Finite Group Databasebeta

← C3×D4×SL(2,ℤ/4) #11922774

Subgroups — C3×D4×SL(2,ℤ/4) #11922774

Order 1152.

1076 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1076 conjugacy classes of subgroups — too many to draw. Only its 140 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×C3×D4×A4C211yesmaximal, characteristic
2C22×C3×SL(2,ℤ/4)C212yesmaximal, complemented
3C3×#5432C212yesmaximal, complemented
4C3×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5C3×C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6D4×SL(2,ℤ/4)C311yesmaximal, characteristic, direct product factor
7C3×D4×C22⋊C4—31nomaximal
8C23×C3×A4C2212yes—
9C3×D4×A4C414yescomplemented
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3×SL(2,ℤ/4)—24no—
12C2×C3×SL(2,ℤ/4)C2211yescharacteristic
13C2×C3×SL(2,ℤ/4)C2212yes—
14C2×C3×SL(2,ℤ/4)C2211yescharacteristic
15C3×D4×Dic3—41nomaximal
16C23×C3×D4D311yescharacteristic
17C2×D4×A4—21no—
18C2×D4×A4C2×C311yescharacteristic
19C22×SL(2,ℤ/4)C2×C312yescomplemented
20C2×C3×C4×D4—31no—
21C2×C3×C4×D4—31no—
22C22×C3×C22⋊C4—32no—
23#5432C2×C312yescomplemented
24C4⋊SL(2,ℤ/4)C2×C311yescharacteristic, complemented
25C4×SL(2,ℤ/4)C2×C311yescharacteristic, complemented
26C3×#390—31no—
27C3×#390—31no—
28C3×#386—32no—
29C3×#386—32no—
30C3×#380—31no—
31C3×#379—32no—
32C3×C4×C22⋊C4—31no—
33C22×C3×A4—22no—
34C22×C3×A4—22no—
35C22×C3×A4C2×C414yescomplemented
36C22×C3×A4C2311yescharacteristic
37C2×C32×D4—41no—
38C22×C3×Dic3—42no—
39C3×C4×A4C2×C412yescomplemented
40C3×SL(2,ℤ/4)—22no—
41C3×SL(2,ℤ/4)—21no—
42C3×SL(2,ℤ/4)D412yesdirect product factor
43C3×#217—42no—
44C3×C4⋊Dic3—41no—
45C3×C4×Dic3—41no—
46D4×C22⋊C4—31noSylow
47C25×C3C2×D312yes—
48C23×A4—22no—
49C23×A4C22×C312yes—
50C22×C3×D4—31no—
51C22×C3×D4—34no—
52C22×C3×D4—32no—
53C22×C3×D4—31no—
54C22×C3×D4Dic314yescomplemented
55C23×C3×C4—32no—
56C23×C3×C4—32no—
57C23×C3×C4C2×D311yescharacteristic
58D4×A4—24no—
59D4×A4C3×C414yescomplemented
60C2×C4×A4—21no—
61C2×C4×A4C22×C311yescharacteristic
62C2×SL(2,ℤ/4)—24no—
63C2×SL(2,ℤ/4)C22×C311yescharacteristic
64C2×SL(2,ℤ/4)C22×C312yes—
65C2×SL(2,ℤ/4)C22×C311yescharacteristic
66C3×C4×D4—64no—
67C3×C4×D4—62no—
68C3×C4×D4—61no—
69C3×C4×D4—61no—
70C2×C3×C4⋊C4—31no—
71C2×C3×C4⋊C4—31no—
72C2×C3×C22⋊C4—31no—
73C2×C3×C22⋊C4—32no—
74C2×C3×C22⋊C4—32no—
75C2×C3×C22⋊C4—32no—
76C2×C3×C22⋊C4—32no—
77C2×C3×C22⋊C4—31no—
78C2×C3×C22⋊C4—32no—
79C2×C3×C22⋊C4—32no—
80C2×C3×C22⋊C4—32no—
81C2×C3×C22⋊C4—64no—
82C2×C3×C22⋊C4—64no—
83C2×C3×C42—31no—
84C2×C3×C42—31no—
85D4×Dic3—41no—
86C3×#95—31no—
87C3×#95—31no—
88C3×#95—32no—
89C23×C32—42no—
90C2×C3×A4—24no—
91C2×C3×A4C22×C411yescharacteristic
92C2×C3×A4C2×D411yescharacteristic
93C2×C3×A4#4111yescharacteristic
94C32×D4—44no—
95C2×C32×C4—41no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—84no—
100C23×D4C3×D311yescharacteristic
101C2×C4×D4—31no—
102C2×C4×D4—31no—
103C22×C22⋊C4—32no—
104#390—31no—
105#390—31no—
106#386—32no—
107#386—32no—
108#380—31no—
