Finite Group Databasebeta

← #12129

Subgroups — #12129

Order 256.

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

normal subgroups only

This group has 228 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#2784C211yesmaximal, characteristic, complemented
2#2006C211yesmaximal, characteristic
3#1616C211yesmaximal, characteristic, complemented
4#1604C211yesmaximal, characteristic, complemented
5#1354C211yesmaximal, characteristic, complemented
6#1353C211yesmaximal, characteristic, complemented
7#1318C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—21no—
9#537C2211yescharacteristic
10#481—21no—
11#448—21no—
12C2×#104—21no—
13#400C2211yescharacteristic
14#396—21no—
15#395C2211yescharacteristic
16#393—21no—
17#390C2211yescharacteristic
18#387—21no—
19#386—21no—
20#385—21no—
21#380—21no—
22#343C2211yescharacteristic
23#327—21no—
24#327—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C2×#41—21no—
28C22×Q8D411yescharacteristic
29C22×D4D411yescharacteristic
30C23×C4—21no—
31AGL(1,ℤ/8)—42no—
32AGL(1,ℤ/8)—42no—
33C2×SD16—41no—
34C2×SD16—41no—
35C2×D8—41no—
36C2×M16—21no—
37C22⋊Q8—41no—
38#121—21no—
39#121—21no—
40#121—41no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4C2311yescharacteristic
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C42D411yescharacteristic
52#106—41no—
53#104—42no—
54#103—41no—
55#102—41no—
56C22⋊C8—21no—
57C22⋊C8—21no—
58#95—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×Q8—41no—
72C2×Q8—81no—
73C2×D4—21no—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—41no—
78C2×D4—41no—
79C2×D4—41no—
80C2×D4SD1612yescomplemented
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—41no—
87C22×C4—42no—
88C22×C4—81no—
89C22×C4—81no—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93SD16—44no—
94SD16—82no—
95D8—44no—
96M16—22no—
97M16—41no—
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—42no—
106C4⋊C4—81no—
107C4⋊C4—81no—
108C22⋊C4—42no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—81no—
112C22⋊C4—81no—
113C22⋊C4—81no—
114C22⋊C4—81no—
115C42—41no—
116C23—22no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23#12011yescharacteristic
125Q8—41no—
126Q8—41no—
127Q8—42no—
128Q8—81no—
129D4—42no—
130D4—41no—
131D4—41no—
132D4—81no—
133D4—81no—
134D4—81no—
135D4—81no—
136D4—81no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—21no—
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—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4—84no—
162C2×C4C2×SD1611yescharacteristic
163C2×C4#13711yescharacteristic
164C8—42no—
165C8—81no—
166C8—81no—
167C22—21no—
168C22—21no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22#44811yescharacteristic
180C22#45311yescharacteristic
181C22B(4,𝔽2)11yescharacteristic
182C4—21no—
183C4—21no—
184C4—42no—
185C4—41no—
186C4—41no—
187C4—81no—
188C4—81no—
189C4—81no—
190C4—81no—
191C4—81no—
192C2—21no—
193C2—21no—
194C2—82no—
195C2—161no—
196C2#159611yescharacteristic
197C2#220011yescharacteristic
198C2#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. #2784C2semidirect
3. #1616C2semidirect
4. #1604C2semidirect
5. #1354C2semidirect
6. #1353C2semidirect
7. #1318C2semidirect
80. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#27842
#20062
#16162
#16042
#13542
#13532
#13182