Finite Group Databasebeta

← #29977

Subgroups — #29977

Order 256.

337 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 119 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#2628C211yesmaximal, characteristic
2#2317C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2088C214yesmaximal, complemented
5#2058C212yesmaximal, complemented
6#1853C212yesmaximal
7#1825C212yesmaximal
8#1798C211yesmaximal, characteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#117C2211yescharacteristic
11C2×#117C2211yescharacteristic
12C22×C4⋊C4C2211yescharacteristic
13C2×#106C2214yes—
14C2×C4⋊C8C2212yes—
15C2×C22⋊C8C2212yes—
16#400—21no—
17#399—21no—
18#395—21no—
19#393—21no—
20#389C2212yes—
21#389C2212yes—
22#389C2212yes—
23#385—21no—
24#385—21no—
25#384—21no—
26#382—21no—
27#382C2211yescharacteristic
28#380C2212yes—
29#340C2214yes—
30#340C2214yes—
31#340C2214yes—
32#336C2212yes—
33C22×Q8—21no—
34C23×C4—21no—
35C23×C4C2311yescharacteristic
36C22×C8C2314yes—
37C4⋊Q8—42no—
38#117—22no—
39#117—42no—
40#117Q814yescomplemented
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—42no—
47C2×C4⋊C4D411yescharacteristic
48C2×C4⋊C4D411yescharacteristic
49C2×C4⋊C4C2312yes—
50C2×C4⋊C4C2312yes—
51C2×C4⋊C4C2312yes—
52C2×C4⋊C4C2312yes—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—22no—
55C2×C22⋊C4D411yescharacteristic
56C2×C22⋊C4D411yescharacteristic
57C2×C42—21no—
58C2×C42C2311yescharacteristic
59C2×C42C2311yescharacteristic
60#106—44no—
61C4⋊C8—42no—
62C22⋊C8—24no—
63#95—22no—
64#95—22no—
65#95—22no—
66#95—21no—
67#95—21no—
68#95—22no—
69C24C2×D411yescharacteristic
70C2×Q8—42no—
71C2×Q8—82no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—42no—
79C22×C4—82no—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4C2×Q812yes—
86C22×C4#4112yes—
87C22×C4C2411yescharacteristic
88C2×C8—24no—
89C2×C8—24no—
90C2×C8—24no—
91C4⋊C4—22no—
92C4⋊C4—24no—
93C4⋊C4—24no—
94C4⋊C4—24no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—42no—
98C4⋊C4—42no—
99C4⋊C4—82no—
100C4⋊C4—82no—
101C4⋊C4—82no—
102C4⋊C4C2×Q814yes—
103C22⋊C4—24no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—82no—
107C42—24no—
108C42—22no—
109C42—22no—
110C23—21no—
111C23—21no—
112C23—22no—
113C23—22no—
114C23C22⋊Q812yes—
115C23C22×D411yescharacteristic
116Q8—84no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—24no—
120C2×C4—24no—
121C2×C4—24no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—22no—
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×C4—42no—
140C2×C4—42no—
141C2×C4—84no—
142C2×C4C22⋊Q812yes—
143C2×C4C22×Q811yescharacteristic
144C2×C4C2×#4111yescharacteristic
145C2×C4H(5,𝔽2)12yes—
146C2×C42−1+412yes—
147C8—44no—
148C22—24no—
149C22—22no—
150C22—41no—
151C22—41no—
152C22—42no—
153C22C2×C22⋊Q811yescharacteristic
154C22#53411yescharacteristic
155C22#53611yescharacteristic
156C22#57712yes—
157C22#57712yes—
158C4—22no—
159C4—44no—
160C4—41no—
161C4—41no—
162C4—41no—
163C4—41no—
164C4—82no—
165C4—82no—
166C4—82no—
167C4#53912yes—
168C4#54412yes—
169C2—42no—
170C2#262711yescharacteristic
171C2#308412yes—
172C2#320011yescharacteristic
173C2#320011yescharacteristic
174C2#324412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2101C2semidirect
4. #2088C2semidirect
5. #2058C2semidirect
40. #117Q8semidirect

Maximal subgroups

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

subgroupindex
#26282
#23172
#21012
#20882
#20582
#18532
#18252
#17982