Finite Group Databasebeta

← C2×D3×C3⋊SL(2,ℤ/4) #13180836

Subgroups — C2×D3×C3⋊SL(2,ℤ/4) #13180836

Order 1728.

1483 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1483 conjugacy classes of subgroups — too many to draw. Only its 182 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×C3×D3×A4C211yesmaximal, characteristic
2C2×C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4D3×C3⋊SL(2,ℤ/4)C214yesmaximal, direct product factor
5C22×C3⋊SL(2,ℤ/4)—31nomaximal
6C2×D3×SL(2,ℤ/4)—33nomaximal
7C2×D3×#217—31nomaximal
8C22×C32×A4C2211yescharacteristic
9C2×C3×D3×A4C414yescomplemented
10C2×C3×D3×A4C2212yes—
11C2×D3×C3⋊Dic3—41nomaximal
12C3⋊(C3⋊SL(2,ℤ/4))C2212yescomplemented
13C3×C3⋊SL(2,ℤ/4)C2212yescomplemented
14C24×C3×D3D311yescharacteristic
15C23×C3×A4—31no—
16C22×D3×A4D313yes—
17C22×D3×Dic3—31no—
18C22×D3×Dic3—31no—
19C2×C3⋊SL(2,ℤ/4)—34no—
20C2×C3⋊SL(2,ℤ/4)—31no—
21C2×C3⋊SL(2,ℤ/4)—33no—
22C2×C3⋊SL(2,ℤ/4)—61no—
23C2×C3⋊SL(2,ℤ/4)—63no—
24C2×C3⋊SL(2,ℤ/4)D311yescharacteristic, direct product factor
25C2×C3×SL(2,ℤ/4)—33no—
26D3×SL(2,ℤ/4)—312no—
27C2×#3757—31no—
28C2×C3×#217—31no—
29D3×#217—34no—
30D3×#217—34no—
31C2×(C2×D3)⋊Dic3—31no—
32C2×(C2×D3)⋊Dic3—31no—
33C22×C32×D3—41no—
34C2×C32×A4C2×C412yescomplemented
35C2×C32×A4C2311yescharacteristic
36C3×D3×A4C2×C414yescomplemented
37C2×C3⋊(C3⋊Dic3)—41no—
38C2×C3×C3⋊Dic3—41no—
39D3×C3⋊Dic3—44no—
40C22×SL(2,ℤ/4)—93no—
41C22×#217—91no—
42C2×D3×C22⋊C4—91no—
43C24×C32C2×D311yescharacteristic
44C23×C3×D3—31no—
45C23×C3×D3—34no—
46C23×C3×D3—31no—
47C23×C3×D3—32no—
48C23×C3×D3Dic314yescomplemented
49C23×C3×D3C2×D312yes—
50C22×C3×A4—21no—
51C22×C3×A4—23no—
52C22×C3×A4—34no—
53C22×C3×A4—32no—
54C22×C3×A4C2×D311yescharacteristic
55C22×C3×A4C2×D313yes—
56C2×D3×A4Dic3112yescomplemented
57C2×D3×A4C2×D316yes—
58C22×C3⋊Dic3—31no—
59C22×C3⋊Dic3—31no—
60C22×C3⋊Dic3—121no—
61C22×C3×Dic3—31no—
62C22×C3×Dic3—31no—
63C2×D3×Dic3—34no—
64C2×D3×Dic3—34no—
65C2×D3×Dic3—64no—
66C2×D3×Dic3—61no—
67C2×D3×Dic3—62no—
68C2×D3×Dic3—61no—
69C2×D3×Dic3—123no—
70C3⋊SL(2,ℤ/4)—32no—
71C3⋊SL(2,ℤ/4)—36no—
72C3⋊SL(2,ℤ/4)—62no—
73C3⋊SL(2,ℤ/4)—66no—
74C3⋊SL(2,ℤ/4)C2×D312yesdirect product factor
75C3×SL(2,ℤ/4)—36no—
76#3757—32no—
77#3757—32no—
78C3×#217—32no—
79C3×#217—32no—
80(C2×D3)⋊Dic3—34no—
81(C2×D3)⋊Dic3—34no—
82C22×C33—41no—
83C2×C32×D3—44no—
84C2×C32×D3—42no—
85C32×A4C22×C411yescharacteristic, complemented
86C3⋊(C3⋊Dic3)—42no—
87C3×C3⋊Dic3—42no—
88C25×C3—31no—
89C24×D3C3⋊D311yescharacteristic
90C23×A4—33no—
91C23×Dic3—91no—
92C23×Dic3—91no—
93C22×C4×D3—91no—
94C22×C4×D3—91no—
95C2×SL(2,ℤ/4)—33no—
96C2×SL(2,ℤ/4)—912no—
97C2×SL(2,ℤ/4)—93no—
