Finite Group Databasebeta

← #6489546

Subgroups — #6489546

Order 512.

3400 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#125C211yesmaximal, characteristic, complemented
2C2×#3234C212yesmaximal, complemented
3#36624C211yesmaximal, characteristic
4#32979C212yesmaximal, complemented
5#32978C212yesmaximal, complemented
6#29640C214yesmaximal, complemented
7#25913C214yesmaximal, complemented
8#25819C214yesmaximal, complemented
9#24305C218yesmaximal, complemented
10#21838C212yesmaximal, complemented
11#20715C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C22×#125—22no—
14C22×#125—21no—
15C22×#125—21no—
16C22×#125—22no—
17C22×#125—216no—
18C22×#125C2214yes—
19C22×#125C2212yes—
20C22×#125C2214yes—
21C22×#125C2212yes—
22C2×#525—21no—
23C23×C4⋊C4—21no—
24C23×C4⋊C4—21no—
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C42C2211yescharacteristic
28#3234—216no—
29C2×#501C2214yescomplemented
30C2×#490C2214yescomplemented
31C2×#481C2214yes—
32C2×#480C2214yes—
33C2×#478C2218yescomplemented
34C2×#432C2212yes—
35#2857C2212yes—
36#2856C2212yes—
37#2854—21no—
38#2854C2212yes—
39#2850—22no—
40#2847—21no—
41#2847C2212yes—
42C4×#121—22no—
43C4×#121C2214yes—
44#2297—21no—
45#2285C2211yescharacteristic
46#2059C22116yescomplemented
47#2025C22116yescomplemented
48#1992C2218yescomplemented
49#1989C2218yescomplemented
50#1874C2218yes—
51#1874C2218yes—
52#1853C2218yes—
53#1852C2218yes—
54#1824C2214yes—
55#1823C2214yes—
56#1817C2212yes—
57#1797C2214yes—
58#1779C2212yes—
59#1749C2214yes—
60C22×#41—21no—
61C24×C4—21no—
62C24×C4—21no—
63C24×C4D411yescharacteristic
64C24×C4D411yescharacteristic
65C24×C4C2311yescharacteristic
66#532—28no—
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#125—216no—
70C2×#125—216no—
71C2×#125—232no—
72C2×#125—232no—
73C2×#125—232no—
74C2×#125—232no—
75C2×#125—28no—
76C2×#125—28no—
77C2×#125—24no—
78C2×#125—24no—
79C2×#125—24no—
80C2×#125—24no—
81C2×#125—24no—
82C2×#125—22no—
83C2×#125—22no—
84C2×#125—28no—
85C2×#125D4116yescomplemented
86C2×#125C2318yes—
87C2×#125C2314yes—
88C2×#125C2314yes—
89C2×#124—21no—
90#525—48no—
91C2×#123—22no—
92C2×C22⋊Q8—22no—
93C2×#121—21no—
94C2×#121—22no—
95C2×#121—22no—
96C2×#121C2312yes—
97C2×#121C2314yes—
98C2×#120—22no—
99C2×C4×D4—24no—
100C2×C4×D4—21no—
101C2×C4×D4C2312yes—
102C2×#117—21no—
103C22×C4⋊C4—216no—
104C22×C4⋊C4—216no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—28no—
118C22×C4⋊C4—28no—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C42—28no—
124C22×C42D412yes—
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D412yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#501—216no—
133#490—216no—
134#481—216no—
135#480—216no—
136#478—232no—
137#432—28no—
138C2×#107C2318yes—
139C2×#106C2318yes—
140C2×C4⋊C8C2314yes—
141C2×#102C2318yes—
142C2×#102C2318yes—
143C2×C22⋊C8C2318yes—
144C2×C22⋊C8C2314yes—
145C2×#97C2314yes—
146#391—22no—
147#390—22no—
148#390—22no—
149#390C2314yes—
150#389—22no—
151#389C2314yes—
152#388—24no—
153#388—24no—
154#386—24no—
155#386—22no—
156#386—24no—
157#386—22no—
158#384C2312yes—
159#382—21no—
160#380—21no—
161#380—22no—
162#380C2312yes—
163#380C2314yes—
164C4×C4⋊C4—21no—
165C4×C4⋊C4C2312yes—
166C4×C22⋊C4—24no—
167#340C23116yes—
168#340C23116yes—
169#336C2318yes—
170C25C2×D411yescharacteristic
171C2×#41—44no—
172C2×#41—44no—
173C2×#41—44no—
174C22×Q8—21no—
175C22×D4—24no—
176C22×D4—22no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4C2412yes—
180C23×C4—24no—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—48no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C22×C8C2418yes—
210C4⋊Q8—24no—
211#127—24no—
212#125—232no—
213#125—232no—
214#125—232no—
215#125—232no—
216#125—232no—
217#125—216no—
218#125—216no—
219#125—216no—
220#125—216no—
221#125—28no—
222#125—28no—
223#125C2×D4116yes—
224#125C2×D4116yes—
225#124—44no—
226#123—48no—
227C22⋊Q8—48no—
228#121—28no—
229#121—216no—
230#121—48no—
231#121—48no—
232#121—44no—
233#120—48no—
234C4×D4—28no—
235C4×D4—416no—
236C4×D4—44no—
237#117—44no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—216no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—232no—
244C2×C4⋊C4—232no—
245C2×C4⋊C4—232no—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—216no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—216no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2414yes—
273C2×C4⋊C4C2414yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—28no—
