Finite Group Databasebeta

← #85513

Subgroups — #85513

Order 384.

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

normal subgroups only

This group has 336 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
1A4×#41C211yesmaximal, characteristic, complemented
2C2×C4⋊S4C211yesmaximal, characteristic, complemented
3#5417C212yesmaximal, complemented
4#5415C212yesmaximal, complemented
5#5409C211yesmaximal, characteristic, complemented
6#3006—31nomaximal, Sylow
7C22×S4—21no—
8Q8×A4C2211yescharacteristic, complemented
9D4×A4—21no—
10D4×A4C2211yescharacteristic, complemented
11C2×C4×A4—21no—
12C2×C4×A4C2211yescharacteristic
13C4⋊S4—21no—
14C4⋊S4C2212yes—
15#949—41nomaximal
16A4⋊C8C2212yescomplemented
17C22×#41D311yescharacteristic, complemented
18C2×AGL(1,ℤ/8)—31no—
19C2×AGL(1,ℤ/8)—31no—
20C2×#121—31no—
21#449—32no—
22#449—32no—
23#448—32no—
24#447—32no—
25#417—31no—
26#417—31no—
27#407—31no—
28C22×A4—21no—
29C22×A4D411yescharacteristic
30C2×S4—22no—
31C2×S4—42no—
32C3×#41—41no—
33C2×D12—41no—
34C4×A4—21no—
35C4×A4D411yescharacteristic
36C4×A4C2311yescharacteristic
37#215—42no—
38#213—42no—
39#208—41no—
40C2×#41—31no—
41C2×#41—31no—
42C2×#41—31no—
43C2×#41—31no—
44C2×#41—61no—
45C2×#41—61no—
46C22×Q8C2×D311yescharacteristic, complemented
47C22×D4—21no—
48C22×D4—31no—
49C22×D4—61no—
50C22×D4C2×D311yescharacteristic, complemented
51C23×C4—21no—
52C23×C4C2×D311yescharacteristic
53AGL(1,ℤ/8)—62no—
54AGL(1,ℤ/8)—62no—
55AGL(1,ℤ/8)—61no—
56AGL(1,ℤ/8)—61no—
57AGL(1,ℤ/8)—61no—
58AGL(1,ℤ/8)—61no—
59C2×SD16—32no—
60C2×SD16—32no—
61C2×D8—32no—
62C2×D8—32no—
63C2×M16—31no—
64C2×M16—31no—
65#121—32no—
66#121—32no—
67#121—61no—
68#121—61no—
69C2×C4⋊C4—31no—
70C2×C22⋊C4—61no—
71#103—32no—
72#103—32no—
73#102—32no—
74#102—32no—
75C22⋊C8—32no—
76C22⋊C8—32no—
77C22×D3—81no—
78C2×A4—21no—
79C2×A4—41no—
80C2×A4C2×D411yescharacteristic
81S4—42no—
82C3×Q8—41no—
83C3×D4—41no—
84C3×D4—81no—
85C2×C3×C4—41no—
86C2×C3×C4—81no—
87D12—42no—
88D12—81no—
89C3⋊C8—42no—
90C24—21no—
91C24—61no—
92C24#6711yescharacteristic
93#41—32no—
94#41—32no—
95#41—31no—
96#41—31no—
97#41—31no—
98#41—61no—
99#41—61no—
100#41—61no—
101#41—62no—
102#41—62no—
103#41S411yescharacteristic, complemented
104C2×Q8—31no—
105C2×Q8—31no—
106C2×Q8—61no—
107C2×D4—31no—
108C2×D4—31no—
109C2×D4—32no—
110C2×D4—61no—
111C2×D4—61no—
112C2×D4—61no—
113C2×D4—61no—
114C2×D4—61no—
115C2×D4—61no—
116C2×D4—61no—
117C2×D4—61no—
118C2×D4—61no—
119C2×D4—62no—
120C2×D4—62no—
121C2×D4—61no—
122C2×D4—122no—
123C2×D4—122no—
124C22×C4—21no—
125C22×C4—31no—
126C22×C4—31no—
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—61no—
