Finite Group Databasebeta

← #73084

Subgroups — #73084

Order 384.

185 conjugacy classes of subgroups; only the 14 normal subgroups are shown.

normal subgroups only

This group has 185 conjugacy classes of subgroups — too many to draw. Only its 14 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
1C2×#127⋊C3C211yesmaximal, characteristic, complemented
2C2×#857C211yesmaximal, characteristic, complemented
3#4623C211yesmaximal, characteristic, complemented
4#2002—31nomaximal, Sylow
5GL(2,ℤ/4)—41nomaximal
6#127⋊C3—21no—
7C2×C42⋊C3C2211yescharacteristic
8#857—21no—
9C2×AGL(1,ℤ/8)—31no—
10#535—31no—
11C2×#127D311yescharacteristic, complemented
12C2×#104—31no—
13#418—31no—
14C2×#100—31no—
15#339—31no—
16C22×A4—41no—
17C2×S4—41no—
18SL(2,ℤ/4)—41no—
19C42⋊C3D411yescharacteristic, complemented
20C2×#41—31no—
21C22×D4—31no—
22C22×D4—31no—
23AGL(1,ℤ/8)—62no—
24AGL(1,ℤ/8)—62no—
25C2×SD16—61no—
26C2×D8—61no—
27C2×M16—31no—
28C2×M16—31no—
29#127—21no—
30#127—61no—
31#127—61no—
32#124—61no—
33#121—61no—
34#121—61no—
35#120—61no—
36#120—61no—
37#117—31no—
38C2×C42C2×D311yescharacteristic
39#104—61no—
40#104—61no—
41#102—61no—
42#102—61no—
43#100—62no—
44C2×A4—41no—
45C2×A4—81no—
46S4—81no—
47#67—161no—
48C24—31no—
49C24S411yescharacteristic
50#41—61no—
51#41—61no—
52#41—61no—
53#41—61no—
54C2×Q8—31no—
55C2×D4—31no—
56C2×D4—32no—
57C2×D4—61no—
58C2×D4—61no—
59C2×D4—61no—
60C2×D4—61no—
61C2×D4—61no—
62C2×D4—61no—
63C2×D4—61no—
64C2×D4—61no—
65C2×D4—61no—
66C2×D4—61no—
67C2×D4—61no—
68C2×D4—61no—
69C2×D4—61no—
70C2×D4—61no—
71C2×D4—61no—
72C2×D4—61no—
73C22×C4—31no—
74C22×C4—61no—
75SD16—64no—
76D8—64no—
77M16—32no—
78M16—61no—
79M16—61no—
80M16—61no—
81C2×C8—61no—
82C2×C8—61no—
83C4⋊C4—31no—
84C4⋊C4—31no—
85C22⋊C4—61no—
86C22⋊C4—61no—
87C22⋊C4—61no—
88C22⋊C4—61no—
89C22⋊C4—61no—
90C42—31no—
91C42—61no—
92C42#6711yescharacteristic, complemented
93C22×C3—161no—
94C2×D3—161no—
95A4—41no—
96Dic3—161no—
97C23—21no—
98C23—61no—
99C23—61no—
100C23—61no—
101C23—61no—
102C23—61no—
103C23—61no—
104C23—121no—
105C23—121no—
106C23C2×S411yescharacteristic
107Q8—61no—
108Q8—61no—
109D4—61no—
110D4—62no—
111D4—62no—
112D4—61no—
113D4—121no—
114D4—121no—
115D4—121no—
116D4—121no—
117D4—121no—
118D4—121no—
119D4—121no—
120D4—121no—
121D4—241no—
122D4—241no—
123D4—241no—
124D4—241no—
125C2×C4—31no—
126C2×C4—31no—
127C2×C4—31no—
128C2×C4—31no—
129C2×C4—31no—
130C2×C4—31no—
131C2×C4—61no—
132C2×C4—61no—
133C2×C4—61no—
134C2×C4—121no—
135C2×C4—121no—
136C8—62no—
137C8—121no—
138C2×C3—161no—
139C2×C3—321no—
140D3—321no—
141C22—31no—
142C22—31no—
143C22—41no—
144C22—121no—
145C22—121no—
146C22—121no—
147C22—122no—
148C22—121no—
149C22—121no—
150C22—121no—
151C22—121no—
152C22—121no—
153C22—241no—
154C22GL(2,ℤ/4)11yescharacteristic
155C4—31no—
156C4—31no—
157C4—31no—
158C4—31no—
159C4—121no—
160C4—121no—
161C4—121no—
162C3—161noSylow
163C2—31no—
164C2—31no—
165C2—81no—
166C2—241no—
167C2—241no—
168C2#857⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2#857⋊C2
commutator, cosocleC2×C42⋊C3C22
Frattini, socleC23C2×S4
FittingC2×#127D3
hypocenterC42⋊C3D4
LCSG ▷ C2×C42⋊C3 ▷ C42⋊C3
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×#127⋊C3C2semidirect
2. C2×#857C2semidirect
3. #4623C2semidirect
11. C2×#127D3semidirect
19. C42⋊C3D4semidirect
92. C42#67semidirect

Maximal subgroups

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

subgroupindex
C2×#127⋊C32
C2×#8572
#46232
#20023
GL(2,ℤ/4)4