Finite Group Databasebeta

← #29275

Subgroups — #29275

Order 256.

342 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 133 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#2101C212yesmaximal
4#2059C212yesmaximal
5#1987C211yesmaximal, characteristic
6#1987C211yesmaximal, characteristic
7#1948C211yesmaximal, characteristic
8#1947C211yesmaximal, characteristic
9#1920C212yesmaximal
10#1918C212yesmaximal
11#1831C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C4×Q8—21no—
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×C4⋊C8C2211yescharacteristic
18C2×C4⋊C8C2211yescharacteristic
19#391—21no—
20#389—21no—
21#389C2211yescharacteristic
22#389C2211yescharacteristic
23#389C2212yes—
24#387—21no—
25#387—21no—
26#384C2211yescharacteristic
27#382C2211yescharacteristic
28C4×C4⋊C4—21no—
29C4×C4⋊C4—21no—
30C4×C4⋊C4—21no—
31C4×C4⋊C4C2211yescharacteristic
32C4×C4⋊C4C2211yescharacteristic
33C4×C4⋊C4C2212yes—
34C43C2211yescharacteristic
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×C8C2312yes—
45C22×C8C2312yes—
46C4⋊Q8—22no—
47C4⋊Q8Q814yes—
48#125—22no—
49#125—22no—
50C4×Q8—44no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C42—22no—
66C2×C42—21no—
67C2×C42—21no—
68C2×C42D411yescharacteristic
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75#107—24no—
76#106—24no—
77C4⋊C8—22no—
78C4⋊C8—22no—
79#95—21no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—22no—
84#95—21no—
85#95—21no—
86#95—22no—
87C2×Q8—24no—
88C2×Q8—42no—
89C2×Q8—42no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2411yescharacteristic
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C4⋊C4—24no—
110C4⋊C4—24no—
111C4⋊C4—24no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—44no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4C2×Q814yes—
127C4⋊C4#4112yes—
128C4⋊C4#4112yes—
129C42—22no—
130C42—22no—
131C42—22no—
132C42—21no—
133C42—21no—
134C42—21no—
135C42—21no—
136C42—41no—
137C42—42no—
138C42—44no—
139C42—42no—
140C42C2×Q812yes—
141C42#4112yes—
142C23C22×D411yescharacteristic
143Q8—48no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—24no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4C22⋊Q812yes—
176C2×C4C22⋊Q812yes—
177C2×C4#13512yes—
178C2×C4AGL(1,ℤ/8)12yes—
179C2×C4#13712yes—
180C2×C4C22×Q811yescharacteristic
181C2×C4C2×#4111yescharacteristic
182C2×C4C2×#4111yescharacteristic
183C2×C4C2×#4111yescharacteristic
184C2×C42−1+411yescharacteristic
185C2×C42−1+411yescharacteristic
186C8—42no—
187C8—42no—
188C22C2×C22⋊Q811yescharacteristic
189C22#52511yescharacteristic
190C22#53611yescharacteristic
191C22C2×#13511yescharacteristic
192C22C2×AGL(1,ℤ/8)11yescharacteristic
193C22C2×#13711yescharacteristic
194C22#57511yescharacteristic
195C4—22no—
196C4—22no—
197C4—44no—
198C4—41no—
199C4—42no—
200C4—41no—
201C4—42no—
202C4—41no—
203C4—82no—
204C4—81no—
205C4—81no—
206C4#47914yes—
207C4#55611yescharacteristic
208C4#55611yescharacteristic
209C4#55711yescharacteristic
210C4#55711yescharacteristic
211C2#258111yescharacteristic
212C2C2×#47911yescharacteristic
213C2#307711yescharacteristic
214C2#310611yescharacteristic
215C2#311611yescharacteristic
216C2#323311yescharacteristic
217C2#323811yescharacteristic
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
C4×C4⋊Q82
C4×#1252
#21012
#20592
#19872
#19482
#19472
#19202
#19182
#18312