Finite Group Databasebeta

← #33271⋊C3 #11709641

Subgroups — #33271⋊C3 #11709641

Order 768.

156 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 23 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
1#2645⋊C3C211yesmaximal, characteristic, complemented
2#2644⋊C3C211yesmaximal, characteristic
3#1726⋊C3C211yesmaximal, characteristic, complemented
4#33271C311yesmaximal, characteristic, complemented, Sylow
5A4×SD16—41nomaximal
6#401⋊C3—21no—
7#396⋊C3—21no—
8#376⋊C3C2211yescharacteristic
9#2645C2×C311yescharacteristic, complemented
10#2644C2×C311yescharacteristic
11#1968—31no—
12#1941—31no—
13#1936—31no—
14#1935—31no—
15#1726C2×C311yescharacteristic, complemented
16Q8×A4—41no—
17D4×A4—41no—
18C8×A4—41no—
19#95⋊C3D411yescharacteristic
20C22×SD16A411yescharacteristic
21C2×#106—31no—
22C2×#103—31no—
23C2×#102—31no—
24C2×#97—31no—
25C2×C4×C8—31no—
26#401—21no—
27#400—61no—
28#397—61no—
29#396—21no—
30#391—31no—
31#390—31no—
32#389—31no—
33#389—31no—
34#376C22×C311yescharacteristic
35#340—32no—
36C22×A4—81no—
37C4×A4—41no—
38C4×A4—81no—
39C3×SD16—161no—
40C22×Q8C2×A411yescharacteristic
41C22×D4C2×A411yescharacteristic
42C2×SD16—61no—
43C2×SD16—61no—
44C22×C8—31no—
45C22×C8C2×A411yescharacteristic
46C2×C4⋊C4—31no—
47C2×C4⋊C4—31no—
48C2×C4⋊C4—61no—
49C2×C4⋊C4—61no—
50C2×C22⋊C4—61no—
51C2×C22⋊C4—61no—
52C2×C42—31no—
53#106—61no—
54#106—61no—
55#103—62no—
56#102—62no—
57#97—62no—
58C4×C8—61no—
59C4×C8—61no—
60#95—31no—
61#95—61no—
62#95—61no—
63#95—61no—
64#95—61no—
65#95C3×D411yescharacteristic
66C2×A4—41no—
67C2×A4—161no—
68C3×Q8—161no—
69C3×D4—161no—
70C3×C8—161no—
71C24—21no—
72C2×Q8—61no—
73C2×Q8—61no—
74C2×D4—61no—
75C2×D4—61no—
76C22×C4—21no—
77C22×C4—31no—
78C22×C4—31no—
79C22×C4—61no—
80C22×C4—61no—
81C22×C4C22×A411yescharacteristic
82SD16—41no—
83SD16—121no—
84C2×C8—31no—
85C2×C8—31no—
86C2×C8—32no—
87C2×C8—61no—
88C2×C8—61no—
89C4⋊C4—61no—
90C4⋊C4—61no—
91C4⋊C4—61no—
92C4⋊C4—61no—
93C4⋊C4—241no—
94C4⋊C4—241no—
95C22⋊C4—241no—
96C22⋊C4—241no—
97C42—61no—
98C42—61no—
99C22×C3—321no—
100A4—41no—
101C3×C4—161no—
102C3×C4—321no—
103C23—41no—
104C23—121no—
105C23—121no—
106C23D4×A411yescharacteristic
107Q8—41no—
108Q8—121no—
109D4—41no—
110D4—121no—
111C2×C4—31no—
112C2×C4—31no—
113C2×C4—61no—
114C2×C4—61no—
115C2×C4—61no—
116C2×C4—61no—
117C2×C4—61no—
118C2×C4—61no—
119C2×C4—121no—
120C2×C4—121no—
121C2×C4—121no—
122C2×C4—121no—
123C2×C4—121no—
124C2×C4—121no—
125C2×C4—121no—
126C8—31no—
127C8—62no—
128C8#126⋊C311yescharacteristic
129C2×C3—161no—
130C2×C3—641no—
131C22—31no—
132C22—31no—
133C22—81no—
134C22—241no—
135C22—241no—
136C22#577⋊C311yescharacteristic
137C4—31no—
138C4—81no—
139C4—121no—
140C4—121no—
141C4—241no—
142C4—241no—
143C4C2×#126⋊C311yescharacteristic
144C3—161noSylow
145C2—31no—
146C2—31no—
147C2—161no—
148C2C4⋊(#126⋊C3)11yescharacteristic
subgroupG/H
centerC2C4⋊(#126⋊C3)
commutator, cosocle#376C22×C3
FrattiniC22×C4C22×A4
Fitting#33271C3
socleC23D4×A4
prime radical#2644⋊C3C2
hypercenterC8#126⋊C3
hypocenter#95C3×D4
LCSG ▷ #376 ▷ #95
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #2645⋊C3C2semidirect
3. #1726⋊C3C2semidirect
4. #33271C3semidirect
9. #2645C2×C3semidirect
15. #1726C2×C3semidirect

Maximal subgroups

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

subgroupindex
#2645⋊C32
#2644⋊C32
#1726⋊C32
#332713
A4×SD164