Finite Group Databasebeta

← #138186

Subgroups — #138186

Order 512.

324 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C2×#2166C211yesmaximal, characteristic
2#33214C211yesmaximal, characteristic
3#12789C211yesmaximal, characteristic
4C2×#1366C212yesmaximal, complemented
5#7268C212yesmaximal, complemented
6C2×#499C2211yescharacteristic
7C2×#405C2211yescharacteristic
8#2629C412yes—
9#2166—24no—
10C2×#364C2212yes—
11C2×#364C2212yes—
12C2×#349—21no—
13#2112—21no—
14C2×#347—21no—
15#1992C412yes—
16#1944—21no—
17#1941—21no—
18#1844C2211yescharacteristic
19#1809—21no—
20#1807—21no—
21#1376—21no—
22#1366—24no—
23C22×M16D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×#125C2×C412yes—
26C2×#117C2×C411yescharacteristic
27C2×M32—22no—
28C2×M32—22no—
29#499—22no—
30#499—22no—
31C2×#107—21no—
32C2×#106—21no—
33C2×#106—21no—
34C2×#106—21no—
35#405—22no—
36#405—22no—
37C2×#97C2×C411yescharacteristic
38C2×#97D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8C2311yescharacteristic
41#398—42no—
42#396—42no—
43#389—22no—
44#384C2×C412yes—
45#380—22no—
46#364—22no—
47#364—22no—
48#364—22no—
49#364D414yes—
50#349—24no—
51C22⋊C16—42no—
52#347—24no—
53#340—22no—
54#340—22no—
55C23×C4C22⋊C411yescharacteristic
56C2×M16—21no—
57C2×M16—21no—
58C2×M16—22no—
59C2×M16—22no—
60C22×C8—22no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—22no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8C2×D411yescharacteristic
67C22×C8C2×D411yescharacteristic
68#125—24no—
69#125—42no—
70#117—22no—
71#117—41no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—42no—
80C2×C42C22⋊C411yescharacteristic
81C2×C42C22×C411yescharacteristic
82M32—24no—
83M32—42no—
84M32—42no—
85M32—42no—
86C2×C16—42no—
87C2×C16—42no—
88#107—41no—
89#107—41no—
90#106—42no—
91#106—42no—
92#106—41no—
93#106—41no—
94#97—21no—
95#97—21no—
96#97—22no—
97C4×C8—22no—
98C4×C8C2×D412yes—
99C4×C8#4112yes—
100#95—42no—
101#95—42no—
102C24#10011yescharacteristic
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4#10011yescharacteristic
114C22×C4C2×C22⋊C411yescharacteristic
115M16—42no—
116M16—42no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—42no—
124C2×C8—42no—
125C2×C8#12011yescharacteristic
126C2×C8#12112yes—
127C2×C8#12311yescharacteristic
128C2×C8#13512yes—
129C2×C8AGL(1,ℤ/8)11yescharacteristic
130C2×C8#13712yes—
131C2×C8#13712yes—
132C2×C8#13711yescharacteristic
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—84no—
139C4⋊C4—162no—
140C4⋊C4—162no—
141C22⋊C4—42no—
142C22⋊C4—162no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42C4×D412yes—
147C16—44no—
148C16—82no—
149C23—21no—
150C23—21no—
151C23—21no—
152C23—42no—
153C23C2×#10011yescharacteristic
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4#38611yescharacteristic
172C2×C4#41311yescharacteristic
173C2×C4#41611yescharacteristic
174C2×C4#41711yescharacteristic
175C2×C4#41911yescharacteristic
176C2×C4#41911yescharacteristic
177C8—41no—
178C8—41no—
179C8#45212yes—
180C8#46812yes—
181C8#47012yes—
182C8#48512yes—
183C22—22no—
184C22—42no—
185C22—41no—
186C22—42no—
187C22—81no—
188C22#155111yescharacteristic
189C22#155811yescharacteristic
190C22#190011yescharacteristic
191C4—42no—
192C4—41no—
193C4—81no—
194C4—162no—
195C4—162no—
196C4#155411yescharacteristic
197C4#155911yescharacteristic
198C4#187311yescharacteristic
199C4#189411yescharacteristic
200C2—21no—
201C2—21no—
202C2—81no—
203C2#913611yescharacteristic
204C2#1199112yes—
subgroupG/H
center, socleC22#1551
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1366C2semidirect
5. #7268C2semidirect

Maximal subgroups

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

subgroupindex
C2×#21662
#332142
#127892
C2×#13662
#72682