Finite Group Databasebeta

← #6510744

Subgroups — #6510744

Order 512.

1918 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic, complemented
2C2×#2501C211yesmaximal, characteristic
3#29110C212yesmaximal, complemented
4#29103C212yesmaximal, complemented
5#27966C212yesmaximal, complemented
6#27960C212yesmaximal, complemented
7#27719C218yesmaximal, complemented
8#27709C218yesmaximal, complemented
9#21767C212yesmaximal
10#20700C212yesmaximal, complemented
11#20524C211yesmaximal, characteristic, complemented
12C22×#123—22no—
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#407C2212yes—
16#2857C2212yes—
17#2851C2212yes—
18#2626—22no—
19#2608—22no—
20#2607—22no—
21#2501—22no—
22#2501—22no—
23#2501C2218yes—
24#2441C2214yescomplemented
25#2439C2214yescomplemented
26C2×#399—22no—
27#2329—22no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32#2280C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#2100C2218yescomplemented
35#2099C2218yescomplemented
36#2073C2214yescomplemented
37#2071C2214yescomplemented
38#2025C2214yescomplemented
39#2023C2214yescomplemented
40#1992C2214yescomplemented
41#1990C2214yescomplemented
42#1936C2218yescomplemented
43#1934C2218yescomplemented
44#1895C2218yes—
45#1875C2214yes—
46#1875C2214yes—
47#1836C2214yescomplemented
48#1834C2214yescomplemented
49#1824C2218yescomplemented
50#1823C2214yes—
51#1823C2214yes—
52#1823C2214yes—
53#1786C2214yes—
54#1753C2218yescomplemented
55#1749C2212yes—
56C4×C22⋊C8C2212yes—
57#1736C2214yescomplemented
58#1736C2214yescomplemented
59C23×D4—21no—
60C23×D4C2311yescharacteristic
61C24×C4—21no—
62C24×C4C2311yescharacteristic
63C22×M16C2314yes—
64C2×C4⋊Q8C2312yes—
65C2×#125C2312yes—
66C2×#123—28no—
67C2×#123—28no—
68C2×#123—416no—
69C2×C22⋊Q8—22no—
70C2×#121—22no—
71C2×#121—28no—
72C2×#121C2314yes—
73C2×#117C2312yes—
74C2×#117C2312yes—
75C22×C4⋊C4—28no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2314yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4D412yes—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#102C2318yes—
91C2×#102C2318yes—
92#407—28no—
93C2×C22⋊C8C2314yes—
94C2×C22⋊C8C2314yes—
95C2×#97C2314yes—
96C2×C4×C8C2314yes—
97#399—24no—
98#399—28no—
99#399—28no—
100#396—24no—
101#396—24no—
102#393—24no—
103#391—22no—
104#391—24no—
105#390—22no—
106#390—21no—
107#390—21no—
108#390C2314yes—
109#390C2314yes—
110#389—22no—
111#389—24no—
112#389C2314yes—
113#389C2314yes—
114#387—24no—
115#387—24no—
116#387—24no—
117#385—24no—
118#384—22no—
119#384—21no—
120#384—21no—
121#384C2312yes—
122#384C2314yes—
123#384C2314yes—
124#382C2312yes—
125#381—22no—
126#380—22no—
127#380C2314yes—
128#380C2314yes—
129#379—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4D412yes—
133C4×C22⋊C4D412yes—
134C4×C22⋊C4C2314yes—
135C2×#95D412yes—
136#340C2318yes—
137#340C2318yes—
138#340C2318yes—
139#340C2318yes—
140#340C2318yes—
141#340C2318yes—
142#336C2314yes—
143#336C2314yes—
144C25—21no—
145C25C2×D411yescharacteristic
146C22×Q8—22no—
147C22×D4—24no—
148C22×D4—21no—
149C22×D4—24no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—48no—
153C22×D4—44no—
154C22×D4C2412yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—24no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—48no—
162C23×C4—44no—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4#4118yes—
168C23×C4C2411yescharacteristic
169C23×C4C2412yes—
170C2×M16—28no—
171C2×M16—44no—
172C2×M16—44no—
173C22×C8C2418yes—
174C4⋊Q8—44no—
175#125—44no—
176#123—432no—
177#123—432no—
178C22⋊Q8—48no—
179#121—28no—
180#121—28no—
181#121—48no—
182#117—44no—
183#117—44no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4C2412yes—
207C2×C4⋊C4C2412yes—
208C2×C4⋊C4C2412yes—
209C2×C4⋊C4C2414yes—
210C2×C4⋊C4C2414yes—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4C2×D414yes—
224C2×C22⋊C4C2×D414yes—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2412yes—
233C2×C42C2412yes—
234#102—48no—
235#102—48no—
236C22⋊C8—28no—
237C22⋊C8—28no—
238#97—44no—
