Finite Group Databasebeta

← C2×C3×#64236 #13181049

Subgroups — C2×C3×#64236 #13181049

Order 1728.

1184 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 178 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
1C22×C3×#625C211yesmaximal, characteristic, complemented
2C2×C3×#3805C211yesmaximal, characteristic, complemented
3C2×C3×#3792C211yesmaximal, characteristic, complemented
4C2×C3×#3775C212yesmaximal, complemented
5C3×#64236C218yesmaximal, direct product factor
6C2×C3×#3774C212yesmaximal, complemented
7C2×#64236C311yesmaximal, characteristic, direct product factor
8C23×C32×D3—21no—
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C32×#67—21no—
11C2×C3×#627C2211yescharacteristic, complemented
12C2×C3×#625—22no—
13C2×C3×#625—21no—
14C2×C3×#625C2212yes—
15C2×C3×#625C2214yescomplemented
16C2×C3×#625C2211yescharacteristic, complemented
17C3×#3805—22no—
18C3×#3805C2214yescomplemented
19C2×C3×D3×Dic3—21no—
20C3×#3792C2214yescomplemented
21C2×C3×#622C2212yescomplemented
22C3×#3775C2218yescomplemented
23C3×#3774C2218yescomplemented
24C22×C3×#67—61no—
25C2×C3×#237—61no—
26C22×#625C2×C311yescharacteristic, complemented
27C2×#3805C2×C311yescharacteristic, complemented
28C2×#3792C2×C311yescharacteristic, complemented
29C2×#3775C2×C312yescomplemented
30#64236C2×C318yesdirect product factor
31C2×#3774C2×C312yescomplemented
32C23×C33D411yescharacteristic
33C22×C32×D3—22no—
34C22×C32×D3—24no—
35C22×C32×D3—21no—
36C22×C32×D3—44no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C3×C3⋊Dic3D412yes—
39C2×C3×C3⋊Dic3C2311yescharacteristic
40C2×C3×C3⋊Dic3C2312yes—
41C32×#67—24no—
42C2×C32×Dic3—21no—
43C3×#627—21no—
44C3×#627C2312yescomplemented
45C3×#625—21no—
46C3×#625—22no—
47C3×#625—24no—
48C3×#625C2314yes—
49C3×#625C2312yescomplemented
50C3×D3×Dic3—24no—
51C3×#622C2314yescomplemented
52C2×C3×AGL(1,ℤ/8)—91nomaximal
53C24×C32—61no—
54C23×C3×D3—21no—
55C23×C3×D3—21no—
56C23×C3×D3—41no—
57C2×C32×D4—61no—
58C22×C3⋊Dic3C22×C311yescharacteristic
59C2×C3×#67—21no—
60C2×C3×#67—21no—
61C2×C3×#67—41no—
62C2×C3×#67—61no—
63C2×C3×#67—62no—
64C2×C3×#67—64no—
65C2×C3×#67—64no—
66C2×C3×#67—62no—
67C22×C3×Dic3—61no—
68C22×C3×Dic3—61no—
69C22×C3×Dic3—61no—
70C3×#237—64no—
71C3×#237—64no—
72C2×C3×C4×D3—61no—
73C2×C3×Dic6—61no—
74C2×#627C22×C311yescharacteristic, complemented
75C2×#625—22no—
76C2×#625—21no—
77C2×#625C22×C312yes—
78C2×#625C22×C314yescomplemented
79C2×#625C22×C311yescharacteristic, complemented
80#3805—22no—
81#3805C22×C314yescomplemented
82C2×D3×Dic3—21no—
83#3792C22×C314yescomplemented
84C2×#622C22×C312yescomplemented
85#3775C22×C318yescomplemented
86#3774C22×C318yescomplemented
87C22×C33—22no—
88C22×C33C2×D412yes—
89C22×C33C2×D411yescharacteristic
90C2×C32×D3—22no—
