Finite Group Databasebeta

← C3×#10596499 #13181018

Subgroups — C3×#10596499 #13181018

Order 1728.

946 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64306C211yesmaximal, characteristic, complemented
2C2×C3×#3798C211yesmaximal, characteristic, complemented
3C3×#64283C211yesmaximal, characteristic, complemented
4C3×#64239C212yesmaximal, complemented
5C3×#64236C212yesmaximal, complemented
6C3×#64226C212yesmaximal, complemented
7C3×#64224C212yesmaximal, complemented
8C3×#64223C214yesmaximal, complemented
9#10596499C311yesmaximal, characteristic, direct product factor
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C32×#235—21no—
12C2×C32×C4×D3—21no—
13C3×#3811—21no—
14C2×C3×#627C2211yescharacteristic, complemented
15C2×C3×#626—21no—
16C2×C3×#625C2211yescharacteristic, complemented
17C3×#3805C2212yes—
18C3×#3802C2211yescharacteristic, complemented
19C3×C4×D32—21no—
20C3×#3798—21no—
21C3×#3798—21no—
22C3×#3798C2212yes—
23C3×#3798C2212yescomplemented
24C3×#3798C2212yescomplemented
25C3×#3797C2211yescharacteristic, complemented
26C3×#3796—21no—
27C3×#3792C2212yescomplemented
28C3×#3788C2212yescomplemented
29C3×#3787C2212yescomplemented
30C3×#3776C2214yescomplemented
31C3×#3775C2214yescomplemented
32C3×#3775C2214yescomplemented
33C3×#3774C2214yescomplemented
34C3×D3×#41—61no—
35C2×C3×#235—61no—
36#64306C2×C311yescharacteristic, complemented
37C2×#3798C2×C311yescharacteristic, complemented
38#64283C2×C311yescharacteristic, complemented
39#64239C2×C312yescomplemented
40#64236C2×C312yescomplemented
41#64226C2×C312yescomplemented
42#64224C2×C312yescomplemented
43#64223C2×C314yescomplemented
44C22×C3×C3⋊D3D411yescharacteristic
45C22×C32×D3—21no—
46C2×C3×D32—21no—
47C2×C33×C4D411yescharacteristic
48C2×C3×C3⋊Dic3C2311yescharacteristic
49C3×C4×C3⋊D3D411yescharacteristic
50C3×C4×C3⋊D3D411yescharacteristic
51C3×C4×C3⋊D3C2311yescharacteristic
52C3×C4×C3⋊D3C2311yescharacteristic
53C32×#67—22no—
54C2×C32×Dic3—21no—
55C32×D12—21no—
56C32×C4×D3—22no—
57C32×C4×D3—22no—
58C32×C4×D3—22no—
59C32×Dic6—21no—
60C3×#627—21no—
61C3×#627C2312yes—
62C3×#627C2312yes—
63C3×#626—22no—
64C3×#626—22no—
65C3×#626—22no—
66C3×#625—21no—
67C3×#625C2312yes—
68C3×#625C2312yes—
69C3×#624—21no—
70C3×D3×Dic3—22no—
71C3×#622C2314yes—
72C3×#575—91nomaximal
73C32×#41—61no—
74C22×C32×C4—61no—
75C2×C4×C3⋊D3C22×C311yescharacteristic
76C2×C3×#67—61no—
77C2×C3×#67—61no—
78C3×#239—61no—
79C3×D3×Q8—61no—
80C3×#237—61no—
81C3×#237—62no—
82C3×D3×D4—61no—
83C3×D3×D4—62no—
84C3×#235—21no—
85C3×#235—21no—
86C3×#235—41no—
87C3×#235—62no—
88C3×#235—62no—
89C3×#235—62no—
90C3×#235—62no—
91C3×#235—62no—
92C2×C3×D12—61no—
93C2×C3×C4×D3—21no—
94C2×C3×C4×D3—21no—
95C2×C3×C4×D3—41no—
96C2×C3×C4×D3—62no—
