Finite Group Databasebeta

← #145158

Subgroups — #145158

Order 512.

339 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 74 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#28249C211yesmaximal, characteristic
2#12822C212yesmaximal, complemented
3C2×#1335C212yesmaximal, complemented
4#8822C211yesmaximal, characteristic, complemented
5#7273C211yesmaximal, characteristic, complemented
6#3067—22no—
7C2×#443C2212yes—
8C2×#364C2212yes—
9#2143—22no—
10#2050—22no—
11#1848C2211yescharacteristic
12#1848C2211yescharacteristic
13#1813—22no—
14#1566—22no—
15C2×C82C2211yescharacteristic
16#1388—22no—
17#1335—22no—
18#1335—22no—
19#575—22no—
20C2×#135D412yes—
21C2×#131—22no—
22C2×M32—22no—
23#473—42no—
24#472—42no—
25#471—44no—
26#443—22no—
27#443—22no—
28#443—22no—
29#443—22no—
30C2×#108—22no—
31C2×#108—22no—
32C2×#108D412yes—
33#423—21no—
34#423D412yes—
35C2×#104—22no—
36#413—24no—
37C2×C4×C8—21no—
38C2×C4×C8C2311yescharacteristic
39#364—22no—
40#364—22no—
41#359—44no—
42#338—22no—
43#334—44no—
44C82—21no—
45C82D411yescharacteristic
46C82D411yescharacteristic
47C2×#41—22no—
48#137—44no—
49AGL(1,ℤ/8)—44no—
50#135—22no—
51#135—22no—
52#135—22no—
53#135—22no—
54#135—82no—
55C2×Q16SD1612yes—
56C2×SD16—22no—
57C2×D8SD1612yes—
58#131—42no—
59#131—42no—
60#131—42no—
61#131—42no—
62C2×M16—22no—
63C2×M16—22no—
64C2×M16—42no—
65C2×M16—41no—
66C22×C8—22no—
67C22×C8—42no—
68C22×C8C2×D412yes—
69C2×C42C2×D411yescharacteristic
70M32—42no—
71M32—42no—
72C2×C16—42no—
73#108—22no—
74#108—22no—
75#108—42no—
76#108—42no—
77#108—42no—
78#108—42no—
79C4⋊C8—42no—
80C4⋊C8—42no—
81#104—42no—
82#104—42no—
83#101—44no—
84#100—44no—
85C4×C8—22no—
86C4×C8—21no—
87C4×C8—21no—
88C4×C8C2×D411yescharacteristic
89C4×C8#4111yescharacteristic
90C4×C8#4112yes—
91#41—42no—
92#41—42no—
93#41—42no—
94#41—42no—
95C2×Q8—22no—
96C2×D4—22no—
97C2×D4—42no—
98C22×C4—21no—
99C22×C4—42no—
100C22×C4#12011yescharacteristic
101Q16—22no—
102Q16—22no—
103Q16—82no—
104SD16—42no—
105SD16—42no—
106SD16—84no—
107D8—22no—
108D8—22no—
109D8—82no—
110M16—42no—
111M16—42no—
112M16—44no—
113M16—82no—
114M16—82no—
115M16—82no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—44no—
119C2×C8—42no—
120C2×C8—42no—
121C2×C8—82no—
122C2×C8—82no—
123C2×C8—81no—
124C2×C8#12112yes—
125C2×C8#12312yes—
126C2×C8C2×SD1612yes—
127C2×C8#13512yes—
128C2×C8AGL(1,ℤ/8)12yes—
129C2×C8AGL(1,ℤ/8)12yes—
130C42—21no—
131C42#12111yescharacteristic
132C42#12411yescharacteristic
133C16—82no—
134C23—42no—
135C23#45611yescharacteristic
136Q8—42no—
137Q8—42no—
138D4—42no—
139D4—42no—
140D4—82no—
141D4—82no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4#39411yescharacteristic
149C2×C4#44912yes—
150C2×C4#45012yes—
151C2×C4#45311yescharacteristic
152C8—24no—
153C8—42no—
154C8—82no—
155C8—82no—
156C8—161no—
157C8#46512yes—
158C8#46912yes—
159C8#48112yes—
160C8#48212yes—
161C22—41no—
162C22—82no—
163C22—162no—
164C22#165412yes—
165C22#203911yescharacteristic
166C4—22no—
167C4—41no—
168C4—82no—
169C4#162411yescharacteristic
170C4#162911yescharacteristic
171C4#203412yes—
172C2—21no—
173C2—22no—
174C2—162no—
175C2#1043211yescharacteristic
subgroupG/H
centerC4#1624
commutator, Frattini, cosocleC2×C4×C8C23
socleC2#10432
prime radical#28249C2
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12822C2semidirect
3. C2×#1335C2semidirect
4. #8822C2semidirect
5. #7273C2semidirect

Maximal subgroups

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

subgroupindex
#282492
#128222
C2×#13352
#88222
#72732