Finite Group Databasebeta

← #11725

Subgroups — #11725

Order 256.

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

normal subgroups only

This group has 324 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×#99C211yesmaximal, characteristic, complemented
3#1460C211yesmaximal, characteristic, complemented
4#1351C212yesmaximal, complemented
5#1343C212yesmaximal, complemented
6C22×#41C411yescharacteristic, complemented
7C2×C4×D4—21no—
8C2×C4×D4C2211yescharacteristic
9#431—21no—
10C2×#99—21no—
11#395C411yescharacteristic, complemented
12#394—21no—
13#394C411yescharacteristic, complemented
14#393C2211yescharacteristic
15#392—21no—
16#392C2211yescharacteristic
17C4×C22⋊C4—21no—
18C4×C22⋊C4—21no—
19C4×C22⋊C4C411yescharacteristic, complemented
20#342C2212yes—
21#323—21no—
22#323C2212yes—
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
34C4×D4—41no—
35C4×D4—41no—
36C4×D4—44no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C22⋊C4—22no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4D411yescharacteristic
46C2×C22⋊C4C2311yescharacteristic
47C2×C42—21no—
48C2×C42—41no—
49C2×C42D411yescharacteristic
50C4⋊C8—41no—
51#99—24no—
52C22⋊C8—22no—
53C4×C8—41no—
54#95—21no—
55#95—21no—
56#95—41no—
57#95C2×C411yescharacteristic
58#95C2×C411yescharacteristic
59C24—21no—
60C24C22⋊C411yescharacteristic
61C24C2×D411yescharacteristic
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×C4—42no—
93C22×C4C22⋊C411yescharacteristic
94C22×C4C22⋊C411yescharacteristic
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22×C411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C2×C8—42no—
99C4⋊C4—41no—
100C4⋊C4—42no—
101C22⋊C4—24no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—42no—
110C42—21no—
111C42—21no—
112C42—42no—
113C42—82no—
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—
130C23C2×C22⋊C411yescharacteristic
131C23#13512yes—
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—44no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—81no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4#9911yescharacteristic
167C2×C4#9911yescharacteristic
168C2×C4#10412yes—
169C8—82no—
170C22—21no—
171C22—21no—
172C22—22no—
173C22—42no—
174C22—42no—
175C22—42no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22C2×#9911yescharacteristic
190C22#41611yescharacteristic
191C22C2×#10411yescharacteristic
192C4—21no—
193C4—22no—
194C4—42no—
195C4—42no—
196C4—41no—
197C4—41no—
198C4—41no—
199C4—81no—
200C4—84no—
201C4AΣL(1,𝔽16)12yes—
202C2—21no—
203C2—21no—
204C2—42no—
205C2—81no—
206C2—82no—
207C2#150111yescharacteristic
208C2C2×AΣL(1,𝔽16)11yescharacteristic
209C2#212611yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
socleC22C2×#99
prime radical#2388C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#99C2semidirect
3. #1460C2semidirect
4. #1351C2semidirect
5. #1343C2semidirect
6. C22×#41C4semidirect
11. #395C4semidirect
13. #394C4semidirect
19. C4×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#23882
C4×#992
#14602
#13512
#13432