Finite Group Databasebeta

← #30098

Subgroups — #30098

Order 256.

304 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 113 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#2859C211yesmaximal, characteristic
2C4×#125C211yesmaximal, characteristic
3#2101C212yesmaximal
4#2100C211yesmaximal, characteristic
5#2100C211yesmaximal, characteristic
6#2084C211yesmaximal, characteristic
7#2083C211yesmaximal, characteristic
8#2059C212yesmaximal
9#1919C212yesmaximal
10#1918C212yesmaximal
11#1839C211yesmaximal, characteristic
12C2×#125—21no—
13C2×#125C2211yescharacteristic
14C2×#107C2212yes—
15C2×#106C2212yes—
16#389—21no—
17#389—21no—
18#389C2211yescharacteristic
19#389C2211yescharacteristic
20#389C2212yes—
21#389C2212yes—
22#387—21no—
23#387—21no—
24#384C2211yescharacteristic
25#384C2211yescharacteristic
26#382—21no—
27#382C2211yescharacteristic
28C4×C4⋊C4—21no—
29C4×C4⋊C4—21no—
30C4×C4⋊C4C2211yescharacteristic
31C4×C4⋊C4C2211yescharacteristic
32C43C2211yescharacteristic
33#340C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#336C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×C8C2312yes—
44C22×C8C2312yes—
45#125—22no—
46#125—22no—
47#125—44no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C42—21no—
65C2×C42—21no—
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42D411yescharacteristic
70C2×C42C2311yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#107—24no—
74#106—24no—
75#95—21no—
76#95—21no—
77#95—21no—
78#95—21no—
79#95—21no—
80#95—21no—
81#95—22no—
82#95—22no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2411yescharacteristic
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—22no—
102C4⋊C4—24no—
103C4⋊C4—24no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—24no—
111C4⋊C4—42no—
112C4⋊C4—42no—
113C4⋊C4—42no—
114C4⋊C4—44no—
115C4⋊C4—42no—
116C4⋊C4—42no—
117C4⋊C4—44no—
118C4⋊C4—42no—
119C4⋊C4#4112yes—
120C4⋊C4#4112yes—
121C42—22no—
122C42—22no—
123C42—22no—
124C42—21no—
125C42—21no—
126C42—22no—
127C42—22no—
128C42—41no—
129C42—42no—
130C42—42no—
131C23C22×D411yescharacteristic
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4#13512yes—
166C2×C4AGL(1,ℤ/8)12yes—
167C2×C4#13712yes—
168C2×C4C2×#4111yescharacteristic
169C2×C4C2×#4111yescharacteristic
170C2×C42−1+412yes—
171C2×C42−1+412yes—
172C8—42no—
173C8—42no—
174C22#52511yescharacteristic
175C22#53611yescharacteristic
176C22#53611yescharacteristic
177C22C2×#13511yescharacteristic
178C22C2×AGL(1,ℤ/8)11yescharacteristic
179C22C2×#13711yescharacteristic
180C22#57511yescharacteristic
181C4—22no—
182C4—22no—
183C4—22no—
184C4—41no—
185C4—42no—
186C4—41no—
187C4—42no—
188C4—41no—
189C4—82no—
190C4—81no—
191C4—82no—
192C4—81no—
193C4#54112yes—
194C4#54112yes—
195C2#263411yescharacteristic
196C2#310611yescharacteristic
197C2#311611yescharacteristic
198C2#323111yescharacteristic
199C2#323211yescharacteristic
200C2#323411yescharacteristic
201C2#323911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#28592
C4×#1252
#21012
#21002
#20842
#20832
#20592
#19192
#19182
#18392