Finite Group Databasebeta

← #6519040

Subgroups — #6519040

Order 512.

1836 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#27602C214yesmaximal, complemented
4#25025C218yesmaximal, complemented
5#24980C218yesmaximal, complemented
6#24010C214yesmaximal, complemented
7#21986C212yesmaximal, complemented
8#21790C211yesmaximal, characteristic
9#20740C212yesmaximal, complemented
10C22×C22⋊Q8C2212yes—
11C22×C4×Q8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C2×#485C2218yescomplemented
14C2×#432C2212yes—
15#2536—24no—
16#2536C2218yes—
17C2×#398—21no—
18C2×#394—21no—
19#2357—24no—
20#2354C2214yes—
21#2354C2214yes—
22#2312—22no—
23#2309—21no—
24C2×#388—22no—
25#2285C2212yes—
26C2×#382C2211yescharacteristic
27#2280C2214yes—
28C2×#380C2212yes—
29#2274—21no—
30#2077C2218yescomplemented
31#2076C2218yescomplemented
32#2036C2218yescomplemented
33#2035C2218yescomplemented
34#1955C2218yes—
35#1949C2218yescomplemented
36#1873C2218yescomplemented
37#1854C2218yescomplemented
38#1853C2214yes—
39#1852C2214yes—
40#1847C2214yescomplemented
41#1836C2214yescomplemented
42#1825C2218yescomplemented
43#1823C2214yes—
44#1803C2214yescomplemented
45#1803C2214yescomplemented
46#1787C2214yes—
47#1779C2214yes—
48#1763C2218yescomplemented
49C23×Q8C2311yescharacteristic
50C23×D4D411yescharacteristic
51C24×C4C2312yes—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—48no—
54C2×C22⋊Q8C2318yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—28no—
57C2×C4×Q8C2314yes—
58C2×C4×D4—28no—
59C2×C4×D4D414yes—
60C2×C4×D4C2314yes—
61C2×#117—22no—
62C2×#117C2314yes—
63C2×#117C2314yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D412yes—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#485—416no—
75#432—28no—
76C2×#107C2318yes—
77C2×#106C2318yes—
78C2×C4⋊C8C2314yes—
79C2×#103C2318yes—
80C2×#103C2318yes—
81C2×C22⋊C8C2314yes—
82C2×C22⋊C8C2318yes—
83C2×C22⋊C8C2318yes—
84C2×#97C2314yes—
85#398—28no—
86#394—28no—
87#390—24no—
88#390—24no—
89#388—24no—
90#388—24no—
91#388—24no—
92#388—24no—
93#388—28no—
94#388—28no—
95#384C2312yes—
96#384C2312yes—
97#382—22no—
98#382—22no—
99#382C2314yes—
100#381—24no—
101#380—22no—
102#380—24no—
103#380C2314yes—
104#380C2318yes—
105#380C2314yes—
106#380C2314yes—
107#379—24no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2314yes—
112C4×C22⋊C4—24no—
113C4×C22⋊C4—24no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—21no—
117#340C2318yes—
118#340C2318yes—
119#340C2318yes—
120#340C2318yes—
121C25C2×D412yes—
122C22×Q8—24no—
123C22×Q8—21no—
124C22×Q8—24no—
125C22×Q8—42no—
126C22×Q8C2412yes—
127C22×D4—24no—
128C22×D4—44no—
129C22×D4C2×D414yes—
130C23×C4—21no—
131C23×C4—28no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D414yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2414yes—
141C23×C4C2411yescharacteristic
142C22×C8C2418yes—
143C22⋊Q8—216no—
144C22⋊Q8—48no—
145C22⋊Q8—416no—
146C4×Q8—28no—
147C4×Q8—28no—
148C4×Q8—48no—
149C4×D4—216no—
150C4×D4—28no—
151C4×D4—28no—
152#117—48no—
153#117—48no—
154#117—48no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2412yes—
170C2×C4⋊C4C2414yes—
171C2×C4⋊C4C2414yes—
172C2×C4⋊C4C2414yes—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—48no—
182C2×C22⋊C4—48no—
183C2×C22⋊C4C2×D414yes—
184C2×C22⋊C4C2×D414yes—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—44no—
190C2×C42C2×D412yes—
191C2×C42#4118yes—
192C2×C42C2412yes—
193#107—48no—
