Finite Group Databasebeta

← #137248

Subgroups — #137248

Order 512.

365 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 86 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×#2089C211yesmaximal, characteristic
2#21971C211yesmaximal, characteristic
3#12871C212yesmaximal
4#11443C211yesmaximal, characteristic
5#7265C212yesmaximal
6#2921—21no—
7C2×#428—21no—
8C22×#107C2211yescharacteristic
9C2×#424C2211yescharacteristic
10#2112C2212yes—
11#2089—22no—
12#2089C414yes—
13#1950—24no—
14#1811—21no—
15#1810C2212yes—
16C2×#343—21no—
17C2×#341C2211yescharacteristic
18#1387—24no—
19#1323—22no—
20C22×M16—21no—
21C22×M16D411yescharacteristic
22C23×C8C2311yescharacteristic
23C2×#117D411yescharacteristic
24C22×C4⋊C4—21no—
25C2×M32—22no—
26#433—42no—
27#433—42no—
28C2×#108—22no—
29C2×#108D414yes—
30#428—24no—
31#428—42no—
32C2×#107—22no—
33C2×#107—22no—
34C2×#107—21no—
35C2×#107—41no—
36C2×#107C2×C412yes—
37C2×#106—41no—
38#424—22no—
39#424C2×C412yes—
40#424D412yes—
41#423—22no—
42C2×C4⋊C8—22no—
43#407—22no—
44C4×M16—22no—
45C22⋊C16—22no—
46C22⋊C16—22no—
47#343—24no—
48#341—22no—
49#341—22no—
50#341—21no—
51#341C2×C412yes—
52C23×C4—41no—
53C23×C4C2×D411yescharacteristic
54C2×M16—21no—
55C2×M16—24no—
56C2×M16—22no—
57C2×M16—24no—
58C2×M16—42no—
59C2×M16—41no—
60C2×M16—42no—
61C2×M16C22⋊C412yes—
62C22×C8—21no—
63C22×C8—42no—
64C22×C8—41no—
65C22×C8C22×C411yescharacteristic
66C22×C8C2×D412yes—
67C22×C8C2×D411yescharacteristic
68C22×C8#4112yes—
69#117—21no—
70#117—21no—
71#117C22⋊C412yes—
72#117D812yes—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4SD1611yescharacteristic
80C2×C4⋊C4SD1611yescharacteristic
81C2×C22⋊C4—21no—
82C2×C42—21no—
83M32—44no—
84C2×C16—42no—
85#108—44no—
86#108—82no—
87#107—24no—
88#107—42no—
89#107—44no—
90#106—44no—
91C4⋊C8—24no—
92C4⋊C8—24no—
93C4⋊C8—44no—
94C22⋊C8—42no—
95C22⋊C8—42no—
96#97—42no—
97C4×C8—42no—
98C24#13711yescharacteristic
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—42no—
104C22×C4—81no—
105C22×C4—81no—
106C22×C4—162no—
107C22×C4C2×C22⋊C411yescharacteristic
108C22×C4#12011yescharacteristic
109C22×C4#12311yescharacteristic
110C22×C4C2×D811yescharacteristic
111C22×C4C2×SD1611yescharacteristic
112C22×C4AGL(1,ℤ/8)11yescharacteristic
113M16—24no—
114M16—44no—
115M16—42no—
116M16—42no—
117M16—44no—
118M16—44no—
119M16—81no—
120C2×C8—24no—
121C2×C8—22no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—42no—
125C2×C8—44no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—44no—
129C2×C8C4×D412yes—
130C2×C8#12112yes—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—81no—
136C4⋊C4—81no—
137C4⋊C4#10212yes—
138C4⋊C4#10212yes—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C42—22no—
142C42—22no—
143C16—82no—
144C23—22no—
145C23—41no—
146C23—41no—
147C23#41711yescharacteristic
148C23#44811yescharacteristic
149C23#48011yescharacteristic
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4—82no—
162C2×C4—162no—
163C2×C4—162no—
164C2×C4—162no—
165C2×C4#38611yescharacteristic
166C2×C4C2×#10211yescharacteristic
167C2×C4#44711yescharacteristic
168C2×C4#45612yes—
169C2×C4#45812yes—
170C2×C4#48111yescharacteristic
171C8—22no—
172C8—22no—
173C8—44no—
174C8—42no—
175C8—84no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—44no—
185C22#161112yes—
186C22#187411yescharacteristic
187C4—21no—
188C4—21no—
189C4—42no—
190C4—42no—
191C4—162no—
192C4#161012yes—
193C4#188611yescharacteristic
194C4#189011yescharacteristic
195C2—22no—
196C2—41no—
197C2—41no—
198C2#902711yescharacteristic
199C2#1300611yescharacteristic
200C2#1300711yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24#137
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#424 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20892
#219712
#128712
#114432
#72652