Finite Group Databasebeta

← #8966

Subgroups — #8966

Order 256.

313 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 85 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#2972C211yesmaximal, characteristic
2C4×#123C211yesmaximal, characteristic
3#1817C212yesmaximal, complemented
4#1459C211yesmaximal, characteristic
5#1282C212yesmaximal, complemented
6C2×#123C411yescharacteristic
7C2×C4×D4C2211yescharacteristic
8C22×C42C2211yescharacteristic
9C8×D4—24no—
10#431—21no—
11C2×C22⋊C8—21no—
12C2×C22⋊C8C2212yes—
13C2×C22⋊C8C2212yes—
14#388—21no—
15#387—21no—
16#386C411yescharacteristic
17#385C411yescharacteristic
18#381C411yescharacteristic
19C4×C4⋊C4—21no—
20C4×C22⋊C4—21no—
21C4×C22⋊C4C2211yescharacteristic
22#336—24no—
23#324—21no—
24#323—21no—
25C22×D4C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4C2×C411yescharacteristic
28C23×C4D411yescharacteristic
29C23×C4C2311yescharacteristic
30C22×C8—22no—
31C22×C8—22no—
32C22×C8D414yes—
33#123—22no—
34#123C814yescomplemented
35C4×D4—22no—
36C4×D4—22no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4D411yescharacteristic
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4C2×C411yescharacteristic
43C2×C22⋊C4C2×C411yescharacteristic
44C2×C22⋊C4C2×C411yescharacteristic
45C2×C22⋊C4D411yescharacteristic
46C2×C42—22no—
47C2×C42—21no—
48C2×C42—21no—
49C2×C42—21no—
50C2×C42—22no—
51C2×C42D411yescharacteristic
52C4⋊C8—22no—
53C22⋊C8—22no—
54C22⋊C8—22no—
55C22⋊C8—22no—
56C22⋊C8—22no—
57C22⋊C8—22no—
58C4×C8—22no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63#95—21no—
64C24C22⋊C411yescharacteristic
65C24C22×C411yescharacteristic
66C2×D4—22no—
67C2×D4—22no—
68C2×D4C2×C812yes—
69C2×D4M1612yes—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—21no—
76C22×C4—22no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4C22⋊C411yescharacteristic
84C22×C4C22⋊C411yescharacteristic
85C22×C4C22⋊C411yescharacteristic
86C22×C4C2×C812yescomplemented
87C22×C4C2×D412yes—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4#4112yes—
91C2×C8—24no—
92C2×C8—44no—
93C2×C8—44no—
94C2×C8—42no—
95C2×C8—82no—
96C4⋊C4—24no—
97C4⋊C4—22no—
98C4⋊C4—42no—
99C22⋊C4—24no—
100C22⋊C4—22no—
101C22⋊C4—21no—
102C22⋊C4—22no—
103C22⋊C4—22no—
104C22⋊C4—42no—
105C22⋊C4C2×C812yescomplemented
106C42—21no—
107C42—21no—
108C42—42no—
109C42—41no—
110C42—42no—
111C42—41no—
112C23—21no—
113C23—21no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23C22⋊C812yes—
119C23C2×C22⋊C411yescharacteristic
120C23C4×D412yes—
121C23C22×C811yescharacteristic
122C23C2×M1611yescharacteristic
123C23#13112yes—
124D4—24no—
125D4—24no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—22no—
129C2×C4—24no—
130C2×C4—22no—
131C2×C4—24no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—24no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—81no—
154C2×C4C22⋊C812yes—
155C2×C4#12011yescharacteristic
156C2×C4#12112yes—
157C2×C4#12311yescharacteristic
158C8—44no—
159C8—82no—
160C22—21no—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—41no—
166C22—41no—
167C22—42no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22#38611yescharacteristic
173C22C2×C22⋊C811yescharacteristic
174C22#40811yescharacteristic
175C22C8×D412yes—
176C22#43512yes—
177C4—22no—
178C4—22no—
179C4—21no—
180C4—42no—
181C4—42no—
182C4—41no—
183C4—42no—
184C4—41no—
185C4—81no—
186C4#45812yes—
187C2—21no—
188C2—21no—
189C2—22no—
190C2—22no—
191C2—42no—
192C2#186911yescharacteristic
193C2#189011yescharacteristic
194C2#189111yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC23C22×C8
Frattini, cosocleC23×C4C23
socleC22#386
prime radicalC22×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1817C2semidirect
5. #1282C2semidirect
34. #123C8semidirect
86. C22×C4C2×C8semidirect
105. C22⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
#29722
C4×#1232
#18172
#14592
#12822