286C2×C42—28no—
287C2×C42—28no—
288C2×C42—22no—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D418yes—
297C2×C42C2412yes—
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300#107—216no—
301#106—216no—
302C4⋊C8—28no—
303#102—216no—
304#102—216no—
305C22⋊C8—28no—
306C22⋊C8—216no—
307#97—28no—
308#95—24no—
309#95—24no—
310#95—22no—
311#95—22no—
312C24—22no—
313C24—24no—
314C24—24no—
315C24—22no—
316C24—22no—
317C24AGL(1,ℤ/8)14yes—
318C24C22×D411yescharacteristic
319C24C22×D412yes—
320#41—816no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×D4—24no—
325C2×D4—28no—
326C2×D4—42no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4—44no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—44no—
333C2×D4—48no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—42no—
337C22×C4—232no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—216no—
349C22×C4—216no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—416no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—416no—
378C22×C4—432no—
379C22×C4—432no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—44no—
385C22×C4—48no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4#12014yes—
389C22×C4#12014yes—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D414yes—
398C22×C4C22×D412yes—
399C22×C42−1+418yes—
400C22×C4C2511yescharacteristic
401C2×C8—28no—
402C2×C8—216no—
403C4⋊C4—216no—
404C4⋊C4—216no—
405C4⋊C4—232no—
406C4⋊C4—232no—
407C4⋊C4—216no—
408C4⋊C4—216no—
409C4⋊C4—216no—
410C4⋊C4—216no—
411C4⋊C4—28no—
412C4⋊C4—28no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—28no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4C22×D418yes—
422C4⋊C4C22×D418yes—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—44no—
426C22⋊C4—48no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—48no—
430C42—24no—
431C42—24no—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—24no—
436C42—44no—
437C42#12014yes—
438C42#12014yes—
439C42C22×D414yes—
440C42C22×D414yes—
441C23—28no—
442C23—28no—
443C23—216no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—216no—
448C23—28no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—42no—
453C23—44no—
454C23—42no—
455C23—44no—
456C23—42no—
457C23C2×#12012yes—
458C23#53614yes—
459C23C2×AGL(1,ℤ/8)14yes—
460C23C2×AGL(1,ℤ/8)12yes—
461C23#57514yes—
462C23#57512yes—
463C23C23×D411yescharacteristic
464Q8—84no—
465D4—48no—
466D4—84no—
467D4—84no—
468D4—816no—
469D4—88no—
470C2×C4—216no—
471C2×C4—28no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—28no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—28no—
483C2×C4—216no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—44no—
493C2×C4—48no—
494C2×C4—48no—
495C2×C4—48no—
496C2×C4—48no—
497C2×C4—432no—
498C2×C4—44no—
499C2×C4—416no—
500C2×C4—432no—
501C2×C4—44no—
502C2×C4—416no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—88no—
513C2×C4C2×#12012yes—
514C2×C4C2×#12012yes—
515C2×C4C2×#12012yes—
516C2×C4C2×#12012yes—
517C2×C4C2×#12012yes—
518C2×C4#53614yes—
519C2×C4C23×D412yes—
520C2×C4C2×2−1+414yes—
521C8—48no—
522C22—232no—
523C22—22no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—22no—
532C22—24no—
533C22—82no—
534C22—84no—
535C22—84no—
536C22—82no—
537C22—82no—
538C22#300614yes—
539C22#323418yescomplemented
540C22C22×#12011yescharacteristic
541C22C2×#53612yes—
542C22C22×AGL(1,ℤ/8)11yescharacteristic
543C22C2×#57512yes—
544C22C2×#57511yescharacteristic
545C4—22no—
546C4—24no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—42no—
552C4—48no—
553C4—42no—
554C4—48no—
555C4—84no—
556C4—82no—
557C4#348614yes—
558C2—28no—
559C2—24no—
560C2—162no—
561C2#3524211yescharacteristic
562C2C2×#300611yescharacteristic
563C2#6035911yescharacteristic
564C2C2×#323412yes—
565C2#6086112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×#125C2semidirect
2. C2×#3234C2semidirect
4. #32979C2semidirect
5. #32978C2semidirect
6. #29640C2semidirect
7. #25913C2semidirect
8. #25819C2semidirect
9. #24305C2semidirect
10. #21838C2semidirect
11. #20715C2semidirect
29. C2×#501C22semidirect
30. C2×#490C22semidirect
33. C2×#478C22semidirect
46. #2059C22semidirect
47. #2025C22semidirect
48. #1992C22semidirect
49. #1989C22semidirect
85. C2×#125D4semidirect
539. C22#3234semidirect

Maximal subgroups

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

subgroupindex
C23×#1252
C2×#32342
#366242
#329792
#329782
#296402
#259132
#258192
#243052
#218382
#207152