Finite Group Databasebeta

← #29909

Subgroups — #29909

Order 256.

411 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 131 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#2855C211yesmaximal, characteristic
2C2×#384C211yesmaximal, characteristic
3#2100C212yesmaximal, complemented
4#2095C212yesmaximal, complemented
5#2083C212yesmaximal, complemented
6#2072C212yesmaximal, complemented
7#1872C212yesmaximal
8#1852C212yesmaximal
9#1817C211yesmaximal, characteristic
10C2×C22⋊Q8C2211yescharacteristic
11C22×C4⋊C4—21no—
12C22×C4⋊C4—21no—
13C22×C4⋊C4C2212yes—
14C22×C42C2211yescharacteristic
15C2×#107C2214yes—
16C2×#103C2214yes—
17C2×C22⋊C8C2212yes—
18#391C2211yescharacteristic
19#391C2211yescharacteristic
20#389C2211yescharacteristic
21#389C2211yescharacteristic
22#387—21no—
23#387—21no—
24#385—21no—
25#384—21no—
26#384—21no—
27#384C2212yes—
28#384C2212yes—
29#381—21no—
30#380C2211yescharacteristic
31#340C2214yes—
32#340C2214yes—
33#336C2212yes—
34#336C2212yes—
35C22×Q8C2311yescharacteristic
36C23×C4—22no—
37C23×C4D411yescharacteristic
38C23×C4D411yescharacteristic
39C23×C4C2311yescharacteristic
40C22×C8C2314yes—
41C22⋊Q8—24no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—44no—
51C2×C4⋊C4—44no—
52C2×C4⋊C4—44no—
53C2×C4⋊C4C2314yes—
54C2×C4⋊C4C2311yescharacteristic
55C2×C4⋊C4C2311yescharacteristic
56C2×C4⋊C4C2311yescharacteristic
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C42—24no—
60C2×C42D411yescharacteristic
61C2×C42D411yescharacteristic
62C2×C42C2311yescharacteristic
63C2×C42C2311yescharacteristic
64#107—44no—
65#103—44no—
66C22⋊C8—24no—
67#95—21no—
68#95—21no—
69#95—21no—
70#95—21no—
71#95—21no—
72#95—21no—
73#95—21no—
74C24C2×D411yescharacteristic
75C2×Q8—22no—
76C2×Q8—22no—
77C2×Q8—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—24no—
90C22×C4—44no—
91C22×C4—44no—
92C22×C4—44no—
93C22×C4Q1614yes—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4#4114yes—
100C22×C4C2411yescharacteristic
101C2×C8—24no—
102C2×C8—24no—
103C2×C8—24no—
104C4⋊C4—22no—
105C4⋊C4—24no—
106C4⋊C4—22no—
107C4⋊C4—24no—
108C4⋊C4—22no—
109C4⋊C4—44no—
110C4⋊C4—44no—
111C4⋊C4—44no—
112C4⋊C4—44no—
113C4⋊C4—42no—
114C4⋊C4—42no—
115C4⋊C4—44no—
116C4⋊C4—42no—
117C22⋊C4—42no—
118C22⋊C4—42no—
119C42—22no—
120C42—22no—
121C42—22no—
122C42—22no—
123C23—21no—
124C23—21no—
125C23—21no—
126C23—21no—
127C23#12312yes—
128C23C2×Q1612yes—
129C23AGL(1,ℤ/8)12yes—
130C23C22×D411yescharacteristic
131Q8—44no—
132C2×C4—28no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—28no—
144C2×C4—28no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—44no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—44no—
155C2×C4—44no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—88no—
161C2×C4#12312yes—
162C2×C4C2×Q1612yes—
163C2×C4C2×Q1612yes—
164C2×C4AGL(1,ℤ/8)12yes—
165C2×C4#13712yes—
166C2×C4C2×#4112yes—
167C2×C4H(5,𝔽2)12yes—
168C2×C42−1+412yes—
169C8—44no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—22no—
174C22—22no—
175C22—22no—
176C22#48414yes—
177C22C2×#12311yescharacteristic
178C22#53511yescharacteristic
179C22#53611yescharacteristic
180C22C22×Q1611yescharacteristic
181C22C2×AGL(1,ℤ/8)11yescharacteristic
182C22C2×AGL(1,ℤ/8)11yescharacteristic
183C22C2×#13711yescharacteristic
184C4—22no—
185C4—22no—
186C4—22no—
187C4—22no—
188C4—22no—
189C4—81no—
190C4—81no—
191C4—81no—
192C4—84no—
193C4—81no—
194C4#53914yes—
195C2—24no—
196C2#262311yescharacteristic
197C2C2×#48411yescharacteristic
198C2#309211yescharacteristic
199C2#320911yescharacteristic
200C2#321711yescharacteristic
201C2#323011yescharacteristic
202C2#323811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2100C2semidirect
4. #2095C2semidirect
5. #2083C2semidirect
6. #2072C2semidirect

Maximal subgroups

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

subgroupindex
#28552
C2×#3842
#21002
#20952
#20832
#20722
#18722
#18522
#18172