Finite Group Databasebeta

← C2×#85291 #11714465

Subgroups — C2×#85291 #11714465

Order 768.

1459 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 1459 conjugacy classes of subgroups — too many to draw. Only its 141 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×C42×A4C211yesmaximal, characteristic, complemented
2C2×(C2×S4)⋊C4C213yesmaximal
3C2×SL(2,ℤ/4)⋊C4C213yesmaximal, complemented
4#85291C218yesmaximal, direct product factor
5C2×#2852—31nomaximal, Sylow
6C23×S4C2211yescharacteristic
7C22×C4×A4C2213yes—
8C22×SL(2,ℤ/4)C2213yes—
9C2×#876—41nomaximal
10C42×A4C2214yescomplemented
11(C2×S4)⋊C4C22112yes—
12SL(2,ℤ/4)⋊C4C22112yescomplemented
13C22×#126—31no—
14C23×C42D311yescharacteristic, complemented
15#2852—38no—
16#2852—38no—
17C2×#388—33no—
18C2×#387—33no—
19C2×#386—33no—
20C2×#385—33no—
21C2×#383—31no—
22C23×A4C2311yescharacteristic
23C22×S4—26no—
24C22×S4C2312yes—
25C2×C4×A4—26no—
26C2×C4×A4C2316yes—
27C2×SL(2,ℤ/4)—26no—
28C2×SL(2,ℤ/4)C2316yes—
29C2×C3×C42—41no—
30C2×(C2×D3)⋊C4—43no—
31C2×Dic3⋊C4—43no—
32#876—48no—
33C23×D4—31no—
34C24×C4C2×D313yes—
35C2×#126—38no—
36C2×#126—63no—
37C2×#126—64no—
38C2×#126—61no—
39C22×C4⋊C4—33no—
40C22×C4⋊C4—33no—
41C22×C22⋊C4—33no—
42C22×C22⋊C4—33no—
43C22×C22⋊C4—33no—
44C22×C42—31no—
45C22×C42—34no—
46C22×C42—33no—
47C22×C42C2×D314yescomplemented
48#388—324no—
49#387—324no—
50#386—312no—
51#386—312no—
52#385—312no—
53#385—312no—
54#383—38no—
55C2×#95—33no—
56C2×#95—33no—
57C2×#95—33no—
58C2×#95—33no—
59C23×D3—41no—
60C22×A4#4116yes—
61C22×A4C2411yescharacteristic
62C2×S4—26no—
63C2×S4—44no—
64C22×C3×C4—43no—
65C22×Dic3—43no—
66C4×A4—26no—
67SL(2,ℤ/4)—26no—
68C3×C42—44no—
69(C2×D3)⋊C4—412no—
70Dic3⋊C4—412no—
71C25—31no—
72C25C22×D311yescharacteristic
73C22×D4—32no—
74C22×D4—32no—
75C22×D4—66no—
76C22×D4—66no—
77C23×C4—26no—
78C23×C4—33no—
79C23×C4—33no—
80C23×C4—36no—
81C23×C4—33no—
82C23×C4—33no—
83C23×C4—31no—
84C23×C4—66no—
85C23×C4C22×D316yes—
86#126—624no—
87#126—68no—
88C2×C4⋊C4—312no—
89C2×C4⋊C4—312no—
90C2×C4⋊C4—612no—
91C2×C4⋊C4—63no—
92C2×C4⋊C4—66no—
93C2×C4⋊C4—63no—
94C2×C22⋊C4—312no—
95C2×C22⋊C4—36no—
96C2×C22⋊C4—312no—
97C2×C22⋊C4—36no—
98C2×C22⋊C4—612no—
99C2×C22⋊C4—63no—
100C2×C22⋊C4—66no—
101C2×C22⋊C4—66no—
102C2×C22⋊C4—66no—
103C2×C22⋊C4—63no—
104C2×C42—33no—
105C2×C42—34no—
106C2×C42—34no—
107C2×C42—312no—
108C2×C42—66no—
109C2×C42—61no—
110C2×C42S411yescharacteristic, complemented
111#95—312no—
112#95—312no—
113#95—312no—
114#95—312no—
115C23×C3—41no—
116C22×D3—42no—
117C22×D3—86no—
118C2×A4#12614yescomplemented
119C2×A4C2×#4113yes—
120S4—42no—
121C2×C3×C4—46no—
122C2×C3×C4—86no—
123C2×Dic3—46no—
124C2×Dic3—86no—
125C24—31no—
126C24—31no—
127C24—36no—
128C24—32no—
129C24—61no—
130C24—66no—
131C24—66no—
132C24—61no—
