Finite Group Databasebeta

← C3×#10596501 #13181020

Subgroups — C3×#10596501 #13181020

Order 1728.

1090 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1090 conjugacy classes of subgroups — too many to draw. Only its 174 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
1C3×#64306C211yesmaximal, characteristic, complemented
2C2×C3×#3802C211yesmaximal, characteristic, complemented
3C3×#64283C211yesmaximal, characteristic, complemented
4C3×#64236C214yesmaximal, complemented
5C3×#64227C212yesmaximal, complemented
6C3×#64225C212yesmaximal, complemented
7C3×#64224C214yesmaximal, complemented
8#10596501C311yesmaximal, characteristic, direct product factor
9C22×C3×D32—21no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C32×#235—21no—
12C2×C32×D12—21no—
13C3×#3811—21no—
14C2×C3×#625C2212yescomplemented
15C3×#3805C2212yescomplemented
16C3×#3802—21no—
17C3×#3802—21no—
18C3×#3802C2212yes—
19C3×#3802C2212yescomplemented
20C3×#3802C2211yescharacteristic, complemented
21C3×C4×D32—21no—
22C3×#3798C2212yescomplemented
23C3×#3797C2211yescharacteristic, complemented
24C3×#3796—21no—
25C3×#3792C2212yescomplemented
26C3×#3788C2212yescomplemented
27C3×#3787C2212yescomplemented
28C3×#3775C2218yescomplemented
29C3×#3774C2218yescomplemented
30C3×D3×#41—61no—
31C2×C3×D3×D4—61no—
32#64306C2×C311yescharacteristic, complemented
33C2×#3802C2×C311yescharacteristic, complemented
34#64283C2×C311yescharacteristic, complemented
35#64236C2×C314yescomplemented
36#64227C2×C312yescomplemented
37#64225C2×C312yescomplemented
38#64224C2×C314yescomplemented
39C22×C3×C3⋊D3D411yescharacteristic
40C22×C32×D3—22no—
41C2×C3×D32—22no—
42C2×C3×D32—22no—
43C2×C3×D32—21no—
44C2×C3×D32—44no—
45C2×C33×C4D411yescharacteristic
46C2×C3×C3⋊Dic3C2311yescharacteristic
47C3×C4×C3⋊D3D411yescharacteristic
48C3×C4×C3⋊D3D411yescharacteristic
49C3×C4×C3⋊D3C2311yescharacteristic
50C3×C4×C3⋊D3C2311yescharacteristic
51C32×#67—22no—
52C32×D12—21no—
53C32×D12—22no—
54C32×D12—22no—
55C32×C4×D3—22no—
56C32×Dic6—21no—
57C3×#627C2312yescomplemented
58C3×#626—22no—
59C3×#625—22no—
60C3×#625C2314yes—
61C3×#625C2312yescomplemented
62C3×#624—21no—
63C3×D3×Dic3—22no—
64C3×#622C2314yescomplemented
65C2×C3×AGL(1,ℤ/8)—91nomaximal
66C23×C3×D3—62no—
67C22×D32—21no—
68C32×#41—61no—
69C2×C32×D4—61no—
70C2×C4×C3⋊D3C22×C311yescharacteristic
71C2×C3×#67—62no—
72C3×#239—61no—
73C3×D3×Q8—61no—
74C3×#237—61no—
75C3×#237—62no—
76C3×D3×D4—62no—
77C3×D3×D4—62no—
78C3×D3×D4—61no—
79C3×D3×D4—62no—
80C3×D3×D4—62no—
81C3×D3×D4—62no—
82C3×#235—21no—
83C3×#235—21no—
84C3×#235—41no—
85C3×#235—62no—
86C2×C3×D12—21no—
87C2×C3×D12—21no—
