Finite Group Databasebeta

← C2×#3189 #60731

Subgroups — C2×#3189 #60731

Order 256.

1177 conjugacy classes of subgroups; only the 495 normal subgroups are shown.

normal subgroups only

This group has 1177 conjugacy classes of subgroups — too many to draw. Only its 495 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
1C22×C22⋊Q8C212yesmaximal
2#3189C2116yesmaximal, direct product factor
3C2×#484C214yesmaximal, complemented
4C2×#467C214yesmaximal, complemented
5C2×#451C214yesmaximal, complemented
6C22×C22⋊C8C211yesmaximal, characteristic
7C23×Q8C2212yes—
8C24×C4C2211yescharacteristic
9C22×Q16C2214yescomplemented
10C23×C8C2212yes—
11C2×C22⋊Q8—22no—
12C2×C22⋊Q8—28no—
13C2×C22⋊Q8—22no—
14C2×C22⋊Q8C2214yes—
15C2×C22⋊Q8C22116yes—
16C2×C22⋊Q8C2214yes—
17C22×C4⋊C4—22no—
18C22×C4⋊C4C2212yes—
19C22×C22⋊C4—22no—
20#484C22132yescomplemented
21#467C22132yescomplemented
22#451C22132yescomplemented
23C2×#107C2214yescomplemented
24C2×#103C2218yescomplemented
25C2×C22⋊C8C2214yes—
26C2×C22⋊C8C2218yes—
27C25D411yescharacteristic
28C22×Q8—24no—
29C22×Q8—22no—
30C22×Q8—48no—
31C22×Q8C2314yes—
32C22×Q8C2318yes—
33C23×C4—22no—
34C23×C4—22no—
35C23×C4—44no—
36C23×C4D412yes—
37C23×C4D414yes—
38C23×C4D411yescharacteristic
39C23×C4C2312yes—
40C23×C4C2311yescharacteristic
41C23×C4C2314yes—
42C2×Q16—48no—
43C2×Q16C23116yescomplemented
44C22×C8—44no—
45C22×C8C2314yes—
46C22×C8C2318yes—
47C22⋊Q8—28no—
48C22⋊Q8—28no—
49C22⋊Q8—216no—
50C22⋊Q8—216no—
51C22⋊Q8C23116yes—
52C22⋊Q8C23116yes—
53C2×C4⋊C4—216no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4C2314yes—
59C2×C4⋊C4C2318yes—
60C2×C22⋊C4—216no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—24no—
63#107C23116yescomplemented
64#103C23132yescomplemented
65C22⋊C8C23116yes—
66C24—42no—
67C24—42no—
68C24Q1618yescomplemented
69C24C2×D411yescharacteristic
70C24C2×D412yes—
71C24C2×D414yes—
72C2×Q8—28no—
73C2×Q8—24no—
74C2×Q8—216no—
75C2×Q8—416no—
76C2×Q8—48no—
77C2×Q8—416no—
78C2×Q8C2418yes—
79C22×C4—24no—
80C22×C4—28no—
81C22×C4—28no—
82C22×C4—28no—
83C22×C4—24no—
84C22×C4—24no—
85C22×C4—48no—
86C22×C4—48no—
87C22×C4—44no—
88C22×C4—42no—
89C22×C4—48no—
90C22×C4—48no—
91C22×C4C2×D414yes—
92C22×C4C2×D412yes—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D412yes—
95C22×C4C2×D418yes—
96C22×C4C2×D414yes—
97C22×C4C2414yes—
98C22×C4C2412yes—
99C22×C4C2411yescharacteristic
100Q16—416no—
101C2×C8—28no—
102C2×C8—48no—
103C2×C8C2418yes—
104C4⋊C4—216no—
105C4⋊C4—216no—
106C4⋊C4—28no—
107C4⋊C4—24no—
108C4⋊C4C2418yes—
109C22⋊C4—216no—
110C22⋊C4—216no—
111C23—24no—
112C23—22no—
113C23—24no—
114C23—22no—
115C23—44no—
116C23—416no—
117C23—44no—
118C23C2×Q1614yes—
119C23C2×Q1618yes—
120C23C2×Q16116yescomplemented
121C23C22×D412yes—
122C23C22×D414yes—
123C23C22×D411yescharacteristic
124Q8—216no—
125Q8—416no—
126Q8—48no—
127C2×C4—22no—
128C2×C4—24no—
129C2×C4—24no—
130C2×C4—28no—
131C2×C4—24no—
132C2×C4—28no—
133C2×C4—28no—
134C2×C4—28no—
135C2×C4—48no—
136C2×C4—44no—
137C2×C4—416no—
138C2×C4—416no—
139C2×C4—48no—
140C2×C4—48no—
141C2×C4C22×D412yes—
142C2×C4C22×D414yes—
143C2×C4C22×D411yescharacteristic
144C2×C4H(5,𝔽2)14yes—
145C2×C4C2511yescharacteristic
146C8—28no—
147C22—216no—
148C22—28no—
149C22—24no—
150C22—22no—
151C22—24no—
152C22—22no—
153C22—416no—
154C22#53412yes—
155C22C22×Q1614yes—
156C22C22×Q1618yes—
157C22C22×Q1612yes—
158C22#57512yes—
159C22C23×D411yescharacteristic
160C4—22no—
161C4—24no—
162C4—48no—
163C4—48no—
164C4#53412yes—
165C4C2×H(5,𝔽2)12yes—
166C2—28no—
167C2—24no—
168C2#318914yesdirect product factor
169C2C2×#53411yescharacteristic
170C2C23×Q1611yescharacteristic
171C2C2×#57511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC25D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3189C2direct (N × H)
3. C2×#484C2semidirect
4. C2×#467C2semidirect
5. C2×#451C2semidirect
9. C22×Q16C22semidirect
20. #484C22semidirect
21. #467C22semidirect
22. #451C22semidirect
23. C2×#107C22semidirect
24. C2×#103C22semidirect
43. C2×Q16C23semidirect
63. #107C23semidirect
64. #103C23semidirect
68. C24Q16semidirect
120. C23C2×Q16semidirect
168. C2#3189direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
#31892
C2×#4842
C2×#4672
C2×#4512
C22×C22⋊C82