Finite Group Databasebeta

← #124126

Subgroups — #124126

Order 512.

448 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 138 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#21301C211yesmaximal, characteristic
2C2×#1377C212yesmaximal
3#7133C214yesmaximal, complemented
4C2×#533C411yescharacteristic
5C22×M32—22no—
6C2×#423C411yescharacteristic
7C22×#97C2211yescharacteristic
8#1985—22no—
9#1985C412yes—
10#1985C2212yes—
11#1729—22no—
12#1377—22no—
13#1377C2214yes—
14C22×M16—21no—
15C23×C8D412yes—
16#533—42no—
17#533—42no—
18C2×C4⋊Q8C811yescharacteristic
19C2×C4⋊Q8C811yescharacteristic
20C2×#125—21no—
21C2×C22⋊Q8—21no—
22C2×C22⋊Q8C812yes—
23C2×C4×Q8—21no—
24C2×C4×Q8C2×C412yes—
25C2×#117—21no—
26C22×C42C2×C411yescharacteristic
27C2×M32—28no—
28C2×M32—24no—
29C2×M32—42no—
30C2×M32—44no—
31C22×C16—42no—
32#423—42no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8C2×C412yes—
35C2×#97—42no—
36C2×#97C2×C411yescharacteristic
37C2×#97D412yes—
38C2×#97C2311yescharacteristic
39#336—24no—
40#336—24no—
41C22×Q8—21no—
42C22×Q8C2×C812yes—
43C23×C4C22⋊C411yescharacteristic
44C23×C4C2×C811yescharacteristic
45C23×C4M1611yescharacteristic
46C2×M16—42no—
47C2×M16—44no—
48C22×C8—22no—
49C22×C8—22no—
50C22×C8—24no—
51C22×C8—22no—
52C22×C8C22⋊C412yes—
53C22×C8SD1614yes—
54C22×C8Q1614yes—
55C22×C8C2×D412yes—
56C4⋊Q8—42no—
57C4⋊Q8—42no—
58#125—44no—
59C22⋊Q8—44no—
60C22⋊Q8—44no—
61C4×Q8—42no—
62C4×Q8—42no—
63C4×Q8—42no—
64C4×Q8—42no—
65#117—42no—
66#117—42no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4C2×C812yes—
73C2×C22⋊C4—22no—
74C2×C42—22no—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42C22⋊C411yescharacteristic
78C2×C42C2×C811yescharacteristic
79C2×C42M1611yescharacteristic
80C2×C42C22×C411yescharacteristic
81M32—28no—
82M32—48no—
83M32—44no—
84C2×C16—48no—
85C2×C16—44no—
86C4⋊C8—42no—
87C4⋊C8—42no—
88#97—41no—
89#97—42no—
90#97—41no—
91C24C22⋊C811yescharacteristic
92C2×Q8—42no—
93C2×Q8—42no—
94C2×Q8—82no—
95C2×Q8—82no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—81no—
101C22×C4—81no—
102C22×C4C22⋊C811yescharacteristic
103C22×C4C22⋊C811yescharacteristic
104C22×C4C22⋊C811yescharacteristic
105C22×C4#9912yes—
106C22×C4#10012yes—
107C22×C4#10314yes—
108C22×C4#10414yes—
109C22×C4C2×C22⋊C411yescharacteristic
110C22×C4C22×C811yescharacteristic
111C22×C4C2×M1611yescharacteristic
112M16—84no—
113C2×C8—24no—
114C2×C8—24no—
115C2×C8—24no—
116C2×C8—24no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—42no—
122C2×C8—44no—
123C2×C8#10314yes—
124C2×C8C2×SD1612yes—
125C2×C8C2×Q1612yes—
126C4⋊C4—42no—
127C4⋊C4—44no—
128C4⋊C4—44no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C22⋊C4—44no—
133C42—22no—
134C42—22no—
135C42—21no—
136C42—21no—
137C42—21no—
138C42—21no—
139C42—42no—
140C42—42no—
141C16—48no—
142C23—41no—
143C23—41no—
144C23#32312yes—
145C23Q8⋊C814yes—
146C23C2×C22⋊C811yescharacteristic
147Q8—84no—
148Q8—84no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—44no—
158C2×C4—44no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—82no—
162C2×C4—82no—
163C2×C4#32312yes—
164C2×C4Q8⋊C814yes—
165C2×C4#32814yes—
166C2×C4#32814yes—
167C2×C4#32914yes—
168C2×C4#33314yes—
169C2×C4C2×#9911yescharacteristic
170C2×C4C2×#10011yescharacteristic
171C2×C4C2×#10312yes—
172C2×C4C2×#10412yes—
173C8—22no—
174C8—24no—
175C8—22no—
176C8—82no—
177C22—21no—
178C22—22no—
179C22—21no—
180C22—22no—
181C22—44no—
182C22#127214yes—
183C22C2×#32311yescharacteristic
184C22C2×Q8⋊C812yes—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—41no—
189C4—82no—
190C4—82no—
191C4#127214yes—
192C4C2×#32812yes—
193C4C2×#32911yescharacteristic
194C4C2×#33311yescharacteristic
195C2—22no—
196C2—24no—
197C2C2×#127211yescharacteristic
198C2#1143611yescharacteristic
199C2#1143811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22C2×#323
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7133C2semidirect

Maximal subgroups

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

subgroupindex
#213012
C2×#13772
#71332