88C2×C3×D12—41no—
89C2×C3×C4×D3—62no—
90C2×C3×C4×D3—61no—
91C2×C3×C4×D3—61no—
92#3811—21no—
93C2×#625C22×C312yescomplemented
94#3805C22×C312yescomplemented
95#3802—21no—
96#3802—21no—
97#3802C22×C312yes—
98#3802C22×C312yescomplemented
99#3802C22×C311yescharacteristic, complemented
100C4×D32—21no—
101#3798C22×C312yescomplemented
102#3797C22×C311yescharacteristic, complemented
103#3796—21no—
104#3792C22×C312yescomplemented
105#3788C22×C312yescomplemented
106#3787C22×C312yescomplemented
107#3775C22×C318yescomplemented
108#3774C22×C318yescomplemented
109C22×C33C2×D411yescharacteristic
110C2×C3×C3⋊D3—22no—
111C2×C3×C3⋊D3C2×D411yescharacteristic
112C2×C3×C3⋊D3C2×D411yescharacteristic
113C2×C32×D3—22no—
114C2×C32×D3—24no—
115C2×C32×D3—44no—
116C3×D32—44no—
117C3×D32—42no—
118C3×D32—44no—
119C33×C4C2×D411yescharacteristic
120C33×C4C2×D411yescharacteristic
121C3×C3⋊Dic3C2×D411yescharacteristic
122C3×C3⋊Dic3C2411yescharacteristic
123C32×Dic3—22no—
124C2×C3×#41—91no—
125C22×C3×D4—91no—
126D3×#41—61no—
127C2×D3×D4—61no—
128C3×AGL(1,ℤ/8)—94no—
129C3×AGL(1,ℤ/8)—94no—
130C2×C3×SD16—92no—
131C2×C3×D8—92no—
132C2×C3×M16—91no—
133C23×C32—62no—
134C22×C3⋊D3C3×D411yescharacteristic
135C22×C3×D3—22no—
136C22×C3×D3—22no—
137C22×C3×D3—42no—
138C22×C3×D3—61no—
139C22×C3×D3—61no—
140C22×C3×D3—64no—
141C22×C3×D3—64no—
142C22×C3×D3—62no—
143C22×C3×D3—128no—
144C2×D32—22no—
145C2×D32—22no—
146C2×D32—21no—
147C2×D32—44no—
148C32×Q8—61no—
149C32×D4—62no—
150C32×D4—62no—
151C32×D4—61no—
152C32×D4—62no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—41no—
156C2×C32×C4—41no—
157C2×C32×C4—62no—
158C2×C32×C4C3×D411yescharacteristic
159C2×C3⋊Dic3C23×C311yescharacteristic
160C4×C3⋊D3C3×D411yescharacteristic
161C4×C3⋊D3C3×D411yescharacteristic
162C4×C3⋊D3C23×C311yescharacteristic
163C4×C3⋊D3C23×C311yescharacteristic
164C3×#67—22no—
165C3×#67—22no—
166C3×#67—42no—
167C3×#67—64no—
168C3×#67—62no—
169C3×#67—62no—
170C3×#67—64no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—62no—
174C3×D12—22no—
175C3×D12—22no—
176C3×D12—21no—
177C3×D12—21no—
178C3×D12—22no—
179C3×D12—22no—
180C3×D12—42no—
181C3×D12—42no—
182C3×D12—41no—
183C3×D12—62no—
184C3×C4×D3—22no—
185C3×C4×D3—22no—
186C3×C4×D3—42no—
187C3×C4×D3—61no—
188C3×C4×D3—61no—
189C3×C4×D3—61no—
190C3×C4×D3—61no—
191C3×C4×D3—61no—
192C3×C4×D3—61no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195C3×C4×D3—62no—
196C3×C4×D3—62no—
197C3×Dic6—21no—
198C3×Dic6—21no—
199C3×Dic6—41no—
200C3×Dic6—62no—
201#627C23×C312yescomplemented
202#626—22no—
203#625—22no—
204#625C23×C314yes—
205#625C23×C312yescomplemented
206#624—21no—
207D3×Dic3—22no—
