Finite Group Databasebeta

← S4×C22⋊C8 #11712887

Subgroups — S4×C22⋊C8 #11712887

Order 768.

1046 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1046 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C22×C4×S4C211yesmaximal, characteristic
2C2×C8×S4C212yesmaximal, complemented
3A4×C22⋊C8C211yesmaximal, characteristic, complemented
4#73141C211yesmaximal, characteristic, complemented
5(C2×S4)⋊C8C212yesmaximal, complemented
6D4×C22⋊C8—31nomaximal, Sylow
7C23×S4C411yescharacteristic
8C22×C4×A4C2211yescharacteristic
9C22×SL(2,ℤ/4)C411yescharacteristic
10C2×C4×S4—24no—
11C2×C4×S4C411yescharacteristic
12C2×C4×S4C411yescharacteristic
13C2×C4×S4C2211yescharacteristic
14C2×C4×S4C2211yescharacteristic
15C2×C8×A4C2212yescomplemented
16C2×A4⋊C8C2212yescomplemented
17C8×S4—24no—
18D3×C22⋊C8—41nomaximal
19C22×C4×D4—31no—
20C2×C8×D4—32no—
21C22×C22⋊C8—31no—
22C22×C22⋊C8D311yescharacteristic, complemented
23#1954—32no—
24#1869—32no—
25#1869—32no—
26#1797—31no—
27#1779—31no—
28#1779—31no—
29C4×C22⋊C8—31no—
30C23×A4C2×C411yescharacteristic
31C22×S4—22no—
32C22×S4—22no—
33C22×S4C814yescomplemented
34C22×S4C2×C411yescharacteristic
35C22×S4C2×C411yescharacteristic
36C22×C4×D3—41no—
37C2×C4×A4—22no—
38C2×C4×A4C2×C411yescharacteristic
39C2×C4×A4C2311yescharacteristic
40C2×SL(2,ℤ/4)—22no—
41C2×SL(2,ℤ/4)C2×C411yescharacteristic
42C2×SL(2,ℤ/4)C2×C411yescharacteristic
43C4×S4—22no—
44C4×S4—22no—
45C4×S4—22no—
46C4×S4D414yes—
47C2×D3×C8—42no—
48C8×A4—22no—
49A4⋊C8—22no—
50C3×C22⋊C8—41no—
51#830—41no—
52(C2×D3)⋊C8—42no—
53C23×D4—31no—
54C24×C4—31no—
55C24×C4C2×D311yescharacteristic
56C23×C8—32no—
57C23×C8C2×D312yescomplemented
58C2×C4×D4—31no—
59C2×C4×D4—31no—
60C2×C4×D4—31no—
61C2×C4×D4—31no—
62C2×C4×D4—31no—
63C2×C4×D4—31no—
64C2×C4×D4—31no—
65C2×C4×D4—31no—
66C2×C4×D4—64no—
67C2×C4×D4—64no—
68C22×C4⋊C4—31no—
69C22×C22⋊C4—31no—
70C22×C22⋊C4—31no—
71C22×C42—31no—
72C8×D4—64no—
73C8×D4—64no—
74C2×C4⋊C8—32no—
75C2×C22⋊C8—31no—
76C2×C22⋊C8—31no—
77C2×C22⋊C8—32no—
78C2×C22⋊C8—31no—
79C2×C22⋊C8—32no—
80C2×C22⋊C8—32no—
81C2×C22⋊C8—31no—
82C2×C22⋊C8—32no—
83C2×C22⋊C8—32no—
84C2×C22⋊C8—32no—
85C2×C22⋊C8—61no—
86C2×C22⋊C8—62no—
87C2×C22⋊C8—61no—
88C2×C4×C8—32no—
89#336—32no—
90#336—32no—
91C23×D3—41no—
92C22×A4—21no—
93C22×A4—21no—
94C22×A4C2×C812yescomplemented
95C22×A4C22×C411yescharacteristic
96C2×S4—21no—
97C2×S4—21no—
98C2×S4—28no—
99C2×S4C22⋊C412yes—
100C2×S4C2×C812yes—
101C2×S4C2×C812yescomplemented
102C2×S4M1612yes—
103C22×C3×C4—41no—
104C22×Dic3—41no—
105C2×C4×D3—41no—
106C2×C4×D3—41no—
107C2×C4×D3—41no—
108C2×C4×D3—41no—
109C2×C4×D3—84no—
110C4×A4—21no—
111C4×A4C2×D412yes—
