Finite Group Databasebeta

← S4×#208 #11921859

Subgroups — S4×#208 #11921859

Order 1152.

614 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 79 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×C3×C4×S4C211yesmaximal, characteristic
2A4×#208C211yesmaximal, characteristic, complemented
3#10593643C211yesmaximal, characteristic, complemented
4#10593175C212yesmaximal, complemented
5C3⋊C8×S4C212yesmaximal, complemented
6M16×S4—31nomaximal
7D4×#208—31nomaximal
8C22×C3×S4C411yescharacteristic
9C2×C3×C4×A4C2211yescharacteristic
10C2×C3×SL(2,ℤ/4)C411yescharacteristic
11C3×C4×S4C411yescharacteristic
12C3×C4×S4C411yescharacteristic
13C3×C4×S4C2211yescharacteristic
14C3×C4×S4C2211yescharacteristic
15D3×#208—41nomaximal
16A4×C3⋊C8C2212yescomplemented
17C3⋊(A4⋊C8)C2212yescomplemented
18C2×C4×S4D311yescharacteristic
19C2×C3×C4×D4—31no—
20C22×#208—31no—
21C22×#208D311yescharacteristic, complemented
22A4×M16—31no—
23#5409—31no—
24#5409—61no—
25#5400—32no—
26C8×S4—32no—
27#5207—31no—
28#5207—31no—
29#5012—32no—
30#5011—32no—
31#5011—32no—
32D4×C3⋊C8—32no—
33#4924—31no—
34C4×#208—31no—
35C22×C3×A4C2×C411yescharacteristic
36C2×C3×S4—22no—
37C2×C3×S4C2×C411yescharacteristic
38C2×C3×S4C2×C411yescharacteristic
39C2×C3×C4×D3—41no—
40C3×C4×A4C2×C411yescharacteristic
41C3×C4×A4C2311yescharacteristic
42C3×SL(2,ℤ/4)C2×C411yescharacteristic
43C3×SL(2,ℤ/4)C2×C411yescharacteristic
44#3748—41no—
45C3×#208—41no—
46#3711—42no—
47D3×C3⋊C8—42no—
48D4×M16—91noSylow
49C22×S4Dic311yescharacteristic
50C22×C3×D4—31no—
51C23×C3×C4—31no—
52C23×C3×C4C2×D311yescharacteristic
53C2×C4×A4—21no—
54C2×C4×A4C2×D311yescharacteristic
55C2×SL(2,ℤ/4)Dic311yescharacteristic
56C4×S4Dic311yescharacteristic
57C4×S4Dic311yescharacteristic
58C4×S4C2×D311yescharacteristic
59C4×S4C2×D311yescharacteristic
60C3×C4×D4—31no—
61C3×C4×D4—31no—
62C3×C4×D4—31no—
63C3×C4×D4—31no—
64C3×C4×D4—31no—
65C3×C4×D4—31no—
66C3×C4×D4—31no—
67C3×C4×D4—31no—
68C2×C3×C4⋊C4—31no—
69C2×C3×C22⋊C4—31no—
70C2×C3×C22⋊C4—31no—
71C2×C3×C42—31no—
72C2×#208—31no—
73C2×#208—31no—
74C2×#208—32no—
75C2×#208—31no—
76C2×#208—31no—
77C2×#208—32no—
78C2×#208—61no—
79C2×#208—62no—
80C2×#208—61no—
81C22×C3⋊C8—32no—
82C22×C3⋊C8C2×D312yescomplemented
83D3×M16—121no—
84C8×A4—32no—
85A4⋊C8—32no—
86A4⋊C8—62no—
87#830—32no—
88#830—32no—
89#830—32no—
90#830—32no—
91C4⋊(C3⋊C8)—32no—
92C4⋊(C3⋊C8)—32no—
93#803—32no—
94C4×C3⋊C8—32no—
95C22×C3×D3—41no—
96C2×C3×A4—21no—
97C2×C3×A4C22×C411yescharacteristic
