Finite Group Databasebeta

← #3428⋊C3 #85579

Subgroups — #3428⋊C3 #85579

Order 384.

365 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 84 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×#41⋊C3C211yesmaximal, characteristic
2#520⋊C3C212yesmaximal, complemented
3#519⋊C3C212yesmaximal, complemented
4C4×#41⋊C3C212yesmaximal, complemented
5#3428C311yesmaximal, characteristic, complemented, Sylow
6C2×#41⋊C3C414yescomplemented
7C2×#41⋊C3C2211yescharacteristic
8C2×#41⋊C3C2211yescharacteristic
9C22×SL(2,𝔽3)C2211yescharacteristic
10C3×#117—41nomaximal
11C4×SL(2,𝔽3)C2214yescomplemented
12C22×#41C2×C311yescharacteristic
13#520—32no—
14#520C2×C312yescomplemented
15#519C2×C312yes—
16#518—32no—
17C4×#41—32no—
18C4×#41C2×C312yescomplemented
19C2×#117—31no—
20C2×#117—31no—
21C22×C3×C4—41no—
22#41⋊C3C2×C412yescomplemented
23#41⋊C3C2×C412yescomplemented
24C2×SL(2,𝔽3)C2×C412yescomplemented
25C2×SL(2,𝔽3)C2311yescharacteristic
26C3×C4⋊C4—42no—
27C3×C22⋊C4—42no—
28C3×C42—42no—
29C2×#41—31no—
30C2×#41—31no—
31C2×#41—34no—
32C2×#41C3×C414yescomplemented
33C2×#41C22×C311yescharacteristic
34C2×#41C22×C311yescharacteristic
35C22×Q8C22×C311yescharacteristic
36C22×D4—31no—
37C23×C4—31no—
38C4×Q8—34no—
39C4×Q8C22×C314yes—
40C4×D4—38no—
41C4×D4—34no—
42C4×D4—34no—
43#117—32no—
44#117—31no—
45#117—32no—
46#117—32no—
47#117—32no—
48#117—32no—
49#117—32no—
50#117A411yescharacteristic
51C2×C4⋊C4—32no—
52C2×C4⋊C4—32no—
53C2×C22⋊C4—32no—
54C2×C22⋊C4—32no—
55C2×C42—32no—
56C2×C42—32no—
57C23×C3—41no—
58C2×C3×C4—44no—
59C2×C3×C4—41no—
60C2×C3×C4—44no—
61C2×C3×C4—41no—
62SL(2,𝔽3)C22×C411yescharacteristic
63C24—31no—
64#41—34no—
65#41—34no—
66#41—34no—
67#41—34no—
68#41—32no—
69#41—32no—
70#41C2×C3×C412yes—
71#41C2×C3×C412yes—
72C2×Q8—31no—
73C2×Q8—32no—
74C2×Q8C2×C3×C412yes—
75C2×Q8C23×C311yescharacteristic
76C2×D4—31no—
77C2×D4—31no—
78C2×D4—34no—
79C2×D4—32no—
80C2×D4—34no—
81C22×C4—31no—
82C22×C4—34no—
83C22×C4—34no—
84C22×C4—31no—
85C22×C4—31no—
86C22×C4—31no—
87C22×C4—31no—
88C22×C4—34no—
89C22×C4—34no—
90C22×C4C2×A411yescharacteristic
91C4⋊C4—34no—
92C4⋊C4—32no—
93C4⋊C4—34no—
94C4⋊C4C2×A412yes—
95C22⋊C4—34no—
96C22⋊C4—32no—
97C22⋊C4—34no—
98C22⋊C4C2×A412yes—
99C42—34no—
100C42—34no—
101C42—32no—
102C42C2×A412yes—
103C22×C3—42no—
104C22×C3—41no—
105C22×C3—82no—
106C3×C4—42no—
107C3×C4—42no—
108C3×C4—84no—
109C23—31no—
110C23—31no—
111C23—34no—
112C23—122no—
113C23C22×A411yescharacteristic
114Q8—32no—
115Q8—31no—
116Q8—32no—
117Q8C22×C3×C411yescharacteristic
118D4—32no—
119D4—32no—
120D4—32no—
121D4—32no—
122D4—38no—
123C2×C4—32no—
124C2×C4—32no—
125C2×C4—31no—
126C2×C4—31no—
127C2×C4—32no—
128C2×C4—34no—
129C2×C4—32no—
130C2×C4—34no—
131C2×C4—32no—
132C2×C4—32no—
133C2×C4—34no—
134C2×C4—124no—
135C2×C4—124no—
136C2×C4C4×A414yes—
137C2×C4C22×A414yes—
138C2×C4C22×A411yescharacteristic
139C2×C4C22×A411yescharacteristic
140C2×C3—41no—
141C2×C3—42no—
142C2×C3—82no—
143C22—22no—
144C22—32no—
145C22—32no—
146C22—62no—
147C22—62no—
148C22—124no—
149C22C2×C4×A412yes—
150C22C23×A411yescharacteristic
151C4—24no—
152C4—31no—
153C4—32no—
154C4—31no—
155C4—64no—
156C4C2×C4×A412yes—
157C4C2×C4×A412yes—
158C3—41noSylow
159C2—22no—
160C2—62no—
161C2—62no—
162C2C22×C4×A411yescharacteristic
163C2#594512yes—
subgroupG/H
centerC2×C4C22×A4
commutator, hypocenterQ8C22×C3×C4
Frattini, socleC22C23×A4
Fitting#3428C3
prime radicalC22×#41⋊C3C2
cosocleC2×Q8C23×C3
hypercenter#117A4
LCSG ▷ Q8
UCSC1 ◁ C2×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #520⋊C3C2semidirect
3. #519⋊C3C2semidirect
4. C4×#41⋊C3C2semidirect
5. #3428C3semidirect
6. C2×#41⋊C3C4semidirect
11. C4×SL(2,𝔽3)C22semidirect
14. #520C2×C3semidirect
18. C4×#41C2×C3semidirect
22. #41⋊C3C2×C4semidirect
23. #41⋊C3C2×C4semidirect
24. C2×SL(2,𝔽3)C2×C4semidirect
32. C2×#41C3×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#41⋊C32
#520⋊C32
#519⋊C32
C4×#41⋊C32
#34283
C3×#1174