Finite Group Databasebeta

← C3×#10593521 #13166886

Subgroups — C3×#10593521 #13166886

Order 1728.

690 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 70 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×C3×#3805C211yesmaximal, characteristic, complemented
2C2×C3×#3775C211yesmaximal, characteristic, complemented
3C2×C3×#3774C211yesmaximal, characteristic, complemented
4C3×#63980C211yesmaximal, characteristic, complemented
5C3×C22⋊#622C211yesmaximal, characteristic, complemented
6C3×#63740C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic, complemented
8#10593521C311yesmaximal, characteristic, direct product factor
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C32×#67—21no—
11C2×C32×#67—21no—
12C2×C3×#627C2211yescharacteristic
13C2×C3×#625C2211yescharacteristic
14C2×C3×#625C2211yescharacteristic
15C3×#3805—22no—
16C3×#3805—22no—
17C2×C3×D3×Dic3—21no—
18C2×C3×#622C2211yescharacteristic
19C2×C3×#622C2211yescharacteristic
20C3×#3775—22no—
21C3×#3774—22no—
22C3×#3724C2211yescharacteristic
23C3×(C2×D3)⋊Dic3—21no—
24C2×C3×#237—61no—
25C2×#3805C2×C311yescharacteristic, complemented
26C2×#3775C2×C311yescharacteristic, complemented
27C2×#3774C2×C311yescharacteristic, complemented
28C3×#939—61no—
29#63980C2×C311yescharacteristic, complemented
30C22⋊#622C2×C311yescharacteristic, complemented
31#63740C2×C311yescharacteristic, complemented
32#63739C2×C311yescharacteristic, complemented
33C23×C33D411yescharacteristic
34C22×C32×D3—21no—
35C22×C32×D3—21no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3D411yescharacteristic
39C2×C3×C3⋊Dic3C2311yescharacteristic
40C32×#67—24no—
41C32×#67—42no—
42C2×C32×Dic3—21no—
43C2×C32×Dic3—21no—
44C3×#627—21no—
45C3×#627D412yescomplemented
46C3×#625—21no—
47C3×#625—21no—
48C3×#625—21no—
49C3×#625D412yescomplemented
50C3×D3×Dic3—24no—
51C3×#622—21no—
52C3×#622—21no—
53C3×#449—91nomaximal
54C2×C32×D4—61no—
55C2×C32×D4—61no—
56C22×C3⋊Dic3C22×C311yescharacteristic
57C2×C3×#67—21no—
58C2×C3×#67—21no—
59C2×C3×#67—21no—
60C2×C3×#67—21no—
61C2×C3×#67—41no—
62C2×C3×#67—41no—
63C2×C3×#67—61no—
64C2×C3×#67—61no—
65C22×C3×Dic3—61no—
66C22×C3×Dic3—61no—
67C22×C3×Dic3—61no—
68C3×#237—64no—
69C3×#237—64no—
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—61no—
72C2×#627C22×C311yescharacteristic
73C2×#625C22×C311yescharacteristic
74C2×#625C22×C311yescharacteristic
75#3805—22no—
76#3805—22no—
77C2×D3×Dic3—21no—
78C2×#622C22×C311yescharacteristic
79C2×#622C22×C311yescharacteristic
80#3775—22no—
81#3774—22no—
82C3×#217—61no—
83C3×(C2×D3)⋊C4—61no—
84C3×Dic3⋊C4—61no—
85#3724C22×C311yescharacteristic
86(C2×D3)⋊Dic3—21no—
87C22×C33—21no—
88C22×C33—21no—
89C22×C33—21no—
90C22×C33C2×D411yescharacteristic
91C2×C32×D3—22no—
92C2×C32×D3—41no—
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C3×C3⋊Dic3—21no—
98C3×C3⋊Dic3C2×D411yescharacteristic
99C3×C3⋊Dic3C2×D411yescharacteristic
100C32×Dic3—22no—
101C32×Dic3—41no—
102C2×C3×#41—91no—
103C2×#237—61no—
104C2×C3×SD16—91no—
105C2×C3×D8—91no—
106C3×#121—91no—
107#939—61no—
108C3×#103—91no—
109C3×#102—91no—
110C3×C22⋊C8—91no—
111C23×C32—21no—
112C23×C32—21no—
113C23×C32—41no—
114C23×C32—41no—
115C23×C32—61no—
116C23×C32—61no—
117C23×C32C3×D411yescharacteristic
118C22×C3×D3—21no—
119C22×C3×D3—21no—
120C22×C3×D3—21no—
121C22×C3×D3—21no—
122C22×C3×D3—41no—
123C22×C3×D3—41no—
124C32×D4—64no—
125C32×D4—122no—
126C2×C32×C4—61no—
127C2×C32×C4—61no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3C3×D411yescharacteristic
131C2×C3⋊Dic3C23×C311yescharacteristic
132C3×#67—24no—
