Finite Group Databasebeta

← C3×#10596575 #13181375

Subgroups — C3×#10596575 #13181375

Order 1728.

1006 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 1006 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
1C2×C32×C4×A4C211yesmaximal, characteristic, complemented
2C3×#64269C212yesmaximal, complemented
3C3×C4⋊(C3⋊S4)C211yesmaximal, characteristic, complemented
4C3×C4×C3⋊S4C212yesmaximal, complemented
5C3×#64259C211yesmaximal, characteristic, complemented
6#10596575C311yesmaximal, characteristic, direct product factor
7C3×#5912—33nomaximal
8C3×#5791—31nomaximal
9C22×C32×A4C2211yescharacteristic
10C2×C3×C3⋊S4C2212yes—
11C3×#3828—41nomaximal
12C32×C4×A4C2211yescharacteristic
13C32×C4×A4C2211yescharacteristic
14C3×C3⋊SL(2,ℤ/4)C2212yes—
15C23×C32×C4D311yescharacteristic, complemented
16C2×C3×#235—31no—
17C2×C3×C4×A4—21no—
18C2×C3×C4×A4—23no—
19C2×C3×C4×A4D313yescomplemented
20C2×C3×C4×A4C2×C311yescharacteristic, complemented
21#64269C2×C312yescomplemented
22C4⋊(C3⋊S4)C2×C311yescharacteristic, complemented
23C4×C3⋊S4C2×C312yescomplemented
24#64259C2×C311yescharacteristic, complemented
25C3×GL(2,ℤ/4)—36no—
26C3×C4⋊S4—33no—
27C3×C4×S4—36no—
28C3×A4⋊Q8—33no—
29C3×#953—32no—
30C3×#930—31no—
31C3×#930—31no—
32C3×#929—32no—
33C3×C4×#67—32no—
34C3×C4×#67—32no—
35C3×#926—31no—
36C3×#924—31no—
37C3×#924—31no—
38C2×C32×A4—21no—
39C2×C32×A4C2311yescharacteristic
40C3×C3⋊S4—22no—
41C2×C33×C4—41no—
42C3×#638—42no—
43C3×C4⋊(C3⋊D3)—41no—
44C3×C4×C3⋊D3—42no—
45C3×#634—41no—
46#5912—33no—
47C3×#525—91no—
48#5791—31no—
49C24×C32C2×D311yescharacteristic
50C22×C3×A4—21no—
51C22×C3×A4—23no—
52C22×C3×A4C2×D313yes—
53C22×C3×A4C22×C311yescharacteristic
54C2×C3⋊S4C22×C312yes—
55C2×C3×S4—36no—
56C22×C32×C4—31no—
57C22×C32×C4—31no—
58C22×C32×C4—31no—
59C22×C32×C4—31no—
60C22×C32×C4C2×D311yescharacteristic
61C22×C32×C4C2×D311yescharacteristic
62#3828—41no—
63C2×C3×#67—32no—
64C2×C3×#67—32no—
65C2×C3×#67—32no—
66C3×#235—61no—
67C3×#235—61no—
68C3×#235—61no—
69C3×#235—61no—
70C3×#235—123no—
71C2×C3×D12—31no—
72C2×C3×C4×D3—32no—
73C2×C3×Dic6—31no—
74C3×C4×A4—21no—
75C3×C4×A4—21no—
76C3×C4×A4—23no—
77C3×C4×A4—23no—
78C3×C4×A4C2×D313yes—
79C3×C4×A4C2×D313yes—
80C3×C4×A4C22×C311yescharacteristic
81C3×C4×A4C22×C311yescharacteristic
82C3⋊SL(2,ℤ/4)C22×C312yes—
83C3×SL(2,ℤ/4)—36no—
84C3×#217—32no—
85C3×#217—32no—
86C3×#217—32no—
87C3×(C2×D3)⋊C4—32no—
88C3×(C2×D3)⋊C4—32no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×Dic3⋊C4—32no—
92C3×Dic3⋊C4—32no—
93C3×C4×Dic3—32no—
94C22×C33—41no—
