Finite Group Databasebeta

← S4×#106 #11713003

Subgroups — S4×#106 #11713003

Order 768.

605 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 77 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×C8×S4C211yesmaximal, characteristic
2C4⋊C4×S4C212yesmaximal
3A4×#106C211yesmaximal, characteristic, complemented
4#73110C211yesmaximal, characteristic, complemented
5#73051C212yesmaximal, complemented
6D4×#106—31nomaximal, Sylow
7C2×C4×S4—22no—
8C2×C4×S4C2211yescharacteristic
9C2×C8×A4C2211yescharacteristic
10A4×C4⋊C4C2212yes—
11C4⋊SL(2,ℤ/4)C2212yes—
12SL(2,ℤ/4)⋊C4—22no—
13C2×A4⋊C8C2211yescharacteristic
14C8×S4C414yescomplemented
15D3×#106—41nomaximal
16C2×C8×D4—31no—
17C22×#106—31no—
18C22×#106D311yescharacteristic, complemented
19D4×C4⋊C4—32no—
20#1948—32no—
21#1853—32no—
22#1853—32no—
23#1844—31no—
24#1807—31no—
25#1807—31no—
26C4×#106—31no—
27C22×S4D411yescharacteristic
28C2×C4×A4—22no—
29C2×C4×A4C2311yescharacteristic
30C2×SL(2,ℤ/4)—22no—
31C2×SL(2,ℤ/4)D411yescharacteristic
32C4×S4—44no—
33C4×S4C2×C412yes—
34C4×S4Q812yes—
35C2×D3×C8—41no—
36D3×C4⋊C4—42no—
37C8×A4C2×C412yescomplemented
38A4⋊C8C2×C412yescomplemented
39C3×#106—41no—
40#817—41no—
41#808—42no—
42C23×C8—31no—
43C23×C8C2×D311yescharacteristic
44C2×C4×D4—31no—
45C2×C4×D4—62no—
46C22×C4⋊C4—32no—
47C22×C4⋊C4C2×D312yes—
48C8×D4—34no—
49C8×D4—34no—
50C2×#106—31no—
51C2×#106—31no—
52C2×#106—32no—
53C2×#106—31no—
54C2×#106—31no—
55C2×#106—32no—
56C2×#106—32no—
57C2×#106—61no—
58C2×#106—62no—
59C2×#106—61no—
60C2×C4⋊C8—31no—
61C2×C22⋊C8—31no—
62C2×C22⋊C8—31no—
63C2×C4×C8—31no—
64#389—62no—
65#385—62no—
66#385—62no—
67#384—32no—
68#380—32no—
69#380—32no—
70C4×C4⋊C4—32no—
71#340—32no—
72#340—32no—
73#340—32no—
74C22×A4C2×D411yescharacteristic
75C2×S4C4⋊C412yes—
76C2×S4SD1614yes—
77C2×C4×D3—41no—
78C2×C4×D3—82no—
79C4×A4—42no—
80C4×A4C22×C411yescharacteristic
81C4×A4C2×Q811yescharacteristic
82SL(2,ℤ/4)—42no—
83SL(2,ℤ/4)C4⋊C412yes—
84C2×C3×C8—41no—
85C3×C4⋊C4—42no—
86C4⋊Dic3—42no—
87Dic3⋊C4—82no—
88C2×C3⋊C8—41no—
89D3×C8—44no—
90C22×D4—31no—
91C23×C4—22no—
92C23×C4—31no—
93C23×C4—62no—
94C23×C4C22×D311yescharacteristic
95C22×C8—31no—
96C22×C8—31no—
97C22×C8—32no—
98C22×C8—31no—
99C22×C8—34no—
100C22×C8—31no—
101C22×C8—62no—
102C22×C8—61no—
103C22×C8—61no—
104C22×C8C4×D312yescomplemented
105C4×D4—32no—
106C4×D4—32no—
107C4×D4—32no—
108C4×D4—32no—
109C4×D4—124no—
110C4×D4—124no—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—62no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4—62no—
121C2×C4⋊C4—62no—
122C2×C4⋊C4—122no—
123C2×C22⋊C4—31no—
124C2×C22⋊C4—31no—
125C2×C22⋊C4—62no—
126C2×C22⋊C4—62no—
127C2×C42—31no—
128C2×C42—62no—
129#106—31no—
130#106—32no—
131#106—61no—
132#106—62no—
133#106—62no—
134#106—61no—
135#106—61no—
136#106—62no—
137#106—122no—
138#106S411yescharacteristic, direct product factor
139C4⋊C8—34no—
140C22⋊C8—34no—
141C22⋊C8—32no—
142C22⋊C8—32no—
143C4×C8—34no—
144#95—62no—
145#95—62no—
146#95—62no—
147C22×D3—41no—
148C2×A4C2×C4⋊C411yescharacteristic
149C2×A4C2×SD1612yes—
150S4#10614yesdirect product factor
151C2×C3×C4—41no—
152C2×C3×C4—82no—
153C2×Dic3—41no—
154C2×Dic3—82no—
155C4×D3—42no—
156C4×D3—42no—
157C4×D3—164no—
158C3×C8—42no—
