Finite Group Databasebeta

← #6532049

Subgroups — #6532049

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35515C211yesmaximal, characteristic
2#34374C211yesmaximal, characteristic, complemented
3#27620C212yesmaximal, complemented
4#27612C212yesmaximal, complemented
5#27594C212yesmaximal
6#27582C212yesmaximal
7#25068C214yesmaximal, complemented
8#25056C214yesmaximal, complemented
9#25007C214yesmaximal, complemented
10#24999C214yesmaximal, complemented
11#21991C212yesmaximal, complemented
12#21824C211yesmaximal, characteristic, complemented
13#20744C212yesmaximal
14C2×#533C2211yescharacteristic
15C2×#518C2211yescharacteristic
16C2×#433C2212yes—
17#2580C2214yes—
18#2538C2214yes—
19#2454—22no—
20#2452—22no—
21#2437—24no—
22#2382—22no—
23#2365—22no—
24#2355C2214yescomplemented
25Q8×C4⋊C4C2214yescomplemented
26#2323—24no—
27#2317C2211yescharacteristic
28#2316C2211yescharacteristic
29#2314C2212yes—
30#2314C2212yes—
31#2309C2211yescharacteristic
32#2296C2211yescharacteristic
33C2×#390C2211yescharacteristic
34#2285C2212yes—
35#2101C2214yescomplemented
36#2100C2212yescomplemented
37#2100C2212yescomplemented
38#2094C2212yes—
39#2093C2214yes—
40#2093C2212yes—
41#2084C2212yescomplemented
42#2083C2212yescomplemented
43#2083C2214yescomplemented
44#2073C2212yes—
45#2073C2212yes—
46#2071C2214yes—
47#1985C2214yescomplemented
48#1984C2214yescomplemented
49#1949C2214yescomplemented
50#1947C2214yescomplemented
51#1926C2214yes—
52#1925C2214yes—
53#1925C2214yes—
54#1925C2214yes—
55#1921C2214yescomplemented
56#1920C2214yescomplemented
57#1918C2214yescomplemented
58#1918C2214yescomplemented
59#1854C2214yes—
60#1852C2214yes—
61#1847C2212yescomplemented
62#1845C2212yescomplemented
63#1836C2212yescomplemented
64#1835C2212yescomplemented
65#1825C2214yes—
66#1818C2212yes—
67#1817C2212yes—
68#1804C2214yes—
69#1802C2214yes—
70#1786C2214yes—
71#1762C2214yes—
72C4×#102C2214yes—
73C23×D4C2311yescharacteristic
74#533—28no—
75C2×C4⋊Q8D412yes—
76C2×#125D412yes—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8D414yes—
79#518—28no—
80C2×C4×Q8—24no—
81C2×C4×Q8C2314yes—
82C2×C4×D4—22no—
83C2×C4×D4C2314yes—
84C2×#117—22no—
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C42C2311yescharacteristic
93#433—28no—
94C2×#107C2314yes—
95C2×#107C2314yes—
96C2×#107C2314yes—
97C2×#106C2314yes—
98C2×C4⋊C8C2314yes—
99C2×#102C2314yes—
100C2×#102C2314yes—
101C2×#102C2314yes—
102C2×#102C2314yes—
103C2×C22⋊C8C2314yes—
104C2×#97C2312yes—
105C2×C4×C8C2312yes—
106#399—24no—
107#396—24no—
108#396—24no—
109#393—24no—
110#391—24no—
111#390—22no—
112#390—21no—
113#390—21no—
114#390C2312yes—
115#390C2312yes—
116#390C2314yes—
117#389—22no—
118#389—22no—
119#389—24no—
120#389—24no—
121#389—24no—
122#389C2314yes—
123#389C2312yes—
124#389C2312yes—
125#388—24no—
126#388—24no—
127#387—24no—
128#387—24no—
129#387—24no—
130#387—24no—
131#385—22no—
132#385—24no—
133#385—24no—
134#385—22no—
135#384C2312yes—
136#384C2312yes—
137#384C2312yes—
138#384C2312yes—
139#384C2312yes—
140#382—22no—
141#382C2312yes—
142#382C2312yes—
143#382C2312yes—
144#380—22no—
145#380—24no—
146#380—22no—
147#380C2314yes—
148#379—22no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4C2314yes—
154C4×C4⋊C4C2314yes—
155C4×C4⋊C4C2314yes—
156#340C2314yes—
157#340C2314yes—
158#340C2314yes—
159#340C2314yes—
160#340C2314yes—
161#340C2314yes—
162#340C2314yes—
163#340C2314yes—
164#336C2314yes—
165#336C2312yes—
166#336C2312yes—
167#336C2312yes—
168#336C2312yes—
169#336C2314yes—
170C25—21no—
171C22×Q8C2×D414yes—
172C22×D4—24no—
173C22×D4—21no—
174C22×D4—24no—
175C22×D4—42no—
176C22×D4C2412yes—
177C23×C4—21no—
178C23×C4—24no—
179C23×C4C2×D412yes—
180C23×C4C2411yescharacteristic
181C22×C8C2414yes—
182C22×C8C2414yes—
183C4⋊Q8—28no—
184#125—28no—
185C22⋊Q8—216no—
186C22⋊Q8—88no—
187C4×Q8—28no—
188C4×Q8—28no—
189C4×Q8—88no—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—48no—
193#117—24no—
194#117—24no—
195#117—42no—
196#117—84no—