109#379—32no—
110C4×C22⋊C4—31no—
111C24×C3—22no—
112C24×C3—22no—
113C24×C3—32no—
114C24×C3—32no—
115C24×C3—34no—
116C24×C3—62no—
117C24×C3—62no—
118C24×C3C2×Dic314yescomplemented
119C24×C3C22×D311yescharacteristic
120C22×A4—22no—
121C22×A4—22no—
122C22×A4—24no—
123C22×A4—21no—
124C22×A4—42no—
125C22×A4—42no—
126C22×A4C2×C3×C414yescomplemented
127C22×A4C23×C311yescharacteristic
128C2×C3×D4—32no—
129C2×C3×D4—34no—
130C2×C3×D4—34no—
131C2×C3×D4—34no—
132C2×C3×D4—38no—
133C2×C3×D4—31no—
134C2×C3×D4—41no—
135C2×C3×D4—62no—
136C2×C3×D4—61no—
137C2×C3×D4—64no—
138C2×C3×D4—81no—
139C2×C3×D4S411yescharacteristic
140C22×C3×C4—31no—
141C22×C3×C4—31no—
142C22×C3×C4—32no—
143C22×C3×C4—31no—
144C22×C3×C4—31no—
145C22×C3×C4—31no—
146C22×C3×C4—31no—
147C22×C3×C4—32no—
148C22×C3×C4—32no—
149C22×C3×C4—64no—
150C22×C3×C4—64no—
151C22×C3×C4—64no—
152C22×C3×C4—62no—
153C22×C3×C4—62no—
154C22×C3×C4—122no—
155C22×C3×C4—122no—
156C22×C3×C4C2×Dic312yescomplemented
157C22×Dic3—42no—
158C4×A4—22no—
159C4×A4C2×C3×C412yescomplemented
160SL(2,ℤ/4)—22no—
161SL(2,ℤ/4)—21no—
162SL(2,ℤ/4)C3×D412yesdirect product factor
163C3×C4⋊C4—61no—
164C3×C4⋊C4—62no—
165C3×C4⋊C4—61no—
166C3×C22⋊C4—32no—
167C3×C22⋊C4—32no—
168C3×C22⋊C4—64no—
169C3×C22⋊C4—64no—
170C3×C22⋊C4—61no—
171C3×C22⋊C4—62no—
172C3×C22⋊C4—64no—
173C3×C22⋊C4—62no—
174C3×C22⋊C4—62no—
175C3×C22⋊C4—61no—
176C3×C22⋊C4—62no—
177C3×C22⋊C4—122no—
178C3×C22⋊C4—122no—
179C3×C42—61no—
180C3×C42—62no—
181C3×C42—61no—
182#217—42no—
183C4⋊Dic3—41no—
184C4×Dic3—41no—
185C22×C32—44no—
186C22×C32—41no—
187C22×C32—82no—
188C22×C32—82no—
189C3×A4C4×D411yescharacteristic, complemented
190C32×C4—42no—
191C3×Dic3—42no—
192C3×Dic3—82no—
193C3×Dic3—81no—
194C25C2×C3×D312yes—
195C22×D4—31no—
196C22×D4—32no—
197C22×D4—31no—
198C22×D4—34no—
199C22×D4C3×Dic314yescomplemented
200C23×C4—32no—
201C23×C4—32no—
202C23×C4C2×C3×D311yescharacteristic
203C4×D4—64no—
204C4×D4—62no—
205C4×D4—61no—
206C4×D4—61no—
207C2×C4⋊C4—31no—
208C2×C4⋊C4—31no—
209C2×C22⋊C4—31no—
210C2×C22⋊C4—32no—
211C2×C22⋊C4—32no—
212C2×C22⋊C4—32no—
213C2×C22⋊C4—32no—
214C2×C22⋊C4—32no—
215C2×C22⋊C4—31no—
216C2×C22⋊C4—32no—
217C2×C22⋊C4—32no—
218C2×C22⋊C4—64no—
219C2×C22⋊C4—64no—
220C2×C42—31no—
221C2×C42—31no—
222#95—31no—
223#95—31no—
224#95—32no—
225C23×C3—24no—
226C23×C3—31no—
227C23×C3—31no—
228C23×C3—31no—
229C23×C3—31no—
230C23×C3—34no—
231C23×C3—34no—
232C23×C3—34no—
233C23×C3—32no—
234C23×C3—42no—
235C23×C3—62no—
236C23×C3—62no—
237C23×C3—62no—
238C23×C3—62no—
239C23×C3—62no—
240C23×C3—64no—
241C23×C3—62no—
242C23×C3—62no—
243C23×C3—62no—
244C23×C3—64no—
245C23×C3—64no—
246C23×C3—62no—
247C23×C3—82no—
248C23×C3—124no—
249C23×C3D3×D411yescharacteristic
250C23×C3#23711yescharacteristic
251C23×C3C22×Dic311yescharacteristic
252C23×C3C2×S412yes—
253C2×A4—24no—
254C2×A4—21no—
255C2×A4—21no—
256C2×A4—21no—
257C2×A4—44no—
258C2×A4C22×C3×C411yescharacteristic
259C2×A4C2×C3×D411yescharacteristic
260C2×A4C3×#4111yescharacteristic
