Finite Group Databasebeta

← #149805

Subgroups — #149805

Order 512.

346 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 66 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×#1836C211yesmaximal, characteristic
2#11512C212yesmaximal
3#7367C214yesmaximal, complemented
4C22×#97C411yescharacteristic
5C2×#384C2211yescharacteristic
6C2×#382C411yescharacteristic
7#1836—22no—
8#1836C412yes—
9#1836C2212yes—
10C2×#340—22no—
11#1463—22no—
12#1325—22no—
13#1325C2214yes—
14C23×C8—21no—
15C22×C4⋊C4—21no—
16C22×C4⋊C4—41no—
17C22×C4⋊C4D412yes—
18C22×C42C2×C411yescharacteristic
19#407—42no—
20C2×#97—21no—
21C2×#97—24no—
22C2×#97C2×C412yes—
23#384—41no—
24#384C2×C411yescharacteristic
25#384D412yes—
26#384C2311yescharacteristic
27#382—21no—
28#382—41no—
29#382C2×C412yes—
30C2×#95—41no—
31#340—24no—
32#340—24no—
33#324—24no—
34#324—42no—
35C23×C4—21no—
36C23×C4—22no—
37C23×C4—41no—
38C23×C4C22⋊C411yescharacteristic
39C2×M16—82no—
40C22×C8—24no—
41C22×C8—22no—
42C22×C8—41no—
43C22×C8—41no—
44C22×C8—42no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—42no—
48C2×C4⋊C4—42no—
49C2×C4⋊C4—44no—
50C2×C4⋊C4—41no—
51C2×C4⋊C4—42no—
52C2×C4⋊C4—41no—
53C2×C4⋊C4—41no—
54C2×C4⋊C4—84no—
55C2×C4⋊C4C22⋊C412yes—
56C2×C4⋊C4C2×D412yes—
57C2×C42—21no—
58C2×C42—22no—
59C2×C42—22no—
60C2×C42C22⋊C411yescharacteristic
61C2×C42C22×C411yescharacteristic
62C22⋊C8—44no—
63C22⋊C8—82no—
64#97—24no—
65#97—22no—
66#97M1614yescomplemented
67#95—42no—
68#95—42no—
69C24#9911yescharacteristic
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—22no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—24no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—84no—
86C22×C4—82no—
87C22×C4—82no—
88C22×C4—82no—
89C22×C4#9911yescharacteristic
90C22×C4C2×C22⋊C411yescharacteristic
91M16—164no—
92C2×C8—24no—
93C2×C8—44no—
94C2×C8—44no—
95C2×C8—42no—
96C2×C8—42no—
97C2×C8—84no—
98C4⋊C4—42no—
99C4⋊C4—42no—
100C4⋊C4—44no—
101C4⋊C4—82no—
102C4⋊C4—82no—
103C4⋊C4—84no—
104C4⋊C4—82no—
105C4⋊C4—82no—
106C42—24no—
107C42—21no—
108C42—21no—
109C42#10112yes—
110C42C2×M1612yes—
111C23—21no—
112C23—22no—
113C23—22no—
114C23—21no—
115C23#35411yescharacteristic
116C23#35411yescharacteristic
117C23C2×#9911yescharacteristic
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—48no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—84no—
136C2×C4—82no—
137C2×C4—82no—
138C2×C4—82no—
139C2×C4—82no—
140C2×C4—82no—
141C2×C4—162no—
142C2×C4—164no—
143C2×C4#35512yes—
144C2×C4#40711yescharacteristic
145C2×C4C2×#10111yescharacteristic
146C8—44no—
147C8—164no—
148C22—24no—
149C22—24no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22#141212yes—
157C22#141312yes—
158C22#146311yescharacteristic
159C22C2×#35411yescharacteristic
160C22C2×#35511yescharacteristic
161C4—21no—
162C4—22no—
163C4—21no—
164C4—42no—
165C4—42no—
166C4—82no—
167C4—82no—
168C4—162no—
169C2—21no—
170C2—21no—
171C2—22no—
172C2—22no—
173C2#1151111yescharacteristic
174C2C2×#141211yescharacteristic
175C2C2×#141311yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7367C2semidirect
66. #97M16semidirect

Maximal subgroups

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

subgroupindex
C2×#18362
#115122
#73672