98C2×SL(2,ℤ/4)—183no—
99C2×C3×C22⋊C4—91no—
100C2×#217—31no—
101C2×#217—94no—
102C2×#217—94no—
103C2×#217—91no—
104C2×#217—91no—
105C2×#217—91no—
106C2×#217—91no—
107C2×#217—181no—
108C2×(C2×D3)⋊C4—91no—
109C2×(C2×D3)⋊C4—91no—
110D3×C22⋊C4—94no—
111D3×C22⋊C4—94no—
112C23×C32—31no—
113C23×C32—32no—
114C23×C32—31no—
115C23×C32—121no—
116C23×C32C2×Dic312yescomplemented
117C23×C32C22×D311yescharacteristic
118C22×C3×D3—31no—
119C22×C3×D3—34no—
120C22×C3×D3—34no—
121C22×C3×D3—32no—
122C22×C3×D3—32no—
123C22×C3×D3—34no—
124C22×C3×D3—38no—
125C22×C3×D3—34no—
126C22×C3×D3—32no—
127C22×C3×D3—43no—
128C22×C3×D3—64no—
129C22×C3×D3—62no—
130C22×C3×D3—61no—
131C22×C3×D3C2×Dic314yescomplemented
132C22×C3×D3S411yescharacteristic
133C2×C3×A4—22no—
134C2×C3×A4—21no—
135C2×C3×A4—26no—
136C2×C3×A4—23no—
137C2×C3×A4—34no—
138C2×C3×A4C4×D312yescomplemented
139C2×C3×A4C2×Dic316yescomplemented
140C2×C3×A4C22×D313yes—
141C2×C3×A4C22×D311yescharacteristic
142D3×A4C2×Dic3112yescomplemented
143C2×C3⋊Dic3—32no—
144C2×C3⋊Dic3—32no—
145C2×C3⋊Dic3—41no—
146C2×C3⋊Dic3—62no—
147C2×C3⋊Dic3—61no—
148C2×C3⋊Dic3—61no—
149C2×C3⋊Dic3—124no—
150C2×C3⋊Dic3—121no—
151C2×C3⋊Dic3—123no—
152C2×C3⋊Dic3—241no—
153C2×C3⋊Dic3—243no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—123no—
160D3×Dic3—68no—
161D3×Dic3—64no—
162D3×Dic3—64no—
163D3×Dic3—1212no—
164C22×C22⋊C4—271noSylow
165C2×C33—42no—
166C2×C33—41no—
167C32×D3—44no—
168C24×C3—21no—
169C24×C3—34no—
170C24×C3—32no—
171C24×C3—91no—
172C24×C3—94no—
173C24×C3—91no—
174C24×C3—92no—
175C24×C3D3211yescharacteristic
176C24×C3C2×C3⋊D311yescharacteristic
177C23×D3—31no—
178C23×D3—32no—
179C23×D3—31no—
180C23×D3—34no—
181C23×D3C3⋊Dic314yescomplemented
182C23×D3C2×C3⋊D312yes—
183C22×A4—23no—
184C22×A4—312no—
185C22×A4—36no—
186C22×A4D3213yes—
187C22×C3×C4—91no—
188C22×C3×C4—91no—
189C22×Dic3—31no—
190C22×Dic3—31no—
191C22×Dic3—94no—
192C22×Dic3—94no—
193C22×Dic3—91no—
194C22×Dic3—91no—
195C22×Dic3—91no—
196C22×Dic3—91no—
197C22×Dic3—181no—
198C22×Dic3—181no—
199C22×Dic3—181no—
200C22×Dic3—182no—
201C22×Dic3—184no—
202C22×Dic3—181no—
203C22×Dic3—363no—
204C2×C4×D3—94no—
205C2×C4×D3—94no—
206C2×C4×D3—184no—
207C2×C4×D3—181no—
208C2×C4×D3—182no—
209C2×C4×D3—181no—
210SL(2,ℤ/4)—36no—
211SL(2,ℤ/4)—96no—
212SL(2,ℤ/4)—186no—
213C3×C22⋊C4—92no—
214C3×C22⋊C4—92no—
215#217—32no—
216#217—32no—
217#217—94no—
218#217—94no—
219#217—92no—
220#217—92no—
221#217—92no—
222#217—92no—
223#217—182no—
224#217—182no—
225(C2×D3)⋊C4—94no—
226(C2×D3)⋊C4—94no—
227C22×C32—32no—
228C22×C32—31no—
229C22×C32—32no—
230C22×C32—32no—
231C22×C32—31no—
232C22×C32—31no—
233C22×C32—41no—
234C22×C32—43no—
