Finite Group Databasebeta

← #32631

Subgroups — #32631

Order 256.

324 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 117 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#2568C211yesmaximal, characteristic
2#2567C211yesmaximal, characteristic
3#2068C212yesmaximal
4#2043C212yesmaximal
5#2024C212yesmaximal
6#1984C211yesmaximal, characteristic
7#1933C212yesmaximal
8#1928C212yesmaximal
9#1921C212yesmaximal
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4×Q8C2211yescharacteristic
13C2×#107C2212yes—
14C2×C4⋊C8C2211yescharacteristic
15C2×#103C2212yes—
16C2×#103C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×C4×C8C2211yescharacteristic
20#400—21no—
21#400—21no—
22#395—21no—
23#395—21no—
24#391—21no—
25#391—21no—
26#391C2212yes—
27#391C2212yes—
28#389C2212yes—
29#389C2212yes—
30#384—21no—
31#382—21no—
32C4×C4⋊C4C2211yescharacteristic
33C4×C4⋊C4C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#336C2212yes—
38#336C2212yes—
39C22×Q8—21no—
40C22×Q8D411yescharacteristic
41C22×Q8C2312yes—
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—41no—
45C4⋊Q8—41no—
46C4⋊Q8—41no—
47C4⋊Q8—41no—
48C4×Q8—41no—
49C4×Q8—41no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4D411yescharacteristic
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C42—21no—
62C2×C42—21no—
63C2×C42C2311yescharacteristic
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66#107—24no—
67C4⋊C8—41no—
68#103—24no—
69#103—24no—
70#103—42no—
71#103—42no—
72C4×C8—41no—
73#95—21no—
74#95—21no—
75#95—22no—
76#95—22no—
77#95—22no—
78#95—22no—
79#95—22no—
80#95—22no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—41no—
88C2×Q8—41no—
89C2×Q8—42no—
90C2×Q8—82no—
91C2×Q8Q1614yes—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2411yescharacteristic
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—82no—
121C4⋊C4—82no—
122C4⋊C4#4114yes—
123C42—21no—
124C42—21no—
125C42—21no—
126C42—21no—
127C42—21no—
128C42—21no—
129C42—42no—
130C42—42no—
131C23C22×D411yescharacteristic
132Q8—44no—
133Q8—42no—
134Q8—42no—
135Q8—42no—
136Q8—42no—
137Q8—84no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4C2×Q1612yes—
163C2×C4C2×Q1612yes—
164C2×C4C2×Q1612yes—
165C2×C4C2×#4112yes—
166C2×C4H(5,𝔽2)11yescharacteristic
167C2×C4H(5,𝔽2)11yescharacteristic
168C2×C42−1+412yes—
169C8—42no—
170C8—42no—
171C22#53411yescharacteristic
172C22#54812yes—
173C22C22×Q1611yescharacteristic
174C22#57511yescharacteristic
175C22#57712yes—
176C4—22no—
177C4—22no—
178C4—22no—
179C4—44no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—82no—
184C4—82no—
185C4—82no—
186C4#54212yes—
187C4#55511yescharacteristic
188C4#56311yescharacteristic
189C2#273111yescharacteristic
190C2#318911yescharacteristic
191C2#320011yescharacteristic
192C2#336312yes—
193C2#337612yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ 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
#25682
#25672
#20682
#20432
#20242
#19842
#19332
#19282
#19212