Finite Group Databasebeta

← #507942

Subgroups — #507942

Order 512.

462 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 118 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#21292C212yesmaximal, complemented
2#13519C211yesmaximal, characteristic, complemented
3#13466C212yesmaximal, complemented
4#13444C211yesmaximal, characteristic
5#13137C212yesmaximal, complemented
6#13117C212yesmaximal, complemented
7#13027C212yesmaximal
8#12951C212yesmaximal, complemented
9#12883C211yesmaximal, characteristic, complemented
10#3414C2211yescharacteristic
11C4×#135C2212yes—
12#2922—22no—
13#2161C2211yescharacteristic
14#2155C2212yes—
15#2150C2211yescharacteristic
16#2150C2211yescharacteristic
17#2148C2212yes—
18#2145C2212yes—
19#2143C2212yes—
20#1974C2212yes—
21#1974C2212yes—
22#1945C2212yes—
23#1945C2212yes—
24#1937—22no—
25#1937—22no—
26C4×#108C2212yes—
27C4×#104—22no—
28C42×C8C2211yescharacteristic
29#1389C2212yes—
30#1388C2212yes—
31#1385C2211yescharacteristic
32#1384C2211yescharacteristic
33#1376C2212yes—
34#1376C2212yes—
35C2×#135C2312yes—
36C4×#41—22no—
37#510—22no—
38#509—22no—
39#508—41no—
40C2×M32C2311yescharacteristic
41C2×M32C2312yes—
42C2×M32C2311yescharacteristic
43C4×Q16—22no—
44C4×SD16—24no—
45C4×D8—22no—
46#433—44no—
47#431—44no—
48C2×#108C2312yes—
49C2×#108C2312yes—
50#427C2312yes—
51#424—22no—
52#424—22no—
53C2×#104—22no—
54C2×#104—22no—
55#416—22no—
56#408—22no—
57#408—22no—
58C4×M16—22no—
59C2×C4×C8D412yes—
60C2×C4×C8D411yescharacteristic
61C2×C4×C8C2312yes—
62C2×C4×C8C2311yescharacteristic
63C43D411yescharacteristic
64#368—41no—
65#365—24no—
66C4⋊C16—41no—
67C4⋊C16—41no—
68#362—22no—
69#361—22no—
70#360—24no—
71#359—42no—
72#341—22no—
73#339—22no—
74#339—22no—
75C2×#41—22no—
76#135—22no—
77#135—22no—
78#135—22no—
79#135—22no—
80C2×Q16#4112yes—
81C2×SD16—22no—
82C2×D8#4112yes—
83C2×M16—22no—
84C2×M16—22no—
85C22×C8C2×D412yes—
86C22×C8C2×D411yescharacteristic
87C22×C8C2411yescharacteristic
88C4×Q8—24no—
89C4×D4—24no—
90C4×D4—44no—
91#117—22no—
92#117—42no—
93C2×C42—41no—
94C2×C42—42no—
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D412yes—
97Q32—41no—
98SD32—42no—
99D16—41no—
100M32—22no—
101M32—22no—
102M32—22no—
103M32—22no—
104C2×C16—22no—
105C2×C16—21no—
106C2×C16—21no—
107#108—22no—
108#108—22no—
109#108—22no—
110#108—22no—
111#107—22no—
112#106#4114yes—
113C4⋊C8—44no—
114C4⋊C8—44no—
115#104—42no—
116#104—42no—
117#104—42no—
118#104—42no—
119#103—24no—
120#102—24no—
121C22⋊C8—44no—
122C22⋊C8—44no—
123#97—42no—
124C4×C8—22no—
125C4×C8—22no—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—21no—
129C4×C8—21no—
130C4×C8—22no—
131C4×C8—22no—
132C4×C8—42no—
133#41—42no—
134#41—42no—
135#41—42no—
136#41—42no—
137C2×Q8—22no—
138C2×D4—22no—
139C2×D4—42no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—44no—
143C22×C4—42no—
144C22×C4C22×D411yescharacteristic
145Q16—22no—
146Q16—22no—
147SD16—42no—
148SD16—42no—
149D8—22no—
150D8—22no—
151M16—42no—
152M16—42no—
153M16—42no—
154M16—42no—
155C2×C8—22no—
156C2×C8—21no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8C2×#4112yes—
163C2×C8C2×#4112yes—
164C2×C8H(5,𝔽2)11yescharacteristic
165C2×C82−1+411yescharacteristic
166C4⋊C4—24no—
167C4⋊C4—44no—
168C22⋊C4—44no—
169C22⋊C4—42no—
170C42—22no—
171C42—21no—
172C42—44no—
173C42—44no—
174C42—42no—
175C42#13511yescharacteristic
176C42#13511yescharacteristic
177C42AGL(1,ℤ/8)12yes—
178C42AGL(1,ℤ/8)12yes—
179C16—41no—
180C16—41no—
181C16—42no—
182C23—42no—
183C23#57511yescharacteristic
184Q8—42no—
185Q8—42no—
186D4—42no—
187D4—42no—
188D4—82no—
189D4—82no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—88no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4#52512yes—
202C2×C4#53711yescharacteristic
203C2×C4C2×#13511yescharacteristic
204C2×C4C2×AGL(1,ℤ/8)12yes—
205C8—42no—
206C8—41no—
207C8—82no—
208C8—82no—
209C8#55111yescharacteristic
210C8#55212yes—
211C8#55511yescharacteristic
212C22—41no—
213C22—84no—
214C22—82no—
215C22#311412yes—
216C22#325011yescharacteristic
217C4—22no—
218C4—24no—
219C4—42no—
220C4—41no—
221C4—84no—
222C4—82no—
223C4#260911yescharacteristic
224C4#310612yes—
225C4#325311yescharacteristic
226C2—22no—
227C2—21no—
228C2—162no—
229C2#2953311yescharacteristic
subgroupG/H
centerC4#3253
commutator, Frattini, cosocleC22×C8C24
socleC2#29533
prime radical#3414C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21292C2semidirect
2. #13519C2semidirect
3. #13466C2semidirect
5. #13137C2semidirect
6. #13117C2semidirect
8. #12951C2semidirect
9. #12883C2semidirect

Maximal subgroups

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

subgroupindex
#212922
#135192
#134662
#134442
#131372
#131172
#130272
#129512
#128832