Finite Group Databasebeta

← #9238

Subgroups — #9238

Order 256.

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

normal subgroups only

This group has 256 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
1#3103C211yesmaximal, characteristic
2#2295C211yesmaximal, characteristic, complemented
3#1962C211yesmaximal, characteristic
4#1502C211yesmaximal, characteristic
5#1487C211yesmaximal, characteristic, complemented
6#1303C211yesmaximal, characteristic, complemented
7#1276C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10#481—21no—
11#475C412yescomplemented
12#460C412yescomplemented
13#448—21no—
14C4×SD16—22no—
15#431C2211yescharacteristic
16#431C2211yescharacteristic
17#423C2211yescharacteristic
18C2×#104—21no—
19C2×#104—21no—
20#391—21no—
21#389—21no—
22#387—22no—
23C4×C4⋊C4—21no—
24C4×C4⋊C4C2211yescharacteristic
25#376—21no—
26C43C2211yescharacteristic
27#327—22no—
28Q8⋊C8—22no—
29C2×#41—21no—
30C22×Q8D411yescharacteristic
31C2×SD16—22no—
32C2×M16—21no—
33C4⋊Q8C2×C411yescharacteristic
34#127C2×C411yescharacteristic
35#124—21no—
36#121—21no—
37#121—21no—
38C4×Q8—21no—
39C4×Q8—21no—
40C4×Q8C2×C412yes—
41C4×Q8D412yes—
42C4×D4—21no—
43C4×D4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4D411yescharacteristic
48C2×C42—21no—
49C2×C42—21no—
50C2×C42—21no—
51C2×C42—21no—
52C2×C42D411yescharacteristic
53C2×C42D411yescharacteristic
54C2×C42C2311yescharacteristic
55#106—22no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8C2×C412yes—
60#104—42no—
61#104—42no—
62#103—22no—
63#102—22no—
64C22⋊C8—21no—
65C22⋊C8—21no—
66C4×C8—21no—
67C4×C8—21no—
68#95—22no—
69#95—22no—
70#95—22no—
71#95D412yes—
72#41—41no—
73#41—41no—
74#41—41no—
75#41—41no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—42no—
79C2×Q8C22⋊C412yes—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—41no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—41no—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93SD16—44no—
94M16—41no—
95M16—41no—
96C2×C8—22no—
97C2×C8—41no—
98C2×C8—41no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—42no—
105C4⋊C4—44no—
106C4⋊C4C22⋊C412yes—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C42—22no—
110C42—22no—
111C42—21no—
112C42—21no—
113C42—21no—
114C42—42no—
115C42—42no—
116C42—41no—
117C42—41no—
118C42SD1612yescomplemented
119C42C22×C411yescharacteristic
120C42C2×D411yescharacteristic
121C42#4111yescharacteristic
122C42#4111yescharacteristic
123C23—41no—
124C23#12011yescharacteristic
125Q8—22no—
126Q8—22no—
127Q8—42no—
128Q8—41no—
129Q8—41no—
130D4—42no—
131D4—41no—
132D4—41no—
133D4—81no—
134D4—81no—
135D4—81no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—81no—
153C2×C4—81no—
154C2×C4C2×C22⋊C411yescharacteristic
155C2×C4C4×D411yescharacteristic
156C2×C4C4×D411yescharacteristic
157C2×C4#12111yescharacteristic
158C2×C4#12111yescharacteristic
159C2×C4#12311yescharacteristic
160C2×C4C2×SD1611yescharacteristic
161C2×C4#13511yescharacteristic
162C2×C4AGL(1,ℤ/8)11yescharacteristic
163C2×C4#13711yescharacteristic
164C8—42no—
165C8—81no—
166C8—81no—
167C22—21no—
168C22—21no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22#38611yescharacteristic
173C22#44811yescharacteristic
174C22#44911yescharacteristic
175C4—21no—
176C4—21no—
177C4—21no—
178C4—42no—
179C4—42no—
180C4—42no—
181C4—42no—
182C4—82no—
183C4—81no—
184C4#42112yes—
185C4C4×SD1611yescharacteristic
186C4#44011yescharacteristic
187C4#46011yescharacteristic
188C4#46411yescharacteristic
189C2—21no—
190C2—21no—
191C2—161no—
192C2#188011yescharacteristic
193C2#189711yescharacteristic
194C2#190611yescharacteristic
subgroupG/H
centerC2×C4#121
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#532C22
LCSG ▷ C42 ▷ 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
2. #2295C2semidirect
5. #1487C2semidirect
6. #1303C2semidirect
7. #1276C2semidirect
11. #475C4semidirect
12. #460C4semidirect
118. C42SD16semidirect

Maximal subgroups

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

subgroupindex
#31032
#22952
#19622
#15022
#14872
#13032
#12762