Finite Group Databasebeta

← C2×C32×A4⋊S4 #72242581

Subgroups — C2×C32×A4⋊S4 #72242581

Order 5184.

2886 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 2886 conjugacy classes of subgroups — too many to draw. Only its 150 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×C32×A42C211yesmaximal, characteristic, complemented
2C32×A4⋊S4C212yesmaximal, direct product factor
3C2×C32×#1020—32nomaximal
4C2×C3×A4⋊S4C314yesmaximal, direct product factor
5C2×C32×GL(2,ℤ/4)—32nomaximal
6C2×C32×C3⋊S4—42nomaximal
7C32×A42C2211yescharacteristic, complemented
8C2×C32×C22⋊A4D312yescomplemented
9C23×C32×A4D312yescomplemented
10C32×#1020—34no—
11C22×C32×S4—32no—
12C2×C3×A42—28no—
13C2×C3×A42—28no—
14C2×C3×A42C2×C314yescomplemented
15C3×A4⋊S4C2×C318yesdirect product factor
16C32×GL(2,ℤ/4)—34no—
17C32×GL(2,ℤ/4)—34no—
18C2×C32×SL(2,ℤ/4)—32no—
19C2×C33×A4—42no—
20C32×C3⋊S4—44no—
21C2×C3×#1020—38no—
22C2×A4⋊S4C3211yescharacteristic, direct product factor
23C2×C3×GL(2,ℤ/4)—38no—
24C2×C32×#120—91no—
25C32×C22⋊A4C2×D312yescomplemented
26C22×C32×A4—32no—
27C22×C32×A4—32no—
28C22×C32×A4C2×D312yescomplemented
29C2×C3×C3⋊S4—48no—
30C2×C32×S4—34no—
31C2×C32×S4—62no—
32C2×C32×S4—62no—
33C2×C32×S4—124no—
34C2×C32×S4—122no—
35C3×A42—28no—
36C3×A42—28no—
37C3×A42C22×C314yescomplemented
38C2×C32×#67—122no—
39C32×SL(2,ℤ/4)—34no—
40C2×C32×C3⋊D3—161no—
41C33×A4—42no—
42C25×C32C3⋊D311yescharacteristic, complemented
43C2×C3×C22⋊A4—28no—
44C2×C3×C22⋊A4C3×D318yescomplemented
45C23×C3×A4—28no—
46C23×C3×A4C3×D318yescomplemented
47C3×#1020—316no—
48C22×C3×S4—38no—
49C2×A42—224no—
50C2×A42—28no—
51C2×A42—28no—
52C2×A42C2×C3211yescharacteristic, complemented
53A4⋊S4C2×C3212yesdirect product factor
54C22×C32×D4—92no—
55C22×C32×D4—91no—
56C3×GL(2,ℤ/4)—316no—
57C3×GL(2,ℤ/4)—316no—
58C2×C3×SL(2,ℤ/4)—38no—
59C32×#120—94no—
60C32×#120—92no—
61C32×#120—92no—
62C2×C32×C22⋊C4—92no—
63C2×C32×C22⋊C4—91no—
64C23×C33—42no—
65C22×C32×D3—122no—
66C2×C32×A4—32no—
67C2×C32×A4—32no—
68C2×C32×A4—44no—
69C2×C32×A4—48no—
70C2×C32×A4—816no—
71C2×C32×A4—88no—
72C2×C32×A4—88no—
73C2×C32×A4—122no—
74C2×C32×A4S412yescomplemented
75C3×C3⋊S4—416no—
76C32×S4—64no—
77C32×S4—128no—
78C32×S4—124no—
79C32×#67—124no—
80C32×#67—124no—
81C2×C32×Dic3—122no—
82C2×#1020—32no—
83C2×GL(2,ℤ/4)—32no—
84C2×C3×#120—94no—
85C2×C34—161no—
86C32×C3⋊D3—162no—
87C24×C32—32no—
88C24×C32—32no—
89C24×C32—91no—
90C24×C32—91no—
