Finite Group Databasebeta

← #326593

Subgroups — #326593

Order 512.

413 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 126 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#22304C211yesmaximal, characteristic, complemented
2#22236C211yesmaximal, characteristic
3#10668C211yesmaximal, characteristic, complemented
4#10667C211yesmaximal, characteristic
5#10373C211yesmaximal, characteristic, complemented
6#10372C211yesmaximal, characteristic
7#9754C212yesmaximal
8#9630C212yesmaximal, complemented
9#8879C212yesmaximal, complemented
10#8878C212yesmaximal
11#8782C211yesmaximal, characteristic, complemented
12#3131C2211yescharacteristic
13#2961C2211yescharacteristic
14#2932C2211yescharacteristic
15#2402C2211yescharacteristic
16#1995C2211yescharacteristic
17#1963C2211yescharacteristic
18#1940—22no—
19#1940C2212yes—
20#1929—22no—
21#1917—22no—
22#1866—22no—
23#1848C2211yescharacteristic
24#1843C2211yescharacteristic
25#1770C2212yes—
26#1746C2211yescharacteristic
27#1571C2212yes—
28#1571C2212yes—
29#1564C2212yes—
30#1503C2212yes—
31#1502C2212yes—
32#1481C2212yes—
33#1480C2212yes—
34#1303C2212yes—
35#1303C2212yes—
36#1281C2212yes—
37#1274C2211yescharacteristic
38#1274C2211yescharacteristic
39C2×#131—21no—
40#533C2311yescharacteristic
41#532C2311yescharacteristic
42#488—21no—
43#488—21no—
44#487Q812yes—
45#486Q812yes—
46#446—42no—
47#435—42no—
48#431C2312yes—
49#431C2312yes—
50C2×#108—22no—
51#424—21no—
52#423C2311yescharacteristic
53#423C2311yescharacteristic
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8C2312yes—
57C2×#104—22no—
58#416—22no—
59#408—21no—
60#405—21no—
61C2×#97—22no—
62C2×#97C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#400—22no—
65#398—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4C2312yes—
68#376C2311yescharacteristic
69#339—22no—
70#336—22no—
71#336—22no—
72#336D412yes—
73#336D412yes—
74#335—22no—
75#335—42no—
76#335—42no—
77#334—22no—
78#328—22no—
79#328—22no—
80#327—22no—
81#327—22no—
82Q8⋊C8—42no—
83D4⋊C8—42no—
84C2×#41—21no—
85#131—82no—
86C2×M16—21no—
87C2×M16—21no—
88C2×M16—41no—
89C22×C8—22no—
90C22×C8—41no—
91C22×C8C2×D411yescharacteristic
92C22×C8C2×D411yescharacteristic
93C22×C8C2×D412yes—
94C4⋊Q8C2×Q811yescharacteristic
95C4⋊Q8C2×Q812yes—
96#127C2×Q811yescharacteristic
97#125—21no—
98#124—21no—
99C22⋊Q8—22no—
100#121—22no—
101C4×Q8—22no—
102C4×D4—22no—
103#117—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C42—22no—
108C2×C42C2×D412yes—
109C2×C42C2411yescharacteristic
110#108—82no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—42no—
118C4⋊C8—82no—
119C4⋊C8—82no—
120#104—82no—
121#103—22no—
122#103—22no—
123#102—22no—
124#102—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—42no—
128#97—22no—
129#97—41no—
130#97—82no—
131C4×C8—22no—
132C4×C8—22no—
133C4×C8—41no—
134C4×C8C2×Q812yes—
135C4×C8#4111yescharacteristic
136C4×C8#4111yescharacteristic
137#95—22no—
138#95—22no—
139#95—22no—
140#95#4112yes—
141#95#4112yes—
142#41—42no—
143#41—42no—
144C2×Q8—21no—
145C2×Q8—22no—
146C2×D4—21no—
147C2×D4—22no—
148C2×D4—41no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—41no—
153C22×C4C22×D411yescharacteristic
154C22×C4C2×#4112yes—
155M16—42no—
156M16—42no—
157M16—82no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—82no—
172C2×C8C22⋊Q812yes—
173C2×C8C22⋊Q812yes—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—82no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C42—22no—
184C42—22no—
185C42—41no—
186C42—42no—
187C42—42no—
188C42C22×Q811yescharacteristic
189C42C2×#4111yescharacteristic
190C42H(5,𝔽2)12yes—
191C23—41no—
192C23#52511yescharacteristic
193Q8—42no—
194Q8—82no—
195D4—42no—
196D4—82no—
197D4—82no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4C2×C22⋊Q811yescharacteristic
213C2×C4#53511yescharacteristic
214C2×C4#55412yes—
215C2×C4#55912yes—
216C2×C4#57612yes—
217C2×C4#57812yes—
218C8—22no—
219C8—22no—
220C8—41no—
221C8—41no—
222C8—82no—
223C8—82no—
224C8—82no—
225C8—81no—
226C8—81no—
227C22—22no—
228C22—81no—
229C22—82no—
230C22#258011yescharacteristic
231C22#312412yes—
232C4—22no—
233C4—22no—
234C4—82no—
235C4—82no—
236C4—82no—
237C4—81no—
238C4—82no—
239C4#308312yes—
240C4#322011yescharacteristic
241C4#322311yescharacteristic
242C2—22no—
243C2—161no—
244C2#2926811yescharacteristic
245C2#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. #22304C2semidirect
3. #10668C2semidirect
5. #10373C2semidirect
8. #9630C2semidirect
9. #8879C2semidirect
11. #8782C2semidirect

Maximal subgroups

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

subgroupindex
#223042
#222362
#106682
#106672
#103732
#103722
#97542
#96302
#88792
#88782
#87822