Finite Group Databasebeta

← C4×D3×SL(2,ℤ/4) #11922003

Subgroups — C4×D3×SL(2,ℤ/4) #11922003

Order 1152.

1188 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1188 conjugacy classes of subgroups — too many to draw. Only its 139 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×C4×D3×A4C211yesmaximal, characteristic
2C2×D3×SL(2,ℤ/4)C212yesmaximal
3C4×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C3×C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5Dic3×SL(2,ℤ/4)C212yesmaximal, complemented
6C2×C4×SL(2,ℤ/4)—31nomaximal
7C4×D3×C22⋊C4—31nomaximal
8C22×D3×A4C2211yescharacteristic
9C2×C3×C4×A4C2211yescharacteristic
10C2×Dic3×A4C2211yescharacteristic
11C4×D3×A4C414yescomplemented
12C2×C3⋊SL(2,ℤ/4)C2212yes—
13C2×C3×SL(2,ℤ/4)C2212yes—
14D3×SL(2,ℤ/4)C418yesdirect product factor
15C4×D3×Dic3—41nomaximal
16C23×C4×D3D311yescharacteristic
17C22×C4×A4—31no—
18C22×SL(2,ℤ/4)—32no—
19C2×D3×C22⋊C4—32no—
20C2×C42×D3—31no—
21C2×C42×D3—31no—
22C4×SL(2,ℤ/4)—31no—
23C4×SL(2,ℤ/4)—32no—
24C4×SL(2,ℤ/4)—61no—
25C4×SL(2,ℤ/4)D311yescharacteristic, direct product factor
26C3×C4×C22⋊C4—31no—
27C4×#217—31no—
28Dic3×C22⋊C4—32no—
29C4×(C2×D3)⋊C4—31no—
30C4×(C2×D3)⋊C4—31no—
31#4666—32no—
32D3×#95—32no—
33C22×C3×A4C2311yescharacteristic
34C2×D3×A4C2×C414yes—
35C2×D3×A4C2×C412yes—
36C2×C3×C4×D3—41no—
37C3×C4×A4C2×C412yescomplemented
38C2×D3×Dic3—42no—
39Dic3×A4C2×C412yescomplemented
40C3⋊SL(2,ℤ/4)C2×C414yescomplemented
41C3×SL(2,ℤ/4)C2×C414yescomplemented
42C4×C3⋊Dic3—41no—
43C3×C4×Dic3—41no—
44Dic32—42no—
45C2×C4×C22⋊C4—91noSylow
46C24×D3C2×D311yescharacteristic
47C23×A4—31no—
48C23×C3×C4C2×D311yescharacteristic
49C23×Dic3C2×D311yescharacteristic
50C22×C4×D3—31no—
51C22×C4×D3—31no—
52C22×C4×D3—31no—
53C22×C4×D3—34no—
54C22×C4×D3—31no—
55C22×C4×D3—32no—
56C22×C4×D3—32no—
57C22×C4×D3Dic314yescomplemented
58C2×C4×A4—21no—
59C2×C4×A4—31no—
60C2×C4×A4—34no—
61C2×C4×A4C2×D311yescharacteristic
62C2×SL(2,ℤ/4)—32no—
63C2×SL(2,ℤ/4)—38no—
64C2×SL(2,ℤ/4)—62no—
65C2×SL(2,ℤ/4)C2×D312yes—
66C2×C3×C22⋊C4—32no—
67C2×C3×C42—31no—
68C2×C3×C42—31no—
69C2×#217—32no—
70C2×(C2×D3)⋊C4—32no—
71C2×(C2×D3)⋊C4—32no—
72C2×C4×Dic3—32no—
73C2×C4×Dic3—32no—
74C2×C4×Dic3—31no—
75C2×C4×Dic3—31no—
76C2×C4×Dic3—121no—
77D3×C22⋊C4—38no—
78D3×C22⋊C4—38no—
79C42×D3—61no—
80C42×D3—64no—
81C42×D3—61no—
82C42×D3—61no—
83C42×D3—61no—
84C3×#95—32no—
85C3⋊#95—32no—
86C3⋊#95—32no—
87C3⋊#95—32no—
88C22×C3×D3—41no—
89C2×C3×A4C22×C412yes—
90C2×C3×A4C22×C411yescharacteristic
91D3×A4C4214yescomplemented
92C2×C32×C4—41no—
93C2×C3⋊Dic3—42no—
94C2×C3×Dic3—42no—
95C2×C3×Dic3—41no—
96C3×C4×D3—44no—
97D3×Dic3—48no—
98C24×C4—31no—
99C22×C22⋊C4—92no—
100C22×C42—91no—
101C22×C42—91no—
