Finite Group Databasebeta

← #20143

Subgroups — #20143

Order 256.

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

normal subgroups only

This group has 307 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#3354C212yesmaximal, complemented
2#3303C212yesmaximal, complemented
3#1716C211yesmaximal, characteristic, complemented
4#1703C212yesmaximal, complemented
5#1671C212yesmaximal, complemented
6#1654C212yesmaximal, complemented
7#1634C211yesmaximal, characteristic, complemented
8#1634C211yesmaximal, characteristic, complemented
9#1586C212yesmaximal, complemented
10#555C2211yescharacteristic
11#555C2211yescharacteristic
12#547C2212yes—
13#498C2212yes—
14#498C2211yescharacteristic
15#494C2211yescharacteristic
16#487C2212yes—
17#485—22no—
18#482—22no—
19#481—22no—
20#476C2212yes—
21#476C2212yes—
22#474C2212yes—
23#469—22no—
24#465—22no—
25#461C2212yes—
26#449—22no—
27C4×D8C2212yes—
28#436C2212yes—
29C8×D4C2212yes—
30#428—22no—
31#416—22no—
32#334C2212yes—
33#333C2211yescharacteristic
34#333C2211yescharacteristic
35#332C2212yes—
36D4⋊C8C2212yes—
37D4⋊C8C2212yes—
38#322C2211yescharacteristic
39C2×#41—22no—
40C2×Q16—41no—
41C2×SD16—22no—
42C2×D8D412yes—
43C2×M16—22no—
44C22×C8—22no—
45C4⋊Q8C2312yes—
46C4⋊Q8C2311yescharacteristic
47C4⋊Q8C2311yescharacteristic
48#124—41no—
49#124—41no—
50#123—42no—
51C22⋊Q8—22no—
52#121—22no—
53#121—22no—
54#121—22no—
55C4×D4C2312yes—
56C4×D4C2312yes—
57#117—22no—
58#117—22no—
59C2×C4⋊C4—22no—
60#107—22no—
61#107D412yes—
62#106—22no—
63#106—22no—
64#106—22no—
65#106—22no—
66#106—41no—
67C4⋊C8C2312yes—
68C4⋊C8C2312yes—
69#103—22no—
70#103—22no—
71#103—22no—
72#102—22no—
73#102—22no—
74#102—22no—
75#102D412yes—
76#102D412yes—
77C22⋊C8—22no—
78C22⋊C8—22no—
79C4×C8C2311yescharacteristic
80C4×C8C2312yes—
81#41—44no—
82#41—82no—
83C2×Q8—22no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×D4—42no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4C2×D412yes—
90C2×D4C2×D412yes—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—42no—
94Q16—82no—
95SD16—82no—
96D8—42no—
97M16—44no—
98C2×C8—21no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—82no—
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—22no—
113C22⋊C4—42no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C42—41no—
117C42—41no—
118C42C2411yescharacteristic
119C23—22no—
120C23—22no—
121Q8—42no—
122Q8—42no—
123Q8—81no—
124Q8—81no—
125D4—22no—
126D4—22no—
127D4—22no—
128D4—22no—
129D4—84no—
130D4—161no—
131D4—161no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—84no—
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—42no—
153C22—42no—
154C22—42no—
155C22D4211yescharacteristic
156C4—21no—
157C4—42no—
158C4—42no—
159C4—82no—
160C4—81no—
161C4—81no—
162C4#57512yes—
163C4#57712yes—
164C2—82no—
165C2—82no—
166C2#326712yes—
167C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
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. #3354C2semidirect
2. #3303C2semidirect
3. #1716C2semidirect
4. #1703C2semidirect
5. #1671C2semidirect
6. #1654C2semidirect
7. #1634C2semidirect
8. #1634C2semidirect
9. #1586C2semidirect

Maximal subgroups

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

subgroupindex
#33542
#33032
#17162
#17032
#16712
#16542
#16342
#15862