112SL(2,ℤ/4)—21no—
113SL(2,ℤ/4)C22⋊C412yes—
114C2×C3×C8—42no—
115C2×C3⋊C8—42no—
116D3×C8—84no—
117C25—31no—
118C25C4×D311yescharacteristic
119C22×D4—31no—
120C22×D4—31no—
121C22×D4—31no—
122C22×D4—34no—
123C22×D4—34no—
124C22×D4—31no—
125C22×D4—62no—
126C22×D4—62no—
127C22×D4—62no—
128C22×D4—62no—
129C23×C4—22no—
130C23×C4—31no—
131C23×C4—31no—
132C23×C4—31no—
133C23×C4—31no—
134C23×C4—31no—
135C23×C4—31no—
136C23×C4—31no—
137C23×C4—31no—
138C23×C4—31no—
139C23×C4—31no—
140C23×C4—31no—
141C23×C4—62no—
142C23×C4—64no—
143C23×C4—61no—
144C23×C4—61no—
145C23×C4—61no—
146C23×C4—61no—
147C23×C4—122no—
148C23×C4C4×D311yescharacteristic
149C23×C4C22×D311yescharacteristic
150C22×C8—22no—
151C22×C8—32no—
152C22×C8—32no—
153C22×C8—32no—
154C22×C8—32no—
155C22×C8—62no—
156C22×C8—62no—
157C22×C8—64no—
158C22×C8—62no—
159C22×C8—122no—
160C4×D4—34no—
161C4×D4—34no—
162C4×D4—64no—
163C4×D4—62no—
164C4×D4—64no—
165C4×D4—62no—
166C4×D4—64no—
167C4×D4—62no—
168C4×D4—62no—
169C4×D4—64no—
170C4×D4—62no—
171C4×D4—62no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—64no—
177C2×C22⋊C4—31no—
178C2×C22⋊C4—31no—
179C2×C22⋊C4—31no—
180C2×C22⋊C4—31no—
181C2×C22⋊C4—31no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—31no—
184C2×C22⋊C4—31no—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—64no—
187C2×C22⋊C4—62no—
188C2×C42—31no—
189C2×C42—31no—
190C2×C42—31no—
191C2×C42—31no—
192C2×C42—64no—
193C4⋊C8—64no—
194C22⋊C8—31no—
195C22⋊C8—32no—
196C22⋊C8—61no—
197C22⋊C8—62no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201C22⋊C8—64no—
202C22⋊C8—62no—
203C22⋊C8—62no—
204C22⋊C8—61no—
205C22⋊C8—62no—
206C22⋊C8—122no—
207C22⋊C8—122no—
208C22⋊C8S411yescharacteristic, direct product factor
209C4×C8—64no—
210C23×C3—41no—
211C22×D3—41no—
212C22×D3—41no—
213C22×D3—44no—
214C22×D3—82no—
215C22×D3—82no—
216C2×A4—22no—
217C2×A4C2×C22⋊C411yescharacteristic
218C2×A4C22×C811yescharacteristic
219C2×A4C2×M1611yescharacteristic
220S4—22no—
221S4C22⋊C814yesdirect product factor
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—82no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—82no—
228C4×D3—44no—
229C4×D3—82no—
230C4×D3—82no—
231C4×D3—82no—
232C3×C8—82no—
233C3⋊C8—82no—
234C24—21no—
235C24—21no—
236C24—31no—
237C24—31no—
238C24—31no—
239C24—32no—
240C24—31no—
241C24—34no—
242C24—61no—
243C24—61no—
244C24—62no—
245C24—61no—
246C24—62no—
247C24—62no—
248C24—61no—
249C24—121no—
250C24—121no—
251C24D3×C812yescomplemented
252C24C2×C4×D311yescharacteristic
253C2×D4—32no—
254C2×D4—32no—
255C2×D4—32no—
256C2×D4—32no—
257C2×D4—32no—
258C2×D4—32no—
259C2×D4—34no—
260C2×D4—32no—
