Finite Group Databasebeta

← #149690

Subgroups — #149690

Order 512.

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

normal subgroups only

This group has 398 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
1C4×#492C211yesmaximal, characteristic
2#11478C212yesmaximal, complemented
3#7371C212yesmaximal, complemented
4#7354C212yesmaximal
5C2×#492C411yescharacteristic
6C2×C4×SD16—22no—
7C4×C4⋊Q8C2211yescharacteristic
8C4×#127C411yescharacteristic
9#1968—22no—
10#1957C2211yescharacteristic
11#1956C411yescharacteristic
12#1844C2211yescharacteristic
13C42×C8C411yescharacteristic
14#1461—22no—
15#1325C2214yes—
16#1319—22no—
17C22×SD16—22no—
18C2×C4⋊Q8C2×C411yescharacteristic
19C2×#127C2×C411yescharacteristic
20C2×C4×Q8D412yes—
21C2×C4×D4—21no—
22C2×C4×D4—41no—
23#492—22no—
24#492C814yescomplemented
25C4×SD16—44no—
26C4×SD16—44no—
27#431—42no—
28C2×#106—22no—
29C2×#103—22no—
30C2×#102—22no—
31C2×C4×C8—21no—
32C2×C4×C8—21no—
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8C2×C411yescharacteristic
35#391D412yes—
36#390—21no—
37#390—41no—
38#389—41no—
39#384C2×C411yescharacteristic
40#384C2311yescharacteristic
41C4×C4⋊C4—41no—
42C43C2×C411yescharacteristic
43#324—24no—
44#324—42no—
45C22×Q8C22⋊C412yes—
46C22×D4—21no—
47C22×D4—41no—
48C23×C4—41no—
49C2×SD16—28no—
50C2×SD16—44no—
51C2×SD16—44no—
52C22×C8—21no—
53C22×C8—21no—
54C4⋊Q8—41no—
55C4⋊Q8C2×C812yes—
56C4⋊Q8M1612yes—
57#127—21no—
58#127—41no—
59#127C2×C812yes—
60C4×Q8—42no—
61C4×Q8—42no—
62C4×D4—42no—
63C4×D4—42no—
64C4×D4—84no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4C2×D412yes—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—41no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—22no—
76C2×C42C22⋊C411yescharacteristic
77C2×C42C22⋊C411yescharacteristic
78C2×C42C22×C411yescharacteristic
79#106—44no—
80C4⋊C8—82no—
81#103—44no—
82#102—44no—
83C22⋊C8—44no—
84C4×C8—22no—
85C4×C8—22no—
86C4×C8—21no—
87C4×C8—41no—
88C4×C8—42no—
89C4×C8—82no—
90C4×C8C2×C812yes—
91#95—42no—
92C24—41no—
93C2×Q8—22no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8C22⋊C814yes—
97C2×D4—22no—
98C2×D4—42no—
99C2×D4—42no—
100C2×D4—42no—
101C2×D4—42no—
102C2×D4—41no—
103C2×D4—82no—
104C2×D4—82no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—41no—
110C22×C4—84no—
111C22×C4—81no—
112C22×C4—81no—
113C22×C4#9911yescharacteristic
114C22×C4#9911yescharacteristic
115C22×C4C2×C22⋊C411yescharacteristic
116SD16—416no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—84no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—82no—
129C4⋊C4—82no—
130C4⋊C4—82no—
131C22⋊C4—82no—
132C22⋊C4—84no—
133C42—24no—
134C42—21no—
135C42—21no—
136C42—21no—
137C42—41no—
138C42—42no—
139C42—42no—
140C42—42no—
141C42C22×C811yescharacteristic
142C42C2×M1611yescharacteristic
143C23—42no—
144C23—82no—
145C23—82no—
146C23—81no—
147C23—81no—
148C23C2×#9911yescharacteristic
149Q8—44no—
150Q8—82no—
151D4—44no—
152D4—82no—
153D4—84no—
154D4—84no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—81no—
174C2×C4—162no—
175C2×C4—162no—
176C2×C4#35311yescharacteristic
177C2×C4#35311yescharacteristic
178C2×C4C2×C22⋊C811yescharacteristic
179C2×C4#41311yescharacteristic
180C8—24no—
181C8—42no—
182C8—164no—
183C22—21no—
184C22—21no—
185C22—84no—
186C22—82no—
187C22—162no—
188C22—161no—
189C22—161no—
190C22#145911yescharacteristic
191C22C2×#35311yescharacteristic
192C22#213211yescharacteristic
193C4—22no—
194C4—22no—
195C4—21no—
196C4—21no—
197C4—21no—
198C4—42no—
199C4—41no—
200C4—44no—
201C4—82no—
202C4—161no—
203C4#141012yes—
204C2—21no—
205C2—21no—
206C2—162no—
207C2#1147511yescharacteristic
208C2#1358311yescharacteristic
209C2C2×#141011yescharacteristic
subgroupG/H
centerC2×C4#353
commutatorC42C22×C8
Frattini, cosocle#384C23
socleC22C2×#353
prime radicalC2×#127C2×C4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11478C2semidirect
3. #7371C2semidirect
24. #492C8semidirect

Maximal subgroups

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

subgroupindex
C4×#4922
#114782
#73712
#73542