133C3×#67—24no—
134C3×#67—44no—
135C3×#67—42no—
136C3×#67—42no—
137C3×#67—44no—
138C3×#67—62no—
139C3×#67—62no—
140C3×#67—121no—
141C3×#67—121no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—41no—
147C2×C3×Dic3—41no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—64no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—121no—
156C2×C3×Dic3—121no—
157C3×C4×D3—64no—
158C3×Dic6—62no—
159C3×Dic6—62no—
160#627—21no—
161#627C3×D412yescomplemented
162#625—21no—
163#625—21no—
164#625—21no—
165#625C3×D412yescomplemented
166D3×Dic3—24no—
167#622—21no—
168#622—21no—
169#449—91noSylow
170C2×C33—41no—
171C2×C33#12011yescharacteristic
172C2×C33#13511yescharacteristic
173C2×C33AGL(1,ℤ/8)11yescharacteristic
174C32×D3—42no—
175C32×D3—81no—
176C3×#41—182no—
177C3×#41—182no—
178C2×C3×Q8—91no—
179C2×C3×D4—61no—
180C2×C3×D4—61no—
181C2×C3×D4—61no—
182C2×C3×D4—61no—
183C2×C3×D4—91no—
184C2×C3×D4—91no—
185C2×C3×D4—121no—
186C2×C3×D4—121no—
187C22×C3×C4—91no—
188C22×C3×C4—181no—
189C2×#67—21no—
190C2×#67—21no—
191C2×#67—61no—
192C2×#67—61no—
193C22×Dic3—61no—
194C22×Dic3—61no—
195C22×Dic3—61no—
196#237—64no—
197#237—64no—
198C2×C4×D3—61no—
199C2×Dic6—61no—
200C3×SD16—182no—
201C3×D8—182no—
202C2×C3×C8—91no—
203C2×C3×C8—91no—
204C3×C4⋊C4—91no—
205C3×C22⋊C4—181no—
206#217—61no—
207(C2×D3)⋊C4—61no—
208Dic3⋊C4—61no—
209C22×C32—22no—
210C22×C32—22no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—42no—
223C22×C32—42no—
224C22×C32—62no—
225C22×C32—81no—
226C22×C32—81no—
227C22×C32—81no—
228C22×C32—81no—
229C22×C32—121no—
230C22×C32—121no—
231C22×C32—121no—
232C22×C32—121no—
233C22×C32—121no—
234C22×C32C2×C3×D411yescharacteristic
235C2×C3×D3—22no—
236C2×C3×D3—22no—
237C2×C3×D3—42no—
238C2×C3×D3—41no—
239C2×C3×D3—41no—
240C2×C3×D3—41no—
241C2×C3×D3—41no—
242C2×C3×D3—41no—
243C2×C3×D3—41no—
244C2×C3×D3—41no—
245C2×C3×D3—42no—
246C2×C3×D3—41no—
247C2×C3×D3—41no—
248C2×C3×D3—41no—
249C2×C3×D3—81no—
250C2×C3×D3—81no—
251C2×C3×D3—81no—
252C2×C3×D3—81no—
253C32×C4—62no—
254C32×C4—121no—
255C3⋊Dic3—21no—
256C3⋊Dic3C2×C3×D411yescharacteristic
257C3⋊Dic3C2×C3×D411yescharacteristic
258C3×Dic3—22no—
259C3×Dic3—22no—
260C3×Dic3—42no—
261C3×Dic3—41no—
262C3×Dic3—42no—
263C3×Dic3—41no—
264C3×Dic3—61no—
265C3×Dic3—61no—
266C3×Dic3—61no—
267C3×Dic3—61no—
268C3×Dic3—62no—
269C3×Dic3—121no—
270C2×#41—91no—
271C2×SD16—91no—
272C2×D8—91no—
273#121—91no—
274#103—91no—
275#102—91no—
276C22⋊C8—91no—
277C33#44911yescharacteristic, complemented, Sylow
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—61no—
283C23×C3—61no—
284C23×C3—61no—
285C23×C3—61no—
286C23×C3—121no—
287C23×C3—121no—
288C23×C3GO+(4,𝔽2)11yescharacteristic
289C22×D3—21no—
290C22×D3—21no—
291C3×Q8—92no—
292C3×Q8—181no—
293C3×D4—64no—
294C3×D4—64no—
295C3×D4—92no—
296C3×D4—124no—
297C3×D4—122no—
298C3×D4—122no—
299C3×D4—124no—
300C3×D4—181no—
301C3×D4—181no—
302C3×D4—181no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—91no—
308C2×C3×C4—91no—
309C2×C3×C4—121no—
310C2×C3×C4—121no—
311C2×C3×C4—184no—
312C2×C3×C4—181no—
313C2×C3×C4—181no—
314#67—24no—
315#67—42no—
316#67—62no—
317#67—62no—
318#67—121no—
319#67—121no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—61no—
323C2×Dic3—61no—
324C2×Dic3—61no—
325C2×Dic3—61no—
326C2×Dic3—64no—
327C2×Dic3—62no—
328C2×Dic3—62no—
329C2×Dic3—121no—
330C2×Dic3—121no—
331C4×D3—64no—
332Dic6—62no—
333Dic6—62no—
334C3×C8—181no—
335C3×C8—181no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—41no—
343C2×C32—41no—
344C2×C32—41no—
345C2×C32—41no—
346C2×C32—41no—