235C22×C32—61no—
236C22×C32—83no—
237C22×C32—81no—
238C22×C32—122no—
239C22×C32—124no—
240C22×C32C22×Dic311yescharacteristic, complemented
241C22×C32C2×S411yescharacteristic
242C2×C3×D3—34no—
243C2×C3×D3—32no—
244C2×C3×D3—34no—
245C2×C3×D3—34no—
246C2×C3×D3—38no—
247C2×C3×D3—34no—
248C2×C3×D3—46no—
249C2×C3×D3—412no—
250C2×C3×D3—62no—
251C2×C3×D3—64no—
252C2×C3×D3SL(2,ℤ/4)14yescomplemented
253C2×C3×D3C2×S412yes—
254C3×A4—21no—
255C3×A4—23no—
256C3×A4C2×C4×D311yescharacteristic, complemented
257C3×A4C22×Dic313yescomplemented
258C3⋊Dic3—42no—
259C3⋊Dic3—62no—
260C3⋊Dic3—62no—
261C3⋊Dic3—122no—
262C3⋊Dic3—126no—
263C3⋊Dic3—242no—
264C3⋊Dic3—246no—
265C3×Dic3—62no—
266C3×Dic3—62no—
267C3×Dic3—126no—
268C25—31no—
269C23×C4—271no—
270C23×C4—271no—
271C2×C22⋊C4—91no—
272C2×C22⋊C4—274no—
273C2×C22⋊C4—274no—
274C2×C22⋊C4—271no—
275C2×C22⋊C4—271no—
276C2×C22⋊C4—271no—
277C33—41noSylow
278C23×C3—22no—
279C23×C3—21no—
280C23×C3—31no—
281C23×C3—31no—
282C23×C3—31no—
283C23×C3—31no—
284C23×C3—31no—
285C23×C3—32no—
286C23×C3—32no—
287C23×C3—34no—
288C23×C3—61no—
289C23×C3—61no—
290C23×C3—62no—
291C23×C3—92no—
292C23×C3—92no—
293C23×C3—94no—
294C23×C3—94no—
295C23×C3—92no—
296C23×C3—98no—
297C23×C3—94no—
298C23×C3—94no—
299C23×C3—91no—
300C23×C3—123no—
301C23×C3—181no—
302C23×C3—182no—
303C23×C3—184no—
304C23×C3D3×Dic312yescomplemented
305C23×C3C2×C3⋊Dic312yescomplemented
306C23×C3C2×D3211yescharacteristic
307C23×C3C22×C3⋊D311yescharacteristic
308C22×D3—31no—
309C22×D3—32no—
310C22×D3—32no—
311C22×D3—34no—
312C22×D3—34no—
313C22×D3—32no—
314C22×D3—34no—
315C22×D3—38no—
316C22×D3—34no—
317C22×D3—64no—
318C22×D3—62no—
319C22×D3—61no—
320C22×D3C2×C3⋊Dic314yescomplemented
321C22×D3C3⋊S411yescharacteristic
322C2×A4—26no—
323C2×A4—23no—
324C2×A4—312no—
325C2×A4D3×Dic316yescomplemented
326C2×A4C2×D3213yes—
327C2×C3×C4—92no—
328C2×C3×C4—92no—
329C2×C3×C4—182no—
330C2×C3×C4—181no—
331C2×C3×C4—181no—
332C2×Dic3—32no—
333C2×Dic3—32no—
334C2×Dic3—62no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—92no—
338C2×Dic3—92no—
339C2×Dic3—92no—
340C2×Dic3—92no—
341C2×Dic3—123no—
342C2×Dic3—182no—
343C2×Dic3—182no—
344C2×Dic3—184no—
345C2×Dic3—188no—
346C2×Dic3—184no—
347C2×Dic3—182no—
348C2×Dic3—181no—
349C2×Dic3—182no—
350C2×Dic3—181no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—3612no—
354C2×Dic3—361no—
355C2×Dic3—363no—
356C2×Dic3—361no—
357C2×Dic3—362no—
358C2×Dic3—723no—
359C4×D3—188no—
360C4×D3—184no—
361C4×D3—184no—
362C2×C32—31no—
363C2×C32—31no—
364C2×C32—32no—
365C2×C32—41no—
366C2×C32—42no—
367C2×C32—46no—
368C2×C32—43no—
369C2×C32—83no—
370C2×C32—86no—
371C2×C32—81no—
