Finite Group Databasebeta

← C4×(C2×S4)⋊C4 #11711684

Subgroups — C4×(C2×S4)⋊C4 #11711684

Order 768.

1477 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 1477 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
1C22×C4×S4C211yesmaximal, characteristic
2C2×C42×A4C211yesmaximal, characteristic, complemented
3C2×(C2×S4)⋊C4C212yesmaximal
4C2×C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5A4⋊#95C212yesmaximal, complemented
6C4×#386—31nomaximal, Sylow
7C23×S4C2211yescharacteristic
8C22×C4×A4C2212yes—
9C22×C4×A4C2211yescharacteristic
10C22×SL(2,ℤ/4)C2211yescharacteristic
11C22×SL(2,ℤ/4)C2212yes—
12C2×C4×S4—22no—
13C2×C4×S4—22no—
14C2×C4×S4C414yescomplemented
15C42×A4—22no—
16(C2×S4)⋊C4C418yesdirect product factor
17C4×SL(2,ℤ/4)—22no—
18C4×(C2×D3)⋊C4—41nomaximal
19C22×C4×D4—31no—
20C23×C42D311yescharacteristic, complemented
21C2×#386—32no—
22C2×C4×C22⋊C4—31no—
23C2×C4×C22⋊C4—31no—
24C2×C4×C22⋊C4—31no—
25#1437—32no—
26#1436—32no—
27C22⋊#95—32no—
28C4×#95—31no—
29C4×#95—31no—
30C23×A4C2311yescharacteristic
31C22×S4—21no—
32C22×S4—21no—
33C22×S4—22no—
34C22×S4C2×C412yes—
35C22×S4C2×C414yes—
36C22×C4×D3—41no—
37C2×C4×A4—22no—
38C2×C4×A4—22no—
39C2×C4×A4C2×C412yes—
40C2×C4×A4C2×C414yescomplemented
41C2×C4×A4D412yes—
42C2×C4×A4D412yes—
43C2×SL(2,ℤ/4)—21no—
44C2×SL(2,ℤ/4)—21no—
45C2×SL(2,ℤ/4)—22no—
46C2×SL(2,ℤ/4)—22no—
47C2×SL(2,ℤ/4)C2×C412yes—
48C2×SL(2,ℤ/4)C2×C414yescomplemented
49C4×S4—28no—
50C2×C3×C42—41no—
51C2×(C2×D3)⋊C4—42no—
52C2×C4×Dic3—41no—
53C3⋊#95—42no—
54C23×D4—31no—
55C24×C4—31no—
56C24×C4C2×D312yes—
57C24×C4C2×D311yescharacteristic
58C2×C4×D4—34no—
59C2×C4×D4—34no—
60C2×C4×D4—62no—
61C2×C4×D4—62no—
62C2×C4×D4—62no—
63C2×C4×D4—62no—
64C22×C4⋊C4—31no—
65C22×C22⋊C4—31no—
66C22×C22⋊C4—32no—
67C22×C22⋊C4—32no—
68C22×C22⋊C4—31no—
69C22×C22⋊C4—32no—
70C22×C42—22no—
71C22×C42—31no—
72C22×C42—31no—
73C22×C42—31no—
74C22×C42—32no—
75C22×C42—31no—
76C22×C42—31no—
77C22×C42—62no—
78#386—38no—
79#386—38no—
80C4×C22⋊C4—64no—
81C4×C22⋊C4—61no—
82C4×C22⋊C4—62no—
83C4×C22⋊C4—62no—
84C4×C22⋊C4—62no—
85C4×C22⋊C4—61no—
86C2×#95—32no—
87C2×#95—32no—
88C2×#95—32no—
89C2×#95—32no—
90C2×#95—32no—
91C2×#95—32no—
92C2×#95—32no—
93C23×D3—41no—
94C22×A4C22×C411yescharacteristic
95C22×A4C22×C412yes—
96C22×A4C2×D411yescharacteristic
97C22×A4C2×D411yescharacteristic
98C22×A4#4112yes—
99C2×S4—22no—
100C2×S4—22no—
101C2×S4—24no—
102C2×S4—24no—
103C2×S4C4214yescomplemented
104C22×C3×C4—42no—
105C22×C3×C4—41no—
106C22×Dic3—41no—
107C22×Dic3—42no—
108C2×C4×D3—44no—
109C2×C4×D3—82no—
110C2×C4×D3—82no—
111C4×A4—24no—
112C4×A4C22⋊C414yescomplemented
113SL(2,ℤ/4)—22no—
114SL(2,ℤ/4)—24no—
115C3×C42—82no—
116(C2×D3)⋊C4—48no—
117C4×Dic3—82no—
118C25—31no—
119C25C22×D311yescharacteristic
120C22×D4—32no—
