Finite Group Databasebeta

← #6510741

Subgroups — #6510741

Order 512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic, complemented
2C2×#2504C211yesmaximal, characteristic
3#29109C212yesmaximal, complemented
4#29104C212yesmaximal, complemented
5#27969C212yesmaximal, complemented
6#27964C212yesmaximal, complemented
7#27725C218yesmaximal, complemented
8#27703C218yesmaximal, complemented
9#21767C212yesmaximal
10#20700C212yesmaximal, complemented
11#20524C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C2×#407C2212yes—
15#2857C2212yes—
16#2851C2212yes—
17#2626—22no—
18#2608—22no—
19#2607—22no—
20#2504—22no—
21#2504—22no—
22#2504C2218yes—
23#2441C2214yescomplemented
24#2439C2214yescomplemented
25C2×#397—22no—
26#2329—22no—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#386—22no—
30C2×#382C2211yescharacteristic
31C2×#382C2211yescharacteristic
32#2280C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#2101C2218yescomplemented
35#2101C2218yescomplemented
36#2073C2214yescomplemented
37#2071C2214yescomplemented
38#2025C2214yescomplemented
39#2023C2214yescomplemented
40#1993C2214yescomplemented
41#1991C2214yescomplemented
42#1936C2218yescomplemented
43#1934C2218yescomplemented
44#1895C2218yes—
45#1875C2214yes—
46#1875C2214yes—
47#1836C2214yescomplemented
48#1835C2214yescomplemented
49#1824C2214yes—
50#1824C2214yes—
51#1823C2214yes—
52#1823C2218yescomplemented
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×C22⋊Q8—22no—
67C2×#121—22no—
68C2×#121—28no—
69C2×#121C2314yes—
70C2×#117C2312yes—
71C2×#117C2312yes—
72C22×C4⋊C4—28no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4D412yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#102C2318yes—
86C2×#102C2318yes—
87#407—28no—
88C2×C22⋊C8C2314yes—
89C2×C22⋊C8C2314yes—
90C2×#97C2314yes—
91C2×C4×C8C2314yes—
92#399—24no—
93#397—28no—
94#397—28no—
95#396—24no—
96#396—24no—
97#393—24no—
98#391—22no—
99#391—24no—
100#390—22no—
101#390—21no—
102#390—21no—
103#390C2314yes—
104#390C2314yes—
105#389—22no—
106#389—24no—
107#389C2314yes—
108#389C2314yes—
109#387—24no—
110#387—24no—
111#387—24no—
112#386—28no—
113#386—28no—
114#385—24no—
115#384C2312yes—
116#382—22no—
117#382—21no—
118#382—21no—
119#382C2312yes—
120#382C2314yes—
121#382C2314yes—
122#381—22no—
123#380—22no—
124#380C2314yes—
125#380C2314yes—
126#379—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
131C4×C22⋊C4C2314yes—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D412yes—
135#340C2318yes—
136#340C2318yes—
137#340C2318yes—
138#340C2318yes—
139#340C2318yes—
140#340C2318yes—
141#336C2314yes—
142#336C2314yes—
143C25—21no—
144C25C2×D411yescharacteristic
145C22×Q8—22no—
146C22×D4—24no—
147C22×D4—21no—
148C22×D4—24no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—48no—
152C22×D4—44no—
153C22×D4C2412yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—24no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—84no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4#4118yes—
167C23×C4C2411yescharacteristic
168C23×C4C2412yes—
169C2×M16—28no—
170C2×M16—44no—
171C2×M16—44no—
172C22×C8C2418yes—
173C4⋊Q8—44no—
174#125—44no—
175C22⋊Q8—48no—
176#121—28no—
177#121—28no—
178#121—48no—
179#117—44no—
180#117—44no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4C2412yes—
199C2×C4⋊C4C2412yes—
200C2×C4⋊C4C2412yes—
201C2×C4⋊C4C2414yes—
202C2×C4⋊C4C2414yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—84no—
214C2×C22⋊C4—84no—
215C2×C22⋊C4C2×D414yes—
216C2×C22⋊C4C2×D414yes—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2412yes—
225C2×C42C2412yes—
226#102—48no—
227#102—48no—
228C22⋊C8—28no—
229C22⋊C8—28no—
230#97—44no—
231C4×C8—44no—
