Finite Group Databasebeta

← #6633283

Subgroups — #6633283

Order 512.

1688 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37421C211yesmaximal, characteristic, complemented
2#35679C211yesmaximal, characteristic, complemented
3#32915C211yesmaximal, characteristic, complemented
4#32912C211yesmaximal, characteristic, complemented
5#31647C214yesmaximal, complemented
6#31643C214yesmaximal, complemented
7#30748C214yesmaximal, complemented
8#30740C214yesmaximal, complemented
9#29723C211yesmaximal, characteristic, complemented
10#29721C211yesmaximal, characteristic
11#29333C212yesmaximal, complemented
12#25724C212yesmaximal, complemented
13#23946C212yesmaximal, complemented
14#23945C212yesmaximal, complemented
15#21913C211yesmaximal, characteristic, complemented
16C2×#542C2212yes—
17C22×#124C2211yescharacteristic
18C2×#519C2211yescharacteristic
19C2×#495C2214yes—
20C2×#462C2212yescomplemented
21C2×#460C2212yescomplemented
22C2×#451C2212yes—
23C2×#448C2212yes—
24C2×#450C2212yes—
25C2×#447C2212yes—
26#2850C2212yes—
27#2712—22no—
28#2702—24no—
29#2693—22no—
30#2621C2211yescharacteristic
31#2619C2211yescharacteristic
32#2568C2212yescomplemented
33#2563C2212yescomplemented
34#2545C2212yescomplemented
35#2544C2212yescomplemented
36#2482—22no—
37#2482—22no—
38#2448—24no—
39#2448C2214yes—
40#2448C2214yes—
41#2310C2211yescharacteristic
42#2309C2211yescharacteristic
43#2309C2211yescharacteristic
44#2296C2211yescharacteristic
45#2296C2211yescharacteristic
46#2101C2214yescomplemented
47#2073C2212yes—
48#2073C2212yes—
49#2055C2214yes—
50#2054C2214yes—
51#2042C2214yescomplemented
52#2024C2214yescomplemented
53#2023C2214yes—
54#2015C2214yes—
55#2015C2214yescomplemented
56#2013C2212yes—
57#2012C2214yes—
58#2012C2214yes—
59#2011C2212yes—
60#1958C2214yes—
61#1934C2214yescomplemented
62#1933C2214yescomplemented
63#1916C2214yescomplemented
64#1916C2214yescomplemented
65#1880C2214yes—
66#1880C2214yescomplemented
67#1879C2214yes—
68#1878C2214yescomplemented
69#1878C2214yescomplemented
70#1877C2214yescomplemented
71#1875C2212yes—
72#1875C2212yes—
73#1873C2212yes—
74#1873C2212yes—
75#1818C2212yes—
76#1805C2212yescomplemented
77#1805C2212yescomplemented
78#1798C2211yescharacteristic
79#1798C2211yescharacteristic
80C4×C22⋊C8C2212yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C22×Q16C2314yes—
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86C22×D8C2314yes—
87#542—44no—
88#542—44no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×#127C2311yescharacteristic
92C2×#126—22no—
93C2×#124—22no—
94C2×#124—22no—
95C2×#124—44no—
96C2×#124C2314yes—
97C2×#124C2312yes—
98C2×#124C2312yes—
99C2×#123—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8C2312yes—
102C2×#121C2312yes—
103#519—44no—
104C2×C4×Q8—22no—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×D4C2312yes—
109C2×#117—22no—
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C22×C22⋊C4—22no—
115C22×C42C2311yescharacteristic
116#495—48no—
117#462—44no—
118#460—44no—
119#451—44no—
120#450—44no—
121#448—44no—
122#447—44no—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2312yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C4×C8C2314yes—
136#400—24no—
137#399—24no—
138#399—24no—
139#398—22no—
140#397—22no—
141#397—24no—
142#397—24no—
143#396—24no—
144#394—22no—
145#394—24no—
146#393—24no—
147#392—22no—
148#391—22no—
149#391—24no—
150#391C2314yes—
151#391C2312yes—
152#391C2312yes—
153#390—22no—
154#390—22no—
155#390C2312yes—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#390C2314yes—
160#389D414yes—
161#389C2314yes—
162#388—24no—
163#388—24no—
164#388—24no—
165#388—24no—
166#388—24no—
167#388—24no—
168#386—24no—
169#386—24no—
170#385—22no—
171#385D412yes—
172#385D412yes—
173#382C2311yescharacteristic
174#382C2312yes—
175#382C2311yescharacteristic
176#381—21no—
177#381—21no—
178#380—24no—
179#379—21no—
180#379—21no—
181#379—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2312yes—
190#336C2312yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—21no—
195C22×Q8—24no—
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—42no—
201C22×Q8C2412yes—
202C22×Q8C2412yes—
203C22×D4—22no—
204C22×D4—21no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4C2412yes—
211C22×D4C2412yes—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4C2×D412yes—
215C23×C4C2411yescharacteristic
216C2×Q16—28no—
217C2×Q16—44no—
218C2×Q16—44no—
219C2×SD16—28no—
220C2×SD16—44no—
221C2×SD16—44no—
222C2×SD16—44no—
223C2×SD16—44no—
224C2×SD16—44no—
225C2×D8—44no—
226C2×D8—44no—
227C2×D8—44no—
