Finite Group Databasebeta

← C2×Dic3⋊SL(2,ℤ/4) #11922507

Subgroups — C2×Dic3⋊SL(2,ℤ/4) #11922507

Order 1152.

1052 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 143 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×Dic3×A4C211yesmaximal, characteristic
2C22×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic
3C22×C3×SL(2,ℤ/4)C211yesmaximal, characteristic
4Dic3⋊SL(2,ℤ/4)C214yesmaximal, direct product factor
5C2×C4⋊SL(2,ℤ/4)—31nomaximal
6C2×#4942—31nomaximal
7C23×C3×A4C2211yescharacteristic
8C2×Dic3×A4C414yescomplemented
9C2×Dic3×A4C2212yes—
10C2×C3⋊SL(2,ℤ/4)—22no—
11C2×C3⋊SL(2,ℤ/4)C2212yes—
12C2×C3×SL(2,ℤ/4)—22no—
13C2×C3×SL(2,ℤ/4)C2212yes—
14C2×Dic3⋊Dic3—41nomaximal
15C24×Dic3D311yescharacteristic
16C22×C4×A4—31no—
17C22×SL(2,ℤ/4)—31no—
18C22×SL(2,ℤ/4)—61no—
19C22×SL(2,ℤ/4)D311yescharacteristic
20C22×C3×C22⋊C4—31no—
21C22×#217—31no—
22C22×Dic3⋊C4—31no—
23C22×Dic3⋊C4—31no—
24C4⋊SL(2,ℤ/4)—34no—
25C2×C3⋊#95—31no—
26C2×C3⋊#95—31no—
27#4942—34no—
28#4942—34no—
29C22×C3×A4C2×C412yes—
30C22×C3×A4D412yes—
31C22×C3×A4Q812yes—
32C22×C3×A4C2311yescharacteristic
33C22×C3⋊Dic3—41no—
34C22×C3×Dic3—41no—
35C22×C3×Dic3—41no—
36Dic3×A4C2×C414yescomplemented
37C3⋊SL(2,ℤ/4)—22no—
38C3×SL(2,ℤ/4)—22no—
39Dic3⋊Dic3—44no—
40C2×#380—91noSylow
41C25×C3C2×D311yescharacteristic
42C23×A4—21no—
43C23×A4C2×D311yescharacteristic
44C23×C3×C4—31no—
45C23×C3×C4—31no—
46C23×Dic3—31no—
47C23×Dic3—32no—
48C23×Dic3—34no—
49C23×Dic3—31no—
50C23×Dic3—31no—
51C23×Dic3—31no—
52C23×Dic3Dic314yescomplemented
53C23×Dic3C2×D312yes—
54C2×C4×A4—34no—
55C2×C4×A4—32no—
56C2×SL(2,ℤ/4)—22no—
57C2×SL(2,ℤ/4)—32no—
58C2×SL(2,ℤ/4)—62no—
59C2×SL(2,ℤ/4)—64no—
60C2×SL(2,ℤ/4)—62no—
61C2×SL(2,ℤ/4)C2×D312yes—
62C2×C3×C22⋊C4—32no—
63C2×C3×C22⋊C4—32no—
64C2×C3×C22⋊C4—62no—
65C2×C3×C22⋊C4—62no—
66C2×#217—32no—
67C2×#217—32no—
68C2×#217—62no—
69C2×#217—62no—
70C2×C4⋊Dic3—121no—
71C2×Dic3⋊C4—34no—
72C2×Dic3⋊C4—34no—
73C2×Dic3⋊C4—61no—
74C2×Dic3⋊C4—64no—
75C2×Dic3⋊C4—62no—
76C2×Dic3⋊C4—61no—
77C3⋊#95—34no—
78C3⋊#95—34no—
79C23×C32—41no—
80C2×C3×A4C4⋊C414yescomplemented
81C2×C3×A4C22×C411yescharacteristic
82C2×C3×A4C2×D411yescharacteristic
83C2×C3×A4C2×Q811yescharacteristic
84C2×C3⋊Dic3—42no—
85C2×C3⋊Dic3—82no—
86C2×C3×Dic3—42no—
87C2×C3×Dic3—44no—
88C2×C3×Dic3—42no—
89C2×C3×Dic3—82no—
90C24×C4—31no—
91C22×C4⋊C4—91no—
92C22×C4⋊C4—91no—
93C22×C22⋊C4—31no—
94C22×C22⋊C4—91no—
95#380—94no—
96#380—94no—
97C2×#95—91no—
98C2×#95—91no—
99C24×C3—31no—
100C24×C3—32no—
101C24×C3—31no—
102C24×C3—32no—
103C24×C3—32no—
104C24×C3Dic612yes—
105C24×C3D1212yes—
106C24×C3C2×Dic312yes—
107C24×C3C22×D311yescharacteristic