261C3×D4—38no—
262C3×D4—34no—
263C3×D4—44no—
264C3×D4—64no—
265C3×D4—84no—
266C3×D4SL(2,ℤ/4)14yesdirect product factor
267C2×C3×C4—31no—
268C2×C3×C4—32no—
269C2×C3×C4—32no—
270C2×C3×C4—32no—
271C2×C3×C4—32no—
272C2×C3×C4—32no—
273C2×C3×C4—41no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—62no—
281C2×C3×C4—62no—
282C2×C3×C4—62no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—62no—
288C2×C3×C4—62no—
289C2×C3×C4—61no—
290C2×C3×C4—62no—
291C2×C3×C4—81no—
292C2×C3×C4—124no—
293C2×C3×C4—128no—
294C2×C3×C4—124no—
295C2×C3×C4C2×S411yescharacteristic
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—84no—
300C2×C32—41no—
301C2×C32—41no—
302C2×C32—41no—
303C2×C32—84no—
304C24—22no—
305C24—22no—
306C24—32no—
307C24—32no—
308C24—34no—
309C24—62no—
310C24—62no—
311C24C2×C3×Dic314yescomplemented
312C24C22×C3×D311yescharacteristic
313C2×D4—32no—
314C2×D4—34no—
315C2×D4—34no—
316C2×D4—34no—
317C2×D4—38no—
318C2×D4—31no—
319C2×D4—62no—
320C2×D4—61no—
321C2×D4—64no—
322C2×D4C3×S411yescharacteristic
323C22×C4—31no—
324C22×C4—31no—
325C22×C4—32no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—31no—
329C22×C4—31no—
330C22×C4—32no—
331C22×C4—32no—
332C22×C4—64no—
333C22×C4—64no—
334C22×C4—64no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—122no—
338C22×C4—122no—
339C22×C4C2×C3×Dic312yescomplemented
340C4⋊C4—61no—
341C4⋊C4—62no—
342C4⋊C4—61no—
343C22⋊C4—32no—
344C22⋊C4—32no—
345C22⋊C4—64no—
346C22⋊C4—64no—
347C22⋊C4—62no—
348C22⋊C4—64no—
349C22⋊C4—62no—
350C22⋊C4—61no—
351C22⋊C4—62no—
352C22⋊C4—62no—
353C22⋊C4—61no—
354C22⋊C4—122no—
355C22⋊C4—122no—
356C42—61no—
357C42—62no—
358C42—61no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—31no—
362C22×C3—31no—
363C22×C3—31no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—31no—
367C22×C3—31no—
368C22×C3—31no—
369C22×C3—31no—
370C22×C3—34no—
371C22×C3—44no—
372C22×C3—41no—
373C22×C3—64no—
374C22×C3—64no—
375C22×C3—64no—
376C22×C3—64no—
377C22×C3—61no—
378C22×C3—64no—
379C22×C3—64no—
380C22×C3—62no—
381C22×C3—62no—
382C22×C3—82no—
383C22×C3—82no—
384C22×C3—84no—
385C22×C3—81no—
386C22×C3—122no—
387C22×C3—122no—
388C22×C3—162no—
389C22×C3—162no—
390C22×C3D4×Dic311yescharacteristic, complemented
391C22×C3C2×SL(2,ℤ/4)14yescomplemented
392C22×C3C22×S411yescharacteristic
393A4—21no—
394A4C3×C4×D411yescharacteristic, complemented
395C3×C4—32no—
396C3×C4—42no—
397C3×C4—62no—
398C3×C4—62no—
399C3×C4—82no—
400C3×C4—121no—
401C3×C4—122no—
402C3×C4—122no—
403C3×C4—121no—
404C3×C4C2×SL(2,ℤ/4)12yescomplemented
405Dic3—42no—
406Dic3—82no—
407Dic3—81no—
408C32—41noSylow
409C23—24no—
410C23—31no—
411C23—31no—
412C23—31no—
413C23—31no—
414C23—34no—
415C23—34no—
416C23—34no—
417C23—32no—
418C23—62no—
419C23—62no—
420C23—62no—
421C23—62no—
422C23—62no—
423C23—64no—
424C23—62no—
425C23—62no—
426C23—62no—
427C23—64no—
428C23—64no—
429C23—62no—
430C23—124no—
431C23C3×D3×D411yescharacteristic
432C23C3×#23711yescharacteristic
