Finite Group Databasebeta

← #10327

Subgroups — #10327

Order 256.

403 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 53 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#3192C211yesmaximal, characteristic
2#2436C212yesmaximal, complemented
3#1784C211yesmaximal, characteristic, complemented
4#1507C212yesmaximal, complemented
5#1297C211yesmaximal, characteristic, complemented
6C23×Q8C2211yescharacteristic
7C2×#124—22no—
8C2×#121C2212yes—
9#481—22no—
10#469—22no—
11#447—22no—
12C2×#101—22no—
13#407C2211yescharacteristic
14#407C2211yescharacteristic
15#393—24no—
16#391—22no—
17#386—24no—
18C4×C22⋊C4C2212yes—
19#327—24no—
20C22×Q8—22no—
21C22×Q8—21no—
22C22×Q8—22no—
23C22×Q8—21no—
24C22×Q8—44no—
25C22×D4—22no—
26C22×D4D412yes—
27C23×C4D412yes—
28C23×C4C2311yescharacteristic
29C2×D8—22no—
30C2×M16—21no—
31C2×M16D412yes—
32#124—48no—
33#121—22no—
34#121—22no—
35#121—22no—
36#121—22no—
37C2×C4⋊C4D412yes—
38C2×C22⋊C4—28no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C42—22no—
42#106—22no—
43#102—24no—
44#101—44no—
45C22⋊C8—22no—
46C22⋊C8—22no—
47#95—28no—
48#95D414yes—
49C24—22no—
50C24C2×D411yescharacteristic
51C2×Q8—24no—
52C2×Q8—28no—
53C2×Q8—44no—
54C2×Q8—42no—
55C2×Q8—48no—
56C2×Q8—44no—
57C2×Q8—41no—
58C2×Q8—42no—
59C2×Q8—41no—
60C2×D4—22no—
61C2×D4—22no—
62C2×D4—44no—
63C2×D4—48no—
64C2×D4—42no—
65C2×D4—42no—
66C22×C4—24no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—24no—
70C22×C4—44no—
71C22×C4—42no—
72C22×C4C2×D412yes—
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D412yes—
75C22×C4#4111yescharacteristic
76D8—82no—
77M16—42no—
78M16—42no—
79C2×C8—22no—
80C2×C8—41no—
81C4⋊C4—22no—
82C4⋊C4—22no—
83C22⋊C4—416no—
84C22⋊C4—44no—
85C22⋊C4—42no—
86C22⋊C4—42no—
87C42—44no—
88C23—24no—
89C23—41no—
90C23—44no—
91C23—44no—
92C23—42no—
93C23—44no—
94C23#12012yes—
95C23#12111yescharacteristic
96Q8—44no—
97Q8—48no—
98Q8—44no—
99D4—44no—
100D4—82no—
101D4—84no—
102D4—84no—
103C2×C4—24no—
104C2×C4—22no—
105C2×C4—28no—
106C2×C4—24no—
107C2×C4—42no—
108C2×C4—41no—
109C2×C4—44no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—44no—
113C2×C4—44no—
114C2×C4—44no—
115C2×C4—42no—
116C2×C4—44no—
117C2×C4#12011yescharacteristic
118C2×C4#12112yes—
119C2×C4#12711yescharacteristic
120C8—42no—
121C8—81no—
122C22—22no—
123C22—22no—
124C22—22no—
125C22—21no—
126C22—44no—
127C22—88no—
128C22—82no—
129C22—82no—
130C22—82no—
131C22#39211yescharacteristic
132C22#45414yes—
133C4—21no—
134C4—22no—
135C4—44no—
136C4—84no—
137C4—82no—
138C4B(4,𝔽2)12yes—
139C2—24no—
140C2—41no—
141C2—162no—
142C2#200312yes—
143C2#202111yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radical#3192C2
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
2. #2436C2semidirect
3. #1784C2semidirect
4. #1507C2semidirect
5. #1297C2semidirect

Maximal subgroups

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

subgroupindex
#31922
#24362
#17842
#15072
#12972