Finite Group Databasebeta

← #9441

Subgroups — #9441

Order 256.

288 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 59 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#3153C211yesmaximal, characteristic
2#2372C211yesmaximal, characteristic, complemented
3#1533C211yesmaximal, characteristic, complemented
4#1505C212yesmaximal, complemented
5#1300C212yesmaximal, complemented
6C2×#137—21no—
7C2×AGL(1,ℤ/8)—21no—
8#532C2211yescharacteristic
9C2×#124C2211yescharacteristic
10#497C412yescomplemented
11#496C412yescomplemented
12#495—22no—
13#432C2212yes—
14C4×M16C2211yescharacteristic
15#391—21no—
16#388—22no—
17#386—21no—
18C4×C22⋊C4—22no—
19#376C2212yes—
20#329—22no—
21#327—22no—
22#327—22no—
23C2×#41—21no—
24C22×Q8D411yescharacteristic
25C22×D4D411yescharacteristic
26C23×C4—21no—
27#137—42no—
28#137—42no—
29AGL(1,ℤ/8)—42no—
30AGL(1,ℤ/8)—42no—
31C2×Q16—22no—
32C2×SD16—22no—
33C2×SD16—22no—
34C2×D8—22no—
35C2×M16—21no—
36C4⋊Q8C2×C411yescharacteristic, complemented
37#127C2×C411yescharacteristic, complemented
38#124—21no—
39#124—22no—
40#124C2×C412yes—
41#124D412yes—
42#121—22no—
43C4×D4—22no—
44C2×C4⋊C4—21no—
45C2×C22⋊C4—22no—
46C2×C22⋊C4—22no—
47C2×C22⋊C4—21no—
48C2×C42—22no—
49C2×C42—21no—
50C2×C42C2311yescharacteristic
51C4⋊C8—22no—
52C22⋊C8—22no—
53#97—22no—
54#97C2×C412yescomplemented
55C4×C8—21no—
56#95—22no—
57#95—22no—
58#95D412yes—
59#95D412yes—
60C24—21no—
61#41—41no—
62#41—41no—
63#41—41no—
64#41—41no—
65C2×Q8—21no—
66C2×Q8—21no—
67C2×Q8—41no—
68C2×Q8—41no—
69C2×Q8C22⋊C412yes—
70C2×D4—21no—
71C2×D4—22no—
72C2×D4—21no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4C22⋊C412yes—
77C22×C4—22no—
78C22×C4—21no—
79C22×C4—22no—
80C22×C4—44no—
81C22×C4—41no—
82C22×C4—81no—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D412yes—
85Q16—44no—
86SD16—44no—
87SD16—44no—
88D8—44no—
89M16—22no—
90M16—22no—
91C2×C8—22no—
92C2×C8—42no—
93C4⋊C4—22no—
94C4⋊C4—81no—
95C22⋊C4—24no—
96C22⋊C4—24no—
97C22⋊C4—44no—
98C22⋊C4—42no—
99C22⋊C4—81no—
100C42—44no—
101C42—81no—
102C42C22×C411yescharacteristic
103C42C2×D411yescharacteristic
104C42#4112yes—
105C23—22no—
106C23—42no—
107C23—42no—
108C23—41no—
109C23—41no—
110C23—41no—
111C23#12011yescharacteristic
112Q8—42no—
113Q8—41no—
114Q8—41no—
115Q8—81no—
116D4—42no—
117D4—41no—
118D4—41no—
119D4—81no—
120D4—82no—
121D4—81no—
122D4—81no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—24no—
126C2×C4—22no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—41no—
135C2×C4—42no—
136C2×C4—81no—
137C2×C4—81no—
138C2×C4—84no—
139C2×C4C2×C22⋊C411yescharacteristic
140C2×C4C4×D412yes—
141C2×C4#12112yes—
142C2×C4#12311yescharacteristic
143C8—24no—
144C8—82no—
145C22—22no—
146C22—42no—
147C22—44no—
148C22—81no—
149C22—81no—
150C22—82no—
151C22—81no—
152C22—82no—
153C22#38611yescharacteristic
154C22#45311yescharacteristic
155C22#45311yescharacteristic
156C4—21no—
157C4—21no—
158C4—22no—
159C4—22no—
160C4—44no—
161C4—42no—
162C4—82no—
163C4—81no—
164C2—21no—
165C2—21no—
166C2—82no—
167C2—161no—
168C2#188711yescharacteristic
169C2#190612yes—
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3153C2
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
2. #2372C2semidirect
3. #1533C2semidirect
4. #1505C2semidirect
5. #1300C2semidirect
10. #497C4semidirect
11. #496C4semidirect
36. C4⋊Q8C2×C4semidirect
37. #127C2×C4semidirect
54. #97C2×C4semidirect

Maximal subgroups

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

subgroupindex
#31532
#23722
#15332
#15052
#13002