Finite Group Databasebeta

← #11733

Subgroups — #11733

Order 256.

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

normal subgroups only

This group has 316 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
1#2388C211yesmaximal, characteristic
2C4×#100C211yesmaximal, characteristic, complemented
3#1459C211yesmaximal, characteristic, complemented
4#1354C212yesmaximal, complemented
5#1345C212yesmaximal, complemented
6C22×#41C411yescharacteristic
7C2×C4×D4—21no—
8C2×C4×D4C2211yescharacteristic
9C2×#100—21no—
10C2×C22⋊C8—21no—
11C4×M16—21no—
12#395C2211yescharacteristic
13#394—21no—
14#394C2211yescharacteristic
15#393C411yescharacteristic
16#392—21no—
17#392C411yescharacteristic
18C4×C22⋊C4—21no—
19C4×C22⋊C4C411yescharacteristic
20#343C2212yes—
21#324C2212yes—
22#323—21no—
23C2×#41—44no—
24C2×#41—41no—
25C2×#41—41no—
26C22×Q8C2×C411yescharacteristic
27C22×D4—21no—
28C22×D4—21no—
29C22×D4C2×C411yescharacteristic
30C22×D4D411yescharacteristic
31C23×C4—21no—
32C23×C4C2×C411yescharacteristic
33C23×C4D411yescharacteristic
34C2×M16—22no—
35C22×C8—41no—
36C4×D4—41no—
37C4×D4—41no—
38C4×D4—44no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4C2311yescharacteristic
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4C2×C411yescharacteristic
47C2×C22⋊C4D411yescharacteristic
48C2×C42—21no—
49C2×C42D411yescharacteristic
50#100—24no—
51C22⋊C8—22no—
52C22⋊C8—41no—
53#97—41no—
54C4×C8—41no—
55#95—21no—
56#95—21no—
57#95C2×C411yescharacteristic
58#95C2×C411yescharacteristic
59C24—21no—
60C24C22⋊C411yescharacteristic
61C24C22⋊C411yescharacteristic
62#41—44no—
63#41—44no—
64#41—44no—
65#41—44no—
66C2×Q8—42no—
67C2×Q8—41no—
68C2×D4—22no—
69C2×D4—22no—
70C2×D4—41no—
71C2×D4—41no—
72C2×D4—42no—
73C2×D4—42no—
74C2×D4—42no—
75C2×D4—42no—
76C2×D4—42no—
77C2×D4—42no—
78C2×D4—42no—
79C22×C4—24no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—42no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4C22⋊C411yescharacteristic
93C22×C4C22⋊C411yescharacteristic
94C22×C4C22×C411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97M16—24no—
98M16—42no—
99C2×C8—42no—
100C2×C8—42no—
101C2×C8—82no—
102C4⋊C4—41no—
103C4⋊C4—42no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—42no—
110C22⋊C4—42no—
111C42—21no—
112C42—21no—
113C42—42no—
114C23—21no—
115C23—21no—
116C23—22no—
117C23—21no—
118C23—21no—
119C23—21no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23#10412yes—
131C23C2×C22⋊C411yescharacteristic
132Q8—42no—
133Q8—42no—
134D4—44no—
135D4—42no—
136D4—48no—
137D4—48no—
138D4—42no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—44no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—81no—
163C2×C4#9911yescharacteristic
164C2×C4#9911yescharacteristic
165C2×C4#13512yes—
166C8—44no—
167C8—82no—
168C22—21no—
169C22—21no—
170C22—22no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—42no—
178C22—41no—
179C22—41no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22C2×#9911yescharacteristic
188C22#41611yescharacteristic
189C22C2×#10411yescharacteristic
190C4—21no—
191C4—22no—
192C4—42no—
193C4—41no—
194C4—42no—
195C4—41no—
196C4—41no—
197C4—81no—
198C4AΣL(1,𝔽16)12yes—
199C2—21no—
200C2—21no—
201C2—42no—
202C2—81no—
203C2—82no—
204C2#150111yescharacteristic
205C2C2×AΣL(1,𝔽16)11yescharacteristic
206C2#213211yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
socleC22C2×#99
prime radical#2388C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#100C2semidirect
3. #1459C2semidirect
4. #1354C2semidirect
5. #1345C2semidirect

Maximal subgroups

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

subgroupindex
#23882
C4×#1002
#14592
#13542
#13452