Finite Group Databasebeta

← #9974

Subgroups — #9974

Order 256.

384 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 51 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#3186C211yesmaximal, characteristic, complemented
2#3006C212yesmaximal, complemented
3#1780C211yesmaximal, characteristic, complemented
4#1471C212yesmaximal, complemented
5#1293C211yesmaximal, characteristic, complemented
6C22×#41C2211yescharacteristic
7C2×AGL(1,ℤ/8)—22no—
8C2×#121C2212yes—
9#480—22no—
10#466—22no—
11#449—24no—
12#448—22no—
13#447—22no—
14#447—22no—
15#417—22no—
16#407C2212yes—
17C2×C22⋊C8—22no—
18C2×C22⋊C8C2211yescharacteristic
19C2×C22⋊C8C2211yescharacteristic
20#323—24no—
21C2×#41—22no—
22C2×#41—22no—
23C2×#41—44no—
24C22×Q8D411yescharacteristic
25C22×D4—21no—
26C22×D4—22no—
27C22×D4D412yes—
28C22×D4D411yescharacteristic
29C23×C4—21no—
30C23×C4C2311yescharacteristic
31AGL(1,ℤ/8)—48no—
32C2×SD16—24no—
33C2×D8—24no—
34C2×D8—22no—
35C2×M16—22no—
36C22×C8—21no—
37C22×C8D412yes—
38#121—22no—
39#121—22no—
40#121—22no—
41#121—22no—
42C2×C4⋊C4D412yes—
43C2×C22⋊C4—22no—
44#107—22no—
45#103—24no—
46#102—24no—
47#102—24no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50C22⋊C8—22no—
51C22⋊C8—44no—
52C22⋊C8D414yes—
53C24—21no—
54C24—22no—
55C24#4111yescharacteristic
56#41—28no—
57#41—44no—
58#41—44no—
59#41—44no—
60C2×Q8—22no—
61C2×Q8—42no—
62C2×D4—22no—
63C2×D4—22no—
64C2×D4—22no—
65C2×D4—42no—
66C2×D4—44no—
67C2×D4—42no—
68C2×D4—42no—
69C2×D4—42no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4—84no—
73C22×C4—24no—
74C22×C4—21no—
75C22×C4—22no—
76C22×C4—42no—
77C22×C4—42no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4C2×D412yes—
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D412yes—
84SD16—48no—
85D8—48no—
86D8—82no—
87M16—44no—
88C2×C8—22no—
89C2×C8—24no—
90C2×C8—42no—
91C2×C8—42no—
92C2×C8—41no—
93C4⋊C4—22no—
94C4⋊C4—22no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C23—22no—
98C23—22no—
99C23—41no—
100C23—41no—
101C23—41no—
102C23—42no—
103C23—44no—
104C23—41no—
105C23—42no—
106C23—42no—
107C23—82no—
108C23—82no—
109C23#12112yes—
110C23#12111yescharacteristic
111Q8—24no—
112Q8—42no—
113D4—24no—
114D4—42no—
115D4—44no—
116D4—48no—
117D4—44no—
118D4—84no—
119D4—84no—
120C2×C4—22no—
121C2×C4—24no—
122C2×C4—22no—
123C2×C4—41no—
124C2×C4—42no—
125C2×C4—44no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—44no—
129C2×C4—42no—
130C2×C4—82no—
131C2×C4#12012yes—
132C2×C4#12011yescharacteristic
133C2×C4#12711yescharacteristic
134C8—44no—
135C8—42no—
136C8—81no—
137C22—21no—
138C22—21no—
139C22—22no—
140C22—22no—
141C22—22no—
142C22—44no—
143C22—42no—
144C22—82no—
145C22—82no—
146C22—81no—
147C22—81no—
148C22—82no—
149C22—82no—
150C22—82no—
151C22—164no—
152C22#39211yescharacteristic
153C22#47212yes—
154C4—22no—
155C4—21no—
156C4—42no—
157C4—82no—
158C4B(4,𝔽2)12yes—
159C2—22no—
160C2—42no—
161C2—82no—
162C2—162no—
163C2#201711yescharacteristic
164C2#201811yescharacteristic
165C2#202111yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3186C2semidirect
2. #3006C2semidirect
3. #1780C2semidirect
4. #1471C2semidirect
5. #1293C2semidirect

Maximal subgroups

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

subgroupindex
#31862
#30062
#17802
#14712
#12932