Finite Group Databasebeta

← #9033

Subgroups — #9033

Order 256.

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

normal subgroups only

This group has 264 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#2905C211yesmaximal, characteristic
2#2396C211yesmaximal, characteristic
3#1487C212yesmaximal, complemented
4#1458C211yesmaximal, characteristic
5#1303C212yesmaximal, complemented
6C22×M16—21no—
7C2×C4×Q8C2211yescharacteristic
8C2×#117C2211yescharacteristic
9#431C2212yes—
10#424—24no—
11#423—21no—
12#423C2212yes—
13#400C411yescharacteristic
14#399C411yescharacteristic
15#398—21no—
16#396—21no—
17#395C411yescharacteristic
18#393C411yescharacteristic
19C4×C4⋊C4—21no—
20C4×C4⋊C4C2211yescharacteristic
21C4×C22⋊C4—21no—
22Q8⋊C8—24no—
23#324—21no—
24#323—21no—
25C22×Q8C2×C411yescharacteristic
26C23×C4D411yescharacteristic
27C2×M16—22no—
28C2×M16—22no—
29C2×M16—42no—
30C2×M16—42no—
31C22×C8—22no—
32C4×Q8—22no—
33C4×Q8—22no—
34#117—22no—
35#117—22no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C4⋊C4D411yescharacteristic
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4C2×C411yescharacteristic
46C2×C22⋊C4D411yescharacteristic
47C2×C42—21no—
48C2×C42—21no—
49C2×C42D411yescharacteristic
50C2×C42C2311yescharacteristic
51C4⋊C8—24no—
52C4⋊C8—22no—
53C4⋊C8—22no—
54C4⋊C8D414yes—
55C22⋊C8—22no—
56C4×C8—22no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95C2×C411yescharacteristic
62#95C2×C411yescharacteristic
63C24C22⋊C411yescharacteristic
64C2×Q8—22no—
65C2×Q8—42no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22⋊C411yescharacteristic
77C22×C4SD1612yes—
78C22×C4Q1612yes—
79C22×C4C22×C411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82M16—44no—
83M16—42no—
84M16—42no—
85C2×C8—24no—
86C2×C8—44no—
87C2×C8—42no—
88C4⋊C4—22no—
89C4⋊C4—22no—
90C4⋊C4—22no—
91C4⋊C4—42no—
92C4⋊C4—42no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C22⋊C4—42no—
98C22⋊C4—42no—
99C42—21no—
100C42—21no—
101C42—41no—
102C42—41no—
103C42—41no—
104C42—41no—
105C42C2×D412yes—
106C42#4112yes—
107C23—21no—
108C23—21no—
109C23—41no—
110C23—41no—
111C23#10312yes—
112C23C2×C22⋊C411yescharacteristic
113Q8—24no—
114Q8—42no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4#10312yes—
140C2×C4C4×D412yes—
141C2×C4#12011yescharacteristic
142C2×C4#12112yes—
143C2×C4#12311yescharacteristic
144C2×C4C2×SD1611yescharacteristic
145C2×C4C2×Q1611yescharacteristic
146C2×C4AGL(1,ℤ/8)11yescharacteristic
147C2×C4#13711yescharacteristic
148C8—44no—
149C8—82no—
150C22—21no—
151C22—21no—
152C22—22no—
153C22—42no—
154C22—41no—
155C22—41no—
156C22—81no—
157C22#38611yescharacteristic
158C22C2×#10311yescharacteristic
159C22#41711yescharacteristic
160C4—22no—
161C4—22no—
162C4—22no—
163C4—42no—
164C4—41no—
165C4—41no—
166C4—42no—
167C4—81no—
168C4—81no—
169C4#45011yescharacteristic
170C4#45111yescharacteristic
171C4#48311yescharacteristic
172C4#48411yescharacteristic
173C2—21no—
174C2—21no—
175C2—42no—
176C2#187211yescharacteristic
177C2#188411yescharacteristic
178C2#189111yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1487C2semidirect
5. #1303C2semidirect

Maximal subgroups

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

subgroupindex
#29052
#23962
#14872
#14582
#13032