208#622C23×C314yescomplemented
209C2×AGL(1,ℤ/8)—91noSylow
210C2×C33—21no—
211C2×C33C22×D411yescharacteristic
212C3×C3⋊D3—21no—
213C3×C3⋊D3AGL(1,ℤ/8)12yescomplemented
214C32×D3—44no—
215C32×D3—42no—
216C24×C3—181no—
217C23×D3—62no—
218C3×#41—61no—
219C3×#41—61no—
220C3×#41—92no—
221C3×#41—92no—
222C3×#41—121no—
223C3×#41—181no—
224C2×C3×Q8—91no—
225C2×C3×D4—61no—
226C2×C3×D4—61no—
227C2×C3×D4—91no—
228C2×C3×D4—92no—
229C2×C3×D4—92no—
230C2×C3×D4—92no—
231C2×C3×D4—121no—
232C2×C3×D4—181no—
233C2×C3×D4—181no—
234C2×C3×D4—181no—
235C22×C3×C4—91no—
236C22×C3×C4—181no—
237C2×#67—62no—
238#239—61no—
239D3×Q8—61no—
240#237—62no—
241#237—61no—
242D3×D4—62no—
243D3×D4—62no—
244D3×D4—61no—
245D3×D4—62no—
246D3×D4—62no—
247D3×D4—62no—
248#235—21no—
249#235—62no—
250C2×D12—21no—
251C2×C4×D3—61no—
252C2×C4×D3—61no—
253C2×C4×D3—62no—
254C3×SD16—98no—
255C3×D8—98no—
256C3×M16—92no—
257C3×M16—92no—
258C2×C3×C8—92no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—41no—
262C22×C32—41no—
263C22×C32—62no—
264C22×C32—64no—
265C22×C32—124no—
266C22×C32C2×C3×D411yescharacteristic
267C2×C3⋊D3—22no—
268C2×C3⋊D3C2×C3×D411yescharacteristic
269C2×C3⋊D3C2×C3×D411yescharacteristic
270C2×C3×D3—22no—
271C2×C3×D3—24no—
272C2×C3×D3—22no—
273C2×C3×D3—24no—
274C2×C3×D3—42no—
275C2×C3×D3—44no—
276C2×C3×D3—44no—
277C2×C3×D3—44no—
278C2×C3×D3—61no—
279C2×C3×D3—61no—
280C2×C3×D3—61no—
281C2×C3×D3—61no—
282C2×C3×D3—64no—
283C2×C3×D3—84no—
284C2×C3×D3—122no—
285C2×C3×D3—124no—
286C2×C3×D3—128no—
287C2×C3×D3—128no—
288D32—44no—
289D32—42no—
290D32—44no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—41no—
296C32×C4—41no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4—62no—
300C32×C4C2×C3×D411yescharacteristic
301C32×C4C2×C3×D411yescharacteristic
302C3⋊Dic3C2×C3×D411yescharacteristic
303C3⋊Dic3C24×C311yescharacteristic
304C3×Dic3—22no—
305C3×Dic3—22no—
306C3×Dic3—42no—
307C3×Dic3—61no—
308C3×Dic3—61no—
309C3×Dic3—61no—
310C3×Dic3—61no—
311C2×#41—91no—
312C22×D4—91no—
313AGL(1,ℤ/8)—94no—
314AGL(1,ℤ/8)—94no—
315C2×SD16—92no—
316C2×D8—92no—
317C2×M16—91no—
318C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
319C23×C3—62no—
320C23×C3—62no—
321C23×C3—91no—
322C23×C3—122no—
323C23×C3—181no—
324C23×C3—182no—
325C23×C3—182no—
326C23×C3—364no—
327C22×D3—22no—
328C22×D3—61no—
329C22×D3—61no—
330C22×D3—62no—
331C22×D3—64no—
332C22×D3—64no—
333C22×D3—128no—
334C3×Q8—61no—
335C3×Q8—61no—
336C3×Q8—92no—
337C3×Q8—121no—
338C3×D4—62no—
339C3×D4—62no—
340C3×D4—61no—
341C3×D4—62no—
342C3×D4—62no—
343C3×D4—62no—
344C3×D4—62no—
345C3×D4—61no—
346C3×D4—92no—
347C3×D4—94no—
348C3×D4—122no—
349C3×D4—122no—
