Finite Group Databasebeta

← #11809

Subgroups — #11809

Order 256.

308 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 53 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×#399C211yesmaximal, characteristic
2#1819C211yesmaximal, characteristic
3#1462C211yesmaximal, characteristic
4#1352C214yesmaximal, complemented
5C22×C4⋊C4—21no—
6C22×C4⋊C4C411yescharacteristic, complemented
7C22×C22⋊C4—21no—
8C22×C22⋊C4C2211yescharacteristic
9#407—21no—
10#399—21no—
11#399—21no—
12#399C412yescomplemented
13#399C2212yes—
14#379—21no—
15C2×#95C411yescharacteristic, complemented
16#342—21no—
17#342C2212yes—
18#323—21no—
19#323C2212yes—
20C25D411yescharacteristic
21C23×C4—21no—
22C23×C4—21no—
23C23×C4C2×C411yescharacteristic
24C23×C4D411yescharacteristic
25C2×M16—41no—
26C2×C4⋊C4—22no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—41no—
30C2×C4⋊C4—41no—
31C2×C4⋊C4—44no—
32C2×C4⋊C4C2×C412yes—
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—41no—
37C2×C22⋊C4—41no—
38C2×C22⋊C4—44no—
39C2×C22⋊C4—41no—
40C2×C22⋊C4—41no—
41C2×C22⋊C4C2×C411yescharacteristic
42C2×C22⋊C4D411yescharacteristic
43C2×C22⋊C4D411yescharacteristic
44C2×C22⋊C4C2311yescharacteristic
45C22⋊C8—22no—
46C22⋊C8—41no—
47#95—21no—
48#95C2×C412yes—
49C24—22no—
50C24—21no—
51C24—21no—
52C24—41no—
53C24—42no—
54C24—42no—
55C24C22⋊C411yescharacteristic
56C24C22⋊C411yescharacteristic
57C24C2×D411yescharacteristic
58C22×C4—24no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—21no—
64C22×C4—22no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4—42no—
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22×C411yescharacteristic
77C22×C4C2×D411yescharacteristic
78M16—82no—
79C2×C8—42no—
80C4⋊C4—42no—
81C4⋊C4—44no—
82C4⋊C4—44no—
83C4⋊C4—42no—
84C22⋊C4—41no—
85C22⋊C4—41no—
86C22⋊C4—41no—
87C22⋊C4—41no—
88C22⋊C4—44no—
89C22⋊C4—44no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C22⋊C4—82no—
93C22⋊C4—82no—
94C23—22no—
95C23—22no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23—21no—
103C23—42no—
104C23—42no—
105C23—41no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23—48no—
110C23—42no—
111C23—42no—
112C23—81no—
113C23—81no—
114C23—84no—
115C23#9911yescharacteristic
116C23#9911yescharacteristic
117C23C2×C22⋊C411yescharacteristic
118C23#13712yes—
119C2×C4—24no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—48no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—81no—
138C2×C4—82no—
139C2×C4—82no—
140C2×C4—82no—
141C2×C4—82no—
142C2×C4—82no—
143C8—82no—
144C22—24no—
145C22—24no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—22no—
153C22—42no—
154C22—42no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22—41no—
159C22—41no—
160C22—81no—
161C22—84no—
162C22—84no—
163C22—81no—
164C22#35212yes—
165C22#35312yes—
166C22C2×#9911yescharacteristic
167C22#41911yescharacteristic
168C22#42111yescharacteristic
169C4—44no—
170C4—41no—
171C4—41no—
172C4—41no—
173C4—41no—
174C4—82no—
175C4—162no—
176C2—21no—
177C2—21no—
178C2—22no—
179C2—22no—
180C2—42no—
181C2—81no—
182C2#151111yescharacteristic
183C2C2×#35211yescharacteristic
184C2C2×#35311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1352C2semidirect
6. C22×C4⋊C4C4semidirect
12. #399C4semidirect
15. C2×#95C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3992
#18192
#14622
#13522