Finite Group Databasebeta

← D12×SL(2,ℤ/4) #11922012

Subgroups — D12×SL(2,ℤ/4) #11922012

Order 1152.

1048 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 99 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×A4×D12C211yesmaximal, characteristic
2C2×D3×SL(2,ℤ/4)C212yesmaximal, complemented
3C4⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
4C3×C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5(C2×D3)⋊SL(2,ℤ/4)C212yesmaximal, complemented
6D4×SL(2,ℤ/4)—31nomaximal
7C22⋊C4×D12—31nomaximal
8C22×D3×A4C2212yes—
9C2×C3×C4×A4C2211yescharacteristic
10A4×D12C414yescomplemented
11C2×C3⋊SL(2,ℤ/4)C2212yes—
12C2×C3×SL(2,ℤ/4)C2211yescharacteristic
13C2×C3×SL(2,ℤ/4)C2211yescharacteristic
14D3×SL(2,ℤ/4)—24no—
15Dic3×D12—41nomaximal
16C23×D12D311yescharacteristic
17C2×D4×A4—31no—
18C22×SL(2,ℤ/4)—32no—
19C2×D3×C22⋊C4—32no—
20C2×C4×D12—31no—
21C2×C4×D12—31no—
22#5432—32no—
23C4⋊SL(2,ℤ/4)—31no—
24C4⋊SL(2,ℤ/4)—61no—
25C4×SL(2,ℤ/4)D311yescharacteristic, complemented
26C3×C4×C22⋊C4—31no—
27#5212—31no—
28#4955—32no—
29#4939—31no—
30#4939—31no—
31#4665—32no—
32#4665—32no—
33C22×C3×A4C2311yescharacteristic
34C2×D3×A4—22no—
35C2×D3×A4—22no—
36C2×D3×A4C2×C414yescomplemented
37C2×C3×D12—41no—
38C3×C4×A4C2×C412yescomplemented
39C2×D3×Dic3—42no—
40C3⋊SL(2,ℤ/4)—22no—
41C3×SL(2,ℤ/4)—21no—
42C3×SL(2,ℤ/4)D412yescomplemented
43C4⋊(C3⋊Dic3)—41no—
44C3×C4×Dic3—41no—
45(C2×D3)⋊Dic3—42no—
46D4×C22⋊C4—91noSylow
47C24×D3C2×D312yes—
48C23×A4—32no—
49C23×C3×C4C2×D311yescharacteristic
50C22×D12—31no—
51C22×D12—32no—
52C22×D12—34no—
53C22×D12—31no—
54C22×D12Dic314yescomplemented
55C22×C4×D3—32no—
56C22×C4×D3—32no—
57D4×A4—34no—
58C2×C4×A4—21no—
59C2×C4×A4C2×D311yescharacteristic
60C2×SL(2,ℤ/4)—32no—
61C2×SL(2,ℤ/4)—62no—
62C2×SL(2,ℤ/4)—64no—
63C2×SL(2,ℤ/4)C2×D311yescharacteristic
64C2×SL(2,ℤ/4)C2×D311yescharacteristic
65C2×C3×C22⋊C4—31no—
66C2×C3×C22⋊C4—31no—
67C2×C3×C42—31no—
68C2×C3×C42—31no—
69C2×#217—32no—
70D4×Dic3—121no—
71C2×(C2×D3)⋊C4—32no—
72C2×(C2×D3)⋊C4—32no—
73C2×(C2×D3)⋊C4—32no—
74C2×(C2×D3)⋊C4—32no—
75C2×(C2×D3)⋊C4—32no—
76C2×(C2×D3)⋊C4—32no—
77C2×C4⋊Dic3—31no—
78C2×C4⋊Dic3—31no—
79D3×C22⋊C4—64no—
80D3×C22⋊C4—64no—
81C4×D12—64no—
82C4×D12—62no—
83C4×D12—61no—
84C4×D12—61no—
85C3×#95—31no—
86C3×#95—31no—
87C3⋊#95—32no—
88C22×C3×D3—42no—
89C2×C3×A4C22×C411yescharacteristic
90C2×C3×A4C2×D411yescharacteristic
91C2×C3×A4#4111yescharacteristic
92D3×A4—24no—
93C2×C32×C4—41no—
94C2×C3⋊Dic3—42no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—41no—
97C3×D12—44no—
98D3×Dic3—84no—
99C23×D4—31no—
100C2×C4×D4—91no—
101C2×C4×D4—91no—
102C22×C22⋊C4—92no—
103#390—91no—
104#390—91no—
105#386—92no—
106#386—92no—
107#380—91no—
108#379—92no—
109C4×C22⋊C4—31no—
