Finite Group Databasebeta

← #82314

Subgroups — #82314

Order 384.

448 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 145 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×C3×C4×D4C211yesmaximal, characteristic
2#5470C211yesmaximal, characteristic, complemented
3#5218C212yesmaximal
4#5019C212yesmaximal, complemented
5C4×#214C214yesmaximal, complemented
6#5008C212yesmaximal, complemented
7#4971C212yesmaximal, complemented
8#4925C211yesmaximal, characteristic, complemented
9#3101—31nomaximal, Sylow
10C22×C3×D4C2211yescharacteristic
11C23×C3×C4—21no—
12C3×C4×D4—21no—
13C3×C4×D4—21no—
14C3×C4×D4C2212yes—
15C3×C4×D4C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4—21no—
18C2×C3×C42C2211yescharacteristic
19C2×#214C2214yes—
20#926—21no—
21#924—21no—
22#924C2212yes—
23#870—21no—
24#869C2211yescharacteristic, complemented
25#869C2211yescharacteristic, complemented
26C4×Dic6—21no—
27C4×Dic6C2212yescomplemented
28#832C2214yes—
29#830C2212yes—
30#810C2214yescomplemented
31#808C2214yescomplemented
32C4⋊(C3⋊C8)C2212yescomplemented
33C4×C3⋊C8C2212yescomplemented
34#533—31no—
35C2×C4×D4D311yescharacteristic
36#483—32no—
37#476—32no—
38#461—32no—
39#450—32no—
40C4×SD16—34no—
41#431—31no—
42C24×C3—21no—
43C2×C3×D4—24no—
44C2×C3×D4—21no—
45C2×C3×D4C2312yes—
46C22×C3×C4—21no—
47C22×C3×C4—22no—
48C22×C3×C4—22no—
49C22×C3×C4—44no—
50C22×C3×C4D411yescharacteristic
51C22×C3×C4D411yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C2×Dic6—21no—
54C2×Dic6C2312yes—
55C3×C4⋊C4—21no—
56C3×C4⋊C4C2312yes—
57C3×C22⋊C4—22no—
58C3×C22⋊C4—22no—
59C3×C42D411yescharacteristic
60C3×C42D411yescharacteristic
61C3×C42C2311yescharacteristic
62C3×C42C2311yescharacteristic
63#217—22no—
64#214—28no—
65C4⋊Dic3—21no—
66C4⋊Dic3—21no—
67C4⋊Dic3—21no—
68C4⋊Dic3C2312yes—
69Dic3⋊C4—22no—
70Dic3⋊C4—22no—
71C4×Dic3—22no—
72C2×C3⋊C8C2314yes—
73C22×D4C2×D311yescharacteristic
74C23×C4—21no—
75C2×SD16—34no—
76C4⋊Q8—31no—
77C4⋊Q8—31no—
78#125—61no—
79C22⋊Q8—32no—
80C22⋊Q8—61no—
81C4×Q8—32no—
82C4×Q8—61no—
83C4×D4—21no—
84C4×D4—21no—
85C4×D4C2×D312yes—
86C4×D4C2×D312yes—
87#117—61no—
88C2×C4⋊C4C2×D311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C42C2×D311yescharacteristic
91#106—34no—
92C4⋊C8—32no—
93#103—34no—
94#102—34no—
95C22⋊C8—32no—
96C4×C8—32no—
97C23×C3—24no—
98C23×C3—22no—
99C23×C3—42no—
100C23×C3—42no—
101C23×C3C2×D411yescharacteristic
102C3×D4—24no—
103C3×D4—22no—
104C3×D4#4114yes—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—41no—
108C2×C3×C4—42no—
109C2×C3×C4—44no—
110C2×C3×C4—44no—
111C2×C3×C4SD1614yescomplemented
112C2×C3×C4C2×D411yescharacteristic
113C2×C3×C4C2×D411yescharacteristic
114C2×C3×C4C2×D411yescharacteristic
115C2×C3×C4C2×D411yescharacteristic
116C2×C3×C4C2×D411yescharacteristic
117C2×C3×C4C2411yescharacteristic
