Finite Group Databasebeta

← C2×#11724201 #13180834

Subgroups — C2×#11724201 #13180834

Order 1728.

1471 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 1471 conjugacy classes of subgroups — too many to draw. Only its 126 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⋊S4)C211yesmaximal, characteristic
2C2×C3×Dic3×A4C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4#11724201C214yesmaximal, direct product factor
5C2×C4×C3⋊S4—31nomaximal
6C2×(C3⋊S4)⋊C4—33nomaximal
7C2×#63952—31nomaximal
8C22×C32×A4C2211yescharacteristic
9C2×C3⋊(C3⋊S4)C414yescomplemented
10C2×C3⋊(C3⋊S4)C2212yes—
11C2×#6289—41nomaximal
12C3×Dic3×A4C2212yescomplemented
13C3×C3⋊SL(2,ℤ/4)C2212yescomplemented
14C22×C3⋊S4—31no—
15C22×C3⋊S4—33no—
16C22×C3⋊S4—61no—
17C22×C3⋊S4—63no—
18C22×#638—31no—
19C23×C3×Dic3D311yescharacteristic, complemented
20C2×C3×C4×A4—31no—
21C22×#624—31no—
22C2×Dic3×A4D313yescomplemented
23C2×C3⋊SL(2,ℤ/4)D311yescharacteristic, complemented
24C4×C3⋊S4—34no—
25C2×C3×SL(2,ℤ/4)—33no—
26(C3⋊S4)⋊C4—312no—
27C2×C3×#217—31no—
28C2×#3724—31no—
29C2×#3722—31no—
30C2×Dic32—31no—
31#63952—34no—
32#63952—34no—
33C22×C3⋊(C3⋊D3)—41no—
34C2×C32×A4C2×C412yescomplemented
35C2×C32×A4C2311yescharacteristic
36C3⋊(C3⋊S4)C2×C414yescomplemented
37C2×C3×C3⋊Dic3—41no—
38C2×C32×Dic3—41no—
39#6289—44no—
40C2×C4×S4—93no—
41C2×C4×#67—91no—
42C2×#67⋊C4—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⋊S4—34no—
50C2×C3⋊S4—312no—
51C2×C3⋊S4—32no—
52C2×C3⋊S4—36no—
53C2×C3⋊S4—64no—
54C2×C3⋊S4—62no—
55C2×C3⋊S4—612no—
56C2×C3⋊S4—66no—
57C2×#638—34no—
58C2×#638—32no—
59C2×#638—34no—
60C2×#638—32no—
61C22×C3⋊Dic3—31no—
62C2×C4×C3⋊D3—121no—
63C22×C3×Dic3—31no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—31no—
66C22×C3×Dic3—31no—
67C22×C3×Dic3—31no—
68C22×C3×Dic3C2×D312yescomplemented
69C3×C4×A4—32no—
70C2×#624—34no—
71C2×#624—61no—
72C2×#624—62no—
73C2×#624—61no—
74C2×#624—123no—
75Dic3×A4C2×D316yescomplemented
76C3⋊SL(2,ℤ/4)C2×D312yescomplemented
77C3×SL(2,ℤ/4)—36no—
78C3×#217—32no—
79C3×#217—32no—
80#3724—34no—
81#3722—34no—
82Dic32—34no—
83C22×C33—41no—
84C2×C3⋊(C3⋊D3)—44no—
85C2×C3⋊(C3⋊D3)—42no—
86C32×A4C22×C411yescharacteristic, complemented
87C3×C3⋊Dic3—42no—
88C32×Dic3—42no—
89C22×S4—93no—
90C22×S4—183no—
91C23×C3×C4—31no—
92C22×#67—91no—
93C22×#67—91no—
94C22×#67—181no—
95C23×Dic3C3⋊D311yescharacteristic, complemented
96C22×C4×D3—91no—
97C22×C4×D3—91no—
