Finite Group Databasebeta

← #9048

Subgroups — #9048

Order 256.

212 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 69 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#3104C211yesmaximal, characteristic
2#2961C211yesmaximal, characteristic
3#1985C211yesmaximal, characteristic
4#1542C211yesmaximal, characteristic
5#1487C211yesmaximal, characteristic
6#1301C211yesmaximal, characteristic
7#1281C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10#484C412yes—
11#477—21no—
12#462—21no—
13#451C412yes—
14#446—22no—
15#446—22no—
16C4×Q16—22no—
17#431C2211yescharacteristic
18#431C2211yescharacteristic
19#424—21no—
20#423C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22#405—21no—
23C2×#97C2211yescharacteristic
24#336—21no—
25#336—21no—
26#333—21no—
27#332—21no—
28Q8⋊C8—22no—
29C22×Q8C2×C411yescharacteristic
30C2×Q16—22no—
31C2×M16—21no—
32C22×C8—21no—
33C22×C8D411yescharacteristic
34C22×C8D411yescharacteristic
35C4⋊Q8D411yescharacteristic
36C4⋊Q8D411yescharacteristic
37#125—21no—
38C22⋊Q8—21no—
39C22⋊Q8C2×C412yes—
40C4×Q8—21no—
41C4×Q8—21no—
42C4×Q8—21no—
43C4×Q8—21no—
44C4×Q8—21no—
45C4×Q8D412yes—
46#117—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4C2×C411yescharacteristic
49C2×C42—21no—
50C2×C42C2311yescharacteristic
51#107—22no—
52C4⋊C8—24no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8D412yes—
59#103—22no—
60#103—22no—
61C22⋊C8—21no—
62C22⋊C8C2×C412yes—
63#97—22no—
64#97—21no—
65#97—21no—
66C4×C8—22no—
67C4×C8—21no—
68C4×C8—21no—
69C2×Q8—21no—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—42no—
73C2×Q8C22⋊C412yes—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4C22×C411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4#4111yescharacteristic
79Q16—44no—
80M16—41no—
81M16—41no—
82C2×C8—21no—
83C2×C8—21no—
84C2×C8—22no—
85C2×C8—22no—
86C2×C8—21no—
87C2×C8—21no—
88C2×C8—21no—
89C2×C8—41no—
90C2×C8—41no—
91C2×C8—41no—
92C2×C8—41no—
93C2×C8Q1612yescomplemented
94C4⋊C4—22no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4—22no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4C22⋊C412yes—
104C22⋊C4—22no—
105C42—22no—
106C42—41no—
107C42—41no—
108C42C2×D411yescharacteristic
109C42C2×D411yescharacteristic
110C42C2×D411yescharacteristic
111C42#4111yescharacteristic
112C23C4×D411yescharacteristic
113Q8—22no—
114Q8—22no—
115Q8—42no—
116Q8—41no—
117Q8—41no—
118Q8—81no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—41no—
128C2×C4C2×C22⋊C411yescharacteristic
129C2×C4C4×D411yescharacteristic
130C2×C4#12011yescharacteristic
131C2×C4#12111yescharacteristic
132C2×C4#12111yescharacteristic
133C2×C4#12311yescharacteristic
134C2×C4C2×Q1611yescharacteristic
135C2×C4#13511yescharacteristic
136C2×C4#13711yescharacteristic
137C2×C4#13711yescharacteristic
138C8—22no—
139C8—42no—
140C8—41no—
141C8—42no—
142C8—81no—
143C8—81no—
144C22—21no—
145C22—21no—
146C22#38611yescharacteristic
147C22C4×Q1611yescharacteristic
148C22#44111yescharacteristic
149C4—21no—
150C4—21no—
151C4—21no—
152C4—21no—
153C4—42no—
154C4—42no—
155C4—42no—
156C4—81no—
157C4#45111yescharacteristic
158C4#45211yescharacteristic
159C4#46211yescharacteristic
160C4#46411yescharacteristic
161C2—21no—
162C2—21no—
163C2#187811yescharacteristic
164C2#188111yescharacteristic
165C2#189111yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radicalC23C4×D4
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
93. C2×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#31042
#29612
#19852
#15422
#14872
#13012
#12812