108C22×A4—22no—
109C22×A4—22no—
110C22×A4—22no—
111C22×A4—21no—
112C22×A4Dic612yes—
113C22×A4C4×D312yes—
114C22×A4#6712yes—
115C22×A4C22×D311yescharacteristic
116C22×C3×C4—32no—
117C22×C3×C4—32no—
118C22×C3×C4—62no—
119C22×C3×C4—61no—
120C22×C3×C4—62no—
121C22×C3×C4—62no—
122C22×C3×C4—62no—
123C22×C3×C4—62no—
124C22×C3×C4—61no—
125C22×C3×C4—121no—
126C22×Dic3—31no—
127C22×Dic3—34no—
128C22×Dic3—34no—
129C22×Dic3—32no—
130C22×Dic3—32no—
131C22×Dic3—32no—
132C22×Dic3—34no—
133C22×Dic3—38no—
134C22×Dic3—34no—
135C22×Dic3—32no—
136C22×Dic3—32no—
137C22×Dic3—41no—
138C22×Dic3—64no—
139C22×Dic3—62no—
140C22×Dic3—61no—
141C22×Dic3—62no—
142C22×Dic3—61no—
143C22×Dic3—62no—
144C22×Dic3—62no—
145C22×Dic3—62no—
146C22×Dic3—62no—
147C22×Dic3—61no—
148C22×Dic3—121no—
149C22×Dic3—241no—
150C22×Dic3C2×Dic314yescomplemented
151C22×Dic3S411yescharacteristic
152C4×A4—34no—
153SL(2,ℤ/4)—22no—
154SL(2,ℤ/4)—64no—
155SL(2,ℤ/4)—62no—
156C3×C22⋊C4—64no—
157C3×C22⋊C4—62no—
158C3×C22⋊C4—62no—
159#217—64no—
160#217—62no—
161#217—62no—
162C4⋊Dic3—124no—
163Dic3⋊C4—64no—
164Dic3⋊C4—68no—
165Dic3⋊C4—64no—
166C22×C32—42no—
167C22×C32—42no—
168C22×C32—42no—
169C22×C32—41no—
170C3×A4C2×C4⋊C411yescharacteristic, complemented
171C3⋊Dic3—82no—
172C3×Dic3—44no—
173C3×Dic3—82no—
174C25D3211yescharacteristic
175C23×C4—31no—
176C23×C4—31no—
177C23×C4—34no—
178C23×C4—32no—
179C23×C4—91no—
180C23×C4—91no—
181C23×C4—92no—
182C23×C4—94no—
183C23×C4—91no—
184C23×C4—91no—
185C2×C4⋊C4—94no—
186C2×C4⋊C4—94no—
187C2×C4⋊C4—181no—
188C2×C4⋊C4—182no—
189C2×C4⋊C4—184no—
190C2×C4⋊C4—181no—
191C2×C22⋊C4—32no—
192C2×C22⋊C4—32no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—92no—
196C2×C22⋊C4—92no—
197C2×C22⋊C4—182no—
198C2×C22⋊C4—182no—
199#95—94no—
200#95—94no—
201C23×C3—32no—
202C23×C3—32no—
203C23×C3—32no—
204C23×C3—32no—
205C23×C3—32no—
206C23×C3—32no—
207C23×C3—31no—
208C23×C3—31no—
209C23×C3—32no—
210C23×C3—31no—
211C23×C3—32no—
212C23×C3—31no—
213C23×C3—32no—
214C23×C3—34no—
215C23×C3—34no—
216C23×C3—34no—
217C23×C3—31no—
218C23×C3—41no—
219C23×C3—61no—
220C23×C3—61no—
221C23×C3—61no—
222C23×C3—64no—
223C23×C3—81no—
224C23×C3C4⋊Dic314yescomplemented
225C23×C3C2×Dic611yescharacteristic
226C23×C3C2×D1211yescharacteristic
227C23×C3C22×Dic311yescharacteristic
228C23×C3C2×S411yescharacteristic
229C2×A4—24no—
230C2×A4—21no—
231C2×A4—21no—
232C2×A4—21no—
233C2×A4Dic3⋊C414yescomplemented
234C2×A4C2×Dic611yescharacteristic
235C2×A4C2×C4×D311yescharacteristic
236C2×A4C2×#6711yescharacteristic
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—122no—
244C2×C3×C4—124no—
245C2×C3×C4—124no—