102C4×C22⋊C4—31no—
103C4×C22⋊C4—92no—
104C4×C22⋊C4—91no—
105C4×C22⋊C4—91no—
106C4×C22⋊C4—92no—
107C4×C22⋊C4—91no—
108C2×#95—92no—
109C24×C3C22×D311yescharacteristic
110C23×D3—31no—
111C23×D3—34no—
112C23×D3—31no—
113C23×D3—32no—
114C23×D3C4×D314yes—
115C23×D3C2×Dic312yes—
116C22×A4—21no—
117C22×A4—34no—
118C22×A4—32no—
119C22×A4C22×D311yescharacteristic
120C22×C3×C4—31no—
121C22×C3×C4—31no—
122C22×C3×C4—32no—
123C22×C3×C4—32no—
124C22×C3×C4—32no—
125C22×C3×C4—121no—
126C22×C3×C4C2×Dic312yescomplemented
127C22×Dic3—31no—
128C22×Dic3—31no—
129C22×Dic3—32no—
130C22×Dic3—32no—
131C22×Dic3—32no—
132C22×Dic3—122no—
133C22×Dic3C2×Dic312yescomplemented
134C2×C4×D3—31no—
135C2×C4×D3—31no—
136C2×C4×D3—31no—
137C2×C4×D3—34no—
138C2×C4×D3—34no—
139C2×C4×D3—34no—
140C2×C4×D3—34no—
141C2×C4×D3—34no—
142C2×C4×D3—38no—
143C2×C4×D3—38no—
144C2×C4×D3—62no—
145C2×C4×D3—62no—
146C2×C4×D3—62no—
147C2×C4×D3—61no—
148C2×C4×D3—64no—
149C2×C4×D3—62no—
150C2×C4×D3—64no—
151C2×C4×D3—64no—
152C2×C4×D3—62no—
153C2×C4×D3S411yescharacteristic
154C4×A4—22no—
155C4×A4—32no—
156C4×A4C4×D312yescomplemented
157SL(2,ℤ/4)—34no—
158SL(2,ℤ/4)—64no—
159SL(2,ℤ/4)C4×D314yesdirect product factor
160C3×C22⋊C4—34no—
161C3×C22⋊C4—34no—
162C3×C42—61no—
163C3×C42—62no—
164C3×C42—61no—
165#217—34no—
166#217—34no—
167(C2×D3)⋊C4—38no—
168(C2×D3)⋊C4—38no—
169C4×Dic3—41no—
170C4×Dic3—62no—
171C4×Dic3—64no—
172C4×Dic3—62no—
173C4×Dic3—61no—
174C4×Dic3—61no—
175C4×Dic3—62no—
176C4×Dic3—122no—
177C4×Dic3—121no—
178C4×Dic3—241no—
179C22×C32—41no—
180C2×C3×D3—44no—
181C2×C3×D3—42no—
182C3×A4C2×C4211yescharacteristic, complemented
183C32×C4—42no—
184C3⋊Dic3—44no—
185C3×Dic3—44no—
186C3×Dic3—42no—
187C25—31no—
188C23×C4—31no—
189C23×C4—34no—
190C23×C4—91no—
191C23×C4—91no—
192C23×C4—91no—
193C23×C4—91no—
194C23×C4—94no—
195C23×C4—92no—
196C23×C4—92no—
197C23×C4D3211yescharacteristic
198C2×C22⋊C4—32no—
199C2×C22⋊C4—98no—
200C2×C22⋊C4—98no—
201C2×C22⋊C4—92no—
202C2×C22⋊C4—92no—
203C2×C22⋊C4—92no—
204C2×C42—31no—
205C2×C42—31no—
206C2×C42—92no—
207C2×C42—91no—
208C2×C42—91no—
209C2×C42—92no—
210C2×C42—181no—
211C2×C42—181no—
212C2×C42—184no—
213C2×C42—181no—
214C2×C42—181no—
215#95—32no—
216#95—92no—
217#95—92no—
218#95—92no—
219C23×C3—31no—
220C23×C3—32no—
221C23×C3—31no—
222C23×C3—121no—
223C23×C3C2×C4×D312yes—
224C23×C3C22×Dic311yescharacteristic
225C22×D3—31no—
226C22×D3—34no—
227C22×D3—34no—
228C22×D3—32no—
229C22×D3—32no—
230C22×D3—34no—
231C22×D3—32no—
232C22×D3—38no—
233C22×D3—34no—
234C22×D3—64no—
235C22×D3—61no—
236C22×D3—62no—
237C22×D3C4×Dic314yescomplemented
238C22×D3C2×S411yescharacteristic
239C2×A4—22no—
240C2×A4—21no—
241C2×A4—34no—
242C2×A4C2×C4×D312yes—