372C2×C32—82no—
373C2×C32—124no—
374C2×C32C2×SL(2,ℤ/4)12yescomplemented
375C2×C32C22×S411yescharacteristic
376C3×D3—34no—
377C3×D3—412no—
378C3×D3C2×SL(2,ℤ/4)14yescomplemented
379C24—34no—
380C24—32no—
381C24—91no—
382C24—91no—
383C24—92no—
384C24—94no—
385C24D3×C3⋊D311yescharacteristic
386C22×C4—91no—
387C22×C4—91no—
388C22×C4—274no—
389C22×C4—274no—
390C22×C4—271no—
391C22×C4—271no—
392C22×C4—541no—
393C22×C4—542no—
394C22×C4—544no—
395C22×C4—541no—
396C22⋊C4—92no—
397C22⋊C4—92no—
398C22⋊C4—274no—
399C22⋊C4—274no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—32no—
405C22×C3—31no—
406C22×C3—32no—
407C22×C3—31no—
408C22×C3—32no—
409C22×C3—32no—
410C22×C3—31no—
411C22×C3—31no—
412C22×C3—32no—
413C22×C3—31no—
414C22×C3—31no—
415C22×C3—32no—
416C22×C3—32no—
417C22×C3—34no—
418C22×C3—43no—
419C22×C3—61no—
420C22×C3—62no—
421C22×C3—61no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—61no—
425C22×C3—61no—
426C22×C3—61no—
427C22×C3—83no—
428C22×C3—94no—
429C22×C3—94no—
430C22×C3—98no—
431C22×C3—94no—
432C22×C3—92no—
433C22×C3—94no—
434C22×C3—126no—
435C22×C3—1212no—
436C22×C3—121no—
437C22×C3—182no—
438C22×C3—184no—
439C22×C3C2×D3×Dic311yescharacteristic, complemented
440C22×C3C22×C3⋊Dic311yescharacteristic, complemented
441C22×C3D3×S411yescharacteristic
442C22×C3C2×C3⋊S411yescharacteristic
443C2×D3—34no—
444C2×D3—32no—
445C2×D3—34no—
446C2×D3—34no—
447C2×D3—38no—
448C2×D3—34no—
449C2×D3—62no—
450C2×D3—64no—
451C2×D3C3⋊SL(2,ℤ/4)14yesdirect product factor
452C2×D3C2×C3⋊S412yes—
453A4—23no—
454A4C2×D3×Dic313yescomplemented
455C3×C4—182no—
456C3×C4—182no—
457Dic3—62no—
458Dic3—62no—
459Dic3—126no—
460Dic3—182no—
461Dic3—182no—
462Dic3—182no—
463Dic3—182no—
464Dic3—362no—
465Dic3—366no—
466Dic3—362no—
467Dic3—726no—
468C32—41no—
469C32—43no—
470C32—83no—
471C32—81no—
472C32C22×SL(2,ℤ/4)11yescharacteristic, complemented
473C23—31no—
474C23—31no—
475C23—31no—
476C23—32no—
477C23—34no—
478C23—92no—
479C23—94no—
480C23—92no—
481C23—94no—
482C23—92no—
483C23—98no—
484C23—94no—
485C23—94no—
486C23—91no—
487C23—182no—
488C23—181no—
489C23—184no—
490C23D3×C3⋊Dic312yescomplemented
491C23C2×D3×C3⋊D311yescharacteristic
492C2×C4—92no—
493C2×C4—92no—
494C2×C4—182no—
495C2×C4—181no—
496C2×C4—181no—
497C2×C4—272no—
498C2×C4—272no—
499C2×C4—544no—
500C2×C4—548no—
501C2×C4—544no—
502C2×C4—542no—
503C2×C4—541no—
504C2×C4—541no—
505C2×C3—22no—
506C2×C3—21no—
507C2×C3—31no—
508C2×C3—31no—
509C2×C3—32no—
510C2×C3—34no—
511C2×C3—31no—
512C2×C3—31no—
513C2×C3—32no—
514C2×C3—46no—
515C2×C3—43no—
516C2×C3—61no—
517C2×C3—61no—
518C2×C3—62no—
519C2×C3—86no—
520C2×C3—83no—
