Finite Group Databasebeta

← #9204

Subgroups — #9204

Order 256.

311 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 75 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#2618C211yesmaximal, characteristic
2#2316C211yesmaximal, characteristic, complemented
3#1519C212yesmaximal
4C2×#324C211yesmaximal, characteristic, complemented
5#1289C212yesmaximal, complemented
6C2×#123—21no—
7C2×#121C2211yescharacteristic
8C2×#117—21no—
9C22×C4⋊C4C2211yescharacteristic
10C2×C22⋊C8C2212yes—
11#398C412yescomplemented
12#396—22no—
13#394C412yescomplemented
14#389—24no—
15#386—21no—
16#380—21no—
17#380C2211yescharacteristic
18#380C2212yes—
19#327—24no—
20#324—22no—
21C22×D4—21no—
22C22×D4C2×C411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4D411yescharacteristic
25C23×C4C2311yescharacteristic
26C22×C8—22no—
27#123—42no—
28#123—42no—
29#121—22no—
30#121—22no—
31#117—42no—
32#117—42no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—22no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4C2×C412yes—
42C2×C4⋊C4D412yes—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—22no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—22no—
48C2×C42—22no—
49C22⋊C8—22no—
50C22⋊C8—22no—
51#95—22no—
52#95—22no—
53#95C2×C412yes—
54#95D414yes—
55C24—21no—
56C24C2×D411yescharacteristic
57C2×D4—22no—
58C2×D4—41no—
59C2×D4—41no—
60C2×D4—41no—
61C2×D4—41no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4—81no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—22no—
73C22×C4—24no—
74C22×C4—81no—
75C22×C4—81no—
76C22×C4C22⋊C412yes—
77C22×C4C22⋊C412yes—
78C22×C4D812yes—
79C22×C4SD1612yes—
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D412yes—
82C22×C4C2×D411yescharacteristic
83C22×C4#4112yes—
84C2×C8—42no—
85C2×C8—82no—
86C4⋊C4—24no—
87C4⋊C4—24no—
88C4⋊C4—22no—
89C4⋊C4—44no—
90C4⋊C4—44no—
91C4⋊C4—81no—
92C4⋊C4—82no—
93C4⋊C4—82no—
94C22⋊C4—42no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—42no—
98C22⋊C4—44no—
99C22⋊C4—42no—
100C22⋊C4—82no—
101C42—44no—
102C23—41no—
103C23—41no—
104C23—42no—
105C23—42no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23C2×C22⋊C411yescharacteristic
110C23#12011yescharacteristic
111C23#12311yescharacteristic
112C23C2×D811yescharacteristic
113C23C2×SD1611yescharacteristic
114C23AGL(1,ℤ/8)11yescharacteristic
115C23#13711yescharacteristic
116D4—82no—
117D4—81no—
118D4—81no—
119D4—81no—
120D4—81no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—41no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—44no—
136C2×C4—44no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—44no—
143C2×C4—81no—
144C2×C4#10214yes—
145C2×C4C4×D412yes—
146C2×C4#12112yes—
147C8—82no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—41no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22—82no—
159C22—82no—
160C22—81no—
161C22—81no—
162C22—81no—
163C22#38611yescharacteristic
164C22C2×#10211yescharacteristic
165C22#41711yescharacteristic
166C22#44711yescharacteristic
167C22#44811yescharacteristic
168C22#48011yescharacteristic
169C22#48111yescharacteristic
170C4—24no—
171C4—22no—
172C4—22no—
173C4—42no—
174C4—84no—
175C4—82no—
176C4—81no—
177C2—21no—
178C2—21no—
179C2—21no—
180C2—21no—
181C2—21no—
182C2—21no—
183C2—161no—
184C2#187411yescharacteristic
185C2#191211yescharacteristic
186C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2316C2semidirect
4. C2×#324C2semidirect
5. #1289C2semidirect
11. #398C4semidirect
13. #394C4semidirect

Maximal subgroups

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

subgroupindex
#26182
#23162
#15192
C2×#3242
#12892