Finite Group Databasebeta

← #6644831

Subgroups — #6644831

Order 512.

1974 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35000C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic, complemented
3#33095C212yesmaximal, complemented
4#32259C214yesmaximal, complemented
5#31322C218yesmaximal, complemented
6#31261C214yesmaximal, complemented
7#29639C212yesmaximal, complemented
8#27601C212yesmaximal, complemented
9#24553C214yesmaximal, complemented
10#24005C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic, complemented
12C2×#534—21no—
13C22×C4⋊Q8C2211yescharacteristic, complemented
14C22×#121C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#481C2214yes—
17C22×C22⋊C8C2212yescomplemented
18#2854C2212yes—
19#2847C2212yes—
20#2618—21no—
21#2617—21no—
22#2563—22no—
23#2563—22no—
24#2563C2214yes—
25#2563C2214yescomplemented
26#2543—22no—
27#2534—22no—
28#2486C2212yes—
29#2416C2214yes—
30#2408C2212yes—
31C2×#393—21no—
32C2×#393—21no—
33#2308—21no—
34C2×#390—21no—
35C2×#382—21no—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2212yescomplemented
39C2×C4×C4⋊C4C2211yescharacteristic, complemented
40#2077C2218yescomplemented
41#2077C2218yescomplemented
42#2058C2218yescomplemented
43#2034C2218yescomplemented
44#2034C2218yescomplemented
45#2023C2218yescomplemented
46#1926C2218yescomplemented
47#1919C2218yescomplemented
48#1893C2218yescomplemented
49#1874C2214yes—
50#1869C2214yes—
51#1869C2214yescomplemented
52#1853C2214yescomplemented
53#1853C2214yescomplemented
54#1853C2214yescomplemented
55#1824C2214yescomplemented
56#1823C2214yescomplemented
57#1817C2212yescomplemented
58#1807C2218yescomplemented
59#1786C2214yescomplemented
60#1779C2212yes—
61C23×Q8—21no—
62C23×D4—21no—
63C23×D4D411yescharacteristic
64C23×D4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C23×C8C2314yes—
68#534—48no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—44no—
71C2×C4⋊Q8C2314yes—
72C2×#123—22no—
73C2×#123—22no—
74C2×#121—24no—
75C2×#121—21no—
76C2×#121—21no—
77C2×#121—44no—
78C2×#121C2312yes—
79C2×#121C2314yes—
80C2×#121C2312yes—
81C2×#120—22no—
82C2×#120—22no—
83C2×C4×D4—44no—
84C2×C4×D4D412yes—
85C2×C4×D4D412yes—
86C2×C4×D4C2312yes—
87C2×C4×D4C2312yes—
88C22×C4⋊C4—21no—
89C22×C4⋊C4D411yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—22no—
97C22×C22⋊C4—22no—
98C22×C22⋊C4D411yescharacteristic
99C22×C22⋊C4D411yescharacteristic
100C22×C42—21no—
101C22×C42C2311yescharacteristic
102#481—48no—
103C2×#106C2318yes—
104C2×#106C2318yes—
105C2×#102C2318yes—
106C2×#102C2318yes—
107C2×C22⋊C8—44no—
108C2×C22⋊C8C2314yes—
109C2×C22⋊C8C2314yes—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112#398—22no—
113#397—22no—
114#396—24no—
115#394—22no—
116#394—24no—
117#393—24no—
118#393—24no—
119#393—24no—
120#393—24no—
121#392—22no—
122#390—24no—
123#390—24no—
124#390C2314yes—
125#389C2314yes—
126#388—24no—
127#386—22no—
128#386—22no—
129#386—24no—
130#386—24no—
131#382—24no—
132#382—24no—
133#381—22no—
134#380—22no—
135#380—22no—
136#380—24no—
137#380—21no—
138#380—21no—
139#380C2314yes—
140#380C2318yes—
141#380C2314yes—
142#380C2314yes—
143#380C2312yes—
144#380C2312yes—
145#379—22no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4C2314yes—
148C2×#95—21no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—21no—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156#336C2318yes—
157C25—21no—
158C25C2×D411yescharacteristic
159C25C2×D411yescharacteristic
160C22×Q8—24no—
161C22×Q8—44no—
162C22×Q8—84no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—24no—
166C22×D4—21no—
167C22×D4—42no—
168C22×D4—44no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4SD1618yes—
175C22×D4C2×D412yes—
176C22×D4C2×D412yes—
177C22×D4C2412yes—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C22×C8—48no—
198C22×C8—44no—
199C22×C8C2418yes—
200C4⋊Q8—44no—
201C4⋊Q8—44no—
202C4⋊Q8—48no—
203#123—48no—
204#123—48no—
205#121—28no—
206#121—44no—
207#121—44no—
208#121—44no—
209#121—416no—
210#121—48no—
211#120—48no—
212#120—48no—
213C4×D4—44no—
214C4×D4—44no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—42no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2412yes—
234C2×C4⋊C4C2414yes—
235C2×C4⋊C4C2414yes—
236C2×C4⋊C4C2414yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—84no—
253C2×C22⋊C4—84no—
254C2×C22⋊C4C2×D412yes—
255C2×C22⋊C4C2×D412yes—
256C2×C22⋊C4C2×D412yes—
257C2×C22⋊C4C2×D412yes—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—44no—
