Finite Group Databasebeta

← #6433989

Subgroups — #6433989

Order 512.

2220 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C212yesmaximal, complemented
2#30364C214yesmaximal, complemented
3#30324C214yesmaximal, complemented
4#30296C211yesmaximal, characteristic, complemented
5#30247C211yesmaximal, characteristic, complemented
6#30197C212yesmaximal, complemented
7#29742C214yesmaximal, complemented
8#23755C214yesmaximal, complemented
9#23567C214yesmaximal, complemented
10C4×#469C214yesmaximal, complemented
11C4×#407C211yesmaximal, characteristic, complemented
12C2×#532C2212yes—
13C2×#525—22no—
14C23×C42C2211yescharacteristic
15C2×#469C2214yes—
16C2×C4×D8C2214yescomplemented
17C2×#407C2212yes—
18C2×C4×M16C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2856C2212yes—
22#2856C2212yes—
23#2854—22no—
24#2854C2214yes—
25#2850—22no—
26#2850—22no—
27#2847—22no—
28#2847C2214yes—
29C4×#121—22no—
30C4×#121—22no—
31C4×#121C2214yes—
32C4×#121C2214yes—
33#2297—22no—
34#2285C2212yes—
35#2100C2214yescomplemented
36#2094C2214yescomplemented
37#2084C2214yescomplemented
38#2077C2218yes—
39#2071C2214yescomplemented
40#2057C2214yescomplemented
41#2041C2214yescomplemented
42#2034C2218yes—
43#2033C2218yes—
44#2023C2214yescomplemented
45#2011C2214yescomplemented
46#1999C2218yes—
47#1972C2214yescomplemented
48#1944C2214yescomplemented
49#1936C2218yescomplemented
50#1895C2218yes—
51#1809C2214yes—
52C4×#106C2214yescomplemented
53C4×#102C2218yescomplemented
54#1749C2214yes—
55C4×C22⋊C8C2214yes—
56#1728C2211yescharacteristic
57#1728C2211yescharacteristic
58C22×#41—22no—
59C24×C4D411yescharacteristic
60C24×C4D411yescharacteristic
61C24×C4C2311yescharacteristic
62C22×D8C2314yes—
63C22×M16C2312yes—
64C22×M16C2312yes—
65#532—48no—
66C2×C4⋊Q8C2312yes—
67C2×#127C2312yes—
68C2×#124—22no—
69#525—416no—
70C2×#123—24no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×#121—22no—
74C2×#121—22no—
75C2×#121—22no—
76C2×#121—22no—
77C2×#121—22no—
78C2×#121C2314yes—
79C2×#121C2314yes—
80C2×#121C2314yes—
81C2×#120—24no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4—22no—
85C2×C4×D4C2314yes—
86C2×#117—22no—
87C22×C42—44no—
88C22×C42D411yescharacteristic
89C22×C42D412yes—
90C22×C42D412yes—
91C22×C42D411yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2312yes—
94C22×C42C2312yes—
95C22×C42C2311yescharacteristic
96#469—216no—
97C4×D8—48no—
98C2×#107C2314yes—
99C2×#106C2314yes—
100C2×#106C2314yes—
101C2×#102C2318yes—
102C2×#102C2318yes—
103C2×#102C2318yes—
104#407—28no—
105C2×C22⋊C8C2318yes—
106C4×M16—44no—
107C2×#97C2314yes—
108C2×C4×C8C2314yes—
109#391—22no—
110#391—22no—
111#390—22no—
112#390—22no—
113#390—22no—
114#390—22no—
115#390C2314yes—
116#390C2314yes—
117#389—22no—
118#389—22no—
119#389C2314yes—
120#389C2314yes—
121#388—24no—
122#388—24no—
123#388—24no—
124#388—24no—
125#386—24no—
126#386—24no—
127#386—24no—
128#386—24no—
129#386—24no—
130#386—24no—
131#384C2312yes—
132#384C2312yes—
133#382—22no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380C2314yes—
138#380C2314yes—
139#380C2314yes—
140C4×C4⋊C4—22no—
141C4×C4⋊C4C2314yes—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—24no—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#336C2314yes—
148#336C2314yes—
149C25C2×D411yescharacteristic
150C2×#41—48no—
151C2×#41—48no—
152C2×#41—48no—
153C22×Q8—22no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4C2414yes—
160C23×C4—28no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—44no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—41no—
167C23×C4—41no—
168C23×C4—42no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2412yes—
180C23×C4C2411yescharacteristic
181C2×D8—28no—
182C2×D8—44no—
183C2×D8—44no—
184C2×M16—24no—
185C2×M16—42no—
186C2×M16—42no—
187C2×M16—42no—
188C2×M16—42no—
189C2×M16#4118yescomplemented
190C22×C8C2414yes—
191C22×C8C2414yes—
192C4⋊Q8—44no—
193#127—44no—
194#124—48no—
195#123—416no—
196C22⋊Q8—48no—
197C22⋊Q8—48no—
198#121—216no—
199#121—44no—
200#121—44no—
201#121—48no—
202#121—48no—
203#121—48no—
204#121—48no—
205#121—48no—
206#121—48no—
207#120—416no—
208C4×D4—216no—
209C4×D4—48no—
210C4×D4—416no—
211C4×D4—416no—
212#117—48no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4C2414yes—
221C2×C4⋊C4C2414yes—
