Finite Group Databasebeta

← #149800

Subgroups — #149800

Order 512.

294 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 56 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#21846C211yesmaximal, characteristic
2#11522C212yesmaximal
3#7361C212yesmaximal, complemented
4#7360C212yesmaximal, complemented
5#3458C411yescharacteristic
6#2921C2211yescharacteristic
7#2298C411yescharacteristic
8#2165—22no—
9#1995C411yescharacteristic
10#1995C2211yescharacteristic
11#1994C411yescharacteristic
12#1994C2211yescharacteristic
13#1986—22no—
14#1826—22no—
15#1327C2214yes—
16#1320—22no—
17#544—22no—
18#544—22no—
19#544—41no—
20#543—41no—
21#533—21no—
22#533—42no—
23#533C2×C411yescharacteristic
24#533C2×C411yescharacteristic
25C2×#117—21no—
26C2×#117C2×C411yescharacteristic
27#433—42no—
28#423C2×C411yescharacteristic
29#423C2311yescharacteristic
30#407D412yes—
31C4×M16D412yes—
32#388—41no—
33#387—41no—
34C4×C4⋊C4—21no—
35C4×C22⋊C4—21no—
36#376C2×C411yescharacteristic
37C43C2×C411yescharacteristic
38#364—42no—
39#349—24no—
40#347—42no—
41#339—22no—
42#339—22no—
43#339—22no—
44#339—22no—
45C23×C4C22⋊C411yescharacteristic
46C2×M16C22⋊C412yes—
47C2×M16C2×D412yes—
48C4⋊Q8—22no—
49C4⋊Q8—21no—
50C4⋊Q8—21no—
51C4⋊Q8—42no—
52C4⋊Q8—42no—
53#125—22no—
54#125—42no—
55#125—42no—
56#125M1612yes—
57#125M1612yes—
58C22⋊Q8—42no—
59C22⋊Q8—42no—
60C22⋊Q8—42no—
61C22⋊Q8—42no—
62C22⋊Q8—42no—
63C22⋊Q8—41no—
64C22⋊Q8—41no—
65C4×Q8—41no—
66C4×Q8—41no—
67C4×Q8—42no—
68#117—22no—
69#117—21no—
70#117—21no—
71#117—22no—
72#117—21no—
73#117—21no—
74#117—41no—
75#117—41no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—41no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4—41no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—41no—
85C2×C42—41no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22×C411yescharacteristic
88M32—44no—
89C4⋊C8—41no—
90C4⋊C8—41no—
91C22⋊C8—42no—
92#97—22no—
93C4×C8—22no—
94#95—41no—
95#95—41no—
96#95—41no—
97C24#10011yescharacteristic
98C2×Q8—42no—
99C2×Q8—41no—
100C2×Q8—41no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—42no—
109C22×C4—41no—
110C22×C4#10011yescharacteristic
111C22×C4C2×C22⋊C411yescharacteristic
112M16—42no—
113M16—42no—
114C2×C8—22no—
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—41no—
127C4⋊C4—41no—
128C4⋊C4—82no—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—42no—
132C22⋊C4—41no—
133C22⋊C4—41no—
134C22⋊C4—82no—
135C42—21no—
136C42—21no—
137C42—42no—
138C42—41no—
139C42—41no—
140C42—41no—
141C42—41no—
142C42—41no—
143C42—41no—
144C42—82no—
145C42#9912yes—
146C42C2×M1611yescharacteristic
147C42C2×M1611yescharacteristic
148C16—84no—
149C23—21no—
150C23—41no—
151C23—81no—
152C23C2×#10011yescharacteristic
153Q8—162no—
154Q8—161no—
155Q8—161no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4—162no—
177C2×C4#40711yescharacteristic
178C2×C4C2×#9911yescharacteristic
179C8—42no—
180C8—42no—
181C22—21no—
182C22—41no—
183C22—42no—
184C22—41no—
185C22—161no—
186C22#146211yescharacteristic
187C4—21no—
188C4—42no—
189C4—41no—
190C4—41no—
191C4—41no—
192C4—82no—
193C4—82no—
194C4—82no—
195C4—161no—
196C4#212911yescharacteristic
197C4#213511yescharacteristic
198C2—21no—
199C2—41no—
200C2—81no—
201C2#1151011yescharacteristic
subgroupG/H
centerC4#2129
commutatorC2×C42C22×C4
Frattini, cosocle#423C23
socleC2#11510
prime radicalC24#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C42 ◁ #2921 ◁ G
LECSG ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7361C2semidirect
4. #7360C2semidirect

Maximal subgroups

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

subgroupindex
#218462
#115222
#73612
#73602