Finite Group Databasebeta

← C2×Dic3×C3⋊S4 #13180831

Subgroups — C2×Dic3×C3⋊S4 #13180831

Order 1728.

1307 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1307 conjugacy classes of subgroups — too many to draw. Only its 134 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×C3⋊S4C211yesmaximal, characteristic
2C2×C3×Dic3×A4C211yesmaximal, characteristic, complemented
3C2×C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
4Dic3×C3⋊S4C214yesmaximal, direct product factor
5C2×C4×C3⋊S4—31nomaximal
6C2×Dic3×S4—33nomaximal
7C2×Dic3×#67—31nomaximal
8C22×C32×A4C2211yescharacteristic
9C2×C3×C3⋊S4C414yescomplemented
10C2×C3×C3⋊S4C2212yes—
11C2×Dic3×C3⋊D3—41nomaximal
12C3×Dic3×A4C2212yescomplemented
13C3⋊(C3⋊SL(2,ℤ/4))C2212yescomplemented
14C22×C3⋊S4D311yescharacteristic
15C22×C3×S4—33no—
16C22×C3×#67—31no—
17C23×C3×Dic3D311yescharacteristic, complemented
18C2×C3×C4×A4—31no—
19C22×D3×Dic3—31no—
20C2×Dic3×A4D313yescomplemented
21C2×C3⋊SL(2,ℤ/4)—31no—
22C2×C3⋊SL(2,ℤ/4)—33no—
23C2×C3⋊SL(2,ℤ/4)—61no—
24C2×C3⋊SL(2,ℤ/4)—63no—
25C4×C3⋊S4—34no—
26Dic3×S4—312no—
27C2×#3757—31no—
28Dic3×#67—34no—
29Dic3×#67—34no—
30C2×Dic3⋊Dic3—31no—
31C2×(C2×D3)⋊Dic3—31no—
32C2×Dic32—31no—
33C22×C3×C3⋊D3—41no—
34C2×C32×A4C2×C412yescomplemented
35C2×C32×A4C2311yescharacteristic
36C3×C3⋊S4C2×C414yescomplemented
37C2×C3⋊(C3⋊Dic3)—41no—
38C2×C32×Dic3—41no—
39Dic3×C3⋊D3—44no—
40C2×C4×S4—93no—
41C2×D4×Dic3—91no—
42C2×C4×#67—91no—
43C24×C32C2×D311yescharacteristic
44C23×C3×D3—31no—
45C22×C3×A4—21no—
46C22×C3×A4—23no—
47C22×C3×A4C2×D313yes—
48C22×C3×A4C2×D311yescharacteristic
49C2×C3⋊S4Dic314yesdirect product factor
50C2×C3⋊S4C2×D312yes—
51C2×C3×S4—312no—
52C2×C3×S4—36no—
53C22×C3⋊Dic3—31no—
54C22×C3⋊Dic3—31no—
55C2×C4×C3⋊D3—121no—
56C2×C3×#67—34no—
57C2×C3×#67—32no—
58C2×C3×#67—34no—
59C2×C3×#67—32no—
60C22×C3×Dic3—31no—
61C22×C3×Dic3—31no—
62C22×C3×Dic3—31no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3C2×D312yescomplemented
65C3×C4×A4—32no—
66C2×D3×Dic3—34no—
67C2×D3×Dic3—61no—
68C2×D3×Dic3—62no—
69C2×D3×Dic3—61no—
70C2×D3×Dic3—123no—
71Dic3×A4C2×D316yescomplemented
72C3⋊SL(2,ℤ/4)—32no—
73C3⋊SL(2,ℤ/4)—36no—
74C3⋊SL(2,ℤ/4)—62no—
75C3⋊SL(2,ℤ/4)—66no—
76#3757—32no—
77#3757—32no—
78Dic3⋊Dic3—34no—
79(C2×D3)⋊Dic3—34no—
80Dic32—34no—
81C22×C33—41no—
82C2×C3×C3⋊D3—44no—
83C2×C3×C3⋊D3—42no—
84C32×A4C22×C411yescharacteristic, complemented
85C3⋊(C3⋊Dic3)—42no—
86C32×Dic3—42no—
87C22×S4—33no—
88C22×C3×D4—91no—
89C23×C3×C4—31no—
