Finite Group Databasebeta

← #6506547

Subgroups — #6506547

Order 512.

2218 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35001C211yesmaximal, characteristic
2C2×#2439C211yesmaximal, characteristic, complemented
3#29334C212yesmaximal, complemented
4#29328C212yesmaximal, complemented
5#27982C212yesmaximal, complemented
6#27719C218yesmaximal, complemented
7#27273C218yesmaximal, complemented
8#25903C212yesmaximal, complemented
9#21764C212yesmaximal, complemented
10#20912C212yesmaximal, complemented
11#20524C211yesmaximal, characteristic, complemented
12C2×#534—21no—
13C2×#534—21no—
14C22×C4⋊Q8C2211yescharacteristic, complemented
15C22×#121C2212yes—
16C22×#117C2211yescharacteristic
17C2×#486C2214yescomplemented
18C2×#418C2214yescomplemented
19C2×#407C2212yes—
20#2856C2212yes—
21#2856C2212yes—
22#2656—21no—
23#2656—21no—
24#2539—24no—
25#2501C2214yes—
26#2439—24no—
27#2439C2218yescomplemented
28#2433C2214yes—
29C2×#396—21no—
30C2×#393—21no—
31C2×#391—21no—
32C2×#389C2211yescharacteristic, complemented
33C2×#389C2211yescharacteristic, complemented
34C2×#387—21no—
35C2×#382C2211yescharacteristic, complemented
36C2×#380C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#2100C2214yescomplemented
39#2099C2214yescomplemented
40#2073C2218yescomplemented
41#2041C2214yescomplemented
42#2041C2214yescomplemented
43#2023C2218yescomplemented
44#1990C2214yescomplemented
45#1961C2214yescomplemented
46#1936C2218yescomplemented
47#1934C2218yescomplemented
48#1895C2218yes—
49#1875C2218yes—
50#1874C2214yescomplemented
51#1874C2214yescomplemented
52#1874C2214yescomplemented
53#1874C2214yescomplemented
54#1836C2214yescomplemented
55#1824C2214yescomplemented
56#1786C2214yescomplemented
57#1753C2218yescomplemented
58#1749C2212yescomplemented
59C4×C22⋊C8C2212yescomplemented
60#1736C2214yescomplemented
61C23×Q8—21no—
62C23×D4—22no—
63C23×D4C2312yes—
64C24×C4C2311yescharacteristic
65C22×M16C2314yes—
66#534—48no—
67#534—48no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—44no—
70C2×C4⋊Q8C2314yes—
71C2×#127C2312yes—
72C2×#127C2312yes—
73C2×#123—24no—
74C2×#123—24no—
75C2×#121—22no—
76C2×#121—22no—
77C2×#121—22no—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121—22no—
81C2×#121—28no—
82C2×#121—28no—
83C2×#121C2314yes—
84C2×#121C2314yes—
85C2×#120—24no—
86C2×#120—24no—
87C2×#117—22no—
88C2×#117—44no—
89C2×#117C2314yes—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4—21no—
99C22×C22⋊C4—22no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4D411yescharacteristic
102C22×C22⋊C4D411yescharacteristic
103C22×C42—21no—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106#486—48no—
107#418—48no—
108C2×#102C2318yes—
109C2×#102C2318yes—
110C2×#102C2318yes—
111C2×#102C2318yes—
112#407—28no—
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C2×#97C2314yes—
116C2×C4×C8C2314yes—
117#399—24no—
118#399—24no—
119#396—28no—
120#393—28no—
121#392—24no—
122#392—24no—
123#391—28no—
124#390—22no—
125#390—22no—
126#390—22no—
127#390—22no—
128#390C2314yes—
129#390C2314yes—
130#389—22no—
131#389—22no—
132#389C2314yes—
133#389C2314yes—
134#387—28no—
135#384C2312yes—
136#384C2312yes—
137#382—22no—
138#382C2314yes—
139#380—22no—
140#380C2314yes—
141C4×C22⋊C4D414yes—
142C4×C22⋊C4C2314yes—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95D411yescharacteristic
149C2×#95D411yescharacteristic
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#336C2314yes—
155#336C2314yes—
156C25—22no—
157C25C2×D411yescharacteristic
158C22×Q8—24no—
159C22×Q8—44no—
160C22×Q8—44no—
161C22×Q8—82no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—48no—
174C22×D4—48no—
175C22×D4—48no—
176C22×D4C2414yes—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D412yes—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4#4118yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C2×M16—28no—
194C2×M16—44no—
195C2×M16—44no—
196C22×C8C2418yes—
197C4⋊Q8—28no—
198C4⋊Q8—44no—
199C4⋊Q8—48no—
200#127—44no—
201#127—44no—
202#123—416no—
203#123—416no—
204#121—28no—
205#121—28no—
206#121—28no—
207#121—28no—
208#121—48no—
209#121—48no—
210#121—48no—
211#121—48no—
212#121—48no—
213#121—48no—
214#120—416no—
215#120—416no—
216#117—28no—
217#117—44no—
218#117—48no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2414yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2412yes—
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—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4C2×D414yes—
259C2×C22⋊C4C2×D414yes—
