Finite Group Databasebeta

← #29183

Subgroups — #29183

Order 256.

321 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 127 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#2859C211yesmaximal, characteristic
2C4×#125C211yesmaximal, characteristic
3#2088C212yesmaximal
4#2087C212yesmaximal
5#1992C211yesmaximal, characteristic
6#1992C211yesmaximal, characteristic
7#1919C212yesmaximal
8#1918C212yesmaximal
9#1847C211yesmaximal, characteristic
10#1836C211yesmaximal, characteristic
11#1767C211yesmaximal, characteristic
12C2×#125—21no—
13C2×#125C2211yescharacteristic
14C2×#107C2212yes—
15C2×#106C2212yes—
16C2×C4⋊C8C2212yes—
17C2×#97C2211yescharacteristic
18C2×#97C2211yescharacteristic
19#389—21no—
20#389—21no—
21#389C2211yescharacteristic
22#389C2211yescharacteristic
23#389C2212yes—
24#389C2212yes—
25#387—21no—
26#387—21no—
27#384C2211yescharacteristic
28#384C2211yescharacteristic
29#382—21no—
30#382C2211yescharacteristic
31C4×C4⋊C4—21no—
32C4×C4⋊C4—21no—
33C4×C4⋊C4C2211yescharacteristic
34C4×C4⋊C4C2211yescharacteristic
35C43C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×C8C2312yes—
44C22×C8C2312yes—
45#125—22no—
46#125—44no—
47#125Q814yes—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C42—21no—
65C2×C42—21no—
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42D411yescharacteristic
70C2×C42C2311yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#107—24no—
74#106—24no—
75C4⋊C8—24no—
76#97—22no—
77#97—22no—
78#95—21no—
79#95—21no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—22no—
85#95—22no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2411yescharacteristic
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8#4114yes—
105C4⋊C4—24no—
106C4⋊C4—24no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—24no—
114C4⋊C4—42no—
115C4⋊C4—42no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—44no—
120C4⋊C4—44no—
121C4⋊C4—42no—
122C4⋊C4C2×Q812yes—
123C4⋊C4C2×Q812yes—
124C42—22no—
125C42—22no—
126C42—22no—
127C42—22no—
128C42—41no—
129C42—41no—
130C42—42no—
131C42—42no—
132C42C2×Q812yes—
133C42#4112yes—
134C23C22×D411yescharacteristic
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4C22⋊Q812yes—
168C2×C4C22⋊Q812yes—
169C2×C4AGL(1,ℤ/8)12yes—
170C2×C4#13712yes—
171C2×C4C22×Q811yescharacteristic
172C2×C4C2×#4111yescharacteristic
173C2×C4C2×#4112yes—
174C2×C42−1+412yes—
175C8—24no—
176C8—42no—
177C22C2×C22⋊Q811yescharacteristic
178C22#53211yescharacteristic
179C22#53611yescharacteristic
180C22C2×AGL(1,ℤ/8)11yescharacteristic
181C22C2×#13711yescharacteristic
182C22#57511yescharacteristic
183C22#57511yescharacteristic
184C4—22no—
185C4—22no—
186C4—22no—
187C4—22no—
188C4—42no—
189C4—42no—
190C4—41no—
191C4—41no—
192C4—81no—
193C4—82no—
194C4—81no—
195C4—82no—
196C4#55212yes—
197C4#55712yes—
198C2#256511yescharacteristic
199C2#307911yescharacteristic
200C2#308011yescharacteristic
201C2#317111yescharacteristic
202C2#318111yescharacteristic
203C2#323411yescharacteristic
204C2#323711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ 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
#28592
C4×#1252
#20882
#20872
#19922
#19192
#19182
#18472
#18362
#17672