Finite Group Databasebeta

← C3×#10596577 #13181377

Subgroups — C3×#10596577 #13181377

Order 1728.

1014 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C32×D4×A4C211yesmaximal, characteristic, complemented
2C3×#64269C212yesmaximal, complemented
3C2×C3×C3⋊SL(2,ℤ/4)C212yesmaximal, complemented
4C3×C4×C3⋊S4C211yesmaximal, characteristic, complemented
5C3×#64259C211yesmaximal, characteristic, complemented
6#10596577C311yesmaximal, characteristic, direct product factor
7C3×#5914—33nomaximal
8C3×#5806—31nomaximal
9C22×C32×A4C2212yes—
10C2×C3×C3⋊S4C2211yescharacteristic
11C3×#3830—41nomaximal
12C32×C4×A4C2211yescharacteristic
13C3×C3⋊SL(2,ℤ/4)C2212yes—
14C3×C3⋊SL(2,ℤ/4)C2211yescharacteristic
15C22×C32×D4D311yescharacteristic, complemented
16C2×C3×#237—31no—
17C3×D4×A4—21no—
18C3×D4×A4—23no—
19C3×D4×A4D313yescomplemented
20C3×D4×A4C2×C311yescharacteristic, complemented
21#64269C2×C312yescomplemented
22C2×C3⋊SL(2,ℤ/4)C2×C312yescomplemented
23C4×C3⋊S4C2×C311yescharacteristic, complemented
24#64259C2×C311yescharacteristic, complemented
25C3×GL(2,ℤ/4)—36no—
26C2×C3×SL(2,ℤ/4)—36no—
27C3×C4×S4—33no—
28C3×A4⋊Q8—33no—
29C3×#953—32no—
30C2×C3×#217—32no—
31C3×#939—32no—
32C3×#938—31no—
33C3×#936—31no—
34C3×#935—32no—
35C3×D4×Dic3—31no—
36C3×C4×#67—31no—
37C3×#924—31no—
38C2×C32×A4—22no—
39C2×C32×A4C2311yescharacteristic
40C3×C3⋊S4—21no—
41C33×D4—41no—
42C3×#638—42no—
43C2×C3×C3⋊Dic3—42no—
44C3×C4×C3⋊D3—41no—
45C3×#634—41no—
46#5914—33no—
47C3×#546—91no—
48#5806—31no—
49C24×C32C2×D312yes—
50C22×C3×A4—22no—
51C22×C3×A4—26no—
52C22×C3×A4C2×D316yes—
53C22×C3×A4C22×C312yes—
54C2×C3⋊S4C22×C311yescharacteristic
55C2×C3×S4—33no—
56C2×C32×D4—31no—
57C2×C32×D4—32no—
58C2×C32×D4—31no—
59C22×C32×C4C2×D311yescharacteristic
60#3830—41no—
61C2×C3×#67—32no—
62C2×C3×#67—31no—
63C2×C3×#67—32no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—32no—
66C3×#237—62no—
67C3×#237—61no—
68C3×#237—61no—
69C3×#237—123no—
70C2×C3×C4×D3—31no—
71C2×C3×Dic6—31no—
72C3×C4×A4—21no—
73C3×C4×A4—23no—
74C3×C4×A4C2×D313yes—
75C3×C4×A4C22×C311yescharacteristic
76C3⋊SL(2,ℤ/4)C22×C312yes—
77C3⋊SL(2,ℤ/4)C22×C311yescharacteristic
78C3×SL(2,ℤ/4)—33no—
79C3×SL(2,ℤ/4)—36no—
80C3×#217—32no—
81C3×#217—32no—
82C3×#217—32no—
83C3×#217—32no—
84C3×#217—32no—
85C3×#217—31no—
86C3×(C2×D3)⋊C4—31no—
87C3×C4⋊Dic3—31no—
88C3×Dic3⋊C4—32no—
89C3×Dic3⋊C4—31no—
90C3×C4×Dic3—31no—
91C22×C33—42no—
92C2×C3×C3⋊D3—41no—
93C32×A4#4111yescharacteristic, complemented
