Finite Group Databasebeta

← S4×#634 #13180733

Subgroups — S4×#634 #13180733

Order 1728.

1032 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 1032 conjugacy classes of subgroups — too many to draw. Only its 101 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
1A4×#634C211yesmaximal, characteristic, complemented
2#11724463C211yesmaximal, characteristic, complemented
3C32×C4×S4C211yesmaximal, characteristic
4S4×C3⋊Dic3C212yesmaximal
5#11724184C212yesmaximal, complemented
6Dic6×S4—34nomaximal
7D4×#634—31nomaximal
8C2×C32×S4C2211yescharacteristic
9C32×C4×A4C2211yescharacteristic
10D3×#634—41nomaximal
11A4×C3⋊Dic3C2212yes—
12C3⋊(C3⋊SL(2,ℤ/4))C2212yes—
13C32×SL(2,ℤ/4)C2211yescharacteristic
14C22×#634—31no—
15C22×#634D311yescharacteristic, complemented
16A4×Dic6—34no—
17#64259—34no—
18#64259—64no—
19C3×C4×S4D314yes—
20Dic3×S4—38no—
21#64204—38no—
22C32×C4×D4—31no—
23D4×C3⋊Dic3—32no—
24#64133—31no—
25#64133—31no—
26#64097—32no—
27#64087—32no—
28#64087—32no—
29#64078—31no—
30C4×#634—31no—
31C2×C32×A4C2311yescharacteristic
32C32×S4Q812yescomplemented
33#6308—41no—
34C3×#634—41no—
35C32×C4×D3—41no—
36#6293—42no—
37D3×C3⋊Dic3—42no—
38Q8×S4—91no—
39D4×Dic6—94no—
40C2×C3×S4C2×D314yes—
41C2×C32×D4—31no—
42C22×C32×C4—31no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C3⋊Dic3—32no—
45C22×C3⋊Dic3C2×D312yes—
46C2×#634—31no—
47C2×#634—31no—
48C2×#634—32no—
49C2×#634—31no—
50C2×#634—32no—
51C2×#634—61no—
52C2×#634—62no—
53C2×#634—61no—
54C3×C4×A4—24no—
55C3×C4×A4C2×D314yes—
56D3×Dic6—124no—
57Dic3×A4—38no—
58C3⋊SL(2,ℤ/4)—38no—
59C3⋊SL(2,ℤ/4)—68no—
60C3×SL(2,ℤ/4)C2×D314yes—
61C32×C4⋊C4—31no—
62C32×C22⋊C4—31no—
63C32×C22⋊C4—31no—
64C32×C42—31no—
65#3757—32no—
66#3757—32no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—32no—
69C4⋊(C3⋊Dic3)—32no—
70#3750—32no—
71#3750—32no—
72#3750—32no—
73C4×C3⋊Dic3—32no—
74C2×C32×D3—41no—
75C32×A4C2×Q811yescharacteristic, complemented
76C33×C4—41no—
77C3⋊(C3⋊Dic3)—42no—
78C3×C3⋊Dic3—42no—
79C32×Dic3—41no—
80C22×Dic6—34no—
81C22×Dic6—94no—
82Q8×A4—91no—
83C4×S4—92no—
84C4×S4C3⋊D311yescharacteristic
85A4⋊Q8—91no—
86A4⋊Q8—92no—
87A4⋊Q8—184no—
88C3×C4×D4—34no—
89D4×Dic3—98no—
90#924—94no—
91#924—94no—
92#888—98no—
93#878—98no—
94#878—98no—
95#869—94no—
96C4×Dic6—94no—
97C23×C32—31no—
98C23×C32C22×D311yescharacteristic
99C2×C3×A4—24no—
100C2×C3×A4C22×D314yes—
101C3×S4Dic618yescomplemented
102C32×D4—32no—
103C32×D4—32no—
104C2×C32×C4—31no—
105C2×C32×C4—31no—
106C2×C32×C4—31no—
107C2×C32×C4—31no—
108C2×C32×C4—31no—
109C2×C32×C4—61no—
