Finite Group Databasebeta

← #11828

Subgroups — #11828

Order 256.

306 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 51 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
2#1800C211yesmaximal, characteristic, complemented
3#1465C211yesmaximal, characteristic, complemented
4#1354C212yesmaximal, complemented
5#1345C212yesmaximal, complemented
6C22×#41C411yescharacteristic
7C2×C4×D4—21no—
8C2×C4×D4C2211yescharacteristic
9#423—21no—
10C2×#100—21no—
11#407—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—
35C2×M16—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
50C4⋊C8—41no—
51C4⋊C8—41no—
52#100—42no—
53C22⋊C8—22no—
54C22⋊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—21no—
69C2×D4—21no—
70C2×D4—41no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—42no—
74C2×D4—42no—
75C2×D4—42no—
76C2×D4—42no—
77C2×D4—42no—
78C2×D4—42no—
79C2×D4—42no—
80C22×C4—24no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4C22⋊C411yescharacteristic
94C22×C4C22⋊C411yescharacteristic
95C22×C4C22×C411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98M16—42no—
99M16—42no—
100M16—82no—
101C2×C8—42no—
102C2×C8—42no—
103C4⋊C4—41no—
104C4⋊C4—42no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—42no—
110C22⋊C4—42no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C42—21no—
114C42—21no—
115C42—42no—
116C23—22no—
117C23—21no—
118C23—21no—
119C23—21no—
120C23—21no—
121C23—21no—
122C23—21no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23C2×C22⋊C411yescharacteristic
134Q8—42no—
135Q8—42no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—48no—
140D4—48no—
141D4—42no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—44no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—81no—
166C2×C4#9911yescharacteristic
167C2×C4#9911yescharacteristic
168C2×C4AGL(1,ℤ/8)11yescharacteristic
169C2×C4#13711yescharacteristic
170C8—82no—
171C8—82no—
172C22—21no—
173C22—21no—
174C22—42no—
175C22—42no—
176C22—42no—
177C22—41no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—41no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22C2×#9911yescharacteristic
192C22#41711yescharacteristic
193C22#42111yescharacteristic
194C4—21no—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—41no—
199C4—42no—
200C4—41no—
201C4—41no—
202C4—81no—
203C4AΣL(1,𝔽16)12yes—
204C2—21no—
205C2—21no—
206C2—81no—
207C2—81no—
208C2—82no—
209C2#151311yescharacteristic
210C2C2×AΣL(1,𝔽16)11yescharacteristic
211C2#213211yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radical#2388C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #1800C2semidirect
3. #1465C2semidirect
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
#18002
#14652
#13542
#13452