Finite Group Databasebeta

← C2×#5914 #87476

Subgroups — C2×#5914 #87476

Order 384.

905 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 905 conjugacy classes of subgroups — too many to draw. Only its 108 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, complemented
2C2×GL(2,ℤ/4)C212yesmaximal, complemented
3C22×SL(2,ℤ/4)C212yesmaximal, complemented
4#5914C218yesmaximal, direct product factor
5C2×C4×S4C211yesmaximal, characteristic, complemented
6C2×A4⋊Q8C211yesmaximal, characteristic, complemented
7C2×#546—31nomaximal, Sylow
8C23×A4C2212yes—
9C22×S4C2211yescharacteristic
10C2×#237—41nomaximal
11D4×A4C2214yescomplemented
12C2×C4×A4C2211yescharacteristic
13GL(2,ℤ/4)C2218yescomplemented
14C2×SL(2,ℤ/4)C2212yes—
15C2×SL(2,ℤ/4)C2211yescharacteristic
16C2×SL(2,ℤ/4)C2218yescomplemented
17C4×S4C2214yescomplemented
18A4⋊Q8C2214yescomplemented
19C22×#41—31no—
20C23×D4D311yescharacteristic, complemented
21#546—38no—
22#546—38no—
23C2×#124—31no—
24C2×#123—32no—
25C2×C22⋊Q8—31no—
26C2×#121—31no—
27C2×#121—32no—
28C2×#120—32no—
29C2×C4×D4—31no—
30C2×C4×D4—31no—
31C22×C22⋊C4—32no—
32C22×A4—24no—
33C22×A4C2314yes—
34C22×A4C2311yescharacteristic
35C2×S4—22no—
36C2×S4C2312yes—
37C2×C3×D4—41no—
38C2×#67—42no—
39C22×Dic3—42no—
40#237—48no—
41C2×C4×D3—41no—
42C2×Dic6—41no—
43C4×A4C2312yes—
44SL(2,ℤ/4)C2314yes—
45SL(2,ℤ/4)C2312yes—
46C25C2×D312yes—
47C2×#41—38no—
48C2×#41—64no—
49C2×#41—62no—
50C2×#41—61no—
51C2×#41—61no—
52C22×Q8—31no—
53C22×D4—31no—
54C22×D4—32no—
55C22×D4—31no—
56C22×D4—34no—
57C22×D4—32no—
58C22×D4—31no—
59C22×D4—32no—
60C22×D4C2×D314yescomplemented
61C23×C4—32no—
62C23×C4—31no—
63C23×C4—32no—
64C23×C4C2×D311yescharacteristic
65#124—38no—
66#123—316no—
67C22⋊Q8—34no—
68C22⋊Q8—34no—
69#121—38no—
70#121—316no—
71#120—38no—
72#120—38no—
73C4×D4—34no—
74C4×D4—38no—
75C4×D4—34no—
76C2×C4⋊C4—31no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—32no—
81C2×C22⋊C4—38no—
82C2×C22⋊C4—32no—
83C2×C22⋊C4—31no—
84C2×C22⋊C4—32no—
85C2×C22⋊C4—31no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—38no—
88C2×C42—31no—
89C23×C3—42no—
90C22×D3—41no—
91C2×A4—24no—
92C2×A4#4112yescomplemented
93C2×A4C2411yescharacteristic
94S4—22no—
95C3×D4—44no—
96C2×C3×C4—41no—
97#67—48no—
98C2×Dic3—42no—
99C2×Dic3—41no—
100C2×Dic3—48no—
101C4×D3—44no—
102Dic6—44no—
103C24—24no—
104C24—32no—
105C24—32no—
106C24—34no—
107C24—31no—
108C24—64no—
109C24C22×D314yes—
110C24C22×D311yescharacteristic
111#41—616no—
112#41—68no—
113#41—68no—
114C2×Q8—34no—
115C2×Q8—61no—
116C2×Q8—62no—
117C2×Q8—61no—
118C2×D4—32no—
119C2×D4—34no—
120C2×D4—34no—
121C2×D4—38no—
122C2×D4—34no—
123C2×D4—32no—
124C2×D4—38no—
125C2×D4—32no—
126C2×D4—38no—
127C2×D4—31no—
128C2×D4—62no—
129C2×D4—61no—
130C2×D4—64no—
131C2×D4—62no—
132C2×D4—64no—
133C2×D4—62no—
134C2×D4—62no—
135C2×D4—62no—
136C2×D4—124no—
137C2×D4S411yescharacteristic, complemented
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—32no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—32no—
