Finite Group Databasebeta

← #134645

Subgroups — #134645

Order 512.

351 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 92 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#21707C211yesmaximal, characteristic
2#12880C211yesmaximal, characteristic, complemented
3#11929C212yesmaximal
4#8780C211yesmaximal, characteristic, complemented
5#7276C212yesmaximal, complemented
6#3147C2211yescharacteristic
7C2×#444C412yes—
8C2×#443—21no—
9#2932—21no—
10C2×C4⋊C16C2212yes—
11#1975C412yes—
12#1974—21no—
13#1963—21no—
14#1848C2211yescharacteristic
15#1726C2211yescharacteristic
16#1564C2212yes—
17#1374—21no—
18#1374—21no—
19#1362—22no—
20#1361—22no—
21#1281—21no—
22#1274—21no—
23C2×#135—21no—
24C2×#135C2×C412yes—
25C2×#131—22no—
26#532—21no—
27C22×C16—22no—
28#495—21no—
29#495Q812yes—
30#494D411yescharacteristic
31#493D411yescharacteristic
32#492—21no—
33#444—44no—
34#443—44no—
35#435—44no—
36#431—22no—
37C2×#108—21no—
38C2×#108C2×C412yes—
39#423—21no—
40C2×#104—22no—
41C2×#104—22no—
42#408—22no—
43C2×#97C2×C411yescharacteristic
44C2×#97C2×C411yescharacteristic
45C2×#97D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8C2311yescharacteristic
49#376D411yescharacteristic
50C4⋊C16—22no—
51C4⋊C16—22no—
52#335—22no—
53#334—22no—
54C2×#41—22no—
55#135—42no—
56#135—42no—
57#135—42no—
58#135—42no—
59C2×Q16—21no—
60C2×Q16C4⋊C412yes—
61C2×SD16—22no—
62C2×SD16—22no—
63C2×D8—21no—
64C2×D8C4⋊C412yes—
65#131—44no—
66#131—44no—
67C2×M16—22no—
68C2×M16—42no—
69C22×C8—42no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C4⋊Q8—21no—
75#127—21no—
76#124—22no—
77#121—42no—
78C4×D4—42no—
79C2×C42C22⋊C411yescharacteristic
80C2×C42C22⋊C411yescharacteristic
81C2×C42C2×D411yescharacteristic
82C2×C16—42no—
83C2×C16—42no—
84C2×C16—42no—
85#108—42no—
86#108—42no—
87C4⋊C8—42no—
88C4⋊C8—42no—
89#104—44no—
90#104—44no—
91C22⋊C8—44no—
92C22⋊C8—42no—
93#97—22no—
94#97—22no—
95#97—41no—
96C4×C8—22no—
97C4×C8—22no—
98C4×C8—42no—
99C4×C8C2×D411yescharacteristic
100C4×C8C2×Q811yescharacteristic
101C4×C8#4111yescharacteristic
102C4×C8#4111yescharacteristic
103#95—22no—
104#41—42no—
105#41—42no—
106#41—42no—
107#41—42no—
108C2×Q8—22no—
109C2×D4—22no—
110C2×D4—42no—
111C2×D4—42no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—42no—
115C22×C4C2×C22⋊C411yescharacteristic
116C22×C4#13512yes—
117Q16—42no—
118Q16—42no—
119SD16—44no—
120SD16—44no—
121D8—42no—
122D8—42no—
123M16—42no—
124M16—42no—
125M16—84no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—42no—
134C2×C8—84no—
135C2×C8—82no—
136C2×C8#10012yes—
137C2×C8C2×C4⋊C411yescharacteristic
138C2×C8#11711yescharacteristic
139C2×C8#12111yescharacteristic
140C2×C8C22⋊Q811yescharacteristic
141C2×C8#13512yes—
142C2×C8AGL(1,ℤ/8)12yes—
143C4⋊C4—42no—
144C22⋊C4—44no—
145C42—21no—
146C42—21no—
147C42—21no—
148C42—21no—
149C42#12111yescharacteristic
150C42C22⋊Q811yescharacteristic
151C42AGL(1,ℤ/8)12yes—
152C16—82no—
153C23—42no—
154C23#41611yescharacteristic
155Q8—42no—
156Q8—42no—
157D4—42no—
158D4—42no—
159D4—82no—
160D4—82no—
161D4—162no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4#38011yescharacteristic
171C2×C4C2×#10011yescharacteristic
172C2×C4#41311yescharacteristic
173C2×C4#41611yescharacteristic
174C2×C4#41811yescharacteristic
175C2×C4#41811yescharacteristic
176C2×C4#46312yes—
177C2×C4#47812yes—
178C8—21no—
179C8—21no—
180C8—22no—
181C8—41no—
182C8—82no—
183C8—162no—
184C8#46312yes—
185C8#47812yes—
186C22—22no—
187C22—84no—
188C22—82no—
189C22#154311yescharacteristic
190C22#155311yescharacteristic
191C22#180411yescharacteristic
192C4—22no—
193C4—22no—
194C4—81no—
195C4—81no—
196C4—82no—
197C4#155311yescharacteristic
198C4#155511yescharacteristic
199C4#180011yescharacteristic
200C4#180411yescharacteristic
201C2—21no—
202C2—21no—
203C2—162no—
204C2#859211yescharacteristic
205C2#1282912yes—
subgroupG/H
centerC2×C4#418
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1804
prime radical#3147C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1726 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12880C2semidirect
4. #8780C2semidirect
5. #7276C2semidirect

Maximal subgroups

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

subgroupindex
#217072
#128802
#119292
#87802
#72762