118C2×Dic3—22no—
119C2×Dic3—22no—
120Dic6—24no—
121Dic6—42no—
122C3⋊C8—24no—
123C24—21no—
124C2×Q8—32no—
125C2×Q8—61no—
126C2×D4—24no—
127C2×D4—21no—
128C2×D4C22×D312yes—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—44no—
133C22×C4#6711yescharacteristic
134C22×C4#6711yescharacteristic
135C22×C4C22×D311yescharacteristic
136SD16—68no—
137C2×C8—34no—
138C4⋊C4—21no—
139C4⋊C4—32no—
140C4⋊C4—61no—
141C4⋊C4—61no—
142C4⋊C4—62no—
143C4⋊C4—62no—
144C4⋊C4—61no—
145C4⋊C4C22×D312yes—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C22⋊C4—62no—
149C42—62no—
150C42#6711yescharacteristic
151C42#6711yescharacteristic
152C42C22×D311yescharacteristic
153C42C22×D311yescharacteristic
154C22×C3—21no—
155C22×C3—21no—
156C22×C3—24no—
157C22×C3—48no—
158C22×C3—42no—
159C22×C3—42no—
160C22×C3C2×SD1612yes—
161C22×C3C22×D411yescharacteristic
162C3×C4—21no—
163C3×C4—21no—
164C3×C4—21no—
165C3×C4—42no—
166C3×C4C2×SD1612yes—
167C3×C4C2×SD1612yes—
168C3×C4C2×#4112yes—
169Dic3—42no—
170Dic3—42no—
171C23—24no—
172C23—22no—
173C23—42no—
174C23—42no—
175C23C2×#6711yescharacteristic
176Q8—64no—
177Q8—122no—
178D4—24no—
179D4—22no—
180D4#23514yes—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—44no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4#21414yescomplemented
190C2×C4C2×#6711yescharacteristic
191C2×C4C2×#6711yescharacteristic
192C2×C4C2×#6711yescharacteristic
193C2×C4C2×#6711yescharacteristic
194C2×C4C2×#6711yescharacteristic
195C2×C4C23×D311yescharacteristic
196C8—64no—
197C2×C3—22no—
198C2×C3—44no—
199C2×C3#52511yescharacteristic
200C2×C3C22×SD1611yescharacteristic
201C2×C3#57511yescharacteristic
202C22—21no—
203C22—21no—
204C22—24no—
205C22—48no—
206C22—42no—
207C22—42no—
208C22C2×#21412yes—
209C22C22×#6711yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—42no—
214C4—122no—
215C4—122no—
216C4C2×#21412yes—
217C4C2×#21412yes—
218C4C2×#23512yes—
219C3#310111yescharacteristic, complemented, Sylow
220C2—22no—
221C2—44no—
222C2#579111yescharacteristic
223C2C22×#21411yescharacteristic
224C2#582211yescharacteristic
subgroupG/H
centerC2×C4C2×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×C4×D4C2
socleC22×C3C22×D4
prime radicalC22×C3×D4C22
hypercenterC2×C4×D4D3
hypocenterC3#3101
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #5470C2semidirect
4. #5019C2semidirect
5. C4×#214C2semidirect
6. #5008C2semidirect
7. #4971C2semidirect
8. #4925C2semidirect
24. #869C22semidirect
25. #869C22semidirect
27. C4×Dic6C22semidirect
30. #810C22semidirect
31. #808C22semidirect
32. C4⋊(C3⋊C8)C22semidirect
33. C4×C3⋊C8C22semidirect
111. C2×C3×C4SD16semidirect
189. C2×C4#214semidirect
219. C3#3101semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
#54702
#52182
#50192
C4×#2142
#50082
#49712
#49252
#31013