Finite Group Databasebeta

← #27730

Subgroups — #27730

Order 256.

313 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 123 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#2571C211yesmaximal, characteristic
2#2568C211yesmaximal, characteristic
3#2101C212yesmaximal
4#2074C212yesmaximal
5#1990C211yesmaximal, characteristic
6#1988C211yesmaximal, characteristic
7#1933C212yesmaximal
8#1835C212yesmaximal
9#1803C211yesmaximal, characteristic
10C8×C4⋊C4C211yesmaximal, characteristic
11C4×#103C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C2×C4⋊C8C2211yescharacteristic
15C2×#103C2212yes—
16C2×#103C2211yescharacteristic
17C2×#103C2211yescharacteristic
18C2×C4×C8C2212yes—
19C2×C4×C8C2211yescharacteristic
20#400—22no—
21#400—22no—
22#391—21no—
23#391—21no—
24#391C2212yes—
25#389—21no—
26#389C2212yes—
27#384C2211yescharacteristic
28#382—21no—
29#382C2212yes—
30#382C2212yes—
31C4×C4⋊C4C2211yescharacteristic
32C4×C4⋊C4C2211yescharacteristic
33#340C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2211yescharacteristic
37#340C2212yes—
38#340C2211yescharacteristic
39#340C2212yes—
40#336C2212yes—
41C22×Q8—21no—
42C22×Q8C2311yescharacteristic
43C22×C8C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2311yescharacteristic
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4×Q8—22no—
49C4×Q8—22no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4D411yescharacteristic
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C42—21no—
63C2×C42—21no—
64C2×C42C2312yes—
65C2×C42C2311yescharacteristic
66C4⋊C8—22no—
67#103—24no—
68#103—22no—
69#103—22no—
70C4×C8—24no—
71C4×C8—22no—
72#95—21no—
73#95—22no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95—22no—
78#95—22no—
79#95—22no—
80#95D412yes—
81C2×Q8—21no—
82C2×Q8—21no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—42no—
87C2×Q8—42no—
88C2×Q8—42no—
89C2×Q8#4112yes—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4C2×D412yes—
97C22×C4C2×D412yes—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2411yescharacteristic
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—22no—
104C2×C8—21no—
105C2×C8—22no—
106C2×C8—21no—
107C2×C8—22no—
108C2×C8—21no—
109C2×C8#4112yes—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—81no—
126C4⋊C4—82no—
127C42—22no—
128C42—22no—
129C42—42no—
130C42—42no—
131C42#4112yes—
132C42#4112yes—
133C23C22×D411yescharacteristic
134Q8—22no—
135Q8—22no—
136Q8—41no—
137Q8—41no—
138Q8—84no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
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—42no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4#12412yes—
166C2×C4#12412yes—
167C2×C4#13512yes—
168C2×C4#13512yes—
169C2×C4C2×#4111yescharacteristic
170C2×C4C2×#4111yescharacteristic
171C2×C4C2×#4111yescharacteristic
172C2×C4C2×#4111yescharacteristic
173C2×C42−1+411yescharacteristic
174C2×C42−1+411yescharacteristic
175C8—22no—
176C8—42no—
177C8—41no—
178C22C2×#12411yescharacteristic
179C22#54711yescharacteristic
180C22#54811yescharacteristic
181C22C2×#13511yescharacteristic
182C22C2×#13511yescharacteristic
183C22#57711yescharacteristic
184C22#57811yescharacteristic
185C4—22no—
186C4—42no—
187C4—42no—
188C4—42no—
189C4—81no—
190C4—81no—
191C4—82no—
192C4—82no—
193C4—81no—
194C4#48814yes—
195C4#52911yescharacteristic
196C4#54111yescharacteristic
197C4#55211yescharacteristic
198C4#55211yescharacteristic
199C2#248911yescharacteristic
200C2C2×#48811yescharacteristic
201C2#313711yescharacteristic
202C2#334011yescharacteristic
203C2#334211yescharacteristic
204C2#336411yescharacteristic
205C2#336511yescharacteristic
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
#25712
#25682
#21012
#20742
#19902
#19882
#19332
#18352
#18032
C8×C4⋊C42
C4×#1032