22×C4C2×#4114yes—
372C22×C4C2511yescharacteristic
373M16—416no—
374C2×C8—28no—
375C2×C8—28no—
376C2×C8—28no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—28no—
382C4⋊C4—24no—
383C4⋊C4—28no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—48no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—84no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4C22×Q814yes—
397C4⋊C4H(5,𝔽2)14yes—
398C22⋊C4—216no—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—48no—
406C22⋊C4—84no—
407C42—24no—
408C42—24no—
409C42—42no—
410C42—42no—
411C42—48no—
412C42—84no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—28no—
424C23—24no—
425C23—44no—
426C23—44no—
427C23—48no—
428C23—48no—
429C23—42no—
430C23—42no—
431C23—48no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—84no—
441C23—84no—
442C23—84no—
443C23—84no—
444C23C2×C22⋊Q814yes—
445C23C2×C22⋊Q812yes—
446C23C2×AGL(1,ℤ/8)12yes—
447C23C2×#13712yes—
448C23C23×D411yescharacteristic
449Q8—164no—
450Q8—164no—
451D4—28no—
452D4—48no—
453D4—44no—
454D4—816no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—28no—
464C2×C4—24no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—44no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—48no—
480C2×C4—48no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—816no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—84no—
498C2×C4—164no—
499C2×C4—164no—
500C2×C4C2×C22⋊Q814yes—
501C2×C4C2×C22⋊Q812yes—
502C2×C4C2×AGL(1,ℤ/8)14yes—
503C2×C4C2×#13714yes—
504C2×C4C23×Q811yescharacteristic
505C2×C4C22×#4111yescharacteristic
506C2×C4C2×H(5,𝔽2)12yes—
507C2×C4#58512yes—
508C8—48no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—88no—
526C22—816no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22#307914yes—
532C22C22×C22⋊Q811yescharacteristic
533C22#348411yescharacteristic
534C22#348511yescharacteristic
535C22C22×AGL(1,ℤ/8)11yescharacteristic
536C22C22×#13711yescharacteristic
537C22#358511yescharacteristic
538C22#358511yescharacteristic
539C4—24no—
540C4—24no—
541C4—24no—
542C4—22no—
543C4—42no—
544C4—44no—
545C4—44no—
546C4—42no—
547C4—84no—
548C4—84no—
549C4—82no—
550C4#345912yes—
551C4#350112yes—
552C2—24no—
553C2—44no—
554C2—84no—
555C2#3498511yescharacteristic
556C2C2×#307911yescharacteristic
557C2#6049011yescharacteristic
558C2#6156911yescharacteristic
559C2#6158811yescharacteristic
560C2#6170811yescharacteristic
561C2#6172011yescharacteristic
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. #33149C2semidirect
4. #32953C2semidirect
5. #29967C2semidirect
6. #29952C2semidirect
7. #26427C2semidirect
8. #26357C2semidirect
9. #21934C2semidirect
11. #20846C2semidirect
16. C2×#474C22semidirect
17. C2×#428C22semidirect
24. #2440C22semidirect
25. #2440C22semidirect
38. #2099C22semidirect
39. #2094C22semidirect
40. #2088C22semidirect
41. #2065C22semidirect
42. #2058C22semidirect
43. #2023C22semidirect
44. #1987C22semidirect
45. #1947C22semidirect
46. #1921C22semidirect
50. #1862C22semidirect
51. #1854C22semidirect
54. #1851C22semidirect
56. #1802C22semidirect
60. #1736C22semidirect
75. C2×#121Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#23632
#331492
#329532
#299672
#299522
#264272
#263572
#219342
#217642
#208462