261C2×D4—34no—
262C2×D4—32no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—68no—
266C2×D4—68no—
267C2×D4—68no—
268C2×D4—68no—
269C2×D4—61no—
270C2×D4—62no—
271C2×D4—61no—
272C2×D4—62no—
273C2×D4—61no—
274C2×D4—61no—
275C22×C4—21no—
276C22×C4—31no—
277C22×C4—31no—
278C22×C4—32no—
279C22×C4—31no—
280C22×C4—34no—
281C22×C4—31no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—34no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—31no—
289C22×C4—31no—
290C22×C4—31no—
291C22×C4—31no—
292C22×C4—62no—
293C22×C4—62no—
294C22×C4—61no—
295C22×C4—62no—
296C22×C4—62no—
297C22×C4—61no—
298C22×C4—62no—
299C22×C4—62no—
300C22×C4—62no—
301C22×C4—62no—
302C22×C4—62no—
303C22×C4—62no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—62no—
307C22×C4—62no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—62no—
314C22×C4—62no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4—61no—
319C22×C4—61no—
320C22×C4—124no—
321C22×C4—121no—
322C22×C4—121no—
323C22×C4—121no—
324C22×C4—121no—
325C22×C4—121no—
326C22×C4—121no—
327C22×C4—124no—
328C22×C4—124no—
329C22×C4—124no—
330C22×C4D3×D412yes—
331C22×C4C2×S411yescharacteristic
332C2×C8—32no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8—62no—
336C2×C8—62no—
337C2×C8—62no—
338C2×C8—62no—
339C2×C8—62no—
340C2×C8—62no—
341C2×C8—62no—
342C2×C8—62no—
343C2×C8—124no—
344C2×C8—124no—
345C2×C8C2×S412yescomplemented
346C4⋊C4—34no—
347C4⋊C4—62no—
348C4⋊C4—62no—
349C4⋊C4—62no—
350C22⋊C4—34no—
351C22⋊C4—32no—
352C22⋊C4—32no—
353C22⋊C4—62no—
354C22⋊C4—62no—
355C22⋊C4—62no—
356C22⋊C4—62no—
357C22⋊C4—62no—
358C22⋊C4—61no—
359C22⋊C4—61no—
360C42—34no—
361C42—62no—
362C42—62no—
363C42—62no—
364C22×C3—41no—
365C22×C3—42no—
366C22×C3—81no—
367C22×C3—81no—
368C2×D3—42no—
369C2×D3—42no—
370C2×D3—42no—
371C2×D3—42no—
372C2×D3—88no—
373C2×D3—81no—
374C2×D3—81no—
375A4C2×C22⋊C811yescharacteristic, complemented
376C3×C4—42no—
377C3×C4—81no—
378Dic3—42no—
379Dic3—81no—
380C23—22no—
381C23—31no—
382C23—31no—
383C23—31no—
384C23—31no—
385C23—32no—
386C23—32no—
387C23—32no—
388C23—32no—
389C23—31no—
390C23—32no—
391C23—32no—
392C23—32no—
393C23—61no—
394C23—61no—
395C23—61no—
396C23—61no—
397C23—61no—
398C23—62no—
399C23—61no—
400C23—61no—
401C23—61no—
402C23—61no—
403C23—61no—
404C23—62no—
405C23—62no—
406C23—61no—
407C23—61no—
408C23—61no—
409C23—61no—
410C23—61no—
411C23—68no—
412C23—61no—
413C23—64no—
414C23—61no—
415C23—61no—
416C23—61no—
417C23—64no—
418C23—122no—
419C23—122no—
420C23—122no—
421C23—124no—
422C23—122no—
423C23—124no—
424C23—124no—
425C23—122no—
426C23D3×C22⋊C411yescharacteristic
427C23C2×D3×C811yescharacteristic
