Finite Group Databasebeta

← #73061

Subgroups — #73061

Order 384.

186 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 36 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×#226C211yesmaximal, characteristic
2C2×C4×SL(2,𝔽3)C211yesmaximal, characteristic
3C2×#859C211yesmaximal, characteristic
4#1878—31nomaximal, Sylow
5C22×SL(2,𝔽3)C2211yescharacteristic
6C2×#226—21no—
7C2×#226—21no—
8C2×#226C412yescomplemented
9C4×SL(2,𝔽3)—21no—
10#859—21no—
11C3⋊#95—41nomaximal
12C22×Q16—31no—
13C2×C4×Q8D311yescharacteristic
14C2×#103—31no—
15C2×#103—31no—
16#391—31no—
17#340—31no—
18#336—31no—
19C22×C3×C4—41no—
20C22×Dic3—41no—
21C22×Dic3—41no—
22C2×SL(2,𝔽3)C2×C411yescharacteristic
23C2×SL(2,𝔽3)D411yescharacteristic
24C2×SL(2,𝔽3)D411yescharacteristic
25#226—22no—
26C22×Q8—31no—
27C22×Q8C2×D311yescharacteristic
28C2×Q16—32no—
29C2×Q16—32no—
30C2×Q16—61no—
31C2×Q16—61no—
32C2×Q16—61no—
33C2×Q16—61no—
34C22×C8—31no—
35C22×C8—31no—
36C4×Q8—21no—
37C4×Q8—61no—
38C2×C4⋊C4—31no—
39C2×C4⋊C4—31no—
40C2×C42—31no—
41#103—62no—
42#103—61no—
43#103—61no—
44#95—61no—
45#95—61no—
46C23×C3—41no—
47C2×C3×C4—42no—
48C2×C3×C4—81no—
49C2×C3×C4—81no—
50C2×Dic3—42no—
51C2×Dic3—42no—
52C2×Dic3—81no—
53C2×Dic3—81no—
54C2×Dic3—81no—
55C2×Dic3—81no—
56SL(2,𝔽3)C22⋊C411yescharacteristic
57C2×Q8—31no—
58C2×Q8—31no—
59C2×Q8—31no—
60C2×Q8—32no—
61C2×Q8—61no—
62C2×Q8—61no—
63C2×Q8—61no—
64C2×Q8—61no—
65C2×Q8—61no—
66C2×Q8C4×D311yescharacteristic
67C2×Q8D1211yescharacteristic
68C2×Q8#6711yescharacteristic
69C22×C4—31no—
70C22×C4—31no—
71C22×C4—61no—
72C22×C4—61no—
73C22×C4S411yescharacteristic
74Q16—62no—
75Q16—62no—
76Q16—62no—
77Q16—62no—
78C2×C8—32no—
79C2×C8—61no—
80C2×C8—61no—
81C2×C8—61no—
82C2×C8—61no—
83C2×C8—61no—
84C4⋊C4—32no—
85C4⋊C4—61no—
86C4⋊C4—61no—
87C4⋊C4—61no—
88C42—61no—
89C42—61no—
90C22×C3—41no—
91C22×C3—41no—
92C22×C3—41no—
93C22×C3—41no—
94C22×C3—41no—
95C22×C3—41no—
96C22×C3—41no—
97C3×C4—82no—
98Dic3—82no—
99Dic3—82no—
100C23C2×S411yescharacteristic
101Q8—31no—
102Q8—31no—
103Q8—31no—
104Q8—61no—
105Q8—62no—
106Q8—62no—
107Q8—121no—
108Q8—121no—
109Q8(C2×D3)⋊C411yescharacteristic
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—31no—
113C2×C4—31no—
114C2×C4—31no—
115C2×C4—31no—
116C2×C4—31no—
117C2×C4—31no—
118C2×C4—62no—
119C2×C4—61no—
120C2×C4—61no—
121C2×C4—61no—
122C2×C4—121no—
123C2×C4—121no—
124C2×C4—121no—
125C2×C4—121no—
126C2×C4—121no—
127C2×C4#22612yescomplemented
128C8—62no—
129C8—121no—
130C2×C3—41no—
131C2×C3—41no—
132C2×C3—41no—
133C2×C3—41no—
134C2×C3—41no—
135C2×C3—41no—
136C2×C3—41no—
137C22C4×S411yescharacteristic
138C22C4⋊S411yescharacteristic
139C22C2×#22611yescharacteristic
140C22#98311yescharacteristic
141C22#98411yescharacteristic
142C22#98511yescharacteristic
143C22GL(2,ℤ/4)11yescharacteristic
144C4—22no—
145C4—31no—
146C4—31no—
147C4—31no—
148C4—31no—
149C4—122no—
150C4—121no—
151C4—241no—
152C3—41noSylow
153C2C4×#22611yescharacteristic
154C2#538811yescharacteristic
155C2C4⋊#22611yescharacteristic
156C2#539511yescharacteristic
157C2(C2×S4)⋊C411yescharacteristic
158C2#541911yescharacteristic
159C2#542511yescharacteristic
subgroupG/H
center, Frattini, socleC23C2×S4
commutatorC2×SL(2,𝔽3)C2×C4
FittingC2×C4×Q8D3
prime radical, cosocleC22×SL(2,𝔽3)C22
hypercenterC22×C4S4
hypocenterSL(2,𝔽3)C22⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
8. C2×#226C4semidirect
127. C2×C4#226semidirect

Maximal subgroups

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

subgroupindex
C22×#2262
C2×C4×SL(2,𝔽3)2
C2×#8592
#18783
C3⋊#954