95C2×C3×C3⋊D3—42no—
96C32×A4#4111yescharacteristic, complemented
97C33×C4—41no—
98C33×C4—41no—
99C3×C3⋊Dic3—42no—
100C2×C3×#41—91no—
101C23×C3×C4—21no—
102C23×C3×C4C3×D311yescharacteristic, complemented
103C23×C3×C4C3⋊D311yescharacteristic, complemented
104C2×#235—31no—
105C2×C4×A4—23no—
106C2×C4×A4C3×D313yescomplemented
107GL(2,ℤ/4)—36no—
108C4⋊S4—33no—
109C4×S4—36no—
110A4⋊Q8—33no—
111C3×#123—92no—
112C3×C22⋊Q8—91no—
113C3×C22⋊Q8—91no—
114C3×#121—91no—
115C3×#121—91no—
116C3×#120—92no—
117C3×C4×D4—92no—
118C3×C4×D4—92no—
119C3×#117—91no—
120#953—32no—
121#930—31no—
122#930—31no—
123#929—32no—
124C4×#67—32no—
125C4×#67—32no—
126#926—31no—
127#924—31no—
128#924—31no—
129C23×C32—21no—
130C23×C32—31no—
131C23×C32—31no—
132C23×C32—61no—
133C23×C32C22×D311yescharacteristic
134C22×C3×D3—32no—
135C2×C3×A4—22no—
136C2×C3×A4—26no—
137C2×C3×A4—23no—
138C2×C3×A4—21no—
139C2×C3×A4—21no—
140C2×C3×A4—23no—
141C2×C3×A4C22×D313yes—
142C2×C3×A4C23×C311yescharacteristic
143C3⋊S4—22no—
144C3×S4—66no—
145C2×C32×C4—31no—
146C2×C32×C4—31no—
147C2×C32×C4—31no—
148C2×C32×C4—31no—
149C2×C32×C4—31no—
150C2×C32×C4—31no—
151C2×C32×C4—31no—
152C2×C32×C4—41no—
153C2×C32×C4—43no—
154C2×C32×C4—61no—
155C2×C32×C4—83no—
156C2×C32×C4—81no—
157C2×C32×C4S411yescharacteristic, complemented
158#638—42no—
159C4⋊(C3⋊D3)—41no—
160C4×C3⋊D3—42no—
161#634—41no—
162C3×#67—62no—
163C3×#67—62no—
164C3×#67—64no—
165C3×#67—62no—
166C3×#67—62no—
167C3×#67—126no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C3×D12—61no—
172C3×D12—61no—
173C3×D12—123no—
174C3×C4×D3—62no—
175C3×C4×D3—62no—
176C3×C4×D3—126no—
177C3×Dic6—61no—
178C3×Dic6—61no—
179C3×Dic6—123no—
180#525—91noSylow
181C2×C33—41no—
182C2×C33—81no—
183C3×C3⋊D3—82no—
184C24×C3—21no—
185C24×C3C2×C3×D311yescharacteristic
186C24×C3C2×C3⋊D311yescharacteristic
187C22×A4—23no—
188C22×A4C2×C3×D313yes—
189C2×S4—36no—
190C3×#41—181no—
191C3×#41—181no—
192C3×#41—181no—
193C3×#41—181no—
194C2×C3×Q8—91no—
195C2×C3×D4—91no—
196C2×C3×D4—92no—
197C2×C3×D4—92no—
198C2×C3×D4—92no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—31no—
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×C3×C4—31no—
205C22×C3×C4—31no—
206C22×C3×C4—31no—
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—61no—
210C22×C3×C4—61no—
211C22×C3×C4—61no—
212C22×C3×C4—61no—
213C22×C3×C4—92no—
214C22×C3×C4C2×C3×D311yescharacteristic
215C22×C3×C4C2×C3×D311yescharacteristic
216C22×C3×C4C2×C3⋊D311yescharacteristic
217C22×C3×C4C2×C3⋊D311yescharacteristic
218C2×#67—32no—
219C2×#67—32no—
220C2×#67—32no—