121C22×D4—34no—
122C22×D4—32no—
123C22×D4—34no—
124C22×D4—61no—
125C22×D4—61no—
126C22×D4—62no—
127C22×D4—61no—
128C22×D4—62no—
129C22×D4—61no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—31no—
133C23×C4—31no—
134C23×C4—32no—
135C23×C4—32no—
136C23×C4—32no—
137C23×C4—32no—
138C23×C4—31no—
139C23×C4—32no—
140C23×C4—34no—
141C23×C4—32no—
142C23×C4—32no—
143C23×C4—31no—
144C23×C4—31no—
145C23×C4—34no—
146C23×C4—62no—
147C23×C4—62no—
148C23×C4—62no—
149C23×C4—62no—
150C23×C4—62no—
151C23×C4—62no—
152C23×C4—61no—
153C23×C4—62no—
154C23×C4—61no—
155C23×C4C4×D312yes—
156C23×C4C4×D314yescomplemented
157C23×C4D1212yes—
158C23×C4#6712yes—
159C4×D4—68no—
160C4×D4—68no—
161C4×D4—68no—
162C4×D4—68no—
163C2×C4⋊C4—34no—
164C2×C4⋊C4—62no—
165C2×C4⋊C4—62no—
166C2×C22⋊C4—34no—
167C2×C22⋊C4—32no—
168C2×C22⋊C4—38no—
169C2×C22⋊C4—38no—
170C2×C22⋊C4—34no—
171C2×C22⋊C4—32no—
172C2×C22⋊C4—34no—
173C2×C22⋊C4—64no—
174C2×C22⋊C4—64no—
175C2×C22⋊C4—62no—
176C2×C22⋊C4—62no—
177C2×C22⋊C4—64no—
178C2×C22⋊C4—62no—
179C2×C22⋊C4—62no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—62no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—62no—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—62no—
188C2×C42—34no—
189C2×C42—34no—
190C2×C42—32no—
191C2×C42—31no—
192C2×C42—62no—
193C2×C42—62no—
194C2×C42—61no—
195C2×C42—62no—
196C2×C42—62no—
197C2×C42—64no—
198C2×C42—64no—
199C2×C42—62no—
200C2×C42—61no—
201C2×C42—62no—
202C2×C42—62no—
203C2×C42—62no—
204C2×C42—62no—
205C2×C42—62no—
206C2×C42—61no—
207C2×C42S411yescharacteristic, complemented
208#95—38no—
209#95—38no—
210#95—64no—
211#95—64no—
212#95—64no—
213#95—64no—
214#95—62no—
215#95—62no—
216C23×C3—41no—
217C22×D3—44no—
218C22×D3—42no—
219C22×D3—81no—
220C22×D3—81no—
221C22×D3—82no—
222C2×A4C2×C4211yescharacteristic
223C2×A4C2×C22⋊C411yescharacteristic
224C2×A4#11711yescharacteristic
225C2×A4C4×D412yes—
226C2×A4C4×D412yes—
227S4—24no—
228C2×C3×C4—42no—
229C2×C3×C4—44no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—82no—
233C2×C3×C4—82no—
234C2×Dic3—44no—
235C2×Dic3—42no—
236C2×Dic3—82no—
237C2×Dic3—81no—
238C2×Dic3—81no—
239C2×Dic3—82no—
240C4×D3—88no—
241C24—31no—
242C24—31no—
243C24—32no—
244C24—31no—
245C24—31no—
246C24—32no—
247C24—34no—
248C24—32no—
249C24—61no—
250C24—61no—
251C24—61no—
252C24—62no—
253C24—62no—
254C24—61no—
255C24—61no—
256C24—62no—
257C24—61no—
258C24C2×C4×D311yescharacteristic
259C24C2×C4×D312yes—
260C24C2×D1211yescharacteristic
261C24#23512yes—
262C24C2×#6711yescharacteristic
263C2×D4—38no—
264C2×D4—34no—
265C2×D4—34no—
266C2×D4—62no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—64no—
270C2×D4—64no—
271C2×D4—64no—
272C2×D4—64no—
273C2×D4—64no—
274C2×D4—64no—
275C2×D4—62no—
276C2×D4—64no—
