Finite Group Databasebeta

← #7748

Subgroups — #7748

Order 256.

268 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 51 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#2806C211yesmaximal, characteristic, complemented
2#2041C211yesmaximal, characteristic
3#2014C211yesmaximal, characteristic
4#2014C211yesmaximal, characteristic
5#1272C211yesmaximal, characteristic, complemented
6#1272C211yesmaximal, characteristic, complemented
7#1271C211yesmaximal, characteristic, complemented
8C22×SD16—21no—
9C22×SD16—21no—
10C2×#127C2211yescharacteristic
11C2×#102—21no—
12C2×#102—21no—
13C2×#102—21no—
14C2×#102—21no—
15#398—21no—
16#398—21no—
17#395—21no—
18#391—21no—
19#391C2211yescharacteristic
20#391C2211yescharacteristic
21#389—21no—
22#389C2211yescharacteristic
23#387—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C22×Q8D411yescharacteristic
29C22×D4—21no—
30C22×D4—21no—
31C22×D4—21no—
32C2×SD16—42no—
33C2×SD16—42no—
34C2×SD16—42no—
35C2×SD16—42no—
36C2×SD16—42no—
37C2×SD16—42no—
38C22×C8—21no—
39C22×C8—21no—
40C22×C8—21no—
41#127—42no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4D411yescharacteristic
48C2×C4⋊C4D411yescharacteristic
49C2×C4⋊C4D411yescharacteristic
50C2×C4⋊C4D411yescharacteristic
51C2×C42—21no—
52C2×C42C2311yescharacteristic
53#102—42no—
54#102—42no—
55#102—42no—
56#102—42no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63#95—21no—
64#95—21no—
65C24—41no—
66C2×Q8—21no—
67C2×Q8—21no—
68C2×Q8—21no—
69C2×Q8—41no—
70C2×Q8—41no—
71C2×Q8—41no—
72C2×Q8—41no—
73C2×Q8SD1612yescomplemented
74C2×D4—22no—
75C2×D4—22no—
76C2×D4—22no—
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—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—41no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101SD16—48no—
102SD16—48no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—41no—
106C2×C8—41no—
107C2×C8—41no—
108C2×C8—41no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C4⋊C4—21no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—81no—
119C4⋊C4—81no—
120C4⋊C4—81no—
121C4⋊C4—81no—
122C4⋊C4SD1612yescomplemented
123C42—41no—
124C42—81no—
125C23—42no—
126C23—81no—
127C23—81no—
128C23—81no—
129C23—81no—
130C23—81no—
131C23—81no—
132C23#12011yescharacteristic
133Q8—42no—
134Q8—42no—
135Q8—81no—
136Q8—81no—
137D4—42no—
138D4—42no—
139D4—42no—
140D4—42no—
141D4—42no—
142D4—42no—
143D4—81no—
144D4—81no—
145D4—81no—
146D4—81no—
147D4—81no—
148D4—81no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
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×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4C2×SD1611yescharacteristic
174C2×C4C2×SD1611yescharacteristic
175C2×C4#13511yescharacteristic
176C2×C4AGL(1,ℤ/8)11yescharacteristic
177C2×C4#13711yescharacteristic
178C2×C4#13711yescharacteristic
179C8—42no—
180C8—42no—
181C8—81no—
182C22—82no—
183C22—82no—
184C22—82no—
185C22—161no—
186C22—161no—
187C22—161no—
188C22—161no—
189C22#44811yescharacteristic
190C22#44811yescharacteristic
191C22#44911yescharacteristic
192C22#45311yescharacteristic
193C22#45311yescharacteristic
194C22#45411yescharacteristic
195C22B(4,𝔽2)11yescharacteristic
196C4—21no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—42no—
203C4—42no—
204C4—81no—
205C4—81no—
206C4—81no—
207C4—81no—
208C2—162no—
209C2#159611yescharacteristic
210C2#159911yescharacteristic
211C2#160011yescharacteristic
212C2#162211yescharacteristic
213C2#162311yescharacteristic
214C2#162611yescharacteristic
215C2#165611yescharacteristic
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. #2806C2semidirect
5. #1272C2semidirect
6. #1272C2semidirect
7. #1271C2semidirect
73. C2×Q8SD16semidirect
122. C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#28062
#20412
#20142
#12722
#12712