Finite Group Databasebeta

← #20294

Subgroups — #20294

Order 256.

304 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 99 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#3373C212yesmaximal, complemented
2#3329C212yesmaximal, complemented
3#1708C211yesmaximal, characteristic, complemented
4#1704C212yesmaximal, complemented
5#1701C212yesmaximal, complemented
6#1651C212yesmaximal, complemented
7#1630C211yesmaximal, characteristic, complemented
8#1624C211yesmaximal, characteristic
9#1590C212yesmaximal, complemented
10#559C2211yescharacteristic
11#555C2211yescharacteristic
12D4×Q8C2212yes—
13#498C2211yescharacteristic
14#492C2211yescharacteristic
15#489—22no—
16#487C2212yes—
17#486C2212yes—
18#485—22no—
19#477C2212yes—
20#476—22no—
21#475C2212yes—
22#468—22no—
23#463—22no—
24#461C2212yes—
25#459C2212yes—
26#448—22no—
27#446C2212yes—
28#442—22no—
29C4×SD16C2212yes—
30#436C2212yes—
31#419—22no—
32#334C2212yes—
33#333C2211yescharacteristic
34#332C2212yes—
35#331C2211yescharacteristic
36Q8⋊C8C2212yes—
37D4⋊C8C2212yes—
38C82C2211yescharacteristic
39C22×Q8—22no—
40C2×SD16—22no—
41C2×SD16—41no—
42C2×SD16D412yes—
43C2×D8—22no—
44C2×M16—22no—
45C4⋊Q8—22no—
46C4⋊Q8C2312yes—
47C4⋊Q8C2311yescharacteristic
48#127C2311yescharacteristic
49#124—22no—
50#124—41no—
51#123—41no—
52C22⋊Q8—22no—
53C22⋊Q8—42no—
54#121—22no—
55C4×Q8C2312yes—
56C4×D4—22no—
57C4×D4—22no—
58C4×D4C2312yes—
59#117—22no—
60#107—22no—
61#106—22no—
62#106—41no—
63#106D412yes—
64C4⋊C8—22no—
65C4⋊C8C2312yes—
66C4⋊C8C2312yes—
67#103—22no—
68#103—22no—
69#103—22no—
70#103—22no—
71#103—22no—
72#103D412yes—
73#102—22no—
74#102—22no—
75#102—22no—
76#102D412yes—
77C22⋊C8—22no—
78#97—22no—
79C4×C8C2311yescharacteristic
80C4×C8C2312yes—
81C2×Q8—22no—
82C2×Q8—21no—
83C2×Q8—42no—
84C2×Q8—44no—
85C2×Q8—82no—
86C2×Q8C2×D412yes—
87C2×D4—22no—
88C2×D4—21no—
89C2×D4—41no—
90C2×D4C2×D412yes—
91C22×C4—22no—
92C22×C4—41no—
93C22×C4—42no—
94SD16—42no—
95SD16—82no—
96SD16—82no—
97D8—44no—
98M16—82no—
99C2×C8—21no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8C2×D412yes—
103C2×C8C2×D412yes—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—42no—
109C4⋊C4C2×D412yes—
110C4⋊C4C2×D412yes—
111C22⋊C4—22no—
112C22⋊C4—42no—
113C22⋊C4—41no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C42—22no—
117C42—41no—
118C42C2411yescharacteristic
119C23—22no—
120C23—41no—
121Q8—22no—
122Q8—22no—
123Q8—42no—
124Q8—42no—
125Q8—81no—
126Q8—84no—
127D4—22no—
128D4—22no—
129D4—42no—
130D4—42no—
131D4—81no—
132D4—161no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—84no—
141C2×C4—161no—
142C2×C4C22×D412yes—
143C2×C4H(5,𝔽2)11yescharacteristic
144C8—22no—
145C8—22no—
146C8—22no—
147C8—42no—
148C8—42no—
149C22—42no—
150C22—42no—
151C22—42no—
152C22—82no—
153C22—81no—
154C22D4211yescharacteristic
155C4—21no—
156C4—42no—
157C4—42no—
158C4—42no—
159C4—82no—
160C4—81no—
161C4#57512yes—
162C4#57712yes—
163C2—82no—
164C2—161no—
165C2#327512yes—
166C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#1624C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3373C2semidirect
2. #3329C2semidirect
3. #1708C2semidirect
4. #1704C2semidirect
5. #1701C2semidirect
6. #1651C2semidirect
7. #1630C2semidirect
9. #1590C2semidirect

Maximal subgroups

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

subgroupindex
#33732
#33292
#17082
#17042
#17012
#16512
#16302
#16242
#15902