Finite Group Databasebeta

← #12211

Subgroups — #12211

Order 256.

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

normal subgroups only

This group has 232 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#2699C211yesmaximal, characteristic
2#2027C211yesmaximal, characteristic, complemented
3#1611C211yesmaximal, characteristic, complemented
4#1599C211yesmaximal, characteristic, complemented
5#1346C211yesmaximal, characteristic, complemented
6#1345C211yesmaximal, characteristic, complemented
7#1318C211yesmaximal, characteristic, complemented
8#544C2211yescharacteristic
9C2×#121—21no—
10#483—21no—
11#450—21no—
12#418—21no—
13#398C2211yescharacteristic
14#394—21no—
15#394C2211yescharacteristic
16#392—21no—
17#390C2211yescharacteristic
18#388—21no—
19#386—21no—
20#381—21no—
21#343C2211yescharacteristic
22#339—21no—
23#328—21no—
24#328—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C22×D4—21no—
28C22×D4—21no—
29C22×D4D411yescharacteristic
30C22×D4D411yescharacteristic
31C23×C4—21no—
32C2×SD16—41no—
33C2×M16—21no—
34C4⋊Q8—41no—
35#125—41no—
36C22⋊Q8—21no—
37C22⋊Q8—21no—
38#121—42no—
39#121—42no—
40#117—21no—
41C2×C4⋊C4—21no—
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×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C42D411yescharacteristic
52#106—41no—
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#95D411yescharacteristic
63#95D411yescharacteristic
64C24—21no—
65C24—21no—
66C2×Q8—21no—
67C2×Q8—21no—
68C2×D4—21no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—41no—
72C2×D4—42no—
73C2×D4—42no—
74C2×D4—42no—
75C2×D4—81no—
76C2×D4—81no—
77C2×D4—81no—
78C2×D4—81no—
79C2×D4SD1612yescomplemented
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—81no—
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90SD16—82no—
91M16—41no—
92M16—41no—
93C2×C8—41no—
94C2×C8—41no—
95C2×C8—41no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—81no—
112C22⋊C4—81no—
113C22⋊C4—81no—
114C22⋊C4—81no—
115C42—41no—
116C42—41no—
117C23—22no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23#12011yescharacteristic
132Q8—81no—
133Q8—81no—
134D4—42no—
135D4—42no—
136D4—81no—
137D4—84no—
138D4—81no—
139D4—84no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—21no—
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—41no—
160C2×C4—84no—
161C2×C4C2×SD1611yescharacteristic
162C2×C4AGL(1,ℤ/8)11yescharacteristic
163C8—81no—
164C8—81no—
165C8—81no—
166C22—21no—
167C22—21no—
168C22—42no—
169C22—42no—
170C22—42no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22#45011yescharacteristic
183C22#45511yescharacteristic
184C22B(4,𝔽2)11yescharacteristic
185C4—21no—
186C4—21no—
187C4—42no—
188C4—41no—
189C4—41no—
190C4—81no—
191C4—81no—
192C4—81no—
193C4—81no—
194C4—81no—
195C2—21no—
196C2—21no—
197C2—82no—
198C2—161no—
199C2#160111yescharacteristic
200C2#220011yescharacteristic
201C2#220411yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radical#2699C2
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
2. #2027C2semidirect
3. #1611C2semidirect
4. #1599C2semidirect
5. #1346C2semidirect
6. #1345C2semidirect
7. #1318C2semidirect
79. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#26992
#20272
#16112
#15992
#13462
#13452
#13182