159C3⋊C8—42no—
160C24—31no—
161C24D3×D411yescharacteristic
162C2×D4—32no—
163C2×D4—34no—
164C2×D4—32no—
165C2×D4—34no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—32no—
174C22×C4—32no—
175C22×C4—42no—
176C22×C4—62no—
177C22×C4—61no—
178C22×C4—61no—
179C22×C4—61no—
180C22×C4—61no—
181C22×C4—62no—
182C22×C4—62no—
183C22×C4—62no—
184C22×C4—122no—
185C22×C4—122no—
186C22×C4—122no—
187C22×C4—124no—
188C22×C4—242no—
189C22×C4C2×C4×D311yescharacteristic
190C22×C4D3×Q811yescharacteristic
191C2×C8—31no—
192C2×C8—32no—
193C2×C8—32no—
194C2×C8—32no—
195C2×C8—32no—
196C2×C8—32no—
197C2×C8—61no—
198C2×C8—64no—
199C2×C8—61no—
200C2×C8—61no—
201C2×C8—61no—
202C2×C8—61no—
203C2×C8—64no—
204C2×C8C2×S411yescharacteristic
205C4⋊C4—32no—
206C4⋊C4—32no—
207C4⋊C4—32no—
208C4⋊C4—62no—
209C4⋊C4—62no—
210C4⋊C4—62no—
211C4⋊C4—62no—
212C4⋊C4—62no—
213C4⋊C4—62no—
214C4⋊C4—122no—
215C4⋊C4—124no—
216C4⋊C4—122no—
217C4⋊C4—242no—
218C4⋊C4C2×S412yes—
219C22⋊C4—32no—
220C22⋊C4—32no—
221C22⋊C4—32no—
222C22⋊C4—32no—
223C22⋊C4—124no—
224C22⋊C4—122no—
225C22⋊C4—122no—
226C42—32no—
227C42—32no—
228C42—124no—
229C22×C3—41no—
230C2×D3—42no—
231C2×D3—44no—
232A4C2×#10611yescharacteristic, complemented
233C3×C4—41no—
234C3×C4—41no—
235C3×C4—162no—
236Dic3—42no—
237Dic3—162no—
238C23—31no—
239C23—31no—
240C23—32no—
241C23—32no—
242C23—34no—
243C23—61no—
244C23—62no—
245C23—61no—
246C23D3×C4⋊C411yescharacteristic
247C23D3×SD1612yes—
248D4—34no—
249D4—38no—
250D4—34no—
251C2×C4—22no—
252C2×C4—32no—
253C2×C4—34no—
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—32no—
262C2×C4—32no—
263C2×C4—61no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—122no—
273C2×C4—122no—
274C2×C4—122no—
275C2×C4—122no—
276C2×C4—122no—
277C2×C4—122no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—244no—
283C2×C4—244no—
284C2×C4C22×S411yescharacteristic
285C8—32no—
286C8—62no—
287C8—62no—
288C8C4×S412yescomplemented
289C2×C3—41no—
290C2×C3—42no—
291D3—44no—
292C22—31no—
293C22—32no—
294C22—31no—
295C22—31no—
296C22—32no—
297C22—32no—
298C22—34no—
299C22—61no—
300C22—62no—
301C22—64no—
302C22—62no—
303C22—64no—
304C22D3×#10611yescharacteristic, complemented
305C22D4×S411yescharacteristic
306C4—34no—
307C4—31no—
308C4—31no—
309C4—42no—
310C4—62no—
311C4—62no—
312C4—122no—
313C4—242no—
314C4—242no—
315C4C2×C4×S411yescharacteristic
316C4Q8×S411yescharacteristic
317C3—41noSylow
318C2—31no—
319C2—31no—
320C2—32no—
321C2—64no—
322C2C4⋊C4×S411yescharacteristic
323C2SD16×S412yes—
subgroupG/H
centerC22D4×S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×#106D3
socleC24D3×D4
prime radicalC22×S4D4
cosocleC2×C4×A4C23
hypercenter#106S4
hypocenterA4C2×#106
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
3. A4×#106C2semidirect
4. #73110C2semidirect
5. #73051C2semidirect
14. C8×S4C4semidirect
18. C22×#106D3semidirect
37. C8×A4C2×C4semidirect
38. A4⋊C8C2×C4semidirect
104. C22×C8C4×D3semidirect
138. #106S4direct (N × H)
150. S4#106direct (N × H)
232. A4C2×#106semidirect
288. C8C4×S4semidirect
304. C22D3×#106semidirect

Maximal subgroups

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

subgroupindex
C2×C8×S42
C4⋊C4×S42
A4×#1062
#731102
#730512
D4×#1063
D3×#1064