98C2×C4×A4—33no—
99C2×SL(2,ℤ/4)—33no—
100C4×S4—912no—
101C2×C3×C22⋊C4—91no—
102C2×#217—31no—
103C4×#67—94no—
104C4×#67—94no—
105C2×(C2×D3)⋊C4—91no—
106C2×(C2×D3)⋊C4—91no—
107C2×Dic3⋊C4—91no—
108C2×Dic3⋊C4—91no—
109C2×C4×Dic3—91no—
110C2×C4×Dic3—91no—
111#67⋊C4—94no—
112#67⋊C4—94no—
113C23×C32—31no—
114C23×C32—31no—
115C23×C32—32no—
116C23×C32C4×D312yescomplemented
117C23×C32C22×D311yescharacteristic
118C22×C3⋊D3—32no—
119C22×C3⋊D3—34no—
120C22×C3⋊D3—62no—
121C22×C3⋊D3—61no—
122C22×C3⋊D3—61no—
123C22×C3⋊D3—121no—
124C22×C3⋊D3—123no—
125C22×C3⋊D3—241no—
126C22×C3⋊D3—243no—
127C2×C3×A4—22no—
128C2×C3×A4—21no—
129C2×C3×A4—26no—
130C2×C3×A4—23no—
131C2×C3×A4C4×D316yescomplemented
132C2×C3×A4C4×D312yescomplemented
133C2×C3×A4C22×D313yes—
134C2×C3×A4C22×D311yescharacteristic
135C3⋊S4—34no—
136C3⋊S4—312no—
137C3⋊S4—64no—
138C3⋊S4—612no—
139C2×C32×C4—121no—
140#638—38no—
141#638—34no—
142#638—34no—
143C2×C3⋊Dic3—32no—
144C2×C3⋊Dic3—34no—
145C2×C3⋊Dic3—41no—
146C4×C3⋊D3—124no—
147C2×C3×Dic3—31no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—43no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—123no—
159C2×C3×Dic3S411yescharacteristic, complemented
160#624—64no—
161#624—64no—
162#624—1212no—
163C2×C4×D4—271noSylow
164C2×C33—42no—
165C2×C33—41no—
166C3⋊(C3⋊D3)—44no—
167C24×C3—21no—
168C24×C3D3211yescharacteristic
169C24×C3C2×C3⋊D311yescharacteristic
170C23×D3—91no—
171C23×D3—91no—
172C23×D3—181no—
173C22×A4—23no—
174C22×A4D3213yes—
175C2×S4—912no—
176C2×S4—96no—
177C2×S4—1812no—
178C2×S4—186no—
179C22×C3×C4—32no—
180C22×C3×C4—91no—
181C22×C3×C4—91no—
182C22×C3×C4—92no—
183C22×C3×C4—91no—
184C22×C3×C4—91no—
185C2×#67—92no—
186C2×#67—92no—
187C2×#67—94no—
188C2×#67—94no—
189C2×#67—94no—
190C2×#67—94no—
191C2×#67—92no—
192C2×#67—92no—
193C2×#67—184no—
194C2×#67—182no—
195C2×#67—182no—
196C2×#67—184no—
197C22×Dic3—31no—
198C22×Dic3—32no—
199C22×Dic3—31no—
200C22×Dic3—31no—
201C22×Dic3—31no—
202C22×Dic3—91no—
203C22×Dic3—91no—
204C22×Dic3—181no—
205C22×Dic3C2×C3⋊D312yescomplemented
206C2×C4×D3—94no—
207C2×C4×D3—94no—
208C2×C4×D3—181no—
209C2×C4×D3—182no—
210C2×C4×D3—181no—
211C2×C4×D3—182no—
212C2×C4×D3—181no—
213C2×C4×D3—181no—
214C2×C4×D3—363no—
215C4×A4—36no—
216SL(2,ℤ/4)—36no—
217C3×C22⋊C4—92no—
218C3×C22⋊C4—92no—
219#217—32no—
220#217—32no—
221(C2×D3)⋊C4—94no—
222(C2×D3)⋊C4—94no—
223Dic3⋊C4—94no—
224Dic3⋊C4—94no—
