Finite Group Databasebeta

← #13609

Subgroups — #13609

Order 256.

174 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 43 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#3220C211yesmaximal, characteristic
2#2128C211yesmaximal, characteristic, complemented
3#2127C211yesmaximal, characteristic, complemented
4#1412C212yesmaximal, complemented
5#1408C212yesmaximal, complemented
6#540C411yescharacteristic, complemented
7#535C2211yescharacteristic
8#496C411yescharacteristic, complemented
9#493C2211yescharacteristic
10#490C2211yescharacteristic
11#488C411yescharacteristic, complemented
12#463—21no—
13#459—21no—
14#449—21no—
15#447—21no—
16#433C411yescharacteristic, complemented
17#410—21no—
18C2×#99—21no—
19#354—21no—
20#353—21no—
21#353C2212yes—
22#353C2212yes—
23C2×#41D411yescharacteristic
24C22×D4D411yescharacteristic
25C2×SD16—21no—
26C2×D8—21no—
27C2×D8—21no—
28C2×D8—21no—
29#127C2×C411yescharacteristic, complemented
30#127D411yescharacteristic
31#127C2311yescharacteristic
32#125C2×C411yescharacteristic, complemented
33#124C2×C411yescharacteristic
34#124D411yescharacteristic
35#123—41no—
36#121—21no—
37#121—41no—
38#121—41no—
39#120—41no—
40C4×D4—21no—
41#117—41no—
42#117C2×C411yescharacteristic
43C2×C22⋊C4—41no—
44C4⋊C8—21no—
45#103—21no—
46#102—21no—
47#102—21no—
48#102—21no—
49#99—22no—
50#99—22no—
51#99—41no—
52#99—41no—
53C22⋊C8—21no—
54#97C2×C411yescharacteristic, complemented
55C4×C8C2×C411yescharacteristic, complemented
56C24—21no—
57#41—81no—
58C2×Q8C22⋊C411yescharacteristic
59C2×D4—21no—
60C2×D4—21no—
61C2×D4—21no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4—41no—
66C2×D4—81no—
67C2×D4C22⋊C411yescharacteristic
68C2×D4C2×D411yescharacteristic
69C2×D4C2×D411yescharacteristic
70C22×C4—21no—
71C22×C4—41no—
72C22×C4—81no—
73C22×C4C22⋊C411yescharacteristic
74SD16—81no—
75D8—81no—
76D8—81no—
77D8—81no—
78C2×C8—21no—
79C2×C8—21no—
80C4⋊C4—21no—
81C4⋊C4—21no—
82C4⋊C4—41no—
83C4⋊C4—81no—
84C22⋊C4—21no—
85C22⋊C4—41no—
86C22⋊C4—41no—
87C22⋊C4—41no—
88C22⋊C4—41no—
89C22⋊C4—42no—
90C22⋊C4—42no—
91C22⋊C4—81no—
92C42—81no—
93C42C22⋊C411yescharacteristic
94C42C22×C411yescharacteristic
95C23—21no—
96C23—21no—
97C23—21no—
98C23—41no—
99C23—41no—
100C23—41no—
101C23—161no—
102C23#9911yescharacteristic
103Q8—41no—
104D4—41no—
105D4—41no—
106D4—41no—
107D4—41no—
108D4—41no—
109D4—81no—
110D4—81no—
111D4—161no—
112D4—161no—
113D4—161no—
114D4—161no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—82no—
123C2×C4—82no—
124C2×C4—161no—
125C2×C4—162no—
126C2×C4#9911yescharacteristic
127C2×C4C2×C22⋊C411yescharacteristic
128C8—41no—
129C8—41no—
130C22—21no—
131C22—41no—
132C22—41no—
133C22—41no—
134C22—41no—
135C22—81no—
136C22—81no—
137C22—81no—
138C22—81no—
139C22—81no—
140C22—82no—
141C22—81no—
142C22—161no—
143C22C2×#9911yescharacteristic
144C4—21no—
145C4—21no—
146C4—41no—
147C4—41no—
148C4—41no—
149C4—81no—
150C4—162no—
151C4—162no—
152C2—21no—
153C2—41no—
154C2—81no—
155C2—81no—
156C2—81no—
157C2—161no—
158C2#212711yescharacteristic
subgroupG/H
center, socleC2#2127
commutatorC42C22×C4
Frattini, cosocle#127C23
prime radical#3220C2
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #535 ◁ G
LECSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2128C2semidirect
3. #2127C2semidirect
4. #1412C2semidirect
5. #1408C2semidirect
6. #540C4semidirect
8. #496C4semidirect
11. #488C4semidirect
16. #433C4semidirect
29. #127C2×C4semidirect
32. #125C2×C4semidirect
54. #97C2×C4semidirect
55. C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#32202
#21282
#21272
#14122
#14082