Finite Group Databasebeta

← #18186

Subgroups — #18186

Order 256.

599 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 117 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#3497C211yesmaximal, characteristic, complemented
2#3109C212yesmaximal, complemented
3#2398C211yesmaximal, characteristic, complemented
4#1616C214yesmaximal, complemented
5#1598C214yesmaximal, complemented
6#1505C212yesmaximal, complemented
7#1467C211yesmaximal, characteristic, complemented
8C2×2−1+4—21no—
9#550—22no—
10D4×Q8—22no—
11#548—24no—
12#537—21no—
13#537C2214yes—
14#532—22no—
15#532C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18#481C2214yes—
19#476—22no—
20#459—22no—
21#447C2214yes—
22#442—24no—
23#432C2212yes—
24#400C2212yes—
25#397—21no—
26#394C2212yes—
27#393—21no—
28C4×C22⋊C4—22no—
29#376C2211yescharacteristic
30#327C2218yes—
31#324C2214yes—
322−1+4—48no—
33C2×#41—24no—
34C2×#41—24no—
35C2×#41—22no—
36C22×Q8—24no—
37C22×Q8D411yescharacteristic
38C22×D4C2311yescharacteristic
39C23×C4—21no—
40C2×D8—24no—
41C4⋊Q8—22no—
42C4⋊Q8D412yes—
43#127—22no—
44#127D412yes—
45#124—24no—
46#124—22no—
47C22⋊Q8—24no—
48C22⋊Q8—28no—
49#121—24no—
50#121—28no—
51#121—22no—
52#121—24no—
53#121C2314yes—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×D4—24no—
57C4×D4—24no—
58C4×D4—22no—
59C4×D4—22no—
60C4×D4—22no—
61C4×D4D414yes—
62C2×C4⋊C4D411yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C42—21no—
69C2×C42D412yes—
70C2×C42C2311yescharacteristic
71#106—24no—
72C4⋊C8—22no—
73#102—24no—
74#102—24no—
75C22⋊C8C2314yes—
76#97—22no—
77#95—21no—
78#95—22no—
79#95—22no—
80#95C2312yes—
81C24—21no—
82#41—48no—
83#41—48no—
84#41—416no—
85#41—48no—
86C2×Q8—24no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—24no—
90C2×Q8—416no—
91C2×D4—24no—
92C2×D4—24no—
93C2×D4—24no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—42no—
98C2×D4C2×D414yes—
99C22×C4—28no—
100C22×C4—24no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—44no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D412yes—
109C22×C4C2×D412yes—
110C22×C4C2411yescharacteristic
111D8—48no—
112C2×C8—24no—
113C4⋊C4—24no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—24no—
117C4⋊C4—22no—
118C4⋊C4C2×D414yes—
119C22⋊C4—28no—
120C22⋊C4—24no—
121C22⋊C4—24no—
122C22⋊C4—44no—
123C22⋊C4—42no—
124C22⋊C4—44no—
125C42—22no—
126C42—22no—
127C42—42no—
128C42C2×D412yes—
129C42C2×D412yes—
130C23—22no—
131C23—24no—
132C23—42no—
133C23—41no—
134C23—41no—
135C23—42no—
136C23C22×D411yescharacteristic
137Q8—28no—
138Q8—44no—
139Q8—48no—
140D4—24no—
141D4—44no—
142D4—42no—
143D4—44no—
144D4—48no—
145D4—48no—
146D4—44no—
147D4—84no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—41no—
155C2×C4—48no—
156C2×C4—416no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—82no—
164C2×C4—82no—
165C2×C4#12011yescharacteristic
166C2×C4#12012yes—
167C2×C4#12011yescharacteristic
168C2×C4AGL(1,ℤ/8)14yes—
169C2×C4C22×D412yes—
170C8—44no—
171C22—21no—
172C22—21no—
173C22—44no—
174C22—44no—
175C22—44no—
176C22—42no—
177C22—44no—
178C22—82no—
179C22—81no—
180C22—81no—
181C22C2×#12011yescharacteristic
182C22C2×AGL(1,ℤ/8)12yes—
183C4—22no—
184C4—24no—
185C4—21no—
186C4—22no—
187C4—44no—
188C4—42no—
189C4—82no—
190C4—81no—
191C4#45412yes—
192C2—22no—
193C2—84no—
194C2—82no—
195C2#300311yescharacteristic
196C2C2×#45411yescharacteristic
197C2#302511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3497C2semidirect
2. #3109C2semidirect
3. #2398C2semidirect
4. #1616C2semidirect
5. #1598C2semidirect
6. #1505C2semidirect
7. #1467C2semidirect

Maximal subgroups

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

subgroupindex
#34972
#31092
#23982
#16162
#15982
#15052
#14672