521C2×C3—94no—
522C2×C3—1212no—
523C2×C3D3×SL(2,ℤ/4)12yescomplemented
524C2×C3C2×C3⋊SL(2,ℤ/4)12yescomplemented
525C2×C3C2×D3×S411yescharacteristic
526C2×C3C22×C3⋊S411yescharacteristic
527D3—34no—
528D3C2×C3⋊SL(2,ℤ/4)14yesdirect product factor
529C22—32no—
530C22—31no—
531C22—32no—
532C22—32no—
533C22—34no—
534C22—31no—
535C22—31no—
536C22—32no—
537C22—61no—
538C22—94no—
539C22—94no—
540C22—98no—
541C22—94no—
542C22—94no—
543C22—92no—
544C22—182no—
545C22—184no—
546C22C2×D3×C3⋊Dic311yescharacteristic, complemented
547C22D3×C3⋊S411yescharacteristic
548C4—182no—
549C4—182no—
550C4—542no—
551C4—542no—
552C3—21no—
553C3—43no—
554C3—83no—
555C3C2×D3×SL(2,ℤ/4)11yescharacteristic, complemented
556C3C22×C3⋊SL(2,ℤ/4)11yescharacteristic, complemented
557C2—31no—
558C2—31no—
559C2—32no—
560C2—34no—
561C2—94no—
562C2D3×C3⋊SL(2,ℤ/4)12yesdirect product factor
563C2C2×D3×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC22D3×C3⋊S4
commutator, hypocenterC32×A4C22×C4
FrattiniC2C2×D3×C3⋊S4
Fitting, socleC24×C32C2×D3
prime radicalC22×C3×D3×A4C2
cosocleC2×C32×A4C23
LCSG ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(C3⋊SL(2,ℤ/4))C2semidirect
3. C2×C3×C3⋊SL(2,ℤ/4)C2semidirect
4. D3×C3⋊SL(2,ℤ/4)C2direct (N × H)
9. C2×C3×D3×A4C4semidirect
12. C3⋊(C3⋊SL(2,ℤ/4))C22semidirect
13. C3×C3⋊SL(2,ℤ/4)C22semidirect
24. C2×C3⋊SL(2,ℤ/4)D3direct (N × H)
34. C2×C32×A4C2×C4semidirect
36. C3×D3×A4C2×C4semidirect
48. C23×C3×D3Dic3semidirect
56. C2×D3×A4Dic3semidirect
74. C3⋊SL(2,ℤ/4)C2×D3direct (N × H)
85. C32×A4C22×C4semidirect
116. C23×C32C2×Dic3semidirect
131. C22×C3×D3C2×Dic3semidirect
138. C2×C3×A4C4×D3semidirect
139. C2×C3×A4C2×Dic3semidirect
142. D3×A4C2×Dic3semidirect
181. C23×D3C3⋊Dic3semidirect
240. C22×C32C22×Dic3semidirect
252. C2×C3×D3SL(2,ℤ/4)semidirect
256. C3×A4C2×C4×D3semidirect
257. C3×A4C22×Dic3semidirect
304. C23×C3D3×Dic3semidirect
305. C23×C3C2×C3⋊Dic3semidirect
320. C22×D3C2×C3⋊Dic3semidirect
325. C2×A4D3×Dic3semidirect
374. C2×C32C2×SL(2,ℤ/4)semidirect
378. C3×D3C2×SL(2,ℤ/4)semidirect
439. C22×C3C2×D3×Dic3semidirect
440. C22×C3C22×C3⋊Dic3semidirect
451. C2×D3C3⋊SL(2,ℤ/4)direct (N × H)
454. A4C2×D3×Dic3semidirect
472. C32C22×SL(2,ℤ/4)semidirect
490. C23D3×C3⋊Dic3semidirect
523. C2×C3D3×SL(2,ℤ/4)semidirect
524. C2×C3C2×C3⋊SL(2,ℤ/4)semidirect
528. D3C2×C3⋊SL(2,ℤ/4)direct (N × H)
546. C22C2×D3×C3⋊Dic3semidirect
555. C3C2×D3×SL(2,ℤ/4)semidirect
556. C3C22×C3⋊SL(2,ℤ/4)semidirect
562. C2D3×C3⋊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×C3×D3×A42
C2×C3⋊(C3⋊SL(2,ℤ/4))2
C2×C3×C3⋊SL(2,ℤ/4)2
D3×C3⋊SL(2,ℤ/4)2
C22×C3⋊SL(2,ℤ/4)3
C2×D3×SL(2,ℤ/4)3
C2×D3×#2173
C2×D3×C3⋊Dic34