260C2×C42—24no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—44no—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2412yes—
267C2×C42C2412yes—
268#102—216no—
269#102—48no—
270#102—48no—
271#102—48no—
272C22⋊C8—28no—
273C22⋊C8—28no—
274#97—44no—
275C4×C8—44no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95C2×D414yes—
282#95C2×D414yes—
283C24—22no—
284C24—22no—
285C24—24no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24—82no—
294C24—82no—
295C24—82no—
296C24—82no—
297C24#12414yes—
298C24C22×D412yes—
299C24C22×D411yescharacteristic
300C2×Q8—48no—
301C2×Q8—48no—
302C2×Q8—84no—
303C2×Q8—816no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—28no—
307C2×D4—28no—
308C2×D4—416no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—88no—
321C2×D4—82no—
322C2×D4—82no—
323C2×D4—82no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4—82no—
328C2×D4—82no—
329C2×D4—82no—
330C2×D4—82no—
331C2×D4—832no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—416no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4#12418yes—
372C22×C4#12414yes—
373C22×C4AGL(1,ℤ/8)14yes—
374C22×C4AGL(1,ℤ/8)14yes—
375C22×C4C22×D412yes—
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C2×#4114yes—
382C22×C4C2×#4114yes—
383C22×C4C2×#4114yes—
384C22×C4C2511yescharacteristic
385M16—88no—
386C2×C8—28no—
387C2×C8—28no—
388C2×C8—28no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—48no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—48no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—84no—
407C4⋊C4H(5,𝔽2)14yes—
408C22⋊C4—28no—
409C22⋊C4—28no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—84no—
418C42—24no—
419C42—24no—
420C42—42no—
421C42—42no—
422C42—48no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—28no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—88no—
448C23—88no—
449C23—88no—
450C23—164no—
451C23—164no—
452C23—164no—
453C23—164no—
454C23C2×#12414yes—
455C23C2×#12412yes—
456C23C2×AGL(1,ℤ/8)12yes—
457C23C2×AGL(1,ℤ/8)12yes—
458C23C23×D411yescharacteristic
459Q8—816no—
460D4—48no—
461D4—48no—
462D4—88no—
463D4—816no—
464D4—816no—
465D4—164no—
466D4—164no—
467D4—164no—
468D4—164no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—48no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—84no—
509C2×C4—84no—
510C2×C4—88no—
511C2×C4—84no—
512C2×C4C2×#12412yes—
513C2×C4C2×#12414yes—
514C2×C4C2×AGL(1,ℤ/8)14yes—
515C2×C4C2×AGL(1,ℤ/8)14yes—
516C2×C4C22×#4112yes—
517C2×C4C2×H(5,𝔽2)12yes—
518C2×C4#58512yes—
519C8—48no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—88no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—1616no—
542C22#313814yes—
543C22C22×#12411yescharacteristic
544C22#348411yescharacteristic
545C22#348511yescharacteristic
546C22C22×AGL(1,ℤ/8)11yescharacteristic
547C22C22×AGL(1,ℤ/8)11yescharacteristic
548C22#358511yescharacteristic
549C22#358611yescharacteristic
550C4—22no—
551C4—24no—
552C4—24no—
553C4—24no—
554C4—42no—
555C4—42no—
556C4—44no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—84no—
561C4#350514yes—
562C2—24no—
563C2—44no—
564C2—164no—
565C2#3537711yescharacteristic
566C2C2×#313811yescharacteristic
567C2#6062111yescharacteristic
568C2#6156911yescharacteristic
569C2#6157211yescharacteristic
570C2#6172011yescharacteristic
571C2#6172211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35001C2
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
2. C2×#2439C2semidirect
3. #29334C2semidirect
4. #29328C2semidirect
5. #27982C2semidirect
6. #27719C2semidirect
7. #27273C2semidirect
8. #25903C2semidirect
9. #21764C2semidirect
10. #20912C2semidirect
11. #20524C2semidirect
14. C22×C4⋊Q8C22semidirect
17. C2×#486C22semidirect
18. C2×#418C22semidirect
27. #2439C22semidirect
32. C2×#389C22semidirect
33. C2×#389C22semidirect
35. C2×#382C22semidirect
38. #2100C22semidirect
39. #2099C22semidirect
40. #2073C22semidirect
41. #2041C22semidirect
42. #2041C22semidirect
43. #2023C22semidirect
44. #1990C22semidirect
45. #1961C22semidirect
46. #1936C22semidirect
47. #1934C22semidirect
50. #1874C22semidirect
51. #1874C22semidirect
52. #1874C22semidirect
53. #1874C22semidirect
54. #1836C22semidirect
55. #1824C22semidirect
56. #1786C22semidirect
57. #1753C22semidirect
58. #1749C22semidirect
59. C4×C22⋊C8C22semidirect
60. #1736C22semidirect

Maximal subgroups

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

subgroupindex
#350012
C2×#24392
#293342
#293282
#279822
#277192
#272732
#259032
#217642
#209122
#205242