91C24×C32—91no—
92C24×C32C2×C3⋊D311yescharacteristic, complemented
93C3×C22⋊A4—28no—
94C3×C22⋊A4C2×C3×D318yescomplemented
95C22×C3×A4—28no—
96C22×C3×A4—38no—
97C22×C3×A4—38no—
98C22×C3×A4—68no—
99C22×C3×A4—68no—
100C22×C3×A4C2×C3×D318yescomplemented
101C2×C3⋊S4—42no—
102C2×C3×S4—316no—
103C2×C3×S4—68no—
104C2×C3×S4—68no—
105C2×C3×S4—1216no—
106C2×C3×S4—128no—
107A42—224no—
108A42—28no—
109A42—28no—
110A42C22×C3211yescharacteristic, complemented
111C2×C32×D4—94no—
112C2×C32×D4—94no—
113C2×C32×D4—92no—
114C2×C32×D4—92no—
115C2×C32×D4—182no—
116C2×C32×D4—182no—
117C2×C32×D4—182no—
118C2×C32×D4—182no—
119C2×C32×D4—182no—
120C2×C32×D4—182no—
121C22×C32×C4—92no—
122C22×C32×C4—91no—
123C2×C3×#67—128no—
124C3×SL(2,ℤ/4)—316no—
125C32×C22⋊C4—94no—
126C32×C22⋊C4—92no—
127C32×C22⋊C4—94no—
128C32×C22⋊C4—92no—
129C22×C33—42no—
130C22×C33—122no—
131C22×C33—122no—
132C2×C3×C3⋊D3—164no—
133C2×C32×D3—124no—
134C2×C32×D3—242no—
135C2×C32×D3—242no—
136C2×C32×D3—482no—
137C32×A4—44no—
138C32×A4—48no—
139C32×A4—816no—
140C32×A4—88no—
141C32×A4—88no—
142C32×A4—122no—
143C32×A4C2×S412yescomplemented
144C32×Dic3—124no—
145C25×C3C3×C3⋊D314yescomplemented
146C2×C22⋊A4—28no—
147C2×C22⋊A4C32×D312yescomplemented
148C23×A4—28no—
149C23×A4C32×D312yescomplemented
150#1020—34no—
151C22×S4—32no—
152C22×C3×D4—98no—
153C22×C3×D4—94no—
154GL(2,ℤ/4)—34no—
155GL(2,ℤ/4)—34no—
156C2×SL(2,ℤ/4)—32no—
157C3×#120—916no—
158C3×#120—98no—
159C3×#120—98no—
160C2×C3×C22⋊C4—98no—
161C2×C3×C22⋊C4—94no—
162C34—161noSylow
163C23×C32—32no—
164C23×C32—32no—
165C23×C32—32no—
166C23×C32—48no—
167C23×C32—88no—
168C23×C32—91no—
169C23×C32—91no—
170C23×C32—92no—
171C23×C32—92no—
172C23×C32—92no—
173C23×C32—92no—
174C23×C32—91no—
175C23×C32—92no—
176C23×C32—182no—
177C23×C32—182no—
178C23×C32—182no—
179C23×C32—181no—
180C23×C32—181no—
181C23×C32C3⋊S412yescomplemented
182C22×C3×D3—128no—
183C2×C3×A4—28no—
184C2×C3×A4—38no—
185C2×C3×A4—38no—
186C2×C3×A4—42no—
187C2×C3×A4—416no—
188C2×C3×A4—68no—
189C2×C3×A4—68no—
190C2×C3×A4—816no—
191C2×C3×A4—88no—
192C2×C3×A4—848no—
193C2×C3×A4—816no—
194C2×C3×A4—88no—
195C2×C3×A4—128no—
196C2×C3×A4—248no—
197C2×C3×A4C3×S418yescomplemented
198C3⋊S4—44no—
199C3×S4—616no—
200C3×S4—1232no—
201C3×S4—1216no—
202C32×D4—184no—
203C32×D4—184no—
204C32×D4—184no—
205C32×D4—184no—
206C32×D4—184no—
207C32×D4—184no—
208C2×C32×C4—94no—
209C2×C32×C4—92no—
210C2×C32×C4—182no—
211C2×C32×C4—182no—
