Finite Group Databasebeta

← #69701

Subgroups — #69701

Order 384.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#5483C211yesmaximal, characteristic
2C3×#393C211yesmaximal, characteristic
3#5228C212yesmaximal, complemented
4#4525C211yesmaximal, characteristic
5#4452C212yesmaximal, complemented
6#1602—31nomaximal, Sylow
7C22×C3×Q8C2211yescharacteristic
8C2×C3×C22⋊C4—21no—
9C2×C3×C22⋊C4C2211yescharacteristic
10C2×#217—21no—
11C2×#216—22no—
12#924—21no—
13#924C2212yes—
14#878—22no—
15#878—22no—
16C3×#95—21no—
17C3×#95C2211yescharacteristic
18#835—22no—
19#830C2212yes—
20#543—31no—
21#451—32no—
22#393D311yescharacteristic
23#328—32no—
24#323—31no—
25C24×C3D411yescharacteristic
26C2×C3×Q8—22no—
27C2×C3×Q8—42no—
28C22×C3×C4—21no—
29C22×C3×C4—21no—
30C22×C3×C4D411yescharacteristic
31C22×C3×C4C2311yescharacteristic
32C22×Dic3—21no—
33C22×Dic3—21no—
34C2×Dic6—21no—
35C2×Dic6D412yes—
36C3×C22⋊C4—22no—
37C3×C22⋊C4—42no—
38#217—22no—
39#217—22no—
40#216—44no—
41C4⋊Dic3—21no—
42C4⋊Dic3D412yes—
43Dic3⋊C4—22no—
44Dic3⋊C4—22no—
45C2×C3⋊C8—22no—
46C22×Q8C2×D311yescharacteristic
47C2×Q16—62no—
48C22⋊Q8—32no—
49C22⋊Q8—62no—
50C22⋊Q8—62no—
51C22⋊Q8—61no—
52C2×C22⋊C4—21no—
53C2×C22⋊C4—61no—
54C2×C22⋊C4C2×D311yescharacteristic
55#103—62no—
56C22⋊C8—32no—
57#95—21no—
58#95C2×D311yescharacteristic
59C23×C3—21no—
60C23×C3—21no—
61C23×C3—41no—
62C23×C3—41no—
63C23×C3Q1612yescomplemented
64C23×C3C2×D411yescharacteristic
65C3×Q8—24no—
66C3×Q8—42no—
67C2×C3×C4—22no—
68C2×C3×C4—21no—
69C2×C3×C4—41no—
70C2×C3×C4—41no—
71C2×C3×C4—41no—
72C2×C3×C4—41no—
73C2×C3×C4—41no—
74C2×C3×C4—41no—
75C2×C3×C4—41no—
76C2×C3×C4C2×D412yes—
77C2×Dic3—22no—
78C2×Dic3—22no—
79C2×Dic3—81no—
80C2×Dic3—81no—
81Dic6—42no—
82Dic6—81no—
83C3⋊C8—42no—
84C24#6711yescharacteristic
85C2×Q8—22no—
86C2×Q8—32no—
87C2×Q8—42no—
88C2×Q8—61no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—61no—
92C22×C4—61no—
93C22×C4#6711yescharacteristic
94C22×C4C22×D311yescharacteristic
95Q16—124no—
96C2×C8—62no—
97C4⋊C4—32no—
98C4⋊C4—62no—
99C4⋊C4—61no—
100C4⋊C4—62no—
101C22⋊C4—22no—
102C22⋊C4—42no—
103C22⋊C4—62no—
104C22⋊C4—62no—
105C22×C3—21no—
106C22×C3—21no—
107C22×C3—22no—
108C22×C3—42no—
109C22×C3—41no—
110C22×C3—41no—
111C22×C3—81no—
112C22×C3#12011yescharacteristic
113C22×C3C2×Q1611yescharacteristic
114C22×C3#13711yescharacteristic
115C3×C4—22no—
116C3×C4—41no—
117C3×C4—42no—
118C3×C4—81no—
119Dic3—42no—
120Dic3—42no—
121C23—21no—
122C23—21no—
123C23—41no—
124C23—41no—
125C23#21612yescomplemented
126C23C2×#6711yescharacteristic
127Q8—24no—
128Q8—42no—
129Q8—122no—
130Q8—241no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—62no—
141C2×C4—62no—
142C2×C4—241no—
143C2×C4—241no—
144C2×C4D3×D412yes—
145C8—122no—
146C2×C3—21no—
147C2×C3—21no—
148C2×C3—42no—
149C2×C3#45111yescharacteristic
150C2×C3#45511yescharacteristic
151C2×C3B(4,𝔽2)11yescharacteristic
152C22—21no—
153C22—21no—
154C22—22no—
155C22—42no—
156C22—41no—
157C22—41no—
158C22—81no—
159C22#93711yescharacteristic
160C22C2×#21611yescharacteristic
161C22#95111yescharacteristic
162C4—22no—
163C4—42no—
164C4—41no—
165C4—81no—
166C4—122no—
167C4—122no—
168C3#160211yescharacteristic, complemented, Sylow
169C2—21no—
170C2—21no—
171C2—42no—
172C2#503211yescharacteristic
173C2#505011yescharacteristic
174C2#506111yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#393C2
socleC22×C3#120
prime radicalC24×C3D4
hypercenter#393D3
hypocenterC3#1602
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
3. #5228C2semidirect
5. #4452C2semidirect
63. C23×C3Q16semidirect
125. C23#216semidirect
168. C3#1602semidirect

Maximal subgroups

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

subgroupindex
#54832
C3×#3932
#52282
#45252
#44522
#16023