194#106—48no—
195C4⋊C8—28no—
196#103—216no—
197#103—48no—
198C22⋊C8—28no—
199C22⋊C8—48no—
200C22⋊C8—48no—
201#97—28no—
202#95—28no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—24no—
209C24—24no—
210C24—24no—
211C24—24no—
212C24—42no—
213C24—42no—
214C24—42no—
215C24—42no—
216C24C22×D414yes—
217C24C22×D411yescharacteristic
218C2×Q8—24no—
219C2×Q8—22no—
220C2×Q8—24no—
221C2×Q8—24no—
222C2×Q8—416no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8H(5,𝔽2)14yes—
226C2×D4—28no—
227C2×D4—44no—
228C2×D4—44no—
229C2×D4—416no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—28no—
237C22×C4—28no—
238C22×C4—28no—
239C22×C4—28no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—816no—
257C22×C4#12314yes—
258C22×C4#12314yes—
259C22×C4C22×D414yes—
260C22×C4C22×D412yes—
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D412yes—
263C22×C4C2×#4114yes—
264C22×C4C2511yescharacteristic
265C2×C8—28no—
266C2×C8—28no—
267C2×C8—28no—
268C4⋊C4—28no—
269C4⋊C4—24no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—28no—
273C4⋊C4—416no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—48no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C22⋊C4—28no—
282C22⋊C4—28no—
283C22⋊C4—416no—
284C22⋊C4—416no—
285C22⋊C4—48no—
286C22⋊C4—84no—
287C22⋊C4—84no—
288C22⋊C4—84no—
289C22⋊C4—84no—
290C42—48no—
291C42—48no—
292C42—84no—
293C42—84no—
294C42#12318yes—
295C42C2×#4114yes—
296C42C2×#4114yes—
297C23—22no—
298C23—21no—
299C23—21no—
300C23—24no—
301C23—24no—
302C23—24no—
303C23—24no—
304C23—24no—
305C23—24no—
306C23—24no—
307C23—48no—
308C23—48no—
309C23—48no—
310C23—84no—
311C23—84no—
312C23—84no—
313C23—84no—
314C23C2×#12312yes—
315C23#57514yes—
316C23C23×D411yescharacteristic
317Q8—28no—
318Q8—48no—
319Q8—44no—
320D4—416no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—28no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—28no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—416no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—88no—
350C2×C4—84no—
351C2×C4—88no—
352C2×C4—84no—
353C2×C4C2×#12314yes—
354C2×C4C2×#12312yes—
355C2×C4C2×#12314yes—
356C2×C4C22×#4112yes—
357C2×C4C2×H(5,𝔽2)12yes—
358C2×C4#58512yes—
359C8—48no—
360C22—24no—
361C22—22no—
362C22—24no—
363C22—22no—
364C22—44no—
365C22—44no—
366C22—44no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—816no—
372C22#309314yes—
373C22C22×#12311yescharacteristic
374C22#348411yescharacteristic
375C22#348511yescharacteristic
376C22C2×#57512yes—
377C22#358612yes—
378C4—24no—
379C4—24no—
380C4—22no—
381C4—44no—
382C4—44no—
383C4—84no—
384C4—82no—
385C4—84no—
386C4—84no—
387C4#349414yes—
388C2—24no—
389C2—84no—
390C2#3554011yescharacteristic
391C2C2×#309311yescharacteristic
392C2#6051911yescharacteristic
393C2#6157812yes—
394C2#6171112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27602C2semidirect
4. #25025C2semidirect
5. #24980C2semidirect
6. #24010C2semidirect
7. #21986C2semidirect
9. #20740C2semidirect
13. C2×#485C22semidirect
30. #2077C22semidirect
31. #2076C22semidirect
32. #2036C22semidirect
33. #2035C22semidirect
35. #1949C22semidirect
36. #1873C22semidirect
37. #1854C22semidirect
40. #1847C22semidirect
41. #1836C22semidirect
42. #1825C22semidirect
44. #1803C22semidirect
45. #1803C22semidirect
48. #1763C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
#341862
#276022
#250252
#249802
#240102
#219862
#217902
#207402