246C2×C3×C4—122no—
247C2×C3×C4—122no—
248C2×Dic3—34no—
249C2×Dic3—32no—
250C2×Dic3—34no—
251C2×Dic3—34no—
252C2×Dic3—38no—
253C2×Dic3—34no—
254C2×Dic3—42no—
255C2×Dic3—62no—
256C2×Dic3—64no—
257C2×Dic3—62no—
258C2×Dic3—62no—
259C2×Dic3—62no—
260C2×Dic3—62no—
261C2×Dic3—62no—
262C2×Dic3—62no—
263C2×Dic3—82no—
264C2×Dic3—122no—
265C2×Dic3—124no—
266C2×Dic3—122no—
267C2×Dic3—122no—
268C2×Dic3—242no—
269C2×Dic3—242no—
270C2×Dic3—244no—
271C2×Dic3SL(2,ℤ/4)14yescomplemented
272C2×Dic3C2×S412yes—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—44no—
276C2×C32—41no—
277C24—31no—
278C24—31no—
279C24—32no—
280C24—32no—
281C24—32no—
282C24D3×Dic312yes—
283C24#62612yes—
284C24#62712yes—
285C24C2×D3211yescharacteristic
286C22×C4—31no—
287C22×C4—32no—
288C22×C4—32no—
289C22×C4—34no—
290C22×C4—62no—
291C22×C4—62no—
292C22×C4—62no—
293C22×C4—61no—
294C22×C4—62no—
295C22×C4—62no—
296C22×C4—61no—
297C22×C4—92no—
298C22×C4—92no—
299C22×C4—94no—
300C22×C4—94no—
301C22×C4—91no—
302C22×C4—92no—
303C22×C4—94no—
304C22×C4—98no—
305C22×C4—94no—
306C22×C4—92no—
307C22×C4—92no—
308C22×C4—184no—
309C22×C4—181no—
310C22×C4—182no—
311C22×C4—182no—
312C22×C4—182no—
313C22×C4—182no—
314C22×C4—181no—
315C22×C4—182no—
316C22×C4—182no—
317C22×C4—181no—
318C4⋊C4—184no—
319C4⋊C4—188no—
320C4⋊C4—184no—
321C22⋊C4—64no—
322C22⋊C4—62no—
323C22⋊C4—62no—
324C22⋊C4—184no—
325C22⋊C4—182no—
326C22⋊C4—182no—
327C22×C3—34no—
328C22×C3—31no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—32no—
334C22×C3—31no—
335C22×C3—31no—
336C22×C3—32no—
337C22×C3—34no—
338C22×C3—34no—
339C22×C3—34no—
340C22×C3—34no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—32no—
344C22×C3—41no—
345C22×C3—42no—
346C22×C3—42no—
347C22×C3—42no—
348C22×C3—62no—
349C22×C3—61no—
350C22×C3—62no—
351C22×C3—62no—
352C22×C3—81no—
353C22×C3—82no—
354C22×C3—82no—
355C22×C3—82no—
356C22×C3C2×C4⋊Dic311yescharacteristic, complemented
357C22×C3A4⋊Q812yes—
358C22×C3C4⋊S412yes—
359C22×C3C2×SL(2,ℤ/4)12yes—
360C22×C3C22×S411yescharacteristic
361A4—21no—
362A4C2×Dic3⋊C411yescharacteristic, complemented
363C3×C4—124no—
364C3×C4—122no—
365C3×C4—122no—
366Dic3—34no—
367Dic3—82no—
368Dic3—122no—
369Dic3—122no—
370Dic3—242no—
371Dic3—244no—
372Dic3C2×SL(2,ℤ/4)14yescomplemented
373C32—41noSylow
374C23—31no—
375C23—31no—
376C23—32no—
377C23—32no—
378C23—32no—
379C23—32no—
380C23—32no—
381C23—32no—
382C23—32no—
383C23—31no—
384C23—32no—
385C23—31no—
386C23—34no—
387C23—34no—
388C23—32no—
389C23—34no—
390C23—31no—
391C23—61no—
392C23—61no—
393C23—61no—
394C23—64no—
395C23Dic3⋊Dic314yescomplemented
396C23C2×D3×Dic311yescharacteristic
