Finite Group Databasebeta

← #9235

Subgroups — #9235

Order 256.

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

normal subgroups only

This group has 275 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#3101C211yesmaximal, characteristic
2#2293C211yesmaximal, characteristic
3#1994C211yesmaximal, characteristic, complemented
4#1503C211yesmaximal, characteristic, complemented
5#1486C211yesmaximal, characteristic
6#1303C211yesmaximal, characteristic, complemented
7#1276C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10C2×#117—21no—
11#483—21no—
12#476C412yescomplemented
13#461C412yescomplemented
14#450—21no—
15C4×SD16—22no—
16#431C2211yescharacteristic
17#431C2211yescharacteristic
18#423C2211yescharacteristic
19#390—21no—
20#389—21no—
21#388—22no—
22C4×C4⋊C4C2211yescharacteristic
23C4×C22⋊C4—21no—
24C43C2211yescharacteristic
25#339—21no—
26#339—21no—
27#328—22no—
28D4⋊C8—22no—
29C22×D4D411yescharacteristic
30C23×C4—21no—
31C2×SD16—22no—
32C2×M16—21no—
33C4⋊Q8C2×C411yescharacteristic
34C4⋊Q8C2×C411yescharacteristic
35#125—21no—
36C22⋊Q8—21no—
37C22⋊Q8—21no—
38C4×Q8—21no—
39C4×Q8—21no—
40C4×D4—21no—
41C4×D4—21no—
42C4×D4C2×C412yes—
43C4×D4D412yes—
44#117—21no—
45#117—42no—
46#117—42no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4D411yescharacteristic
50C2×C22⋊C4—21no—
51C2×C22⋊C4—22no—
52C2×C22⋊C4—22no—
53C2×C42—21no—
54C2×C42—21no—
55C2×C42—21no—
56C2×C42D411yescharacteristic
57C2×C42D411yescharacteristic
58C2×C42C2311yescharacteristic
59#106—22no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8C2×C412yes—
64#103—22no—
65#102—22no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68C4×C8—21no—
69C4×C8—21no—
70#95—22no—
71#95D412yes—
72C24—21no—
73C2×Q8—21no—
74C2×Q8—21no—
75C2×D4—21no—
76C2×D4—42no—
77C2×D4C22⋊C412yes—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91SD16—44no—
92M16—41no—
93M16—41no—
94C2×C8—22no—
95C2×C8—41no—
96C2×C8—41no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—42no—
103C4⋊C4—44no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4C22⋊C412yes—
107C22⋊C4—22no—
108C22⋊C4—22no—
109C22⋊C4—44no—
110C22⋊C4—44no—
111C22⋊C4—41no—
112C42—21no—
113C42—21no—
114C42—21no—
115C42—42no—
116C42—42no—
117C42—41no—
118C42—41no—
119C42—41no—
120C42SD1612yescomplemented
121C42C22×C411yescharacteristic
122C42C2×D411yescharacteristic
123C42#4111yescharacteristic
124C42#4111yescharacteristic
125C23—22no—
126C23—42no—
127C23—42no—
128C23—41no—
129C23—41no—
130C23#12011yescharacteristic
131Q8—42no—
132Q8—81no—
133D4—22no—
134D4—22no—
135D4—42no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4C2×C22⋊C411yescharacteristic
157C2×C4C4×D411yescharacteristic
158C2×C4C4×D411yescharacteristic
159C2×C4#12111yescharacteristic
160C2×C4#12111yescharacteristic
161C2×C4#12311yescharacteristic
162C2×C4C2×SD1611yescharacteristic
163C2×C4#13511yescharacteristic
164C2×C4AGL(1,ℤ/8)11yescharacteristic
165C2×C4#13711yescharacteristic
166C8—42no—
167C8—81no—
168C8—81no—
169C22—21no—
170C22—21no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22—82no—
175C22—82no—
176C22#38611yescharacteristic
177C22#45011yescharacteristic
178C22#45211yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—42no—
183C4—42no—
184C4—42no—
185C4—82no—
186C4—81no—
187C4—81no—
188C4#42112yes—
189C4C4×SD1611yescharacteristic
190C4#44011yescharacteristic
191C4#46111yescharacteristic
192C4#46311yescharacteristic
193C2—21no—
194C2—21no—
195C2—82no—
196C2#187711yescharacteristic
197C2#189711yescharacteristic
198C2#190611yescharacteristic
subgroupG/H
centerC2×C4#121
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radicalC22×D4D4
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
3. #1994C2semidirect
4. #1503C2semidirect
6. #1303C2semidirect
7. #1276C2semidirect
12. #476C4semidirect
13. #461C4semidirect
120. C42SD16semidirect

Maximal subgroups

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

subgroupindex
#31012
#22932
#19942
#15032
#14862
#13032
#12762