212C2×C32×C4—181no—
213C2×C32×C4—181no—
214C3×#67—1216no—
215C3×#67—1216no—
216C2×C3×Dic3—128no—
217C2×#120—91noSylow
218C2×C33—42no—
219C2×C33—122no—
220C2×C33—122no—
221C2×C33—164no—
222C2×C33—162no—
223C2×C33—328no—
224C2×C33—328no—
225C3×C3⋊D3—168no—
226C32×D3—244no—
227C32×D3—484no—
228C24×C3—38no—
229C24×C3—38no—
230C24×C3—94no—
231C24×C3—94no—
232C24×C3—94no—
233C24×C3C2×C3×C3⋊D314yescomplemented
234C22⋊A4—28no—
235C22⋊A4C2×C32×D312yescomplemented
236C22×A4—28no—
237C22×A4—32no—
238C22×A4—32no—
239C22×A4—68no—
240C22×A4—68no—
241C22×A4C2×C32×D312yescomplemented
242C2×S4—34no—
243C2×S4—62no—
244C2×S4—62no—
245C2×S4—124no—
246C2×S4—122no—
247C2×C3×D4—916no—
248C2×C3×D4—916no—
249C2×C3×D4—98no—
250C2×C3×D4—98no—
251C2×C3×D4—188no—
252C2×C3×D4—188no—
253C2×C3×D4—188no—
254C2×C3×D4—188no—
255C2×C3×D4—188no—
256C2×C3×D4—188no—
257C22×C3×C4—98no—
258C22×C3×C4—94no—
259C2×#67—122no—
260SL(2,ℤ/4)—34no—
261C3×C22⋊C4—916no—
262C3×C22⋊C4—98no—
263C3×C22⋊C4—916no—
264C3×C22⋊C4—98no—
265C22×C32—32no—
266C22×C32—32no—
267C22×C32—32no—
268C22×C32—48no—
269C22×C32—88no—
270C22×C32—91no—
271C22×C32—92no—
272C22×C32—91no—
273C22×C32—92no—
274C22×C32—92no—
275C22×C32—92no—
276C22×C32—91no—
277C22×C32—128no—
278C22×C32—128no—
279C22×C32—182no—
280C22×C32—184no—
281C22×C32—182no—
282C22×C32—248no—
283C22×C32—248no—
284C22×C32—362no—
285C22×C32—361no—
286C22×C32—361no—
287C22×C32C2×C3⋊S412yescomplemented
288C2×C3⋊D3—161no—
289C2×C3×D3—1216no—
290C2×C3×D3—248no—
291C2×C3×D3—248no—
292C2×C3×D3—488no—
293C3×A4—28no—
294C3×A4—42no—
295C3×A4—416no—
296C3×A4—816no—
297C3×A4—88no—
298C3×A4—848no—
299C3×A4—816no—
300C3×A4—88no—
301C3×A4—128no—
302C3×A4—248no—
303C3×A4C2×C3×S418yescomplemented
304C32×C4—184no—
305C32×C4—182no—
306C3×Dic3—1216no—
307C25C32×C3⋊D311yescharacteristic, complemented
308C22×D4—92no—
309C22×D4—91no—
310#120—94no—
311#120—92no—
312#120—92no—
313C2×C22⋊C4—92no—
314C2×C22⋊C4—91no—
315C33—42no—
316C33—164no—
317C33—162no—
318C33—328no—
319C33—328no—
320C23×C3—38no—
321C23×C3—38no—
322C23×C3—38no—
323C23×C3—42no—
324C23×C3—88no—
325C23×C3—94no—
326C23×C3—94no—
327C23×C3—98no—
328C23×C3—98no—
329C23×C3—98no—
330C23×C3—98no—
331C23×C3—94no—
332C23×C3—98no—
333C23×C3—188no—
334C23×C3—188no—
335C23×C3—188no—
336C23×C3—184no—
337C23×C3—184no—
338C23×C3C3×C3⋊S418yescomplemented
339C22×D3—122no—
340C2×A4—28no—
341C2×A4—32no—
342C2×A4—32no—
343C2×A4—44no—
