Finite Group Databasebeta

← #9684

Subgroups — #9684

Order 256.

385 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 79 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#2856C211yesmaximal, characteristic
2#1749C211yesmaximal, characteristic, complemented
3C2×#327C214yesmaximal, complemented
4#1434C211yesmaximal, characteristic, complemented
5C2×#127C411yescharacteristic, complemented
6C2×#121—21no—
7C2×#121—21no—
8C2×#121C2212yes—
9C22×C42C2211yescharacteristic
10C2×C22⋊C8C2212yes—
11C2×#97—21no—
12#390—21no—
13#390—21no—
14#390C412yescomplemented
15#384C411yescharacteristic, complemented
16C2×#95—21no—
17C2×#95—21no—
18C2×#95C2212yes—
19#336—21no—
20#327—28no—
21C22×D4—22no—
22C22×D4—21no—
23C22×D4—21no—
24C22×D4C2×C412yes—
25C23×C4—22no—
26C23×C4D411yescharacteristic
27C23×C4D411yescharacteristic
28C23×C4C2311yescharacteristic
29C22×C8—22no—
30#127—42no—
31#121—24no—
32#121—24no—
33#121—44no—
34#121—44no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4C2×C412yes—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C42—42no—
42C2×C42C2×C411yescharacteristic
43C2×C42C2×C411yescharacteristic
44C2×C42D411yescharacteristic
45C2×C42D411yescharacteristic
46C22⋊C8—22no—
47C22⋊C8D414yescomplemented
48#97—42no—
49#95—24no—
50#95—24no—
51#95—42no—
52C24—22no—
53C24C2×D411yescharacteristic
54C2×D4—24no—
55C2×D4—42no—
56C2×D4—42no—
57C2×D4—42no—
58C2×D4—42no—
59C2×D4—42no—
60C2×D4—42no—
61C2×D4—42no—
62C2×D4—42no—
63C2×D4—44no—
64C2×D4—44no—
65C2×D4—42no—
66C2×D4—42no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—24no—
72C22×C4—22no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—42no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4C22⋊C411yescharacteristic
84C22×C4C22⋊C411yescharacteristic
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22×C411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D412yes—
90C22×C4#4112yes—
91C2×C8—24no—
92C2×C8—42no—
93C2×C8—42no—
94C4⋊C4—24no—
95C4⋊C4—42no—
96C4⋊C4—42no—
97C22⋊C4—44no—
98C22⋊C4—44no—
99C22⋊C4—44no—
100C42—22no—
101C42—22no—
102C42—41no—
103C42—41no—
104C23—21no—
105C23—21no—
106C23—21no—
107C23—21no—
108C23—42no—
109C23—42no—
110C23—42no—
111C23—42no—
112C23—42no—
113C23—42no—
114C23—42no—
115C23C2×C22⋊C411yescharacteristic
116C23#12411yescharacteristic
117C23#12711yescharacteristic
118C23AGL(1,ℤ/8)12yes—
119C23AGL(1,ℤ/8)12yes—
120D4—84no—
121D4—84no—
122D4—84no—
123D4—84no—
124D4—84no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—44no—
142C2×C4—44no—
143C2×C4—44no—
144C2×C4—44no—
145C2×C4—44no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4—82no—
153C2×C4#9912yes—
154C2×C4#10412yes—
155C2×C4C4×D412yes—
156C2×C4#12112yes—
157C8—44no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—82no—
167C22—82no—
168C22—82no—
169C22—82no—
170C22—82no—
171C22—82no—
172C22—82no—
173C22#39011yescharacteristic
174C22C2×#9911yescharacteristic
175C22#41011yescharacteristic
176C22#41811yescharacteristic
177C22#41811yescharacteristic
178C22C2×#10411yescharacteristic
179C22#42111yescharacteristic
180C22#49012yes—
181C22#49612yes—
182C4—22no—
183C4—22no—
184C4—22no—
185C4—22no—
186C4—41no—
187C4—41no—
188C4—84no—
189C4—82no—
190C2—22no—
191C2—22no—
192C2—162no—
193C2#150511yescharacteristic
194C2#150711yescharacteristic
195C2#151011yescharacteristic
196C2#151411yescharacteristic
197C2#196111yescharacteristic
198C2#196411yescharacteristic
199C2#198011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#2856C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1749C2semidirect
3. C2×#327C2semidirect
4. #1434C2semidirect
5. C2×#127C4semidirect
14. #390C4semidirect
15. #384C4semidirect
47. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#28562
#17492
C2×#3272
#14342