Finite Group Databasebeta

← #125556

Subgroups — #125556

Order 512.

265 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 72 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#33297C211yesmaximal, characteristic
2#12896C211yesmaximal, characteristic
3#12882C211yesmaximal, characteristic
4#7130C214yesmaximal
5#2868C411yescharacteristic
6#2859C411yescharacteristic
7C2×#364—21no—
8#1993—21no—
9#1993—21no—
10#1992C412yes—
11#1992C2212yes—
12#1834—21no—
13#1834—21no—
14C4×#97C2211yescharacteristic
15#1377—21no—
16#1377—21no—
17#1376—21no—
18#1375C2212yes—
19#1375C2212yes—
20C2×C4⋊Q8—21no—
21C2×C4⋊Q8—41no—
22C2×#125C2×C412yes—
23C2×M32—22no—
24C2×M32—22no—
25C2×#97C2×C411yescharacteristic
26C2×#97D411yescharacteristic
27C2×#97D411yescharacteristic
28C2×#97C2311yescharacteristic
29C2×C4×C8D411yescharacteristic
30C2×C4×C8D411yescharacteristic
31#389—41no—
32#389—41no—
33#384—21no—
34#384—21no—
35#384—41no—
36#384C2×C412yes—
37#382—21no—
38C43C2×C411yescharacteristic
39#364—42no—
40#340—22no—
41#340—22no—
42#340—22no—
43#340—22no—
44C22×Q8—41no—
45C22×C8C22⋊C411yescharacteristic
46C22×C8C22⋊C411yescharacteristic
47C22×C8C2×D411yescharacteristic
48C22×C8C2×D411yescharacteristic
49C4⋊Q8—42no—
50C4⋊Q8—42no—
51C4⋊Q8—84no—
52#125—24no—
53#125—42no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—41no—
62C2×C42—21no—
63C2×C42—21no—
64C2×C42C22⋊C411yescharacteristic
65C2×C42C22⋊C411yescharacteristic
66C2×C42C22×C411yescharacteristic
67M32—42no—
68M32—42no—
69M32—42no—
70M32—42no—
71C2×C16—42no—
72C2×C16—42no—
73#97—22no—
74#97—41no—
75#97—41no—
76#97—41no—
77C4×C8—22no—
78C4×C8SD1612yes—
79C4×C8Q1612yes—
80#95—41no—
81#95—41no—
82C2×Q8—82no—
83C2×Q8—84no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4#10011yescharacteristic
89C22×C4#10011yescharacteristic
90C22×C4C2×C22⋊C411yescharacteristic
91C2×C8—21no—
92C2×C8—21no—
93C2×C8—21no—
94C2×C8—21no—
95C2×C8—21no—
96C2×C8—21no—
97C2×C8—21no—
98C2×C8—21no—
99C2×C8—21no—
100C2×C8#10312yes—
101C2×C8C2×SD1611yescharacteristic
102C2×C8C2×Q1611yescharacteristic
103C2×C8#13712yes—
104C4⋊C4—42no—
105C4⋊C4—42no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—84no—
109C4⋊C4—82no—
110C4⋊C4—82no—
111C4⋊C4—82no—
112C4⋊C4—84no—
113C4⋊C4—84no—
114C42—21no—
115C42—21no—
116C42—21no—
117C42—21no—
118C42—42no—
119C42—42no—
120C42#10312yes—
121C42#13112yes—
122C16—82no—
123C16—82no—
124C23C2×#10011yescharacteristic
125Q8—164no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—84no—
137C2×C4—82no—
138C2×C4—82no—
139C2×C4—82no—
140C2×C4—82no—
141C2×C4#40811yescharacteristic
142C2×C4#41011yescharacteristic
143C2×C4C2×#10311yescharacteristic
144C2×C4#41911yescharacteristic
145C2×C4#42111yescharacteristic
146C2×C4#42111yescharacteristic
147C8—22no—
148C8—22no—
149C8—21no—
150C8—21no—
151C8—41no—
152C22—21no—
153C22—21no—
154C22#146911yescharacteristic
155C22#153311yescharacteristic
156C22#154811yescharacteristic
157C4—24no—
158C4—42no—
159C4—41no—
160C4—41no—
161C4—162no—
162C4—162no—
163C4#148111yescharacteristic
164C4#149711yescharacteristic
165C4#151511yescharacteristic
166C4#151811yescharacteristic
167C2—21no—
168C2—21no—
169C2#772211yescharacteristic
170C2#1191511yescharacteristic
171C2#1191611yescharacteristic
subgroupG/H
center, socleC22#1533
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC23C2×#100
LCSG ▷ C2×C42 ▷ 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
#332972
#128962
#128822
#71302