110C2×C32×C4—61no—
111C2×C3⋊Dic3—32no—
112C2×C3⋊Dic3—32no—
113C2×C3⋊Dic3—32no—
114C2×C3⋊Dic3—32no—
115C2×C3⋊Dic3—62no—
116C2×C3⋊Dic3—62no—
117#634—31no—
118#634—32no—
119#634—61no—
120#634—62no—
121#634—61no—
122#634—61no—
123#634—62no—
124#634—124no—
125#634—244no—
126#634S411yescharacteristic, direct product factor
127C3×C4×D3—44no—
128C3×Dic6—124no—
129#627—128no—
130D3×Dic3—128no—
131D4×Q8—271noSylow
132C2×C33—41no—
133C32×D3—42no—
134C2×S4C2×C3⋊D311yescharacteristic
135C2×C3×D4—34no—
136C22×C3×C4—34no—
137C22×C3×C4D3214yes—
138C22×Dic3—38no—
139C22×Dic3—98no—
140D3×Q8—361no—
141C2×Dic6—94no—
142C2×Dic6—94no—
143C2×Dic6—98no—
144C2×Dic6—94no—
145C2×Dic6—98no—
146C2×Dic6—184no—
147C2×Dic6—188no—
148C2×Dic6—184no—
149C4×A4—24no—
150C4×A4—92no—
151C4×A4C2×C3⋊D311yescharacteristic
152SL(2,ℤ/4)—92no—
153SL(2,ℤ/4)—188no—
154SL(2,ℤ/4)C2×C3⋊D311yescharacteristic
155C3×C4⋊C4—34no—
156C3×C22⋊C4—34no—
157C3×C22⋊C4—34no—
158C3×C42—34no—
159#217—98no—
160#217—98no—
161C4⋊Dic3—94no—
162C4⋊Dic3—98no—
163C4⋊Dic3—98no—
164Dic3⋊C4—98no—
165Dic3⋊C4—98no—
166Dic3⋊C4—98no—
167C4×Dic3—98no—
168C22×C32—31no—
169C22×C32—31no—
170C22×C32—32no—
171C22×C32—61no—
172C22×C32—61no—
173C22×C32D3×Q811yescharacteristic, complemented
174C2×C3×D3—44no—
175C3×A4—24no—
176C3×A4C2×Dic614yescomplemented
177C32×C4—32no—
178C32×C4—31no—
179C32×C4—44no—
180C32×C4—61no—
181C32×C4—61no—
182C32×C4—84no—
183C32×C4C2×S411yescharacteristic
184C3⋊Dic3—32no—
185C3⋊Dic3—62no—
186C3⋊Dic3—62no—
187C3⋊Dic3—128no—
188C3⋊Dic3—248no—
189C3⋊Dic3C2×S412yes—
190C3×Dic3—44no—
191C3×Dic3—128no—
192C22×Q8—91no—
193C22×Q8—271no—
194C4⋊Q8—271no—
195C4⋊Q8—272no—
196C22⋊Q8—272no—
197C22⋊Q8—272no—
198C22⋊Q8—271no—
199C22⋊Q8—271no—
200C4×Q8—271no—
201C4×D4—31no—
202C4×D4—272no—
203C33—41noSylow
204C23×C3—34no—
205C23×C3C2×D3214yes—
206C2×A4—24no—
207C2×A4C22×C3⋊D311yescharacteristic
208S4#63412yesdirect product factor
209C3×Q8—361no—
210C3×D4—38no—
211C3×D4—38no—
212C2×C3×C4—34no—
213C2×C3×C4—34no—
214C2×C3×C4—34no—
215C2×C3×C4—34no—
216C2×C3×C4—34no—
217C2×C3×C4—64no—
218C2×C3×C4—64no—
219C2×Dic3—98no—
220C2×Dic3—98no—
221C2×Dic3—98no—
222C2×Dic3—98no—
223C2×Dic3—188no—
224C2×Dic3—188no—
225C4×D3—41no—
226C4×D3—362no—
227Dic6—34no—
228Dic6—98no—
229Dic6—94no—
230Dic6—188no—
231Dic6—184no—
232Dic6—188no—
233Dic6—184no—
234Dic6—184no—
235Dic6—362no—
236Dic6—361no—
237Dic6—724no—
238C2×C32—31no—
239C2×C32—31no—
240C2×C32—44no—
241C2×C32—62no—
242C2×C32—84no—
243C2×C32C22×S411yescharacteristic
244C3×D3—48no—
245C2×Q8—271no—
