Finite Group Databasebeta

← #18213

Subgroups — #18213

Order 256.

487 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 127 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#3501C211yesmaximal, characteristic
2#3104C212yesmaximal, complemented
3#2580C211yesmaximal, characteristic
4#1614C214yesmaximal, complemented
5#1602C214yesmaximal, complemented
6#1518C212yesmaximal, complemented
7#1460C211yesmaximal, characteristic
8#554—22no—
9#554—24no—
10D4×Q8—22no—
11#544—21no—
12#543C2214yes—
13#533C2212yes—
14C2×C22⋊Q8—22no—
15#518—21no—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18#484C2214yes—
19#477—22no—
20#462—22no—
21#451C2214yes—
22C4×Q16—24no—
23#431C2212yes—
24#399C2212yes—
25#393C2212yes—
26#390—21no—
27#389—21no—
28#385—22no—
29#382C2211yescharacteristic
30#328C2218yes—
31#323C2214yes—
32C22×Q8—22no—
33C22×Q8C2311yescharacteristic
34C22×D4D411yescharacteristic
35C23×C4D412yes—
36C2×Q16—24no—
37C4⋊Q8—24no—
38C4⋊Q8D412yes—
39C4⋊Q8D412yes—
40#125—24no—
41#125—22no—
42C22⋊Q8—24no—
43C22⋊Q8—24no—
44C22⋊Q8—28no—
45C22⋊Q8—22no—
46C22⋊Q8—24no—
47C22⋊Q8—28no—
48C22⋊Q8C2314yes—
49C4×Q8—22no—
50C4×Q8—22no—
51C4×Q8—22no—
52C4×Q8D414yes—
53C4×D4—24no—
54C4×D4—24no—
55C4×D4—22no—
56C4×D4—22no—
57#117—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C22⋊C4—24no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4C2312yes—
69C2×C42—21no—
70C2×C42C2311yescharacteristic
71#107—24no—
72C4⋊C8—22no—
73#103—24no—
74#103—24no—
75C22⋊C8C2314yes—
76C4×C8—22no—
77#95—21no—
78#95—22no—
79#95C2312yes—
80C24C2×D412yes—
81C2×Q8—24no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—42no—
85C2×Q8—84no—
86C2×Q8C2×D414yes—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4Q1614yescomplemented
90C22×C4—21no—
91C22×C4—24no—
92C22×C4—24no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—82no—
98C22×C4C2×D412yes—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2411yescharacteristic
102Q16—48no—
103C2×C8—24no—
104C4⋊C4—24no—
105C4⋊C4—28no—
106C4⋊C4—22no—
107C4⋊C4—24no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—24no—
112C4⋊C4—22no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—42no—
116C4⋊C4—82no—
117C4⋊C4C2×D414yes—
118C22⋊C4—28no—
119C22⋊C4—24no—
120C22⋊C4—24no—
121C22⋊C4—44no—
122C22⋊C4—82no—
123C42—24no—
124C42—42no—
125C42C2×D412yes—
126C42C2×D412yes—
127C23—22no—
128C23—22no—
129C23—42no—
130C23—42no—
131C23C2×Q1614yes—
132C23C22×D411yescharacteristic
133Q8—24no—
134Q8—44no—
135Q8—42no—
136Q8—44no—
137Q8—84no—
138D4—28no—
139C2×C4—22no—
140C2×C4—24no—
141C2×C4—24no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—44no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—82no—
156C2×C4—84no—
157C2×C4—84no—
158C2×C4#12011yescharacteristic
159C2×C4#12011yescharacteristic
160C2×C4#12012yes—
161C2×C4C2×Q1612yes—
162C2×C4#13712yes—
163C2×C4C22×D412yes—
164C8—44no—
165C22—21no—
166C22—21no—
167C22—24no—
168C22—44no—
169C22—42no—
170C22—42no—
171C22—82no—
172C22C2×#12011yescharacteristic
173C22C22×Q1611yescharacteristic
174C22C2×#13711yescharacteristic
175C4—21no—
176C4—21no—
177C4—22no—
178C4—42no—
179C4—44no—
180C4—42no—
181C4—44no—
182C4—82no—
183C4—81no—
184C4#45114yes—
185C2—21no—
186C2—21no—
187C2—44no—
188C2C2×#45111yescharacteristic
189C2#301911yescharacteristic
190C2#302511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3104C2semidirect
4. #1614C2semidirect
5. #1602C2semidirect
6. #1518C2semidirect
89. C2×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#35012
#31042
#25802
#16142
#16022
#15182
#14602