Finite Group Databasebeta

← #12760

Subgroups — #12760

Order 256.

438 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#559C211yesmaximal, characteristic
2C2×#353C211yesmaximal, characteristic
3C2×#352C212yesmaximal, complemented
4#1890C212yesmaximal, complemented
5#1887C211yesmaximal, characteristic
6#559—22no—
7#559—24no—
8#559C414yescomplemented
9C2×#127C2211yescharacteristic
10C2×#124—22no—
11C2×#123—21no—
12C2×#123C2212yes—
13#518C2211yescharacteristic
14C2×C4×Q8—21no—
15C2×C4×D4—22no—
16#418—21no—
17C2×#100C2211yescharacteristic
18C2×#99C2212yes—
19#381—22no—
20#353—22no—
21#352—24no—
22#342—22no—
23#339—21no—
24C22×Q8—21no—
25C22×D4—22no—
26C22×D4D411yescharacteristic
27C22×D4C2311yescharacteristic
28C23×C4—21no—
29C23×C4D412yes—
30C2×M16—21no—
31#127—22no—
32#127C2×C412yes—
33#127D412yes—
34#124—28no—
35#124—28no—
36#123—24no—
37#123—24no—
38#123—22no—
39#123—22no—
40#123C2×C414yes—
41#123D414yes—
42C4×Q8—24no—
43C4×Q8—24no—
44C4×D4—28no—
45C4×D4—28no—
46C4×D4—42no—
47#117—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—22no—
50C2×C22⋊C4—22no—
51C2×C22⋊C4—21no—
52C2×C22⋊C4—22no—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—22no—
55C2×C22⋊C4D412yes—
56C2×C42—22no—
57C2×C42D411yescharacteristic
58#102—42no—
59#100—22no—
60#99—22no—
61#99—22no—
62#95—44no—
63C24—21no—
64C24C2×D412yes—
65C2×Q8—24no—
66C2×Q8—44no—
67C2×D4—24no—
68C2×D4—24no—
69C2×D4—21no—
70C2×D4—41no—
71C2×D4—41no—
72C2×D4—48no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4C22⋊C412yes—
76C2×D4C22×C411yescharacteristic
77C2×D4C2×D411yescharacteristic
78C2×D4C2×D412yes—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—24no—
83C22×C4—22no—
84C22×C4—24no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4—81no—
89C22×C4—81no—
90C22×C4C22⋊C414yes—
91C22×C4C2×D411yescharacteristic
92M16—41no—
93M16—41no—
94C2×C8—41no—
95C4⋊C4—24no—
96C4⋊C4—24no—
97C4⋊C4—24no—
98C4⋊C4—22no—
99C22⋊C4—28no—
100C22⋊C4—24no—
101C22⋊C4—24no—
102C22⋊C4—24no—
103C22⋊C4—22no—
104C22⋊C4—42no—
105C22⋊C4—41no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4C22⋊C414yes—
109C42—28no—
110C42—21no—
111C42—41no—
112C42C22⋊C412yes—
113C23—22no—
114C23—22no—
115C23—22no—
116C23—44no—
117C23—44no—
118C23—41no—
119C23—41no—
120C23C2×C22⋊C412yes—
121C23#12012yes—
122C23#12011yescharacteristic
123Q8—48no—
124D4—41no—
125D4—42no—
126D4—41no—
127D4—42no—
128D4—48no—
129D4—48no—
130D4—81no—
131C2×C4—24no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—24no—
136C2×C4—24no—
137C2×C4—44no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—44no—
141C2×C4—42no—
142C2×C4—82no—
143C2×C4—84no—
144C2×C4—84no—
145C2×C4—82no—
146C2×C4—84no—
147C2×C4C2×C22⋊C411yescharacteristic
148C2×C4#12011yescharacteristic
149C8—81no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—42no—
154C22—42no—
155C22—44no—
156C22—44no—
157C22—42no—
158C22—82no—
159C22—82no—
160C22—82no—
161C22#37911yescharacteristic
162C22B(4,𝔽2)11yescharacteristic
163C22#45811yescharacteristic
164C4—24no—
165C4—21no—
166C4—21no—
167C4—44no—
168C4—44no—
169C4—82no—
170C4—162no—
171C2—21no—
172C2—21no—
173C2—42no—
174C2—42no—
175C2—82no—
176C2#179011yescharacteristic
177C2#219812yes—
subgroupG/H
center, socleC22B(4,𝔽2)
commutatorC2×D4C22×C4
Frattini, cosocleC22×D4C23
prime radicalC2×#127C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#352C2semidirect
4. #1890C2semidirect
8. #559C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5592
C2×#3532
C2×#3522
#18902
#18872