261C2×C42C2×D412yes—
262C2×C42C2412yes—
263#106—48no—
264#106—48no—
265#102—48no—
266#102—48no—
267C22⋊C8—28no—
268C22⋊C8—44no—
269C22⋊C8—44no—
270C22⋊C8—44no—
271#95—24no—
272#95—28no—
273#95—24no—
274#95—24no—
275#95—22no—
276#95—22no—
277#95—24no—
278#95—24no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—82no—
292C24—82no—
293C24C2×SD1614yes—
294C24C2×SD1614yes—
295C24C22×D411yescharacteristic
296C24C22×D412yes—
297C24C22×D411yescharacteristic
298C24C22×D411yescharacteristic
299C2×Q8—48no—
300C2×Q8—88no—
301C2×Q8—816no—
302C2×D4—24no—
303C2×D4—28no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—416no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—48no—
317C2×D4—82no—
318C2×D4—88no—
319C2×D4—84no—
320C2×D4—84no—
321C2×D4—816no—
322C2×D4C2×SD1618yes—
323C2×D4C2×SD1618yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—28no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4C2×SD1614yes—
365C22×C4C22×D412yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4H(5,𝔽2)14yes—
372C22×C4C2511yescharacteristic
373C2×C8—28no—
374C2×C8—28no—
375C2×C8—28no—
376C2×C8—88no—
377C4⋊C4—28no—
378C4⋊C4—24no—
379C4⋊C4—28no—
380C4⋊C4—28no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—48no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—82no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—48no—
397C22⋊C4—44no—
398C22⋊C4—48no—
399C22⋊C4—84no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C22⋊C4—88no—
403C22⋊C4—84no—
404C42—24no—
405C42—22no—
406C42—22no—
407C42—48no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—24no—
412C23—22no—
413C23—21no—
414C23—21no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—22no—
419C23—21no—
420C23—21no—
421C23—22no—
422C23—28no—
423C23—28no—
424C23—48no—
425C23—48no—
426C23—44no—
427C23—42no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—82no—
432C23—82no—
433C23—84no—
434C23—84no—
435C23—82no—
436C23—82no—
437C23—84no—
438C23—84no—
439C23—84no—
440C23—164no—
441C23—164no—
442C23#53412yes—
443C23C22×SD1612yes—
444C23C22×SD1614yes—
445C23C22×SD1612yes—
446C23C22×SD1612yes—
447C23#57512yes—
448C23C23×D411yescharacteristic
449Q8—816no—
450D4—48no—
451D4—48no—
452D4—48no—
453D4—84no—
454D4—88no—
455D4—88no—
456D4—168no—
457C2×C4—28no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—28no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—84no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—88no—
493C2×C4—84no—
494C2×C4—88no—
495C2×C4—164no—
496C2×C4#53412yes—
497C2×C4C22×SD1614yes—
498C2×C4C2×H(5,𝔽2)12yes—
499C2×C4#58514yes—
500C8—48no—
501C22—24no—
502C22—24no—
503C22—22no—
504C22—22no—
505C22—24no—
506C22—28no—
507C22—24no—
508C22—22no—
509C22—22no—
510C22—24no—
511C22—42no—
512C22—44no—
513C22—42no—
514C22—41no—
515C22—42no—
516C22—41no—
517C22—42no—
518C22—41no—
519C22—41no—
520C22—88no—
521C22—88no—
522C22—84no—
523C22—84no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—168no—
528C22#318714yes—
529C22C2×#53411yescharacteristic
530C22#348512yes—
531C22C23×SD1611yescharacteristic
532C22C2×#57511yescharacteristic
533C22#358512yes—
534C4—24no—
535C4—22no—
536C4—42no—
537C4—41no—
538C4—42no—
539C4—41no—
540C4—84no—
541C4—84no—
542C4—82no—
543C4—84no—
544C4#354514yes—
545C2—24no—
546C2—42no—
547C2—44no—
548C2—42no—
549C2—162no—
550C2#3711911yescharacteristic
551C2C2×#318711yescharacteristic
552C2#6075311yescharacteristic
553C2#6169712yes—
554C2#6171012yes—
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×#2563C2semidirect
3. #33095C2semidirect
4. #32259C2semidirect
5. #31322C2semidirect
6. #31261C2semidirect
7. #29639C2semidirect
8. #27601C2semidirect
9. #24553C2semidirect
10. #24005C2semidirect
11. #21738C2semidirect
13. C22×C4⋊Q8C22semidirect
17. C22×C22⋊C8C22semidirect
25. #2563C22semidirect
38. C2×#380C22semidirect
39. C2×C4×C4⋊C4C22semidirect
40. #2077C22semidirect
41. #2077C22semidirect
42. #2058C22semidirect
43. #2034C22semidirect
44. #2034C22semidirect
45. #2023C22semidirect
46. #1926C22semidirect
47. #1919C22semidirect
48. #1893C22semidirect
51. #1869C22semidirect
52. #1853C22semidirect
53. #1853C22semidirect
54. #1853C22semidirect
55. #1824C22semidirect
56. #1823C22semidirect
57. #1817C22semidirect
58. #1807C22semidirect
59. #1786C22semidirect

Maximal subgroups

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

subgroupindex
#350002
C2×#25632
#330952
#322592
#313222
#312612
#296392
#276012
#245532
#240052
#217382