Finite Group Databasebeta

← #30310

Subgroups — #30310

Order 256.

325 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 131 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#2862C211yesmaximal, characteristic
2#2859C211yesmaximal, characteristic
3#2085C214yesmaximal
4#2058C212yesmaximal
5#2057C212yesmaximal
6#1847C212yesmaximal
7#1836C212yesmaximal
8#1745C211yesmaximal, characteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×#125—21no—
12C2×#107C2214yes—
13C2×#106C2214yes—
14C2×#97C2212yes—
15#391—21no—
16#391—21no—
17#391—21no—
18#391—21no—
19#389—21no—
20#389—21no—
21#389C2212yes—
22#389C2212yes—
23#384C2211yescharacteristic
24#384C2211yescharacteristic
25#382—21no—
26#382C2212yes—
27#382C2212yes—
28C43C2211yescharacteristic
29#340C2214yes—
30#340C2214yes—
31#336C2212yes—
32#336C2212yes—
33C22×Q8—21no—
34C22×Q8—21no—
35C22×C8C2314yes—
36C4⋊Q8—42no—
37C4⋊Q8—42no—
38#125—44no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4C2312yes—
48C2×C4⋊C4C2312yes—
49C2×C4⋊C4C2312yes—
50C2×C4⋊C4C2312yes—
51C2×C42D411yescharacteristic
52C2×C42D411yescharacteristic
53C2×C42D411yescharacteristic
54C2×C42D411yescharacteristic
55C2×C42C2311yescharacteristic
56C2×C42C2311yescharacteristic
57C2×C42C2311yescharacteristic
58#107—44no—
59#106—44no—
60#97Q818yescomplemented
61#95—22no—
62#95—22no—
63#95—22no—
64#95—22no—
65#95—22no—
66#95—22no—
67C2×Q8—42no—
68C2×Q8—42no—
69C2×Q8—42no—
70C2×Q8—42no—
71C2×Q8—42no—
72C2×Q8—42no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2411yescharacteristic
84C2×C8—24no—
85C2×C8—24no—
86C2×C8C2×Q818yes—
87C4⋊C4—24no—
88C4⋊C4—24no—
89C4⋊C4—24no—
90C4⋊C4—24no—
91C4⋊C4—42no—
92C4⋊C4—42no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C4⋊C4—44no—
96C4⋊C4—42no—
97C4⋊C4—42no—
98C4⋊C4—44no—
99C42—22no—
100C42—22no—
101C42—22no—
102C42—22no—
103C42—41no—
104C42—41no—
105C42C2×Q814yes—
106C23C22×D411yescharacteristic
107Q8—84no—
108Q8—84no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4C4⋊Q812yes—
136C2×C4C4⋊Q812yes—
137C2×C4AGL(1,ℤ/8)12yes—
138C2×C4#13712yes—
139C2×C4C22×Q812yes—
140C2×C4H(5,𝔽2)12yes—
141C2×C42−1+412yes—
142C8—28no—
143C22C2×C4⋊Q811yescharacteristic
144C22#53511yescharacteristic
145C22#53611yescharacteristic
146C22C2×AGL(1,ℤ/8)11yescharacteristic
147C22C2×#13711yescharacteristic
148C22#57511yescharacteristic
149C22#57511yescharacteristic
150C4—22no—
151C4—22no—
152C4—22no—
153C4—22no—
154C4—41no—
155C4—41no—
156C4—82no—
157C4—82no—
158C4—82no—
159C4—82no—
160C4#54414yes—
161C2#266011yescharacteristic
162C2#316511yescharacteristic
163C2#316611yescharacteristic
164C2#321311yescharacteristic
165C2#321511yescharacteristic
166C2#323511yescharacteristic
167C2#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

Complements

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

NH ≅ G/Ntype
60. #97Q8semidirect

Maximal subgroups

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

subgroupindex
#28622
#28592
#20852
#20582
#20572
#18472
#18362
#17452