Finite Group Databasebeta

← #85486

Subgroups — #85486

Order 384.

497 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 50 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
1C22×C4×A4C211yesmaximal, characteristic, complemented
2C2×GL(2,ℤ/4)C212yesmaximal, complemented
3C2×C4⋊S4C211yesmaximal, characteristic, complemented
4(C2×S4)⋊C4C212yesmaximal, complemented
5C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6#2847—31nomaximal, Sylow
7C23×A4C2211yescharacteristic
8C22×S4C2212yes—
9C2×C4×A4—22no—
10C2×C4×A4C2211yescharacteristic
11C2×C4×A4C2211yescharacteristic
12GL(2,ℤ/4)—24no—
13C2×SL(2,ℤ/4)C2212yes—
14C4⋊S4—22no—
15#930—41nomaximal
16C24×C4D311yescharacteristic, complemented
17C2×#121—31no—
18C2×#121—31no—
19C2×#121—31no—
20C2×#120—32no—
21#386—32no—
22#386—32no—
23#386—32no—
24#380—31no—
25#380—31no—
26#380—31no—
27C22×A4—21no—
28C22×A4—21no—
29C22×A4D412yescomplemented
30C22×A4C2311yescharacteristic
31C2×S4—22no—
32C2×S4—22no—
33C2×S4—22no—
34C22×C3×C4—41no—
35C2×#67—42no—
36C2×D12—41no—
37C4×A4—21no—
38C4×A4D412yescomplemented
39SL(2,ℤ/4)—22no—
40(C2×D3)⋊C4—42no—
41C4⋊Dic3—41no—
42C25C2×D311yescharacteristic
43C22×D4—32no—
44C22×D4—32no—
45C22×D4—31no—
46C22×D4—32no—
47C23×C4—22no—
48C23×C4—31no—
49C23×C4—31no—
50C23×C4—31no—
51C23×C4—31no—
52C23×C4—62no—
53C23×C4C2×D311yescharacteristic
54C23×C4C2×D311yescharacteristic
55#121—62no—
56#121—64no—
57#121—61no—
58#121—61no—
59#121—61no—
60#121—62no—
61#121—61no—
62#120—64no—
63#120—64no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—31no—
66C2×C4⋊C4—31no—
67C2×C22⋊C4—32no—
68C2×C22⋊C4—32no—
69C2×C22⋊C4—32no—
70C2×C22⋊C4—32no—
71C2×C22⋊C4—32no—
72C2×C22⋊C4—32no—
73#95—32no—
74#95—32no—
75#95—32no—
76C23×C3—41no—
77C22×D3—42no—
78C2×A4—22no—
79C2×A4C2×D411yescharacteristic
80C2×A4C2×D411yescharacteristic
81C2×A4#4111yescharacteristic
82S4—42no—
83C2×C3×C4—41no—
84C2×C3×C4—41no—
85C2×C3×C4—82no—
86#67—84no—
87C2×Dic3—42no—
88D12—82no—
89C24—21no—
90C24—21no—
91C24—31no—
92C24—31no—
93C24—32no—
94C24—32no—
95C24—61no—
96C24—61no—
97C24D1212yescomplemented
98C24C22×D311yescharacteristic
99C2×D4—61no—
100C2×D4—62no—
101C2×D4—62no—
102C2×D4—64no—
103C2×D4—62no—
104C2×D4—64no—
105C2×D4—62no—
106C2×D4—64no—
107C2×D4—62no—
108C2×D4—62no—
109C2×D4—64no—
110C2×D4—64no—
111C2×D4—62no—
112C2×D4—62no—
113C2×D4—62no—
114C2×D4—62no—
115C2×D4—61no—
116C22×C4—21no—
117C22×C4—31no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—32no—
121C22×C4—34no—
122C22×C4—31no—
123C22×C4—31no—
124C22×C4—31no—
125C22×C4—31no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4—32no—
129C22×C4—62no—
130C22×C4—62no—
131C22×C4—61no—
132C22×C4—61no—
133C22×C4—62no—
134C22×C4—62no—
135C22×C4—62no—
136C22×C4—62no—
137C22×C4—62no—
138C22×C4#6712yescomplemented
139C22×C4S411yescharacteristic, complemented
140C4⋊C4—61no—
141C4⋊C4—62no—