428C23D3×M1611yescharacteristic
429C23C4×S411yescharacteristic
430D4—34no—
431D4—34no—
432D4—34no—
433D4—34no—
434D4—68no—
435D4—62no—
436D4—62no—
437D4—62no—
438D4—68no—
439D4—62no—
440C2×C4—22no—
441C2×C4—32no—
442C2×C4—32no—
443C2×C4—32no—
444C2×C4—31no—
445C2×C4—32no—
446C2×C4—32no—
447C2×C4—32no—
448C2×C4—32no—
449C2×C4—32no—
450C2×C4—32no—
451C2×C4—31no—
452C2×C4—62no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—62no—
457C2×C4—64no—
458C2×C4—68no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—64no—
466C2×C4—62no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—124no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4C4×S411yescharacteristic
486C2×C4C22×S411yescharacteristic
487C8—22no—
488C8—62no—
489C8—122no—
490C8—122no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—82no—
495D3—44no—
496D3—82no—
497C22—21no—
498C22—21no—
499C22—31no—
500C22—31no—
501C22—31no—
502C22—31no—
503C22—31no—
504C22—31no—
505C22—31no—
506C22—31no—
507C22—31no—
508C22—34no—
509C22—32no—
510C22—62no—
511C22—62no—
512C22—62no—
513C22—62no—
514C22—61no—
515C22—62no—
516C22—62no—
517C22—62no—
518C22—62no—
519C22—62no—
520C22—62no—
521C22—62no—
522C22—61no—
523C22—62no—
524C22—62no—
525C22—61no—
526C22—62no—
527C22—62no—
528C22—128no—
529C22—121no—
530C22—121no—
531C22—121no—
532C22—121no—
533C22—128no—
534C22—121no—
535C22—121no—
536C22D3×C22⋊C811yescharacteristic, complemented
537C22C8×S412yescomplemented
538C22C2×C4×S411yescharacteristic
539C4—21no—
540C4—34no—
541C4—32no—
542C4—62no—
543C4—62no—
544C4—61no—
545C4—62no—
546C4—121no—
547C4—121no—
548C4D4×S412yes—
549C3—41noSylow
550C2—22no—
551C2—31no—
552C2—31no—
553C2—31no—
554C2—31no—
555C2—62no—
556C2—64no—
557C2—122no—
558C2C22⋊C4×S411yescharacteristic
559C2C2×C8×S411yescharacteristic
560C2M16×S411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×A4C22×C8
FittingC22×C22⋊C8D3
socleC24C2×C4×D3
prime radicalC23×S4C4
cosocleC2×C4×A4C23
hypercenterC22⋊C8S4
hypocenterA4C2×C22⋊C8
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C8×S4C2semidirect
3. A4×C22⋊C8C2semidirect
4. #73141C2semidirect
5. (C2×S4)⋊C8C2semidirect
15. C2×C8×A4C22semidirect
16. C2×A4⋊C8C22semidirect
22. C22×C22⋊C8D3semidirect
33. C22×S4C8semidirect
57. C23×C8C2×D3semidirect
94. C22×A4C2×C8semidirect
101. C2×S4C2×C8semidirect
208. C22⋊C8S4direct (N × H)
221. S4C22⋊C8direct (N × H)
251. C24D3×C8semidirect
345. C2×C8C2×S4semidirect
375. A4C2×C22⋊C8semidirect
536. C22D3×C22⋊C8semidirect
537. C22C8×S4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×S42
C2×C8×S42
A4×C22⋊C82
#731412
(C2×S4)⋊C82
D4×C22⋊C83
D3×C22⋊C84