225C4×Dic3—94no—
226C4×Dic3—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×C32C2×C4×D311yescharacteristic, complemented
239C22×C32C2×S411yescharacteristic
240C2×C3⋊D3—34no—
241C2×C3⋊D3—62no—
242C2×C3⋊D3—64no—
243C2×C3⋊D3—64no—
244C2×C3⋊D3—62no—
245C2×C3⋊D3—122no—
246C2×C3⋊D3—124no—
247C2×C3⋊D3—126no—
248C2×C3⋊D3—1212no—
249C2×C3⋊D3—242no—
250C2×C3⋊D3—244no—
251C2×C3⋊D3—246no—
252C2×C3⋊D3—2412no—
253C3×A4—21no—
254C3×A4—23no—
255C3×A4C2×C4×D313yescomplemented
256C3×A4C2×C4×D311yescharacteristic, complemented
257C32×C4—122no—
258C3⋊Dic3—34no—
259C3⋊Dic3—42no—
260C3×Dic3—32no—
261C3×Dic3—46no—
262C3×Dic3—62no—
263C3×Dic3—62no—
264C3×Dic3—126no—
265C3×Dic3C2×S412yescomplemented
266C22×D4—271no—
267C23×C4—31no—
268C23×C4—271no—
269C4×D4—274no—
270C4×D4—274no—
271C2×C4⋊C4—271no—
272C2×C22⋊C4—91no—
273C2×C22⋊C4—271no—
274C2×C42—271no—
275C33—41noSylow
276C23×C3—22no—
277C23×C3—21no—
278C23×C3—31no—
279C23×C3—31no—
280C23×C3—31no—
281C23×C3—31no—
282C23×C3—32no—
283C23×C3—32no—
284C23×C3—61no—
285C23×C3—61no—
286C23×C3—62no—
287C23×C3#62412yescomplemented
288C23×C3C4×C3⋊D312yescomplemented
289C23×C3C2×D3211yescharacteristic
290C23×C3C22×C3⋊D311yescharacteristic
291C22×D3—92no—
292C22×D3—92no—
293C22×D3—94no—
294C22×D3—94no—
295C22×D3—182no—
296C22×D3—184no—
297C22×D3—181no—
298C22×D3—181no—
299C22×D3—182no—
300C22×D3—182no—
301C22×D3—181no—
302C22×D3—181no—
303C22×D3—361no—
304C22×D3—363no—
305C22×D3—361no—
306C22×D3—362no—
307C22×D3—723no—
308C2×A4—26no—
309C2×A4—23no—
310C2×A4#62416yescomplemented
311C2×A4C2×D3213yes—
312S4—912no—
313S4—1812no—
314C2×C3×C4—31no—
315C2×C3×C4—92no—
316C2×C3×C4—92no—
317C2×C3×C4—91no—
318C2×C3×C4—92no—
319C2×C3×C4—92no—
320C2×C3×C4—92no—
321C2×C3×C4—123no—
322C2×C3×C4—181no—
323C2×C3×C4—181no—
324C2×C3×C4—182no—
325C2×C3×C4—181no—
326#67—98no—
327#67—98no—
328#67—94no—
329#67—94no—
330#67—94no—
331#67—94no—
332#67—184no—
333#67—188no—
334#67—184no—
335C2×Dic3—31no—
336C2×Dic3—32no—
337C2×Dic3—32no—
338C2×Dic3—32no—
339C2×Dic3—32no—
340C2×Dic3—32no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—62no—
344C2×Dic3—61no—
345C2×Dic3—92no—
346C2×Dic3—92no—
347C2×Dic3—94no—
348C2×Dic3—94no—
349C2×Dic3—123no—
350C2×Dic3—182no—
351C2×Dic3—184no—
352C2×Dic3C3⋊S411yescharacteristic, complemented
353C4×D3—184no—
354C4×D3—184no—
355C4×D3—184no—
356C4×D3—184no—
357C4×D3—3612no—
358C2×C32—31no—
359C2×C32—31no—