433C23C22×C3×Dic311yescharacteristic
434C23C2×C3×S412yes—
435D4—38no—
436D4—34no—
437D4—64no—
438D4C3×SL(2,ℤ/4)14yesdirect product factor
439C2×C4—31no—
440C2×C4—32no—
441C2×C4—32no—
442C2×C4—32no—
443C2×C4—32no—
444C2×C4—32no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—61no—
461C2×C4—62no—
462C2×C4—124no—
463C2×C4—128no—
464C2×C4—124no—
465C2×C4C2×C3×S411yescharacteristic
466C2×C3—24no—
467C2×C3—31no—
468C2×C3—31no—
469C2×C3—31no—
470C2×C3—31no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—64no—
475C2×C3—84no—
476C2×C3—81no—
477C2×C3—81no—
478C2×C3—81no—
479C2×C3—164no—
480C2×C3D4×S411yescharacteristic
481C2×C3#591411yescharacteristic
482C2×C3C22×SL(2,ℤ/4)11yescharacteristic
483C22—22no—
484C22—22no—
485C22—31no—
486C22—31no—
487C22—31no—
488C22—31no—
489C22—31no—
490C22—31no—
491C22—31no—
492C22—31no—
493C22—31no—
494C22—34no—
495C22—64no—
496C22—64no—
497C22—64no—
498C22—64no—
499C22—61no—
500C22—64no—
501C22—64no—
502C22—62no—
503C22—62no—
504C22—122no—
505C22—122no—
506C22C3×D4×Dic311yescharacteristic, complemented
507C22C2×C3×SL(2,ℤ/4)14yescomplemented
508C22C22×C3×S411yescharacteristic
509C4—32no—
510C4—62no—
511C4—62no—
512C4—122no—
513C4—121no—
514C4—122no—
515C4—121no—
516C4C2×C3×SL(2,ℤ/4)12yescomplemented
517C3—41no—
518C3—81no—
519C3D4×SL(2,ℤ/4)11yescharacteristic, direct product factor
520C2—24no—
521C2—31no—
522C2—31no—
523C2—31no—
524C2—31no—
525C2—64no—
526C2C3×D4×S411yescharacteristic
527C2C3×#591411yescharacteristic
528C2C22×C3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutatorC2×A4C22×C3×C4
FrattiniC22C22×C3×S4
FittingC23×C3×D4D3
socleC24×C3C22×D3
prime radicalC2×C3×D4×A4C2
cosocleC22×A4C23×C3
hypercenterC2×C3×D4S4
hypocenterA4C3×C4×D4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C22×C3×SL(2,ℤ/4)C2semidirect
3. C3×#5432C2semidirect
4. C3×C4⋊SL(2,ℤ/4)C2semidirect
5. C3×C4×SL(2,ℤ/4)C2semidirect
6. D4×SL(2,ℤ/4)C3direct (N × H)
9. C3×D4×A4C4semidirect
19. C22×SL(2,ℤ/4)C2×C3semidirect
23. #5432C2×C3semidirect
24. C4⋊SL(2,ℤ/4)C2×C3semidirect
25. C4×SL(2,ℤ/4)C2×C3semidirect
35. C22×C3×A4C2×C4semidirect
39. C3×C4×A4C2×C4semidirect
42. C3×SL(2,ℤ/4)D4direct (N × H)
54. C22×C3×D4Dic3semidirect
59. D4×A4C3×C4semidirect
118. C24×C3C2×Dic3semidirect
126. C22×A4C2×C3×C4semidirect
156. C22×C3×C4C2×Dic3semidirect
159. C4×A4C2×C3×C4semidirect
162. SL(2,ℤ/4)C3×D4direct (N × H)
189. C3×A4C4×D4semidirect
199. C22×D4C3×Dic3semidirect
266. C3×D4SL(2,ℤ/4)direct (N × H)
311. C24C2×C3×Dic3semidirect
339. C22×C4C2×C3×Dic3semidirect
390. C22×C3D4×Dic3semidirect
391. C22×C3C2×SL(2,ℤ/4)semidirect
394. A4C3×C4×D4semidirect
404. C3×C4C2×SL(2,ℤ/4)semidirect
438. D4C3×SL(2,ℤ/4)direct (N × H)
506. C22C3×D4×Dic3semidirect
507. C22C2×C3×SL(2,ℤ/4)semidirect
516. C4C2×C3×SL(2,ℤ/4)semidirect
519. C3D4×SL(2,ℤ/4)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×A42
C22×C3×SL(2,ℤ/4)2
C3×#54322
C3×C4⋊SL(2,ℤ/4)2
C3×C4×SL(2,ℤ/4)2
D4×SL(2,ℤ/4)3
C3×D4×C22⋊C43
C3×D4×Dic34