Finite Group Databasebeta

← #12336

Subgroups — #12336

Order 256.

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

normal subgroups only

This group has 168 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, complemented
2#2250C211yesmaximal, characteristic, complemented
3#2206C211yesmaximal, characteristic, complemented
4#1688C211yesmaximal, characteristic, complemented
5#1360C211yesmaximal, characteristic, complemented
6#1338C211yesmaximal, characteristic, complemented
7#1333C211yesmaximal, characteristic, complemented
8#559—21no—
9C2×SD32—21no—
10C2×D16—21no—
11C2×D16—21no—
12#496—21no—
13#495C2211yescharacteristic
14#493C2211yescharacteristic
15#464—21no—
16#463—21no—
17#460—21no—
18#459—21no—
19#446C2211yescharacteristic
20#440—21no—
21C4×D8C2211yescharacteristic
22C4⋊C16C2211yescharacteristic
23#357—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#331—21no—
27C2×Q16D411yescharacteristic
28C2×SD16—21no—
29C2×SD16—21no—
30C2×SD16—41no—
31C2×D8—21no—
32C2×D8—21no—
33C2×D8—21no—
34C2×D8D411yescharacteristic
35C2×D8D411yescharacteristic
36#127—21no—
37#127—21no—
38#124—21no—
39#124—21no—
40#124—41no—
41#123—41no—
42#123—41no—
43C4×Q8—21no—
44C4×Q8D411yescharacteristic
45C4×D4—21no—
46C4×D4—21no—
47C4×D4—41no—
48SD32—42no—
49D16—42no—
50D16—42no—
51C2×C16—21no—
52C2×C16—21no—
53#107D411yescharacteristic
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8D411yescharacteristic
57#103—21no—
58#102—21no—
59#102—21no—
60#102—41no—
61#97—21no—
62C4×C8C2311yescharacteristic
63C2×Q8—21no—
64C2×Q8D811yescharacteristic
65C2×D4—21no—
66C2×D4—21no—
67C2×D4—41no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—41no—
72C22×C4—41no—
73C22×C4—41no—
74Q16—41no—
75SD16—41no—
76SD16—41no—
77SD16—81no—
78SD16—82no—
79D8—21no—
80D8—21no—
81D8—41no—
82D8—41no—
83D8—41no—
84D8—41no—
85D8—41no—
86D8—81no—
87C2×C8—21no—
88C2×C8—41no—
89C2×C8C2×D411yescharacteristic
90C2×C8C2×D411yescharacteristic
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—41no—
94C4⋊C4D811yescharacteristic
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C42—21no—
102C42—41no—
103C42C2×D411yescharacteristic
104C16—22no—
105C16—41no—
106C23—41no—
107C23—41no—
108C23—81no—
109Q8—41no—
110Q8—41no—
111Q8—41no—
112D4—41no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—81no—
117D4—81no—
118D4—82no—
119D4—81no—
120D4—81no—
121D4—161no—
122D4—161no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—161no—
128C2×C4—161no—
129C2×C4#12011yescharacteristic
130C2×C4C2×D811yescharacteristic
131C2×C4AGL(1,ℤ/8)11yescharacteristic
132C8—21no—
133C8—41no—
134C8—81no—
135C8#13511yescharacteristic
136C8AGL(1,ℤ/8)11yescharacteristic
137C22—81no—
138C22—81no—
139C22—81no—
140C22—81no—
141C22—81no—
142C22—81no—
143C22—161no—
144C22—161no—
145C22—161no—
146C22#44711yescharacteristic
147C4—21no—
148C4—21no—
149C4—41no—
150C4—41no—
151C4—81no—
152C4—81no—
153C4#44911yescharacteristic
154C4#45311yescharacteristic
155C2—161no—
156C2—161no—
157C2—321no—
158C2#162111yescharacteristic
159C2#219011yescharacteristic
160C2#219211yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
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
1. #3403C2semidirect
2. #2250C2semidirect
3. #2206C2semidirect
4. #1688C2semidirect
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
#22502
#22062
#16882
#13602
#13382
#13332