Finite Group Databasebeta

← #26027

Subgroups — #26027

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#2552C211yesmaximal, characteristic
2#2355C211yesmaximal, characteristic
3#2066C212yesmaximal
4#2026C212yesmaximal
5#1989C212yesmaximal
6#1948C211yesmaximal, characteristic
7#1939C211yesmaximal, characteristic
8#1920C212yesmaximal
9#1834C211yesmaximal, characteristic
10#1803C211yesmaximal, characteristic
11C4×#103C211yesmaximal, characteristic
12C2×#125C2212yes—
13C2×C4×Q8C2211yescharacteristic
14C2×#106C2212yes—
15C2×C4⋊C8C2211yescharacteristic
16C2×C4⋊C8C2212yes—
17C2×#103C2211yescharacteristic
18C2×#103C2211yescharacteristic
19C2×#103C2212yes—
20C2×C4×C8C2211yescharacteristic
21#398—22no—
22#391C2212yes—
23#389—21no—
24#389C2212yes—
25#387—22no—
26#384—21no—
27#384C2211yescharacteristic
28#384C2211yescharacteristic
29#382—21no—
30C4×C4⋊C4—21no—
31C4×C4⋊C4C2212yes—
32C4×C4⋊C4C2211yescharacteristic
33C4×C4⋊C4C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#336C2212yes—
41C22×Q8C2311yescharacteristic
42C22×C8C2311yescharacteristic
43C22×C8C2312yes—
44C22×C8C2311yescharacteristic
45#125—24no—
46#125—42no—
47C4×Q8—22no—
48C4×Q8Q814yes—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4C2312yes—
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#106—24no—
70C4⋊C8—22no—
71C4⋊C8—42no—
72#103—22no—
73#103—22no—
74#103—42no—
75C4×C8—22no—
76#95—22no—
77#95—22no—
78#95—21no—
79#95—22no—
80#95—22no—
81#95D412yes—
82C2×Q8—21no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8C2×Q812yes—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4C2×D412yes—
95C22×C4C2×D412yes—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2411yescharacteristic
99C2×C8—21no—
100C2×C8—21no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—21no—
107C2×C8—21no—
108C4⋊C4—24no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—44no—
119C4⋊C4—44no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—81no—
123C4⋊C4—81no—
124C4⋊C4—82no—
125C4⋊C4C2×Q812yes—
126C4⋊C4#4112yes—
127C4⋊C4#4112yes—
128C42—24no—
129C42—22no—
130C42—22no—
131C42—42no—
132C42—82no—
133C42C2×Q812yes—
134C42#4112yes—
135C23C22×D411yescharacteristic
136Q8—22no—
137Q8—22no—
138Q8—41no—
139Q8—41no—
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—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4C22⋊Q812yes—
167C2×C4C22⋊Q812yes—
168C2×C4#13512yes—
169C2×C4#13712yes—
170C2×C4C22×Q811yescharacteristic
171C2×C4C2×#4111yescharacteristic
172C2×C4C2×#4111yescharacteristic
173C2×C4C2×#4111yescharacteristic
174C2×C4H(5,𝔽2)11yescharacteristic
175C2×C42−1+411yescharacteristic
176C8—41no—
177C8—42no—
178C8—41no—
179C22C2×C22⋊Q811yescharacteristic
180C22#54611yescharacteristic
181C22#54811yescharacteristic
182C22C2×#13511yescharacteristic
183C22C2×#13711yescharacteristic
184C22#57711yescharacteristic
185C22#57711yescharacteristic
186C4—22no—
187C4—42no—
188C4—42no—
189C4—42no—
190C4—42no—
191C4—42no—
192C4—42no—
193C4—82no—
194C4—82no—
195C4#47914yes—
196C4#53311yescharacteristic
197C4#54411yescharacteristic
198C4#55511yescharacteristic
199C4#55611yescharacteristic
200C2#243911yescharacteristic
201C2C2×#47911yescharacteristic
202C2#308411yescharacteristic
203C2#330411yescharacteristic
204C2#333111yescharacteristic
205C2#336511yescharacteristic
206C2#337211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ 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
#25522
#23552
#20662
#20262
#19892
#19482
#19392
#19202
#18342
#18032
C4×#1032