221#235—61no—
222#235—61no—
223#235—61no—
224#235—61no—
225#235—123no—
226C2×D12—31no—
227C2×C4×D3—32no—
228C2×Dic6—31no—
229C4×A4—23no—
230C4×A4—23no—
231C4×A4C2×C3×D313yes—
232C4×A4C2×C3×D313yes—
233SL(2,ℤ/4)—36no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—91no—
238C3×C22⋊C4—92no—
239C3×C22⋊C4—92no—
240C3×C22⋊C4—92no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4—92no—
243C3×C42—92no—
244#217—32no—
245#217—32no—
246#217—32no—
247(C2×D3)⋊C4—32no—
248(C2×D3)⋊C4—32no—
249C4⋊Dic3—31no—
250C4⋊Dic3—31no—
251Dic3⋊C4—32no—
252Dic3⋊C4—32no—
253C4×Dic3—32no—
254C22×C32—31no—
255C22×C32—31no—
256C22×C32—32no—
257C22×C32—31no—
258C22×C32—41no—
259C22×C32—43no—
260C22×C32—61no—
261C22×C32—61no—
262C22×C32—61no—
263C22×C32—83no—
264C22×C32—81no—
265C22×C32#23511yescharacteristic, complemented
266C22×C32C2×S411yescharacteristic
267C2×C3⋊D3—42no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—126no—
272C3×A4—21no—
273C3×A4—23no—
274C3×A4#23513yescomplemented
275C3×A4C3×#4111yescharacteristic, complemented
276C32×C4—31no—
277C32×C4—31no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4—43no—
281C32×C4—43no—
282C32×C4—83no—
283C32×C4—83no—
284C32×C4—81no—
285C32×C4—81no—
286C32×C4C2×S411yescharacteristic
287C32×C4C2×S411yescharacteristic
288C3⋊Dic3—42no—
289C3×Dic3—62no—
290C3×Dic3—62no—
291C3×Dic3—62no—
292C3×Dic3—126no—
293C2×#41—91no—
294C23×C4C3×C3⋊D311yescharacteristic, complemented
295#123—92no—
296C22⋊Q8—91no—
297C22⋊Q8—91no—
298#121—91no—
299#121—91no—
300#120—92no—
301C4×D4—92no—
302C4×D4—92no—
303#117—91no—
304C33—41noSylow
305C23×C3—22no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—31no—
310C23×C3—31no—
311C23×C3—31no—
312C23×C3—31no—
313C23×C3—61no—
314C23×C3—61no—
315C23×C3—61no—
316C23×C3—61no—
317C23×C3—62no—
318C23×C3—92no—
319C23×C3C22×C3×D311yescharacteristic
320C23×C3C22×C3⋊D311yescharacteristic
321C22×D3—32no—
322C2×A4—26no—
323C2×A4—23no—
324C2×A4—23no—
325C2×A4C22×C3×D313yes—
326S4—66no—
327C3×Q8—181no—
328C3×Q8—181no—
329C3×D4—181no—
330C3×D4—182no—
331C3×D4—184no—
332C3×D4—182no—
333C3×D4—182no—
334C3×D4—182no—
335C3×D4—181no—
336C2×C3×C4—21no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—31no—
340C2×C3×C4—31no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—31no—
349C2×C3×C4—31no—
350C2×C3×C4—31no—
351C2×C3×C4—43no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—62no—
362C2×C3×C4—83no—
363C2×C3×C4—92no—
364C2×C3×C4—92no—
365C2×C3×C4—92no—
366C2×C3×C4—182no—
367C2×C3×C4—182no—
368C2×C3×C4C3×S411yescharacteristic, complemented