91C2×C32×D3—24no—
92C2×C32×D3—48no—
93C2×C32×D3—44no—
94C2×C32×D3—42no—
95C3×C3⋊Dic3C2×D412yes—
96C3×C3⋊Dic3C2×D411yescharacteristic
97C3×C3⋊Dic3C2411yescharacteristic
98C32×Dic3—22no—
99C2×C3×#41—91no—
100C22×C3×D4—91no—
101C22×#67—61no—
102C2×#237—61no—
103C3×AGL(1,ℤ/8)—98no—
104C2×C3×SD16—92no—
105C2×C3×D8—92no—
106C2×C3×M16—91no—
107C23×C32—21no—
108C23×C32—21no—
109C23×C32—41no—
110C23×C32—41no—
111C23×C32—64no—
112C23×C32—62no—
113C23×C32—61no—
114C23×C32—124no—
115C23×C32C3×D411yescharacteristic
116C22×C3×D3—24no—
117C22×C3×D3—22no—
118C22×C3×D3—21no—
119C22×C3×D3—22no—
120C22×C3×D3—24no—
121C22×C3×D3—21no—
122C22×C3×D3—42no—
123C22×C3×D3—44no—
124C22×C3×D3—41no—
125C22×C3×D3—44no—
126C22×C3×D3—44no—
127C22×C3×D3—48no—
128C32×D4—64no—
129C2×C32×C4—61no—
130C2×C3⋊Dic3C3×D411yescharacteristic
131C2×C3⋊Dic3C3×D412yes—
132C2×C3⋊Dic3C23×C311yescharacteristic
133C2×C3⋊Dic3C23×C312yes—
134C3×#67—24no—
135C3×#67—24no—
136C3×#67—44no—
137C3×#67—64no—
138C3×#67—62no—
139C3×#67—62no—
140C3×#67—64no—
141C3×#67—68no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—41no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—64no—
154C3×C4×D3—64no—
155C3×Dic6—62no—
156C3×Dic6—62no—
157#627—21no—
158#627C23×C312yescomplemented
159#625—21no—
160#625—22no—
161#625—24no—
162#625C23×C314yes—
163#625C23×C312yescomplemented
164D3×Dic3—24no—
165#622C23×C314yescomplemented
166C2×AGL(1,ℤ/8)—91noSylow
167C2×C33—22no—
168C2×C33AGL(1,ℤ/8)12yescomplemented
169C2×C33C22×D411yescharacteristic
170C32×D3—44no—
171C32×D3—42no—
172C24×C3—61no—
173C24×C3—61no—
174C24×C3—121no—
175C23×D3—21no—
176C3×#41—94no—
177C3×#41—182no—
178C2×C3×Q8—91no—
179C2×C3×D4—61no—
180C2×C3×D4—61no—
181C2×C3×D4—91no—
182C2×C3×D4—92no—
183C2×C3×D4—94no—
184C2×C3×D4—121no—
185C2×C3×D4—182no—
186C2×C3×D4—181no—
187C22×C3×C4—91no—
188C22×C3×C4—181no—
189C2×#67—21no—
190C2×#67—62no—
191C2×#67—61no—
192C2×#67—64no—
193C2×#67—62no—
194C2×#67—64no—
195C22×Dic3—61no—
196C22×Dic3—61no—
197C22×Dic3—61no—
198#237—64no—
199#237—64no—
200C2×C4×D3—61no—
201C2×Dic6—61no—
202C3×SD16—98no—
203C3×D8—98no—
204C3×M16—94no—
205C2×C3×C8—92no—
206C22×C32—24no—
207C22×C32—22no—
208C22×C32—22no—
209C22×C32—22no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—42no—
215C22×C32—42no—
216C22×C32—42no—
217C22×C32—44no—
218C22×C32—62no—
219C22×C32—64no—
220C22×C32—82no—
221C22×C32—128no—
222C22×C32—124no—
223C22×C32—122no—
224C22×C32C2×C3×D412yes—
225C22×C32C2×C3×D411yescharacteristic
226C2×C3×D3—22no—
227C2×C3×D3—24no—
228C2×C3×D3—22no—
229C2×C3×D3—24no—
230C2×C3×D3—44no—
231C2×C3×D3—42no—
232C2×C3×D3—42no—
233C2×C3×D3—48no—
234C2×C3×D3—48no—
235C2×C3×D3—44no—
236C2×C3×D3—48no—
237C2×C3×D3—416no—
238C2×C3×D3—42no—
239C2×C3×D3—44no—
240C2×C3×D3—82no—
