Finite Group Databasebeta

← #73048

Subgroups — #73048

Order 384.

167 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 15 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×C42⋊C3C211yesmaximal, characteristic, complemented
2C2×#857C211yesmaximal, characteristic, complemented
3#4623C211yesmaximal, characteristic, complemented
4#1896—31nomaximal, Sylow
5GL(2,ℤ/4)—41nomaximal
6C2×C42⋊C3—21no—
7C2×C42⋊C3C2211yescharacteristic
8#857—21no—
9#525—31no—
10C22×C42D311yescharacteristic, complemented
11C2×#104—31no—
12C2×#104—31no—
13#407—31no—
14#339—31no—
15#339—31no—
16C22×A4—41no—
17C2×S4—41no—
18SL(2,ℤ/4)—41no—
19C42⋊C3D411yescharacteristic, complemented
20C2×#41—31no—
21C23×C4—31no—
22C2×M16—31no—
23C2×M16—31no—
24#123—61no—
25C22⋊Q8—31no—
26C22⋊Q8—31no—
27#121—31no—
28#121—31no—
29#120—61no—
30C4×D4—61no—
31C4×D4—61no—
32#117—31no—
33C2×C42—21no—
34C2×C42—31no—
35C2×C42—61no—
36C2×C42C2×D311yescharacteristic
37#104—61no—
38#104—62no—
39#104—61no—
40C22⋊C8—61no—
41C22⋊C8—61no—
42C2×A4—41no—
43C2×A4—81no—
44S4—81no—
45#67—161no—
46C24S411yescharacteristic
47#41—61no—
48#41—61no—
49#41—61no—
50#41—61no—
51C2×Q8—31no—
52C2×D4—31no—
53C2×D4—61no—
54C2×D4—61no—
55C2×D4—61no—
56C22×C4—31no—
57C22×C4—31no—
58C22×C4—31no—
59C22×C4—31no—
60C22×C4—31no—
61C22×C4—31no—
62C22×C4—61no—
63C22×C4—61no—
64C22×C4—61no—
65C22×C4—61no—
66C22×C4—61no—
67M16—61no—
68M16—61no—
69M16—61no—
70M16—61no—
71C2×C8—61no—
72C2×C8—61no—
73C4⋊C4—31no—
74C4⋊C4—31no—
75C4⋊C4—61no—
76C4⋊C4—61no—
77C22⋊C4—61no—
78C22⋊C4—61no—
79C22⋊C4—61no—
80C22⋊C4—61no—
81C22⋊C4—61no—
82C42—31no—
83C42—32no—
84C42—61no—
85C42—61no—
86C42#6711yescharacteristic, complemented
87C22×C3—161no—
88C2×D3—161no—
89A4—41no—
90Dic3—161no—
91C23—21no—
92C23—31no—
93C23—31no—
94C23—61no—
95C23—61no—
96C23C2×S411yescharacteristic
97Q8—61no—
98Q8—61no—
99D4—61no—
100D4—61no—
101D4—122no—
102D4—121no—
103D4—121no—
104D4—121no—
105D4—121no—
106C2×C4—31no—
107C2×C4—31no—
108C2×C4—31no—
109C2×C4—31no—
110C2×C4—31no—
111C2×C4—31no—
112C2×C4—31no—
113C2×C4—31no—
114C2×C4—61no—
115C2×C4—61no—
116C2×C4—61no—
117C2×C4—61no—
118C2×C4—61no—
119C2×C4—61no—
120C2×C4—61no—
121C2×C4—61no—
122C2×C4—61no—
123C2×C4—61no—
124C2×C4—61no—
125C2×C4—61no—
126C2×C4—61no—
127C2×C4—121no—
128C2×C4—121no—
129C8—121no—
130C8—121no—
131C2×C3—161no—
132C2×C3—321no—
133D3—321no—
134C22—31no—
135C22—31no—
136C22—32no—
137C22—31no—
138C22—61no—
139C22—61no—
140C22—61no—
141C22—121no—
142C22—121no—
143C22—121no—
144C22#85711yescharacteristic, complemented
145C22GL(2,ℤ/4)11yescharacteristic
146C4—31no—
147C4—31no—
148C4—31no—
149C4—31no—
150C4—61no—
151C4—61no—
152C4—121no—
153C4—121no—
154C4—121no—
155C3—161noSylow
156C2—21no—
157C2—31no—
158C2—31no—
159C2—61no—
160C2—241no—
161C2C2×#85711yescharacteristic
subgroupG/H
centerC2C2×#857
commutator, cosocleC2×C42⋊C3C22
Frattini, socleC23C2×S4
FittingC22×C42D3
hypercenterC22#857
hypocenterC42⋊C3D4
LCSG ▷ C2×C42⋊C3 ▷ C42⋊C3
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C42⋊C3C2semidirect
2. C2×#857C2semidirect
3. #4623C2semidirect
10. C22×C42D3semidirect
19. C42⋊C3D4semidirect
86. C42#67semidirect
144. C22#857semidirect

Maximal subgroups

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

subgroupindex
C22×C42⋊C32
C2×#8572
#46232
#18963
GL(2,ℤ/4)4