Finite Group Databasebeta

← D4×SL(2,ℤ/4) #85490

Subgroups — D4×SL(2,ℤ/4) #85490

Order 384.

511 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 70 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×D4×A4C211yesmaximal, characteristic
2C22×SL(2,ℤ/4)C212yesmaximal, complemented
3#5432C212yesmaximal, complemented
4C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6D4×C22⋊C4—31nomaximal, Sylow
7C23×A4C2212yes—
8D4×A4C414yescomplemented
9C2×C4×A4C2211yescharacteristic
10C2×SL(2,ℤ/4)—24no—
11C2×SL(2,ℤ/4)C2211yescharacteristic
12C2×SL(2,ℤ/4)C2212yes—
13C2×SL(2,ℤ/4)C2211yescharacteristic
14D4×Dic3—41nomaximal
15C23×D4D311yescharacteristic
16C2×C4×D4—31no—
17C2×C4×D4—31no—
18C22×C22⋊C4—32no—
19#390—31no—
20#390—31no—
21#386—32no—
22#386—32no—
23#380—31no—
24#379—32no—
25C4×C22⋊C4—31no—
26C22×A4—22no—
27C22×A4—22no—
28C22×A4C2×C414yescomplemented
29C22×A4C2311yescharacteristic
30C2×C3×D4—41no—
31C22×Dic3—42no—
32C4×A4C2×C412yescomplemented
33SL(2,ℤ/4)—22no—
34SL(2,ℤ/4)—21no—
35SL(2,ℤ/4)D412yesdirect product factor
36#217—42no—
37C4⋊Dic3—41no—
38C4×Dic3—41no—
39C25C2×D312yes—
40C22×D4—31no—
41C22×D4—34no—
42C22×D4—32no—
43C22×D4—31no—
44C22×D4Dic314yescomplemented
45C23×C4—32no—
46C23×C4—32no—
47C23×C4C2×D311yescharacteristic
48C4×D4—64no—
49C4×D4—62no—
50C4×D4—61no—
51C4×D4—61no—
52C2×C4⋊C4—31no—
53C2×C4⋊C4—31no—
54C2×C22⋊C4—31no—
55C2×C22⋊C4—32no—
56C2×C22⋊C4—32no—
57C2×C22⋊C4—32no—
58C2×C22⋊C4—32no—
59C2×C22⋊C4—32no—
60C2×C22⋊C4—31no—
61C2×C22⋊C4—32no—
62C2×C22⋊C4—32no—
63C2×C22⋊C4—64no—
64C2×C22⋊C4—64no—
65C2×C42—31no—
66C2×C42—31no—
67#95—31no—
68#95—31no—
69#95—32no—
70C23×C3—42no—
71C2×A4—24no—
72C2×A4C22×C411yescharacteristic
73C2×A4C2×D411yescharacteristic
74C2×A4#4111yescharacteristic
75C3×D4—44no—
76C2×C3×C4—41no—
77C2×Dic3—41no—
78C2×Dic3—41no—
79C2×Dic3—42no—
80C2×Dic3—84no—
81C24—22no—
82C24—22no—
83C24—32no—
84C24—32no—
85C24—34no—
86C24—62no—
87C24—62no—
88C24C2×Dic314yescomplemented
89C24C22×D311yescharacteristic
90C2×D4—32no—
91C2×D4—34no—
92C2×D4—34no—
93C2×D4—34no—
94C2×D4—38no—
95C2×D4—31no—
96C2×D4—62no—
97C2×D4—61no—
98C2×D4—64no—
99C2×D4S411yescharacteristic
100C22×C4—31no—
101C22×C4—31no—
102C22×C4—32no—
103C22×C4—31no—
104C22×C4—31no—
105C22×C4—31no—
106C22×C4—31no—
107C22×C4—32no—
108C22×C4—32no—
109C22×C4—64no—
110C22×C4—64no—
111C22×C4—64no—
112C22×C4—62no—
113C22×C4—62no—
114C22×C4—122no—
115C22×C4—122no—
116C22×C4C2×Dic312yescomplemented
117C4⋊C4—61no—
118C4⋊C4—62no—
119C4⋊C4—61no—
120C22⋊C4—32no—
121C22⋊C4—32no—
122C22⋊C4—64no—
123C22⋊C4—64no—
124C22⋊C4—61no—
125C22⋊C4—62no—
126C22⋊C4—64no—
127C22⋊C4—62no—
128C22⋊C4—62no—
