Finite Group Databasebeta

← #6506536

Subgroups — #6506536

Order 512.

2250 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×#2442C211yesmaximal, characteristic, complemented
3#29334C212yesmaximal, complemented
4#29334C212yesmaximal, complemented
5#27982C212yesmaximal, complemented
6#27717C218yesmaximal, complemented
7#27264C218yesmaximal, complemented
8#25913C212yesmaximal, complemented
9#21758C212yesmaximal, complemented
10C2×#1874C212yesmaximal, complemented
11#20526C211yesmaximal, characteristic, complemented
12C2×#534—21no—
13C2×#534—21no—
14C22×#125C2211yescharacteristic, complemented
15C22×#121C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17C2×#490C2214yescomplemented
18C22×#102C2214yescomplemented
19C22×C22⋊C8C2212yes—
20#2856C2212yes—
21#2856C2212yes—
22#2656—21no—
23#2656—21no—
24#2539—24no—
25#2501C2214yes—
26#2442—24no—
27#2442C2218yescomplemented
28#2433C2214yes—
29C2×#400—21no—
30C2×#398—21no—
31C2×#389C2211yescharacteristic, complemented
32C2×#389C2211yescharacteristic, complemented
33C2×#385—21no—
34C2×#382C2211yescharacteristic, complemented
35C2×#380—21no—
36C2×#380C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#2100C2214yescomplemented
39#2100C2214yescomplemented
40#2073C2218yescomplemented
41#2041C2214yescomplemented
42#2041C2214yescomplemented
43#2025C2218yescomplemented
44#1992C2214yescomplemented
45#1961C2214yescomplemented
46#1936C2218yescomplemented
47#1936C2218yescomplemented
48#1893C2218yes—
49#1874C2218yes—
50#1874C2214yescomplemented
51#1874C2214yescomplemented
52#1874C2214yescomplemented
53#1874C2214yescomplemented
54#1836C2214yescomplemented
55#1823C2214yescomplemented
56#1786C2214yescomplemented
57#1752C2218yescomplemented
58#1749C2212yescomplemented
59#1749C2212yescomplemented
60C2×#340C2214yescomplemented
61C23×D4—22no—
62C23×D4C2312yes—
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C23×C8C2314yes—
66#534—48no—
67#534—48no—
68C2×#127C2312yes—
69C2×#127C2312yes—
70C2×#125—22no—
71C2×#125—44no—
72C2×#125C2314yes—
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—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4—21no—
101C22×C22⋊C4—22no—
102C22×C22⋊C4D411yescharacteristic
103C22×C22⋊C4D411yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106#490—48no—
107C2×#102—48no—
108C2×#102C2318yes—
109C2×#102C2318yes—
110C2×#102C2318yes—
111C2×#102C2318yes—
112C2×C22⋊C8—28no—
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C2×#97C2314yes—
116C2×#97C2314yes—
117#400—28no—
118#399—24no—
119#399—24no—
120#398—28no—
121#392—24no—
122#392—24no—
123#390—22no—
124#390—22no—
125#390—22no—
126#390—22no—
127#390C2314yes—
128#390C2314yes—
129#389—22no—
130#389—22no—
131#389C2314yes—
132#389C2314yes—
133#385—28no—
134#384C2312yes—
135#384C2312yes—
136#382—22no—
137#382C2314yes—
138#380—22no—
139#380—28no—
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×D4—24no—
159C22×D4—24no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—48no—
170C22×D4—48no—
171C22×D4—48no—
172C22×D4C2414yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—24no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—82no—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4#4118yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C22×C8—28no—
194C22×C8—44no—
195C22×C8—44no—
196C22×C8C2418yes—
197#127—44no—
198#127—44no—
199#125—28no—
200#125—44no—
201#125—48no—
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—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4—82no—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2412yes—
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—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4C2×D414yes—
261C2×C22⋊C4C2×D414yes—
262C2×C42—42no—
263C2×C42—42no—
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—
275#97—44no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95C2×D414yes—
282#95C2×D414yes—
283C24—24no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—44no—
289C24—44no—
290C24—44no—
291C24—82no—
292C24—82no—
293C24—82no—
294C24—82no—
295C24#12414yes—