344C2×A4—68no—
345C2×A4—68no—
346C2×A4—816no—
347C2×A4—122no—
348C2×A4—248no—
349C2×A4C32×S412yescomplemented
350S4—64no—
351S4—128no—
352S4—124no—
353C3×D4—1816no—
354C3×D4—1816no—
355C3×D4—1816no—
356C3×D4—1816no—
357C3×D4—1816no—
358C3×D4—1816no—
359C2×C3×C4—916no—
360C2×C3×C4—98no—
361C2×C3×C4—188no—
362C2×C3×C4—188no—
363C2×C3×C4—184no—
364C2×C3×C4—184no—
365#67—124no—
366#67—124no—
367C2×Dic3—122no—
368C2×C32—32no—
369C2×C32—32no—
370C2×C32—48no—
371C2×C32—88no—
372C2×C32—91no—
373C2×C32—91no—
374C2×C32—128no—
375C2×C32—128no—
376C2×C32—161no—
377C2×C32—168no—
378C2×C32—248no—
379C2×C32—248no—
380C2×C32—328no—
381C2×C32—328no—
382C2×C32—3224no—
383C2×C32—328no—
384C2×C32—362no—
385C2×C32A4⋊S411yescharacteristic, direct product factor
386C3⋊D3—162no—
387C3×D3—2416no—
388C3×D3—4816no—
389C24—32no—
390C24—32no—
391C24—91no—
392C24—91no—
393C24—91no—
394C24C2×C32×C3⋊D311yescharacteristic, complemented
395C2×D4—94no—
396C2×D4—94no—
397C2×D4—92no—
398C2×D4—92no—
399C2×D4—182no—
400C2×D4—182no—
401C2×D4—182no—
402C2×D4—182no—
403C2×D4—182no—
404C2×D4—182no—
405C22×C4—92no—
406C22×C4—91no—
407C22⋊C4—94no—
408C22⋊C4—92no—
409C22⋊C4—94no—
410C22⋊C4—92no—
411C22×C3—38no—
412C22×C3—38no—
413C22×C3—38no—
414C22×C3—42no—
415C22×C3—88no—
416C22×C3—94no—
417C22×C3—98no—
418C22×C3—94no—
419C22×C3—98no—
420C22×C3—98no—
421C22×C3—98no—
422C22×C3—94no—
423C22×C3—122no—
424C22×C3—122no—
425C22×C3—188no—
426C22×C3—1816no—
427C22×C3—188no—
428C22×C3—248no—
429C22×C3—248no—
430C22×C3—368no—
431C22×C3—364no—
432C22×C3—364no—
433C22×C3C2×C3×C3⋊S418yescomplemented
434C2×D3—124no—
435C2×D3—242no—
436C2×D3—242no—
437C2×D3—482no—
438A4—28no—
439A4—44no—
440A4—816no—
441A4—122no—
442A4—248no—
443A4C2×C32×S412yescomplemented
444C3×C4—1816no—
445C3×C4—188no—
446Dic3—124no—
447C32—48no—
448C32—88no—
449C32—161no—
450C32—168no—
451C32—328no—
452C32—328no—
453C32—3224no—
454C32—328no—
455C32C2×A4⋊S411yescharacteristic, direct product factor
456C23—32no—
457C23—32no—
458C23—32no—
459C23—91no—
460C23—91no—
461C23—92no—
462C23—92no—
463C23—92no—
464C23—92no—
465C23—91no—
466C23—92no—
467C23—182no—
468C23—182no—
469C23—182no—
470C23—181no—
471C23—181no—
472C23C32×C3⋊S412yescomplemented
473D4—184no—
474D4—184no—
475D4—184no—
476D4—184no—
477D4—184no—
478D4—184no—
479C2×C4—94no—
480C2×C4—92no—
481C2×C4—182no—
482C2×C4—182no—
483C2×C4—181no—
484C2×C4—181no—
485C2×C3—38no—