133C24#23516yes—
134C24C23×D311yescharacteristic
135C2×D4—612no—
136C2×D4—66no—
137C2×D4—66no—
138C2×D4—1212no—
139C2×D4—124no—
140C22×C4—26no—
141C22×C4—36no—
142C22×C4—312no—
143C22×C4—33no—
144C22×C4—36no—
145C22×C4—36no—
146C22×C4—32no—
147C22×C4—36no—
148C22×C4—36no—
149C22×C4—66no—
150C22×C4—66no—
151C22×C4—63no—
152C22×C4—66no—
153C22×C4—612no—
154C22×C4—612no—
155C22×C4—66no—
156C22×C4—63no—
157C22×C4—66no—
158C22×C4—66no—
159C22×C4—66no—
160C22×C4—66no—
161C22×C4—66no—
162C22×C4—66no—
163C22×C4—63no—
164C22×C4C2×S413yes—
165C4⋊C4—624no—
166C4⋊C4—612no—
167C4⋊C4—612no—
168C22⋊C4—624no—
169C22⋊C4—612no—
170C22⋊C4—612no—
171C22⋊C4—66no—
172C22⋊C4—66no—
173C22⋊C4—612no—
174C42—312no—
175C42—64no—
176C42C2×S414yescomplemented
177C22×C3—46no—
178C22×C3—41no—
179C2×D3—86no—
180C2×D3—164no—
181A4C2×#12611yescharacteristic, complemented
182C3×C4—86no—
183Dic3—86no—
184C23—36no—
185C23—36no—
186C23—31no—
187C23—31no—
188C23—36no—
189C23—33no—
190C23—312no—
191C23—63no—
192C23—64no—
193C23—66no—
194C23—66no—
195C23—62no—
196C23—124no—
197C23—1212no—
198C23—126no—
199C23—126no—
200C23#87614yescomplemented
201C23C2×#23513yes—
202C23C22×S411yescharacteristic
203D4—128no—
204D4—126no—
205D4—122no—
206C2×C4—26no—
207C2×C4—312no—
208C2×C4—36no—
209C2×C4—66no—
210C2×C4—66no—
211C2×C4—612no—
212C2×C4—612no—
213C2×C4—66no—
214C2×C4—66no—
215C2×C4—66no—
216C2×C4—66no—
217C2×C4—66no—
218C2×C4—66no—
219C2×C4—66no—
220C2×C4—66no—
221C2×C4—66no—
222C2×C4—1212no—
223C2×C4—126no—
224C2×C4—124no—
225C2×C4—126no—
226C2×C4C22×S416yes—
227C2×C3—44no—
228C2×C3—43no—
229D3—162no—
230C22—34no—
231C22—33no—
232C22—34no—
233C22—312no—
234C22—33no—
235C22—33no—
236C22—36no—
237C22—61no—
238C22—66no—
239C22—122no—
240C22—126no—
241C22—126no—
242C22—244no—
243C22—244no—
244C22C2×#87611yescharacteristic, complemented
245C22#591216yes—
246C22C23×S411yescharacteristic
247C4—26no—
248C4—66no—
249C4—126no—
250C4—122no—
251C4—126no—
252C3—41noSylow
253C2—31no—
254C2—33no—
255C2—34no—
256C2—242no—
257C2#8529114yesdirect product factor
258C2C2×#591213yes—
subgroupG/H
centerC23C22×S4
commutator, cosocleC22×A4C24
FrattiniC22C23×S4
FittingC23×C42D3
socleC25C22×D3
prime radicalC23×S4C22
hypercenterC2×C42S4
hypocenterA4C2×#126
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×C42×A4C2semidirect
3. C2×SL(2,ℤ/4)⋊C4C2semidirect
4. #85291C2direct (N × H)
10. C42×A4C22semidirect
12. SL(2,ℤ/4)⋊C4C22semidirect
14. C23×C42D3semidirect
47. C22×C42C2×D3semidirect
110. C2×C42S4semidirect
118. C2×A4#126semidirect
176. C42C2×S4semidirect
181. A4C2×#126semidirect
200. C23#876semidirect
244. C22C2×#876semidirect
257. C2#85291direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C42×A42
C2×(C2×S4)⋊C42
C2×SL(2,ℤ/4)⋊C42
#852912
C2×#28523
C2×#8764