Finite Group Databasebeta

← #69704

Subgroups — #69704

Order 384.

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

normal subgroups only

This group has 266 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#5500C211yesmaximal, characteristic, complemented
2C3×#394C211yesmaximal, characteristic
3#5218C212yesmaximal
4#4528C211yesmaximal, characteristic, complemented
5#4452C212yesmaximal, complemented
6#1599—31nomaximal, Sylow
7C22×C3×D4C2211yescharacteristic
8C2×C3×C4⋊C4C2211yescharacteristic
9C2×C3×C22⋊C4—21no—
10C2×C3×C22⋊C4—21no—
11C2×#214—22no—
12#926—21no—
13#924—21no—
14#924C2212yes—
15#890—22no—
16#888—22no—
17C3×#95C2211yescharacteristic
18#832—22no—
19#830C2212yes—
20#544—31no—
21#450—32no—
22#394D311yescharacteristic
23#328—32no—
24#324—31no—
25C24×C3—21no—
26C2×C3×D4—22no—
27C2×C3×D4—42no—
28C22×C3×C4—21no—
29C22×C3×C4D411yescharacteristic
30C22×C3×C4D411yescharacteristic
31C22×C3×C4C2311yescharacteristic
32C2×Dic6—21no—
33C2×Dic6D412yes—
34C3×C4⋊C4—22no—
35C3×C22⋊C4—42no—
36C3×C22⋊C4—42no—
37#217—22no—
38#214—44no—
39C4⋊Dic3—21no—
40C4⋊Dic3—21no—
41C4⋊Dic3—21no—
42C4⋊Dic3D412yes—
43Dic3⋊C4—22no—
44Dic3⋊C4—22no—
45C4×Dic3—22no—
46C2×C3⋊C8—22no—
47C22×D4C2×D311yescharacteristic
48C2×SD16—62no—
49C4⋊Q8—62no—
50#125—62no—
51C22⋊Q8—32no—
52C22⋊Q8—61no—
53#117—61no—
54C2×C4⋊C4C2×D311yescharacteristic
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57#102—62no—
58C22⋊C8—32no—
59#95C2×D311yescharacteristic
60C23×C3—22no—
61C23×C3—41no—
62C23×C3—41no—
63C23×C3—41no—
64C23×C3—41no—
65C23×C3—41no—
66C23×C3—41no—
67C23×C3C2×D411yescharacteristic
68C3×D4—24no—
69C3×D4—42no—
70C2×C3×C4—21no—
71C2×C3×C4—21no—
72C2×C3×C4—21no—
73C2×C3×C4—41no—
74C2×C3×C4—41no—
75C2×C3×C4—41no—
76C2×C3×C4—41no—
77C2×C3×C4—41no—
78C2×C3×C4SD1612yescomplemented
79C2×C3×C4C2×D412yes—
80C2×Dic3—22no—
81C2×Dic3—22no—
82Dic6—42no—
83Dic6—81no—
84C3⋊C8—42no—
85C24—21no—
86C2×Q8—32no—
87C2×Q8—61no—
88C2×D4—22no—
89C2×D4—42no—
90C22×C4—21no—
91C22×C4#6711yescharacteristic
92C22×C4#6711yescharacteristic
93C22×C4C22×D311yescharacteristic
94SD16—124no—
95C2×C8—62no—
96C4⋊C4—22no—
97C4⋊C4—32no—
98C4⋊C4—61no—
99C4⋊C4—61no—
100C4⋊C4—61no—
101C4⋊C4—62no—
102C4⋊C4—62no—
103C22⋊C4—42no—
104C22⋊C4—42no—
105C22⋊C4—62no—
106C42—62no—
107C22×C3—21no—
108C22×C3—21no—
109C22×C3—42no—
110C22×C3—42no—
111C22×C3—42no—
112C22×C3—81no—
113C22×C3—81no—
114C22×C3—81no—
115C22×C3—81no—
116C22×C3#12011yescharacteristic
117C22×C3C2×SD1611yescharacteristic
118C22×C3#13711yescharacteristic
119C3×C4—22no—
120C3×C4—22no—
121C3×C4—41no—
122C3×C4—81no—
123Dic3—42no—
124Dic3—42no—
125C23—22no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23C2×#6711yescharacteristic
133Q8—122no—
134Q8—241no—
135D4—24no—
136D4—42no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—62no—
146C2×C4—62no—
147C2×C4#21412yescomplemented
148C2×C4D3×D412yes—
149C8—122no—
150C2×C3—21no—
151C2×C3—21no—
152C2×C3—82no—
153C2×C3#44811yescharacteristic
154C2×C3#45411yescharacteristic
155C2×C3B(4,𝔽2)11yescharacteristic
156C22—21no—
157C22—21no—
158C22—42no—
159C22—42no—
160C22—42no—
161C22—81no—
162C22—81no—
163C22—81no—
164C22—81no—
165C22C2×#21411yescharacteristic
166C22#93711yescharacteristic
167C22#95111yescharacteristic
168C4—22no—
169C4—22no—
170C4—41no—
171C4—81no—
172C4—122no—
173C4—122no—
174C3#159911yescharacteristic, complemented, Sylow
175C2—21no—
176C2—21no—
177C2—82no—
178C2#503211yescharacteristic
179C2#504011yescharacteristic
180C2#506211yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#394C2
socleC22×C3#120
prime radicalC22×C3×D4C22
hypercenter#394D3
hypocenterC3#1599
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #5500C2semidirect
4. #4528C2semidirect
5. #4452C2semidirect
78. C2×C3×C4SD16semidirect
147. C2×C4#214semidirect
174. C3#1599semidirect

Maximal subgroups

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

subgroupindex
#55002
C3×#3942
#52182
#45282
#44522
#15993