Finite Group Databasebeta

← #9047

Subgroups — #9047

Order 256.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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#3103C211yesmaximal, characteristic
2#2932C211yesmaximal, characteristic
3#1985C211yesmaximal, characteristic, complemented
4#1540C211yesmaximal, characteristic
5#1487C211yesmaximal, characteristic, complemented
6#1301C211yesmaximal, characteristic, complemented
7#1281C211yesmaximal, characteristic, complemented
8C2×#131—21no—
9#532C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11#481C412yes—
12#475—21no—
13#460—21no—
14#448C412yes—
15C4×SD16—22no—
16#435—22no—
17#435—22no—
18#431C2211yescharacteristic
19#431C2211yescharacteristic
20#423C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22#408—21no—
23C2×#97C2211yescharacteristic
24#336—21no—
25#336—21no—
26#332—21no—
27#331—21no—
28Q8⋊C8—22no—
29C2×#41—21no—
30C22×Q8C2×C411yescharacteristic
31C2×SD16—22no—
32#131—42no—
33#131—42no—
34C2×M16—22no—
35C2×M16—21no—
36C22×C8—22no—
37C22×C8—21no—
38C22×C8D411yescharacteristic
39C22×C8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41#127D411yescharacteristic
42#124—21no—
43#121—21no—
44#121C2×C412yes—
45C4×Q8—21no—
46C4×Q8—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49C4×D4—21no—
50C4×D4D412yes—
51C2×C4⋊C4—21no—
52C2×C4⋊C4C2×C411yescharacteristic
53C2×C42—21no—
54C2×C42C2311yescharacteristic
55#106—22no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8D412yes—
62#103—22no—
63#102—22no—
64C22⋊C8—24no—
65C22⋊C8—21no—
66C22⋊C8C2×C412yes—
67#97—21no—
68#97—21no—
69C4×C8—21no—
70C4×C8—21no—
71#41—41no—
72#41—41no—
73#41—41no—
74#41—41no—
75C2×Q8—21no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—42no—
79C2×D4—22no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4C22⋊C412yes—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4C22×C411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4#4111yescharacteristic
89SD16—44no—
90M16—44no—
91M16—41no—
92M16—41no—
93C2×C8—21no—
94C2×C8—21no—
95C2×C8—22no—
96C2×C8—21no—
97C2×C8—22no—
98C2×C8—21no—
99C2×C8—21no—
100C2×C8—44no—
101C2×C8—41no—
102C2×C8—41no—
103C2×C8—41no—
104C2×C8—41no—
105C2×C8SD1612yescomplemented
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4C22⋊C412yes—
112C22⋊C4—22no—
113C22⋊C4—22no—
114C42—41no—
115C42—41no—
116C42C2×D411yescharacteristic
117C42C2×D411yescharacteristic
118C42C2×D411yescharacteristic
119C42#4111yescharacteristic
120C23—22no—
121C23C4×D411yescharacteristic
122Q8—22no—
123Q8—22no—
124Q8—41no—
125Q8—41no—
126Q8—41no—
127Q8—41no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—41no—
132D4—41no—
133D4—81no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—41no—
144C2×C4C2×C22⋊C411yescharacteristic
145C2×C4C4×D411yescharacteristic
146C2×C4#12011yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12111yescharacteristic
149C2×C4#12311yescharacteristic
150C2×C4C2×SD1611yescharacteristic
151C2×C4#13511yescharacteristic
152C2×C4AGL(1,ℤ/8)11yescharacteristic
153C2×C4#13711yescharacteristic
154C8—22no—
155C8—42no—
156C8—42no—
157C8—41no—
158C8—81no—
159C8—81no—
160C22—21no—
161C22—21no—
162C22—42no—
163C22—42no—
164C22—42no—
165C22#38611yescharacteristic
166C22C4×SD1611yescharacteristic
167C22#44011yescharacteristic
168C4—21no—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—42no—
173C4—42no—
174C4—81no—
175C4#45011yescharacteristic
176C4#45211yescharacteristic
177C4#46111yescharacteristic
178C4#46311yescharacteristic
179C2—21no—
180C2—21no—
181C2—82no—
182C2#187711yescharacteristic
183C2#188111yescharacteristic
184C2#189111yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#532C22
LCSG ▷ C22×C4 ▷ 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
3. #1985C2semidirect
5. #1487C2semidirect
6. #1301C2semidirect
7. #1281C2semidirect
105. C2×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#31032
#29322
#19852
#15402
#14872
#13012
#12812