360C2×C32—32no—
361C2×C32—41no—
362C2×C32—42no—
363C2×C32—46no—
364C2×C32—43no—
365C2×C32—86no—
366C2×C32—83no—
367C2×C32—81no—
368C2×C32—82no—
369C2×C32C4×S412yescomplemented
370C2×C32C22×S411yescharacteristic
371C3⋊D3—64no—
372C3⋊D3—124no—
373C3⋊D3—1212no—
374C3⋊D3—244no—
375C3⋊D3—2412no—
376C24—271no—
377C24D3×C3⋊D311yescharacteristic
378C2×D4—272no—
379C2×D4—274no—
380C2×D4—274no—
381C2×D4—272no—
382C22×C4—32no—
383C22×C4—91no—
384C22×C4—91no—
385C22×C4—92no—
386C22×C4—91no—
387C22×C4—91no—
388C22×C4—274no—
389C22×C4—271no—
390C22×C4—541no—
391C22×C4—542no—
392C22×C4—541no—
393C4⋊C4—274no—
394C22⋊C4—92no—
395C22⋊C4—92no—
396C22⋊C4—274no—
397C42—274no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—32no—
401C22×C3—32no—
402C22×C3—31no—
403C22×C3—31no—
404C22×C3—32no—
405C22×C3—32no—
406C22×C3—31no—
407C22×C3—31no—
408C22×C3—31no—
409C22×C3—31no—
410C22×C3—32no—
411C22×C3—32no—
412C22×C3—43no—
413C22×C3—62no—
414C22×C3—61no—
415C22×C3—62no—
416C22×C3—61no—
417C22×C3—62no—
418C22×C3—61no—
419C22×C3—61no—
420C22×C3—61no—
421C22×C3—83no—
422C22×C3—121no—
423C22×C3C2×#62411yescharacteristic, complemented
424C22×C3C2×C4×C3⋊D311yescharacteristic, complemented
425C22×C3D3×S411yescharacteristic
426C22×C3C2×C3⋊S411yescharacteristic
427C2×D3—94no—
428C2×D3—94no—
429C2×D3—184no—
430C2×D3—182no—
431C2×D3—184no—
432C2×D3—182no—
433C2×D3—184no—
434C2×D3—184no—
435C2×D3—182no—
436C2×D3—182no—
437C2×D3—184no—
438C2×D3—364no—
439C2×D3—362no—
440C2×D3—366no—
441C2×D3—3612no—
442C2×D3—362no—
443C2×D3—364no—
444C2×D3—7212no—
445C2×D3—726no—
446A4—23no—
447A4C2×#62413yescomplemented
448C3×C4—32no—
449C3×C4—92no—
450C3×C4—126no—
451C3×C4—182no—
452C3×C4—182no—
453Dic3—32no—
454Dic3—62no—
455Dic3—62no—
456Dic3—94no—
457Dic3—94no—
458Dic3—126no—
459Dic3—184no—
460Dic3C2×C3⋊S412yescomplemented
461C32—41no—
462C32—43no—
463C32—83no—
464C32—81no—
465C32C2×C4×S411yescharacteristic, complemented
466C23—31no—
467C23—31no—
468C23—32no—
469C23—272no—
470C23—274no—
471C23—541no—
472C23—542no—
473C23—541no—
474C23#628912yescomplemented
475C23C2×D3×C3⋊D311yescharacteristic
476D4—278no—
477D4—274no—
478D4—274no—
479C2×C4—31no—
480C2×C4—92no—
481C2×C4—92no—
482C2×C4—91no—
483C2×C4—92no—
484C2×C4—92no—
485C2×C4—92no—
486C2×C4—181no—
487C2×C4—181no—
488C2×C4—182no—
489C2×C4—181no—
490C2×C4—272no—
491C2×C4—274no—
492C2×C4—544no—
493C2×C4—544no—
494C2×C3—22no—