239C4×C8—44no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95C2×D414yes—
246#95C2×D414yes—
247C24—24no—
248C24—22no—
249C24—41no—
250C24—41no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—44no—
255C24—42no—
256C24—44no—
257C24#12414yes—
258C24C22×D412yes—
259C24C22×D411yescharacteristic
260C2×Q8—44no—
261C2×Q8—82no—
262C2×Q8—82no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—416no—
269C2×D4—48no—
270C2×D4—42no—
271C2×D4—88no—
272C2×D4—816no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—416no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—416no—
293C22×C4—42no—
294C22×C4—48no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—816no—
301C22×C4#12414yes—
302C22×C4#12418yes—
303C22×C4AGL(1,ℤ/8)18yes—
304C22×C4C22×D414yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D412yes—
307C22×C4C22×D412yes—
308C22×C4C22×D412yes—
309C22×C4C2×#4114yes—
310C22×C4C2×#4114yes—
311C22×C4C2×#4114yes—
312C22×C4C2511yescharacteristic
313M16—88no—
314C2×C8—28no—
315C2×C8—28no—
316C2×C8—28no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—28no—
321C4⋊C4—28no—
322C4⋊C4—416no—
323C4⋊C4—48no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—48no—
328C4⋊C4—48no—
329C4⋊C4—44no—
330C4⋊C4—48no—
331C4⋊C4—82no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C4⋊C4—82no—
335C22⋊C4—28no—
336C22⋊C4—28no—
337C22⋊C4—416no—
338C22⋊C4—48no—
339C22⋊C4—48no—
340C22⋊C4—48no—
341C22⋊C4—48no—
342C22⋊C4—84no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C42—24no—
346C42—24no—
347C42—41no—
348C42—41no—
349C42—42no—
350C42—82no—
351C42—82no—
352C23—22no—
353C23—21no—
354C23—21no—
355C23—24no—
356C23—28no—
357C23—22no—
358C23—22no—
359C23—24no—
360C23—44no—
361C23—44no—
362C23—48no—
363C23—48no—
364C23—44no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—44no—
369C23—84no—
370C23—88no—
371C23—88no—
372C23—84no—
373C23—84no—
374C23C2×#12414yes—
375C23C2×#12412yes—
376C23C2×AGL(1,ℤ/8)14yes—
377C23C23×D411yescharacteristic
378Q8—164no—
379D4—48no—
380D4—84no—
381D4—816no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—28no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—44no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—48no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—816no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4C2×#12414yes—
417C2×C4C2×#12412yes—
418C2×C4C2×AGL(1,ℤ/8)14yes—
419C2×C4C2×AGL(1,ℤ/8)14yes—
420C2×C4C22×#4112yes—
421C2×C4#58512yes—
422C2×C4#58512yes—
423C8—48no—
424C22—24no—
425C22—24no—
426C22—22no—
427C22—22no—
428C22—24no—
429C22—24no—
430C22—24no—
431C22—416no—
432C22—44no—
433C22—44no—
434C22—84no—
435C22—84no—
436C22—82no—
437C22—82no—
438C22—82no—
439C22—168no—
440C22#313814yes—
441C22C22×#12411yescharacteristic
442C22#348511yescharacteristic
443C22#348511yescharacteristic
444C22C22×AGL(1,ℤ/8)12yes—
445C22#358612yes—
446C4—24no—
447C4—24no—
448C4—24no—
449C4—22no—
450C4—44no—
451C4—84no—
452C4—84no—
453C4—82no—
454C4—82no—
455C4—82no—
456C4#349114yes—
457C2—24no—
458C2—44no—
459C2—162no—
460C2#3545511yescharacteristic
461C2C2×#313811yescharacteristic
462C2#6062011yescharacteristic
463C2#6170612yes—
464C2#6171712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35033C2semidirect
3. #29110C2semidirect
4. #29103C2semidirect
5. #27966C2semidirect
6. #27960C2semidirect
7. #27719C2semidirect
8. #27709C2semidirect
10. #20700C2semidirect
11. #20524C2semidirect
24. #2441C22semidirect
25. #2439C22semidirect
34. #2100C22semidirect
35. #2099C22semidirect
36. #2073C22semidirect
37. #2071C22semidirect
38. #2025C22semidirect
39. #2023C22semidirect
40. #1992C22semidirect
41. #1990C22semidirect
42. #1936C22semidirect
43. #1934C22semidirect
47. #1836C22semidirect
48. #1834C22semidirect
49. #1824C22semidirect
54. #1753C22semidirect
57. #1736C22semidirect
58. #1736C22semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#25012
#291102
#291032
#279662
#279602
#277192
#277092
#217672
#207002
#205242