Finite Group Databasebeta

← #85443

Subgroups — #85443

Order 384.

299 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 38 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#5912C211yesmaximal, characteristic, complemented
2C2×C4⋊S4C211yesmaximal, characteristic, complemented
3A4×M16C211yesmaximal, characteristic, complemented
4#5402C212yesmaximal, complemented
5#5401C212yesmaximal, complemented
6#3055—31nomaximal, Sylow
7C22×S4—21no—
8C2×C4×A4C2211yescharacteristic
9GL(2,ℤ/4)—21no—
10C4⋊S4—21no—
11C4⋊S4C2212yes—
12C4⋊S4C2211yescharacteristic, complemented
13C4×S4—21no—
14A4⋊Q8C2211yescharacteristic, complemented
15#908—41nomaximal
16C8×A4C2212yescomplemented
17C2×AGL(1,ℤ/8)—31no—
18C22×M16D311yescharacteristic, complemented
19#525—31no—
20C2×#121—31no—
21#469—32no—
22#468—32no—
23#466—32no—
24#465—32no—
25#428—31no—
26#418—31no—
27#417—31no—
28C22×A4D411yescharacteristic
29C2×S4—21no—
30C2×S4—22no—
31C2×S4—42no—
32#235—41no—
33C2×D12—41no—
34C4×A4D411yescharacteristic
35C4×A4C2311yescharacteristic
36SL(2,ℤ/4)—21no—
37C3×M16—41no—
38D24—42no—
39#204—42no—
40C2×#41—31no—
41C22×D4—31no—
42C22×D4—61no—
43C23×C4C2×D311yescharacteristic
44AGL(1,ℤ/8)—61no—
45AGL(1,ℤ/8)—62no—
46AGL(1,ℤ/8)—61no—
47C2×SD16—32no—
48C2×D8—32no—
49C2×M16—31no—
50C2×M16—31no—
51C2×M16—32no—
52C22×C8C2×D312yescomplemented
53#123—61no—
54C22⋊Q8—31no—
55C22⋊Q8—31no—
56#121—32no—
57#121—31no—
58#121—32no—
59#121—31no—
60#121—61no—
61#121—61no—
62#120—61no—
63C4×D4—61no—
64C4×D4—61no—
65#117—31no—
66C2×C4⋊C4—31no—
67C2×C22⋊C4—61no—
68#107—32no—
69#106—32no—
70#103—32no—
71#102—32no—
72#102—32no—
73#102—32no—
74C22×D3—81no—
75C2×A4—21no—
76C2×A4C2×D411yescharacteristic
77S4—41no—
78S4—42no—
79C2×C3×C4—41no—
80#67—81no—
81D12—42no—
82D12—41no—
83D12—81no—
84C4×D3—81no—
85Dic6—41no—
86C3×C8—42no—
87C24—61no—
88C24D1211yescharacteristic
89#41—61no—
90#41—61no—
91#41—61no—
92#41—61no—
93C2×Q8—31no—
94C2×D4—31no—
95C2×D4—32no—
96C2×D4—61no—
97C2×D4—61no—
98C2×D4—61no—
99C2×D4—61no—
100C2×D4—61no—
101C2×D4—61no—
102C2×D4—62no—
103C2×D4—61no—
104C2×D4—62no—
105C2×D4—61no—
106C2×D4—122no—
107C2×D4—122no—
108C22×C4—31no—
109C22×C4—31no—
110C22×C4—31no—
111C22×C4—31no—
112C22×C4—61no—
113C22×C4—61no—
114C22×C4D1211yescharacteristic
115C22×C4C22×D311yescharacteristic
116SD16—62no—
117SD16—62no—
118D8—62no—
119D8—62no—
120M16—31no—
121M16—32no—
122M16—61no—
123M16S411yescharacteristic, complemented
124C2×C8—32no—
125C2×C8—32no—
126C4⋊C4—31no—
127C4⋊C4—31no—
128C4⋊C4—32no—
129C4⋊C4—61no—
130C4⋊C4—61no—
131C4⋊C4—61no—
132C22⋊C4—62no—
133C22⋊C4—62no—
134C22⋊C4—61no—
135C22⋊C4—61no—
136C22⋊C4—61no—
137C22⋊C4—61no—
138C22⋊C4—61no—
139C42—61no—
140C22×C3—41no—
141C2×D3—81no—
142C2×D3—82no—
143C2×D3—162no—
144A4AGL(1,ℤ/8)11yescharacteristic, complemented
145C3×C4—41no—
146C3×C4—41no—
147Dic3—81no—
148C23—21no—
149C23—31no—
150C23—31no—
151C23—61no—
152C23—61no—
153C23—62no—
154C23—121no—
155C23—122no—
156C23—122no—
157C23—121no—
158C23C2×D1211yescharacteristic
159Q8—61no—
160Q8—61no—
161D4—62no—
162D4—61no—
163D4—61no—
164D4—62no—
165D4—121no—
166D4—121no—
167D4—122no—
168D4—121no—
169D4—122no—
170D4—124no—
171D4—122no—
172D4—121no—
173D4—121no—
174D4—121no—
175C2×C4—31no—
176C2×C4—31no—
177C2×C4—31no—
178C2×C4—31no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—31no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—62no—
186C2×C4—61no—
187C2×C4—121no—
188C2×C4—122no—
189C2×C4—121no—
190C2×C4C2×S411yescharacteristic
191C8—32no—
192C8C2×S412yescomplemented
193C2×C3—41no—
194C2×C3—81no—
195D3—162no—
196D3—161no—
197C22—31no—
198C22—31no—
199C22—31no—
200C22—61no—
201C22—61no—
202C22—61no—
203C22—61no—
204C22—122no—
205C22—121no—
206C22—121no—
207C22—122no—
208C22—121no—
209C22—122no—
210C22—242no—
211C22—242no—
212C22#90811yescharacteristic, complemented
213C22C4⋊S411yescharacteristic
214C4—31no—
215C4—31no—
216C4—122no—
217C4—121no—
218C4—121no—
219C4—121no—
220C4C4⋊S411yescharacteristic
221C4C22×S411yescharacteristic
222C3—41noSylow
223C2—21no—
224C2—31no—
225C2—31no—
226C2—61no—
227C2—242no—
228C2—241no—
229C2C2×C4⋊S411yescharacteristic
subgroupG/H
centerC2C2×C4⋊S4
commutator, cosocleC4×A4C23
FrattiniC4C22×S4
FittingC22×M16D3
socleC23C2×D12
hypercenterM16S4
hypocenterA4AGL(1,ℤ/8)
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #5912C2semidirect
2. C2×C4⋊S4C2semidirect
3. A4×M16C2semidirect
4. #5402C2semidirect
5. #5401C2semidirect
12. C4⋊S4C22semidirect
14. A4⋊Q8C22semidirect
16. C8×A4C22semidirect
18. C22×M16D3semidirect
52. C22×C8C2×D3semidirect
123. M16S4semidirect
144. A4AGL(1,ℤ/8)semidirect
192. C8C2×S4semidirect
212. C22#908semidirect

Maximal subgroups

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

subgroupindex
#59122
C2×C4⋊S42
A4×M162
#54022
#54012
#30553
#9084