277C2×D4—62no—
278C2×D4—64no—
279C2×D4—62no—
280C2×D4—64no—
281C22×C4—24no—
282C22×C4—31no—
283C22×C4—32no—
284C22×C4—32no—
285C22×C4—32no—
286C22×C4—32no—
287C22×C4—32no—
288C22×C4—32no—
289C22×C4—34no—
290C22×C4—32no—
291C22×C4—32no—
292C22×C4—32no—
293C22×C4—34no—
294C22×C4—34no—
295C22×C4—34no—
296C22×C4—34no—
297C22×C4—34no—
298C22×C4—34no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—34no—
302C22×C4—34no—
303C22×C4—34no—
304C22×C4—34no—
305C22×C4—32no—
306C22×C4—32no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—64no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—62no—
316C22×C4—64no—
317C22×C4—68no—
318C22×C4—68no—
319C22×C4—64no—
320C22×C4—68no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—64no—
326C22×C4—64no—
327C22×C4—62no—
328C22×C4—61no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—62no—
350C22×C4—64no—
351C22×C4—122no—
352C22×C4—122no—
353C22×C4—124no—
354C22×C4—124no—
355C22×C4—122no—
356C22×C4—122no—
357C22×C4(C2×D3)⋊C414yescomplemented
358C22×C4C2×S412yes—
359C22×C4C2×S411yescharacteristic
360C4⋊C4—68no—
361C22⋊C4—616no—
362C22⋊C4—68no—
363C22⋊C4—68no—
364C22⋊C4—64no—
365C22⋊C4—64no—
366C22⋊C4—64no—
367C22⋊C4—62no—
368C22⋊C4—62no—
369C22⋊C4—62no—
370C22⋊C4—64no—
371C22⋊C4—62no—
372C22⋊C4—68no—
373C42—22no—
374C42—64no—
375C42—64no—
376C42—68no—
377C42—62no—
378C42—122no—
379C42—124no—
380C42—122no—
381C22×C3—41no—
382C22×C3—42no—
383C22×C3—41no—
384C22×C3—42no—
385C22×C3—41no—
386C2×D3—44no—
387C2×D3—82no—
388C2×D3—82no—
389C2×D3—84no—
390C2×D3—84no—
391A4C4×C22⋊C411yescharacteristic, complemented
392C3×C4—44no—
393C3×C4—84no—
394Dic3—82no—
395Dic3—84no—
396C23—31no—
397C23—32no—
398C23—31no—
399C23—32no—
400C23—31no—
401C23—31no—
402C23—32no—
403C23—32no—
404C23—31no—
405C23—31no—
406C23—31no—
407C23—31no—
408C23—31no—
409C23—32no—
410C23—32no—
411C23—32no—
412C23—32no—
413C23—32no—
414C23—32no—
415C23—31no—
416C23—32no—
417C23—31no—
418C23—32no—
419C23—34no—
420C23—61no—
421C23—62no—
422C23—61no—
423C23—61no—
424C23—62no—
425C23—62no—
426C23—64no—
427C23—62no—
428C23—62no—
429C23—64no—
430C23—64no—
431C23—64no—
432C23—62no—
433C23—62no—
434C23—64no—
435C23—64no—
436C23—62no—
437C23—121no—
438C23—121no—
439C23—122no—
440C23—122no—
441C23—122no—
442C23—122no—
443C23—122no—
444C23—121no—
445C23—122no—
446C23—121no—
447C23C42×D311yescharacteristic
448C23#87211yescharacteristic
449C23C4×D1212yes—
450C23C2×(C2×D3)⋊C411yescharacteristic
451C23C4×#6712yes—
452C23C22×S411yescharacteristic
453D4—64no—
454D4—64no—
455D4—68no—
456D4—64no—
457D4—68no—
458D4—64no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—32no—
462C2×C4—32no—
463C2×C4—32no—
464C2×C4—34no—
465C2×C4—34no—
466C2×C4—34no—
467C2×C4—34no—
468C2×C4—34no—
469C2×C4—38no—
