Finite Group Databasebeta

← #11821

Subgroups — #11821

Order 256.

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

normal subgroups only

This group has 291 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#2397C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic, complemented
3#1465C211yesmaximal, characteristic, complemented
4#1352C212yesmaximal, complemented
5#1343C212yesmaximal, complemented
6C2×C4×D4—21no—
7C2×C4×D4—21no—
8C2×C4×D4C411yescharacteristic, complemented
9C2×C4×D4C2211yescharacteristic
10C2×#99—21no—
11#407—21no—
12#399—21no—
13#399C411yescharacteristic, complemented
14#399C2211yescharacteristic
15#392—21no—
16#392C411yescharacteristic, complemented
17#392C2211yescharacteristic
18#380—21no—
19#376C411yescharacteristic, complemented
20#342C2212yes—
21#324—21no—
22#323C2212yes—
23C22×D4—21no—
24C22×D4—21no—
25C22×D4C2×C411yescharacteristic
26C22×D4D411yescharacteristic
27C23×C4—21no—
28C23×C4D411yescharacteristic
29C23×C4D411yescharacteristic
30C2×M16—41no—
31C4×D4—41no—
32C4×D4—41no—
33C4×D4—44no—
34C4×D4—44no—
35C4×D4—41no—
36C4×D4—41no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—41no—
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4D411yescharacteristic
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—22no—
48C2×C22⋊C4C2×C411yescharacteristic
49C2×C22⋊C4C2311yescharacteristic
50C2×C42—21no—
51C2×C42C2×C411yescharacteristic
52#99—42no—
53C22⋊C8—22no—
54C22⋊C8—41no—
55#95—21no—
56#95—41no—
57#95C2×C411yescharacteristic
58#95C2×C411yescharacteristic
59C24—21no—
60C24C22⋊C411yescharacteristic
61C24C2×D411yescharacteristic
62C2×D4—21no—
63C2×D4—21no—
64C2×D4—41no—
65C2×D4—41no—
66C2×D4—41no—
67C2×D4—42no—
68C2×D4—42no—
69C2×D4—42no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—42no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—42no—
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22⋊C411yescharacteristic
92C22×C4C22×C411yescharacteristic
93C22×C4C2×D411yescharacteristic
94M16—82no—
95C2×C8—42no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—42no—
99C4⋊C4—42no—
100C4⋊C4—82no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—42no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—42no—
110C42—24no—
111C42—21no—
112C42—21no—
113C23—21no—
114C23—21no—
115C23—22no—
116C23—21no—
117C23—21no—
118C23—21no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23C2×C22⋊C411yescharacteristic
130C23AGL(1,ℤ/8)11yescharacteristic
131C23#13711yescharacteristic
132D4—42no—
133D4—42no—
134D4—42no—
135D4—44no—
136D4—44no—
137D4—48no—
138D4—42no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—81no—
161C2×C4—81no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4#9911yescharacteristic
168C2×C4#9911yescharacteristic
169C8—82no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—41no—
175C22—41no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—42no—
182C22—41no—
183C22—41no—
184C22—41no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22C2×#9911yescharacteristic
192C22#41711yescharacteristic
193C22#42111yescharacteristic
194C4—21no—
195C4—42no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4—41no—
200C4—41no—
201C4—81no—
202C4—162no—
203C4#35312yes—
204C2—21no—
205C2—21no—
206C2—41no—
207C2—41no—
208C2—81no—
209C2—82no—
210C2#151311yescharacteristic
211C2#212011yescharacteristic
212C2C2×#35311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#2397C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #1799C2semidirect
3. #1465C2semidirect
4. #1352C2semidirect
5. #1343C2semidirect
8. C2×C4×D4C4semidirect
13. #399C4semidirect
16. #392C4semidirect
19. #376C4semidirect

Maximal subgroups

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

subgroupindex
#23972
#17992
#14652
#13522
#13432