222C2×C4⋊C4C2414yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C42—28no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—44no—
242C2×C42—42no—
243C2×C42C2×D412yes—
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D412yes—
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2412yes—
251C2×C42C2411yescharacteristic
252C2×C42C2411yescharacteristic
253#107—44no—
254#106—28no—
255#106—44no—
256#102—216no—
257#102—48no—
258#102—48no—
259C22⋊C8—216no—
260#97—44no—
261C4×C8—28no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268C24—22no—
269C24—24no—
270C24—24no—
271C24—41no—
272C24—41no—
273C24—42no—
274C24C22×D411yescharacteristic
275C24C22×D412yes—
276#41—832no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×D4—28no—
281C2×D4—28no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—44no—
288C2×D4—48no—
289C2×D4—48no—
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—48no—
293C2×D4—48no—
294C2×D4—48no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—44no—
298C2×D4—44no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—416no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—416no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4AGL(1,ℤ/8)18yes—
342C22×C4C22×D412yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4114yes—
351C22×C4C2511yescharacteristic
352D8—416no—
353M16—48no—
354M16—48no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8C2×#4118yes—
361C4⋊C4—28no—
362C4⋊C4—28no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—48no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—44no—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—48no—
376C22⋊C4—48no—
377C22⋊C4—48no—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C42—22no—
384C42—22no—
385C42—24no—
386C42—24no—
387C42—22no—
388C42—24no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—48no—
393C42H(5,𝔽2)14yes—
394C23—22no—
395C23—22no—
396C23—21no—
397C23—22no—
398C23—21no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—44no—
405C23—48no—
406C23—44no—
407C23—48no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23#53212yes—
417C23C2×AGL(1,ℤ/8)14yes—
418C23C23×D411yescharacteristic
419Q8—88no—
420D4—48no—
421D4—48no—
422D4—88no—
423D4—88no—
424D4—816no—
425D4—816no—
426D4—816no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—216no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—44no—
445C2×C4—48no—
446C2×C4—44no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—44no—
450C2×C4—48no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—816no—
465C2×C4#53212yes—
466C2×C4#53412yes—
467C2×C4#53412yes—
468C2×C4C2×AGL(1,ℤ/8)14yes—
469C2×C4C2×AGL(1,ℤ/8)14yes—
470C2×C4#57514yes—
471C2×C4C22×#4112yes—
472C2×C4C2×H(5,𝔽2)12yes—
473C2×C4#58511yescharacteristic
474C2×C4#58511yescharacteristic
475C8—28no—
476C8—44no—
477C22—22no—
478C22—24no—
479C22—24no—
480C22—24no—
481C22—24no—
482C22—416no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—41no—
488C22—42no—
489C22—41no—
490C22—84no—
491C22—84no—
492C22—84no—
493C22—84no—
494C22—84no—
495C22—84no—
496C22—84no—
497C22C2×#53211yescharacteristic
498C22C2×#53411yescharacteristic
499C22#344911yescharacteristic
500C22C22×AGL(1,ℤ/8)12yes—
501C22C2×#57512yes—
502C4—22no—
503C4—24no—
504C4—22no—
505C4—24no—
506C4—42no—
507C4—41no—
508C4—41no—
509C4—84no—
510C4—84no—
511C4—84no—
512C4#319214yes—
513C4#345912yes—
514C4#346012yes—
515C2—42no—
516C2—44no—
517C2—164no—
518C2#3457511yescharacteristic
519C2#6070012yes—
520C2C2×#319211yescharacteristic
521C2#6074111yescharacteristic
522C2#6095112yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36624C2semidirect
2. #30364C2semidirect
3. #30324C2semidirect
4. #30296C2semidirect
5. #30247C2semidirect
6. #30197C2semidirect
7. #29742C2semidirect
8. #23755C2semidirect
9. #23567C2semidirect
10. C4×#469C2semidirect
11. C4×#407C2semidirect
16. C2×C4×D8C22semidirect
35. #2100C22semidirect
36. #2094C22semidirect
37. #2084C22semidirect
39. #2071C22semidirect
40. #2057C22semidirect
41. #2041C22semidirect
44. #2023C22semidirect
45. #2011C22semidirect
47. #1972C22semidirect
48. #1944C22semidirect
49. #1936C22semidirect
52. C4×#106C22semidirect
53. C4×#102C22semidirect
189. C2×M16#41semidirect

Maximal subgroups

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

subgroupindex
#366242
#303642
#303242
#302962
#302472
#301972
#297422
#237552
#235672
C4×#4692
C4×#4072