347C2×C32—41no—
348C2×C32—41no—
349C2×C32—42no—
350C2×C32—42no—
351C2×C32—82no—
352C2×C32—82no—
353C2×C32—122no—
354C2×C32—241no—
355C2×C32C3×#12011yescharacteristic
356C2×C32C3×#13511yescharacteristic
357C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
358C3×D3—42no—
359C3×D3—42no—
360C3×D3—82no—
361C3×D3—81no—
362C3×D3—82no—
363C3×D3—81no—
364#41—182no—
365#41—182no—
366C2×Q8—91no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—91no—
370C2×D4—91no—
371C22×C4—91no—
372C22×C4—181no—
373SD16—182no—
374D8—182no—
375C2×C8—91no—
376C2×C8—91no—
377C4⋊C4—91no—
378C22⋊C4—181no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—62no—
395C22×C3—62no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3—122no—
401C22×C3—121no—
402C22×C3—121no—
403C22×C3—121no—
404C22×C3—121no—
405C22×C3—121no—
406C22×C3—122no—
407C22×C3—121no—
408C22×C3—121no—
409C22×C3—121no—
410C22×C3—121no—
411C22×C3—121no—
412C22×C3—241no—
413C22×C3—241no—
414C22×C3—241no—
415C22×C3—241no—
416C22×C3C2×GO+(4,𝔽2)11yescharacteristic
417C2×D3—22no—
418C2×D3—41no—
419C2×D3—41no—
420C2×D3—41no—
421C2×D3—41no—
422C2×D3—41no—
423C3×C4—62no—
424C3×C4—62no—
425C3×C4—91no—
426C3×C4—91no—
427C3×C4—122no—
428C3×C4—121no—
429C3×C4—122no—
430C3×C4—121no—
431C3×C4—181no—
432Dic3—22no—
433Dic3—41no—
434Dic3—61no—
435Dic3—61no—
436Dic3—61no—
437Dic3—61no—
438Dic3—62no—
439Dic3—121no—
440C32—21no—
441C32—21no—
442C32—41no—
443C32—41no—
444C32C3×#44911yescharacteristic, complemented
445C23—61no—
446C23—61no—
447C23C3×GO+(4,𝔽2)11yescharacteristic
448Q8—92no—
449Q8—181no—
450D4—64no—
451D4—92no—
452D4—122no—
453D4—181no—
454D4—181no—
455D4—181no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—91no—
459C2×C4—91no—
460C2×C4—184no—
461C2×C4—181no—
462C2×C4—181no—
463C8—181no—
464C8—181no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—42no—
472C2×C3—42no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—41no—
476C2×C3—41no—
477C2×C3—41no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—82no—
481C2×C3—82no—
482C2×C3—122no—
483C2×C3—122no—
484C2×C3—242no—
485C2×C3—241no—
486C2×C3—242no—
487C2×C3—241no—
488C2×C3#6423611yescharacteristic
489C2×C3#6423711yescharacteristic
490C2×C3#6424111yescharacteristic
491D3—42no—
492D3—81no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—62no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22—121no—
501C22—121no—
502C22C2×C3×GO+(4,𝔽2)11yescharacteristic
503C4—62no—
504C4—91no—
505C4—91no—
506C4—121no—
507C4—181no—
508C3—21no—
509C3—21no—
510C3—41no—
511C3—41no—
512C3#1059352111yescharacteristic, direct product factor
513C2—41no—
514C2—122no—
515C2—241no—
516C2C3×#6423611yescharacteristic
517C2C3×#6423711yescharacteristic
518C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#449
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3805C2semidirect
2. C2×C3×#3775C2semidirect
3. C2×C3×#3774C2semidirect
4. C3×#63980C2semidirect
5. C3×C22⋊#622C2semidirect
6. C3×#63740C2semidirect
7. C3×#63739C2semidirect
8. #10593521C3direct (N × H)
25. C2×#3805C2×C3semidirect
26. C2×#3775C2×C3semidirect
27. C2×#3774C2×C3semidirect
29. #63980C2×C3semidirect
30. C22⋊#622C2×C3semidirect
31. #63740C2×C3semidirect
32. #63739C2×C3semidirect
45. C3×#627D4semidirect
49. C3×#625D4semidirect
161. #627C3×D4semidirect
165. #625C3×D4semidirect
277. C33#449semidirect
444. C32C3×#449semidirect
512. C3#10593521direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38052
C2×C3×#37752
C2×C3×#37742
C3×#639802
C3×C22⋊#6222
C3×#637402
C3×#637392
#105935213
C3×#4499