241C32×C4—62no—
242C3⋊Dic3C2×C3×D412yes—
243C3⋊Dic3C2×C3×D411yescharacteristic
244C3⋊Dic3C24×C311yescharacteristic
245C3×Dic3—22no—
246C3×Dic3—22no—
247C3×Dic3—42no—
248C3×Dic3—61no—
249C3×Dic3—62no—
250C3×Dic3—62no—
251C3×Dic3—61no—
252C3×Dic3—61no—
253C3×Dic3—61no—
254C2×#41—91no—
255C22×D4—91no—
256AGL(1,ℤ/8)—98no—
257C2×SD16—92no—
258C2×D8—92no—
259C2×M16—91no—
260C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—61no—
266C23×C3—62no—
267C23×C3—64no—
268C23×C3—64no—
269C23×C3—61no—
270C23×C3—62no—
271C23×C3—124no—
272C23×C3—122no—
273C23×C3—121no—
274C23×C3—124no—
275C23×C3—124no—
276C23×C3—128no—
277C23×C3GO+(4,𝔽2)11yescharacteristic
278C22×D3—21no—
279C22×D3—22no—
280C22×D3—24no—
281C22×D3—44no—
282C3×Q8—92no—
283C3×Q8—181no—
284C3×D4—64no—
285C3×D4—64no—
286C3×D4—92no—
287C3×D4—94no—
288C3×D4—124no—
289C3×D4—184no—
290C3×D4—182no—
291C3×D4—181no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—92no—
295C2×C3×C4—92no—
296C2×C3×C4—91no—
297C2×C3×C4—91no—
298C2×C3×C4—121no—
299C2×C3×C4—184no—
300#67—24no—
301#67—64no—
302#67—62no—
303#67—62no—
304#67—64no—
305#67—68no—
306C2×Dic3—21no—
307C2×Dic3—61no—
308C2×Dic3—62no—
309C2×Dic3—61no—
310C2×Dic3—62no—
311C2×Dic3—61no—
312C2×Dic3—62no—
313C2×Dic3—61no—
314C2×Dic3—62no—
315C2×Dic3—64no—
316C4×D3—64no—
317Dic6—62no—
318Dic6—62no—
319C3×C8—94no—
320C2×C32—24no—
321C2×C32—22no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—22no—
325C2×C32—22no—
326C2×C32—42no—
327C2×C32—42no—
328C2×C32—41no—
329C2×C32—41no—
330C2×C32—42no—
331C2×C32—44no—
332C2×C32—82no—
333C2×C32—124no—
334C2×C32—122no—
335C2×C32C3×AGL(1,ℤ/8)12yescomplemented
336C2×C32C22×C3×D411yescharacteristic
337C3×D3—42no—
338C3×D3—44no—
339C3×D3—48no—
340C3×D3—42no—
341C3×D3—44no—
342C3×D3—82no—
343C24—61no—
344#41—94no—
345#41—182no—
346C2×Q8—91no—
347C2×D4—61no—
348C2×D4—91no—
349C2×D4—92no—
350C2×D4—94no—
351C2×D4—181no—
352C2×D4—182no—
353C22×C4—91no—
354C22×C4—181no—
355SD16—98no—
356D8—98no—
357M16—94no—
358C2×C8—92no—
359C22×C3—24no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—22no—
363C22×C3—21no—
364C22×C3—22no—
365C22×C3—42no—
366C22×C3—44no—
367C22×C3—41no—
368C22×C3—42no—
369C22×C3—41no—
370C22×C3—42no—
371C22×C3—62no—
372C22×C3—64no—
373C22×C3—62no—
374C22×C3—64no—
375C22×C3—82no—
376C22×C3—124no—
377C22×C3—122no—
378C22×C3—128no—
379C22×C3—128no—
380C22×C3—124no—
381C22×C3—122no—
382C22×C3—1216no—
383C22×C3—128no—
384C22×C3—124no—
385C22×C3—122no—
386C22×C3—242no—
387C22×C3C2×GO+(4,𝔽2)12yes—
388C22×C3C2×GO+(4,𝔽2)11yescharacteristic
389C2×D3—22no—
390C2×D3—24no—
391C2×D3—48no—
392C2×D3—44no—
393C2×D3—42no—
394C3×C4—62no—
395C3×C4—62no—
396C3×C4—91no—
397C3×C4—92no—
398C3×C4—91no—
399C3×C4—122no—
400Dic3—22no—
401Dic3—61no—