90C22×#67—31no—
91C23×Dic3—91no—
92C23×Dic3C3⋊D311yescharacteristic, complemented
93C22×C4×D3—91no—
94C2×C4×A4—33no—
95C2×SL(2,ℤ/4)—93no—
96C2×SL(2,ℤ/4)—183no—
97C4×S4—912no—
98C2×#217—91no—
99C2×#217—91no—
100C2×#217—91no—
101C2×#217—181no—
102D4×Dic3—94no—
103D4×Dic3—94no—
104C4×#67—94no—
105C4×#67—94no—
106C2×(C2×D3)⋊C4—91no—
107C2×C4⋊Dic3—91no—
108C2×Dic3⋊C4—91no—
109C2×C4×Dic3—91no—
110C2×C4×Dic3—91no—
111C23×C32—31no—
112C23×C32—31no—
113C23×C32—32no—
114C23×C32C4×D312yescomplemented
115C23×C32C22×D311yescharacteristic
116C22×C3⋊D3—41no—
117C22×C3×D3—32no—
118C22×C3×D3—34no—
119C22×C3×D3—62no—
120C22×C3×D3—61no—
121C22×C3×D3—61no—
122C22×C3×D3—123no—
123C2×C3×A4—22no—
124C2×C3×A4—21no—
125C2×C3×A4—26no—
126C2×C3×A4—23no—
127C2×C3×A4C4×D316yescomplemented
128C2×C3×A4C2×Dic312yescomplemented
129C2×C3×A4C22×D313yes—
130C2×C3×A4C22×D311yescharacteristic
131C3⋊S4C2×Dic314yesdirect product factor
132C3×S4—312no—
133C2×C32×C4—121no—
134C2×C3⋊Dic3—32no—
135C2×C3⋊Dic3—32no—
136C2×C3⋊Dic3—61no—
137C2×C3⋊Dic3—62no—
138C2×C3⋊Dic3—61no—
139C2×C3⋊Dic3—121no—
140C2×C3⋊Dic3—123no—
141C2×C3⋊Dic3—241no—
142C2×C3⋊Dic3—243no—
143C4×C3⋊D3—124no—
144C3×#67—38no—
145C3×#67—34no—
146C3×#67—34no—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—31no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—43no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3S411yescharacteristic, complemented
156D3×Dic3—64no—
157D3×Dic3—64no—
158D3×Dic3—1212no—
159C2×C4×D4—271noSylow
160C2×C33—42no—
161C2×C33—41no—
162C3×C3⋊D3—44no—
163C24×C3—21no—
164C24×C3—91no—
165C24×C3D3211yescharacteristic
166C24×C3C2×C3⋊D311yescharacteristic
167C23×D3—31no—
168C22×A4—23no—
169C22×A4D3213yes—
170C2×S4—312no—
171C2×S4—36no—
172C2×C3×D4—92no—
173C2×C3×D4—94no—
174C2×C3×D4—94no—
175C2×C3×D4—92no—
176C22×C3×C4—32no—
177C22×C3×C4—91no—
178C22×C3×C4—91no—
179C22×C3×C4—91no—
180C22×C3×C4—92no—
181C2×#67—32no—
182C2×#67—34no—
183C2×#67—34no—
184C2×#67—32no—
185C22×Dic3—31no—
186C22×Dic3—31no—
187C22×Dic3—31no—
188C22×Dic3—32no—
189C22×Dic3—94no—
190C22×Dic3—91no—
191C22×Dic3—91no—
192C22×Dic3—91no—
193C22×Dic3—91no—
194C22×Dic3—181no—
195C22×Dic3—181no—
196C22×Dic3—181no—
197C22×Dic3—182no—
198C22×Dic3—181no—
199C22×Dic3C2×C3⋊D312yescomplemented
200C2×C4×D3—94no—
201C2×C4×D3—181no—
202C2×C4×D3—182no—
203C2×C4×D3—181no—
204C2×C4×D3—363no—
205C4×A4—36no—
206SL(2,ℤ/4)—96no—
207SL(2,ℤ/4)—186no—
208#217—94no—
209#217—92no—
210#217—92no—
211#217—92no—