97C2×C3×C4×D3—61no—
98C2×C3×C4×D3—61no—
99C2×C3×Dic6—61no—
100#3811—21no—
101C2×#627C22×C311yescharacteristic, complemented
102C2×#626—21no—
103C2×#625C22×C311yescharacteristic, complemented
104#3805C22×C312yes—
105#3802C22×C311yescharacteristic, complemented
106C4×D32—21no—
107#3798—21no—
108#3798—21no—
109#3798C22×C312yes—
110#3798C22×C312yescomplemented
111#3798C22×C312yescomplemented
112#3797C22×C311yescharacteristic, complemented
113#3796—21no—
114#3792C22×C312yescomplemented
115#3788C22×C312yescomplemented
116#3787C22×C312yescomplemented
117#3776C22×C314yescomplemented
118#3775C22×C314yescomplemented
119#3775C22×C314yescomplemented
120#3774C22×C314yescomplemented
121C22×C33C2×D411yescharacteristic
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3C2×D411yescharacteristic
124C2×C3×C3⋊D3C2×D411yescharacteristic
125C2×C32×D3—22no—
126C2×C32×D3—22no—
127C2×C32×D3—42no—
128C3×D32—42no—
129C33×C4C2×D411yescharacteristic
130C33×C4C2×D411yescharacteristic
131C3×C3⋊Dic3C2×D411yescharacteristic
132C3×C3⋊Dic3C2411yescharacteristic
133C32×Dic3—22no—
134C32×Dic3—22no—
135C2×C3×#41—91no—
136C2×C3×#41—91no—
137D3×#41—61no—
138C2×#235—61no—
139C3×#137—92no—
140C3×#137—92no—
141C3×AGL(1,ℤ/8)—92no—
142C3×AGL(1,ℤ/8)—92no—
143C3×#135—94no—
144C2×C3×M16—91no—
145C23×C32—61no—
146C22×C3⋊D3C3×D411yescharacteristic
147C22×C3×D3—21no—
148C22×C3×D3—21no—
149C22×C3×D3—41no—
150C22×C3×D3—61no—
151C22×C3×D3—61no—
152C22×C3×D3—62no—
153C2×D32—21no—
154C32×Q8—61no—
155C32×D4—61no—
156C32×D4—62no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—41no—
160C2×C32×C4—41no—
161C2×C32×C4—62no—
162C2×C32×C4—62no—
163C2×C32×C4—62no—
164C2×C32×C4—61no—
165C2×C32×C4C3×D411yescharacteristic
166C2×C3⋊Dic3C23×C311yescharacteristic
167C4×C3⋊D3C3×D411yescharacteristic
168C4×C3⋊D3C3×D411yescharacteristic
169C4×C3⋊D3C23×C311yescharacteristic
170C4×C3⋊D3C23×C311yescharacteristic
171C3×#67—22no—
172C3×#67—22no—
173C3×#67—42no—
174C3×#67—62no—
175C3×#67—62no—
176C3×#67—62no—
177C3×#67—62no—
178C3×#67—62no—
179C3×#67—62no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—41no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—62no—
186C3×D12—21no—
187C3×D12—21no—
188C3×D12—41no—
189C3×D12—62no—
190C3×D12—62no—
191C3×D12—62no—
192C3×C4×D3—22no—
193C3×C4×D3—22no—
194C3×C4×D3—22no—
195C3×C4×D3—22no—
196C3×C4×D3—22no—
197C3×C4×D3—22no—
198C3×C4×D3—42no—
199C3×C4×D3—42no—
200C3×C4×D3—42no—
201C3×C4×D3—61no—
202C3×C4×D3—61no—
203C3×C4×D3—61no—
204C3×C4×D3—61no—
205C3×C4×D3—61no—
206C3×C4×D3—61no—
207C3×C4×D3—61no—
208C3×C4×D3—61no—
209C3×C4×D3—62no—
210C3×C4×D3—62no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—41no—
214C3×Dic6—62no—
215C3×Dic6—62no—
216C3×Dic6—62no—
217#627—21no—
