Finite Group Databasebeta

← #142200

Subgroups — #142200

Order 512.

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

normal subgroups only

This group has 346 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#10814C211yesmaximal, characteristic
4#7254C214yesmaximal
5C2×#499C411yescharacteristic, complemented
6C2×#405C2211yescharacteristic
7#2629—21no—
8#2150—21no—
9C2×#349C2212yes—
10#1992—21no—
11#1944C412yescomplemented
12#1941—21no—
13#1844C411yescharacteristic, complemented
14#1809C2212yes—
15#1807—21no—
16#1740—21no—
17#1735—21no—
18C2×H(3,ℤ/4)C2212yes—
19#1377—21no—
20C22×M16C2311yescharacteristic
21C23×C8D411yescharacteristic
22C2×#125—21no—
23C2×#117—22no—
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#342—42no—
44#340—22no—
45#340—22no—
46#340—42no—
47H(3,ℤ/4)—22no—
48H(3,ℤ/4)Q814yes—
49C23×C4C2×D411yescharacteristic
50C2×M16—41no—
51C2×M16—41no—
52C2×M16C2×Q812yes—
53C2×M16#4112yes—
54C22×C8—22no—
55C22×C8—22no—
56C22×C8—41no—
57C22×C8C22⋊C411yescharacteristic
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22×C411yescharacteristic
60C22×C8C2×D411yescharacteristic
61#125—42no—
62#125—42no—
63#117—24no—
64#117—41no—
65#117—41no—
66#117—42no—
67#117—42no—
68#117—42no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4SD1612yes—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4D812yes—
79C2×C42—42no—
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×D811yescharacteristic
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4—82no—
103C22×C4#10212yes—
104C22×C4#10212yes—
105C22×C4C2×C22⋊C411yescharacteristic
106C22×C4C22⋊Q812yes—
107C22×C4C2×SD1611yescharacteristic
108C22×C4#13712yes—
109M16—42no—
110M16—42no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—44no—
121C2×C8#9912yes—
122C2×C8C4×Q812yes—
123C2×C8#12411yescharacteristic
124C2×C8C4⋊Q811yescharacteristic
125C4⋊C4—42no—
126C4⋊C4—44no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—82no—
132C4⋊C4—84no—
133C4⋊C4—162no—
134C4⋊C4—162no—
135C22⋊C4—42no—
136C22⋊C4—44no—
137C22⋊C4—82no—
138C22⋊C4—162no—
139C42—21no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—44no—
144C42—82no—
145C16—82no—
146C23—21no—
147C23—21no—
148C23—42no—
149C23C2×#10211yescharacteristic
150C23#47412yes—
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—84no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4—164no—
169C2×C4#39111yescharacteristic
170C2×C4C2×#9911yescharacteristic
171C2×C4#41011yescharacteristic
172C2×C4#41911yescharacteristic
173C2×C4#42111yescharacteristic
174C2×C4#42111yescharacteristic
175C2×C4#47512yes—
176C8—21no—
177C8—22no—
178C8—21no—
179C8—22no—
180C8—41no—
181C22—22no—
182C22—22no—
183C22—42no—
184C22—42no—
185C22—81no—
186C22#151711yescharacteristic
187C22#151911yescharacteristic
188C22#198811yescharacteristic
189C4—22no—
190C4—42no—
191C4—42no—
192C4—162no—
193C4—164no—
194C4—162no—
195C4#151111yescharacteristic
196C4#151511yescharacteristic
197C4#199511yescharacteristic
198C4#199611yescharacteristic
199C2—21no—
200C2—21no—
201C2—42no—
202C2#991211yescharacteristic
203C2#1186712yes—
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#499C4semidirect
11. #1944C4semidirect
13. #1844C4semidirect

Maximal subgroups

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

subgroupindex
#332142
#127862
#108142
#72542