Finite Group Databasebeta

← #10683039

Subgroups — #10683039

Order 768.

164 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 27 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×#1721C211yesmaximal, characteristic
2#67789C211yesmaximal, characteristic, complemented
3#67788C211yesmaximal, characteristic, complemented
4#7527—31nomaximal, Sylow
5C3×#493—21no—
6C3×#493C411yescharacteristic, complemented
7C3×#492C2211yescharacteristic
8C3×#486—21no—
9C3×#322C411yescharacteristic, complemented
10#4548—21no—
11#4545—21no—
12#1721D311yescharacteristic
13#1410—31no—
14#1409—31no—
15C2×C3×SD16—41no—
16C2×C3×D8—41no—
17C2×C3×D8—42no—
18C3×C4⋊Q8D411yescharacteristic
19C3×#127—21no—
20C3×#127D411yescharacteristic
21C3×#102—42no—
22C3×C4×C8—21no—
23C3×C4×C8C2×C411yescharacteristic
24#836—41no—
25#834—41no—
26#493—21no—
27#493Dic311yescharacteristic, complemented
28#492C2×D311yescharacteristic
29#486—21no—
30#356—61no—
31#353—61no—
32#322Dic311yescharacteristic, complemented
33C2×C3×Q8—21no—
34C2×C3×D4—21no—
35C2×C3×D4—41no—
36C2×C3×D4—41no—
37C2×C3×D4—41no—
38C2×C3×D4—41no—
39C3×SD16—82no—
40C3×D8—82no—
41C3×D8—84no—
42C2×C3×C8—21no—
43C2×C3×C8—22no—
44C3×C4⋊C4—41no—
45C3×C42C22⋊C411yescharacteristic
46#217—81no—
47#208—81no—
48C2×SD16—41no—
49C2×D8—41no—
50C2×D8—42no—
51C4⋊Q8#6711yescharacteristic
52#127—21no—
53#127#6711yescharacteristic
54#102—42no—
55#101—121no—
56#99—121no—
57C4×C8—21no—
58C4×C8C2×Dic311yescharacteristic
59C23×C3—41no—
60C23×C3—81no—
61C3×Q8—81no—
62C3×D4—81no—
63C3×D4—82no—
64C3×D4—82no—
65C3×D4—82no—
66C3×D4—82no—
67C2×C3×C4—21no—
68C2×C3×C4—41no—
69C2×C3×C4#9911yescharacteristic
70C2×Dic3—161no—
71C3×C8—22no—
72C3×C8—42no—
73C3⋊C8—161no—
74C2×Q8—21no—
75C2×D4—21no—
76C2×D4—41no—
77C2×D4—41no—
78C2×D4—41no—
79C2×D4—41no—
80SD16—82no—
81D8—82no—
82D8—84no—
83M16—241no—
84C2×C8—21no—
85C2×C8—22no—
86C4⋊C4—41no—
87C22⋊C4—241no—
88C42#21711yescharacteristic
89C22×C3—81no—
90C22×C3—161no—
91C22×C3—161no—
92C22×C3—161no—
93C22×C3—161no—
94C22×C3#35311yescharacteristic
95C3×C4—22no—
96C3×C4—21no—
97C3×C4—81no—
98Dic3—321no—
99C23—41no—
100C23—81no—
101Q8—81no—
102D4—81no—
103D4—82no—
104D4—82no—
105D4—82no—
106D4—82no—
107C2×C4—21no—
108C2×C4—41no—
109C2×C4—481no—
110C2×C4#83411yescharacteristic
111C8—22no—
112C8—42no—
113C8—481no—
114C2×C3—21no—
115C2×C3—161no—
116C2×C3—321no—
117C2×C3#140811yescharacteristic
118C22—81no—
119C22—161no—
120C22—161no—
121C22—161no—
122C22—161no—
123C22#454511yescharacteristic
124C4—22no—
125C4—21no—
126C4—81no—
127C4—961no—
128C3#752711yescharacteristic, complemented, Sylow
129C2—21no—
130C2—161no—
131C2—321no—
132C2#6779211yescharacteristic
subgroupG/H
centerC2#67792
commutatorC3×C4×C8C2×C4
Frattini#492C2×D3
Fitting, prime radicalC3×#1721C2
socleC2×C3#1408
cosocleC3×#492C22
hypercenter#1721D3
hypocenterC3#7527
LCSG ▷ C3×C4×C8 ▷ C3×C42 ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1721

Complements

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

NH ≅ G/Ntype
2. #67789C2semidirect
3. #67788C2semidirect
6. C3×#493C4semidirect
9. C3×#322C4semidirect
27. #493Dic3semidirect
32. #322Dic3semidirect
128. C3#7527semidirect

Maximal subgroups

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

subgroupindex
C3×#17212
#677892
#677882
#75273