Finite Group Databasebeta

← #6531996

Subgroups — #6531996

Order 512.

1770 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic, complemented
3#27616C214yesmaximal, complemented
4#27588C214yesmaximal
5#25059C214yesmaximal, complemented
6#25057C214yesmaximal, complemented
7#24993C214yesmaximal, complemented
8#24991C214yesmaximal, complemented
9#21982C211yesmaximal, characteristic, complemented
10#21981C211yesmaximal, characteristic, complemented
11#21818C211yesmaximal, characteristic, complemented
12#20732C211yesmaximal, characteristic
13#20732C211yesmaximal, characteristic
14C22×C22⋊Q8—21no—
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2538—22no—
20#2538—22no—
21#2538C2214yes—
22#2538C2214yes—
23C2×#396—22no—
24#2356—22no—
25#2356—22no—
26#2355C2212yescomplemented
27#2355C2212yescomplemented
28#2355C2212yescomplemented
29#2355C2212yescomplemented
30#2313—22no—
31#2310—21no—
32C2×#390C2212yes—
33C2×#389C2212yes—
34#2285C2211yescharacteristic
35#2285C2211yescharacteristic
36C2×#384C2211yescharacteristic
37#2283C2214yes—
38C2×#380—21no—
39#2274—21no—
40#2101C2218yescomplemented
41#2092C2218yes—
42#2085C2218yescomplemented
43#2073C2218yes—
44#1984C2214yescomplemented
45#1984C2214yescomplemented
46#1948C2214yescomplemented
47#1947C2214yescomplemented
48#1926C2218yes—
49#1925C2218yes—
50#1921C2218yescomplemented
51#1920C2218yescomplemented
52#1853C2214yes—
53#1852C2214yes—
54#1845C2212yescomplemented
55#1844C2212yescomplemented
56#1834C2212yescomplemented
57#1834C2212yescomplemented
58#1823C2212yes—
59#1823C2212yes—
60#1817C2214yes—
61#1802C2212yes—
62#1802C2212yes—
63#1802C2212yes—
64#1802C2212yes—
65#1786C2212yes—
66#1786C2212yes—
67C4×#102C2214yes—
68C4×#102C2214yes—
69C23×Q8D411yescharacteristic
70C23×D4C2311yescharacteristic
71C24×C4—21no—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8—24no—
74C2×C22⋊Q8—48no—
75C2×C4×Q8—24no—
76C2×C4×Q8—24no—
77C2×C4×Q8D412yes—
78C2×C4×Q8D412yes—
79C2×C4×Q8C2312yes—
80C2×C4×Q8C2312yes—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4C2312yes—
86C2×C4×D4C2312yes—
87C2×#117—21no—
88C2×#117—21no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4D412yes—
93C22×C4⋊C4D411yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—22no—
98C22×C22⋊C4—22no—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101#431—24no—
102#431—24no—
103C2×#107C2318yes—
104C2×#106C2318yes—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107C2×#102C2314yes—
108C2×#102C2314yes—
109C2×#102C2314yes—
110C2×#102C2314yes—
111C2×C22⋊C8C2312yes—
112C2×C22⋊C8C2312yes—
113C2×C4×C8C2312yes—
114C2×C4×C8C2312yes—
115#396—28no—
116#396—28no—
117#391—22no—
118#391—22no—
119#391—22no—
120#391—22no—
121#390—24no—
122#390C2318yes—
123#389—24no—
124#389C2318yes—
125#387—24no—
126#387—24no—
127#387—24no—
128#387—24no—
129#384—21no—
130#384—21no—
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#384C2312yes—
136#384C2312yes—
137#383—24no—
138#382—21no—
139#382—21no—
140#382C2314yes—
141#382C2314yes—
142#381—24no—
143#380—21no—
144#380—21no—
145#380—24no—
146#380—24no—
147#380C2312yes—
148#380C2312yes—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4C2314yes—
156C4×C4⋊C4C2314yes—
157C4×C4⋊C4C2312yes—
158C4×C4⋊C4C2312yes—
159C4×C22⋊C4—24no—
160#376—24no—
161C2×#95—22no—
162#340C2314yes—
163#340C2314yes—
164#340C2314yes—
165#340C2314yes—
166#340C2318yes—
167#340C2318yes—
168#336C2318yes—
169#336C2318yes—
170C25—21no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8C2×D412yes—
175C22×Q8C2×D412yes—
176C22×D4—24no—
177C22×D4—21no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—42no—
181C22×D4C2412yes—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2411yescharacteristic
194C22×C8C2414yes—
195C22×C8C2414yes—
196C22⋊Q8—416no—
197C22⋊Q8—416no—
198C4×Q8—28no—
199C4×Q8—28no—
200C4×Q8—24no—
201C4×Q8—24no—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—44no—
209C4×D4—44no—
210#117—44no—
211#117—44no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2412yes—
240C2×C4⋊C4C2414yes—
