Finite Group Databasebeta

← #29939

Subgroups — #29939

Order 256.

370 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 129 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#2857C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic
3#2100C212yesmaximal, complemented
4#2083C212yesmaximal, complemented
5#2076C214yesmaximal
6#1987C212yesmaximal, complemented
7#1947C212yesmaximal, complemented
8#1797C211yesmaximal, characteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C22⋊Q8—21no—
11C2×#117—21no—
12C22×C42C2211yescharacteristic
13C2×#107C2214yes—
14C2×C4⋊C8C2212yes—
15C2×C22⋊C8C2212yes—
16#391—21no—
17#389—21no—
18#389C2212yes—
19#384C2211yescharacteristic
20#384C2211yescharacteristic
21#384C2211yescharacteristic
22#382—21no—
23#380—21no—
24#380—21no—
25#380C2212yes—
26#380C2212yes—
27C4×C4⋊C4—21no—
28C4×C4⋊C4C2212yes—
29#340C2214yes—
30#340C2214yes—
31#340C2214yes—
32#336C2212yes—
33C22×Q8—21no—
34C23×C4D411yescharacteristic
35C23×C4D411yescharacteristic
36C23×C4C2311yescharacteristic
37C22×C8C2314yes—
38C4⋊Q8—24no—
39C22⋊Q8—44no—
40#117—44no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4C2312yes—
50C2×C4⋊C4C2312yes—
51C2×C4⋊C4C2312yes—
52C2×C4⋊C4C2312yes—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—22no—
55C2×C42—24no—
56C2×C42—22no—
57C2×C42D411yescharacteristic
58C2×C42D411yescharacteristic
59C2×C42C2311yescharacteristic
60C2×C42C2311yescharacteristic
61#107—44no—
62C4⋊C8—24no—
63C22⋊C8—42no—
64#95—22no—
65#95—22no—
66#95—22no—
67#95—22no—
68C24C2×D411yescharacteristic
69C2×Q8—42no—
70C2×Q8—42no—
71C2×Q8—42no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—21no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—41no—
85C22×C4Q1614yes—
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2411yescharacteristic
92C2×C8—24no—
93C2×C8—24no—
94C2×C8—24no—
95C4⋊C4—24no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—24no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—42no—
102C4⋊C4—42no—
103C4⋊C4—42no—
104C4⋊C4—42no—
105C4⋊C4—42no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—44no—
109C22⋊C4—44no—
110C22⋊C4—44no—
111C42—22no—
112C42—22no—
113C42—22no—
114C42—44no—
115C42#4114yes—
116C23—21no—
117C23—21no—
118C23—21no—
119C23—21no—
120C23—21no—
121C23—21no—
122C23C2×Q1612yes—
123C23C22×D411yescharacteristic
124Q8—84no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—28no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—44no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4#12312yes—
151C2×C4#12312yes—
152C2×C4C2×Q1612yes—
153C2×C4C2×Q1612yes—
154C2×C4AGL(1,ℤ/8)12yes—
155C2×C4AGL(1,ℤ/8)12yes—
156C2×C4C2×#4112yes—
157C2×C4H(5,𝔽2)11yescharacteristic
158C2×C42−1+411yescharacteristic
159C2×C42−1+412yes—
160C8—44no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22C2×#12311yescharacteristic
169C22#53611yescharacteristic
170C22#53711yescharacteristic
171C22C22×Q1611yescharacteristic
172C22C2×AGL(1,ℤ/8)11yescharacteristic
173C22C2×AGL(1,ℤ/8)11yescharacteristic
174C22#57511yescharacteristic
175C4—22no—
176C4—22no—
177C4—22no—
178C4—41no—
179C4—82no—
180C4—82no—
181C4—82no—
182C4—82no—
183C4#48414yes—
184C4#54112yes—
185C4#54212yes—
186C2—42no—
187C2#262511yescharacteristic
188C2C2×#48411yescharacteristic
189C2#309411yescharacteristic
190C2#323011yescharacteristic
191C2#323511yescharacteristic
192C2#324911yescharacteristic
193C2#325311yescharacteristic
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. #2083C2semidirect
6. #1987C2semidirect
7. #1947C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#22852
#21002
#20832
#20762
#19872
#19472
#17972