Finite Group Databasebeta

← #10532

Subgroups — #10532

Order 256.

190 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 49 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#3254C211yesmaximal, characteristic
2#1848C211yesmaximal, characteristic, complemented
3#1561C211yesmaximal, characteristic, complemented
4#1560C211yesmaximal, characteristic
5#1493C211yesmaximal, characteristic, complemented
6#1492C211yesmaximal, characteristic
7#1309C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10#477—21no—
11#474—21no—
12#468—22no—
13#461—21no—
14#460—21no—
15C2×#108—22no—
16#423C2211yescharacteristic
17#423C2211yescharacteristic
18#423C2211yescharacteristic
19#423C2211yescharacteristic
20C2×C4×C8C2211yescharacteristic
21#333—21no—
22#332—21no—
23#332—21no—
24#331—21no—
25Q8⋊C8—22no—
26D4⋊C8—22no—
27C2×#41—21no—
28C2×SD16—22no—
29C2×M16—21no—
30C2×M16—21no—
31C2×M16—21no—
32C2×M16D412yes—
33C22×C8—21no—
34C4⋊Q8D411yescharacteristic
35C4⋊Q8D411yescharacteristic
36C4⋊Q8D411yescharacteristic
37#127D411yescharacteristic
38#125—21no—
39#124—21no—
40C22⋊Q8—21no—
41C22⋊Q8—21no—
42#121—21no—
43#121—21no—
44C4×Q8—21no—
45C4×Q8—21no—
46C4×D4—21no—
47C4×D4—21no—
48#117—21no—
49C2×C42C2311yescharacteristic
50#108—44no—
51#107—22no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60#103—22no—
61#102—22no—
62C4×C8—21no—
63C4×C8D412yes—
64#41—81no—
65#41—81no—
66C2×Q8—21no—
67C2×Q8—21no—
68C2×Q8—21no—
69C2×D4—21no—
70C2×D4—21no—
71C2×D4—21no—
72C2×D4—41no—
73C22×C4—41no—
74C22×C4C2×D411yescharacteristic
75C22×C4#4111yescharacteristic
76C22×C4#4111yescharacteristic
77SD16—82no—
78M16—42no—
79M16—42no—
80M16—81no—
81M16—81no—
82C2×C8—22no—
83C2×C8—22no—
84C2×C8—22no—
85C2×C8—22no—
86C2×C8—41no—
87C2×C8—41no—
88C2×C8—41no—
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C42—41no—
101C42C2×D411yescharacteristic
102C42C2×D411yescharacteristic
103C42C2×D411yescharacteristic
104C42#4111yescharacteristic
105C23—41no—
106C23#12411yescharacteristic
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—41no—
111Q8—81no—
112D4—41no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—81no—
117D4—82no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—81no—
128C2×C4—81no—
129C2×C4#12011yescharacteristic
130C2×C4#12111yescharacteristic
131C2×C4#12111yescharacteristic
132C2×C4#12111yescharacteristic
133C2×C4#12311yescharacteristic
134C2×C4#12311yescharacteristic
135C8—24no—
136C8—42no—
137C8—81no—
138C8—81no—
139C8—81no—
140C22—21no—
141C22—21no—
142C22—21no—
143C22—81no—
144C22—81no—
145C22—81no—
146C22#39411yescharacteristic
147C4—21no—
148C4—21no—
149C4—21no—
150C4—81no—
151C4—81no—
152C4—81no—
153C4#45311yescharacteristic
154C4#45611yescharacteristic
155C4#47111yescharacteristic
156C4#47111yescharacteristic
157C4#47211yescharacteristic
158C4#47311yescharacteristic
159C2—41no—
160C2—161no—
161C2#203911yescharacteristic
162C2#205111yescharacteristic
163C2#205211yescharacteristic
subgroupG/H
center, socleC22#394
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1848C2semidirect
3. #1561C2semidirect
5. #1493C2semidirect
7. #1309C2semidirect

Maximal subgroups

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

subgroupindex
#32542
#18482
#15612
#15602
#14932
#14922
#13092