Finite Group Databasebeta

← #12340

Subgroups — #12340

Order 256.

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

normal subgroups only

This group has 173 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#3334C211yesmaximal, characteristic, complemented
2#2240C211yesmaximal, characteristic, complemented
3#2206C211yesmaximal, characteristic, complemented
4#1667C211yesmaximal, characteristic, complemented
5#1364C211yesmaximal, characteristic, complemented
6#1331C211yesmaximal, characteristic, complemented
7D4⋊C16C211yesmaximal, characteristic, complemented
8#547—21no—
9C2×D16—21no—
10#500C2211yescharacteristic
11#493C2211yescharacteristic
12#480—22no—
13#474—21no—
14#466—21no—
15#459—21no—
16C4×D8C2211yescharacteristic
17C8×D4C2211yescharacteristic
18C2×#107—21no—
19C4⋊C16C2211yescharacteristic
20#357—21no—
21#357—22no—
22C4×C16C2211yescharacteristic
23#335C2211yescharacteristic
24#331—21no—
25C2×#41—41no—
26C2×D8—21no—
27C2×D8—21no—
28C2×D8—41no—
29C2×D8D411yescharacteristic
30C22×C8—21no—
31C4⋊Q8—21no—
32#127—21no—
33#123—42no—
34C22⋊Q8—41no—
35#121—22no—
36C4×D4—21no—
37C4×D4D411yescharacteristic
38C2×C4⋊C4—22no—
39D16—42no—
40C2×C16—21no—
41C2×C16—21no—
42#107—22no—
43#107—21no—
44#107D411yescharacteristic
45#107D412yes—
46C4⋊C8—21no—
47C4⋊C8D411yescharacteristic
48#102—22no—
49#102—21no—
50#102—41no—
51C22⋊C8—21no—
52C4×C8C2311yescharacteristic
53#41—82no—
54#41—82no—
55C2×Q8—41no—
56C2×D4—21no—
57C2×D4—42no—
58C2×D4—41no—
59C2×D4—41no—
60C2×D4—41no—
61C2×D4D811yescharacteristic
62C22×C4—21no—
63C22×C4—42no—
64C22×C4—41no—
65D8—21no—
66D8—21no—
67D8—42no—
68D8—41no—
69D8—41no—
70D8—82no—
71C2×C8—21no—
72C2×C8—41no—
73C2×C8—41no—
74C2×C8—41no—
75C2×C8C2×D411yescharacteristic
76C2×C8C2×D411yescharacteristic
77C4⋊C4—21no—
78C4⋊C4—22no—
79C4⋊C4—42no—
80C4⋊C4—41no—
81C4⋊C4D811yescharacteristic
82C22⋊C4—21no—
83C22⋊C4—42no—
84C22⋊C4—41no—
85C42C2×D411yescharacteristic
86C16—41no—
87C16—41no—
88C23—21no—
89C23—41no—
90C23—81no—
91Q8—81no—
92Q8—81no—
93D4—21no—
94D4—41no—
95D4—41no—
96D4—84no—
97D4—81no—
98D4—81no—
99D4—81no—
100D4—81no—
101D4—161no—
102D4D1612yescomplemented
103C2×C4—21no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—42no—
108C2×C4—84no—
109C2×C4—81no—
110C2×C4—81no—
111C2×C4#12011yescharacteristic
112C2×C4C2×D811yescharacteristic
113C2×C4AGL(1,ℤ/8)11yescharacteristic
114C8—21no—
115C8—41no—
116C8—81no—
117C8#13511yescharacteristic
118C8AGL(1,ℤ/8)11yescharacteristic
119C22—22no—
120C22—41no—
121C22—41no—
122C22—81no—
123C22—81no—
124C22—81no—
125C22—161no—
126C22—161no—
127C22—161no—
128C22#44711yescharacteristic
129C4—21no—
130C4—41no—
131C4—81no—
132C4—82no—
133C4#44911yescharacteristic
134C4#45411yescharacteristic
135C4C2×D1611yescharacteristic
136C4#50911yescharacteristic
137C2—42no—
138C2—161no—
139C2—321no—
140C2#162511yescharacteristic
141C2#218411yescharacteristic
142C2#219311yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3334C2semidirect
2. #2240C2semidirect
3. #2206C2semidirect
4. #1667C2semidirect
5. #1364C2semidirect
6. #1331C2semidirect
7. D4⋊C16C2semidirect
102. D4D16semidirect

Maximal subgroups

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

subgroupindex
#33342
#22402
#22062
#16672
#13642
#13312
D4⋊C162