98C3×S4—21no—
99C3×S4M1612yescomplemented
100C2×C32×C4—41no—
101C2×C3×Dic3—41no—
102C3×C4×D3—41no—
103C3×C4×D3—41no—
104C3×C4×D3—41no—
105C3×C4×D3—41no—
106C3⋊(C3⋊C8)—42no—
107C3×C3⋊C8—42no—
108C22×M16—31no—
109C22×M16—91no—
110C2×C4×D4—31no—
111#436—92no—
112#435—92no—
113#435—92no—
114C8×D4—92no—
115#423—91no—
116#407—91no—
117#407—91no—
118C4×M16—91no—
119C24×C3—31no—
120C24×C3C4×D311yescharacteristic
121C22×A4—21no—
122C22×A4C2×Dic311yescharacteristic
123C2×S4—22no—
124C2×S4C2×Dic311yescharacteristic
125C2×S4C2×Dic311yescharacteristic
126C2×C3×D4—31no—
127C2×C3×D4—31no—
128C2×C3×D4—31no—
129C2×C3×D4—31no—
130C2×C3×D4—62no—
131C2×C3×D4—62no—
132C22×C3×C4—31no—
133C22×C3×C4—31no—
134C22×C3×C4—31no—
135C22×C3×C4—31no—
136C22×C3×C4—31no—
137C22×C3×C4—31no—
138C22×C3×C4—31no—
139C22×C3×C4—31no—
140C22×C3×C4—31no—
141C22×C3×C4—31no—
142C22×C3×C4—31no—
143C22×C3×C4—61no—
144C22×C3×C4—61no—
145C22×C3×C4—61no—
146C22×C3×C4—61no—
147C22×C3×C4C4×D311yescharacteristic
148C22×C3×C4C22×D311yescharacteristic
149C2×C4×D3—41no—
150C4×A4—21no—
151C4×A4—21no—
152C4×A4C2×Dic311yescharacteristic
153C4×A4C22×D311yescharacteristic
154SL(2,ℤ/4)C2×Dic311yescharacteristic
155SL(2,ℤ/4)C2×Dic311yescharacteristic
156C3×M16—121no—
157C3×C4⋊C4—31no—
158C3×C4⋊C4—31no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C22⋊C4—31no—
162C3×C22⋊C4—31no—
163C3×C22⋊C4—31no—
164C3×C22⋊C4—31no—
165C3×C22⋊C4—31no—
166C3×C22⋊C4—31no—
167C3×C22⋊C4—31no—
168C3×C22⋊C4—31no—
169C3×C42—31no—
170C3×C42—31no—
171C3×C42—31no—
172C3×C42—31no—
173#208—31no—
174#208—32no—
175#208—61no—
176#208—62no—
177#208—61no—
178#208—61no—
179#208—61no—
180#208—61no—
181#208—62no—
182#208—121no—
183#208—241no—
184#208S411yescharacteristic, direct product factor
185C2×C3⋊C8—32no—
186C2×C3⋊C8—32no—
187C2×C3⋊C8—32no—
188C2×C3⋊C8—32no—
189C2×C3⋊C8—62no—
190C2×C3⋊C8—62no—
191ΓL(1,𝔽25)—122no—
192D3×C8—122no—
193C22×C32—41no—
194C2×C3×D3—41no—
195C2×C3×D3—41no—
196C2×C3×D3—82no—
197C3×A4C2×M1611yescharacteristic, complemented
198C32×C4—41no—
199C32×C4—41no—
200C3×Dic3—41no—
201C3×Dic3—41no—
202C22×D4—31no—
203C23×C4—31no—
204C23×C4D3211yescharacteristic
205C2×M16—91no—
206C2×M16—91no—
207C2×M16—92no—
208C2×M16—91no—
209C2×M16—92no—
210C2×M16—91no—
211C2×M16—181no—
212C2×M16—182no—
213C2×M16—181no—
214C22×C8—32no—
215C22×C8—92no—
216C4×D4—31no—
217C4×D4—31no—
218C4×D4—31no—
219C4×D4—31no—
220C4×D4—31no—
221C4×D4—31no—
222C4×D4—31no—
223C4×D4—31no—
224C2×C4⋊C4—31no—
225C2×C22⋊C4—31no—
