Finite Group Databasebeta

← #26283

Subgroups — #26283

Order 256.

329 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 131 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#2546C211yesmaximal, characteristic
2#2355C211yesmaximal, characteristic
3#2101C212yesmaximal
4#2085C212yesmaximal
5#1990C211yesmaximal, characteristic
6#1988C211yesmaximal, characteristic
7#1984C211yesmaximal, characteristic
8#1948C211yesmaximal, characteristic
9#1921C212yesmaximal
10#1920C212yesmaximal
11C4×#107C211yesmaximal, characteristic
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×#107C2211yescharacteristic
16C2×#107C2211yescharacteristic
17C2×#106C2212yes—
18C2×C4⋊C8C2211yescharacteristic
19C2×C4×C8C2211yescharacteristic
20#398—22no—
21#391—21no—
22#391—21no—
23#389C2212yes—
24#389C2212yes—
25#387—22no—
26#384C2211yescharacteristic
27#382C2212yes—
28C4×C4⋊C4—21no—
29C4×C4⋊C4C2212yes—
30C4×C4⋊C4C2211yescharacteristic
31C4×C4⋊C4C2211yescharacteristic
32#340C2212yes—
33#340C2212yes—
34#340C2211yescharacteristic
35#340C2212yes—
36#340C2212yes—
37#340C2211yescharacteristic
38#340C2212yes—
39#336C2212yes—
40#336C2212yes—
41C22×Q8—21no—
42C22×Q8D411yescharacteristic
43C22×C8C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2311yescharacteristic
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4⋊Q8—44no—
49C4×Q8—22no—
50C4×Q8—22no—
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⋊C4D412yes—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C42—21no—
65C2×C42—21no—
66C2×C42—22no—
67C2×C42C2312yes—
68C2×C42C2311yescharacteristic
69#107—22no—
70#107Q814yescomplemented
71#106—24no—
72C4⋊C8—22no—
73C4×C8—22no—
74#95—22no—
75#95—21no—
76#95—22no—
77#95—21no—
78#95—22no—
79#95—22no—
80#95—22no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—42no—
85C2×Q8—42no—
86C2×Q8—42no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4C2×D412yes—
94C22×C4C2×D412yes—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2411yescharacteristic
98C2×C8—21no—
99C2×C8—21no—
100C2×C8—21no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—22no—
106C2×C8C2×Q812yes—
107C4⋊C4—22no—
108C4⋊C4—24no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—44no—
119C4⋊C4—44no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—81no—
123C4⋊C4C2×Q812yes—
124C4⋊C4C2×Q812yes—
125C4⋊C4#4112yes—
126C42—22no—
127C42—22no—
128C42—42no—
129C42—42no—
130C42—82no—
131C42#4112yes—
132C42#4112yes—
133C23C22×D411yescharacteristic
134Q8—44no—
135Q8—84no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4#12312yes—
163C2×C4#12312yes—
164C2×C4#13512yes—
165C2×C4AGL(1,ℤ/8)12yes—
166C2×C4C22×Q811yescharacteristic
167C2×C4C2×#4111yescharacteristic
168C2×C4C2×#4111yescharacteristic
169C2×C4C2×#4111yescharacteristic
170C2×C42−1+411yescharacteristic
171C2×C42−1+411yescharacteristic
172C8—22no—
173C8—41no—
174C8—42no—
175C22C2×#12311yescharacteristic
176C22#54811yescharacteristic
177C22D4×Q811yescharacteristic
178C22C2×#13511yescharacteristic
179C22C2×AGL(1,ℤ/8)11yescharacteristic
180C22#57711yescharacteristic
181C22#57811yescharacteristic
182C4—22no—
183C4—42no—
184C4—42no—
185C4—42no—
186C4—42no—
187C4—42no—
188C4—82no—
189C4—82no—
190C4—81no—
191C4#48214yes—
192C4#53311yescharacteristic
193C4#54111yescharacteristic
194C4#55611yescharacteristic
195C4#55711yescharacteristic
196C2#244611yescharacteristic
197C2C2×#48211yescharacteristic
198C2#309711yescharacteristic
199C2#336511yescharacteristic
200C2#336711yescharacteristic
201C2#338311yescharacteristic
202C2#338411yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
70. #107Q8semidirect

Maximal subgroups

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

subgroupindex
#25462
#23552
#21012
#20852
#19902
#19882
#19842
#19482
#19212
#19202
C4×#1072