144C22×C4—31no—
145C22×C4—32no—
146C22×C4—38no—
147C22×C4—34no—
148C22×C4—38no—
149C22×C4—61no—
150C22×C4—62no—
151C22×C4—64no—
152C22×C4—62no—
153C22×C4—62no—
154C22×C4—61no—
155C22×C4—124no—
156C22×C4C22×D312yes—
157C4⋊C4—34no—
158C4⋊C4—34no—
159C4⋊C4—38no—
160C22⋊C4—32no—
161C22⋊C4—34no—
162C22⋊C4—38no—
163C22⋊C4—38no—
164C22⋊C4—38no—
165C22⋊C4—34no—
166C22⋊C4—38no—
167C22⋊C4—32no—
168C22⋊C4—34no—
169C42—34no—
170C22×C3—44no—
171C22×C3—41no—
172C22×C3—84no—
173C2×D3—42no—
174C2×D3—82no—
175A4C2×#4111yescharacteristic, complemented
176C3×C4—42no—
177Dic3—44no—
178Dic3—42no—
179C23—24no—
180C23—31no—
181C23—31no—
182C23—32no—
183C23—34no—
184C23—34no—
185C23—34no—
186C23—32no—
187C23—32no—
188C23—64no—
189C23—64no—
190C23—62no—
191C23—64no—
192C23—64no—
193C23—68no—
194C23—64no—
195C23—61no—
196C23—62no—
197C23—62no—
198C23—61no—
199C23—124no—
200C23#23712yescomplemented
201C23C2×S412yes—
202C23C23×D311yescharacteristic
203Q8—64no—
204Q8—64no—
205D4—38no—
206D4—34no—
207D4—64no—
208D4—64no—
209D4—62no—
210D4—68no—
211D4—62no—
212D4—68no—
213D4—128no—
214D4C2×S414yescomplemented
215C2×C4—31no—
216C2×C4—32no—
217C2×C4—32no—
218C2×C4—32no—
219C2×C4—32no—
220C2×C4—32no—
221C2×C4—34no—
222C2×C4—32no—
223C2×C4—34no—
224C2×C4—61no—
225C2×C4—64no—
226C2×C4—61no—
227C2×C4—62no—
228C2×C4—68no—
229C2×C4—62no—
230C2×C4—64no—
231C2×C4—62no—
232C2×C4—68no—
233C2×C4—61no—
234C2×C4—62no—
235C2×C4—64no—
236C2×C4—128no—
237C2×C4C2×S411yescharacteristic
238C2×C3—42no—
239C2×C3—41no—
240C2×C3—84no—
241D3—82no—
242C22—24no—
243C22—32no—
244C22—31no—
245C22—32no—
246C22—32no—
247C22—31no—
248C22—31no—
249C22—34no—
250C22—64no—
251C22—64no—
252C22—68no—
253C22—61no—
254C22—64no—
255C22—64no—
256C22—62no—
257C22—62no—
258C22—64no—
259C22—62no—
260C22—124no—
261C22—122no—
262C22—122no—
263C22C2×#23711yescharacteristic, complemented
264C22C22×S414yes—
265C22C22×S411yescharacteristic
266C4—32no—
267C4—62no—
268C4—64no—
269C4—62no—
270C4—62no—
271C4—64no—
272C4C22×S412yes—
273C3—41noSylow
274C2—24no—
275C2—31no—
276C2—31no—
277C2—32no—
278C2—64no—
279C2—122no—
280C2#591412yesdirect product factor
281C2C23×S411yescharacteristic
subgroupG/H
centerC22C22×S4
commutator, cosocleC2×A4C24
FrattiniC2C23×S4
FittingC23×D4D3
socleC24C22×D3
hypercenterC2×D4S4
hypocenterA4C2×#41
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×A4C2semidirect
2. C2×GL(2,ℤ/4)C2semidirect
3. C22×SL(2,ℤ/4)C2semidirect
4. #5914C2direct (N × H)
5. C2×C4×S4C2semidirect
6. C2×A4⋊Q8C2semidirect
11. D4×A4C22semidirect
13. GL(2,ℤ/4)C22semidirect
16. C2×SL(2,ℤ/4)C22semidirect
17. C4×S4C22semidirect
18. A4⋊Q8C22semidirect
20. C23×D4D3semidirect
60. C22×D4C2×D3semidirect
92. C2×A4#41semidirect
137. C2×D4S4semidirect
175. A4C2×#41semidirect
200. C23#237semidirect
214. D4C2×S4semidirect
263. C22C2×#237semidirect
280. C2#5914direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×A42
C2×GL(2,ℤ/4)2
C22×SL(2,ℤ/4)2
#59142
C2×C4×S42
C2×A4⋊Q82
C2×#5463
C2×#2374