94C33×C4—41no—
95C3×C3⋊Dic3—42no—
96C3×C3⋊Dic3—41no—
97C2×C3×#41—91no—
98C22×C3×D4—21no—
99C22×C3×D4C3×D311yescharacteristic, complemented
100C22×C3×D4C3⋊D311yescharacteristic, complemented
101C2×#237—31no—
102D4×A4—23no—
103D4×A4C3×D313yescomplemented
104GL(2,ℤ/4)—36no—
105C2×SL(2,ℤ/4)—36no—
106C4×S4—33no—
107A4⋊Q8—33no—
108C3×#124—91no—
109C3×#123—92no—
110C3×C22⋊Q8—91no—
111C3×#121—91no—
112C3×#121—92no—
113C3×#120—92no—
114C3×C4×D4—91no—
115C3×C4×D4—91no—
116C2×C3×C22⋊C4—92no—
117#953—32no—
118C2×#217—32no—
119#939—32no—
120#938—31no—
121#936—31no—
122#935—32no—
123D4×Dic3—31no—
124C4×#67—31no—
125#924—31no—
126C23×C32—22no—
127C23×C32—32no—
128C23×C32—32no—
129C23×C32—62no—
130C23×C32C22×D311yescharacteristic
131C22×C3×D3—31no—
132C2×C3×A4—22no—
133C2×C3×A4—21no—
134C2×C3×A4—23no—
135C2×C3×A4—26no—
136C2×C3×A4—46no—
137C2×C3×A4—42no—
138C2×C3×A4C22×D313yes—
139C2×C3×A4C23×C311yescharacteristic
140C3⋊S4—21no—
141C3×S4—63no—
142C32×D4—32no—
143C32×D4—31no—
144C32×D4—41no—
145C32×D4—43no—
146C32×D4—61no—
147C32×D4—83no—
148C32×D4—81no—
149C32×D4S411yescharacteristic, complemented
150C2×C32×C4—31no—
151C2×C32×C4—31no—
152#638—42no—
153C2×C3⋊Dic3—42no—
154C4×C3⋊D3—41no—
155#634—41no—
156C3×#67—62no—
157C3×#67—61no—
158C3×#67—61no—
159C3×#67—62no—
160C3×#67—126no—
161C3×#67—122no—
162C2×C3×Dic3—31no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—31no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—126no—
170C2×C3×Dic3—122no—
171C3×C4×D3—61no—
172C3×C4×D3—61no—
173C3×C4×D3—123no—
174C3×Dic6—61no—
175C3×Dic6—61no—
176C3×Dic6—123no—
177#546—91noSylow
178C2×C33—41no—
179C2×C33—82no—
180C3×C3⋊D3—81no—
181C24×C3—22no—
182C24×C3C2×C3×D312yes—
183C24×C3C2×C3⋊D312yes—
184C22×A4—26no—
185C22×A4C2×C3×D316yes—
186C2×S4—33no—
187C3×#41—182no—
188C3×#41—181no—
189C3×#41—181no—
190C2×C3×Q8—91no—
191C2×C3×D4—31no—
192C2×C3×D4—31no—
193C2×C3×D4—32no—
194C2×C3×D4—31no—
195C2×C3×D4—32no—
196C2×C3×D4—31no—
197C2×C3×D4—61no—
198C2×C3×D4—62no—
199C2×C3×D4—61no—
200C2×C3×D4—92no—
201C2×C3×D4—91no—
202C2×C3×D4—92no—
203C22×C3×C4—21no—
204C22×C3×C4—91no—
205C22×C3×C4—92no—
206C22×C3×C4—92no—
207C22×C3×C4C2×C3×D311yescharacteristic
208C22×C3×C4C2×C3⋊D311yescharacteristic
209C2×#67—32no—
210C2×#67—31no—
211C2×#67—32no—
212C22×Dic3—32no—
213C22×Dic3—32no—
214#237—62no—
215#237—61no—
216#237—61no—
217#237—123no—
218C2×C4×D3—31no—
219C2×Dic6—31no—
220C4×A4—23no—
221C4×A4C2×C3×D313yes—
222SL(2,ℤ/4)—36no—