142C4⋊C4—62no—
143C4⋊C4—61no—
144C22⋊C4—64no—
145C22⋊C4—64no—
146C22⋊C4—62no—
147C22⋊C4—64no—
148C22⋊C4—62no—
149C22⋊C4—64no—
150C22⋊C4—62no—
151C22⋊C4—62no—
152C22×C3—42no—
153C22×C3—41no—
154C22×C3—81no—
155C22×C3—81no—
156C2×D3—82no—
157C2×D3—82no—
158C2×D3—82no—
159A4#12111yescharacteristic, complemented
160C3×C4—42no—
161C3×C4—81no—
162Dic3—82no—
163C23—22no—
164C23—31no—
165C23—32no—
166C23—31no—
167C23—31no—
168C23—31no—
169C23—32no—
170C23—32no—
171C23—32no—
172C23—32no—
173C23—31no—
174C23—61no—
175C23—61no—
176C23—61no—
177C23—61no—
178C23—62no—
179C23—61no—
180C23—61no—
181C23—62no—
182C23—62no—
183C23—61no—
184C23—64no—
185C23—62no—
186C23—62no—
187C23—62no—
188C23—62no—
189C23—62no—
190C23—62no—
191C23—62no—
192C23C2×D1211yescharacteristic
193C23#23511yescharacteristic
194C23C2×#6711yescharacteristic
195C23C2×S411yescharacteristic
196D4—122no—
197D4—122no—
198D4—122no—
199D4—122no—
200D4—128no—
201D4—124no—
202D4—122no—
203D4—124no—
204D4—122no—
205C2×C4—22no—
206C2×C4—31no—
207C2×C4—32no—
208C2×C4—32no—
209C2×C4—32no—
210C2×C4—32no—
211C2×C4—31no—
212C2×C4—62no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—62no—
217C2×C4—64no—
218C2×C4—62no—
219C2×C4—61no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4C2×S411yescharacteristic
230C2×C4C2×S411yescharacteristic
231C2×C3—41no—
232C2×C3—41no—
233C2×C3—41no—
234C2×C3—82no—
235D3—162no—
236C22—21no—
237C22—21no—
238C22—31no—
239C22—31no—
240C22—31no—
241C22—31no—
242C22—31no—
243C22—31no—
244C22—31no—
245C22—31no—
246C22—31no—
247C22—34no—
248C22—32no—
249C22—62no—
250C22—62no—
251C22—62no—
252C22—62no—
253C22—61no—
254C22—62no—
255C22—62no—
256C22—62no—
257C22—61no—
258C22—61no—
259C22—122no—
260C22—122no—
261C22—122no—
262C22—122no—
263C22—122no—
264C22—122no—
265C22—122no—
266C22#93011yescharacteristic, complemented
267C22C4⋊S412yescomplemented
268C22C22×S411yescharacteristic
269C4—21no—
270C4—32no—
271C4—61no—
272C4—122no—
273C4—122no—
274C4—122no—
275C4GL(2,ℤ/4)12yescomplemented
276C3—41noSylow
277C2—22no—
278C2—31no—
279C2—31no—
280C2—31no—
281C2—31no—
282C2—62no—
283C2—242no—
284C2C2×C4⋊S411yescharacteristic
285C2#591211yescharacteristic
286C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, cosocleC22×A4C23
FittingC24×C4D3
socleC24C22×D3
hypercenterC22×C4S4
hypocenterA4#121
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×A4C2semidirect
2. C2×GL(2,ℤ/4)C2semidirect
3. C2×C4⋊S4C2semidirect
4. (C2×S4)⋊C4C2semidirect
5. C4⋊SL(2,ℤ/4)C2semidirect
16. C24×C4D3semidirect
29. C22×A4D4semidirect
38. C4×A4D4semidirect
97. C24D12semidirect
138. C22×C4#67semidirect
139. C22×C4S4semidirect
159. A4#121semidirect
266. C22#930semidirect
267. C22C4⋊S4semidirect
275. C4GL(2,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
C22×C4×A42
C2×GL(2,ℤ/4)2
C2×C4⋊S42
(C2×S4)⋊C42
C4⋊SL(2,ℤ/4)2
#28473
#9304