Finite Group Databasebeta

← #496805

Subgroups — #496805

Order 512.

645 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33432C211yesmaximal, characteristic, complemented
2#33340C211yesmaximal, characteristic, complemented
3#21483C211yesmaximal, characteristic
4#13200C212yesmaximal, complemented
5#13126C212yesmaximal, complemented
6#13123C212yesmaximal, complemented
7#12983C212yesmaximal, complemented
8#12981C211yesmaximal, characteristic, complemented
9#12916C211yesmaximal, characteristic, complemented
10#12685C211yesmaximal, characteristic, complemented
11#12617C211yesmaximal, characteristic, complemented
12#3573C2211yescharacteristic
13#3407C2211yescharacteristic
14C2×#504C2211yescharacteristic
15C23×C16C2211yescharacteristic, complemented
16#3025C411yescharacteristic
17#2929C2211yescharacteristic
18#2929C2212yes—
19#2168—22no—
20D4×C16—22no—
21#2153C2212yescomplemented
22#2151C2211yescharacteristic, complemented
23#2117—22no—
24#2117C2212yes—
25#2117C2212yes—
26C2×#350—22no—
27C2×#350C2212yes—
28C2×#350C2212yes—
29#2115C2211yescharacteristic
30#2113C2211yescharacteristic
31#2112C2211yescharacteristic, complemented
32#2112C2212yescomplemented
33#2110C2212yescomplemented
34#2108C2211yescharacteristic, complemented
35#1891C411yescharacteristic
36#1891C412yes—
37#1868C411yescharacteristic
38#1868C412yes—
39#1791C411yescharacteristic
40#1757C2212yes—
41#1757C2211yescharacteristic
42#1379C2212yescomplemented
43#1379C2212yescomplemented
44#1378C2212yescomplemented
45#585C2×C411yescharacteristic
46#568—22no—
47#568—42no—
48#568C2×C411yescharacteristic
49#568C2×C412yes—
50C2×#131—22no—
51C2×#131—42no—
52C2×#131C2312yes—
53C2×#131C2311yescharacteristic
54C23×C8C2311yescharacteristic
55#525C2×C411yescharacteristic
56#525C2×C412yes—
57#504—42no—
58#504—42no—
59C2×M32—22no—
60C2×M32D412yescomplemented
61C2×M32C2311yescharacteristic
62C2×M32C2312yes—
63C2×M32C2312yes—
64C2×M32C2311yescharacteristic
65C22×C16—21no—
66C22×C16—21no—
67C22×C16—42no—
68C22×C16D412yescomplemented
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#458—22no—
72B(4,𝔽2)—22no—
73#435—22no—
74#435—42no—
75C8×D4—22no—
76C8×D4—42no—
77#424C2×C411yescharacteristic
78#424C2×C412yes—
79#413C2×C412yes—
80#413C2×C411yescharacteristic
81#410C2×C411yescharacteristic
82#410C2×C412yes—
83#408C2×C411yescharacteristic
84#408C2×C412yes—
85#407C2×C411yescharacteristic
86#407C2×C412yes—
87#405C2311yescharacteristic
88#405C2312yes—
89#364—42no—
90C4⋊C16—22no—
91#350—22no—
92#350—24no—
93#350—22no—
94#350—42no—
95#350—42no—
96#349—22no—
97C22⋊C16—22no—
98C22⋊C16—22no—
99C22⋊C16—42no—
100#347—42no—
101#346—21no—
102C4×C16—21no—
103#344C2×C412yes—
104#344C2×C412yes—
105#344C2×C412yes—
1062−1+4—22no—
1072−1+4—21no—
108H(5,𝔽2)—22no—
109H(5,𝔽2)—22no—
110H(5,𝔽2)—21no—
111C2×#41—22no—
112C2×#41—42no—
113C2×#41C22×C412yes—
114C2×#41C22×C411yescharacteristic
115C23×C4C22×C411yescharacteristic
116#131—22no—
117#131—22no—
118#131—24no—
119#131—22no—
120#131—22no—
121#131—22no—
122#131—22no—
123#131—42no—
124#131—42no—
125#131—42no—
126#131—42no—
127#131—44no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—42no—
131C2×M16—42no—
132C2×M16—44no—
133C2×M16C22×C412yes—
134C2×M16C22×C411yescharacteristic
135C2×M16C22×C412yes—
136C2×M16C22×C411yescharacteristic
137C2×M16#4112yes—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—42no—
144C22×C8C22×C411yescharacteristic
145C22×C8C2×D412yes—
146C22×C8C2411yescharacteristic
147#123—22no—
148#123M1612yes—
149C22⋊Q8—21no—
150C22⋊Q8—22no—
151#121—21no—
152#121—22no—
153#120—22no—
154#120M1612yes—
155C4×D4—22no—
156C4×D4—42no—
157#117C22×C412yes—
158#117C22×C411yescharacteristic
159M32—22no—
160M32—22no—
161M32—22no—
162M32—22no—
