Finite Group Databasebeta

← #149784

Subgroups — #149784

Order 512.

366 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 62 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#21321C211yesmaximal, characteristic
2#11513C212yesmaximal
3#7358C214yesmaximal, complemented
4C2×#533C411yescharacteristic
5C22×#97C411yescharacteristic
6#2285C2211yescharacteristic
7#1993—22no—
8#1992C412yes—
9#1992C2212yes—
10#1735—22no—
11#1464—22no—
12#1325C2214yes—
13#1324—22no—
14C23×C8—21no—
15#533—24no—
16#533—42no—
17C2×C4⋊Q8—21no—
18C2×#125C2×C412yes—
19C2×C22⋊Q8—41no—
20C2×C4×Q8—41no—
21C2×#117—21no—
22C2×#117D412yes—
23C22×C42C2×C411yescharacteristic
24#432—42no—
25C2×#97—21no—
26C2×#97—42no—
27C2×#97C2×C412yes—
28#384C2×C411yescharacteristic
29#384C2311yescharacteristic
30#382D412yes—
31#380—41no—
32C4×C4⋊C4—41no—
33#340—24no—
34#340—24no—
35#324—24no—
36#323—42no—
37C22×Q8—41no—
38C23×C4—21no—
39C23×C4C22⋊C411yescharacteristic
40C22×C8—24no—
41C22×C8—22no—
42C22×C8—41no—
43C22×C8—41no—
44C22×C8—42no—
45C4⋊Q8—24no—
46C4⋊Q8—42no—
47#125—22no—
48#125—42no—
49#125M1614yes—
50C22⋊Q8—48no—
51C4×Q8—48no—
52#117—24no—
53#117—42no—
54#117—42no—
55#117—42no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4C22⋊C412yes—
62C2×C4⋊C4C2×D412yes—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—41no—
65C2×C42—21no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42—22no—
69C2×C42—41no—
70C2×C42C22⋊C411yescharacteristic
71C2×C42C22×C411yescharacteristic
72C4⋊C8—82no—
73C22⋊C8—44no—
74#97—42no—
75#97—42no—
76#97—82no—
77#95—42no—
78C24#9911yescharacteristic
79C2×Q8—44no—
80C2×Q8—82no—
81C2×Q8—82no—
82C22×C4—22no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—24no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—42no—
93C22×C4—81no—
94C22×C4—81no—
95C22×C4#9911yescharacteristic
96C22×C4C2×C22⋊C411yescharacteristic
97C2×C8—24no—
98C2×C8—44no—
99C2×C8—44no—
100C2×C8—42no—
101C2×C8—42no—
102C2×C8—84no—
103C4⋊C4—24no—
104C4⋊C4—44no—
105C4⋊C4—42no—
106C4⋊C4—42no—
107C4⋊C4—44no—
108C4⋊C4—44no—
109C4⋊C4—42no—
110C4⋊C4—82no—
111C4⋊C4—82no—
112C22⋊C4—44no—
113C22⋊C4—82no—
114C42—24no—
115C42—21no—
116C42—21no—
117C42—44no—
118C42—42no—
119C42—42no—
120C42#10012yes—
121C42C2×M1612yes—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#35612yes—
127C23C2×#9911yescharacteristic
128Q8—88no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—44no—
146C2×C4—84no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4#35411yescharacteristic
152C2×C4#35411yescharacteristic
153C2×C4#40711yescharacteristic
154C2×C4C2×#10011yescharacteristic
155C8—44no—
156C8—164no—
157C22—21no—
158C22—21no—
159C22—22no—
160C22—42no—
161C22—42no—
162C22—41no—
163C22—41no—
164C22#146211yescharacteristic
165C22C2×#35411yescharacteristic
166C22C2×#35611yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—42no—
172C4—42no—
173C4—84no—
174C4—82no—
175C4—82no—
176C2—21no—
177C2—21no—
178C2—42no—
179C2#1150611yescharacteristic
180C2#1359411yescharacteristic
181C2#1359811yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7358C2semidirect

Maximal subgroups

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

subgroupindex
#213212
#115132
#73582