223SL(2,ℤ/4)—33no—
224C3×C4⋊C4—91no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—91no—
227C3×C22⋊C4—91no—
228C3×C22⋊C4—92no—
229C3×C22⋊C4—92no—
230C3×C22⋊C4—92no—
231C3×C22⋊C4—92no—
232C3×C22⋊C4—91no—
233C3×C22⋊C4—92no—
234C3×C42—91no—
235#217—32no—
236#217—32no—
237#217—32no—
238#217—32no—
239#217—32no—
240#217—31no—
241(C2×D3)⋊C4—31no—
242C4⋊Dic3—31no—
243Dic3⋊C4—32no—
244Dic3⋊C4—31no—
245C4×Dic3—31no—
246C22×C32—31no—
247C22×C32—31no—
248C22×C32—32no—
249C22×C32—42no—
250C22×C32—46no—
251C22×C32—62no—
252C22×C32—62no—
253C22×C32—62no—
254C22×C32—62no—
255C22×C32—86no—
256C22×C32—82no—
257C22×C32#23711yescharacteristic, complemented
258C22×C32C2×S412yes—
259C2×C3⋊D3—41no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C2×C3×D3—61no—
263C2×C3×D3—123no—
264C3×A4—21no—
265C3×A4—23no—
266C3×A4#23713yescomplemented
267C3×A4C3×#4111yescharacteristic, complemented
268C32×C4—31no—
269C32×C4—41no—
270C32×C4—43no—
271C32×C4—83no—
272C32×C4—81no—
273C32×C4C2×S411yescharacteristic
274C3⋊Dic3—42no—
275C3⋊Dic3—41no—
276C3×Dic3—61no—
277C3×Dic3—62no—
278C3×Dic3—61no—
279C3×Dic3—62no—
280C3×Dic3—61no—
281C3×Dic3—126no—
282C3×Dic3—123no—
283C2×#41—91no—
284C22×D4C3×C3⋊D311yescharacteristic, complemented
285#124—91no—
286#123—92no—
287C22⋊Q8—91no—
288#121—91no—
289#121—92no—
290#120—92no—
291C4×D4—91no—
292C4×D4—91no—
293C2×C22⋊C4—92no—
294C33—41noSylow
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—32no—
299C23×C3—32no—
300C23×C3—32no—
301C23×C3—32no—
302C23×C3—42no—
303C23×C3—62no—
304C23×C3—62no—
305C23×C3—62no—
306C23×C3—62no—
307C23×C3—91no—
308C23×C3—122no—
309C23×C3C22×C3×D311yescharacteristic
310C23×C3C22×C3⋊D311yescharacteristic
311C22×D3—31no—
312C2×A4—26no—
313C2×A4—23no—
314C2×A4—46no—
315C2×A4C22×C3×D313yes—
316S4—63no—
317C3×Q8—181no—
318C3×Q8—181no—
319C3×D4—21no—
320C3×D4—32no—
321C3×D4—32no—
322C3×D4—31no—
323C3×D4—31no—
324C3×D4—43no—
325C3×D4—61no—
326C3×D4—62no—
327C3×D4—61no—
328C3×D4—61no—
329C3×D4—62no—
330C3×D4—83no—
331C3×D4—181no—
332C3×D4—182no—
333C3×D4—182no—
334C3×D4—181no—
335C3×D4—362no—
336C3×D4C3×S411yescharacteristic, complemented
337C3×D4C3⋊S411yescharacteristic, complemented
338C2×C3×C4—31no—
339C2×C3×C4—31no—
340C2×C3×C4—31no—
341C2×C3×C4—31no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—91no—
345C2×C3×C4—91no—
346C2×C3×C4—92no—
347C2×C3×C4—91no—
348C2×C3×C4—92no—
349C2×C3×C4—181no—
350C2×C3×C4—182no—
351C2×C3×C4—182no—
352C2×C3×C4—181no—
353C2×C3×C4—362no—