197#117#4118yes—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2412yes—
216C2×C4⋊C4C2412yes—
217C2×C4⋊C4C2411yescharacteristic
218C2×C4⋊C4C2411yescharacteristic
219C2×C4⋊C4C2414yes—
220C2×C4⋊C4C2414yes—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
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—48no—
230C2×C22⋊C4—84no—
231C2×C22⋊C4C2×D412yes—
232C2×C22⋊C4C2×D412yes—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2412yes—
242C2×C42C2412yes—
243C2×C42C2412yes—
244#107—44no—
245#107—44no—
246#107—44no—
247#106—44no—
248C4⋊C8—28no—
249#102—28no—
250#102—28no—
251#102—44no—
252#102—44no—
253C22⋊C8—28no—
254#97—24no—
255C4×C8—24no—
256#95—22no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—22no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266C24—24no—
267C24—22no—
268C24—42no—
269C24—42no—
270C24—82no—
271C24—82no—
272C24C22×D411yescharacteristic
273C2×Q8—28no—
274C2×Q8—28no—
275C2×Q8—44no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4—416no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D42−1+414yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—48no—
297C22×C4—84no—
298C22×C4—84no—
299C22×C4—82no—
300C22×C4C22×D414yes—
301C22×C4C22×D412yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D414yes—
304C22×C4C22×D412yes—
305C22×C4C2×#4114yes—
306C22×C4C2511yescharacteristic
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C4⋊C4—24no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C4⋊C4—84no—
335C4⋊C4—84no—
336C4⋊C4#12318yes—
337C22⋊C4—24no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—48no—
341C22⋊C4—48no—
342C22⋊C4—84no—
343C22⋊C4—84no—
344C22⋊C4—84no—
345C22⋊C4—84no—
346C22⋊C4#12318yes—
347C42—22no—
348C42—22no—
349C42—24no—
350C42—24no—
351C42—42no—
352C42—42no—
353C42—48no—
354C42—84no—
355C42—84no—
356C42C2×#4114yes—
357C42C2×#4114yes—
358C23—22no—
359C23—21no—
360C23—21no—
361C23—22no—
362C23—24no—
363C23—48no—
364C23—44no—
365C23—42no—
366C23—42no—
367C23—44no—
368C23—88no—
369C23—164no—
370C23—162no—
371C23—162no—
372C23C2×#12312yes—
373C23C23×D411yescharacteristic
374Q8—416no—
375D4—28no—
376D4—48no—
377D4—44no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—24no—
384C2×C4—28no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—48no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—168no—
408C2×C4—162no—
409C2×C4C2×#12312yes—
410C2×C4C2×#12318yes—
411C2×C4#57614yes—
412C2×C4C22×#4112yes—
413C2×C4C2×2−1+412yes—
414C2×C4#58512yes—
415C8—44no—
416C8—44no—
417C22—22no—
418C22—24no—
419C22—42no—
420C22—41no—
421C22—41no—
422C22—44no—
423C22—48no—
424C22—84no—
425C22—82no—
426C22—82no—
427C22—168no—
428C22C22×#12311yescharacteristic
429C22#348611yescharacteristic
430C22#348711yescharacteristic
431C22C2×#57612yes—
432C22#358612yes—
433C4—22no—
434C4—24no—
435C4—28no—
436C4—42no—
437C4—44no—
438C4—82no—
439C4—84no—
440C4—81no—
441C4—84no—
442C4—82no—
443C4—81no—
444C4#349514yes—
445C2—42no—
446C2—84no—
447C2#3567211yescharacteristic
448C2#6052112yes—
449C2#6184912yes—
450C2#6193211yescharacteristic
451C2#6194311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34374C2semidirect
3. #27620C2semidirect
4. #27612C2semidirect
7. #25068C2semidirect
8. #25056C2semidirect
9. #25007C2semidirect
10. #24999C2semidirect
11. #21991C2semidirect
12. #21824C2semidirect
24. #2355C22semidirect
25. Q8×C4⋊C4C22semidirect
35. #2101C22semidirect
36. #2100C22semidirect
37. #2100C22semidirect
41. #2084C22semidirect
42. #2083C22semidirect
43. #2083C22semidirect
47. #1985C22semidirect
48. #1984C22semidirect
49. #1949C22semidirect
50. #1947C22semidirect
55. #1921C22semidirect
56. #1920C22semidirect
57. #1918C22semidirect
58. #1918C22semidirect
61. #1847C22semidirect
62. #1845C22semidirect
63. #1836C22semidirect
64. #1835C22semidirect

Maximal subgroups

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

subgroupindex
#355152
#343742
#276202
#276122
#275942
#275822
#250682
#250562
#250072
#249992
#219912
#218242
#207442