Finite Group Databasebeta

← #12154

Subgroups — #12154

Order 256.

229 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 37 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#2609C211yesmaximal, characteristic, complemented
2#2040C211yesmaximal, characteristic, complemented
3#1610C211yesmaximal, characteristic, complemented
4#1598C211yesmaximal, characteristic
5#1346C211yesmaximal, characteristic, complemented
6#1345C211yesmaximal, characteristic, complemented
7#1324C211yesmaximal, characteristic, complemented
8#537C2211yescharacteristic
9C2×#117—21no—
10#480—21no—
11#447—21no—
12C2×#104—21no—
13#417—21no—
14#398C2211yescharacteristic
15#396—21no—
16#394C2211yescharacteristic
17#388—21no—
18#387—21no—
19#386—21no—
20#385—21no—
21#382C2211yescharacteristic
22#343C2211yescharacteristic
23#327—21no—
24#327—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C2×#41—21no—
28C22×D4D411yescharacteristic
29C23×C4—21no—
30C2×D8—41no—
31C2×M16—21no—
32C22⋊Q8—41no—
33#121—21no—
34#121—21no—
35#121—41no—
36#117—42no—
37#117—42no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4D411yescharacteristic
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4C2311yescharacteristic
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C42—21no—
50C2×C42D411yescharacteristic
51#107—41no—
52#104—42no—
53#103—41no—
54#102—41no—
55#102—41no—
56#102—41no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63#95D411yescharacteristic
64#95D411yescharacteristic
65C24—21no—
66#41—41no—
67#41—41no—
68#41—41no—
69#41—41no—
70C2×Q8—21no—
71C2×D4—21no—
72C2×D4—21no—
73C2×D4—21no—
74C2×D4—41no—
75C2×D4—42no—
76C2×D4—41no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—41no—
85C22×C4—42no—
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89D8—82no—
90M16—41no—
91M16—41no—
92C2×C8—41no—
93C2×C8—41no—
94C2×C8—41no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—42no—
101C4⋊C4—42no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—42no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—81no—
112C42—21no—
113C42—42no—
114C42D812yescomplemented
115C23—22no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23#12011yescharacteristic
124Q8—41no—
125Q8—41no—
126D4—41no—
127D4—41no—
128D4—42no—
129D4—81no—
130D4—81no—
131D4—81no—
132D4—81no—
133D4—81no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—48no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
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—81no—
158C2×C4—81no—
159C2×C4C2×D811yescharacteristic
160C2×C4AGL(1,ℤ/8)11yescharacteristic
161C8—81no—
162C8—81no—
163C8—81no—
164C22—21no—
165C22—21no—
166C22—42no—
167C22—42no—
168C22—42no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22#44711yescharacteristic
177C22#45411yescharacteristic
178C22B(4,𝔽2)11yescharacteristic
179C4—21no—
180C4—21no—
181C4—42no—
182C4—42no—
183C4—41no—
184C4—41no—
185C4—81no—
186C4—81no—
187C4—81no—
188C4—81no—
189C2—21no—
190C2—21no—
191C2—82no—
192C2—161no—
193C2#159811yescharacteristic
194C2#219711yescharacteristic
195C2#220411yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radical#1598C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2609C2semidirect
2. #2040C2semidirect
3. #1610C2semidirect
5. #1346C2semidirect
6. #1345C2semidirect
7. #1324C2semidirect
114. C42D8semidirect

Maximal subgroups

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

subgroupindex
#26092
#20402
#16102
#15982
#13462
#13452
#13242