369C2×C3×C4C3⋊S411yescharacteristic, complemented
370#67—62no—
371#67—62no—
372#67—64no—
373#67—62no—
374#67—62no—
375#67—126no—
376C2×Dic3—32no—
377C2×Dic3—32no—
378C2×Dic3—32no—
379D12—61no—
380D12—61no—
381D12—123no—
382C4×D3—62no—
383C4×D3—62no—
384C4×D3—126no—
385Dic6—61no—
386Dic6—61no—
387Dic6—123no—
388C2×C32—21no—
389C2×C32—31no—
390C2×C32—31no—
391C2×C32—41no—
392C2×C32—43no—
393C2×C32—61no—
394C2×C32—83no—
395C2×C32—81no—
396C2×C32—83no—
397C2×C32—86no—
398C2×C32—82no—
399C2×C32—81no—
400C2×C32C22×S411yescharacteristic
401C3⋊D3—82no—
402C3×D3—122no—
403C3×D3—246no—
404C24C2×C3×C3⋊D311yescharacteristic
405#41—181no—
406#41—181no—
407#41—181no—
408#41—181no—
409C2×Q8—91no—
410C2×D4—91no—
411C2×D4—92no—
412C2×D4—92no—
413C2×D4—92no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—92no—
419C22×C4C2×C3×C3⋊D311yescharacteristic
420C22×C4C2×C3×C3⋊D311yescharacteristic
421C4⋊C4—91no—
422C4⋊C4—91no—
423C4⋊C4—92no—
424C4⋊C4—92no—
425C22⋊C4—92no—
426C22⋊C4—92no—
427C22⋊C4—92no—
428C22⋊C4—92no—
429C22⋊C4—92no—
430C42—92no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—31no—
434C22×C3—31no—
435C22×C3—31no—
436C22×C3—31no—
437C22×C3—32no—
438C22×C3—32no—
439C22×C3—31no—
440C22×C3—31no—
441C22×C3—43no—
442C22×C3—61no—
443C22×C3—61no—
444C22×C3—61no—
445C22×C3—62no—
446C22×C3—61no—
447C22×C3—61no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3—62no—
451C22×C3—62no—
452C22×C3—61no—
453C22×C3—61no—
454C22×C3—62no—
455C22×C3—83no—
456C22×C3—182no—
457C22×C3—182no—
458C22×C3—182no—
459C22×C3C3×#23511yescharacteristic, complemented
460C22×C3#382811yescharacteristic, complemented
461C22×C3C2×C3×S411yescharacteristic
462C22×C3C2×C3⋊S411yescharacteristic
463C2×D3—62no—
464C2×D3—62no—
465C2×D3—62no—
466C2×D3—126no—
467A4—23no—
468A4C3×#23513yescomplemented
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—31no—
472C3×C4—31no—
473C3×C4—31no—
474C3×C4—31no—
475C3×C4—43no—
476C3×C4—43no—
477C3×C4—61no—
478C3×C4—61no—
479C3×C4—83no—
480C3×C4—83no—
481C3×C4—182no—
482C3×C4—182no—
483C3×C4—182no—
484C3×C4C2×C3×S411yescharacteristic
485C3×C4C2×C3×S411yescharacteristic
486C3×C4C2×C3⋊S411yescharacteristic
487C3×C4C2×C3⋊S411yescharacteristic
488Dic3—62no—
489Dic3—62no—
490Dic3—62no—
491Dic3—126no—
492C32—41no—
493C32—43no—
494C32—83no—
495C32—81no—
496C32#591211yescharacteristic, complemented
497C23—21no—
498C23—31no—
499C23—31no—
500C23—61no—
501C23—92no—
502C23C22×C3×C3⋊D311yescharacteristic
503Q8—181no—
504Q8—181no—
505D4—181no—
506D4—182no—
507D4—182no—
