Finite Group Databasebeta

← #29220

Subgroups — #29220

Order 256.

345 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 143 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×C4⋊Q8C211yesmaximal, characteristic
3#2101C214yesmaximal
4#1990C211yesmaximal, characteristic
5#1990C211yesmaximal, characteristic
6#1989C214yesmaximal
7#1834C211yesmaximal, characteristic
8#1834C211yesmaximal, characteristic
9C4×C4⋊C8C211yesmaximal, characteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×#125C2212yes—
12C2×C4×Q8—21no—
13C2×C4⋊C8C2212yes—
14C2×C4×C8C2211yescharacteristic
15C2×C4×C8C2211yescharacteristic
16#391—21no—
17#389—21no—
18#389—21no—
19#389—21no—
20#389—21no—
21#389—21no—
22#389C2212yes—
23#384C2211yescharacteristic
24#384C2211yescharacteristic
25#384C2211yescharacteristic
26#382C2212yes—
27C4×C4⋊C4—21no—
28C4×C4⋊C4C2212yes—
29C43C2211yescharacteristic
30#340C2214yes—
31#340C2214yes—
32#340C2214yes—
33#340C2214yes—
34#336C2212yes—
35C22×Q8—21no—
36C22×C8C2312yes—
37C22×C8C2312yes—
38C4⋊Q8—22no—
39C4⋊Q8Q814yes—
40#125—28no—
41C4×Q8—44no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4C2312yes—
53C2×C4⋊C4C2312yes—
54C2×C4⋊C4C2312yes—
55C2×C4⋊C4C2312yes—
56C2×C42—22no—
57C2×C42D411yescharacteristic
58C2×C42D411yescharacteristic
59C2×C42D411yescharacteristic
60C2×C42D411yescharacteristic
61C2×C42C2311yescharacteristic
62C2×C42C2311yescharacteristic
63C2×C42C2311yescharacteristic
64C4⋊C8—24no—
65C4×C8—22no—
66C4×C8—22no—
67#95—22no—
68#95—22no—
69#95—22no—
70#95—22no—
71C2×Q8—24no—
72C2×Q8—42no—
73C2×Q8—42no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4C2×D411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2411yescharacteristic
85C2×C8—22no—
86C2×C8—22no—
87C2×C8—22no—
88C2×C8—22no—
89C2×C8—22no—
90C2×C8—22no—
91C4⋊C4—24no—
92C4⋊C4—22no—
93C4⋊C4—24no—
94C4⋊C4—24no—
95C4⋊C4—24no—
96C4⋊C4—44no—
97C4⋊C4—44no—
98C4⋊C4—44no—
99C4⋊C4—42no—
100C4⋊C4—42no—
101C4⋊C4—42no—
102C4⋊C4—42no—
103C4⋊C4—42no—
104C4⋊C4—42no—
105C4⋊C4C2×Q814yes—
106C42—22no—
107C42—22no—
108C42—22no—
109C42—22no—
110C42—22no—
111C42—44no—
112C42C2×Q812yes—
113C42#4112yes—
114C42#4114yes—
115C23C22×D411yescharacteristic
116Q8—48no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—24no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4C22⋊Q812yes—
140C2×C4C22⋊Q812yes—
141C2×C4#12412yes—
142C2×C4#12412yes—
143C2×C4#13512yes—
144C2×C4#13512yes—
145C2×C4AGL(1,ℤ/8)12yes—
146C2×C4#13712yes—
147C2×C4C22×Q811yescharacteristic
148C2×C4C2×#4111yescharacteristic
149C2×C4C2×#4112yes—
150C2×C42−1+412yes—
151C8—42no—
152C8—42no—
153C22C2×C22⋊Q811yescharacteristic
154C22C2×#12411yescharacteristic
155C22#53611yescharacteristic
156C22C2×#13511yescharacteristic
157C22C2×#13511yescharacteristic
158C22C2×AGL(1,ℤ/8)11yescharacteristic
159C22C2×#13711yescharacteristic
160C4—22no—
161C4—22no—
162C4—44no—
163C4—41no—
164C4—41no—
165C4—82no—
166C4—82no—
167C4—82no—
168C4#47814yes—
169C4#47914yes—
170C4#55612yes—
171C4#55712yes—
172C2#256811yescharacteristic
173C2C2×#47811yescharacteristic
174C2C2×#47911yescharacteristic
175C2#313111yescharacteristic
176C2#314111yescharacteristic
177C2#322811yescharacteristic
178C2#323111yescharacteristic
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×C4⋊Q82
#21012
#19902
#19892
#18342
C4×C4⋊C82