Finite Group Databasebeta

← #142199

Subgroups — #142199

Order 512.

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

normal subgroups only

This group has 338 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
1#33214C211yesmaximal, characteristic
2#12786C211yesmaximal, characteristic
3#10815C211yesmaximal, characteristic
4#7255C214yesmaximal
5C2×#499C411yescharacteristic
6C2×#405C2211yescharacteristic
7#2629—21no—
8#2150—21no—
9C2×#349C2212yes—
10#1992—21no—
11#1944C412yes—
12#1941—21no—
13#1844C411yescharacteristic
14#1809C2212yes—
15#1807—21no—
16#1742—21no—
17C2×#341—21no—
18C2×#338C2212yes—
19#1377—21no—
20C22×M16—22no—
21C22×M16C2311yescharacteristic
22C23×C8D411yescharacteristic
23C2×#125—21no—
24C2×#117D411yescharacteristic
25C2×M32—22no—
26#499—42no—
27C2×#107C2×C412yes—
28C2×#106—21no—
29C2×#106—21no—
30C2×#106C2×C412yes—
31#405—42no—
32C2×#97C2×C411yescharacteristic
33C2×#97D411yescharacteristic
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8D411yescharacteristic
36#398—42no—
37#396—42no—
38#389—22no—
39#384—21no—
40#380—22no—
41C4⋊C16—42no—
42#349—24no—
43#343—42no—
44#341—42no—
45#340—22no—
46#340—22no—
47#338—22no—
48#338Q814yes—
49C23×C4C2×D411yescharacteristic
50C2×M16—24no—
51C2×M16—41no—
52C2×M16—41no—
53C2×M16—42no—
54C2×M16—42no—
55C2×M16—42no—
56C2×M16—42no—
57C2×M16—42no—
58C2×M16C2×Q812yes—
59C2×M16#4112yes—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—41no—
63C22×C8—42no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22×C411yescharacteristic
67C22×C8C2×D411yescharacteristic
68#125—42no—
69#125—42no—
70#117—41no—
71#117—41no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4D812yes—
78C2×C22⋊C4—42no—
79C2×C22⋊C4SD1612yes—
80C2×C42C22⋊C411yescharacteristic
81C2×C42C22⋊C411yescharacteristic
82M32—42no—
83M32—42no—
84C2×C16—42no—
85#107—42no—
86#106—42no—
87#106—42no—
88#106—42no—
89#97—41no—
90#97—41no—
91C4×C8—22no—
92C4×C8—22no—
93#95—42no—
94#95—42no—
95C24C2×SD1611yescharacteristic
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4#10212yes—
103C22×C4#10212yes—
104C22×C4C2×C22⋊C411yescharacteristic
105C22×C4C22⋊Q812yes—
106C22×C4C2×D811yescharacteristic
107C22×C4#13712yes—
108M16—42no—
109M16—42no—
110M16—44no—
111M16—44no—
112M16—82no—
113M16—82no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—44no—
124C2×C8—44no—
125C2×C8—82no—
126C2×C8—82no—
127C2×C8#9912yes—
128C2×C8C4×Q812yes—
129C2×C8#12411yescharacteristic
130C2×C8C4⋊Q811yescharacteristic
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—84no—
137C4⋊C4—162no—
138C4⋊C4—162no—
139C22⋊C4—42no—
140C22⋊C4—162no—
141C42—21no—
142C42—21no—
143C42—21no—
144C42—21no—
145C16—82no—
146C23—21no—
147C23—21no—
148C23—42no—
149C23C2×#10211yescharacteristic
150C23#47512yes—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—82no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4#39111yescharacteristic
168C2×C4C2×#9911yescharacteristic
169C2×C4#41011yescharacteristic
170C2×C4#41911yescharacteristic
171C2×C4#42111yescharacteristic
172C2×C4#42111yescharacteristic
173C2×C4#47412yes—
174C8—21no—
175C8—22no—
176C8—21no—
177C8—22no—
178C8—41no—
179C8—84no—
180C22—22no—
181C22—22no—
182C22—42no—
183C22—42no—
184C22—81no—
185C22#151711yescharacteristic
186C22#151911yescharacteristic
187C22#198811yescharacteristic
188C4—22no—
189C4—42no—
190C4—42no—
191C4—162no—
192C4—162no—
193C4#151111yescharacteristic
194C4#151511yescharacteristic
195C4#199511yescharacteristic
196C4#199611yescharacteristic
197C2—21no—
198C2—21no—
199C2—42no—
200C2#991211yescharacteristic
201C2#1187212yes—
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332142
#127862
#108152
#72552