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2412yes—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—48no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—84no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42C2×D414yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42#4114yes—
264C2×C42#4114yes—
265C2×C42C2414yes—
266C2×C42C2411yescharacteristic
267C2×C42C2411yescharacteristic
268#107—48no—
269#106—48no—
270C4⋊C8—24no—
271C4⋊C8—24no—
272#102—28no—
273#102—28no—
274#102—44no—
275#102—44no—
276C22⋊C8—24no—
277C22⋊C8—24no—
278C4×C8—24no—
279C4×C8—24no—
280#95—28no—
281#95—22no—
282#95—22no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—22no—
288#95—22no—
289#95—24no—
290#95—24no—
291C24—24no—
292C24—22no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24C22×D411yescharacteristic
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—416no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D42−1+414yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—816no—
347C22×C4#12314yes—
348C22×C4#12314yes—
349C22×C4#13514yes—
350C22×C4AGL(1,ℤ/8)14yes—
351C22×C4C22×D414yes—
352C22×C4C22×D414yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4C2511yescharacteristic
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—28no—
378C4⋊C4—44no—
379C4⋊C4—416no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—81no—
388C4⋊C4—81no—
389C4⋊C4—81no—
390C4⋊C4—81no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C22⋊C4—416no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—88no—
399C42—28no—
400C42—24no—
401C42—24no—
402C42—44no—
403C42—44no—
404C42—84no—
405C42—84no—
406C42#12314yes—
407C42#12314yes—
408C42C2×#4112yes—
409C42C2×#4112yes—
410C42C2×#4112yes—
411C42C2×#4112yes—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—24no—
417C23—48no—
418C23—48no—
419C23—48no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—84no—
427C23—84no—
428C23—84no—
429C23—84no—
430C23C2×#12312yes—
431C23C2×#13512yes—
432C23C2×AGL(1,ℤ/8)12yes—
433C23C23×D411yescharacteristic
434Q8—48no—
435Q8—48no—
436D4—24no—
437D4—24no—
438D4—48no—
439D4—42no—
440D4—42no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—44no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—84no—
476C2×C4—88no—
477C2×C4—84no—
478C2×C4—88no—
479C2×C4—88no—
480C2×C4#482116yes—
481C2×C4C2×#12312yes—
482C2×C4C2×#12312yes—
483C2×C4C2×#12312yes—
484C2×C4C2×#12312yes—
485C2×C4C2×#12312yes—
486C2×C4C2×#13512yes—
487C2×C4C2×#13512yes—
488C2×C4C2×AGL(1,ℤ/8)12yes—
489C2×C4C2×AGL(1,ℤ/8)12yes—
490C2×C4C22×#4111yescharacteristic
491C2×C4C22×#4111yescharacteristic
492C2×C4C2×2−1+412yes—
493C2×C4#58512yes—
494C8—44no—
495C8—44no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—816no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22C2×#48214yes—
511C22C22×#12311yescharacteristic
512C22#348611yescharacteristic
513C22#348711yescharacteristic
514C22C22×#13511yescharacteristic
515C22C22×AGL(1,ℤ/8)11yescharacteristic
516C22#358511yescharacteristic
517C22#358611yescharacteristic
518C4—22no—
519C4—22no—
520C4—22no—
521C4—44no—
522C4—44no—
523C4—82no—
524C4—82no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—84no—
529C4C2×#48214yes—
530C4C2×#48214yes—
531C4#349512yes—
532C4#349512yes—
533C2—24no—
534C2—84no—
535C2#3567211yescharacteristic
536C2C22×#48211yescharacteristic
537C2#6052011yescharacteristic
538C2#6181911yescharacteristic
539C2#6182711yescharacteristic
540C2#6188211yescharacteristic
541C2#6189011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34187C2semidirect
3. #27616C2semidirect
5. #25059C2semidirect
6. #25057C2semidirect
7. #24993C2semidirect
8. #24991C2semidirect
9. #21982C2semidirect
10. #21981C2semidirect
11. #21818C2semidirect
26. #2355C22semidirect
27. #2355C22semidirect
28. #2355C22semidirect
29. #2355C22semidirect
40. #2101C22semidirect
42. #2085C22semidirect
44. #1984C22semidirect
45. #1984C22semidirect
46. #1948C22semidirect
47. #1947C22semidirect
50. #1921C22semidirect
51. #1920C22semidirect
54. #1845C22semidirect
55. #1844C22semidirect
56. #1834C22semidirect
57. #1834C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
#341872
#276162
#275882
#250592
#250572
#249932
#249912
#219822
#219812
#218182
#207322