354#67—62no—
355#67—61no—
356#67—62no—
357#67—61no—
358#67—126no—
359#67—122no—
360C2×Dic3—31no—
361C2×Dic3—31no—
362C2×Dic3—32no—
363C2×Dic3—32no—
364C2×Dic3—31no—
365C2×Dic3—62no—
366C2×Dic3—62no—
367C2×Dic3—126no—
368C2×Dic3—122no—
369C4×D3—61no—
370C4×D3—61no—
371C4×D3—123no—
372Dic6—61no—
373Dic6—61no—
374Dic6—123no—
375C2×C32—22no—
376C2×C32—31no—
377C2×C32—31no—
378C2×C32—41no—
379C2×C32—43no—
380C2×C32—62no—
381C2×C32—83no—
382C2×C32—82no—
383C2×C32—86no—
384C2×C32—81no—
385C2×C32—162no—
386C2×C32—166no—
387C2×C32C22×S411yescharacteristic
388C3⋊D3—81no—
389C3×D3—121no—
390C3×D3—243no—
391C24C2×C3×C3⋊D312yes—
392#41—182no—
393#41—181no—
394#41—181no—
395C2×Q8—91no—
396C2×D4—31no—
397C2×D4—32no—
398C2×D4—31no—
399C2×D4—92no—
400C2×D4—91no—
401C2×D4—92no—
402C22×C4—91no—
403C22×C4—92no—
404C22×C4—92no—
405C22×C4C2×C3×C3⋊D311yescharacteristic
406C4⋊C4—91no—
407C4⋊C4—91no—
408C4⋊C4—92no—
409C22⋊C4—91no—
410C22⋊C4—92no—
411C22⋊C4—92no—
412C22⋊C4—92no—
413C22⋊C4—92no—
414C22⋊C4—91no—
415C22⋊C4—92no—
416C42—91no—
417C22×C3—22no—
418C22×C3—21no—
419C22×C3—31no—
420C22×C3—31no—
421C22×C3—31no—
422C22×C3—31no—
423C22×C3—32no—
424C22×C3—32no—
425C22×C3—46no—
426C22×C3—61no—
427C22×C3—62no—
428C22×C3—61no—
429C22×C3—62no—
430C22×C3—62no—
431C22×C3—62no—
432C22×C3—62no—
433C22×C3—62no—
434C22×C3—62no—
435C22×C3—62no—
436C22×C3—62no—
437C22×C3—86no—
438C22×C3—122no—
439C22×C3—122no—
440C22×C3—122no—
441C22×C3—122no—
442C22×C3—181no—
443C22×C3—181no—
444C22×C3—181no—
445C22×C3C3×#23711yescharacteristic, complemented
446C22×C3#383011yescharacteristic, complemented
447C22×C3C2×C3×S412yes—
448C22×C3C2×C3⋊S412yes—
449C2×D3—61no—
450C2×D3—61no—
451C2×D3—61no—
452C2×D3—123no—
453A4—23no—
454A4C3×#23713yescomplemented
455C3×C4—21no—
456C3×C4—31no—
457C3×C4—31no—
458C3×C4—43no—
459C3×C4—61no—
460C3×C4—83no—
461C3×C4—181no—
462C3×C4—182no—
463C3×C4—181no—
464C3×C4—181no—
465C3×C4—182no—
466C3×C4C2×C3×S411yescharacteristic
467C3×C4C2×C3⋊S411yescharacteristic
468Dic3—61no—
469Dic3—62no—
470Dic3—61no—
471Dic3—62no—
472Dic3—61no—
473Dic3—126no—
474Dic3—123no—
475C32—41no—
476C32—43no—
477C32—83no—
478C32—81no—
479C32#591411yescharacteristic, complemented
480C23—22no—
481C23—32no—
482C23—32no—
483C23—62no—
484C23—91no—
485C23C22×C3×C3⋊D311yescharacteristic
486Q8—181no—
487Q8—181no—
488D4—32no—
489D4—31no—
490D4—61no—
491D4—182no—
492D4—181no—
493D4—182no—
494D4—181no—
495D4—362no—
496D4C3×C3⋊S411yescharacteristic, complemented
