Finite Group Databasebeta

← #32644

Subgroups — #32644

Order 256.

331 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 111 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#2546C211yesmaximal, characteristic
3#2068C212yesmaximal
4#2043C212yesmaximal
5#2024C212yesmaximal
6#1985C211yesmaximal, characteristic
7#1935C212yesmaximal
8#1928C212yesmaximal
9#1918C212yesmaximal
10C2×C4⋊Q8—21no—
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C2×#107C2212yes—
15C2×C4⋊C8C2211yescharacteristic
16C2×#103C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#97C2211yescharacteristic
21#400—21no—
22#400—21no—
23#398—21no—
24#398—21no—
25#391—21no—
26#391—21no—
27#391C2212yes—
28#391C2212yes—
29#389C2212yes—
30#389C2212yes—
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—44no—
49C4×Q8—41no—
50C4×Q8—41no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C42—21no—
63C2×C42—21no—
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66C2×C42C2311yescharacteristic
67#107—24no—
68C4⋊C8—41no—
69#103—24no—
70#103—24no—
71#103—42no—
72#103—42no—
73#97—41no—
74#95—21no—
75#95—21no—
76#95—22no—
77#95—22no—
78#95—22no—
79#95—22no—
80#95—22no—
81#95—22no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—41no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8—42no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2411yescharacteristic
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—44no—
123C4⋊C4—44no—
124C4⋊C4#4114yes—
125C42—21no—
126C42—21no—
127C42—21no—
128C42—21no—
129C42—21no—
130C42—21no—
131C42—42no—
132C42—42no—
133C23C22×D411yescharacteristic
134Q8—44no—
135Q8—42no—
136Q8—42no—
137Q8—42no—
138Q8—42no—
139Q8—84no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—24no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
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×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4AGL(1,ℤ/8)12yes—
166C2×C4#13712yes—
167C2×C4C2×#4112yes—
168C2×C4H(5,𝔽2)11yescharacteristic
169C2×C4H(5,𝔽2)11yescharacteristic
170C2×C42−1+412yes—
171C8—42no—
172C8—42no—
173C22#53411yescharacteristic
174C22#54812yes—
175C22C2×AGL(1,ℤ/8)11yescharacteristic
176C22C2×#13711yescharacteristic
177C22#57812yes—
178C4—22no—
179C4—22no—
180C4—44no—
181C4—41no—
182C4—41no—
183C4—41no—
184C4—41no—
185C4—82no—
186C4—82no—
187C4—82no—
188C4#54212yes—
189C4#55511yescharacteristic
190C4#56311yescharacteristic
191C2#273111yescharacteristic
192C2#319311yescharacteristic
193C2#320211yescharacteristic
194C2#336712yes—
195C2#337412yes—
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
#25462
#20682
#20432
#20242
#19852
#19352
#19282
#19182