218#627C23×C312yes—
219#627C23×C312yes—
220#626—22no—
221#626—22no—
222#626—22no—
223#625—21no—
224#625C23×C312yes—
225#625C23×C312yes—
226#624—21no—
227D3×Dic3—22no—
228#622C23×C314yes—
229#575—91noSylow
230C2×C33—21no—
231C2×C33C22×D411yescharacteristic
232C3×C3⋊D3—21no—
233C3×C3⋊D3—21no—
234C32×D3—42no—
235C32×D3—42no—
236C3×#41—61no—
237C3×#41—61no—
238C3×#41—92no—
239C3×#41—92no—
240C3×#41—92no—
241C3×#41—92no—
242C3×#41—121no—
243C3×#41—181no—
244C3×#41—181no—
245C3×#41—181no—
246C2×C3×Q8—91no—
247C2×C3×Q8—91no—
248C2×C3×D4—91no—
249C2×C3×D4—91no—
250C2×C3×D4—181no—
251C2×C3×D4—181no—
252C22×C3×C4—61no—
253C22×C3×C4—61no—
254C22×C3×C4—91no—
255C22×C3×C4—121no—
256C22×C3×C4—181no—
257C2×#67—61no—
258C2×#67—61no—
259#239—61no—
260D3×Q8—61no—
261#237—62no—
262#237—61no—
263D3×D4—61no—
264D3×D4—62no—
265#235—21no—
266#235—62no—
267#235—62no—
268#235—62no—
269#235—62no—
270#235—62no—
271C2×D12—61no—
272C2×C4×D3—21no—
273C2×C4×D3—61no—
274C2×C4×D3—61no—
275C2×C4×D3—62no—
276C2×Dic6—61no—
277C3×Q16—94no—
278C3×SD16—94no—
279C3×SD16—94no—
280C3×D8—94no—
281C3×M16—92no—
282C3×M16—92no—
283C2×C3×C8—92no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—41no—
287C22×C32—41no—
288C22×C32—62no—
289C22×C32—62no—
290C22×C32—122no—
291C22×C32C2×C3×D411yescharacteristic
292C2×C3⋊D3—41no—
293C2×C3⋊D3C2×C3×D411yescharacteristic
294C2×C3⋊D3C2×C3×D411yescharacteristic
295C2×C3×D3—22no—
296C2×C3×D3—22no—
297C2×C3×D3—22no—
298C2×C3×D3—22no—
299C2×C3×D3—42no—
300C2×C3×D3—42no—
301C2×C3×D3—42no—
302C2×C3×D3—42no—
303C2×C3×D3—44no—
304C2×C3×D3—61no—
305C2×C3×D3—61no—
306C2×C3×D3—61no—
307C2×C3×D3—61no—
308C2×C3×D3—124no—
309C2×C3×D3—122no—
310C2×C3×D3—122no—
311D32—42no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—41no—
317C32×C4—41no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4—62no—
321C32×C4—62no—
322C32×C4C2×C3×D411yescharacteristic
323C32×C4C2×C3×D411yescharacteristic
324C3⋊Dic3C2×C3×D411yescharacteristic
325C3⋊Dic3C24×C311yescharacteristic
326C3×Dic3—22no—
327C3×Dic3—22no—
328C3×Dic3—22no—
329C3×Dic3—22no—
330C3×Dic3—42no—
331C3×Dic3—42no—
332C3×Dic3—61no—
333C3×Dic3—61no—
334C3×Dic3—61no—
335C3×Dic3—61no—
336C2×#41—91no—
337C2×#41—91no—
338#137—92no—
339#137—92no—
340AGL(1,ℤ/8)—92no—
341AGL(1,ℤ/8)—92no—
342#135—94no—
343C2×M16—91no—
344C33#57511yescharacteristic, complemented, Sylow
345C23×C3—61no—
346C23×C3—61no—
347C23×C3—91no—
348C23×C3—121no—
349C23×C3—181no—
350C22×D3—21no—
351C22×D3—61no—
352C22×D3—61no—
353C22×D3—62no—
354C3×Q8—61no—
355C3×Q8—61no—
356C3×Q8—92no—
357C3×Q8—92no—
358C3×Q8—121no—
359C3×Q8—181no—
360C3×D4—61no—
361C3×D4—62no—
362C3×D4—62no—
363C3×D4—61no—
364C3×D4—92no—
365C3×D4—92no—
366C3×D4—121no—
367C3×D4—122no—
368C3×D4—182no—
