Finite Group Databasebeta

← #9232

Subgroups — #9232

Order 256.

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

normal subgroups only

This group has 283 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#3104C211yesmaximal, characteristic
2C42×Q8C211yesmaximal, characteristic
3#1994C211yesmaximal, characteristic
4#1503C211yesmaximal, characteristic
5#1487C211yesmaximal, characteristic
6#1303C211yesmaximal, characteristic
7#1276C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4×Q8—21no—
10C2×C4×Q8C2211yescharacteristic
11#484—21no—
12#477C412yescomplemented
13#462C412yescomplemented
14#451—21no—
15C4×Q16—22no—
16#431C2211yescharacteristic
17#431C2211yescharacteristic
18#423C2211yescharacteristic
19C4×C4⋊C4—21no—
20C4×C4⋊C4—21no—
21C4×C4⋊C4—22no—
22C4×C4⋊C4C2211yescharacteristic
23C43—21no—
24C43C2211yescharacteristic
25#339—21no—
26#339—21no—
27#328—22no—
28Q8⋊C8—22no—
29C22×Q8D411yescharacteristic
30C2×Q16—22no—
31C2×M16—21no—
32C4⋊Q8C2×C411yescharacteristic
33C4⋊Q8C2×C411yescharacteristic
34#125—21no—
35C22⋊Q8—21no—
36C22⋊Q8—21no—
37C4×Q8—28no—
38C4×Q8—21no—
39C4×Q8—21no—
40C4×Q8—21no—
41C4×Q8—21no—
42C4×Q8C2×C412yes—
43C4×Q8D412yes—
44#117—21no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4D411yescharacteristic
49C2×C42—22no—
50C2×C42—21no—
51C2×C42—21no—
52C2×C42—22no—
53C2×C42—21no—
54C2×C42—21no—
55C2×C42D411yescharacteristic
56C2×C42D411yescharacteristic
57C2×C42C2311yescharacteristic
58#107—22no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8C2×C412yes—
63#103—22no—
64#103—22no—
65C22⋊C8—21no—
66C22⋊C8—21no—
67C4×C8—21no—
68C4×C8—21no—
69#95—22no—
70#95D412yes—
71C2×Q8—24no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8—21no—
75C2×Q8C22⋊C412yes—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84Q16—44no—
85M16—41no—
86M16—41no—
87C2×C8—22no—
88C2×C8—41no—
89C2×C8—41no—
90C4⋊C4—28no—
91C4⋊C4—24no—
92C4⋊C4—22no—
93C4⋊C4—22no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4C22⋊C412yes—
102C22⋊C4—41no—
103C42—28no—
104C42—24no—
105C42—22no—
106C42—22no—
107C42—21no—
108C42—21no—
109C42—21no—
110C42—42no—
111C42—44no—
112C42—41no—
113C42—41no—
114C42—41no—
115C42Q1612yescomplemented
116C42C22×C411yescharacteristic
117C42C2×D411yescharacteristic
118C42#4111yescharacteristic
119C42#4111yescharacteristic
120C23#12011yescharacteristic
121Q8—24no—
122Q8—22no—
123Q8—42no—
124Q8—81no—
125Q8#10414yes—
126C2×C4—22no—
127C2×C4—24no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—24no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4C2×C22⋊C411yescharacteristic
144C2×C4C4×D411yescharacteristic
145C2×C4C4×D411yescharacteristic
146C2×C4#12111yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12311yescharacteristic
149C2×C4C2×Q1611yescharacteristic
150C2×C4#13511yescharacteristic
151C2×C4#13711yescharacteristic
152C2×C4#13711yescharacteristic
153C8—42no—
154C8—81no—
155C8—81no—
156C22—21no—
157C22—21no—
158C22#38611yescharacteristic
159C22#45111yescharacteristic
160C22#45211yescharacteristic
161C4—24no—
162C4—24no—
163C4—21no—
164C4—21no—
165C4—21no—
166C4—42no—
167C4—44no—
168C4—42no—
169C4—81no—
170C4—81no—
171C4C2×#10412yes—
172C4C4×Q1611yescharacteristic
173C4#44111yescharacteristic
174C4#46211yescharacteristic
175C4#46411yescharacteristic
176C2—21no—
177C2—21no—
178C2#187811yescharacteristic
179C2#189611yescharacteristic
180C2#190611yescharacteristic
subgroupG/H
centerC2×C4#121
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radicalC23#120
LCSG ▷ C42 ▷ 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
12. #477C4semidirect
13. #462C4semidirect
115. C42Q16semidirect

Maximal subgroups

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

subgroupindex
#31042
C42×Q82
#19942
#15032
#14872
#13032
#12762