228C22×C8C2414yes—
229C22×C8C2414yes—
230C4⋊Q8—42no—
231C4⋊Q8—42no—
232C4⋊Q8—42no—
233#127—42no—
234#126—48no—
235#124—28no—
236#124—44no—
237#124—42no—
238#124—42no—
239#124—44no—
240#124—48no—
241#124—48no—
242#123—48no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—48no—
246#121—24no—
247#121—24no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×Q8—42no—
251C4×Q8—42no—
252C4×Q8—42no—
253C4×Q8—42no—
254C4×Q8—48no—
255C4×D4—42no—
256C4×D4—42no—
257#117—22no—
258#117—22no—
259#117—42no—
260#117—42no—
261#117—42no—
262#117—48no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4C2×D411yescharacteristic
296C2×C22⋊C4C2×D411yescharacteristic
297C2×C22⋊C4C2×D412yes—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—22no—
303C2×C42—42no—
304C2×C42C2×D412yes—
305C2×C42C2412yes—
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308C2×C42C2412yes—
309C4⋊C8—42no—
310C4⋊C8—42no—
311#103—44no—
312#103—44no—
313#103—44no—
314#103—44no—
315#102—44no—
316#102—44no—
317#102—44no—
318#102—44no—
319C22⋊C8—24no—
320C22⋊C8—24no—
321C4×C8—44no—
322#95—24no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95C2×D414yes—
331#95C2×D414yes—
332C24—22no—
333C24—22no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24—84no—
338C24C22×D411yescharacteristic
339C2×Q8—24no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—24no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—24no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—42no—
352C2×Q8—44no—
353C2×Q8—42no—
354C2×Q8—84no—
355C2×Q8H(5,𝔽2)14yes—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—44no—
367C2×D4—42no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4—82no—
371C2×D4—84no—
372C2×D4—84no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—84no—
388C22×C4C22×D412yes—
389C22×C4C22×D414yes—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4H(5,𝔽2)12yes—
395C22×C4H(5,𝔽2)12yes—
396C22×C4C2511yescharacteristic
397Q16—88no—
398SD16—88no—
399SD16—88no—
400D8—88no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—84no—
424C4⋊C4—82no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—42no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C42—24no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—42no—
450C42—44no—
451C42—44no—
452C42—42no—
453C42—42no—
454C42—44no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—84no—
469C23—88no—
470C23—164no—
471C23—164no—
472C23#53512yes—
473C23C23×D411yescharacteristic
474Q8—24no—
475Q8—24no—
476Q8—48no—
477Q8—42no—
478Q8—42no—
479Q8—44no—
480Q8—44no—
481Q8—164no—
482D4—44no—
483D4—44no—
484D4—44no—
485D4—44no—
486D4—84no—
487D4—164no—
488D4—164no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4—164no—
521C2×C4#53512yes—
522C2×C4C2×H(5,𝔽2)11yescharacteristic
523C2×C4C2×H(5,𝔽2)11yescharacteristic
524C2×C4C2×H(5,𝔽2)12yes—
525C2×C4#58512yes—
526C8—44no—
527C8—44no—
528C22—22no—
529C22—24no—
530C22—42no—
531C22—42no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22—82no—
541C22—82no—
542C22—168no—
543C22C2×#53511yescharacteristic
544C22#348411yescharacteristic
545C22#348711yescharacteristic
546C22#358414yes—
547C4—22no—
548C4—44no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—44no—
553C4—42no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4—82no—
558C4—82no—
559C4#353412yes—
560C4#353412yes—
561C2—42no—
562C2—162no—
563C2—162no—
564C2#3671011yescharacteristic
565C2#6081912yes—
566C2#6163812yes—
567C2#6193112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#29721C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #37421C2semidirect
2. #35679C2semidirect
3. #32915C2semidirect
4. #32912C2semidirect
5. #31647C2semidirect
6. #31643C2semidirect
7. #30748C2semidirect
8. #30740C2semidirect
9. #29723C2semidirect
11. #29333C2semidirect
12. #25724C2semidirect
13. #23946C2semidirect
14. #23945C2semidirect
15. #21913C2semidirect
20. C2×#462C22semidirect
21. C2×#460C22semidirect
32. #2568C22semidirect
33. #2563C22semidirect
34. #2545C22semidirect
35. #2544C22semidirect
46. #2101C22semidirect
51. #2042C22semidirect
52. #2024C22semidirect
55. #2015C22semidirect
61. #1934C22semidirect
62. #1933C22semidirect
63. #1916C22semidirect
64. #1916C22semidirect
66. #1880C22semidirect
68. #1878C22semidirect
69. #1878C22semidirect
70. #1877C22semidirect
76. #1805C22semidirect
77. #1805C22semidirect

Maximal subgroups

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

subgroupindex
#374212
#356792
#329152
#329122
#316472
#316432
#307482
#307402
#297232
#297212
#293332
#257242
#239462
#239452
#219132