369C3×D4—182no—
370C3×D4—182no—
371C3×D4—181no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—62no—
377C2×C3×C4—62no—
378C2×C3×C4—62no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—62no—
382C2×C3×C4—62no—
383C2×C3×C4—62no—
384C2×C3×C4—91no—
385C2×C3×C4—91no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—91no—
389C2×C3×C4—122no—
390C2×C3×C4—122no—
391C2×C3×C4—122no—
392C2×C3×C4—121no—
393C2×C3×C4—181no—
394C2×C3×C4—182no—
395C2×C3×C4GO+(4,𝔽2)11yescharacteristic
396#67—22no—
397#67—62no—
398#67—62no—
399#67—62no—
400#67—62no—
401#67—62no—
402#67—62no—
403C2×Dic3—21no—
404C2×Dic3—61no—
405C2×Dic3—61no—
406C2×Dic3—62no—
407D12—21no—
408D12—62no—
409D12—62no—
410D12—62no—
411C4×D3—22no—
412C4×D3—22no—
413C4×D3—22no—
414C4×D3—61no—
415C4×D3—61no—
416C4×D3—61no—
417C4×D3—61no—
418C4×D3—61no—
419C4×D3—61no—
420C4×D3—61no—
421C4×D3—61no—
422C4×D3—62no—
423C4×D3—62no—
424Dic6—21no—
425Dic6—62no—
426Dic6—62no—
427Dic6—62no—
428C3×C8—94no—
429C2×C32—22no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—42no—
437C2×C32—81no—
438C2×C32—122no—
439C2×C32—122no—
440C2×C32C22×C3×D411yescharacteristic
441C3⋊D3—21no—
442C3⋊D3—21no—
443C3×D3—42no—
444C3×D3—42no—
445C3×D3—44no—
446C3×D3—42no—
447C3×D3—42no—
448C3×D3—62no—
449C3×D3—82no—
450C3×D3—121no—
451C3×D3—121no—
452C3×D3—121no—
453#41—61no—
454#41—92no—
455#41—92no—
456#41—92no—
457#41—92no—
458#41—181no—
459#41—181no—
460#41—181no—
461C2×Q8—91no—
462C2×Q8—91no—
463C2×D4—91no—
464C2×D4—91no—
465C2×D4—181no—
466C2×D4—181no—
467C22×C4—61no—
468C22×C4—91no—
469C22×C4—181no—
470Q16—94no—
471SD16—94no—
472SD16—94no—
473D8—94no—
474M16—92no—
475M16—92no—
476C2×C8—92no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—62no—
482C22×C3—62no—
483C22×C3—62no—
484C22×C3—62no—
485C22×C3—91no—
486C22×C3—91no—
487C22×C3—122no—
488C22×C3—122no—
489C22×C3—122no—
490C22×C3—122no—
491C22×C3—124no—
492C22×C3—361no—
493C22×C3—362no—
494C22×C3C2×GO+(4,𝔽2)11yescharacteristic
495C2×D3—22no—
496C2×D3—22no—
497C2×D3—42no—
498C2×D3—61no—
499C2×D3—61no—
500C2×D3—61no—
501C2×D3—61no—
502C2×D3—124no—
503C2×D3—122no—
504C2×D3—122no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—41no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—62no—
514C3×C4—62no—
515C3×C4—62no—
516C3×C4—62no—
517C3×C4—91no—
518C3×C4—91no—
519C3×C4—122no—
520C3×C4—122no—
521C3×C4C2×GO+(4,𝔽2)11yescharacteristic
522C3×C4C2×GO+(4,𝔽2)11yescharacteristic
523Dic3—22no—
524Dic3—22no—
525Dic3—61no—
526Dic3—61no—
527Dic3—61no—
528Dic3—61no—
529C32—21no—
530C32—21no—
531C32—41no—
532C32—41no—
533C32C3×#57511yescharacteristic, complemented
534C23—61no—
535C23—91no—
536C23—181no—
537Q8—61no—
538Q8—92no—
539Q8—92no—
540Q8—181no—
541D4—62no—
542D4—61no—
