Finite Group Databasebeta

← #26092

Subgroups — #26092

Order 256.

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

normal subgroups only

This group has 337 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#2552C211yesmaximal, characteristic
2#2355C211yesmaximal, characteristic
3#2095C212yesmaximal
4#2068C212yesmaximal
5#1947C212yesmaximal
6#1947C211yesmaximal, characteristic
7#1939C211yesmaximal, characteristic
8#1921C212yesmaximal
9#1845C211yesmaximal, characteristic
10#1803C211yesmaximal, characteristic
11C4×#103C211yesmaximal, characteristic
12C2×#125—21no—
13C2×C4×Q8C2211yescharacteristic
14C2×#107C2212yes—
15C2×#107C2212yes—
16C2×#107C2212yes—
17C2×C4⋊C8C2211yescharacteristic
18C2×C4⋊C8C2212yes—
19C2×#103C2211yescharacteristic
20C2×#103C2211yescharacteristic
21C2×#103C2212yes—
22C2×C4×C8C2211yescharacteristic
23#398—22no—
24#391C2212yes—
25#389—21no—
26#389C2212yes—
27#387—22no—
28#384C2211yescharacteristic
29#384C2212yes—
30#384C2211yescharacteristic
31#382—21no—
32C4×C4⋊C4—21no—
33C4×C4⋊C4C2212yes—
34C4×C4⋊C4C2211yescharacteristic
35C4×C4⋊C4C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#340C2211yescharacteristic
39#340C2211yescharacteristic
40#336C2212yes—
41C22×Q8C2311yescharacteristic
42C22×C8C2312yes—
43C22×C8C2311yescharacteristic
44C22×C8C2311yescharacteristic
45#125—44no—
46C4×Q8—22no—
47C4×Q8Q814yes—
48C2×C4⋊C4—21no—
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⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C42—21no—
64C2×C42—21no—
65C2×C42—22no—
66C2×C42C2312yes—
67C2×C42C2311yescharacteristic
68#107—24no—
69#107—42no—
70#107—42no—
71C4⋊C8—24no—
72C4⋊C8—22no—
73#103—22no—
74#103—22no—
75#103—42no—
76C4×C8—22no—
77#95—22no—
78#95—22no—
79#95—21no—
80#95—22no—
81#95—22no—
82#95D412yes—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8C2×Q812yes—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4C2×D412yes—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D412yes—
98C22×C4C2×D411yescharacteristic
99C22×C4C2411yescharacteristic
100C2×C8—21no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8#4112yes—
108C2×C8#4112yes—
109C4⋊C4—24no—
110C4⋊C4—22no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—21no—
118C4⋊C4—22no—
119C4⋊C4—21no—
120C4⋊C4—44no—
121C4⋊C4—44no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—82no—
125C4⋊C4—81no—
126C4⋊C4—81no—
127C4⋊C4Q1614yes—
128C4⋊C4C2×Q812yes—
129C42—24no—
130C42—22no—
131C42—22no—
132C42—42no—
133C42—82no—
134C42C2×Q812yes—
135C42#4112yes—
136C23C22×D411yescharacteristic
137Q8—22no—
138Q8—22no—
139Q8—41no—
140Q8—41no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4C22⋊Q812yes—
167C2×C4C22⋊Q812yes—
168C2×C4C2×Q1614yes—
169C2×C4C2×Q1612yes—
170C2×C4AGL(1,ℤ/8)12yes—
171C2×C4C22×Q811yescharacteristic
172C2×C4C2×#4111yescharacteristic
173C2×C4C2×#4111yescharacteristic
174C2×C4C2×#4111yescharacteristic
175C2×C4H(5,𝔽2)11yescharacteristic
176C2×C42−1+411yescharacteristic
177C8—22no—
178C8—22no—
179C8—42no—
180C22C2×C22⋊Q811yescharacteristic
181C22#54711yescharacteristic
182C22#55011yescharacteristic
183C22C22×Q1611yescharacteristic
184C22C2×AGL(1,ℤ/8)11yescharacteristic
185C22#57611yescharacteristic
186C22#57811yescharacteristic
187C4—24no—
188C4—22no—
189C4—42no—
190C4—42no—
191C4—42no—
192C4—81no—
193C4—81no—
194C4—82no—
195C4—82no—
196C4#47714yes—
197C4#53311yescharacteristic
198C4#54411yescharacteristic
199C4#55211yescharacteristic
200C4#55311yescharacteristic
201C2#244111yescharacteristic
202C2C2×#47711yescharacteristic
203C2#308311yescharacteristic
204C2#333811yescharacteristic
205C2#335311yescharacteristic
206C2#339511yescharacteristic
207C2#339911yescharacteristic
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
#20952
#20682
#19472
#19392
#19212
#18452
#18032
C4×#1032