350C3×D4—121no—
351C3×D4—122no—
352C3×D4—182no—
353C3×D4—182no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4—91no—
361C2×C3×C4—91no—
362C2×C3×C4—91no—
363C2×C3×C4—91no—
364C2×C3×C4—91no—
365C2×C3×C4—122no—
366C2×C3×C4—181no—
367C2×C3×C4—182no—
368C2×C3×C4GO+(4,𝔽2)11yescharacteristic
369#67—22no—
370#67—64no—
371#67—62no—
372#67—62no—
373#67—64no—
374C2×Dic3—61no—
375C2×Dic3—61no—
376C2×Dic3—62no—
377D12—22no—
378D12—22no—
379D12—21no—
380D12—62no—
381C4×D3—22no—
382C4×D3—61no—
383C4×D3—61no—
384C4×D3—61no—
385C4×D3—61no—
386C4×D3—61no—
387C4×D3—61no—
388C4×D3—61no—
389C4×D3—61no—
390C4×D3—62no—
391C4×D3—62no—
392Dic6—21no—
393Dic6—62no—
394C3×C8—94no—
395C2×C32—22no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—41no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—42no—
403C2×C32—81no—
404C2×C32—124no—
405C2×C32—122no—
406C2×C32C22×C3×D411yescharacteristic
407C3⋊D3—21no—
408C3⋊D3C3×AGL(1,ℤ/8)12yescomplemented
409C3×D3—44no—
410C3×D3—42no—
411C3×D3—44no—
412C3×D3—42no—
413C3×D3—62no—
414C3×D3—62no—
415C3×D3—62no—
416C3×D3—82no—
417C3×D3—84no—
418C3×D3—121no—
419C24—181no—
420#41—61no—
421#41—92no—
422#41—92no—
423#41—181no—
424C2×Q8—91no—
425C2×D4—61no—
426C2×D4—91no—
427C2×D4—92no—
428C2×D4—92no—
429C2×D4—92no—
430C2×D4—181no—
431C2×D4—181no—
432C2×D4—181no—
433C22×C4—91no—
434C22×C4—181no—
435SD16—98no—
436D8—98no—
437M16—92no—
438M16—92no—
439C2×C8—92no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—41no—
443C22×C3—41no—
444C22×C3—62no—
445C22×C3—64no—
446C22×C3—62no—
447C22×C3—64no—
448C22×C3—91no—
449C22×C3—91no—
450C22×C3—122no—
451C22×C3—124no—
452C22×C3—124no—
453C22×C3—124no—
454C22×C3—182no—
455C22×C3—244no—
456C22×C3—364no—
457C22×C3—362no—
458C22×C3—364no—
459C22×C3C2×GO+(4,𝔽2)11yescharacteristic
460C2×D3—22no—
461C2×D3—24no—
462C2×D3—44no—
463C2×D3—61no—
464C2×D3—61no—
465C2×D3—61no—
466C2×D3—61no—
467C2×D3—64no—
468C2×D3—122no—
469C2×D3—128no—
470C2×D3—128no—
471C2×D3—124no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—62no—
481C3×C4—62no—
482C3×C4—91no—
483C3×C4—91no—
484C3×C4—122no—
485C3×C4C2×GO+(4,𝔽2)11yescharacteristic
486C3×C4C2×GO+(4,𝔽2)11yescharacteristic
487Dic3—22no—
488Dic3—61no—
489Dic3—61no—
490Dic3—61no—
491Dic3—61no—
492C32—21no—
493C32—21no—
494C32—41no—
495C32—41no—
496C32C2×C3×AGL(1,ℤ/8)11yescharacteristic, complemented
497C23—62no—
498C23—91no—
499C23—181no—
500C23—182no—
501C23—182no—
502C23—364no—
503Q8—61no—
504Q8—92no—
505D4—62no—
506D4—62no—
507D4—62no—
508D4—61no—
509D4—92no—
510D4—94no—
511D4—182no—
512D4—182no—
513C2×C4—62no—
514C2×C4—91no—
515C2×C4—91no—
516C2×C4—91no—
