Finite Group Databasebeta

← #9122

Subgroups — #9122

Order 256.

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

normal subgroups only

This group has 307 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
1C2×C4×M16C211yesmaximal, characteristic
2#2851C211yesmaximal, characteristic
3#1749C211yesmaximal, characteristic
4C2×#330C212yesmaximal, complemented
5#1295C212yesmaximal, complemented
6C22×M16—21no—
7C2×#125C2211yescharacteristic
8C22×C42C2211yescharacteristic
9C2×C22⋊C8—21no—
10C4×M16—24no—
11C2×#97C2212yes—
12C2×#97C2212yes—
13C2×C4×C8—21no—
14#389C411yescharacteristic
15#389C411yescharacteristic
16#387—21no—
17#387—21no—
18#385—21no—
19#385—21no—
20#382C2211yescharacteristic
21#380C411yescharacteristic
22#380C411yescharacteristic
23#336—21no—
24#330—24no—
25C23×C4C2×C411yescharacteristic
26C23×C4D411yescharacteristic
27C23×C4D411yescharacteristic
28C2×M16—24no—
29C2×M16—24no—
30C2×M16—24no—
31C22×C8—22no—
32C22×C8—22no—
33#125—22no—
34#125—22no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C4⋊C4C2×C411yescharacteristic
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C42—22no—
46C2×C42—22no—
47C2×C42C2×C411yescharacteristic
48C2×C42D411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42C2311yescharacteristic
51C22⋊C8—24no—
52#97—22no—
53#97—22no—
54#97—22no—
55#97D414yes—
56C4×C8—24no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63C24C22⋊C411yescharacteristic
64C22×C4—22no—
65C22×C4—22no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—22no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22⋊C411yescharacteristic
81C22×C4C22×C411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84M16—216no—
85C2×C8—24no—
86C2×C8—24no—
87C2×C8—24no—
88C2×C8—24no—
89C2×C8—42no—
90C2×C8—42no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—42no—
96C4⋊C4—42no—
97C4⋊C4—41no—
98C4⋊C4—42no—
99C4⋊C4—41no—
100C22⋊C4—42no—
101C22⋊C4—42no—
102C42—22no—
103C42—21no—
104C42—22no—
105C42—21no—
106C42C2×D412yes—
107C42#4112yes—
108C23—21no—
109C23—21no—
110C23—21no—
111C23—21no—
112C23—21no—
113C23—21no—
114C23#10412yes—
115C23C2×C22⋊C411yescharacteristic
116C2×C4—24no—
117C2×C4—24no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—44no—
131C2×C4—44no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4#10112yes—
145C2×C4#10412yes—
146C2×C4#10412yes—
147C2×C4C4×D412yes—
148C2×C4#12011yescharacteristic
149C2×C4#12112yes—
150C2×C4#12311yescharacteristic
151C8—28no—
152C8—44no—
153C22—22no—
154C22—22no—
155C22—22no—
156C22—41no—
157C22—41no—
158C22—41no—
159C22—41no—
160C22#38611yescharacteristic
161C22C2×#10111yescharacteristic
162C22#41311yescharacteristic
163C22C2×#10411yescharacteristic
164C22C2×#10411yescharacteristic
165C22C2×#10411yescharacteristic
166C22#42111yescharacteristic
167C4—22no—
168C4—22no—
169C4—21no—
170C4—22no—
171C4—21no—
172C4—22no—
173C4—22no—
174C4—81no—
175C4—81no—
176C4—81no—
177C4—81no—
178C2—42no—
179C2#152411yescharacteristic
180C2#152611yescharacteristic
181C2#152811yescharacteristic
182C2#152911yescharacteristic
183C2#187111yescharacteristic
184C2#189611yescharacteristic
185C2#190111yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#330C2semidirect
5. #1295C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×M162
#28512
#17492
C2×#3302
#12952