110C24×C3C22×D311yescharacteristic
111C23×D3—22no—
112C23×D3—22no—
113C23×D3—32no—
114C23×D3—32no—
115C23×D3—34no—
116C23×D3—62no—
117C23×D3—62no—
118C23×D3C2×Dic314yescomplemented
119C22×A4—21no—
120C22×A4—34no—
121C22×A4—62no—
122C22×A4—62no—
123C22×A4C22×D311yescharacteristic
124C2×C3×D4—121no—
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C22×C3×C4—32no—
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—31no—
132C22×C3×C4C2×Dic312yescomplemented
133C22×Dic3—32no—
134C22×Dic3—32no—
135C22×Dic3—122no—
136C2×D12—32no—
137C2×D12—31no—
138C2×D12—34no—
139C2×D12—34no—
140C2×D12—38no—
141C2×D12—34no—
142C2×D12—62no—
143C2×D12—64no—
144C2×D12—61no—
145C2×D12S411yescharacteristic
146C2×C4×D3—64no—
147C2×C4×D3—64no—
148C2×C4×D3—64no—
149C2×C4×D3—62no—
150C2×C4×D3—62no—
151C2×C4×D3—122no—
152C2×C4×D3—122no—
153C4×A4—22no—
154C4×A4C4×D312yescomplemented
155SL(2,ℤ/4)—21no—
156SL(2,ℤ/4)—62no—
157SL(2,ℤ/4)—122no—
158SL(2,ℤ/4)D1212yesdirect product factor
159C3×C22⋊C4—32no—
160C3×C22⋊C4—32no—
161C3×C22⋊C4—61no—
162C3×C22⋊C4—61no—
163C3×C42—61no—
164C3×C42—62no—
165C3×C42—61no—
166#217—62no—
167#217—62no—
168#217—122no—
169(C2×D3)⋊C4—64no—
170(C2×D3)⋊C4—64no—
171(C2×D3)⋊C4—64no—
172(C2×D3)⋊C4—62no—
173(C2×D3)⋊C4—62no—
174(C2×D3)⋊C4—122no—
175(C2×D3)⋊C4—122no—
176C4⋊Dic3—62no—
177C4⋊Dic3—61no—
178C4⋊Dic3—61no—
179C4⋊Dic3—121no—
180C4⋊Dic3—241no—
181C4×Dic3—41no—
182C22×C32—41no—
183C2×C3×D3—44no—
184C2×C3×D3—82no—
185C2×C3×D3—82no—
186C3×A4C4×D411yescharacteristic, complemented
187C32×C4—42no—
188C3⋊Dic3—82no—
189C3×Dic3—42no—
190C3×Dic3—81no—
191C25—32no—
192C22×D4—34no—
193C22×D4—91no—
194C22×D4—92no—
195C22×D4—91no—
196C22×D4—94no—
197C23×C4—92no—
198C23×C4—92no—
199C23×C4D3211yescharacteristic
200C4×D4—181no—
201C4×D4—182no—
202C4×D4—184no—
203C4×D4—181no—
204C2×C4⋊C4—91no—
205C2×C4⋊C4—91no—
206C2×C22⋊C4—31no—
207C2×C22⋊C4—31no—
208C2×C22⋊C4—92no—
209C2×C22⋊C4—92no—
210C2×C22⋊C4—92no—
211C2×C22⋊C4—92no—
212C2×C22⋊C4—92no—
213C2×C22⋊C4—92no—
214C2×C22⋊C4—92no—
215C2×C22⋊C4—184no—
216C2×C22⋊C4—184no—
217C2×C42—31no—
218C2×C42—31no—
219#95—31no—
220#95—31no—
221#95—92no—
222C23×C3—31no—
223C23×C3—31no—
224C23×C3—31no—
225C23×C3—31no—
226C23×C3—122no—
227C23×C3D3×D411yescharacteristic
228C23×C3#23711yescharacteristic
229C23×C3C22×Dic311yescharacteristic
230C22×D3—24no—
231C22×D3—32no—
232C22×D3—34no—
233C22×D3—34no—
234C22×D3—34no—
235C22×D3—62no—
236C22×D3—62no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—62no—
240C22×D3—64no—
241C22×D3—62no—
242C22×D3—62no—
243C22×D3—62no—
244C22×D3—64no—
245C22×D3—64no—
246C22×D3—62no—
247C22×D3—124no—
248C22×D3C2×S412yes—
249C2×A4—21no—
250C2×A4—21no—
251C2×A4—21no—
252C2×A4—64no—