495C2×C3—21no—
496C2×C3—31no—
497C2×C3—31no—
498C2×C3—32no—
499C2×C3—31no—
500C2×C3—31no—
501C2×C3—32no—
502C2×C3—46no—
503C2×C3—43no—
504C2×C3—61no—
505C2×C3—61no—
506C2×C3—62no—
507C2×C3—86no—
508C2×C3—83no—
509C2×C3(C3⋊S4)⋊C412yescomplemented
510C2×C3C4×C3⋊S412yescomplemented
511C2×C3C2×D3×S411yescharacteristic
512C2×C3C22×C3⋊S411yescharacteristic
513D3—184no—
514D3—184no—
515D3—364no—
516D3—3612no—
517D3—7212no—
518C22—32no—
519C22—31no—
520C22—32no—
521C22—32no—
522C22—31no—
523C22—31no—
524C22—61no—
525C22—274no—
526C22—542no—
527C22—544no—
528C22—544no—
529C22—542no—
530C22C2×#628911yescharacteristic, complemented
531C22D3×C3⋊S411yescharacteristic
532C4—32no—
533C4—92no—
534C4—182no—
535C4—182no—
536C4—274no—
537C3—21no—
538C3—43no—
539C3—83no—
540C3C2×(C3⋊S4)⋊C411yescharacteristic, complemented
541C3C2×C4×C3⋊S411yescharacteristic, complemented
542C2—31no—
543C2—31no—
544C2—32no—
545C2—544no—
546C2#1172420112yesdirect product factor
547C2C2×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⋊S4)C2
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. #11724201C2direct (N × H)
9. C2×C3⋊(C3⋊S4)C4semidirect
12. C3×Dic3×A4C22semidirect
13. C3×C3⋊SL(2,ℤ/4)C22semidirect
19. C23×C3×Dic3D3semidirect
22. C2×Dic3×A4D3semidirect
23. C2×C3⋊SL(2,ℤ/4)D3semidirect
34. C2×C32×A4C2×C4semidirect
36. C3⋊(C3⋊S4)C2×C4semidirect
68. C22×C3×Dic3C2×D3semidirect
75. Dic3×A4C2×D3semidirect
76. C3⋊SL(2,ℤ/4)C2×D3semidirect
86. C32×A4C22×C4semidirect
95. C23×Dic3C3⋊D3semidirect
116. C23×C32C4×D3semidirect
131. C2×C3×A4C4×D3semidirect
132. C2×C3×A4C4×D3semidirect
159. C2×C3×Dic3S4semidirect
205. C22×Dic3C2×C3⋊D3semidirect
238. C22×C32C2×C4×D3semidirect
255. C3×A4C2×C4×D3semidirect
256. C3×A4C2×C4×D3semidirect
265. C3×Dic3C2×S4semidirect
287. C23×C3#624semidirect
288. C23×C3C4×C3⋊D3semidirect
310. C2×A4#624semidirect
352. C2×Dic3C3⋊S4semidirect
369. C2×C32C4×S4semidirect
423. C22×C3C2×#624semidirect
424. C22×C3C2×C4×C3⋊D3semidirect
447. A4C2×#624semidirect
460. Dic3C2×C3⋊S4semidirect
465. C32C2×C4×S4semidirect
474. C23#6289semidirect
509. C2×C3(C3⋊S4)⋊C4semidirect
510. C2×C3C4×C3⋊S4semidirect
530. C22C2×#6289semidirect
540. C3C2×(C3⋊S4)⋊C4semidirect
541. C3C2×C4×C3⋊S4semidirect
546. C2#11724201direct (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⋊S4)2
C2×C3×Dic3×A42
C2×C3×C3⋊SL(2,ℤ/4)2
#117242012
C2×C4×C3⋊S43
C2×(C3⋊S4)⋊C43
C2×#639523
C2×#62894