Finite Group Databasebeta

← #124804

Subgroups — #124804

Order 512.

302 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 78 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#21727C211yesmaximal, characteristic
2#12659C211yesmaximal, characteristic
3#8801C211yesmaximal, characteristic
4#7127C212yesmaximal
5#7126C212yesmaximal
6C2×#498C411yescharacteristic
7C4×C4⋊Q8—21no—
8C2×#364—21no—
9C2×#347—21no—
10#1990—21no—
11#1944—21no—
12#1944C2212yes—
13#1845C411yescharacteristic
14#1776—21no—
15#1776C412yes—
16C4×#97C2211yescharacteristic
17#1377C2212yes—
18#1281—21no—
19#1276—21no—
20#1275C2211yescharacteristic
21#1275C2211yescharacteristic
22C2×C4⋊Q8—21no—
23C2×C4×Q8—41no—
24C2×M32—22no—
25#498—24no—
26#431—21no—
27#431—21no—
28C2×#107—21no—
29C2×#107C2×C412yes—
30C2×#106—21no—
31C2×#106C2×C412yes—
32C2×#97C2×C411yescharacteristic
33C2×#97D411yescharacteristic
34C2×#97D411yescharacteristic
35C2×#97C2311yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8D411yescharacteristic
38#391—41no—
39#389—22no—
40#389—41no—
41#384—21no—
42C4×C4⋊C4—22no—
43C4×C4⋊C4—41no—
44C43D411yescharacteristic
45#364—24no—
46#347—42no—
47#340—22no—
48#340—22no—
49C22×Q8—41no—
50C22×C8C22⋊C411yescharacteristic
51C22×C8C22⋊C411yescharacteristic
52C22×C8C22×C411yescharacteristic
53C22×C8C2×D411yescharacteristic
54C4⋊Q8—24no—
55C4⋊Q8—42no—
56C4×Q8—84no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—42no—
63C2×C42—21no—
64C2×C42—21no—
65C2×C42—42no—
66C2×C42C22⋊C411yescharacteristic
67C2×C42C22⋊C411yescharacteristic
68C2×C42C2×D411yescharacteristic
69M32—24no—
70M32—42no—
71C2×C16—42no—
72#107—42no—
73#107—42no—
74#106—24no—
75#106—22no—
76#106M1614yes—
77C4⋊C8—41no—
78C4⋊C8—41no—
79C22⋊C8—41no—
80C22⋊C8—41no—
81#97—22no—
82#97—41no—
83#97—41no—
84#97SD1612yes—
85#97Q1612yes—
86C4×C8—22no—
87C4×C8—22no—
88C4×C8—41no—
89C4×C8—41no—
90#95—42no—
91#95—42no—
92C2×Q8—44no—
93C2×Q8—82no—
94C2×Q8—82no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4C2×C22⋊C411yescharacteristic
100C22×C4AGL(1,ℤ/8)11yescharacteristic
101C22×C4AGL(1,ℤ/8)11yescharacteristic
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—81no—
111C2×C8—81no—
112C2×C8#10112yes—
113C2×C8#10312yes—
114C2×C8C2×M1612yes—
115C2×C8C2×SD1611yescharacteristic
116C2×C8C2×Q1611yescharacteristic
117C4⋊C4—24no—
118C4⋊C4—44no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—82no—
123C4⋊C4—82no—
124C4⋊C4—84no—
125C42—21no—
126C42—21no—
127C42—21no—
128C42—21no—
129C42—42no—
130C42—42no—
131C42—84no—
132C42#10312yes—
133C42#10412yes—
134C16—44no—
135C23#41811yescharacteristic
136Q8—88no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—44no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4—82no—
153C2×C4#40711yescharacteristic
154C2×C4#41011yescharacteristic
155C2×C4C2×#10111yescharacteristic
156C2×C4C2×#10311yescharacteristic
157C2×C4C2×#10411yescharacteristic
158C2×C4#42111yescharacteristic
159C8—22no—
160C8—22no—
161C8—21no—
162C8—21no—
163C8—22no—
164C8—161no—
165C8—161no—
166C22—21no—
167C22—21no—
168C22#149011yescharacteristic
169C22#151411yescharacteristic
170C22#154911yescharacteristic
171C4—24no—
172C4—42no—
173C4—41no—
174C4—41no—
175C4—84no—
176C4—162no—
177C4#146711yescharacteristic
178C4#148711yescharacteristic
179C4#150311yescharacteristic
180C4#153111yescharacteristic
181C2—21no—
182C2—21no—
183C2#762211yescharacteristic
184C2#1183711yescharacteristic
185C2#1183811yescharacteristic
subgroupG/H
center, socleC22#1514
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC23#418
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#217272
#126592
#88012
#71272
#71262