Finite Group Databasebeta

← #10154

Subgroups — #10154

Order 256.

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

normal subgroups only

This group has 287 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#3206C211yesmaximal, characteristic
2C2×#494C211yesmaximal, characteristic, complemented
3#3141C211yesmaximal, characteristic
4#1997C211yesmaximal, characteristic, complemented
5#1549C211yesmaximal, characteristic, complemented
6#1492C211yesmaximal, characteristic
7#1278C211yesmaximal, characteristic, complemented
8C22×Q16—22no—
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12#494—22no—
13#494—22no—
14#492—21no—
15#489—22no—
16#487—21no—
17#487—21no—
18#486—21no—
19#461—22no—
20#448—22no—
21#431C2211yescharacteristic
22#423C2211yescharacteristic
23#417—22no—
24C4×M16C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#343—22no—
27#332—22no—
28D4⋊C8—22no—
29C2×#41—21no—
30C22×Q8—21no—
31C22×Q8D412yes—
32C2×Q16—28no—
33C2×Q16—48no—
34C2×SD16—24no—
35C2×M16—21no—
36C2×M16D412yes—
37C22×C8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8—21no—
45C4⋊Q8D412yes—
46C4⋊Q8D411yescharacteristic
47#127D411yescharacteristic
48#124—21no—
49#121—21no—
50#121—21no—
51C4×D4—21no—
52C4×D4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C42C2311yescharacteristic
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8D412yes—
61#103—24no—
62#103—24no—
63#102—24no—
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—22no—
75C2×Q8—22no—
76C2×Q8—44no—
77C2×Q8—44no—
78C2×Q8—41no—
79C2×D4—21no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—41no—
83C22×C4—22no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4#4111yescharacteristic
89Q16—416no—
90SD16—84no—
91M16—22no—
92M16—22no—
93M16—42no—
94C2×C8—24no—
95C2×C8—22no—
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—41no—
99C4⋊C4—22no—
100C4⋊C4—21no—
101C4⋊C4—22no—
102C4⋊C4—21no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C42C2×D411yescharacteristic
112C42C2×D411yescharacteristic
113C42C2×D411yescharacteristic
114C42C2×D411yescharacteristic
115C23—41no—
116C23#12111yescharacteristic
117Q8—44no—
118Q8—44no—
119Q8—42no—
120Q8—44no—
121Q8—82no—
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—82no—
132C2×C4—81no—
133C2×C4—81no—
134C2×C4—82no—
135C2×C4#12011yescharacteristic
136C2×C4#12011yescharacteristic
137C2×C4#12011yescharacteristic
138C2×C4#12111yescharacteristic
139C2×C4#12111yescharacteristic
140C2×C4#12711yescharacteristic
141C2×C4#13512yes—
142C2×C4#13712yes—
143C8—24no—
144C8—42no—
145C8—42no—
146C8—81no—
147C22—21no—
148C22—21no—
149C22—81no—
150C22—81no—
151C22—81no—
152C22#39211yescharacteristic
153C22#46412yes—
154C4—21no—
155C4—21no—
156C4—82no—
157C4—82no—
158C4—81no—
159C4#45212yes—
160C4#45312yes—
161C4#47312yes—
162C4#49511yescharacteristic
163C4#49711yescharacteristic
164C2—22no—
165C2—161no—
166C2#200211yescharacteristic
167C2#201511yescharacteristic
168C2#202011yescharacteristic
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×#494C2semidirect
4. #1997C2semidirect
5. #1549C2semidirect
7. #1278C2semidirect

Maximal subgroups

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

subgroupindex
#32062
C2×#4942
#31412
#19972
#15492
#14922
#12782