163M32—22no—
164M32—22no—
165M32—42no—
166M32—42no—
167C2×C16—22no—
168C2×C16—21no—
169C2×C16—21no—
170C2×C16—22no—
171C2×C16—22no—
172C2×C16—22no—
173C2×C16—42no—
174C2×C16—42no—
175C2×C16—42no—
176C2×C16—42no—
177C2×C16C2×D411yescharacteristic
178C2×C16C2×D411yescharacteristic
179C2×C16C2×D412yes—
180C4⋊C8—22no—
181C4⋊C8—42no—
182#101—22no—
183#101—21no—
184#100—22no—
185#100—21no—
186#99—22no—
187#99—24no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—42no—
191C22⋊C8—42no—
192#97—21no—
193#97—22no—
194C4×C8—21no—
195C4×C8—22no—
196C24C2×M1611yescharacteristic
197#41—22no—
198#41—22no—
199#41—24no—
200#41—22no—
201#41—22no—
202#41—22no—
203#41—22no—
204#41—44no—
205#41—42no—
206#41—42no—
207#41—42no—
208#41—42no—
209C2×Q8—22no—
210C2×Q8—42no—
211C2×Q8#13112yes—
212C2×Q8#13111yescharacteristic
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—42no—
217C2×D4—44no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4C2×M1612yes—
221C2×D4#13112yes—
222C2×D4#13111yescharacteristic
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4C4×D412yes—
230C22×C4C2×M1611yescharacteristic
231C22×C4C23×C411yescharacteristic
232M16—22no—
233M16—24no—
234M16—22no—
235M16—22no—
236M16—22no—
237M16—22no—
238M16—22no—
239M16—44no—
240M16—42no—
241M16—42no—
242M16—42no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—44no—
251C2×C8—42no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8C4×D412yes—
258C2×C8C22×D411yescharacteristic
259C2×C8C2×#4111yescharacteristic
260C2×C8H(5,𝔽2)12yes—
261C2×C8H(5,𝔽2)12yes—
262C4⋊C4—21no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—42no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4C2×M1612yes—
272C42—21no—
273C42—22no—
274C16—21no—
275C16—21no—
276C16—22no—
277C16—41no—
278C16—42no—
279C16—41no—
280C16—42no—
281C23—21no—
282C23—22no—
283C23—22no—
284C23—42no—
285C23—42no—
286C23#43512yes—
287C23C22×M1611yescharacteristic
288Q8—22no—
289Q8—22no—
290Q8—22no—
291Q8—42no—
292D4—24no—
293D4—22no—
294D4—22no—
295D4—22no—
296D4—42no—
297D4—44no—
298D4—44no—
299D4—42no—
300D4—42no—
301D4—84no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4#43512yes—
317C2×C4C2×C4×D411yescharacteristic
318C2×C4#51812yes—
319C2×C4C2×#13112yes—
320C2×C4C2×#13111yescharacteristic
321C8—22no—
322C8—41no—
323C8—42no—
324C8—41no—
325C8—42no—
326C8#53411yescharacteristic
327C8#53511yescharacteristic
328C8#53812yes—
329C22—22no—
330C22—24no—
331C22—22no—
332C22—41no—
333C22—42no—
334C22—41no—
335C22—42no—
336C22—41no—
337C22—42no—
338C22—82no—
339C22—162no—
340C22C2×#43511yescharacteristic
341C22#297412yes—
342C4—22no—
343C4—42no—
344C4—41no—
345C4—42no—
346C4—82no—
347C4#232111yescharacteristic
348C4#292911yescharacteristic
349C4#297712yes—
350C2—21no—
351C2—22no—
352C2—42no—
353C2—42no—
354C2—82no—
355C2#2203011yescharacteristic
subgroupG/H
centerC8#535
commutatorC22×C4C23×C4
Frattini, cosocleC22×C8C24
socleC2#22030
prime radical#3025C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C22×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33432C2semidirect
2. #33340C2semidirect
4. #13200C2semidirect
5. #13126C2semidirect
6. #13123C2semidirect
7. #12983C2semidirect
8. #12981C2semidirect
9. #12916C2semidirect
10. #12685C2semidirect
11. #12617C2semidirect
15. C23×C16C22semidirect
21. #2153C22semidirect
22. #2151C22semidirect
31. #2112C22semidirect
32. #2112C22semidirect
33. #2110C22semidirect
34. #2108C22semidirect
42. #1379C22semidirect
43. #1379C22semidirect
44. #1378C22semidirect
60. C2×M32D4semidirect
68. C22×C16D4semidirect

Maximal subgroups

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

subgroupindex
#334322
#333402
#214832
#132002
#131262
#131232
#129832
#129812
#129162
#126852
#126172