Finite Group Databasebeta

← #13379

Subgroups — #13379

Order 256.

270 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 45 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#2809C211yesmaximal, characteristic, complemented
2#1793C211yesmaximal, characteristic
3#1793C212yesmaximal
4#1394C211yesmaximal, characteristic, complemented
5#1394C212yesmaximal, complemented
6C2×2−1+4C2211yescharacteristic
7C2×#104—22no—
8C2×#104—21no—
9#417—22no—
10#417—21no—
11C2×#101C2212yes—
12C2×#101C2211yescharacteristic
13#400—21no—
14#398—22no—
15#391C2211yescharacteristic
16#391C2212yes—
17#389—21no—
18#387—22no—
192−1+4—22no—
202−1+4—21no—
212−1+4—42no—
222−1+4Q812yes—
23C2×#41—22no—
24C2×#41—21no—
25C2×#41—41no—
26C22×Q8—41no—
27C22×Q8C2311yescharacteristic
28C2×M16—22no—
29C2×M16—21no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4D412yes—
35C2×C4⋊C4D411yescharacteristic
36C2×C42D412yes—
37C2×C42D411yescharacteristic
38#104—42no—
39#104—44no—
40#103—42no—
41#103—41no—
42#102—42no—
43#102—41no—
44#101—22no—
45#101—22no—
46#101—22no—
47#95—21no—
48#95—22no—
49#95—22no—
50#95—22no—
51#95—22no—
52#41—22no—
53#41—24no—
54#41—24no—
55#41—22no—
56#41—41no—
57#41—42no—
58#41—42no—
59#41—44no—
60#41—42no—
61#41—42no—
62#41—41no—
63C2×Q8—22no—
64C2×Q8—21no—
65C2×Q8—41no—
66C2×Q8—41no—
67C2×Q8—42no—
68C2×Q8—44no—
69C2×Q8—42no—
70C2×Q8C2×Q811yescharacteristic
71C2×Q8#4112yes—
72C2×Q8#4111yescharacteristic
73C2×D4—22no—
74C2×D4—21no—
75C2×D4—41no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—42no—
81C22×C4—41no—
82C22×C4C2×D412yes—
83C22×C4C2×D411yescharacteristic
84M16—41no—
85M16—42no—
86M16—42no—
87M16—41no—
88C2×C8—42no—
89C2×C8—41no—
90C4⋊C4—42no—
91C4⋊C4—41no—
92C4⋊C4—44no—
93C4⋊C4—42no—
94C4⋊C4—81no—
95C4⋊C4—81no—
96C42—41no—
97C42—42no—
98C23—41no—
99C23#12011yescharacteristic
100Q8—22no—
101Q8—24no—
102Q8—41no—
103Q8—42no—
104Q8—41no—
105Q8—41no—
106Q8—42no—
107D4—22no—
108D4—24no—
109D4—41no—
110D4—42no—
111D4—42no—
112D4—42no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—21no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—42no—
121C2×C4—44no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4C22⋊Q812yes—
134C2×C4C22⋊Q811yescharacteristic
135C2×C4#12412yes—
136C2×C4#12411yescharacteristic
137C8—81no—
138C8—82no—
139C22—21no—
140C22—21no—
141C22—42no—
142C22—81no—
143C22—81no—
144C22#39311yescharacteristic
145C22B(4,𝔽2)11yescharacteristic
146C22B(4,𝔽2)11yescharacteristic
147C4—21no—
148C4—22no—
149C4—21no—
150C4—22no—
151C4—42no—
152C4—42no—
153C4—82no—
154C4—81no—
155C4—82no—
156C2—21no—
157C2—21no—
158C2—82no—
159C2#202911yescharacteristic
160C2#220311yescharacteristic
161C2#220411yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC22×Q8C23
prime radicalC2×2−1+4C22
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2809C2semidirect
4. #1394C2semidirect
5. #1394C2semidirect

Maximal subgroups

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

subgroupindex
#28092
#17932
#13942