Finite Group Databasebeta

← #9324

Subgroups — #9324

Order 256.

268 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 65 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#3113C211yesmaximal, characteristic
2#2276C211yesmaximal, characteristic
3#1986C211yesmaximal, characteristic
4#1506C211yesmaximal, characteristic
5#1491C211yesmaximal, characteristic
6#1300C211yesmaximal, characteristic
7#1285C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4×Q8—21no—
10C2×C4×Q8C2211yescharacteristic
11#483—21no—
12#475C412yescomplemented
13#462C412yescomplemented
14#451—21no—
15#441—22no—
16#432C2211yescharacteristic
17#432C2211yescharacteristic
18C4×M16C2211yescharacteristic
19C4×C4⋊C4—21no—
20C4×C4⋊C4—22no—
21C4×C4⋊C4—21no—
22C4×C4⋊C4C2211yescharacteristic
23#376—21no—
24#376C2211yescharacteristic
25#339—22no—
26#328—21no—
27#328—21no—
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×C42D412yes—
54C2×C42C2311yescharacteristic
55#106—22no—
56C4⋊C8—21no—
57C4⋊C8C2×C412yes—
58#103—22no—
59#103—22no—
60C22⋊C8—21no—
61C22⋊C8—21no—
62#97—21no—
63#97—21no—
64#97—21no—
65C4×C8—21no—
66#95—22no—
67#95—21no—
68#95—21no—
69#95D411yescharacteristic
70#95D411yescharacteristic
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—22no—
80C22×C4—21no—
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84Q16—44no—
85M16—22no—
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—42no—
108C42—41no—
109C42—82no—
110C42C22×C411yescharacteristic
111C42C2×D411yescharacteristic
112C42#4111yescharacteristic
113C42#4111yescharacteristic
114C23#12011yescharacteristic
115Q8—24no—
116Q8—22no—
117Q8—42no—
118Q8—81no—
119Q8#10414yes—
120C2×C4—24no—
121C2×C4—22no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—24no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4C2×C22⋊C411yescharacteristic
141C2×C4C4×D411yescharacteristic
142C2×C4C4×D411yescharacteristic
143C2×C4#12111yescharacteristic
144C2×C4#12111yescharacteristic
145C2×C4#12311yescharacteristic
146C8—42no—
147C8—42no—
148C8—81no—
149C22—21no—
150C22—21no—
151C22#38611yescharacteristic
152C22#45511yescharacteristic
153C22#45611yescharacteristic
154C4—21no—
155C4—21no—
156C4—24no—
157C4—21no—
158C4—21no—
159C4—21no—
160C4—44no—
161C4—42no—
162C4—42no—
163C4—42no—
164C4—81no—
165C4—81no—
166C4C2×#10412yes—
167C2—21no—
168C2—21no—
169C2#188211yescharacteristic
170C2#189611yescharacteristic
171C2#190511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC23#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #475C4semidirect
13. #462C4semidirect

Maximal subgroups

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

subgroupindex
#31132
#22762
#19862
#15062
#14912
#13002
#12852