Finite Group Databasebeta

← #12116

Subgroups — #12116

Order 256.

213 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 35 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#3197C211yesmaximal, characteristic, complemented
2#2191C211yesmaximal, characteristic, complemented
3#2190C211yesmaximal, characteristic, complemented
4#2010C211yesmaximal, characteristic, complemented
5#1340C211yesmaximal, characteristic, complemented
6#1339C211yesmaximal, characteristic, complemented
7#1320C211yesmaximal, characteristic, complemented
8#575C2211yescharacteristic
9#525—21no—
10#525—21no—
11#510—21no—
12#509—21no—
13#485—21no—
14#482—21no—
15#470C2211yescharacteristic
16#469C2211yescharacteristic
17#468—21no—
18#452—21no—
19#449—21no—
20#421—21no—
21#407C2211yescharacteristic
22#360—21no—
23#360—21no—
24#349C2211yescharacteristic
25#349C2211yescharacteristic
26H(3,ℤ/4)C2211yescharacteristic
27C2×#41—21no—
28C2×#41—21no—
29C23×C4D411yescharacteristic
30#137—41no—
31AGL(1,ℤ/8)—41no—
32#135—21no—
33#135—21no—
34C2×Q16D411yescharacteristic, complemented
35C2×SD16—21no—
36C2×D8D411yescharacteristic, complemented
37C2×M16D411yescharacteristic
38C2×M16C2311yescharacteristic
39#123—41no—
40#123—41no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43C22⋊Q8—41no—
44#121—21no—
45#121—21no—
46#121—41no—
47#120—41no—
48#120—41no—
49C4×D4—42no—
50C4×D4—41no—
51C4×D4—41no—
52#117—21no—
53#117—21no—
54Q32—41no—
55SD32—41no—
56SD32—41no—
57D16—41no—
58M32—21no—
59M32—21no—
60#107—21no—
61#106D411yescharacteristic, complemented
62#106D411yescharacteristic, complemented
63#104—42no—
64#103—21no—
65#103—21no—
66#102—21no—
67#102—21no—
68C22⋊C8—21no—
69C22⋊C8—21no—
70C24D811yescharacteristic
71#41—41no—
72#41—41no—
73#41—41no—
74#41—41no—
75#41—81no—
76#41—81no—
77C2×Q8—21no—
78C2×Q8—21no—
79C2×D4—21no—
80C2×D4—21no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4—42no—
84C2×D4—41no—
85C2×D4—41no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4D811yescharacteristic
93C22×C4C2×D411yescharacteristic
94Q16—21no—
95Q16—21no—
96Q16—41no—
97SD16—41no—
98SD16—41no—
99SD16—81no—
100D8—21no—
101D8—21no—
102D8—41no—
103M16—41no—
104M16—41no—
105C2×C8—21no—
106C2×C8C2×D411yescharacteristic
107C2×C8C2×D411yescharacteristic
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—42no—
113C4⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—41no—
117C22⋊C4—42no—
118C22⋊C4—42no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C42—22no—
123C42—41no—
124C16—41no—
125C16—41no—
126C23—21no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—81no—
131C23C2×D811yescharacteristic
132Q8—41no—
133Q8—41no—
134Q8—41no—
135Q8—41no—
136D4—41no—
137D4—41no—
138D4—41no—
139D4—41no—
140D4—81no—
141D4—82no—
142D4—82no—
143D4—84no—
144D4—82no—
145D4—81no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—81no—
158C2×C4—81no—
159C2×C4—82no—
160C2×C4—81no—
161C2×C4#12011yescharacteristic
162C2×C4AGL(1,ℤ/8)11yescharacteristic
163C8—21no—
164C8—21no—
165C8—41no—
166C22—21no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—42no—
171C22—41no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—82no—
176C22—81no—
177C22—81no—
178C22#44711yescharacteristic
179C4—21no—
180C4—41no—
181C4—81no—
182C4—81no—
183C4—82no—
184C4—81no—
185C4—81no—
186C4#45311yescharacteristic
187C4B(4,𝔽2)11yescharacteristic
188C2—21no—
189C2—41no—
190C2—81no—
191C2—161no—
192C2—161no—
193C2#159511yescharacteristic
subgroupG/H
center, socleC2#1595
commutator, Frattini, cosocleC2×M16C23
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3197C2semidirect
2. #2191C2semidirect
3. #2190C2semidirect
4. #2010C2semidirect
5. #1340C2semidirect
6. #1339C2semidirect
7. #1320C2semidirect
34. C2×Q16D4semidirect
36. C2×D8D4semidirect
61. #106D4semidirect
62. #106D4semidirect

Maximal subgroups

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

subgroupindex
#31972
#21912
#21902
#20102
#13402
#13392
#13202