212#217—92no—
213#217—182no—
214#217—182no—
215(C2×D3)⋊C4—94no—
216C4⋊Dic3—94no—
217Dic3⋊C4—94no—
218C4×Dic3—94no—
219C4×Dic3—94no—
220C22×C32—32no—
221C22×C32—31no—
222C22×C32—32no—
223C22×C32—32no—
224C22×C32—31no—
225C22×C32—31no—
226C22×C32—41no—
227C22×C32—43no—
228C22×C32—61no—
229C22×C32—83no—
230C22×C32—81no—
231C22×C32C2×C4×D311yescharacteristic, complemented
232C22×C32C2×S411yescharacteristic
233C2×C3⋊D3—44no—
234C2×C3⋊D3—42no—
235C2×C3×D3—34no—
236C2×C3×D3—62no—
237C2×C3×D3—64no—
238C2×C3×D3—64no—
239C2×C3×D3—62no—
240C2×C3×D3—126no—
241C2×C3×D3—1212no—
242C3×A4—21no—
243C3×A4—23no—
244C3×A4C2×C4×D313yescomplemented
245C3×A4C22×Dic311yescharacteristic, complemented
246C32×C4—122no—
247C3⋊Dic3—62no—
248C3⋊Dic3—62no—
249C3⋊Dic3—122no—
250C3⋊Dic3—126no—
251C3⋊Dic3—242no—
252C3⋊Dic3—246no—
253C3×Dic3—34no—
254C3×Dic3—32no—
255C3×Dic3—46no—
256C3×Dic3C2×S412yescomplemented
257C22×D4—91no—
258C23×C4—31no—
259C23×C4—271no—
260C4×D4—274no—
261C4×D4—274no—
262C2×C4⋊C4—271no—
263C2×C22⋊C4—271no—
264C2×C22⋊C4—271no—
265C2×C42—271no—
266C33—41noSylow
267C23×C3—22no—
268C23×C3—21no—
269C23×C3—31no—
270C23×C3—31no—
271C23×C3—31no—
272C23×C3—31no—
273C23×C3—32no—
274C23×C3—32no—
275C23×C3—61no—
276C23×C3—61no—
277C23×C3—62no—
278C23×C3—92no—
279C23×C3—94no—
280C23×C3—181no—
281C23×C3—182no—
282C23×C3—181no—
283C23×C3D3×Dic312yescomplemented
284C23×C3C4×C3⋊D312yescomplemented
285C23×C3C2×D3211yescharacteristic
286C23×C3C22×C3⋊D311yescharacteristic
287C22×D3—32no—
288C22×D3—34no—
289C22×D3—61no—
290C22×D3—62no—
291C22×D3—61no—
292C22×D3—123no—
293C2×A4—26no—
294C2×A4—23no—
295C2×A4D3×Dic316yescomplemented
296C2×A4C2×D3213yes—
297S4—312no—
298C3×D4—98no—
299C3×D4—94no—
300C3×D4—94no—
301C2×C3×C4—31no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4—92no—
305C2×C3×C4—94no—
306C2×C3×C4—91no—
307C2×C3×C4—92no—
308C2×C3×C4—123no—
309C2×C3×C4—181no—
310#67—38no—
311#67—34no—
312#67—34no—
313C2×Dic3—32no—
314C2×Dic3—34no—
315C2×Dic3—31no—
316C2×Dic3—32no—
317C2×Dic3—32no—
318C2×Dic3—32no—
319C2×Dic3—61no—
320C2×Dic3—92no—
321C2×Dic3—92no—
322C2×Dic3—92no—
323C2×Dic3—92no—
324C2×Dic3—182no—
325C2×Dic3—182no—
326C2×Dic3—184no—
327C2×Dic3—181no—
328C2×Dic3—182no—
329C2×Dic3—182no—
330C2×Dic3—181no—
331C2×Dic3—181no—
332C2×Dic3—181no—
333C2×Dic3—184no—
334C2×Dic3—361no—
335C2×Dic3—363no—
336C2×Dic3—361no—
337C2×Dic3—362no—
338C2×Dic3—723no—
339C2×Dic3C3⋊S411yescharacteristic, direct product factor
340C4×D3—184no—