253C2×A4C2×C4×D311yescharacteristic
254C2×A4C2×D1211yescharacteristic
255C2×A4#23511yescharacteristic
256C3×D4—124no—
257C2×C3×C4—31no—
258C2×C3×C4—32no—
259C2×C3×C4—32no—
260C2×C3×C4—32no—
261C2×C3×C4—32no—
262C2×C3×C4—32no—
263C2×C3×C4—41no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—81no—
276C2×C3×C4C2×S411yescharacteristic
277C2×Dic3—41no—
278C2×Dic3—41no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3—62no—
282C2×Dic3—62no—
283C2×Dic3—62no—
284C2×Dic3—62no—
285C2×Dic3—122no—
286C2×Dic3—242no—
287C2×Dic3—244no—
288D12—38no—
289D12—34no—
290D12—64no—
291D12SL(2,ℤ/4)14yesdirect product factor
292C4×D3—124no—
293C4×D3—128no—
294C4×D3—124no—
295C2×C32—41no—
296C2×C32—41no—
297C2×C32—41no—
298C3×D3—84no—
299C24—34no—
300C24—62no—
301C24—62no—
302C24—92no—
303C24—92no—
304C24—94no—
305C24—182no—
306C24—182no—
307C24C2×D3211yescharacteristic
308C2×D4—31no—
309C2×D4—94no—
310C2×D4—94no—
311C2×D4—92no—
312C2×D4—94no—
313C2×D4—98no—
314C2×D4—91no—
315C2×D4—182no—
316C2×D4—181no—
317C2×D4—184no—
318C22×C4—31no—
319C22×C4—31no—
320C22×C4—32no—
321C22×C4—31no—
322C22×C4—31no—
323C22×C4—31no—
324C22×C4—31no—
325C22×C4—92no—
326C22×C4—92no—
327C22×C4—184no—
328C22×C4—184no—
329C22×C4—182no—
330C22×C4—184no—
331C22×C4—182no—
332C22×C4—362no—
333C22×C4—362no—
334C22×C4D3×Dic312yescomplemented
335C4⋊C4—181no—
336C4⋊C4—182no—
337C4⋊C4—181no—
338C22⋊C4—32no—
339C22⋊C4—32no—
340C22⋊C4—61no—
341C22⋊C4—61no—
342C22⋊C4—184no—
343C22⋊C4—184no—
344C22⋊C4—182no—
345C22⋊C4—184no—
346C22⋊C4—182no—
347C22⋊C4—182no—
348C22⋊C4—182no—
349C22⋊C4—362no—
350C22⋊C4—362no—
351C42—61no—
352C42—62no—
353C42—61no—
354C22×C3—31no—
355C22×C3—31no—
356C22×C3—31no—
357C22×C3—31no—
358C22×C3—31no—
359C22×C3—31no—
360C22×C3—31no—
361C22×C3—31no—
362C22×C3—31no—
363C22×C3—41no—
364C22×C3—61no—
365C22×C3—81no—
366C22×C3—124no—
367C22×C3—242no—
368C22×C3—242no—
369C22×C3D4×Dic311yescharacteristic, complemented
370C22×C3C22×S411yescharacteristic
371C2×D3—22no—
372C2×D3—22no—
373C2×D3—34no—
374C2×D3—64no—
375C2×D3—64no—
376C2×D3—64no—
377C2×D3—64no—
378C2×D3—62no—
379C2×D3—62no—
380C2×D3—64no—
381C2×D3—64no—
382C2×D3—122no—
383C2×D3—122no—
384C2×D3C2×SL(2,ℤ/4)14yescomplemented
385A4—21no—
386A4C4×D1211yescharacteristic, complemented
387C3×C4—32no—
388C3×C4—42no—
389C3×C4—62no—
390C3×C4—62no—
391C3×C4—82no—
392C3×C4—121no—
393C3×C4—121no—
394C3×C4C2×SL(2,ℤ/4)12yescomplemented
395Dic3—42no—
396Dic3—81no—
397Dic3—122no—
398Dic3—122no—
399Dic3—242no—
400Dic3—482no—
401C32—41noSylow
402C23—31no—
403C23—31no—
404C23—31no—
405C23—31no—
406C23—32no—
407C23—64no—
408C23—94no—
409C23—94no—
410C23—94no—
411C23—92no—
412C23—182no—
413C23—182no—
414C23—182no—
415C23—182no—