397C23C2×#62611yescharacteristic
398C23C2×#62711yescharacteristic
399C23D3×S411yescharacteristic
400C2×C4—32no—
401C2×C4—34no—
402C2×C4—62no—
403C2×C4—62no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—94no—
409C2×C4—94no—
410C2×C4—98no—
411C2×C4—92no—
412C2×C4—94no—
413C2×C4—94no—
414C2×C4—124no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4—182no—
418C2×C4—184no—
419C2×C4—182no—
420C2×C4—182no—
421C2×C4—182no—
422C2×C4—182no—
423C2×C4—182no—
424C2×C4—182no—
425C2×C4—364no—
426C2×C4—362no—
427C2×C4—362no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—34no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—41no—
434C2×C3—41no—
435C2×C3—41no—
436C2×C3—44no—
437C2×C3—81no—
438C2×C3—81no—
439C2×C3—81no—
440C2×C3—84no—
441C2×C3C4⋊SL(2,ℤ/4)14yescomplemented
442C2×C3C2×A4⋊Q811yescharacteristic
443C2×C3C2×C4⋊S411yescharacteristic
444C2×C3C22×SL(2,ℤ/4)11yescharacteristic
445C22—34no—
446C22—31no—
447C22—31no—
448C22—31no—
449C22—31no—
450C22—31no—
451C22—32no—
452C22—31no—
453C22—31no—
454C22—32no—
455C22—31no—
456C22—31no—
457C22—32no—
458C22—34no—
459C22—34no—
460C22—34no—
461C22—34no—
462C22—62no—
463C22—61no—
464C22—62no—
465C22—62no—
466C22C2×Dic3⋊Dic311yescharacteristic, complemented
467C22#6420412yes—
468C22#6420712yes—
469C22D3×SL(2,ℤ/4)12yes—
470C22C2×D3×S411yescharacteristic
471C4—34no—
472C4—94no—
473C4—122no—
474C4—122no—
475C4—362no—
476C4—362no—
477C3—41no—
478C3—81no—
479C3C2×C4⋊SL(2,ℤ/4)11yescharacteristic, complemented
480C2—31no—
481C2—31no—
482C2—34no—
483C2—31no—
484C2—31no—
485C2Dic3⋊SL(2,ℤ/4)14yesdirect product factor
486C2C2×#6420411yescharacteristic
487C2C2×#6420711yescharacteristic
488C2C2×D3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC23D3×S4
commutatorC2×C3×A4C22×C4
FrattiniC22C2×D3×S4
Fitting, socleC25×C3C2×D3
prime radicalC23×C3×A4C22
cosocleC22×C3×A4C23
hypocenterC3×A4C2×C4⋊C4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. Dic3⋊SL(2,ℤ/4)C2direct (N × H)
8. C2×Dic3×A4C4semidirect
36. Dic3×A4C2×C4semidirect
52. C23×Dic3Dic3semidirect
80. C2×C3×A4C4⋊C4semidirect
150. C22×Dic3C2×Dic3semidirect
170. C3×A4C2×C4⋊C4semidirect
224. C23×C3C4⋊Dic3semidirect
233. C2×A4Dic3⋊C4semidirect
271. C2×Dic3SL(2,ℤ/4)semidirect
356. C22×C3C2×C4⋊Dic3semidirect
362. A4C2×Dic3⋊C4semidirect
372. Dic3C2×SL(2,ℤ/4)semidirect
395. C23Dic3⋊Dic3semidirect
441. C2×C3C4⋊SL(2,ℤ/4)semidirect
466. C22C2×Dic3⋊Dic3semidirect
479. C3C2×C4⋊SL(2,ℤ/4)semidirect
485. C2Dic3⋊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
C22×Dic3×A42
C22×C3⋊SL(2,ℤ/4)2
C22×C3×SL(2,ℤ/4)2
Dic3⋊SL(2,ℤ/4)2
C2×C4⋊SL(2,ℤ/4)3
C2×#49423
C2×Dic3⋊Dic34