341C4×D3—184no—
342C4×D3—3612no—
343C2×C32—31no—
344C2×C32—31no—
345C2×C32—32no—
346C2×C32—41no—
347C2×C32—42no—
348C2×C32—46no—
349C2×C32—43no—
350C2×C32—83no—
351C2×C32—86no—
352C2×C32—81no—
353C2×C32—82no—
354C2×C32C4×S412yescomplemented
355C2×C32C22×S411yescharacteristic
356C3⋊D3—44no—
357C3×D3—64no—
358C3×D3—1212no—
359C24—91no—
360C24D3×C3⋊D311yescharacteristic
361C2×D4—92no—
362C2×D4—94no—
363C2×D4—94no—
364C2×D4—92no—
365C22×C4—32no—
366C22×C4—91no—
367C22×C4—91no—
368C22×C4—91no—
369C22×C4—92no—
370C22×C4—274no—
371C22×C4—271no—
372C22×C4—271no—
373C22×C4—541no—
374C22×C4—542no—
375C22×C4—541no—
376C4⋊C4—274no—
377C22⋊C4—274no—
378C22⋊C4—272no—
379C22⋊C4—272no—
380C42—274no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—32no—
384C22×C3—31no—
385C22×C3—32no—
386C22×C3—31no—
387C22×C3—32no—
388C22×C3—32no—
389C22×C3—31no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—32no—
394C22×C3—32no—
395C22×C3—43no—
396C22×C3—61no—
397C22×C3—62no—
398C22×C3—61no—
399C22×C3—62no—
400C22×C3—62no—
401C22×C3—61no—
402C22×C3—61no—
403C22×C3—61no—
404C22×C3—83no—
405C22×C3—94no—
406C22×C3—121no—
407C22×C3—182no—
408C22×C3—184no—
409C22×C3—184no—
410C22×C3—182no—
411C22×C3C2×D3×Dic311yescharacteristic, complemented
412C22×C3C2×C4×C3⋊D311yescharacteristic, complemented
413C22×C3D3×S411yescharacteristic
414C22×C3C2×C3⋊S411yescharacteristic
415C2×D3—34no—
416C2×D3—62no—
417C2×D3—64no—
418C2×D3—62no—
419C2×D3—64no—
420C2×D3—126no—
421C2×D3—1212no—
422A4—23no—
423A4C2×D3×Dic313yescomplemented
424C3×C4—32no—
425C3×C4—94no—
426C3×C4—92no—
427C3×C4—126no—
428Dic3—34no—
429Dic3—32no—
430Dic3—182no—
431Dic3—182no—
432Dic3—182no—
433Dic3—182no—
434Dic3—362no—
435Dic3—366no—
436Dic3—362no—
437Dic3—726no—
438Dic3C2×C3⋊S412yesdirect product factor
439C32—41no—
440C32—43no—
441C32—83no—
442C32—81no—
443C32C2×C4×S411yescharacteristic, complemented
444C23—31no—
445C23—31no—
446C23—32no—
447C23—92no—
448C23—94no—
449C23—181no—
450C23—182no—
451C23—181no—
452C23Dic3×C3⋊D312yescomplemented
453C23C2×D3×C3⋊D311yescharacteristic
454D4—98no—
455D4—94no—
456D4—94no—
457C2×C4—31no—
458C2×C4—92no—
459C2×C4—92no—
460C2×C4—92no—
461C2×C4—94no—
462C2×C4—91no—
463C2×C4—92no—
464C2×C4—181no—
465C2×C4—272no—
466C2×C4—272no—
467C2×C4—544no—
468C2×C4—544no—
469C2×C4—541no—
470C2×C4—542no—
471C2×C4—541no—
472C2×C3—22no—
473C2×C3—21no—
474C2×C3—31no—
475C2×C3—31no—
476C2×C3—32no—
477C2×C3—31no—
478C2×C3—31no—
479C2×C3—32no—