470C2×C4—34no—
471C2×C4—34no—
472C2×C4—64no—
473C2×C4—64no—
474C2×C4—64no—
475C2×C4—64no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—64no—
481C2×C4—64no—
482C2×C4—62no—
483C2×C4—64no—
484C2×C4—64no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—64no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—64no—
491C2×C4—64no—
492C2×C4—64no—
493C2×C4—64no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—64no—
497C2×C4—62no—
498C2×C4—64no—
499C2×C4—62no—
500C2×C4—128no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—122no—
512C2×C4—121no—
513C2×C4—122no—
514C2×C4—121no—
515C2×C4—128no—
516C2×C4C4×S412yes—
517C2×C4C4×S414yescomplemented
518C2×C4C4⋊S412yes—
519C2×C4GL(2,ℤ/4)12yes—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—42no—
523C2×C3—41no—
524C2×C3—42no—
525D3—84no—
526C22—31no—
527C22—31no—
528C22—32no—
529C22—31no—
530C22—32no—
531C22—31no—
532C22—32no—
533C22—31no—
534C22—31no—
535C22—31no—
536C22—32no—
537C22—32no—
538C22—32no—
539C22—32no—
540C22—32no—
541C22—31no—
542C22—31no—
543C22—31no—
544C22—31no—
545C22—32no—
546C22—32no—
547C22—32no—
548C22—32no—
549C22—61no—
550C22—61no—
551C22—62no—
552C22—61no—
553C22—62no—
554C22—64no—
555C22—64no—
556C22—64no—
557C22—122no—
558C22—124no—
559C22—122no—
560C22—124no—
561C22—122no—
562C22—124no—
563C22—122no—
564C22—124no—
565C22C4×(C2×D3)⋊C411yescharacteristic, complemented
566C22C2×C4×S411yescharacteristic
567C22C2×C4×S412yes—
568C22C2×C4⋊S411yescharacteristic
569C22#591212yes—
570C22C2×GL(2,ℤ/4)11yescharacteristic
571C4—24no—
572C4—34no—
573C4—64no—
574C4—64no—
575C4—122no—
576C4—124no—
577C4—124no—
578C4—122no—
579C4(C2×S4)⋊C414yesdirect product factor
580C3—41noSylow
581C2—31no—
582C2—32no—
583C2—31no—
584C2—31no—
585C2—31no—
586C2—32no—
587C2—124no—
588C2C42×S411yescharacteristic
589C2#8528711yescharacteristic
590C2C4×C4⋊S412yes—
591C2C2×(C2×S4)⋊C411yescharacteristic
592C2C4×GL(2,ℤ/4)12yes—
subgroupG/H
centerC22×C4C2×S4
commutatorC2×A4C2×C42
FrattiniC23C22×S4
FittingC23×C42D3
socleC25C22×D3
prime radicalC23×S4C22
cosocleC23×A4C23
hypercenterC2×C42S4
hypocenterA4C4×C22⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×C42×A4C2semidirect
4. C2×C4×SL(2,ℤ/4)C2semidirect
5. A4⋊#95C2semidirect
14. C2×C4×S4C4semidirect
16. (C2×S4)⋊C4C4direct (N × H)
20. C23×C42D3semidirect
40. C2×C4×A4C2×C4semidirect
48. C2×SL(2,ℤ/4)C2×C4semidirect
103. C2×S4C42semidirect
112. C4×A4C22⋊C4semidirect
156. C23×C4C4×D3semidirect
207. C2×C42S4semidirect
357. C22×C4(C2×D3)⋊C4semidirect
391. A4C4×C22⋊C4semidirect
517. C2×C4C4×S4semidirect
565. C22C4×(C2×D3)⋊C4semidirect
579. C4(C2×S4)⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×S42
C2×C42×A42
C2×(C2×S4)⋊C42
C2×C4×SL(2,ℤ/4)2
A4⋊#952
C4×#3863
C4×(C2×D3)⋊C44