129C22⋊C4—61no—
130C22⋊C4—62no—
131C22⋊C4—122no—
132C22⋊C4—122no—
133C42—61no—
134C42—62no—
135C42—61no—
136C22×C3—44no—
137C22×C3—41no—
138C22×C3—82no—
139C22×C3—82no—
140A4C4×D411yescharacteristic, complemented
141C3×C4—42no—
142Dic3—42no—
143Dic3—82no—
144Dic3—81no—
145C23—24no—
146C23—31no—
147C23—31no—
148C23—31no—
149C23—31no—
150C23—34no—
151C23—34no—
152C23—34no—
153C23—32no—
154C23—62no—
155C23—62no—
156C23—62no—
157C23—62no—
158C23—62no—
159C23—64no—
160C23—62no—
161C23—62no—
162C23—62no—
163C23—64no—
164C23—64no—
165C23—62no—
166C23—124no—
167C23D3×D411yescharacteristic
168C23#23711yescharacteristic
169C23C22×Dic311yescharacteristic
170C23C2×S412yes—
171D4—38no—
172D4—34no—
173D4—64no—
174D4SL(2,ℤ/4)14yesdirect product factor
175C2×C4—31no—
176C2×C4—32no—
177C2×C4—32no—
178C2×C4—32no—
179C2×C4—32no—
180C2×C4—32no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—61no—
186C2×C4—61no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—61no—
197C2×C4—62no—
198C2×C4—124no—
199C2×C4—128no—
200C2×C4—124no—
201C2×C4C2×S411yescharacteristic
202C2×C3—41no—
203C2×C3—41no—
204C2×C3—41no—
205C2×C3—84no—
206C22—22no—
207C22—22no—
208C22—31no—
209C22—31no—
210C22—31no—
211C22—31no—
212C22—31no—
213C22—31no—
214C22—31no—
215C22—31no—
216C22—31no—
217C22—34no—
218C22—64no—
219C22—64no—
220C22—64no—
221C22—64no—
222C22—61no—
223C22—64no—
224C22—64no—
225C22—62no—
226C22—62no—
227C22—122no—
228C22—122no—
229C22D4×Dic311yescharacteristic, complemented
230C22C2×SL(2,ℤ/4)14yescomplemented
231C22C22×S411yescharacteristic
232C4—32no—
233C4—62no—
234C4—62no—
235C4—121no—
236C4—122no—
237C4—122no—
238C4—121no—
239C4C2×SL(2,ℤ/4)12yescomplemented
240C3—41noSylow
241C2—24no—
242C2—31no—
243C2—31no—
244C2—31no—
245C2—31no—
246C2—64no—
247C2D4×S411yescharacteristic
248C2#591411yescharacteristic
249C2C22×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutatorC2×A4C22×C4
FittingC23×D4D3
socleC24C22×D3
prime radicalC2×D4×A4C2
cosocleC22×A4C23
hypercenterC2×D4S4
hypocenterA4C4×D4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×SL(2,ℤ/4)C2semidirect
3. #5432C2semidirect
4. C4⋊SL(2,ℤ/4)C2semidirect
5. C4×SL(2,ℤ/4)C2semidirect
8. D4×A4C4semidirect
28. C22×A4C2×C4semidirect
32. C4×A4C2×C4semidirect
35. SL(2,ℤ/4)D4direct (N × H)
44. C22×D4Dic3semidirect
88. C24C2×Dic3semidirect
116. C22×C4C2×Dic3semidirect
140. A4C4×D4semidirect
174. D4SL(2,ℤ/4)direct (N × H)
229. C22D4×Dic3semidirect
230. C22C2×SL(2,ℤ/4)semidirect
239. C4C2×SL(2,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
C2×D4×A42
C22×SL(2,ℤ/4)2
#54322
C4⋊SL(2,ℤ/4)2
C4×SL(2,ℤ/4)2
D4×C22⋊C43
D4×Dic34