232#95—28no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95C2×D414yes—
241#95C2×D414yes—
242C24—24no—
243C24—22no—
244C24—41no—
245C24—41no—
246C24—42no—
247C24—44no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—44no—
252C24#12414yes—
253C24C22×D412yes—
254C24C22×D411yescharacteristic
255C2×Q8—44no—
256C2×Q8—82no—
257C2×Q8—82no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—416no—
264C2×D4—48no—
265C2×D4—42no—
266C2×D4—88no—
267C2×D4—816no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—416no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—48no—
294C22×C4—44no—
295C22×C4—88no—
296C22×C4—816no—
297C22×C4#12414yes—
298C22×C4#12418yes—
299C22×C4C22×D414yes—
300C22×C4C22×D412yes—
301C22×C4C22×D412yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D412yes—
304C22×C4C2×#4114yes—
305C22×C4C2×#4114yes—
306C22×C4C2×#4114yes—
307C22×C4C2511yescharacteristic
308M16—88no—
309C2×C8—28no—
310C2×C8—28no—
311C2×C8—28no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C4⋊C4—48no—
318C4⋊C4—44no—
319C4⋊C4—48no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—82no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C4⋊C4—82no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—48no—
330C22⋊C4—48no—
331C22⋊C4—88no—
332C22⋊C4—84no—
333C22⋊C4—84no—
334C22⋊C4—84no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C42—24no—
338C42—24no—
339C42—42no—
340C42—41no—
341C42—41no—
342C42—82no—
343C42—82no—
344C23—22no—
345C23—21no—
346C23—21no—
347C23—24no—
348C23—28no—
349C23—22no—
350C23—22no—
351C23—24no—
352C23—44no—
353C23—44no—
354C23—48no—
355C23—48no—
356C23—44no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—44no—
361C23—84no—
362C23—88no—
363C23—88no—
364C23—84no—
365C23—84no—
366C23C2×#12414yes—
367C23C2×#12412yes—
368C23#57514yes—
369C23C23×D411yescharacteristic
370Q8—164no—
371D4—48no—
372D4—84no—
373D4—816no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—28no—
379C2×C4—28no—
380C2×C4—28no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—44no—
391C2×C4—48no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4—816no—
409C2×C4—84no—
410C2×C4C2×#12414yes—
411C2×C4C2×#12412yes—
412C2×C4C22×#4112yes—
413C2×C4#58512yes—
414C2×C4#58512yes—
415C8—48no—
416C22—24no—
417C22—24no—
418C22—22no—
419C22—22no—
420C22—24no—
421C22—24no—
422C22—24no—
423C22—416no—
424C22—44no—
425C22—44no—
426C22—84no—
427C22—84no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—168no—
432C22#314014yes—
433C22C22×#12411yescharacteristic
434C22#348511yescharacteristic
435C22#348511yescharacteristic
436C22C2×#57512yes—
437C22#358512yes—
438C4—24no—
439C4—22no—
440C4—44no—
441C4—42no—
442C4—42no—
443C4—84no—
444C4—84no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4#349114yes—
449C2—24no—
450C2—44no—
451C2—162no—
452C2#3545511yescharacteristic
453C2C2×#314011yescharacteristic
454C2#6061911yescharacteristic
455C2#6171312yes—
456C2#6171812yes—
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. #29109C2semidirect
4. #29104C2semidirect
5. #27969C2semidirect
6. #27964C2semidirect
7. #27725C2semidirect
8. #27703C2semidirect
10. #20700C2semidirect
11. #20524C2semidirect
23. #2441C22semidirect
24. #2439C22semidirect
34. #2101C22semidirect
35. #2101C22semidirect
36. #2073C22semidirect
37. #2071C22semidirect
38. #2025C22semidirect
39. #2023C22semidirect
40. #1993C22semidirect
41. #1991C22semidirect
42. #1936C22semidirect
43. #1934C22semidirect
47. #1836C22semidirect
48. #1835C22semidirect
52. #1823C22semidirect
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×#25042
#291092
#291042
#279692
#279642
#277252
#277032
#217672
#207002
#205242