Finite Group Databasebeta

← #12388

Subgroups — #12388

Order 256.

148 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 39 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#3403C211yesmaximal, characteristic
2#2243C211yesmaximal, characteristic, complemented
3#2224C211yesmaximal, characteristic, complemented
4#1696C211yesmaximal, characteristic, complemented
5#1360C211yesmaximal, characteristic, complemented
6#1338C211yesmaximal, characteristic, complemented
7#1333C211yesmaximal, characteristic, complemented
8#559—21no—
9#500C2211yescharacteristic
10#496—21no—
11#495C2211yescharacteristic
12#477—21no—
13#464—21no—
14#463—21no—
15#459—21no—
16#446C2211yescharacteristic
17#440—21no—
18C4×D8C2211yescharacteristic
19#366—21no—
20C4⋊C16C2211yescharacteristic
21#358—21no—
22#357—21no—
23#357—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#333—21no—
27C2×Q16D411yescharacteristic
28C2×SD16—21no—
29C2×SD16—21no—
30C2×D8—21no—
31C2×D8D411yescharacteristic
32C2×D8D411yescharacteristic
33C4⋊Q8—21no—
34#127—21no—
35#124—21no—
36#124—21no—
37#124—41no—
38#123—41no—
39#123—41no—
40C4×Q8—21no—
41C4×Q8D411yescharacteristic
42C4×D4—21no—
43C4×D4—21no—
44C4×D4—41no—
45C2×C16—21no—
46C2×C16—21no—
47#107—21no—
48#107—21no—
49#107—41no—
50#107D411yescharacteristic
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8D411yescharacteristic
54#103—21no—
55#103—41no—
56#102—21no—
57#102—21no—
58#97—21no—
59C4×C8C2311yescharacteristic
60C2×Q8—21no—
61C2×Q8—41no—
62C2×Q8D811yescharacteristic
63C2×D4—21no—
64C2×D4—21no—
65C2×D4—41no—
66C2×D4—41no—
67C22×C4—41no—
68C22×C4—41no—
69Q16—41no—
70SD16—41no—
71SD16—41no—
72SD16—81no—
73D8—21no—
74D8—21no—
75D8—41no—
76D8—81no—
77C2×C8—21no—
78C2×C8—41no—
79C2×C8C2×D411yescharacteristic
80C2×C8C2×D411yescharacteristic
81C4⋊C4—21no—
82C4⋊C4—21no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4D811yescharacteristic
87C22⋊C4—41no—
88C22⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C42—21no—
94C42—41no—
95C42C2×D411yescharacteristic
96C16—41no—
97C16—41no—
98C23—41no—
99C23—41no—
100Q8—41no—
101Q8—41no—
102Q8—41no—
103Q8—161no—
104D4—41no—
105D4—41no—
106D4—41no—
107D4—41no—
108D4—82no—
109D4—161no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—81no—
115C2×C4—161no—
116C2×C4—161no—
117C2×C4#12011yescharacteristic
118C2×C4C2×D811yescharacteristic
119C2×C4AGL(1,ℤ/8)11yescharacteristic
120C8—21no—
121C8—41no—
122C8—81no—
123C8#13511yescharacteristic
124C8#13711yescharacteristic
125C22—81no—
126C22—81no—
127C22—81no—
128C22—81no—
129C22—81no—
130C22—81no—
131C22#44711yescharacteristic
132C4—21no—
133C4—21no—
134C4—41no—
135C4—41no—
136C4—81no—
137C4—81no—
138C4—161no—
139C4#45211yescharacteristic
140C4#45511yescharacteristic
141C2—161no—
142C2—161no—
143C2#164111yescharacteristic
144C2#219111yescharacteristic
145C2#219211yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#3403C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ #446 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2243C2semidirect
3. #2224C2semidirect
4. #1696C2semidirect
5. #1360C2semidirect
6. #1338C2semidirect
7. #1333C2semidirect

Maximal subgroups

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

subgroupindex
#34032
#22432
#22242
#16962
#13602
#13382
#13332