Finite Group Databasebeta

← #24996

Subgroups — #24996

Order 256.

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

normal subgroups only

This group has 341 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#2582C211yesmaximal, characteristic
2Q8×C4⋊C4C211yesmaximal, characteristic
3#2100C212yesmaximal
4#2072C212yesmaximal
5#1984C211yesmaximal, characteristic
6#1947C211yesmaximal, characteristic
7#1928C212yesmaximal
8#1921C212yesmaximal
9#1834C211yesmaximal, characteristic
10#1803C211yesmaximal, characteristic
11C4×#103C211yesmaximal, characteristic
12C2×C4×Q8—21no—
13C2×C4×Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×#107C2212yes—
16C2×C4⋊C8C2211yescharacteristic
17C2×#103C2212yes—
18C2×#103C2211yescharacteristic
19C2×#103C2211yescharacteristic
20C2×C4×C8C2211yescharacteristic
21#398—22no—
22#391—21no—
23#391C2212yes—
24#389—21no—
25#389—21no—
26#389—22no—
27#389C2212yes—
28#384C2211yescharacteristic
29#384C2211yescharacteristic
30#384C2212yes—
31C4×C4⋊C4C2212yes—
32C4×C4⋊C4C2211yescharacteristic
33#340C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2211yescharacteristic
38#340C2211yescharacteristic
39#336C2212yes—
40#336C2212yes—
41C22×Q8D411yescharacteristic
42C22×Q8C2311yescharacteristic
43C22×C8C2311yescharacteristic
44C22×C8C2311yescharacteristic
45C22×C8C2312yes—
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—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4D411yescharacteristic
60C2×C4⋊C4D412yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C42—22no—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42C2312yes—
70C2×C42C2311yescharacteristic
71#107—24no—
72C4⋊C8—22no—
73#103—24no—
74#103—22no—
75#103—22no—
76C4×C8—22no—
77#95—22no—
78#95—21no—
79#95—22no—
80#95—22no—
81#95—22no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8Q1614yes—
90C2×Q8#4112yes—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4C2×D412yes—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D412yes—
100C22×C4C2×D411yescharacteristic
101C22×C4C2411yescharacteristic
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—22no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—21no—
110C2×C8—21no—
111C4⋊C4—24no—
112C4⋊C4—22no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—44no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—81no—
127C4⋊C4—81no—
128C4⋊C4—82no—
129C4⋊C4—82no—
130C4⋊C4#4112yes—
131C42—22no—
132C42—22no—
133C42—44no—
134C42—42no—
135C42—42no—
136C42#4112yes—
137C42#4112yes—
138C23C22×D411yescharacteristic
139Q8—28no—
140Q8—22no—
141Q8—22no—
142Q8—41no—
143Q8—41no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4#12312yes—
170C2×C4#12312yes—
171C2×C4C2×Q1614yes—
172C2×C4C2×Q1612yes—
173C2×C4AGL(1,ℤ/8)12yes—
174C2×C4C2×#4111yescharacteristic
175C2×C4C2×#4111yescharacteristic
176C2×C4C2×#4111yescharacteristic
177C2×C4C2×#4111yescharacteristic
178C2×C42−1+411yescharacteristic
179C2×C42−1+411yescharacteristic
180C8—42no—
181C8—41no—
182C8—41no—
183C22C2×#12311yescharacteristic
184C22#54811yescharacteristic
185C22#54811yescharacteristic
186C22C22×Q1611yescharacteristic
187C22C2×AGL(1,ℤ/8)11yescharacteristic
188C22#57711yescharacteristic
189C22#57711yescharacteristic
190C4—24no—
191C4—22no—
192C4—42no—
193C4—42no—
194C4—42no—
195C4—81no—
196C4—82no—
197C4—81no—
198C4—82no—
199C4#48414yes—
200C4#52911yescharacteristic
201C4#54111yescharacteristic
202C4#55211yescharacteristic
203C4#55611yescharacteristic
204C2#242111yescharacteristic
205C2C2×#48411yescharacteristic
206C2#309911yescharacteristic
207C2#336311yescharacteristic
208C2#336311yescharacteristic
209C2#336811yescharacteristic
210C2#337311yescharacteristic
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
#25822
Q8×C4⋊C42
#21002
#20722
#19842
#19472
#19282
#19212
#18342
#18032
C4×#1032