243C2×A4C2×C4×D311yescharacteristic
244C2×C3×C4—31no—
245C2×C3×C4—32no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—34no—
249C2×C3×C4—34no—
250C2×C3×C4—41no—
251C2×C3×C4—61no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—81no—
257C2×C3×C4—124no—
258C2×C3×C4—121no—
259C2×C3×C4C2×S411yescharacteristic
260C2×Dic3—31no—
261C2×Dic3—32no—
262C2×Dic3—32no—
263C2×Dic3—32no—
264C2×Dic3—34no—
265C2×Dic3—34no—
266C2×Dic3—42no—
267C2×Dic3—61no—
268C2×Dic3—62no—
269C2×Dic3—62no—
270C2×Dic3—62no—
271C2×Dic3—62no—
272C2×Dic3—128no—
273C2×Dic3—122no—
274C2×Dic3—242no—
275C2×Dic3C2×S411yescharacteristic
276C4×D3—34no—
277C4×D3—34no—
278C4×D3—34no—
279C4×D3—64no—
280C4×D3—616no—
281C4×D3—68no—
282C4×D3—68no—
283C4×D3SL(2,ℤ/4)14yesdirect product factor
284C2×C32—42no—
285C2×C32—41no—
286C3×D3—44no—
287C24—34no—
288C24—32no—
289C24—91no—
290C24—91no—
291C24—94no—
292C24—92no—
293C24C2×D3211yescharacteristic
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—32no—
298C22×C4—32no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—91no—
302C22×C4—94no—
303C22×C4—94no—
304C22×C4—91no—
305C22×C4—91no—
306C22×C4—94no—
307C22×C4—94no—
308C22×C4—94no—
309C22×C4—98no—
310C22×C4—98no—
311C22×C4—91no—
312C22×C4—91no—
313C22×C4—92no—
314C22×C4—92no—
315C22×C4—92no—
316C22×C4—182no—
317C22×C4—181no—
318C22×C4—182no—
319C22×C4—182no—
320C22×C4—182no—
321C22×C4—184no—
322C22×C4—182no—
323C22×C4—184no—
324C22×C4—184no—
325C22×C4D3×Dic312yescomplemented
326C22⋊C4—34no—
327C22⋊C4—34no—
328C22⋊C4—98no—
329C22⋊C4—98no—
330C22⋊C4—94no—
331C22⋊C4—94no—
332C42—61no—
333C42—62no—
334C42—61no—
335C42—182no—
336C42—181no—
337C42—182no—
338C42—184no—
339C42—182no—
340C42—181no—
341C22×C3—32no—
342C22×C3—31no—
343C22×C3—32no—
344C22×C3—32no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—41no—
348C22×C3—61no—
349C22×C3—81no—
350C22×C3—124no—
351C22×C3—122no—
352C22×C3C2×C4×Dic311yescharacteristic, complemented
353C22×C3C22×S411yescharacteristic
354C2×D3—34no—
355C2×D3—32no—
356C2×D3—38no—
357C2×D3—34no—
358C2×D3—34no—
359C2×D3—34no—
360C2×D3—64no—
361C2×D3—62no—
362C2×D3C4×S414yes—
363C2×D3C2×SL(2,ℤ/4)12yes—
364A4—21no—
365A4C42×D311yescharacteristic, complemented
366C3×C4—32no—
367C3×C4—42no—
368C3×C4—64no—
369C3×C4—64no—
370C3×C4—82no—
371C3×C4—122no—
372C3×C4C2×SL(2,ℤ/4)12yescomplemented
373Dic3—32no—
374Dic3—44no—
375Dic3—64no—
376Dic3—64no—
377Dic3—124no—
378Dic3—244no—
379Dic3C2×SL(2,ℤ/4)12yescomplemented
380C32—41noSylow
381C23—31no—
382C23—31no—
383C23—32no—
384C23—31no—
385C23—34no—
386C23—94no—
387C23—92no—
388C23—92no—
389C23—94no—
390C23—94no—
391C23—92no—
392C23—98no—
393C23—94no—
394C23—91no—
395C23—181no—
396C23—182no—
397C23—184no—
