Finite Group Databasebeta

← #7747

Subgroups — #7747

Order 256.

246 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 49 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#2805C211yesmaximal, characteristic, complemented
2#2041C211yesmaximal, characteristic
3#2041C211yesmaximal, characteristic
4#2014C211yesmaximal, characteristic
5#1272C211yesmaximal, characteristic, complemented
6#1271C211yesmaximal, characteristic, complemented
7#1271C211yesmaximal, characteristic, complemented
8C22×SD16—21no—
9C2×#127C2211yescharacteristic
10C2×#102—21no—
11C2×#102—21no—
12C2×#102—21no—
13C2×#102—21no—
14C2×#102—21no—
15#400—21no—
16#400—21no—
17#395—21no—
18#391C2211yescharacteristic
19#389—21no—
20#389C2211yescharacteristic
21#389C2211yescharacteristic
22#387—21no—
23#387—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C22×D4—21no—
29C22×D4—21no—
30C22×D4—21no—
31C2×SD16—42no—
32C2×SD16—42no—
33C2×SD16—42no—
34C22×C8—21no—
35C22×C8—21no—
36C22×C8—21no—
37#127—42no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4D411yescharacteristic
46C2×C4⋊C4D411yescharacteristic
47C2×C42—21no—
48C2×C42C2311yescharacteristic
49#102—42no—
50#102—42no—
51#102—42no—
52#102—42no—
53#102—42no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63C24—41no—
64C2×Q8—21no—
65C2×Q8—21no—
66C2×Q8—41no—
67C2×Q8—41no—
68C2×D4—22no—
69C2×D4—22no—
70C2×D4—22no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4—41no—
77C2×D4—41no—
78C2×D4—41no—
79C2×D4—41no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4—41no—
84C2×D4—41no—
85C2×D4—41no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95SD16—48no—
96C2×C8—22no—
97C2×C8—41no—
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C2×C8—41no—
102C2×C8—41no—
103C2×C8—41no—
104C2×C8—41no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—81no—
116C4⋊C4—81no—
117C4⋊C4SD1612yescomplemented
118C42—41no—
119C42—81no—
120C23—42no—
121C23—81no—
122C23—81no—
123C23—81no—
124C23—81no—
125C23—81no—
126C23—81no—
127C23#12011yescharacteristic
128Q8—42no—
129Q8—81no—
130D4—42no—
131D4—42no—
132D4—42no—
133D4—42no—
134D4—42no—
135D4—42no—
136D4—81no—
137D4—81no—
138D4—81no—
139D4—81no—
140D4—81no—
141D4—81no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4C2×SD1611yescharacteristic
167C2×C4#13511yescharacteristic
168C2×C4#13511yescharacteristic
169C2×C4AGL(1,ℤ/8)11yescharacteristic
170C2×C4AGL(1,ℤ/8)11yescharacteristic
171C2×C4#13711yescharacteristic
172C8—42no—
173C8—81no—
174C8—81no—
175C22—82no—
176C22—82no—
177C22—82no—
178C22—161no—
179C22—161no—
180C22—161no—
181C22—161no—
182C22#44811yescharacteristic
183C22#44911yescharacteristic
184C22#44911yescharacteristic
185C22#45311yescharacteristic
186C22#45411yescharacteristic
187C22#45411yescharacteristic
188C22B(4,𝔽2)11yescharacteristic
189C4—21no—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—42no—
196C4—81no—
197C4—81no—
198C4—81no—
199C4—81no—
200C4—81no—
201C2—162no—
202C2#159711yescharacteristic
203C2#159911yescharacteristic
204C2#160011yescharacteristic
205C2#162211yescharacteristic
206C2#162611yescharacteristic
207C2#162811yescharacteristic
208C2#165711yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radicalC2×#127C22
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2805C2semidirect
5. #1272C2semidirect
6. #1271C2semidirect
7. #1271C2semidirect
117. C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#28052
#20412
#20142
#12722
#12712