Finite Group Databasebeta

← C3⋊#7317 #10682488

Subgroups — C3⋊#7317 #10682488

Order 768.

432 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C3×#1784C211yesmaximal, characteristic
2#72133C212yesmaximal
3#7317—31nomaximal, Sylow
4C23×C3×Q8C2211yescharacteristic
5C2×C3×#101—22no—
6C3×#407C412yes—
7C2×#836—22no—
8C2×#836—22no—
9#5207C414yes—
10#1784—32no—
11#1784D311yescharacteristic
12C22×C3×Q8—44no—
13C22×C3×Q8—42no—
14C22×C3×Q8D411yescharacteristic
15C22×C3×Q8D412yes—
16C22×C3×Q8Q811yescharacteristic
17C23×C3×C4C2×C412yes—
18C23×C3×C4C2×C411yescharacteristic
19C2×C3×M16—22no—
20C2×#208—24no—
21C3×#101—44no—
22C3×C22⋊C8—42no—
23#836—44no—
24#836—44no—
25#830—44no—
26C23×Q8C2×D311yescharacteristic
27C2×#101—22no—
28C2×#101—62no—
29C2×#101—62no—
30#407—34no—
31#407Dic312yes—
32C24×C3C4211yescharacteristic
33C2×C3×Q8—24no—
34C2×C3×Q8—24no—
35C2×C3×Q8—24no—
36C2×C3×Q8—41no—
37C2×C3×Q8—42no—
38C2×C3×Q8—41no—
39C2×C3×Q8—42no—
40C2×C3×Q8—416no—
41C22×C3×C4—24no—
42C22×C3×C4—22no—
43C22×C3×C4—42no—
44C22×C3×C4—42no—
45C22×C3×C4—42no—
46C22×C3×C4C22⋊C412yes—
47C22×C3×C4C22⋊C411yescharacteristic
48C22×C3×C4C4⋊C412yes—
49C22×C3×C4C4⋊C411yescharacteristic
50C3×M16—44no—
51C2×C3×C8—42no—
52#208—44no—
53#208—44no—
54C2×C3⋊C8—44no—
55C22×Q8—44no—
56C22×Q8—42no—
57C22×Q8Dic611yescharacteristic
58C22×Q8D1211yescharacteristic
59C22×Q8#6712yes—
60C23×C4C4×D312yes—
61C23×C4C2×Dic311yescharacteristic
62C2×M16—22no—
63C2×M16—64no—
64#101—44no—
65#101—124no—
66#101—124no—
67C22⋊C8—42no—
68C22⋊C8—124no—
69C23×C3—21no—
70C23×C3—22no—
71C23×C3—42no—
72C23×C3#9511yescharacteristic
73C3×Q8—44no—
74C3×Q8—48no—
75C3×Q8—44no—
76C2×C3×C4—22no—
77C2×C3×C4—22no—
78C2×C3×C4—22no—
79C2×C3×C4—21no—
80C2×C3×C4—22no—
81C2×C3×C4—22no—
82C2×C3×C4—22no—
83C2×C3×C4—22no—
84C2×C3×C4—48no—
85C2×C3×C4—44no—
86C2×C3×C4#9914yes—
87C2×C3×C4#9912yes—
88C3×C8—82no—
89C3⋊C8—84no—
90C24C4×Dic311yescharacteristic
91C2×Q8—24no—
92C2×Q8—24no—
93C2×Q8—24no—
94C2×Q8—41no—
95C2×Q8—42no—
96C2×Q8—41no—
97C2×Q8—42no—
98C2×Q8—416no—
99C22×C4—24no—
100C22×C4—22no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4Dic3⋊C412yes—
105C22×C4C4⋊Dic311yescharacteristic
106C22×C4(C2×D3)⋊C412yes—
107C22×C4#21711yescharacteristic
108M16—44no—
109M16—124no—
110M16—124no—
111C2×C8—42no—
112C2×C8—124no—
113C22×C3—22no—
114C22×C3—24no—
115C22×C3—41no—
116C22×C3—41no—
117C22×C3—42no—
118C22×C3—41no—
119C22×C3#34211yescharacteristic
120C22×C3#34212yes—
121C3×C4—24no—
122C3×C4—22no—
123C3×C4—41no—
124C3×C4—42no—
125C23—21no—
126C23—22no—
127C23—42no—
128C23C3⋊#9511yescharacteristic
129Q8—48no—
130Q8—44no—
131Q8—44no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—44no—
141C2×C4—48no—
142C2×C4#80614yes—
143C2×C4#83412yes—
144C8—82no—
145C8—244no—
146C2×C3—21no—
147C2×C3—22no—
148C2×C3—42no—
149C2×C3#130411yescharacteristic
150C22—22no—
151C22—24no—
152C22—41no—
153C22—41no—
154C22—42no—
155C22—41no—
156C22#452611yescharacteristic
157C22#452712yes—
158C4—22no—
159C4—24no—
160C4—41no—
161C4—42no—
162C3#731711yescharacteristic, complemented, Sylow
163C2—21no—
164C2—22no—
165C2—42no—
166C2C3⋊#130411yescharacteristic
subgroupG/H
centerC2C3⋊#1304
commutator, prime radicalC24×C3C42
FrattiniC23×Q8C2×D3
FittingC3×#1784C2
socleC2×C3#1304
cosocleC23×C3×Q8C22
hypercenter#1784D3
hypocenterC3#7317
LCSG ▷ C24×C3 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ C23×Q8 ◁ #1784

Complements

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

NH ≅ G/Ntype
162. C3#7317semidirect

Maximal subgroups

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

subgroupindex
C3×#17842
#721332
#73173