Finite Group Databasebeta

← #18211

Subgroups — #18211

Order 256.

627 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 127 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#3103C212yesmaximal, complemented
3#2580C211yesmaximal, characteristic, complemented
4#1613C214yesmaximal, complemented
5#1601C214yesmaximal, complemented
6#1517C212yesmaximal, complemented
7#1460C211yesmaximal, 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#481C2214yescomplemented
19#475—22no—
20#460—22no—
21#448C2214yescomplemented
22C4×SD16—24no—
23#431C2212yescomplemented
24#399C2212yescomplemented
25#393C2212yescomplemented
26#390—21no—
27#389—21no—
28#385—22no—
29#382C2211yescharacteristic, complemented
30#327C2218yescomplemented
31#323C2214yescomplemented
32H(5,𝔽2)—44no—
33H(5,𝔽2)—44no—
34C2×#41—24no—
35C2×#41—22no—
36C22×Q8C2311yescharacteristic
37C22×D4—24no—
38C22×D4—24no—
39C22×D4—22no—
40C22×D4D411yescharacteristic
41C23×C4D412yes—
42C2×SD16—24no—
43C4⋊Q8D412yes—
44#127D412yes—
45#124—22no—
46#123—28no—
47C22⋊Q8—24no—
48#121—24no—
49#121—24no—
50#121—24no—
51#121—22no—
52#121—24no—
53#121C2314yes—
54#120—28no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×D4—22no—
58C4×D4—22no—
59C4×D4—22no—
60C4×D4D414yes—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—24no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4C2312yes—
71C2×C42—21no—
72C2×C42C2311yescharacteristic
73#106—24no—
74C4⋊C8—22no—
75#103—24no—
76#102—24no—
77C22⋊C8C2314yes—
78C4×C8—22no—
79#95—21no—
80#95—22no—
81#95C2312yes—
82C24—24no—
83C24C2×D412yes—
84#41—416no—
85#41—44no—
86#41—44no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—42no—
90C2×D4—28no—
91C2×D4—24no—
92C2×D4—24no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—416no—
98C2×D4—48no—
99C2×D4—48no—
100C2×D4—84no—
101C2×D4SD1614yescomplemented
102C2×D4C2×D414yes—
103C22×C4—24no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—24no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—82no—
111C22×C4C2×D412yes—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2411yescharacteristic
115SD16—48no—
116C2×C8—24no—
117C4⋊C4—24no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C4⋊C4—42no—
124C4⋊C4—82no—
125C4⋊C4C2×D414yes—
126C22⋊C4—28no—
127C22⋊C4—24no—
128C22⋊C4—24no—
129C22⋊C4—24no—
130C22⋊C4—44no—
131C22⋊C4—82no—
132C42—42no—
133C42C2×D412yes—
134C42C2×D412yes—
135C23—22no—
136C23—22no—
137C23—24no—
138C23—42no—
139C23—42no—
140C23—48no—
141C23—84no—
142C23—84no—
143C23C2×SD1614yes—
144C23C22×D411yescharacteristic
145Q8—24no—
146Q8—42no—
147Q8—44no—
148D4—28no—
149D4—44no—
150D4—44no—
151D4—416no—
152D4—44no—
153D4—44no—
154D4—44no—
155D4—84no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—44no—
162C2×C4—48no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—48no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—82no—
174C2×C4#12011yescharacteristic
175C2×C4#12011yescharacteristic
176C2×C4#12012yes—
177C2×C4C2×SD1612yes—
178C2×C4AGL(1,ℤ/8)12yes—
179C2×C4C22×D412yes—
180C8—44no—
181C22—21no—
182C22—21no—
183C22—24no—
184C22—44no—
185C22—42no—
186C22—42no—
187C22—44no—
188C22—44no—
189C22—44no—
190C22—82no—
191C22—168no—
192C22C2×#12011yescharacteristic
193C22C22×SD1611yescharacteristic
194C22C2×AGL(1,ℤ/8)11yescharacteristic
195C4—21no—
196C4—21no—
197C4—22no—
198C4—44no—
199C4—42no—
200C4—42no—
201C4—82no—
202C4—81no—
203C4#45014yes—
204C2—21no—
205C2—21no—
206C2—44no—
207C2—84no—
208C2C2×#45011yescharacteristic
209C2#301911yescharacteristic
210C2#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. #3103C2semidirect
3. #2580C2semidirect
4. #1613C2semidirect
5. #1601C2semidirect
6. #1517C2semidirect
7. #1460C2semidirect
16. C2×C4×Q8C22semidirect
18. #481C22semidirect
21. #448C22semidirect
23. #431C22semidirect
24. #399C22semidirect
25. #393C22semidirect
29. #382C22semidirect
30. #327C22semidirect
31. #323C22semidirect
101. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#34962
#31032
#25802
#16132
#16012
#15172
#14602