Finite Group Databasebeta

← #27614

Subgroups — #27614

Order 256.

408 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#389C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic
3#2100C212yesmaximal, complemented
4#2083C212yesmaximal, complemented
5#1921C214yesmaximal, complemented
6#1852C212yesmaximal
7#1824C212yesmaximal
8#1817C211yesmaximal, characteristic
9C2×#117—21no—
10C22×C4⋊C4—21no—
11C22×C4⋊C4C2211yescharacteristic
12C22×C4⋊C4C2211yescharacteristic
13C22×C42C2211yescharacteristic
14C2×#107C2214yes—
15C2×C22⋊C8C2212yes—
16#389—21no—
17#389—21no—
18#389C2212yes—
19#389C2212yes—
20#384C2211yescharacteristic
21#384C2211yescharacteristic
22#382—21no—
23#380—21no—
24#380C2212yes—
25C4×C4⋊C4—21no—
26C4×C4⋊C4C2212yes—
27C2×#95—21no—
28#340C2214yes—
29#340C2214yes—
30#340C2214yes—
31#336C2212yes—
32#336C2212yes—
33C23×C4—21no—
34C23×C4—21no—
35C23×C4D411yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4C2311yescharacteristic
38C22×C8C2314yes—
39#117—44no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—24no—
42C2×C4⋊C4—24no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—44no—
51C2×C4⋊C4C2312yes—
52C2×C4⋊C4C2312yes—
53C2×C4⋊C4C2312yes—
54C2×C4⋊C4C2312yes—
55C2×C22⋊C4—22no—
56C2×C42—24no—
57C2×C42—22no—
58C2×C42D411yescharacteristic
59C2×C42D411yescharacteristic
60C2×C42C2311yescharacteristic
61C2×C42C2311yescharacteristic
62#107—28no—
63C22⋊C8—24no—
64#95—22no—
65#95—22no—
66#95—22no—
67#95—22no—
68C24C2×D411yescharacteristic
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—22no—
72C22×C4—22no—
73C22×C4—24no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—41no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4Q1614yes—
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4#4114yes—
93C22×C4C2411yescharacteristic
94C2×C8—24no—
95C2×C8—24no—
96C2×C8—24no—
97C4⋊C4—24no—
98C4⋊C4—22no—
99C4⋊C4—24no—
100C4⋊C4—24no—
101C4⋊C4—24no—
102C4⋊C4—42no—
103C4⋊C4—44no—
104C4⋊C4—42no—
105C4⋊C4—42no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—44no—
109C4⋊C4#4114yes—
110C22⋊C4—44no—
111C42—22no—
112C42—22no—
113C42—22no—
114C42—22no—
115C42—44no—
116C23—21no—
117C23—21no—
118C23—21no—
119C23—21no—
120C23#12312yes—
121C23C2×Q1612yes—
122C23AGL(1,ℤ/8)12yes—
123C23C22×D411yescharacteristic
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—28no—
133C2×C4—28no—
134C2×C4—21no—
135C2×C4—24no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—44no—
139C2×C4—48no—
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—42no—
151C2×C4—84no—
152C2×C4#12312yes—
153C2×C4C2×Q1612yes—
154C2×C4C2×Q1612yes—
155C2×C4AGL(1,ℤ/8)12yes—
156C2×C4C2×#4112yes—
157C2×C4C2×#4112yes—
158C2×C42−1+412yes—
159C8—44no—
160C22—22no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22#48414yes—
167C22C2×#12311yescharacteristic
168C22#52511yescharacteristic
169C22#53611yescharacteristic
170C22C22×Q1611yescharacteristic
171C22C2×AGL(1,ℤ/8)11yescharacteristic
172C22C2×AGL(1,ℤ/8)11yescharacteristic
173C22#57511yescharacteristic
174C4—22no—
175C4—22no—
176C4—22no—
177C4—22no—
178C4—41no—
179C4—44no—
180C4—82no—
181C4—82no—
182C4—82no—
183C4#55214yes—
184C2—24no—
185C2#248411yescharacteristic
186C2C2×#48411yescharacteristic
187C2#309411yescharacteristic
188C2#310411yescharacteristic
189C2#310911yescharacteristic
190C2#323011yescharacteristic
191C2#323411yescharacteristic
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
5. #1921C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3892
#22852
#21002
#20832
#19212
#18522
#18242
#18172