226C2×C22⋊C4—31no—
227C2×C42—31no—
228C4⋊C8—92no—
229C4⋊C8—92no—
230C22⋊C8—92no—
231C22⋊C8—92no—
232C22⋊C8—92no—
233C22⋊C8—92no—
234#97—92no—
235C4×C8—92no—
236C23×C3—21no—
237C23×C3—31no—
238C23×C3—31no—
239C23×C3—31no—
240C23×C3—31no—
241C23×C3—61no—
242C23×C3—61no—
243C23×C3—62no—
244C23×C3—61no—
245C23×C3—121no—
246C23×C3C2×C4×D311yescharacteristic
247C22×D3—41no—
248C2×A4—21no—
249C2×A4—21no—
250C2×A4—41no—
251C2×A4C22×Dic311yescharacteristic
252S4—21no—
253S4#20812yesdirect product factor
254C3×D4—32no—
255C3×D4—32no—
256C3×D4—62no—
257C3×D4—61no—
258C3×D4—62no—
259C3×D4—61no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—41no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—62no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—81no—
287C2×C3×C4—121no—
288C2×C3×C4C2×S411yescharacteristic
289C2×Dic3—41no—
290C4×D3—41no—
291C4×D3—41no—
292C4×D3—41no—
293C4×D3—41no—
294C3×C8—122no—
295C3⋊C8—32no—
296C3⋊C8—62no—
297C3⋊C8—62no—
298C3⋊C8—122no—
299C3⋊C8—242no—
300C3⋊C8C2×S412yescomplemented
301C2×C32—41no—
302C2×C32—81no—
303C3×D3—42no—
304C3×D3—81no—
305C24—31no—
306C24D3×Dic311yescharacteristic
307C2×D4—31no—
308C2×D4—31no—
309C2×D4—31no—
310C2×D4—31no—
311C2×D4—62no—
312C2×D4—62no—
313C22×C4—31no—
314C22×C4—31no—
315C22×C4—31no—
316C22×C4—31no—
317C22×C4—31no—
318C22×C4—31no—
319C22×C4—31no—
320C22×C4—31no—
321C22×C4—31no—
322C22×C4—31no—
323C22×C4—31no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4D3×Dic311yescharacteristic
329C22×C4C2×D3211yescharacteristic
330M16—31no—
331M16—92no—
332M16—91no—
333M16—182no—
334M16—181no—
335M16—182no—
336M16—181no—
337M16—181no—
338M16—181no—
339M16—181no—
340C2×C8—92no—
341C2×C8—92no—
342C2×C8—92no—
343C2×C8—92no—
344C2×C8—182no—
345C2×C8—182no—
346C4⋊C4—31no—
347C4⋊C4—31no—
348C4⋊C4—31no—
349C4⋊C4—31no—
350C22⋊C4—31no—
351C22⋊C4—31no—
352C22⋊C4—31no—
353C22⋊C4—31no—
354C22⋊C4—31no—
355C22⋊C4—31no—
356C22⋊C4—31no—
357C22⋊C4—31no—
358C42—31no—
359C42—31no—
360C42—31no—
361C42—31no—
362C22×C3—31no—
363C22×C3—31no—
364C22×C3—32no—
365C22×C3—31no—
366C22×C3—41no—
367C22×C3—61no—
368C22×C3—61no—
369C22×C3—61no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3—61no—
373C22×C3—61no—
374C22×C3—61no—
375C22×C3—61no—
376C22×C3—81no—
377C22×C3—122no—
378C22×C3—122no—
379C22×C3D3×M1611yescharacteristic, complemented
380C22×C3C4×S411yescharacteristic
381C2×D3—41no—
382C2×D3—41no—
383C2×D3—82no—
384A4—21no—
385A4C2×#20811yescharacteristic, complemented
