Finite Group Databasebeta

← #326568

Subgroups — #326568

Order 512.

456 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21716C211yesmaximal, characteristic, complemented
2#21659C211yesmaximal, characteristic
3#10670C211yesmaximal, characteristic, complemented
4#10667C211yesmaximal, characteristic
5#10371C211yesmaximal, characteristic, complemented
6#10370C211yesmaximal, characteristic
7#9754C212yesmaximal
8#9632C212yesmaximal, complemented
9#8880C212yesmaximal
10#8879C212yesmaximal, complemented
11#8781C211yesmaximal, characteristic, complemented
12#3131C2211yescharacteristic
13#2961C2211yescharacteristic
14C2×#446—21no—
15#2932C2211yescharacteristic
16#2399C2211yescharacteristic
17#1995C2211yescharacteristic
18#1985—21no—
19#1963C2211yescharacteristic
20#1940—21no—
21#1940C2212yes—
22#1938—21no—
23#1937—21no—
24#1843C2211yescharacteristic
25#1837C2211yescharacteristic
26#1770—21no—
27#1770C2212yes—
28#1765—21no—
29#1764—21no—
30#1726C2211yescharacteristic
31#1571C2212yes—
32#1570C2212yes—
33#1564C2212yes—
34#1503C2212yes—
35#1502C2212yes—
36#1481C2212yes—
37#1480C2212yes—
38#1303C2212yes—
39#1303C2212yes—
40#1281C2212yes—
41#1274C2211yescharacteristic
42#1274C2211yescharacteristic
43C2×#131—21no—
44#533C2311yescharacteristic
45#532C2311yescharacteristic
46C2×C4×Q8—21no—
47#488—22no—
48#487Q812yescomplemented
49#486Q812yescomplemented
50#446—44no—
51#446—42no—
52#435—42no—
53#431C2312yes—
54#431C2312yes—
55#424—21no—
56#423C2311yescharacteristic
57#423C2311yescharacteristic
58C2×C4⋊C8—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2312yes—
62C2×#104—21no—
63C2×#104—21no—
64#416—21no—
65#416—21no—
66#408—21no—
67#405—21no—
68C2×#97—22no—
69C2×#97D411yescharacteristic
70C2×#97D411yescharacteristic
71C2×#97C2311yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#398—22no—
76#395—22no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4C2312yes—
79#376C2311yescharacteristic
80#341—21no—
81#341—21no—
82#339—21no—
83#339—21no—
84#336—22no—
85#336—22no—
86#335—22no—
87#335—42no—
88#334—22no—
89#334—42no—
90#328—22no—
91#328—22no—
92#327—22no—
93#327—22no—
94Q8⋊C8—42no—
95D4⋊C8—42no—
96C2×#41—21no—
97C22×Q8—21no—
98#131—42no—
99#131—42no—
100C2×M16—21no—
101C2×M16—21no—
102C2×M16—41no—
103C22×C8—22no—
104C22×C8—41no—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C4⋊Q8C2×Q811yescharacteristic
110C4⋊Q8C2×Q812yes—
111#127C2×Q811yescharacteristic
112#125—21no—
113#124—21no—
114C22⋊Q8—22no—
115#121—22no—
116C4×Q8—22no—
117C4×Q8—42no—
118C4×Q8—42no—
119C4×D4—22no—
120#117—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C42—22no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2411yescharacteristic
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—44no—
136C4⋊C8—42no—
137C4⋊C8—42no—
138C4⋊C8—42no—
139#104—42no—
140#104—42no—
141#103—22no—
142#103—22no—
143#102—22no—
144#102—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—42no—
148#97—22no—
149#97—22no—
150#97—41no—
151#97—44no—
152#97—41no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—41no—
158C4×C8C2×Q812yes—
159C4×C8#4112yes—
160#95—21no—
161#95—21no—
162#95—22no—
163#95#4112yes—
164#95#4112yes—
165#41—41no—
166#41—41no—
167#41—41no—
168#41—41no—
169C2×Q8—21no—
170C2×Q8—22no—
171C2×Q8—42no—
172C2×Q8—82no—
173C2×D4—21no—
174C2×D4—22no—
175C2×D4—41no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—41no—
181C22×C4C22×D411yescharacteristic
182C22×C4C2×#4112yes—
183M16—41no—
184M16—41no—
185M16—41no—
186M16—41no—
187M16—82no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—82no—
205C2×C8C22⋊Q812yes—
206C2×C8C22⋊Q812yes—
207C2×C8#13712yes—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—42no—
212C4⋊C4—44no—
213C4⋊C4—42no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C22⋊C4—41no—
217C22⋊C4—42no—
218C42—21no—
219C42—21no—
220C42—41no—
221C42—42no—
222C42—42no—
223C42#13512yes—
224C42AGL(1,ℤ/8)12yes—
225C42C22×Q811yescharacteristic
226C42C2×#4111yescharacteristic
227C42H(5,𝔽2)12yes—
228C23—41no—
229C23#52511yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—82no—
233Q8—84no—
234D4—41no—
235D4—41no—
236D4—82no—
237D4—81no—
238D4—81no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4C2×C22⋊Q811yescharacteristic
255C2×C4#53511yescharacteristic
256C2×C4#55412yes—
257C2×C4#55912yes—
258C2×C4C2×#13511yescharacteristic
259C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
260C2×C4C2×#13711yescharacteristic
261C2×C4#57511yescharacteristic
262C8—22no—
263C8—22no—
264C8—22no—
265C8—41no—
266C8—82no—
267C8—82no—
268C8—82no—
269C8—81no—
270C8—81no—
271C22—22no—
272C22—81no—
273C22—82no—
274C22#258011yescharacteristic
275C22#310811yescharacteristic
276C22#311411yescharacteristic
277C4—22no—
278C4—22no—
279C4—42no—
280C4—42no—
281C4—82no—
282C4—82no—
283C4—81no—
284C4—82no—
285C4#307611yescharacteristic
286C4#307911yescharacteristic
287C4#320711yescharacteristic
288C4#321511yescharacteristic
289C2—21no—
290C2—21no—
291C2—161no—
292C2#2926211yescharacteristic
293C2#2927112yes—
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2580
prime radical#532C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21716C2semidirect
3. #10670C2semidirect
5. #10371C2semidirect
8. #9632C2semidirect
10. #8879C2semidirect
11. #8781C2semidirect
48. #487Q8semidirect
49. #486Q8semidirect

Maximal subgroups

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

subgroupindex
#217162
#216592
#106702
#106672
#103712
#103702
#97542
#96322
#88802
#88792
#87812