Finite Group Databasebeta

← #18364

Subgroups — #18364

Order 256.

461 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 115 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#3236C212yesmaximal, complemented
3#2633C211yesmaximal, characteristic
4#1614C214yesmaximal, complemented
5#1608C214yesmaximal, complemented
6#1515C212yesmaximal, complemented
7#1466C211yesmaximal, characteristic
8#554—22no—
9#554—24no—
10D4×Q8—22no—
11#544—21no—
12#543C2214yes—
13#533C2212yes—
14C2×#125C2211yescharacteristic
15C2×C22⋊Q8—22no—
16#518—21no—
17C2×C4×D4C2211yescharacteristic
18#501—22no—
19#491—22no—
20#484C2214yes—
21#483C2214yes—
22#479—24no—
23#432C2212yes—
24#399C2212yes—
25#398—21no—
26#397C2212yes—
27#394—21no—
28#385—22no—
29#383C2211yescharacteristic
30#328C2218yes—
31#323C2214yes—
32C22×Q8—22no—
33C22×D4D411yescharacteristic
34C23×C4D412yes—
35C4⋊Q8—24no—
36C4⋊Q8D412yes—
37C4⋊Q8D412yes—
38#125—24no—
39#125—22no—
40#125—22no—
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×Q8D414yes—
51C4×D4—24no—
52C4×D4—24no—
53C4×D4—22no—
54C4×D4—22no—
55#117—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C22⋊C4—24no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4C2312yes—
68C2×C42C2311yescharacteristic
69#107—24no—
70#106—24no—
71C4⋊C8—22no—
72#103—24no—
73#103—24no—
74C22⋊C8C2314yes—
75#97—22no—
76#95—21no—
77#95—21no—
78#95—22no—
79#95C2312yes—
80C24C2×D412yes—
81C2×Q8—24no—
82C2×Q8—22no—
83C2×Q8—84no—
84C2×Q8C2×D414yes—
85C2×D4—24no—
86C2×D4—22no—
87C22×C4—24no—
88C22×C4—24no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—82no—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D412yes—
97C22×C4C2×D411yescharacteristic
98C22×C4C2411yescharacteristic
99C2×C8—24no—
100C4⋊C4—24no—
101C4⋊C4—28no—
102C4⋊C4—22no—
103C4⋊C4—24no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—24no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—42no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—82no—
114C4⋊C4C2×D414yes—
115C22⋊C4—28no—
116C22⋊C4—24no—
117C22⋊C4—24no—
118C22⋊C4—44no—
119C22⋊C4—82no—
120C42—24no—
121C42C2×D412yes—
122C42C2×D412yes—
123C23—22no—
124C23—22no—
125C23—42no—
126C23—42no—
127C23#13714yes—
128C23C22×D411yescharacteristic
129Q8—44no—
130Q8—44no—
131Q8—84no—
132D4—28no—
133C2×C4—22no—
134C2×C4—24no—
135C2×C4—24no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—44no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—82no—
150C2×C4—84no—
151C2×C4—84no—
152C2×C4#12012yes—
153C2×C4#12011yescharacteristic
154C2×C4#12011yescharacteristic
155C2×C4C22×D412yes—
156C8—44no—
157C22—21no—
158C22—21no—
159C22—24no—
160C22—44no—
161C22—42no—
162C22—42no—
163C22—82no—
164C22C2×#12011yescharacteristic
165C22C2×#13712yes—
166C4—22no—
167C4—21no—
168C4—21no—
169C4—22no—
170C4—42no—
171C4—44no—
172C4—44no—
173C4—82no—
174C4—81no—
175C4—81no—
176C2—22no—
177C2—44no—
178C2#300411yescharacteristic
179C2#301911yescharacteristic
180C2#302811yescharacteristic
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. #3236C2semidirect
4. #1614C2semidirect
5. #1608C2semidirect
6. #1515C2semidirect

Maximal subgroups

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

subgroupindex
#35012
#32362
#26332
#16142
#16082
#15152
#14662