Finite Group Databasebeta

← #13057

Subgroups — #13057

Order 256.

350 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C2×#454C211yesmaximal, characteristic
2#2372C211yesmaximal, characteristic, complemented
3C2×#355C211yesmaximal, characteristic, complemented
4#1793C212yesmaximal, complemented
5#1394C212yesmaximal, complemented
6C2×2−1+4C2211yescharacteristic
7C2×AGL(1,ℤ/8)—21no—
8C2×#124C2211yescharacteristic
9#454—22no—
10#454C414yescomplemented
11C2×#104—21no—
12C2×#104—22no—
13#417—22no—
14C2×#101C2212yes—
15C2×#101C2211yescharacteristic
16#391C2212yes—
17#388—22no—
18#386—21no—
19C4×C22⋊C4—22no—
20#376—21no—
21#355—22no—
222−1+4—21no—
232−1+4—22no—
242−1+4—42no—
252−1+4C2×C412yescomplemented
26C2×#41—21no—
27C2×#41—22no—
28C2×#41—41no—
29C22×Q8—41no—
30C22×Q8C2311yescharacteristic
31C22×D4D411yescharacteristic
32C23×C4—21no—
33AGL(1,ℤ/8)—24no—
34AGL(1,ℤ/8)—42no—
35C2×SD16—41no—
36C2×D8—41no—
37C2×M16—21no—
38C2×M16—22no—
39#124—22no—
40#124C2×C412yes—
41#124D412yes—
42C2×C4⋊C4D412yes—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—21no—
46C2×C42—22no—
47C2×C42D411yescharacteristic
48C2×C42D412yes—
49#104—24no—
50#104—44no—
51#103—42no—
52#102—42no—
53#101—21no—
54#101—22no—
55#101—22no—
56#101C2×C412yescomplemented
57#95—21no—
58#95—21no—
59#95—22no—
60#95—22no—
61C24—21no—
62#41—24no—
63#41—22no—
64#41—22no—
65#41—24no—
66#41—42no—
67#41—41no—
68#41—42no—
69#41—42no—
70#41—44no—
71#41—42no—
72#41—41no—
73C2×Q8—22no—
74C2×Q8—21no—
75C2×Q8—41no—
76C2×Q8—41no—
77C2×Q8—42no—
78C2×Q8—42no—
79C2×Q8—44no—
80C2×Q8C22×C411yescharacteristic
81C2×Q8C2×D411yescharacteristic
82C2×Q8#4112yes—
83C2×D4—22no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4C22⋊C412yes—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—42no—
94C22×C4—41no—
95C22×C4—44no—
96C22×C4—81no—
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D412yes—
99SD16—44no—
100D8—44no—
101M16—22no—
102M16—41no—
103M16—42no—
104M16—42no—
105C2×C8—41no—
106C2×C8—42no—
107C4⋊C4—42no—
108C22⋊C4—24no—
109C22⋊C4—24no—
110C22⋊C4—44no—
111C22⋊C4—81no—
112C42—21no—
113C42—42no—
114C42—44no—
115C42C22⋊C412yes—
116C23—22no—
117C23—42no—
118C23—42no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23#12011yescharacteristic
123Q8—22no—
124Q8—24no—
125Q8—41no—
126Q8—41no—
127Q8—42no—
128Q8—42no—
129Q8—41no—
130D4—22no—
131D4—24no—
132D4—42no—
133D4—42no—
134D4—41no—
135D4—42no—
136D4—42no—
137D4—81no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—44no—
146C2×C4—44no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—84no—
156C2×C4C2×C22⋊C411yescharacteristic
157C2×C4C4×D412yes—
158C2×C4#12112yes—
159C2×C4#12311yescharacteristic
160C8—42no—
161C8—82no—
162C22—22no—
163C22—44no—
164C22—42no—
165C22—42no—
166C22—81no—
167C22—81no—
168C22—82no—
169C22—82no—
170C22#38611yescharacteristic
171C22B(4,𝔽2)11yescharacteristic
172C22B(4,𝔽2)11yescharacteristic
173C4—21no—
174C4—22no—
175C4—22no—
176C4—21no—
177C4—42no—
178C4—44no—
179C4—42no—
180C4—82no—
181C2—21no—
182C2—21no—
183C2—82no—
184C2—82no—
185C2#188911yescharacteristic
186C2#220412yes—
subgroupG/H
center, socleC22B(4,𝔽2)
commutatorC2×Q8C22×C4
Frattini, cosocleC22×Q8C23
prime radicalC2×#454C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2372C2semidirect
3. C2×#355C2semidirect
4. #1793C2semidirect
5. #1394C2semidirect
10. #454C4semidirect
25. 2−1+4C2×C4semidirect
56. #101C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4542
#23722
C2×#3552
#17932
#13942