508D4—184no—
509D4—182no—
510D4—182no—
511D4—181no—
512C2×C4—31no—
513C2×C4—31no—
514C2×C4—31no—
515C2×C4—31no—
516C2×C4—31no—
517C2×C4—31no—
518C2×C4—31no—
519C2×C4—61no—
520C2×C4—92no—
521C2×C4—92no—
522C2×C4—92no—
523C2×C4—182no—
524C2×C4—182no—
525C2×C4C3×C3⋊S411yescharacteristic, complemented
526C2×C3—22no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—31no—
531C2×C3—31no—
532C2×C3—31no—
533C2×C3—31no—
534C2×C3—43no—
535C2×C3—61no—
536C2×C3—61no—
537C2×C3—62no—
538C2×C3—61no—
539C2×C3—61no—
540C2×C3—83no—
541C2×C3—86no—
542C2×C3—83no—
543C2×C3—362no—
544C2×C3C22×C3×S411yescharacteristic
545C2×C3C22×C3⋊S411yescharacteristic
546D3—122no—
547D3—246no—
548C22—31no—
549C22—31no—
550C22—32no—
551C22—31no—
552C22—61no—
553C22—61no—
554C22—61no—
555C22—182no—
556C22—182no—
557C22—182no—
558C22C3×#382811yescharacteristic, complemented
559C22C2×C3×C3⋊S411yescharacteristic
560C4—31no—
561C4—31no—
562C4—182no—
563C4—182no—
564C4—182no—
565C4C2×C3×C3⋊S411yescharacteristic
566C4C2×C3×C3⋊S411yescharacteristic
567C3—21no—
568C3—43no—
569C3—83no—
570C3C3×#591211yescharacteristic, complemented
571C3#1059657511yescharacteristic, direct product factor
572C2—21no—
573C2—31no—
574C2—31no—
575C2—61no—
576C2—362no—
577C2C22×C3×C3⋊S411yescharacteristic
subgroupG/H
centerC3×C4C2×C3⋊S4
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2C22×C3×C3⋊S4
FittingC23×C32×C4D3
socleC23×C32C22×D3
hypercenterC2×C3×C4C3⋊S4
hypocenterC3×A4C3×#41
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×A4C2semidirect
2. C3×#64269C2semidirect
3. C3×C4⋊(C3⋊S4)C2semidirect
4. C3×C4×C3⋊S4C2semidirect
5. C3×#64259C2semidirect
6. #10596575C3direct (N × H)
15. C23×C32×C4D3semidirect
19. C2×C3×C4×A4D3semidirect
20. C2×C3×C4×A4C2×C3semidirect
21. #64269C2×C3semidirect
22. C4⋊(C3⋊S4)C2×C3semidirect
23. C4×C3⋊S4C2×C3semidirect
24. #64259C2×C3semidirect
96. C32×A4#41semidirect
102. C23×C3×C4C3×D3semidirect
103. C23×C3×C4C3⋊D3semidirect
106. C2×C4×A4C3×D3semidirect
157. C2×C32×C4S4semidirect
265. C22×C32#235semidirect
274. C3×A4#235semidirect
275. C3×A4C3×#41semidirect
294. C23×C4C3×C3⋊D3semidirect
368. C2×C3×C4C3×S4semidirect
369. C2×C3×C4C3⋊S4semidirect
459. C22×C3C3×#235semidirect
460. C22×C3#3828semidirect
468. A4C3×#235semidirect
496. C32#5912semidirect
525. C2×C4C3×C3⋊S4semidirect
558. C22C3×#3828semidirect
570. C3C3×#5912semidirect
571. C3#10596575direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C4×A42
C3×#642692
C3×C4⋊(C3⋊S4)2
C3×C4×C3⋊S42
C3×#642592
#105965753
C3×#59123
C3×#57913
C3×#38284