Finite Group Databasebeta

← #68144

Subgroups — #68144

Order 384.

312 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×D3×C4⋊C4C211yesmaximal, characteristic
2#4726C212yesmaximal
3C3×#324C211yesmaximal, characteristic, complemented
4#4528C211yesmaximal, characteristic, complemented
5#4469C212yesmaximal, complemented
6#1463—31nomaximal, Sylow
7C22×C4×D3C411yescharacteristic
8C22×C4×D3C411yescharacteristic
9C22×C4×D3C2211yescharacteristic
10C2×C3×C4⋊C4C2211yescharacteristic
11C2×C4⋊Dic3C411yescharacteristic
12C2×Dic3⋊C4C411yescharacteristic
13C2×Dic3⋊C4C2211yescharacteristic
14C2×ΓL(1,𝔽25)—22no—
15D3×C4⋊C4—22no—
16D3×C4⋊C4—22no—
17C3×C22⋊C8C2212yes—
18#830C2212yes—
19(C2×D3)⋊C8—22no—
20C22×C4⋊C4—31no—
21#407—32no—
22#324—31no—
23#324—32no—
24#324D311yescharacteristic, complemented
25C23×D3C2×C411yescharacteristic
26C22×C3×C4C2×C411yescharacteristic
27C22×C3×C4C2×C411yescharacteristic
28C22×C3×C4C2311yescharacteristic
29C22×Dic3C2×C411yescharacteristic
30C22×Dic3C2×C411yescharacteristic
31C22×Dic3C2×C411yescharacteristic
32C2×C4×D3—21no—
33C2×C4×D3—21no—
34C2×C4×D3—22no—
35C2×C4×D3—22no—
36C2×C4×D3—21no—
37C2×C4×D3—21no—
38C2×C4×D3—42no—
39C2×C4×D3—42no—
40C2×C4×D3D412yes—
41C2×C4×D3D412yes—
42C2×C3×C8—22no—
43C3×C4⋊C4—22no—
44C4⋊Dic3—22no—
45Dic3⋊C4—22no—
46Dic3⋊C4—22no—
47C2×C3⋊C8—22no—
48ΓL(1,𝔽25)—44no—
49C23×C4—31no—
50C23×C4—31no—
51C23×C4—31no—
52C2×M16—62no—
53C2×C4⋊C4—31no—
54C2×C4⋊C4—31no—
55C2×C4⋊C4—31no—
56C2×C4⋊C4—62no—
57C2×C4⋊C4—62no—
58C2×C4⋊C4C2×D311yescharacteristic
59C22⋊C8—32no—
60C22⋊C8—62no—
61C22⋊C8C2×D312yes—
62C23×C3C22×C411yescharacteristic
63C22×D3—21no—
64C22×D3—21no—
65C22×D3—22no—
66C22×D3—22no—
67C22×D3C22⋊C411yescharacteristic
68C22×D3C22⋊C411yescharacteristic
69C2×C3×C4—21no—
70C2×C3×C4—21no—
71C2×C3×C4—21no—
72C2×C3×C4—41no—
73C2×C3×C4—41no—
74C2×C3×C4M1612yescomplemented
75C2×C3×C4C2×D412yes—
76C2×Dic3—21no—
77C2×Dic3—21no—
78C2×Dic3—21no—
79C2×Dic3—41no—
80C2×Dic3—41no—
81C2×Dic3C22⋊C412yes—
82C2×Dic3M1612yescomplemented
83C4×D3—24no—
84C4×D3—24no—
85C4×D3—24no—
86C4×D3—42no—
87C4×D3—42no—
88C4×D3—42no—
89C3×C8—42no—
90C3⋊C8—42no—
91C24—31no—
92C22×C4—31no—
93C22×C4—31no—
94C22×C4—32no—
95C22×C4—32no—
96C22×C4—31no—
97C22×C4—62no—
98C22×C4—61no—
99C22×C4—61no—
100C22×C4—61no—
101C22×C4—61no—
102C22×C4—62no—
103C22×C4—122no—
104C22×C4—122no—
105C22×C4C4×D311yescharacteristic
106C22×C4C4×D311yescharacteristic
107C22×C4C22×D311yescharacteristic
108M16—124no—
109C2×C8—22no—
110C2×C8—62no—
111C4⋊C4—22no—
112C4⋊C4—62no—
113C4⋊C4—62no—
114C4⋊C4—62no—
115C22×C3—21no—
116C22×C3—21no—
117C22×C3C2×C22⋊C411yescharacteristic
118C22×C3C2×M1611yescharacteristic
119C22×C3C2×M1611yescharacteristic
120C2×D3—24no—
121C2×D3—24no—
122C2×D3—21no—
123C2×D3—21no—
124C2×D3—21no—
125C2×D3—21no—
126C2×D3#9912yes—
127C2×D3#10112yes—
128C3×C4—22no—
129C3×C4—22no—
130C3×C4—41no—
131Dic3—22no—
132Dic3—22no—
133Dic3—41no—
134C23—31no—
135C23—31no—
136C23—62no—
137C23—62no—
138C23—61no—
139C23—61no—
140C23C2×C4×D311yescharacteristic
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—32no—
145C2×C4—32no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—64no—
149C2×C4—64no—
150C2×C4—64no—
151C2×C4—61no—
152C2×C4—61no—
153C2×C4—61no—
154C2×C4—122no—
155C2×C4—122no—
156C2×C4—122no—
157C2×C4—121no—
158C2×C4—121no—
159C2×C4ΓL(1,𝔽25)12yescomplemented
160C2×C4D3×D412yes—
161C8—42no—
162C8—122no—
163C2×C3—21no—
164C2×C3—21no—
165C2×C3#40711yescharacteristic
166C2×C3C2×#9911yescharacteristic
167C2×C3C2×#10111yescharacteristic
168D3—22no—
169D3—22no—
170C22—21no—
171C22—21no—
172C22—32no—
173C22—32no—
174C22—64no—
175C22—64no—
176C22—61no—
177C22—61no—
178C22—61no—
179C22—61no—
180C22D3×C22⋊C411yescharacteristic
181C22C2×ΓL(1,𝔽25)11yescharacteristic
182C22D3×M1611yescharacteristic
183C4—22no—
184C4—22no—
185C4—41no—
186C4—62no—
187C4—62no—
188C4—121no—
189C3#146311yescharacteristic, complemented, Sylow
190C2—21no—
191C2—21no—
192C2—62no—
193C2—62no—
194C2#472611yescharacteristic
195C2D3×#9911yescharacteristic
196C2D3×#10111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#324C2
socleC22×C3C2×C22⋊C4
prime radicalC23×D3C2×C4
cosocleC22×C3×C4C23
hypercenter#324D3
hypocenterC3#1463
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
3. C3×#324C2semidirect
4. #4528C2semidirect
5. #4469C2semidirect
24. #324D3semidirect
74. C2×C3×C4M16semidirect
82. C2×Dic3M16semidirect
159. C2×C4ΓL(1,𝔽25)semidirect
189. C3#1463semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊C42
#47262
C3×#3242
#45282
#44692
#14633