398C23C4×D3212yes—
399C23C2×D3×Dic311yescharacteristic
400C2×C4—31no—
401C2×C4—31no—
402C2×C4—32no—
403C2×C4—32no—
404C2×C4—34no—
405C2×C4—32no—
406C2×C4—34no—
407C2×C4—34no—
408C2×C4—61no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—94no—
414C2×C4—92no—
415C2×C4—92no—
416C2×C4—94no—
417C2×C4—91no—
418C2×C4—94no—
419C2×C4—92no—
420C2×C4—94no—
421C2×C4—94no—
422C2×C4—184no—
423C2×C4—188no—
424C2×C4—1816no—
425C2×C4—188no—
426C2×C4—181no—
427C2×C4—182no—
428C2×C4—182no—
429C2×C4—182no—
430C2×C4—182no—
431C2×C4D3×S411yescharacteristic
432C2×C3—32no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—42no—
436C2×C3—41no—
437C2×C3—82no—
438C2×C3—81no—
439C2×C3—124no—
440C2×C3C2×C4×S412yes—
441C2×C3C22×SL(2,ℤ/4)11yescharacteristic
442D3—34no—
443D3C4×SL(2,ℤ/4)14yesdirect product factor
444C22—32no—
445C22—31no—
446C22—32no—
447C22—32no—
448C22—34no—
449C22—31no—
450C22—31no—
451C22—32no—
452C22—61no—
453C22—98no—
454C22—94no—
455C22—94no—
456C22—94no—
457C22—94no—
458C22—92no—
459C22—184no—
460C22—182no—
461C22C4×D3×Dic311yescharacteristic, complemented
462C22C2×D3×S411yescharacteristic
463C4—32no—
464C4—32no—
465C4—64no—
466C4—64no—
467C4—92no—
468C4—184no—
469C4—184no—
470C4D3×SL(2,ℤ/4)12yesdirect product factor
471C3—41no—
472C3—81no—
473C3C2×C4×SL(2,ℤ/4)11yescharacteristic, complemented
474C2—32no—
475C2—31no—
476C2—31no—
477C2—34no—
478C2—94no—
479C2C4×D3×S412yes—
480C2C2×D3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×S4
commutator, hypocenterC3×A4C2×C42
FrattiniC22C2×D3×S4
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC22×D3×A4C22
cosocleC22×C3×A4C23
LCSG ▷ C3×A4
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C4×C3⋊SL(2,ℤ/4)C2semidirect
4. C3×C4×SL(2,ℤ/4)C2semidirect
5. Dic3×SL(2,ℤ/4)C2semidirect
11. C4×D3×A4C4semidirect
14. D3×SL(2,ℤ/4)C4direct (N × H)
25. C4×SL(2,ℤ/4)D3direct (N × H)
37. C3×C4×A4C2×C4semidirect
39. Dic3×A4C2×C4semidirect
40. C3⋊SL(2,ℤ/4)C2×C4semidirect
41. C3×SL(2,ℤ/4)C2×C4semidirect
57. C22×C4×D3Dic3semidirect
91. D3×A4C42semidirect
126. C22×C3×C4C2×Dic3semidirect
133. C22×Dic3C2×Dic3semidirect
156. C4×A4C4×D3semidirect
159. SL(2,ℤ/4)C4×D3direct (N × H)
182. C3×A4C2×C42semidirect
237. C22×D3C4×Dic3semidirect
283. C4×D3SL(2,ℤ/4)direct (N × H)
325. C22×C4D3×Dic3semidirect
352. C22×C3C2×C4×Dic3semidirect
365. A4C42×D3semidirect
372. C3×C4C2×SL(2,ℤ/4)semidirect
379. Dic3C2×SL(2,ℤ/4)semidirect
443. D3C4×SL(2,ℤ/4)direct (N × H)
461. C22C4×D3×Dic3semidirect
470. C4D3×SL(2,ℤ/4)direct (N × H)
473. C3C2×C4×SL(2,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×A42
C2×D3×SL(2,ℤ/4)2
C4×C3⋊SL(2,ℤ/4)2
C3×C4×SL(2,ℤ/4)2
Dic3×SL(2,ℤ/4)2
C2×C4×SL(2,ℤ/4)3
C4×D3×C22⋊C43
C4×D3×Dic34