Finite Group Databasebeta

← #69971

Subgroups — #69971

Order 384.

312 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 81 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×Dic3⋊C4C211yesmaximal, characteristic
2C3×#407C211yesmaximal, characteristic
3#5207C211yesmaximal, characteristic
4#4469C214yesmaximal, complemented
5#1463—31nomaximal, Sylow
6C23×C3×C4C2211yescharacteristic
7C23×Dic3C411yescharacteristic
8C23×Dic3C411yescharacteristic
9C2×C3×M16—21no—
10C2×Dic3⋊C4—21no—
11C2×Dic3⋊C4—21no—
12C2×Dic3⋊C4—21no—
13C2×Dic3⋊C4—21no—
14C2×Dic3⋊C4C412yes—
15C2×Dic3⋊C4C2212yes—
16C2×#208—21no—
17C3×C22⋊C8—21no—
18C3×C22⋊C8C2212yes—
19#830—21no—
20#830C2212yes—
21C22×C4⋊C4—31no—
22#407—31no—
23#407D311yescharacteristic
24#324—34no—
25C24×C3C2×C411yescharacteristic
26C22×C3×C4—42no—
27C22×C3×C4C2×C411yescharacteristic
28C22×C3×C4D411yescharacteristic
29C22×C3×C4D411yescharacteristic
30C22×C3×C4D411yescharacteristic
31C22×C3×C4D411yescharacteristic
32C22×C3×C4C2311yescharacteristic
33C22×Dic3—21no—
34C22×Dic3—21no—
35C22×Dic3—24no—
36C22×Dic3—21no—
37C22×Dic3—21no—
38C22×Dic3—42no—
39C22×Dic3C2×C412yes—
40C22×Dic3C2×C412yes—
41C3×M16—42no—
42C2×C3×C8—22no—
43Dic3⋊C4—22no—
44Dic3⋊C4—22no—
45Dic3⋊C4—22no—
46Dic3⋊C4—22no—
47#208—42no—
48C2×C3⋊C8—22no—
49C23×C4—31no—
50C23×C4—31no—
51C23×C4C2×D311yescharacteristic
52C2×M16—21no—
53C2×M16—61no—
54C2×C4⋊C4—32no—
55C2×C4⋊C4—32no—
56C2×C4⋊C4—61no—
57C2×C4⋊C4—61no—
58C2×C4⋊C4—61no—
59C2×C4⋊C4—61no—
60C22⋊C8—21no—
61C22⋊C8—32no—
62C22⋊C8—61no—
63C22⋊C8C2×D312yes—
64C23×C3—22no—
65C23×C3—22no—
66C23×C3—22no—
67C23×C3C22⋊C411yescharacteristic
68C23×C3C22⋊C411yescharacteristic
69C23×C3C22×C411yescharacteristic
70C2×C3×C4—22no—
71C2×C3×C4—22no—
72C2×C3×C4—41no—
73C2×C3×C4—41no—
74C2×C3×C4—41no—
75C2×C3×C4—41no—
76C2×C3×C4C22⋊C411yescharacteristic
77C2×C3×C4C22⋊C411yescharacteristic
78C2×C3×C4C2×D411yescharacteristic
79C2×C3×C4C2×D411yescharacteristic
80C2×Dic3—24no—
81C2×Dic3—24no—
82C2×Dic3—22no—
83C2×Dic3—22no—
84C2×Dic3—22no—
85C2×Dic3—42no—
86C2×Dic3—42no—
87C2×Dic3—42no—
88C2×Dic3M1614yescomplemented
89C3×C8—42no—
90C3⋊C8—42no—
91C24C4×D311yescharacteristic
92C22×C4—32no—
93C22×C4—32no—
94C22×C4—42no—
95C22×C4—64no—
96C22×C4—61no—
97C22×C4—61no—
98C22×C4—61no—
99C22×C4—61no—
100C22×C4—122no—
101C22×C4C4×D311yescharacteristic
102C22×C4D1211yescharacteristic
103C22×C4D1211yescharacteristic
104C22×C4#6711yescharacteristic
105C22×C4#6711yescharacteristic
106C22×C4C22×D311yescharacteristic
107M16—42no—
108M16—122no—
109C2×C8—22no—
110C2×C8—62no—
111C4⋊C4—62no—
112C4⋊C4—62no—
113C4⋊C4—62no—
114C4⋊C4—62no—
115C22×C3—28no—
116C22×C3—21no—
117C22×C3—21no—
118C22×C3—21no—
119C22×C3—21no—
120C22×C3—21no—
121C22×C3—21no—
122C22×C3#9912yes—
123C22×C3#10112yes—
124C22×C3C2×C22⋊C411yescharacteristic
125C22×C3C2×M1612yes—
126C3×C4—21no—
127C3×C4—21no—
128C3×C4—21no—
129C3×C4—21no—
130Dic3—24no—
131Dic3—42no—
132C23—22no—
133C23—22no—
134C23—22no—
135C23(C2×D3)⋊C411yescharacteristic
136C23(C2×D3)⋊C411yescharacteristic
137C23C2×C4×D311yescharacteristic
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—34no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—64no—
146C2×C4—64no—
147C2×C4—62no—
148C2×C4—62no—
149C2×C4—62no—
150C2×C4—122no—
151C2×C4—122no—
152C2×C4—122no—
153C2×C4(C2×D3)⋊C411yescharacteristic
154C2×C4(C2×D3)⋊C411yescharacteristic
155C2×C4C2×D1211yescharacteristic
156C2×C4C2×#6711yescharacteristic
157C8—42no—
158C8—122no—
159C2×C3—22no—
160C2×C3—22no—
161C2×C3—22no—
162C2×C3#40711yescharacteristic
163C2×C3C2×#9911yescharacteristic
164C2×C3C2×#10111yescharacteristic
165C22—28no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22#80612yes—
173C22#82412yes—
174C22D3×M1612yes—
175C22C2×(C2×D3)⋊C411yescharacteristic
176C4—21no—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—64no—
181C4—122no—
182C3#146311yescharacteristic, complemented, Sylow
183C2—22no—
184C2—22no—
185C2—22no—
186C2C2×#80611yescharacteristic
187C2#512611yescharacteristic
188C2C2×#82411yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#407C2
socleC22×C3C2×C22⋊C4
prime radicalC24×C3C2×C4
cosocleC22×C3×C4C23
hypercenter#407D3
hypocenterC3#1463
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
4. #4469C2semidirect
88. C2×Dic3M16semidirect
182. C3#1463semidirect

Maximal subgroups

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

subgroupindex
C22×Dic3⋊C42
C3×#4072
#52072
#44692
#14633