402Dic3—61no—
403Dic3—62no—
404Dic3—62no—
405Dic3—61no—
406Dic3—61no—
407C32—21no—
408C32—21no—
409C32—41no—
410C32—41no—
411C32C2×C3×AGL(1,ℤ/8)11yescharacteristic, complemented
412C23—61no—
413C23—64no—
414C23—62no—
415C23—124no—
416C23C3×GO+(4,𝔽2)11yescharacteristic
417Q8—92no—
418Q8—181no—
419D4—64no—
420D4—92no—
421D4—94no—
422D4—184no—
423D4—182no—
424D4—181no—
425C2×C4—61no—
426C2×C4—91no—
427C2×C4—92no—
428C2×C4—91no—
429C2×C4—92no—
430C2×C4—184no—
431C8—94no—
432C2×C3—24no—
433C2×C3—22no—
434C2×C3—22no—
435C2×C3—22no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—42no—
439C2×C3—44no—
440C2×C3—42no—
441C2×C3—42no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—82no—
445C2×C3—124no—
446C2×C3—122no—
447C2×C3—128no—
448C2×C3—124no—
449C2×C3—122no—
450C2×C3—242no—
451C2×C3#6423612yesdirect product factor
452C2×C3C22×GO+(4,𝔽2)11yescharacteristic
453D3—44no—
454D3—42no—
455C22—22no—
456C22—62no—
457C22—64no—
458C22—128no—
459C22—124no—
460C22—122no—
461C22C2×C3×GO+(4,𝔽2)12yes—
462C22C2×C3×GO+(4,𝔽2)11yescharacteristic
463C4—62no—
464C4—91no—
465C4—92no—
466C4—91no—
467C3—21no—
468C3—21no—
469C3—41no—
470C3—41no—
471C3C2×#6423611yescharacteristic, direct product factor
472C2—22no—
473C2—124no—
474C2—122no—
475C2C3×#6423612yesdirect product factor
476C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C2×C3×AGL(1,ℤ/8)
LCSG ▷ 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. C22×C3×#625C2semidirect
2. C2×C3×#3805C2semidirect
3. C2×C3×#3792C2semidirect
4. C2×C3×#3775C2semidirect
5. C3×#64236C2direct (N × H)
6. C2×C3×#3774C2semidirect
7. C2×#64236C3direct (N × H)
11. C2×C3×#627C22semidirect
15. C2×C3×#625C22semidirect
16. C2×C3×#625C22semidirect
18. C3×#3805C22semidirect
20. C3×#3792C22semidirect
21. C2×C3×#622C22semidirect
22. C3×#3775C22semidirect
23. C3×#3774C22semidirect
26. C22×#625C2×C3semidirect
27. C2×#3805C2×C3semidirect
28. C2×#3792C2×C3semidirect
29. C2×#3775C2×C3semidirect
30. #64236C2×C3direct (N × H)
31. C2×#3774C2×C3semidirect
44. C3×#627C23semidirect
49. C3×#625C23semidirect
51. C3×#622C23semidirect
74. C2×#627C22×C3semidirect
78. C2×#625C22×C3semidirect
79. C2×#625C22×C3semidirect
81. #3805C22×C3semidirect
83. #3792C22×C3semidirect
84. C2×#622C22×C3semidirect
85. #3775C22×C3semidirect
86. #3774C22×C3semidirect
158. #627C23×C3semidirect
163. #625C23×C3semidirect
165. #622C23×C3semidirect
168. C2×C33AGL(1,ℤ/8)semidirect
260. C33C2×AGL(1,ℤ/8)semidirect
335. C2×C32C3×AGL(1,ℤ/8)semidirect
411. C32C2×C3×AGL(1,ℤ/8)semidirect
451. C2×C3#64236direct (N × H)
471. C3C2×#64236direct (N × H)
475. C2C3×#64236direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6252
C2×C3×#38052
C2×C3×#37922
C2×C3×#37752
C3×#642362
C2×C3×#37742
C2×#642363
C2×C3×AGL(1,ℤ/8)9