517C2×C4—91no—
518C2×C4—91no—
519C2×C4—181no—
520C2×C4—182no—
521C2×C4C3×GO+(4,𝔽2)11yescharacteristic
522C8—94no—
523C2×C3—22no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—41no—
528C2×C3—42no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—81no—
532C2×C3—92no—
533C2×C3—124no—
534C2×C3—122no—
535C2×C3—124no—
536C2×C3—122no—
537C2×C3—181no—
538C2×C3—244no—
539C2×C3—242no—
540C2×C3C22×GO+(4,𝔽2)11yescharacteristic
541D3—44no—
542D3—42no—
543D3—62no—
544D3—62no—
545D3—62no—
546D3—121no—
547C22—62no—
548C22—64no—
549C22—91no—
550C22—91no—
551C22—124no—
552C22—182no—
553C22—364no—
554C22—362no—
555C22—364no—
556C22C2×C3×GO+(4,𝔽2)11yescharacteristic
557C4—62no—
558C4—91no—
559C4—91no—
560C4C2×C3×GO+(4,𝔽2)11yescharacteristic
561C4C2×C3×GO+(4,𝔽2)11yescharacteristic
562C3—21no—
563C3—21no—
564C3—41no—
565C3—41no—
566C3#1059650111yescharacteristic, direct product factor
567C2—21no—
568C2—92no—
569C2—124no—
570C2—122no—
571C2—181no—
572C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C22×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C22×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×AGL(1,ℤ/8)
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#64306C2semidirect
2. C2×C3×#3802C2semidirect
3. C3×#64283C2semidirect
4. C3×#64236C2semidirect
5. C3×#64227C2semidirect
6. C3×#64225C2semidirect
7. C3×#64224C2semidirect
8. #10596501C3direct (N × H)
14. C2×C3×#625C22semidirect
15. C3×#3805C22semidirect
19. C3×#3802C22semidirect
20. C3×#3802C22semidirect
22. C3×#3798C22semidirect
23. C3×#3797C22semidirect
25. C3×#3792C22semidirect
26. C3×#3788C22semidirect
27. C3×#3787C22semidirect
28. C3×#3775C22semidirect
29. C3×#3774C22semidirect
32. #64306C2×C3semidirect
33. C2×#3802C2×C3semidirect
34. #64283C2×C3semidirect
35. #64236C2×C3semidirect
36. #64227C2×C3semidirect
37. #64225C2×C3semidirect
38. #64224C2×C3semidirect
57. C3×#627C23semidirect
61. C3×#625C23semidirect
64. C3×#622C23semidirect
93. C2×#625C22×C3semidirect
94. #3805C22×C3semidirect
98. #3802C22×C3semidirect
99. #3802C22×C3semidirect
101. #3798C22×C3semidirect
102. #3797C22×C3semidirect
104. #3792C22×C3semidirect
105. #3788C22×C3semidirect
106. #3787C22×C3semidirect
107. #3775C22×C3semidirect
108. #3774C22×C3semidirect
201. #627C23×C3semidirect
205. #625C23×C3semidirect
208. #622C23×C3semidirect
213. C3×C3⋊D3AGL(1,ℤ/8)semidirect
318. C33C2×AGL(1,ℤ/8)semidirect
408. C3⋊D3C3×AGL(1,ℤ/8)semidirect
496. C32C2×C3×AGL(1,ℤ/8)semidirect
566. C3#10596501direct (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×#38022
C3×#642832
C3×#642362
C3×#642272
C3×#642252
C3×#642242
#105965013
C2×C3×AGL(1,ℤ/8)9