Finite Group Databasebeta

← #18179

Subgroups — #18179

Order 256.

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

normal subgroups only

This group has 625 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#3496C211yesmaximal, characteristic
2#3112C212yesmaximal, complemented
3#2398C211yesmaximal, characteristic, complemented
4#1607C214yesmaximal, complemented
5#1601C214yesmaximal, complemented
6#1505C212yesmaximal, complemented
7#1466C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)—21no—
9#547—22no—
10#546—24no—
11D42—22no—
12#537—21no—
13#534C2214yes—
14#532C2212yes—
15C2×#121—22no—
16C2×C4×Q8C2211yescharacteristic, complemented
17C2×C4×D4C2211yescharacteristic
18#480C2214yescomplemented
19#477—22no—
20#460—22no—
21#448C2214yescomplemented
22#440—24no—
23#432C2212yescomplemented
24#400—21no—
25#397C2212yescomplemented
26#394—21no—
27#393C2212yescomplemented
28C4×C22⋊C4—22no—
29#376C2211yescharacteristic, complemented
30#327C2218yescomplemented
31#323C2214yescomplemented
32H(5,𝔽2)—48no—
33C2×#41—24no—
34C2×#41—22no—
35C22×Q8C2311yescharacteristic
36C22×D4—24no—
37C22×D4—24no—
38C22×D4—22no—
39C22×D4D411yescharacteristic
40C23×C4D412yes—
41C2×SD16—24no—
42C4⋊Q8D412yes—
43#127D412yes—
44#124—22no—
45#123—28no—
46C22⋊Q8—24no—
47#121—24no—
48#121—24no—
49#121—24no—
50#121—22no—
51#121—24no—
52#121C2314yes—
53#120—28no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×D4—22no—
57C4×D4—22no—
58C4×D4—22no—
59C4×D4D414yes—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4D411yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C22⋊C4—24no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4C2312yes—
68C2×C42—21no—
69C2×C42—21no—
70C2×C42C2311yescharacteristic
71#107—24no—
72C4⋊C8—22no—
73#103—24no—
74#102—24no—
75C22⋊C8C2314yes—
76#97—22no—
77#95—21no—
78#95—22no—
79#95—22no—
80#95C2312yes—
81C24—24no—
82C24C2×D412yes—
83#41—48no—
84#41—416no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—42no—
88C2×D4—24no—
89C2×D4—28no—
90C2×D4—24no—
91C2×D4—24no—
92C2×D4—22no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—416no—
96C2×D4—48no—
97C2×D4—48no—
98C2×D4—84no—
99C2×D4C2×D414yes—
100C22×C4—24no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—24no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—44no—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D412yes—
111C22×C4C2411yescharacteristic
112SD16—48no—
113C2×C8—24no—
114C4⋊C4—24no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—42no—
119C4⋊C4C2×D414yes—
120C22⋊C4—28no—
121C22⋊C4—24no—
122C22⋊C4—24no—
123C22⋊C4—24no—
124C22⋊C4—44no—
125C22⋊C4—44no—
126C42—42no—
127C42—41no—
128C42—41no—
129C42C2×D412yes—
130C42C2×D412yes—
131C23—22no—
132C23—22no—
133C23—24no—
134C23—42no—
135C23—42no—
136C23—48no—
137C23—84no—
138C23—84no—
139C23AGL(1,ℤ/8)14yes—
140C23C22×D411yescharacteristic
141Q8—24no—
142Q8—42no—
143Q8—44no—
144D4—28no—
145D4—44no—
146D4—44no—
147D4—416no—
148D4—48no—
149D4—44no—
150D4—84no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—41no—
157C2×C4—48no—
158C2×C4—48no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—82no—
169C2×C4#12011yescharacteristic
170C2×C4#12012yes—
171C2×C4#12011yescharacteristic
172C2×C4C22×D412yes—
173C8—44no—
174C22—21no—
175C22—21no—
176C22—24no—
177C22—44no—
178C22—42no—
179C22—42no—
180C22—44no—
181C22—44no—
182C22—44no—
183C22—82no—
184C22—168no—
185C22C2×#12011yescharacteristic
186C22C2×AGL(1,ℤ/8)12yes—
187C4—22no—
188C4—21no—
189C4—22no—
190C4—44no—
191C4—42no—
192C4—42no—
193C4—82no—
194C4—81no—
195C4#45512yes—
196C2—22no—
197C2—44no—
198C2—84no—
199C2#300311yescharacteristic
200C2C2×#45511yescharacteristic
201C2#302511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3496C2
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
2. #3112C2semidirect
3. #2398C2semidirect
4. #1607C2semidirect
5. #1601C2semidirect
6. #1505C2semidirect
7. #1466C2semidirect
16. C2×C4×Q8C22semidirect
18. #480C22semidirect
21. #448C22semidirect
23. #432C22semidirect
25. #397C22semidirect
27. #393C22semidirect
29. #376C22semidirect
30. #327C22semidirect
31. #323C22semidirect

Maximal subgroups

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

subgroupindex
#34962
#31122
#23982
#16072
#16012
#15052
#14662