Finite Group Databasebeta

← #10166

Subgroups — #10166

Order 256.

257 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 61 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#3226C211yesmaximal, characteristic
2C2×#498C211yesmaximal, characteristic, complemented
3#3141C211yesmaximal, characteristic
4#1966C211yesmaximal, characteristic, complemented
5#1548C211yesmaximal, characteristic, complemented
6#1492C211yesmaximal, characteristic
7#1278C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#500—21no—
12#498—22no—
13#498—22no—
14#489—22no—
15#487—21no—
16#486—21no—
17#486—21no—
18#480—22no—
19#474—22no—
20#431C2211yescharacteristic
21C2×#106—22no—
22#423C2211yescharacteristic
23#417—22no—
24C2×#101—22no—
25C4×M16C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27#333—22no—
28D4⋊C8—22no—
29C2×#41—21no—
30C22×Q8—21no—
31C22×Q8—21no—
32C2×M16—21no—
33C2×M16D412yes—
34C22×C8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8D412yes—
43C4⋊Q8D411yescharacteristic
44#127D411yescharacteristic
45#124—21no—
46#121—21no—
47#121—21no—
48C4×D4—21no—
49C4×D4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4D412yes—
54C2×C42C2311yescharacteristic
55#107—24no—
56#106—28no—
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8D412yes—
60#103—24no—
61#102—24no—
62#102—24no—
63#101—44no—
64C22⋊C8—21no—
65#97—21no—
66C4×C8—21no—
67C4×C8—21no—
68C4×C8—21no—
69C4×C8D412yes—
70#41—42no—
71#41—42no—
72C2×Q8—21no—
73C2×Q8—22no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8—41no—
77C2×Q8—42no—
78C2×Q8—42no—
79C2×Q8—82no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—41no—
84C22×C4—22no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4#4111yescharacteristic
90M16—22no—
91M16—22no—
92M16—42no—
93C2×C8—24no—
94C2×C8—22no—
95C2×C8—22no—
96C2×C8—22no—
97C2×C8—41no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C42C2×D411yescharacteristic
111C42C2×D411yescharacteristic
112C42C2×D411yescharacteristic
113C42C2×D411yescharacteristic
114C23—41no—
115C23#12111yescharacteristic
116Q8—42no—
117Q8—82no—
118Q8—82no—
119Q8—82no—
120Q8—81no—
121Q8—81no—
122D4—42no—
123D4—42no—
124D4—42no—
125D4—82no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—41no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—44no—
132C2×C4—82no—
133C2×C4—81no—
134C2×C4—81no—
135C2×C4#12011yescharacteristic
136C2×C4#12011yescharacteristic
137C2×C4#12011yescharacteristic
138C2×C4#12111yescharacteristic
139C2×C4#12111yescharacteristic
140C2×C4#12711yescharacteristic
141C2×C4#13512yes—
142C2×C4AGL(1,ℤ/8)12yes—
143C8—24no—
144C8—42no—
145C8—42no—
146C8—81no—
147C22—21no—
148C22—21no—
149C22—81no—
150C22—81no—
151C22—81no—
152C22#39211yescharacteristic
153C22#46312yes—
154C4—21no—
155C4—21no—
156C4—82no—
157C4—81no—
158C4—82no—
159C4#44912yes—
160C4#45612yes—
161C4#47112yes—
162C4#49511yescharacteristic
163C4#49611yescharacteristic
164C2—22no—
165C2—161no—
166C2#200511yescharacteristic
167C2#201211yescharacteristic
168C2#201911yescharacteristic
subgroupG/H
center, socleC22#392
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. C2×#498C2semidirect
4. #1966C2semidirect
5. #1548C2semidirect
7. #1278C2semidirect

Maximal subgroups

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

subgroupindex
#32262
C2×#4982
#31412
#19662
#15482
#14922
#12782