130C22×C4—61no—
131C22×C4—61no—
132C22×C4—61no—
133C22×C4—61no—
134C22×C4#6711yescharacteristic
135C22×C4C22×D311yescharacteristic
136SD16—64no—
137SD16—62no—
138SD16—62no—
139D8—64no—
140D8—62no—
141D8—62no—
142M16—61no—
143M16—61no—
144M16—61no—
145M16—61no—
146C2×C8—32no—
147C2×C8—32no—
148C4⋊C4—32no—
149C4⋊C4—61no—
150C22⋊C4—62no—
151C22⋊C4—62no—
152C22×C3—41no—
153C22×C3—81no—
154C2×D3—82no—
155C2×D3—162no—
156A4AGL(1,ℤ/8)11yescharacteristic, complemented
157C3×C4—41no—
158C3×C4—41no—
159C3×C4—81no—
160C23—21no—
161C23—31no—
162C23—31no—
163C23—41no—
164C23—61no—
165C23—61no—
166C23—61no—
167C23—62no—
168C23—121no—
169C23—121no—
170C23—122no—
171C23—122no—
172C23—121no—
173C23C2×#6711yescharacteristic
174Q8—32no—
175Q8—31no—
176Q8—61no—
177Q8C2×S411yescharacteristic, complemented
178D4—21no—
179D4—32no—
180D4—31no—
181D4—61no—
182D4—61no—
183D4—62no—
184D4—61no—
185D4—62no—
186D4—61no—
187D4—62no—
188D4—121no—
189D4—122no—
190D4—124no—
191D4—122no—
192D4—121no—
193D4C2×S411yescharacteristic, complemented
194C2×C4—21no—
195C2×C4—31no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—61no—
208C2×C4—61no—
209C2×C4—62no—
210C2×C4—122no—
211C2×C4C2×S411yescharacteristic
212C8—62no—
213C8—62no—
214C2×C3—41no—
215C2×C3—81no—
216C2×C3—161no—
217D3—162no—
218C22—21no—
219C22—31no—
220C22—31no—
221C22—31no—
222C22—61no—
223C22—61no—
224C22—61no—
225C22—61no—
226C22—61no—
227C22—121no—
228C22—121no—
229C22—121no—
230C22—121no—
231C22—122no—
232C22—122no—
233C22—122no—
234C22—242no—
235C22—242no—
236C22#94911yescharacteristic, complemented
237C22GL(2,ℤ/4)11yescharacteristic
238C4—21no—
239C4—31no—
240C4—31no—
241C4—61no—
242C4—122no—
243C4GL(2,ℤ/4)11yescharacteristic
244C4C22×S411yescharacteristic
245C3—41noSylow
246C2—21no—
247C2—31no—
248C2—31no—
249C2—41no—
250C2—61no—
251C2—121no—
252C2—242no—
253C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2C2×GL(2,ℤ/4)
commutator, cosocleC4×A4C23
FrattiniC4C22×S4
FittingC22×#41D3
socleC23C2×#67
hypercenter#41S4
hypocenterA4AGL(1,ℤ/8)
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. A4×#41C2semidirect
2. C2×C4⋊S4C2semidirect
3. #5417C2semidirect
4. #5415C2semidirect
5. #5409C2semidirect
8. Q8×A4C22semidirect
10. D4×A4C22semidirect
16. A4⋊C8C22semidirect
17. C22×#41D3semidirect
46. C22×Q8C2×D3semidirect
50. C22×D4C2×D3semidirect
103. #41S4semidirect
156. A4AGL(1,ℤ/8)semidirect
177. Q8C2×S4semidirect
193. D4C2×S4semidirect
236. C22#949semidirect

Maximal subgroups

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

subgroupindex
A4×#412
C2×C4⋊S42
#54172
#54152
#54092
#30063
#9494