Finite Group Databasebeta

← #2341⋊C3 #73175

Subgroups — #2341⋊C3 #73175

Order 384.

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

normal subgroups only

This group has 237 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
1C22×#41⋊C3C211yesmaximal, characteristic
2C2×C4×SL(2,𝔽3)C212yesmaximal, complemented
3#2341C311yesmaximal, characteristic, complemented, Sylow
4C2×#41⋊C3—22no—
5C2×#41⋊C3C412yescomplemented
6C22×SL(2,𝔽3)C2211yescharacteristic
7C4×SL(2,𝔽3)—22no—
8C3×#95—41nomaximal
9C22×#41C2×C311yescharacteristic
10C2×C4×Q8C2×C312yes—
11#390—32no—
12#380—31no—
13C4×C22⋊C4—31no—
14C22×C3×C4—41no—
15C22×C3×C4—42no—
16#41⋊C3—22no—
17C2×SL(2,𝔽3)C2×C411yescharacteristic
18C2×SL(2,𝔽3)D412yes—
19C2×#41—22no—
20C2×#41—32no—
21C2×#41—62no—
22C2×#41C3×C412yescomplemented
23C22×Q8C22×C311yescharacteristic
24C22×D4—31no—
25C23×C4—31no—
26C4×Q8—22no—
27C4×Q8—62no—
28C2×C4⋊C4—32no—
29C2×C22⋊C4—32no—
30C2×C22⋊C4—32no—
31C2×C42—32no—
32#95—31no—
33#95A411yescharacteristic
34C23×C3—41no—
35C2×C3×C4—42no—
36C2×C3×C4—44no—
37C2×C3×C4—82no—
38C2×C3×C4—82no—
39C2×C3×C4—82no—
40SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
41C24—31no—
42#41—22no—
43#41—64no—
44#41—64no—
45#41—62no—
46C2×Q8—32no—
47C2×Q8—31no—
48C2×Q8C2×C3×C411yescharacteristic
49C2×Q8C3×D412yes—
50C2×D4—32no—
51C2×D4—32no—
52C2×D4—62no—
53C2×D4—62no—
54C22×C4—31no—
55C22×C4—32no—
56C22×C4—32no—
57C22×C4—32no—
58C22×C4—62no—
59C22×C4—62no—
60C22×C4C2×A411yescharacteristic
61C22×C4C2×A412yes—
62C4⋊C4—62no—
63C4⋊C4—62no—
64C22⋊C4—64no—
65C22⋊C4—122no—
66C42—62no—
67C42—62no—
68C22×C3—41no—
69C22×C3—42no—
70C22×C3—42no—
71C22×C3—41no—
72C22×C3—41no—
73C3×C4—84no—
74C3×C4—82no—
75C23—32no—
76C23—62no—
77C23—62no—
78C23—61no—
79C23—61no—
80C23C22×A411yescharacteristic
81Q8—31no—
82Q8—32no—
83Q8—32no—
84Q8C3×C22⋊C411yescharacteristic
85D4—62no—
86D4—62no—
87D4—64no—
88C2×C4—22no—
89C2×C4—22no—
90C2×C4—22no—
91C2×C4—31no—
92C2×C4—31no—
93C2×C4—32no—
94C2×C4—32no—
95C2×C4—62no—
96C2×C4—64no—
97C2×C4—62no—
98C2×C4—62no—
99C2×C4—62no—
100C2×C4—62no—
101C2×C4C4×A412yes—
102C2×C4#41⋊C314yescomplemented
103C2×C3—41no—
104C2×C3—41no—
105C2×C3—41no—
106C2×C3—42no—
107C2×C3—42no—
108C22—62no—
109C22—62no—
110C22—62no—
111C22—122no—
112C22—122no—
113C22C2×C4×A411yescharacteristic
114C22D4×A412yes—
115C22C2×#41⋊C312yes—
116C22#99411yescharacteristic
117C222−1+4⋊C311yescharacteristic
118C4—22no—
119C4—24no—
120C4—32no—
121C4—31no—
122C4—31no—
123C4—122no—
124C3—41noSylow
125C2—122no—
126C2A4×C22⋊C411yescharacteristic
127C2C4×#41⋊C311yescharacteristic
128C2#520⋊C311yescharacteristic
129C2#548⋊C312yes—
130C2C4⋊(#41⋊C3)12yes—
subgroupG/H
center, Frattini, socleC23C22×A4
commutatorC2×Q8C2×C3×C4
Fitting#2341C3
prime radicalC22×#41⋊C3C2
cosocleC22×Q8C22×C3
hypercenter#95A4
hypocenterQ8C3×C22⋊C4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
2. C2×C4×SL(2,𝔽3)C2semidirect
3. #2341C3semidirect
5. C2×#41⋊C3C4semidirect
22. C2×#41C3×C4semidirect
40. SL(2,𝔽3)C22⋊C4semidirect
102. C2×C4#41⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×#41⋊C32
C2×C4×SL(2,𝔽3)2
#23413
C3×#954