Finite Group Databasebeta

← C4×#986 #85342

Subgroups — C4×#986 #85342

Order 384.

304 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 48 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×#986C211yesmaximal, characteristic
2C4×#41⋊C3C211yesmaximal, characteristic, complemented
3#5426C211yesmaximal, characteristic, complemented
4#5394C212yesmaximal, complemented
5C4×GL(2,𝔽3)C212yesmaximal, complemented
6C4×AGL(1,ℤ/8)—31nomaximal, Sylow
7C2×#41⋊C3C2211yescharacteristic
8#986C414yesdirect product factor
9C2×GL(2,𝔽3)C2212yes—
10C4×D12—41nomaximal
11C4×SL(2,𝔽3)C2211yescharacteristic, complemented
12C4×SL(2,𝔽3)C2211yescharacteristic, complemented
13#859C2212yescomplemented
14C2×AGL(1,ℤ/8)—31no—
15C4×#41D311yescharacteristic, complemented
16C2×C4×D4—31no—
17#442—32no—
18#440—32no—
19C4×SD16—32no—
20C4×D8—32no—
21#428—31no—
22#418—31no—
23#417—31no—
24C4×M16—31no—
25C2×D12—41no—
26C2×C4×D3—42no—
27#41⋊C3C2×C412yescomplemented
28C2×SL(2,𝔽3)C2311yescharacteristic
29GL(2,𝔽3)C2×C414yescomplemented
30C3×C42—41no—
31(C2×D3)⋊C4—42no—
32C4⋊Dic3—41no—
33C2×#41C2×D311yescharacteristic
34C22×D4—31no—
35C23×C4—61no—
36AGL(1,ℤ/8)—34no—
37AGL(1,ℤ/8)—34no—
38C2×SD16—32no—
39C2×D8—32no—
40C2×M16—31no—
41C2×M16—31no—
42C4×Q8C2×D311yescharacteristic
43C4×Q8C2×D311yescharacteristic
44C4×D4—31no—
45C4×D4—31no—
46C4×D4—32no—
47C4×D4—32no—
48C4×D4—61no—
49C4×D4—61no—
50#117—31no—
51C2×C4⋊C4—31no—
52C2×C22⋊C4—61no—
53C2×C42—31no—
54#107—32no—
55#106—32no—
56#103—32no—
57#102—32no—
58#102—32no—
59#102—32no—
60#97—32no—
61C4×C8—32no—
62C22×D3—42no—
63C2×C3×C4—41no—
64C2×C3×C4—41no—
65C2×C3×C4—41no—
66C2×Dic3—42no—
67D12—44no—
68C4×D3—84no—
69SL(2,𝔽3)C22×C411yescharacteristic, complemented
70C24—61no—
71#41—32no—
72#41C4×D312yescomplemented
73C2×Q8C22×D311yescharacteristic
74C2×D4—31no—
75C2×D4—32no—
76C2×D4—34no—
77C2×D4—62no—
78C2×D4—61no—
79C22×C4—31no—
80C22×C4—31no—
81C22×C4—31no—
82C22×C4—62no—
83C22×C4—61no—
84C22×C4—62no—
85C22×C4—124no—
86SD16—34no—
87SD16—34no—
88D8—38no—
89M16—34no—
90M16—62no—
91C2×C8—32no—
92C2×C8—32no—
93C4⋊C4—31no—
94C4⋊C4—31no—
95C4⋊C4—32no—
96C4⋊C4—61no—
97C22⋊C4—61no—
98C22⋊C4—62no—
99C22⋊C4—62no—
100C42—31no—
101C42—31no—
102C42—31no—
103C42S411yescharacteristic
104C22×C3—41no—
105C2×D3—44no—
106C2×D3—82no—
107C2×D3—82no—
108C3×C4—42no—
109C3×C4—42no—
110C3×C4—81no—
111Dic3—82no—
112C23—31no—
113C23—64no—
114C23—62no—
115C23—122no—
116C23—122no—
117Q8—31no—
118Q8C2×C4×D311yescharacteristic
119D4—32no—
120D4—34no—
121D4—61no—
122D4—62no—
123D4—64no—
124C2×C4—31no—
125C2×C4—32no—
126C2×C4—32no—
127C2×C4—31no—
128C2×C4—31no—
129C2×C4—62no—
130C2×C4—62no—
131C2×C4—121no—
132C2×C4—124no—
133C2×C4—122no—
134C2×C4—124no—
135C2×C4C2×S411yescharacteristic
136C2×C4C2×S411yescharacteristic
137C2×C4C2×S411yescharacteristic
138C8—34no—
139C8—62no—
140C2×C3—41no—
141C2×C3—41no—
142C2×C3—41no—
143D3—84no—
144C22—32no—
145C22—61no—
146C22—61no—
147C22—64no—
148C22—128no—
149C22—122no—
150C22—122no—
151C22C22×S411yescharacteristic
152C4—21no—
153C4—31no—
154C4—31no—
155C4—61no—
156C4—61no—
157C4—122no—
158C4C4×S412yes—
159C4#98612yesdirect product factor
160C3—41noSylow
161C2—62no—
162C2—124no—
163C2C2×C4×S411yescharacteristic
164C2C2×#98611yescharacteristic
165C2#592311yescharacteristic
subgroupG/H
centerC2×C4C2×S4
commutator, hypocenterSL(2,𝔽3)C22×C4
Frattini, socleC22C22×S4
FittingC4×#41D3
prime radicalC2×#986C2
cosocleC2×SL(2,𝔽3)C23
hypercenterC42S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#41⋊C3C2semidirect
3. #5426C2semidirect
4. #5394C2semidirect
5. C4×GL(2,𝔽3)C2semidirect
8. #986C4direct (N × H)
11. C4×SL(2,𝔽3)C22semidirect
12. C4×SL(2,𝔽3)C22semidirect
13. #859C22semidirect
15. C4×#41D3semidirect
27. #41⋊C3C2×C4semidirect
29. GL(2,𝔽3)C2×C4semidirect
69. SL(2,𝔽3)C22×C4semidirect
72. #41C4×D3semidirect
159. C4#986direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#9862
C4×#41⋊C32
#54262
#53942
C4×GL(2,𝔽3)2
C4×AGL(1,ℤ/8)3
C4×D124