Finite Group Databasebeta

← #12196

Subgroups — #12196

Order 256.

217 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 35 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#2593C211yesmaximal, characteristic, complemented
2#2006C211yesmaximal, characteristic
3#1616C211yesmaximal, characteristic, complemented
4#1610C211yesmaximal, characteristic, complemented
5#1353C211yesmaximal, characteristic, complemented
6#1346C211yesmaximal, characteristic, complemented
7#1318C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—21no—
9#537C2211yescharacteristic
10#481—21no—
11#480—21no—
12C2×#104—21no—
13#400C2211yescharacteristic
14#398C2211yescharacteristic
15#394—21no—
16#390C2211yescharacteristic
17#388—21no—
18#388—21no—
19#388—21no—
20#388—21no—
21C4×C4⋊C4—21no—
22#343C2211yescharacteristic
23#327—21no—
24#327—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C2×#41—21no—
28C22×D4D411yescharacteristic
29AGL(1,ℤ/8)—42no—
30AGL(1,ℤ/8)—42no—
31C2×SD16—41no—
32C2×D8—41no—
33C2×M16—21no—
34C22⋊Q8—41no—
35#121—21no—
36#121—21no—
37#121—41no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4D411yescharacteristic
41C2×C4⋊C4D411yescharacteristic
42C2×C4⋊C4C2311yescharacteristic
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C42—21no—
50C2×C42—21no—
51C2×C42D411yescharacteristic
52#107—41no—
53#106—41no—
54#104—42no—
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—21no—
75C2×D4—21no—
76C2×D4—41no—
77C2×D4—41no—
78C2×D4—41no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—41no—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88SD16—44no—
89D8—44no—
90M16—22no—
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—41no—
102C4⋊C4—41no—
103C22⋊C4—42no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C22⋊C4—42no—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C22⋊C4—81no—
110C22⋊C4—81no—
111C42—41no—
112C42—42no—
113C42—81no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23#12011yescharacteristic
123Q8—41no—
124Q8—41no—
125D4—41no—
126D4—42no—
127D4—41no—
128D4—81no—
129D4—81no—
130D4—81no—
131D4—81no—
132D4—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
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—81no—
156C2×C4—81no—
157C2×C4#13511yescharacteristic
158C2×C4AGL(1,ℤ/8)11yescharacteristic
159C8—42no—
160C8—81no—
161C8—81no—
162C22—21no—
163C22—21no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22#44911yescharacteristic
175C22#45311yescharacteristic
176C22B(4,𝔽2)11yescharacteristic
177C4—21no—
178C4—21no—
179C4—42no—
180C4—42no—
181C4—41no—
182C4—41no—
183C4—81no—
184C4—81no—
185C4—81no—
186C4—81no—
187C2—21no—
188C2—21no—
189C2—161no—
190C2—161no—
191C2#160011yescharacteristic
192C2#219811yescharacteristic
193C2#220411yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radical#2006C2
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. #2593C2semidirect
3. #1616C2semidirect
4. #1610C2semidirect
5. #1353C2semidirect
6. #1346C2semidirect
7. #1318C2semidirect

Maximal subgroups

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

subgroupindex
#25932
#20062
#16162
#16102
#13532
#13462
#13182