Finite Group Databasebeta

← #11517

Subgroups — #11517

Order 256.

375 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 59 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×#385C211yesmaximal, characteristic
2#1462C211yesmaximal, characteristic
3#1462C211yesmaximal, characteristic
4#1326C214yesmaximal, complemented
5C24×C4C411yescharacteristic
6C22×C4⋊C4—21no—
7C22×C22⋊C4C411yescharacteristic
8C22×C22⋊C4C2211yescharacteristic
9#407—21no—
10#407—21no—
11#385—21no—
12#385—21no—
13#385C412yes—
14#385C2212yes—
15C2×#95—21no—
16#323—21no—
17#323—21no—
18#323C2212yes—
19#323C2212yes—
20C25C2×C411yescharacteristic
21C23×C4—21no—
22C23×C4—21no—
23C23×C4—21no—
24C23×C4—44no—
25C23×C4—41no—
26C23×C4—41no—
27C23×C4C2×C412yes—
28C23×C4D411yescharacteristic
29C23×C4D411yescharacteristic
30C2×M16—41no—
31C2×M16—41no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—44no—
35C2×C22⋊C4—21no—
36C2×C22⋊C4—24no—
37C2×C22⋊C4—41no—
38C2×C22⋊C4—41no—
39C2×C22⋊C4C2×C412yes—
40C2×C22⋊C4C2×C411yescharacteristic
41C2×C22⋊C4D411yescharacteristic
42C2×C22⋊C4D411yescharacteristic
43C2×C22⋊C4C2311yescharacteristic
44C22⋊C8—22no—
45C22⋊C8—22no—
46C22⋊C8—41no—
47C22⋊C8—41no—
48#95—22no—
49#95—22no—
50C24—21no—
51C24—21no—
52C24—22no—
53C24—22no—
54C24—42no—
55C24—42no—
56C24C22⋊C411yescharacteristic
57C24C22⋊C411yescharacteristic
58C24C22×C411yescharacteristic
59C22×C4—24no—
60C22×C4—24no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—44no—
68C22×C4—44no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4—48no—
74C22×C4—44no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—42no—
78C22×C4—44no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22⋊C411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85M16—82no—
86M16—82no—
87C2×C8—42no—
88C2×C8—42no—
89C4⋊C4—44no—
90C4⋊C4—44no—
91C22⋊C4—24no—
92C22⋊C4—22no—
93C22⋊C4—41no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4M1614yescomplemented
98C23—22no—
99C23—22no—
100C23—21no—
101C23—21no—
102C23—24no—
103C23—24no—
104C23—21no—
105C23—21no—
106C23—21no—
107C23—21no—
108C23—41no—
109C23—41no—
110C23—41no—
111C23—41no—
112C23—42no—
113C23—48no—
114C23—48no—
115C23—42no—
116C23—42no—
117C23—42no—
118C23#9911yescharacteristic
119C23#9911yescharacteristic
120C23#10112yes—
121C23C2×C22⋊C411yescharacteristic
122C23C2×M1612yes—
123C2×C4—24no—
124C2×C4—24no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—42no—
130C2×C4—48no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—44no—
134C2×C4—48no—
135C2×C4—44no—
136C2×C4—42no—
137C2×C4—44no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—44no—
145C2×C4—81no—
146C2×C4—81no—
147C8—82no—
148C8—82no—
149C22—24no—
150C22—24no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22—48no—
163C22—48no—
164C22—42no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22AΣL(1,𝔽16)12yes—
170C22#35212yes—
171C22#40711yescharacteristic
172C22C2×#9911yescharacteristic
173C22C2×#10111yescharacteristic
174C4—44no—
175C4—41no—
176C4—44no—
177C4—41no—
178C4—41no—
179C4—41no—
180C2—21no—
181C2—22no—
182C2—22no—
183C2—21no—
184C2—42no—
185C2—42no—
186C2#146311yescharacteristic
187C2C2×AΣL(1,𝔽16)11yescharacteristic
188C2C2×#35211yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC25C2×C4
LCSG ▷ C24 ▷ C22 ▷ 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. #1326C2semidirect
97. C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#3852
#14622
#13262