246C2×Q8—271no—
247C2×Q8—272no—
248C2×Q8—271no—
249C2×Q8—272no—
250C2×Q8—541no—
251C2×Q8—542no—
252C2×Q8—541no—
253C2×D4—31no—
254C22×C4—31no—
255C22×C4—92no—
256C22×C4—272no—
257C22×C4D3×C3⋊D311yescharacteristic
258C4⋊C4—31no—
259C4⋊C4—272no—
260C4⋊C4—272no—
261C4⋊C4—271no—
262C4⋊C4—272no—
263C4⋊C4—272no—
264C4⋊C4—272no—
265C22⋊C4—31no—
266C22⋊C4—31no—
267C22⋊C4—272no—
268C22⋊C4—272no—
269C42—31no—
270C42—272no—
271C22×C3—34no—
272C22×C3—34no—
273C22×C3—38no—
274C22×C3—64no—
275C22×C3—64no—
276C22×C3D3×Dic614yescomplemented
277C2×D3—41no—
278A4—24no—
279A4C2×#63411yescharacteristic, complemented
280C3×C4—38no—
281C3×C4—34no—
282C3×C4—41no—
283C3×C4—64no—
284C3×C4—64no—
285C3×C4—84no—
286C3×C4—362no—
287C3×C4D3×S414yes—
288Dic3—38no—
289Dic3—41no—
290Dic3—98no—
291Dic3—188no—
292Dic3—188no—
293Dic3—362no—
294Dic3—728no—
295C32—44no—
296C32—84no—
297C32Q8×S411yescharacteristic, complemented
298C23—31no—
299C23C2×D3×C3⋊D311yescharacteristic
300Q8—91no—
301Q8—272no—
302Q8—271no—
303Q8—541no—
304Q8—541no—
305Q8—542no—
306Q8—542no—
307Q8—541no—
308D4—32no—
309D4—32no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—272no—
318C2×C4—272no—
319C2×C4—272no—
320C2×C4—272no—
321C2×C4—542no—
322C2×C4—542no—
323C2×C3—34no—
324C2×C3—34no—
325C2×C3—41no—
326C2×C3—68no—
327C2×C3—84no—
328C2×C3C2×D3×S414yes—
329D3—42no—
330C22—31no—
331C22—31no—
332C22—32no—
333C22—61no—
334C22—61no—
335C22D3×#63411yescharacteristic, complemented
336C4—32no—
337C4—31no—
338C4—61no—
339C4—61no—
340C4—92no—
341C4—272no—
342C4—542no—
343C4—542no—
344C4C3⋊D3×S411yescharacteristic
345C3—41no—
346C3—84no—
347C3Dic6×S414yescomplemented
348C2—31no—
349C2—31no—
350C2—62no—
351C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊D3×S4
commutator, cosocleC2×C32×A4C23
FittingC22×C32×C4C2×D3
socleC23×C32C22×D3
prime radicalC2×C32×S4C22
hypercenterC4C3⋊D3×S4
hypocenterC32×A4C2×Q8
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. A4×#634C2semidirect
2. #11724463C2semidirect
5. #11724184C2semidirect
15. C22×#634D3semidirect
32. C32×S4Q8semidirect
75. C32×A4C2×Q8semidirect
101. C3×S4Dic6semidirect
126. #634S4direct (N × H)
173. C22×C32D3×Q8semidirect
176. C3×A4C2×Dic6semidirect
208. S4#634direct (N × H)
276. C22×C3D3×Dic6semidirect
279. A4C2×#634semidirect
297. C32Q8×S4semidirect
335. C22D3×#634semidirect
347. C3Dic6×S4semidirect

Maximal subgroups

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

subgroupindex
A4×#6342
#117244632
C32×C4×S42
S4×C3⋊Dic32
#117241842
Dic6×S43
D4×#6343
D3×#6344