Finite Group Databasebeta

← #26282

Subgroups — #26282

Order 256.

341 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 139 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#2567C211yesmaximal, characteristic
2Q8×C4⋊C4C211yesmaximal, characteristic
3#2100C212yesmaximal
4#2083C212yesmaximal
5#1990C211yesmaximal, characteristic
6#1987C211yesmaximal, characteristic
7#1984C211yesmaximal, characteristic
8#1947C211yesmaximal, characteristic
9#1921C212yesmaximal
10#1921C212yesmaximal
11C4×#107C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8—21no—
14C2×C4×Q8C2211yescharacteristic
15C2×#107C2211yescharacteristic
16C2×#107C2212yes—
17C2×#107C2211yescharacteristic
18C2×C4⋊C8C2211yescharacteristic
19C2×C4×C8C2211yescharacteristic
20#395—22no—
21#391—21no—
22#389—21no—
23#389—22no—
24#389C2212yes—
25#389C2212yes—
26#384—21no—
27#384C2212yes—
28#384C2211yescharacteristic
29C4×C4⋊C4C2211yescharacteristic
30C4×C4⋊C4C2212yes—
31C4×C4⋊C4C2211yescharacteristic
32#340C2212yes—
33#340C2212yes—
34#340C2211yescharacteristic
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2211yescharacteristic
39#336C2212yes—
40#336C2212yes—
41C22×Q8—21no—
42C22×Q8D411yescharacteristic
43C22×C8C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2311yescharacteristic
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4×Q8—22no—
49C4×Q8—22no—
50C4×Q8—44no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4D412yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C42—22no—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42C2312yes—
70C2×C42C2311yescharacteristic
71#107—22no—
72#107—24no—
73#107Q814yescomplemented
74C4⋊C8—22no—
75C4×C8—22no—
76#95—22no—
77#95—21no—
78#95—21no—
79#95—22no—
80#95—22no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—42no—
84C2×Q8—81no—
85C2×Q8—81no—
86C2×Q8Q1614yes—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4C2×D412yes—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D412yes—
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—24no—
108C4⋊C4—22no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—44no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—81no—
122C4⋊C4—82no—
123C4⋊C4—82no—
124C4⋊C4—82no—
125C4⋊C4C2×Q812yes—
126C4⋊C4C2×Q812yes—
127C4⋊C4#4112yes—
128C42—22no—
129C42—22no—
130C42—44no—
131C42—42no—
132C42—42no—
133C42#4112yes—
134C42#4112yes—
135C23C22×D411yescharacteristic
136Q8—28no—
137Q8—84no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4#12312yes—
164C2×C4#12312yes—
165C2×C4C2×Q1614yes—
166C2×C4C2×Q1612yes—
167C2×C4AGL(1,ℤ/8)12yes—
168C2×C4C22×Q811yescharacteristic
169C2×C4C2×#4111yescharacteristic
170C2×C4C2×#4111yescharacteristic
171C2×C4C2×#4111yescharacteristic
172C2×C42−1+411yescharacteristic
173C2×C42−1+411yescharacteristic
174C8—22no—
175C8—41no—
176C8—42no—
177C22C2×#12311yescharacteristic
178C22#54811yescharacteristic
179C22D4×Q811yescharacteristic
180C22C22×Q1611yescharacteristic
181C22C2×AGL(1,ℤ/8)11yescharacteristic
182C22#57611yescharacteristic
183C22#57711yescharacteristic
184C4—24no—
185C4—22no—
186C4—42no—
187C4—42no—
188C4—42no—
189C4—42no—
190C4—82no—
191C4—82no—
192C4—81no—
193C4#48414yes—
194C4#53311yescharacteristic
195C4#54111yescharacteristic
196C4#55611yescharacteristic
197C4#55711yescharacteristic
198C2#244611yescharacteristic
199C2C2×#48411yescharacteristic
200C2#309611yescharacteristic
201C2#336311yescharacteristic
202C2#336611yescharacteristic
203C2Q8×Q1611yescharacteristic
204C2#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
73. #107Q8semidirect

Maximal subgroups

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

subgroupindex
#25672
Q8×C4⋊C42
#21002
#20832
#19902
#19872
#19842
#19472
#19212
C4×#1072