480C2×C3—46no—
481C2×C3—43no—
482C2×C3—61no—
483C2×C3—61no—
484C2×C3—62no—
485C2×C3—86no—
486C2×C3—83no—
487C2×C3—184no—
488C2×C3Dic3×S412yescomplemented
489C2×C3C4×C3⋊S412yescomplemented
490C2×C3C2×D3×S411yescharacteristic
491C2×C3C22×C3⋊S411yescharacteristic
492D3—64no—
493D3—1212no—
494C22—32no—
495C22—31no—
496C22—32no—
497C22—32no—
498C22—31no—
499C22—31no—
500C22—61no—
501C22—94no—
502C22—182no—
503C22—184no—
504C22—184no—
505C22—182no—
506C22C2×Dic3×C3⋊D311yescharacteristic, complemented
507C22D3×C3⋊S411yescharacteristic
508C4—32no—
509C4—94no—
510C4—92no—
511C4—542no—
512C4—542no—
513C3—21no—
514C3—43no—
515C3—83no—
516C3C2×Dic3×S411yescharacteristic, complemented
517C3C2×C4×C3⋊S411yescharacteristic, complemented
518C2—31no—
519C2—31no—
520C2—32no—
521C2—184no—
522C2Dic3×C3⋊S412yesdirect product factor
523C2C2×D3×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC22D3×C3⋊S4
commutator, hypocenterC32×A4C22×C4
FrattiniC2C2×D3×C3⋊S4
Fitting, socleC24×C32C2×D3
prime radicalC22×C3×C3⋊S4C2
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×Dic3×A4C2semidirect
3. C2×C3⋊(C3⋊SL(2,ℤ/4))C2semidirect
4. Dic3×C3⋊S4C2direct (N × H)
9. C2×C3×C3⋊S4C4semidirect
12. C3×Dic3×A4C22semidirect
13. C3⋊(C3⋊SL(2,ℤ/4))C22semidirect
17. C23×C3×Dic3D3semidirect
20. C2×Dic3×A4D3semidirect
34. C2×C32×A4C2×C4semidirect
36. C3×C3⋊S4C2×C4semidirect
49. C2×C3⋊S4Dic3direct (N × H)
64. C22×C3×Dic3C2×D3semidirect
71. Dic3×A4C2×D3semidirect
84. C32×A4C22×C4semidirect
92. C23×Dic3C3⋊D3semidirect
114. C23×C32C4×D3semidirect
127. C2×C3×A4C4×D3semidirect
128. C2×C3×A4C2×Dic3semidirect
131. C3⋊S4C2×Dic3direct (N × H)
155. C2×C3×Dic3S4semidirect
199. C22×Dic3C2×C3⋊D3semidirect
231. C22×C32C2×C4×D3semidirect
244. C3×A4C2×C4×D3semidirect
245. C3×A4C22×Dic3semidirect
256. C3×Dic3C2×S4semidirect
283. C23×C3D3×Dic3semidirect
284. C23×C3C4×C3⋊D3semidirect
295. C2×A4D3×Dic3semidirect
339. C2×Dic3C3⋊S4direct (N × H)
354. C2×C32C4×S4semidirect
411. C22×C3C2×D3×Dic3semidirect
412. C22×C3C2×C4×C3⋊D3semidirect
423. A4C2×D3×Dic3semidirect
438. Dic3C2×C3⋊S4direct (N × H)
443. C32C2×C4×S4semidirect
452. C23Dic3×C3⋊D3semidirect
488. C2×C3Dic3×S4semidirect
489. C2×C3C4×C3⋊S4semidirect
506. C22C2×Dic3×C3⋊D3semidirect
516. C3C2×Dic3×S4semidirect
517. C3C2×C4×C3⋊S4semidirect
522. C2Dic3×C3⋊S4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊S42
C2×C3×Dic3×A42
C2×C3⋊(C3⋊SL(2,ℤ/4))2
Dic3×C3⋊S42
C2×C4×C3⋊S43
C2×Dic3×S43
C2×Dic3×#673
C2×Dic3×C3⋊D34