296C24AGL(1,ℤ/8)14yes—
297C24AGL(1,ℤ/8)14yes—
298C24C22×D412yes—
299C24C22×D411yescharacteristic
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—28no—
303C2×D4—28no—
304C2×D4—416no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—44no—
316C2×D4—88no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4—82no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—82no—
323C2×D4—82no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4—832no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—416no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—48no—
356C22×C4—42no—
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—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—48no—
370C22×C4—42no—
371C22×C4—84no—
372C22×C4—816no—
373C22×C4#12418yes—
374C22×C4#12414yes—
375C22×C4C22×D412yes—
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4114yes—
382C22×C4C2×#4114yes—
383C22×C4C2×#4114yes—
384C22×C4C2511yescharacteristic
385C2×C8—28no—
386C2×C8—28no—
387C2×C8—28no—
388C2×C8—88no—
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—48no—
398C4⋊C4—48no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4H(5,𝔽2)14yes—
410C22⋊C4—28no—
411C22⋊C4—28no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C42—24no—
420C42—24no—
421C42—42no—
422C42—42no—
423C23—21no—
424C23—22no—
425C23—21no—
426C23—22no—
427C23—21no—
428C23—22no—
429C23—21no—
430C23—22no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—88no—
445C23—88no—
446C23—88no—
447C23—164no—
448C23—164no—
449C23—164no—
450C23—164no—
451C23#490116yescomplemented
452C23C2×#12414yes—
453C23C2×#12412yes—
454C23C2×AGL(1,ℤ/8)14yes—
455C23C2×AGL(1,ℤ/8)14yes—
456C23C2×AGL(1,ℤ/8)12yes—
457C23C2×AGL(1,ℤ/8)12yes—
458C23C23×D411yescharacteristic
459D4—48no—
460D4—48no—
461D4—88no—
462D4—816no—
463D4—816no—
464D4—164no—
465D4—164no—
466D4—164no—
467D4—164no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—48no—
489C2×C4—44no—
490C2×C4—48no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—816no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4—88no—
514C2×C4C2×#12412yes—
515C2×C4C2×#12414yes—
516C2×C4C22×#4112yes—
517C2×C4C2×H(5,𝔽2)12yes—
518C2×C4#58512yes—
519C8—48no—
520C22—22no—
521C22—24no—
522C22—22no—
523C22—24no—
524C22—24no—
525C22—22no—
526C22—22no—
527C22—24no—
528C22—22no—
529C22—24no—
530C22—24no—
531C22—22no—
532C22—416no—
533C22—84no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22—1616no—
541C22C2×#49018yes—
542C22C2×#49014yes—
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—42no—
553C4—42no—
554C4—42no—
555C4—44no—
556C4—42no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—84no—
561C4#350514yes—
562C2—28no—
563C2—24no—
564C2—164no—
565C2#3537711yescharacteristic
566C2C22×#49011yescharacteristic
567C2#6062111yescharacteristic
568C2#6156911yescharacteristic
569C2#6157211yescharacteristic
570C2#6170411yescharacteristic
571C2#6170711yescharacteristic
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×#2442C2semidirect
3. #29334C2semidirect
4. #29334C2semidirect
5. #27982C2semidirect
6. #27717C2semidirect
7. #27264C2semidirect
8. #25913C2semidirect
9. #21758C2semidirect
10. C2×#1874C2semidirect
11. #20526C2semidirect
14. C22×#125C22semidirect
17. C2×#490C22semidirect
18. C22×#102C22semidirect
27. #2442C22semidirect
31. C2×#389C22semidirect
32. C2×#389C22semidirect
34. C2×#382C22semidirect
38. #2100C22semidirect
39. #2100C22semidirect
40. #2073C22semidirect
41. #2041C22semidirect
42. #2041C22semidirect
43. #2025C22semidirect
44. #1992C22semidirect
45. #1961C22semidirect
46. #1936C22semidirect
47. #1936C22semidirect
50. #1874C22semidirect
51. #1874C22semidirect
52. #1874C22semidirect
53. #1874C22semidirect
54. #1836C22semidirect
55. #1823C22semidirect
56. #1786C22semidirect
57. #1752C22semidirect
58. #1749C22semidirect
59. #1749C22semidirect
60. C2×#340C22semidirect
451. C23#490semidirect

Maximal subgroups

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

subgroupindex
#350012
C2×#24422
#293342
#279822
#277172
#272642
#259132
#217582
C2×#18742
#205262