386C3×C4—32no—
387C3×C4—31no—
388C3×C4—31no—
389C3×C4—41no—
390C3×C4—41no—
391C3×C4—61no—
392C3×C4—61no—
393C3×C4—61no—
394C3×C4—61no—
395C3×C4—61no—
396C3×C4—81no—
397C3×C4—81no—
398C3×C4C4×S411yescharacteristic
399C3×C4C22×S411yescharacteristic
400Dic3—41no—
401Dic3—41no—
402C32—41noSylow
403C23—21no—
404C23—31no—
405C23—31no—
406C23—31no—
407C23—31no—
408C23—61no—
409C23—61no—
410C23—62no—
411C23—61no—
412C23—121no—
413C23C2×D3×Dic311yescharacteristic
414D4—32no—
415D4—32no—
416D4—62no—
417D4—61no—
418D4—62no—
419D4—61no—
420C2×C4—31no—
421C2×C4—31no—
422C2×C4—31no—
423C2×C4—31no—
424C2×C4—31no—
425C2×C4—31no—
426C2×C4—31no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—62no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—121no—
446C2×C4D3×S411yescharacteristic
447C8—32no—
448C8—92no—
449C8—182no—
450C8—182no—
451C2×C3—21no—
452C2×C3—31no—
453C2×C3—31no—
454C2×C3—41no—
455C2×C3—61no—
456C2×C3—62no—
457C2×C3—81no—
458C2×C3—81no—
459C2×C3—121no—
460C2×C3—161no—
461C2×C3C2×C4×S411yescharacteristic
462D3—42no—
463D3—81no—
464C22—31no—
465C22—31no—
466C22—32no—
467C22—31no—
468C22—61no—
469C22—61no—
470C22—61no—
471C22—61no—
472C22—61no—
473C22—61no—
474C22—61no—
475C22—61no—
476C22—61no—
477C22—122no—
478C22—122no—
479C22D3×#20811yescharacteristic, complemented
480C22Dic3×S411yescharacteristic
481C4—32no—
482C4—31no—
483C4—31no—
484C4—61no—
485C4—61no—
486C4—61no—
487C4—61no—
488C4—61no—
489C4Dic3×S411yescharacteristic
490C4C2×D3×S411yescharacteristic
491C3—41no—
492C3—81no—
493C3M16×S411yescharacteristic, complemented
494C2—21no—
495C2—31no—
496C2—31no—
497C2—61no—
498C2—62no—
499C2—121no—
500C2C2×Dic3×S411yescharacteristic
subgroupG/H
center, FrattiniC4C2×D3×S4
commutatorC2×C3×A4C22×C4
FittingC23×C3×C4C2×D3
socleC23×C3C2×C4×D3
prime radicalC22×C3×S4C4
cosocleC3×C4×A4C23
hypercenterC2×C4D3×S4
hypocenterC3×A4C2×M16
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. A4×#208C2semidirect
3. #10593643C2semidirect
4. #10593175C2semidirect
5. C3⋊C8×S4C2semidirect
16. A4×C3⋊C8C22semidirect
17. C3⋊(A4⋊C8)C22semidirect
21. C22×#208D3semidirect
82. C22×C3⋊C8C2×D3semidirect
99. C3×S4M16semidirect
184. #208S4direct (N × H)
197. C3×A4C2×M16semidirect
253. S4#208direct (N × H)
300. C3⋊C8C2×S4semidirect
379. C22×C3D3×M16semidirect
385. A4C2×#208semidirect
479. C22D3×#208semidirect
493. C3M16×S4semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×S42
A4×#2082
#105936432
#105931752
C3⋊C8×S42
M16×S43
D4×#2083
D3×#2084