Finite Group Databasebeta

← #29280

Subgroups — #29280

Order 256.

342 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C4×C4⋊Q8C211yesmaximal, characteristic
2C4×#125C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic
4#2099C211yesmaximal, characteristic
5#2058C211yesmaximal, characteristic
6#2057C211yesmaximal, characteristic
7#1989C212yesmaximal
8#1949C212yesmaximal
9#1921C212yesmaximal
10#1920C212yesmaximal
11#1831C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C4×Q8—21no—
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×C4⋊C8C2212yes—
18#391—21no—
19#389—21no—
20#389C2211yescharacteristic
21#389C2211yescharacteristic
22#389C2212yes—
23#387—21no—
24#387—21no—
25#384C2211yescharacteristic
26#382C2211yescharacteristic
27C4×C4⋊C4—21no—
28C4×C4⋊C4—21no—
29C4×C4⋊C4—21no—
30C4×C4⋊C4C2211yescharacteristic
31C4×C4⋊C4C2211yescharacteristic
32C4×C4⋊C4C2212yes—
33C43C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×C8C2312yes—
45C22×C8C2312yes—
46C4⋊Q8—24no—
47#125—22no—
48#125Q814yescomplemented
49C4×Q8—44no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C42—22no—
65C2×C42—21no—
66C2×C42—21no—
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#107—24no—
75#106—24no—
76C4⋊C8—24no—
77#95—21no—
78#95—21no—
79#95—21no—
80#95—21no—
81#95—22no—
82#95—21no—
83#95—21no—
84#95—22no—
85C2×Q8—24no—
86C2×Q8—42no—
87C2×Q8—42no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2411yescharacteristic
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C4⋊C4—24no—
108C4⋊C4—24no—
109C4⋊C4—24no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—24no—
116C4⋊C4—44no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4C2×Q812yes—
122C4⋊C4C2×Q812yes—
123C4⋊C4#4114yes—
124C42—22no—
125C42—22no—
126C42—22no—
127C42—22no—
128C42—22no—
129C42—41no—
130C42—42no—
131C42—44no—
132C42—42no—
133C42C2×Q812yes—
134C42#4112yes—
135C23C22×D411yescharacteristic
136Q8—48no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—24no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4C22⋊Q812yes—
169C2×C4C22⋊Q812yes—
170C2×C4#13512yes—
171C2×C4AGL(1,ℤ/8)12yes—
172C2×C4#13712yes—
173C2×C4C22×Q811yescharacteristic
174C2×C4C2×#4111yescharacteristic
175C2×C4C2×#4112yes—
176C2×C42−1+412yes—
177C8—42no—
178C8—42no—
179C22C2×C22⋊Q811yescharacteristic
180C22#52511yescharacteristic
181C22#53611yescharacteristic
182C22C2×#13511yescharacteristic
183C22C2×AGL(1,ℤ/8)11yescharacteristic
184C22C2×#13711yescharacteristic
185C22#57511yescharacteristic
186C4—22no—
187C4—22no—
188C4—22no—
189C4—22no—
190C4—22no—
191C4—42no—
192C4—42no—
193C4—44no—
194C4—41no—
195C4—81no—
196C4—81no—
197C4—82no—
198C4#55612yes—
199C4#55712yes—
200C2#258111yescharacteristic
201C2#307611yescharacteristic
202C2#307911yescharacteristic
203C2#310811yescharacteristic
204C2#311411yescharacteristic
205C2#322811yescharacteristic
206C2#323611yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
48. #125Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4×C4⋊Q82
C4×#1252
#20992
#20582
#20572
#19892
#19492
#19212
#19202
#18312