Finite Group Databasebeta

← #19933

Subgroups — #19933

Order 256.

312 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 103 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#3367C212yesmaximal
2#3342C211yesmaximal, characteristic, complemented
3D4×Q16C211yesmaximal, characteristic, complemented
4#1707C211yesmaximal, characteristic, complemented
5#1700C212yesmaximal, complemented
6#1673C211yesmaximal, characteristic, complemented
7#1669C211yesmaximal, characteristic, complemented
8#1652C212yesmaximal, complemented
9#1651C212yesmaximal, complemented
10#1591C211yesmaximal, characteristic, complemented
11C8×D8C211yesmaximal, characteristic
12C22×Q16—21no—
13#556C2212yes—
14D4×Q8C2212yes—
15#500C2211yescharacteristic
16#498C2212yes—
17#494C2211yescharacteristic
18#489—22no—
19#487C2212yes—
20#485—22no—
21#477—22no—
22#475C2212yes—
23#474C2212yes—
24#467—21no—
25#467—21no—
26#463—22no—
27#462C2212yes—
28#461C2212yes—
29#451—22no—
30#446C2212yes—
31#440—22no—
32C4×Q16C2211yescharacteristic
33C4×D8C2211yescharacteristic
34C8×D4C2211yescharacteristic
35C8×D4C2211yescharacteristic
36#427—21no—
37#335C2211yescharacteristic
38#335C2211yescharacteristic
39#332C2212yes—
40#332C2212yes—
41Q8⋊C8C2212yes—
42D4⋊C8C2211yescharacteristic
43D4⋊C8C2211yescharacteristic
44C82C2211yescharacteristic
45C22×Q8—22no—
46C2×Q16—22no—
47C2×Q16—21no—
48C2×Q16—21no—
49C2×Q16—44no—
50C2×Q16D411yescharacteristic
51C2×SD16—22no—
52C2×D8D411yescharacteristic
53C22×C8—21no—
54C22×C8—21no—
55C4⋊Q8—22no—
56C4⋊Q8C2312yes—
57C4⋊Q8C2312yes—
58#126—42no—
59#124—22no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—42no—
63C4×Q8—22no—
64C4×Q8C2312yes—
65C4×D4—22no—
66C4×D4C2311yescharacteristic
67C4×D4C2311yescharacteristic
68#117—22no—
69#107—21no—
70#107—21no—
71#107—22no—
72#107D411yescharacteristic
73#107D411yescharacteristic
74#106—22no—
75#106—42no—
76C4⋊C8—22no—
77C4⋊C8C2311yescharacteristic
78C4⋊C8C2311yescharacteristic
79C4⋊C8C2312yes—
80#103—22no—
81#103—22no—
82#103—22no—
83#103—22no—
84#103—22no—
85#103D412yes—
86#102—22no—
87#102D411yescharacteristic
88#102D411yescharacteristic
89C22⋊C8—21no—
90C22⋊C8—21no—
91#97—22no—
92C4×C8C2311yescharacteristic
93C4×C8C2311yescharacteristic
94C4×C8C2311yescharacteristic
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—42no—
98C2×Q8—44no—
99C2×Q8—82no—
100C2×Q8C2×D412yes—
101C2×D4C2×D411yescharacteristic
102C2×D4C2×D411yescharacteristic
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—42no—
106Q16—22no—
107Q16—44no—
108Q16—42no—
109Q16—81no—
110SD16—44no—
111D8—22no—
112C2×C8—21no—
113C2×C8—22no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—42no—
117C2×C8—42no—
118C2×C8C2×D411yescharacteristic
119C2×C8C2×D411yescharacteristic
120C2×C8C2×D411yescharacteristic
121C2×C8C2×D411yescharacteristic
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4C2×D411yescharacteristic
130C4⋊C4C2×D411yescharacteristic
131C4⋊C4C2×D412yes—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—42no—
137C42—22no—
138C42—42no—
139C42C2411yescharacteristic
140C23—21no—
141C23—21no—
142Q8—22no—
143Q8—22no—
144Q8—42no—
145Q8—42no—
146Q8—42no—
147Q8—42no—
148Q8—84no—
149D4—21no—
150D4—21no—
151D4—21no—
152D4#13512yes—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—84no—
164C2×C4C22×D411yescharacteristic
165C2×C4C22×D411yescharacteristic
166C2×C4H(5,𝔽2)11yescharacteristic
167C8—22no—
168C8—21no—
169C8—21no—
170C8—21no—
171C8—42no—
172C8—41no—
173C8—41no—
174C8AGL(1,ℤ/8)12yes—
175C22—22no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22D4211yescharacteristic
182C4—21no—
183C4—41no—
184C4—41no—
185C4—42no—
186C4—42no—
187C4—82no—
188C4—82no—
189C4C2×#13511yescharacteristic
190C4C2×AGL(1,ℤ/8)11yescharacteristic
191C4#57711yescharacteristic
192C4#57811yescharacteristic
193C2—42no—
194C2—81no—
195C2#327211yescharacteristic
196C2#328511yescharacteristic
197C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radicalC2×D8D4
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
2. #3342C2semidirect
3. D4×Q16C2semidirect
4. #1707C2semidirect
5. #1700C2semidirect
6. #1673C2semidirect
7. #1669C2semidirect
8. #1652C2semidirect
9. #1651C2semidirect
10. #1591C2semidirect

Maximal subgroups

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

subgroupindex
#33672
#33422
D4×Q162
#17072
#17002
#16732
#16692
#16522
#16512
#15912
C8×D82