Finite Group Databasebeta

← #6479536

Subgroups — #6479536

Order 512.

1970 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#33095C212yesmaximal, complemented
4#33094C212yesmaximal, complemented
5#29912C214yesmaximal, complemented
6#27865C218yesmaximal, complemented
7#27617C214yesmaximal, complemented
8#27588C214yesmaximal, complemented
9#21942C212yesmaximal, complemented
10#21764C212yesmaximal
11#20682C211yesmaximal, characteristic, complemented
12C22×#123—21no—
13C22×#121C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#407C2212yes—
16#2854—21no—
17#2854—21no—
18#2854—22no—
19#2854C2212yes—
20#2854C2212yes—
21#2854C2214yes—
22C2×#390C2212yes—
23C2×#389C2212yes—
24C2×#388—22no—
25C2×#386—21no—
26#2283—22no—
27#2283—22no—
28#2283C2214yes—
29#2283C2214yescomplemented
30C2×#383—21no—
31C2×#382C2212yescomplemented
32C2×#381—21no—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic, complemented
36C2×C4×C4⋊C4C2211yescharacteristic, complemented
37C2×C4×C22⋊C4—21no—
38C2×#376—21no—
39#2101C2218yescomplemented
40#2092C2218yescomplemented
41#2085C2218yescomplemented
42#2073C2218yescomplemented
43#1926C2218yescomplemented
44#1925C2218yescomplemented
45#1919C2218yescomplemented
46#1918C2218yescomplemented
47#1895C2218yes—
48#1874C2214yes—
49#1874C2214yes—
50#1853C2214yes—
51#1853C2214yescomplemented
52#1852C2214yes—
53#1852C2214yescomplemented
54#1824C2214yes—
55#1817C2214yes—
56#1817C2212yescomplemented
57#1817C2212yescomplemented
58#1809C2218yescomplemented
59#1786C2214yes—
60#1728C2214yescomplemented
61C23×D4—21no—
62C23×D4C2311yescharacteristic
63C24×C4D411yescharacteristic
64C24×C4D411yescharacteristic
65C24×C4C2311yescharacteristic
66C22×M16C2314yes—
67C2×#123—22no—
68C2×#123—24no—
69C2×#123—22no—
70C2×#123—48no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121—28no—
74C2×#121C2312yes—
75C2×#121C2312yes—
76C22×C4⋊C4—22no—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C42—28no—
87C22×C42—21no—
88C22×C42—21no—
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D412yes—
92C22×C42D412yes—
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2312yes—
97C2×#107C2318yes—
98C2×#106C2318yes—
99C2×#102C2318yes—
100C2×#102C2318yes—
101#407—28no—
102C2×C22⋊C8C2314yes—
103C2×C22⋊C8C2314yes—
104#390—22no—
105#390—22no—
106#390C2314yes—
107#390C2314yes—
108#389—22no—
109#389—22no—
110#389C2314yes—
111#389C2314yes—
112#388—24no—
113#388—24no—
114#388—24no—
115#388—24no—
116#386—22no—
117#386—22no—
118#386—24no—
119#383—24no—
120#383—24no—
121#382—22no—
122#382—22no—
123#382C2314yes—
124#382C2314yes—
125#381—24no—
126#381—24no—
127#380—21no—
128#380—21no—
129#380—22no—
130#380—22no—
131#380C2312yes—
132#380C2314yes—
133#380C2314yes—
134#380C2312yes—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C4×C22⋊C4—24no—
140C4×C22⋊C4—24no—
141#376—24no—
142#376—24no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95—21no—
149C2×#95—22no—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#336C2318yes—
155#336C2318yes—
156C25—21no—
157C25C2×D411yescharacteristic
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—21no—
161C22×D4—24no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4C2412yes—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—44no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4C2×D412yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4#4118yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C2×M16—28no—
193C2×M16—44no—
194C2×M16—44no—
195C22×C8C2418yes—
196#123—48no—
197#123—416no—
198#123—48no—
199#121—28no—
200#121—28no—
201#121—44no—
202#121—44no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4C2412yes—
215C2×C4⋊C4C2414yes—
216C2×C4⋊C4C2414yes—
217C2×C4⋊C4C2414yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—48no—
229C2×C22⋊C4—48no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C42—24no—
234C2×C42—28no—
235C2×C42—24no—
236C2×C42—28no—
237C2×C42—24no—
238C2×C42—28no—
239C2×C42—24no—
