Finite Group Databasebeta

← #9328

Subgroups — #9328

Order 256.

222 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 61 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#2320C211yesmaximal, characteristic
3#1986C211yesmaximal, characteristic
4#1506C211yesmaximal, characteristic
5#1491C211yesmaximal, characteristic
6#1300C211yesmaximal, characteristic
7#1285C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10#483—21no—
11#475C412yescomplemented
12#462C412yescomplemented
13#451—21no—
14#441—22no—
15#432C2211yescharacteristic
16#432C2211yescharacteristic
17C4×M16C2211yescharacteristic
18#391—21no—
19#389—21no—
20#387—22no—
21#383—21no—
22#382—21no—
23C4×C4⋊C4C2211yescharacteristic
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—21no—
38C4×Q8—21no—
39C4×Q8—21no—
40C4×Q8—21no—
41C4×Q8C2×C412yes—
42C4×Q8D412yes—
43#117—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4D411yescharacteristic
48C2×C42—21no—
49C2×C42—21no—
50C2×C42D412yes—
51C2×C42C2311yescharacteristic
52#106—22no—
53C4⋊C8—21no—
54C4⋊C8C2×C412yes—
55#103—22no—
56#103—22no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59#97—21no—
60#97—21no—
61#97—21no—
62C4×C8—21no—
63#95—22no—
64#95—22no—
65#95—22no—
66#95—21no—
67#95—21no—
68#95D411yescharacteristic
69#95D411yescharacteristic
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8—42no—
74C2×Q8C22⋊C412yes—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83Q16—44no—
84M16—22no—
85M16—41no—
86C2×C8—22no—
87C2×C8—41no—
88C2×C8—41no—
89C4⋊C4—22no—
90C4⋊C4—22no—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—42no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—82no—
100C4⋊C4C22⋊C412yes—
101C22⋊C4—41no—
102C42—22no—
103C42—22no—
104C42—42no—
105C42—42no—
106C42—41no—
107C42C22×C411yescharacteristic
108C42C2×D411yescharacteristic
109C42#4111yescharacteristic
110C42#4111yescharacteristic
111C23#12011yescharacteristic
112Q8—22no—
113Q8—22no—
114Q8—42no—
115Q8—42no—
116Q8—81no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—41no—
123C2×C4—42no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—41no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4C2×C22⋊C411yescharacteristic
138C2×C4C4×D411yescharacteristic
139C2×C4C4×D411yescharacteristic
140C2×C4#12111yescharacteristic
141C2×C4#12111yescharacteristic
142C2×C4#12311yescharacteristic
143C8—42no—
144C8—42no—
145C8—81no—
146C22—21no—
147C22—21no—
148C22#38611yescharacteristic
149C22#45511yescharacteristic
150C22#45611yescharacteristic
151C4—21no—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—81no—
159C4—82no—
160C4—81no—
161C4—81no—
162C4—81no—
163C4#42112yes—
164C2—21no—
165C2—21no—
166C2#188211yescharacteristic
167C2#189711yescharacteristic
168C2#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
11. #475C4semidirect
12. #462C4semidirect

Maximal subgroups

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

subgroupindex
#31132
#23202
#19862
#15062
#14912
#13002
#12852