486C2×C3—38no—
487C2×C3—42no—
488C2×C3—88no—
489C2×C3—94no—
490C2×C3—94no—
491C2×C3—122no—
492C2×C3—122no—
493C2×C3—162no—
494C2×C3—248no—
495C2×C3—248no—
496C2×C3—328no—
497C2×C3—368no—
498C2×C3C3×A4⋊S414yesdirect product factor
499D3—244no—
500D3—484no—
501C22—32no—
502C22—32no—
503C22—32no—
504C22—91no—
505C22—92no—
506C22—91no—
507C22—92no—
508C22—92no—
509C22—92no—
510C22—91no—
511C22—182no—
512C22—184no—
513C22—182no—
514C22—362no—
515C22—361no—
516C22—361no—
517C22C2×C32×C3⋊S412yescomplemented
518C4—184no—
519C4—182no—
520C3—42no—
521C3—88no—
522C3—162no—
523C3—328no—
524C3C2×C3×A4⋊S414yesdirect product factor
525C2—32no—
526C2—32no—
527C2—91no—
528C2—91no—
529C2—362no—
530C2C32×A4⋊S411yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C32A4⋊S4
commutator, cosocle, hypocenterA42C22×C32
FrattiniC1G
Fitting, socleC25×C32C3⋊D3
LCSG ▷ A42
UCSC1 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. C2×C32×A42C2semidirect
2. C32×A4⋊S4C2direct (N × H)
4. C2×C3×A4⋊S4C3direct (N × H)
7. C32×A42C22semidirect
8. C2×C32×C22⋊A4D3semidirect
9. C23×C32×A4D3semidirect
14. C2×C3×A42C2×C3semidirect
15. C3×A4⋊S4C2×C3direct (N × H)
22. C2×A4⋊S4C32direct (N × H)
25. C32×C22⋊A4C2×D3semidirect
28. C22×C32×A4C2×D3semidirect
37. C3×A42C22×C3semidirect
42. C25×C32C3⋊D3semidirect
44. C2×C3×C22⋊A4C3×D3semidirect
46. C23×C3×A4C3×D3semidirect
52. C2×A42C2×C32semidirect
53. A4⋊S4C2×C32direct (N × H)
74. C2×C32×A4S4semidirect
92. C24×C32C2×C3⋊D3semidirect
94. C3×C22⋊A4C2×C3×D3semidirect
100. C22×C3×A4C2×C3×D3semidirect
110. A42C22×C32semidirect
143. C32×A4C2×S4semidirect
145. C25×C3C3×C3⋊D3semidirect
147. C2×C22⋊A4C32×D3semidirect
149. C23×A4C32×D3semidirect
181. C23×C32C3⋊S4semidirect
197. C2×C3×A4C3×S4semidirect
233. C24×C3C2×C3×C3⋊D3semidirect
235. C22⋊A4C2×C32×D3semidirect
241. C22×A4C2×C32×D3semidirect
287. C22×C32C2×C3⋊S4semidirect
303. C3×A4C2×C3×S4semidirect
307. C25C32×C3⋊D3semidirect
338. C23×C3C3×C3⋊S4semidirect
349. C2×A4C32×S4semidirect
385. C2×C32A4⋊S4direct (N × H)
394. C24C2×C32×C3⋊D3semidirect
433. C22×C3C2×C3×C3⋊S4semidirect
443. A4C2×C32×S4semidirect
455. C32C2×A4⋊S4direct (N × H)
472. C23C32×C3⋊S4semidirect
498. C2×C3C3×A4⋊S4direct (N × H)
517. C22C2×C32×C3⋊S4semidirect
524. C3C2×C3×A4⋊S4direct (N × H)
530. C2C32×A4⋊S4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×A422
C32×A4⋊S42
C2×C32×#10203
C2×C3×A4⋊S43
C2×C32×GL(2,ℤ/4)3
C2×C32×C3⋊S44