240C2×C42—28no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2412yes—
252C2×C42C2412yes—
253#107—48no—
254#106—48no—
255#102—48no—
256#102—48no—
257C22⋊C8—28no—
258C22⋊C8—28no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—22no—
268#95—22no—
269#95—24no—
270#95—24no—
271C24—22no—
272C24—24no—
273C24—22no—
274C24—22no—
275C24—42no—
276C24—42no—
277C24—44no—
278C24—42no—
279C24—42no—
280C24—44no—
281C24#12314yes—
282C24C22×D411yescharacteristic
283C24C22×D411yescharacteristic
284C24C22×D411yescharacteristic
285C2×D4—24no—
286C2×D4—28no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—42no—
295C2×D4—88no—
296C2×D4—816no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—416no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—416no—
330C22×C4—416no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4#12314yes—
344C22×C4#12314yes—
345C22×C4#12314yes—
346C22×C4AGL(1,ℤ/8)14yes—
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C22×D412yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4C2×#4114yes—
355C22×C4C2×#4114yes—
356C22×C4C2×#4114yes—
357C22×C4C2511yescharacteristic
358M16—88no—
359C2×C8—28no—
360C2×C8—28no—
361C2×C8—28no—
362C4⋊C4—24no—
363C4⋊C4—28no—
364C4⋊C4—28no—
365C4⋊C4—28no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—48no—
370C4⋊C4—42no—
371C4⋊C4—48no—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—48no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—48no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—48no—
382C42—24no—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—48no—
391C42—48no—
392C23—22no—
393C23—22no—
394C23—21no—
395C23—22no—
396C23—21no—
397C23—28no—
398C23—28no—
399C23—22no—
400C23—21no—
401C23—22no—
402C23—21no—
403C23—44no—
404C23—48no—
405C23—48no—
406C23—42no—
407C23—44no—
408C23—42no—
409C23—44no—
410C23—42no—
411C23—84no—
412C23—84no—
413C23—88no—
414C23—88no—
415C23—84no—
416C23C2×#12312yes—
417C23C2×#12312yes—
418C23C2×#12312yes—
419C23C2×AGL(1,ℤ/8)12yes—
420C23#57514yes—
421C23#57512yes—
422C23C23×D411yescharacteristic
423D4—48no—
424D4—84no—
425D4—88no—
426D4—88no—
427C2×C4—28no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—48no—
453C2×C4—48no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—88no—
468C2×C4—88no—
469C2×C4—88no—
470C2×C4—84no—
471C2×C4C2×#12312yes—
472C2×C4C2×#12312yes—
473C2×C4C2×#12312yes—
474C2×C4C2×AGL(1,ℤ/8)14yes—
475C2×C4C22×#4112yes—
476C2×C4#58514yes—
477C8—48no—
478C22—24no—
479C22—24no—
480C22—22no—
481C22—22no—
482C22—24no—
483C22—24no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—416no—
490C22—42no—
491C22—42no—
492C22—84no—
493C22—82no—
494C22—84no—
495C22—82no—
496C22—82no—
497C22—168no—
498C22#309314yes—
499C22#309414yes—
500C22C22×#12311yescharacteristic
501C22#344912yes—
502C22C22×AGL(1,ℤ/8)11yescharacteristic
503C22C2×#57512yes—
504C22C2×#57511yescharacteristic
505C4—22no—
506C4—22no—
507C4—22no—
508C4—24no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—84no—
513C4—84no—
514C4—84no—
515C4—82no—
516C4#349114yes—
517C2—24no—
518C2—24no—
519C2—42no—
520C2—162no—
521C2#3509711yescharacteristic
522C2C2×#309311yescharacteristic
523C2C2×#309411yescharacteristic
524C2#6094912yes—
525C2#6096412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
2. C2×#2283C2semidirect
3. #33095C2semidirect
4. #33094C2semidirect
5. #29912C2semidirect
6. #27865C2semidirect
7. #27617C2semidirect
8. #27588C2semidirect
9. #21942C2semidirect
11. #20682C2semidirect
29. #2283C22semidirect
31. C2×#382C22semidirect
35. C2×C4×C4⋊C4C22semidirect
36. C2×C4×C4⋊C4C22semidirect
39. #2101C22semidirect
40. #2092C22semidirect
41. #2085C22semidirect
42. #2073C22semidirect
43. #1926C22semidirect
44. #1925C22semidirect
45. #1919C22semidirect
46. #1918C22semidirect
51. #1853C22semidirect
53. #1852C22semidirect
56. #1817C22semidirect
57. #1817C22semidirect
58. #1809C22semidirect
60. #1728C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#22832
#330952
#330942
#299122
#278652
#276172
#275882
#219422
#217642
#206822