416C23—182no—
417C23—184no—
418C23—182no—
419C23—182no—
420C23—182no—
421C23—184no—
422C23—184no—
423C23—182no—
424C23—364no—
425C23#379511yescharacteristic
426C23D3×D1211yescharacteristic
427C23C2×D3×Dic311yescharacteristic
428D4—34no—
429D4—98no—
430D4—94no—
431D4—184no—
432C2×C4—31no—
433C2×C4—32no—
434C2×C4—32no—
435C2×C4—32no—
436C2×C4—32no—
437C2×C4—32no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—182no—
450C2×C4—182no—
451C2×C4—182no—
452C2×C4—182no—
453C2×C4—182no—
454C2×C4—182no—
455C2×C4—364no—
456C2×C4—368no—
457C2×C4—364no—
458C2×C4D3×S411yescharacteristic
459C2×C3—31no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—81no—
467C2×C3—81no—
468C2×C3—81no—
469C2×C3—244no—
470C2×C3D4×S411yescharacteristic
471C2×C3#591411yescharacteristic
472C2×C3C22×SL(2,ℤ/4)11yescharacteristic
473D3—24no—
474D3—64no—
475C22—31no—
476C22—31no—
477C22—31no—
478C22—31no—
479C22—31no—
480C22—31no—
481C22—31no—
482C22—31no—
483C22—31no—
484C22—34no—
485C22—61no—
486C22—62no—
487C22—62no—
488C22—94no—
489C22—184no—
490C22—184no—
491C22—184no—
492C22—184no—
493C22—184no—
494C22—184no—
495C22—182no—
496C22—182no—
497C22—362no—
498C22—362no—
499C22Dic3×D1211yescharacteristic, complemented
500C22C2×D3×S411yescharacteristic
501C4—32no—
502C4—62no—
503C4—62no—
504C4—121no—
505C4—121no—
506C4—362no—
507C4—362no—
508C4D3×SL(2,ℤ/4)12yescomplemented
509C3—41no—
510C3—81no—
511C3D4×SL(2,ℤ/4)11yescharacteristic, complemented
512C2—31no—
513C2—31no—
514C2—31no—
515C2—31no—
516C2—64no—
517C2—184no—
518C2#1059642411yescharacteristic
519C2D12×S411yescharacteristic
520C2C2×D3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutatorC2×C3×A4C22×C4
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC2×A4×D12C2
cosocleC22×C3×A4C23
hypercenterC2×C4D3×S4
hypocenterC3×A4C4×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×D3×SL(2,ℤ/4)C2semidirect
3. C4⋊(C3⋊SL(2,ℤ/4))C2semidirect
4. C3×C4×SL(2,ℤ/4)C2semidirect
5. (C2×D3)⋊SL(2,ℤ/4)C2semidirect
10. A4×D12C4semidirect
25. C4×SL(2,ℤ/4)D3semidirect
36. C2×D3×A4C2×C4semidirect
38. C3×C4×A4C2×C4semidirect
42. C3×SL(2,ℤ/4)D4semidirect
54. C22×D12Dic3semidirect
118. C23×D3C2×Dic3semidirect
132. C22×C3×C4C2×Dic3semidirect
154. C4×A4C4×D3semidirect
158. SL(2,ℤ/4)D12direct (N × H)
186. C3×A4C4×D4semidirect
291. D12SL(2,ℤ/4)direct (N × H)
334. C22×C4D3×Dic3semidirect
369. C22×C3D4×Dic3semidirect
384. C2×D3C2×SL(2,ℤ/4)semidirect
386. A4C4×D12semidirect
394. C3×C4C2×SL(2,ℤ/4)semidirect
499. C22Dic3×D12semidirect
508. C4D3×SL(2,ℤ/4)semidirect
511. C3D4×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×A4×D122
C2×D3×SL(2,ℤ/4)2
C4⋊(C3⋊SL(2,ℤ/4))2
C3×C4×SL(2,ℤ/4)2
(C2×D3)⋊SL(2,ℤ/4)2
D4×SL(2,ℤ/4)3
C22⋊C4×D123
Dic3×D124