543D4—92no—
544D4—92no—
545D4—182no—
546D4—182no—
547D4—182no—
548D4—181no—
549C2×C4—61no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—91no—
554C2×C4—91no—
555C2×C4—91no—
556C2×C4—91no—
557C2×C4—91no—
558C2×C4—181no—
559C2×C4—182no—
560C2×C4C3×GO+(4,𝔽2)11yescharacteristic
561C8—94no—
562C2×C3—22no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—41no—
567C2×C3—42no—
568C2×C3—41no—
569C2×C3—41no—
570C2×C3—81no—
571C2×C3—122no—
572C2×C3—122no—
573C2×C3—124no—
574C2×C3—122no—
575C2×C3—122no—
576C2×C3—181no—
577C2×C3—181no—
578C2×C3—242no—
579C2×C3C22×GO+(4,𝔽2)11yescharacteristic
580D3—42no—
581D3—42no—
582D3—62no—
583D3—121no—
584D3—121no—
585D3—121no—
586C22—62no—
587C22—62no—
588C22—91no—
589C22—91no—
590C22—122no—
591C22—361no—
592C22—362no—
593C22C2×C3×GO+(4,𝔽2)11yescharacteristic
594C4—62no—
595C4—62no—
596C4—91no—
597C4—91no—
598C4C2×C3×GO+(4,𝔽2)11yescharacteristic
599C4C2×C3×GO+(4,𝔽2)11yescharacteristic
600C3—21no—
601C3—21no—
602C3—41no—
603C3—41no—
604C3#1059649911yescharacteristic, direct product factor
605C2—21no—
606C2—122no—
607C2—122no—
608C2—181no—
609C2—181no—
610C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC3×C4C2×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C22×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#575
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#64306C2semidirect
2. C2×C3×#3798C2semidirect
3. C3×#64283C2semidirect
4. C3×#64239C2semidirect
5. C3×#64236C2semidirect
6. C3×#64226C2semidirect
7. C3×#64224C2semidirect
8. C3×#64223C2semidirect
9. #10596499C3direct (N × H)
14. C2×C3×#627C22semidirect
16. C2×C3×#625C22semidirect
18. C3×#3802C22semidirect
23. C3×#3798C22semidirect
24. C3×#3798C22semidirect
25. C3×#3797C22semidirect
27. C3×#3792C22semidirect
28. C3×#3788C22semidirect
29. C3×#3787C22semidirect
30. C3×#3776C22semidirect
31. C3×#3775C22semidirect
32. C3×#3775C22semidirect
33. C3×#3774C22semidirect
36. #64306C2×C3semidirect
37. C2×#3798C2×C3semidirect
38. #64283C2×C3semidirect
39. #64239C2×C3semidirect
40. #64236C2×C3semidirect
41. #64226C2×C3semidirect
42. #64224C2×C3semidirect
43. #64223C2×C3semidirect
101. C2×#627C22×C3semidirect
103. C2×#625C22×C3semidirect
105. #3802C22×C3semidirect
110. #3798C22×C3semidirect
111. #3798C22×C3semidirect
112. #3797C22×C3semidirect
114. #3792C22×C3semidirect
115. #3788C22×C3semidirect
116. #3787C22×C3semidirect
117. #3776C22×C3semidirect
118. #3775C22×C3semidirect
119. #3775C22×C3semidirect
120. #3774C22×C3semidirect
344. C33#575semidirect
533. C32C3×#575semidirect
604. C3#10596499direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643062
C2×C3×#37982
C3×#642832
C3×#642392
C3×#642362
C3×#642262
C3×#642242
C3×#642232
#105964993
C3×#5759