497C2×C4—31no—
498C2×C4—31no—
499C2×C4—91no—
500C2×C4—91no—
501C2×C4—92no—
502C2×C4—91no—
503C2×C4—92no—
504C2×C4—181no—
505C2×C4—182no—
506C2×C4—182no—
507C2×C4—181no—
508C2×C4—362no—
509C2×C3—22no—
510C2×C3—22no—
511C2×C3—21no—
512C2×C3—31no—
513C2×C3—31no—
514C2×C3—31no—
515C2×C3—31no—
516C2×C3—42no—
517C2×C3—43no—
518C2×C3—61no—
519C2×C3—61no—
520C2×C3—62no—
521C2×C3—62no—
522C2×C3—86no—
523C2×C3—83no—
524C2×C3—122no—
525C2×C3—166no—
526C2×C3—361no—
527C2×C3C22×C3×S411yescharacteristic
528C2×C3C22×C3⋊S411yescharacteristic
529D3—121no—
530D3—243no—
531C22—31no—
532C22—31no—
533C22—32no—
534C22—62no—
535C22—62no—
536C22—62no—
537C22—62no—
538C22—181no—
539C22—181no—
540C22—181no—
541C22C3×#383011yescharacteristic, complemented
542C22C2×C3×C3⋊S412yes—
543C4—31no—
544C4—181no—
545C4—182no—
546C4—181no—
547C4—181no—
548C4—182no—
549C4C2×C3×C3⋊S411yescharacteristic
550C3—21no—
551C3—43no—
552C3—83no—
553C3C3×#591411yescharacteristic, complemented
554C3#1059657711yescharacteristic, direct product factor
555C2—22no—
556C2—31no—
557C2—31no—
558C2—62no—
559C2—361no—
560C2C22×C3×C3⋊S411yescharacteristic
subgroupG/H
centerC2×C3C22×C3⋊S4
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2C22×C3×C3⋊S4
FittingC22×C32×D4D3
socleC23×C32C22×D3
hypercenterC3×D4C3⋊S4
hypocenterC3×A4C3×#41
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C32×D4×A4C2semidirect
2. C3×#64269C2semidirect
3. C2×C3×C3⋊SL(2,ℤ/4)C2semidirect
4. C3×C4×C3⋊S4C2semidirect
5. C3×#64259C2semidirect
6. #10596577C3direct (N × H)
15. C22×C32×D4D3semidirect
19. C3×D4×A4D3semidirect
20. C3×D4×A4C2×C3semidirect
21. #64269C2×C3semidirect
22. C2×C3⋊SL(2,ℤ/4)C2×C3semidirect
23. C4×C3⋊S4C2×C3semidirect
24. #64259C2×C3semidirect
93. C32×A4#41semidirect
99. C22×C3×D4C3×D3semidirect
100. C22×C3×D4C3⋊D3semidirect
103. D4×A4C3×D3semidirect
149. C32×D4S4semidirect
257. C22×C32#237semidirect
266. C3×A4#237semidirect
267. C3×A4C3×#41semidirect
284. C22×D4C3×C3⋊D3semidirect
336. C3×D4C3×S4semidirect
337. C3×D4C3⋊S4semidirect
445. C22×C3C3×#237semidirect
446. C22×C3#3830semidirect
454. A4C3×#237semidirect
479. C32#5914semidirect
496. D4C3×C3⋊S4semidirect
541. C22C3×#3830semidirect
553. C3C3×#5914semidirect
554. C3#10596577direct (N × H)

Maximal subgroups

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

subgroupindex
C32×D4×A42
C3×#642692
C2×C3×C3⋊SL(2,ℤ/4)2
C3×C4×C3⋊S42
C3×#642592
#105965773
C3×#59143
C3×#58063
C3×#38304