Finite Group Databasebeta

← #2351⋊C3 #73177

Subgroups — #2351⋊C3 #73177

Order 384.

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

normal subgroups only

This group has 177 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×C4×SL(2,𝔽3)C212yesmaximal
2C2×C4×SL(2,𝔽3)C211yesmaximal, characteristic
3#2351C311yesmaximal, characteristic, complemented, Sylow
4C22×SL(2,𝔽3)C2211yescharacteristic
5C4×SL(2,𝔽3)—22no—
6C4×SL(2,𝔽3)C412yescomplemented
7C3×#95—41nomaximal
8C2×C4×Q8C2×C312yes—
9C2×C4×Q8C2×C311yescharacteristic
10#389—32no—
11#382—31no—
12C4×C4⋊C4—31no—
13C22×C3×C4—41no—
14C22×C3×C4—42no—
15C2×SL(2,𝔽3)C2×C411yescharacteristic
16C2×SL(2,𝔽3)D411yescharacteristic
17C2×SL(2,𝔽3)Q811yescharacteristic
18C22×Q8C22×C311yescharacteristic
19C4×Q8—22no—
20C4×Q8—32no—
21C4×Q8—62no—
22C4×Q8C3×C412yescomplemented
23C2×C4⋊C4—31no—
24C2×C4⋊C4—32no—
25C2×C4⋊C4—32no—
26C2×C42—31no—
27C2×C42—32no—
28#95—32no—
29#95—31no—
30#95A411yescharacteristic
31C23×C3—41no—
32C2×C3×C4—42no—
33C2×C3×C4—44no—
34C2×C3×C4—82no—
35C2×C3×C4—82no—
36C2×C3×C4—82no—
37SL(2,𝔽3)C4⋊C411yescharacteristic, complemented
38C2×Q8—31no—
39C2×Q8—31no—
40C2×Q8—31no—
41C2×Q8C2×C3×C411yescharacteristic
42C2×Q8C3×D411yescharacteristic
43C2×Q8C3×Q811yescharacteristic
44C22×C4—31no—
45C22×C4—31no—
46C22×C4—32no—
47C22×C4C2×A411yescharacteristic
48C22×C4C2×A412yes—
49C4⋊C4—32no—
50C4⋊C4—32no—
51C4⋊C4—62no—
52C4⋊C4—62no—
53C4⋊C4—64no—
54C42—32no—
55C42—32no—
56C42—62no—
57C42—62no—
58C22×C3—41no—
59C22×C3—41no—
60C22×C3—41no—
61C22×C3—42no—
62C22×C3—42no—
63C3×C4—82no—
64C3×C4—84no—
65C23C22×A411yescharacteristic
66Q8—31no—
67Q8—31no—
68Q8—31no—
69Q8—32no—
70Q8C3×C4⋊C411yescharacteristic
71C2×C4—22no—
72C2×C4—22no—
73C2×C4—22no—
74C2×C4—31no—
75C2×C4—31no—
76C2×C4—31no—
77C2×C4—31no—
78C2×C4—31no—
79C2×C4—31no—
80C2×C4—32no—
81C2×C4—62no—
82C2×C4—62no—
83C2×C4—62no—
84C2×C4—62no—
85C2×C4C4×A412yes—
86C2×C4#41⋊C314yes—
87C2×C3—41no—
88C2×C3—41no—
89C2×C3—41no—
90C2×C3—42no—
91C2×C3—42no—
92C22C2×C4×A411yescharacteristic
93C22D4×A411yescharacteristic
94C22Q8×A411yescharacteristic
95C22C2×#41⋊C312yes—
96C222−1+4⋊C312yes—
97C4—22no—
98C4—24no—
99C4—31no—
100C4—31no—
101C4—31no—
102C4—31no—
103C4—62no—
104C4—122no—
105C3—41noSylow
106C2A4×C4⋊C411yescharacteristic
107C2C4×#41⋊C311yescharacteristic
108C2#519⋊C311yescharacteristic
109C2#548⋊C312yes—
110C2#557⋊C312yes—
subgroupG/H
center, Frattini, socleC23C22×A4
commutatorC2×Q8C2×C3×C4
Fitting#2351C3
prime radicalC22×SL(2,𝔽3)C22
cosocleC22×Q8C22×C3
hypercenter#95A4
hypocenterQ8C3×C4⋊C4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
3. #2351C3semidirect
6. C4×SL(2,𝔽3)C4semidirect
22. C4×Q8C3×C4semidirect
37. SL(2,𝔽3)C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×SL(2,𝔽3)2
#23513
C3×#954