Finite Group Databasebeta

← C2×C3×#64224 #13181016

Subgroups — C2×C3×#64224 #13181016

Order 1728.

1104 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1104 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
1C2×C3×#3802C211yesmaximal, characteristic, complemented
2C2×C3×#3798C211yesmaximal, characteristic, complemented
3C2×C3×#3788C211yesmaximal, characteristic, complemented
4C2×C3×#3775C212yesmaximal, complemented
5C2×C3×#3774C212yesmaximal, complemented
6C3×#64224C218yesmaximal, direct product factor
7C2×#64224C311yesmaximal, characteristic, direct product factor
8C22×C3×D32—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×D12—21no—
11C2×C32×C4×D3—21no—
12C2×C3×#627C2211yescharacteristic, complemented
13C2×C3×#626—21no—
14C2×C3×#625C2212yes—
15C2×C3×#625C2211yescharacteristic, complemented
16C3×#3802—22no—
17C3×#3802C2214yescomplemented
18C3×#3798—22no—
19C3×#3798C2214yescomplemented
20C2×C3×#622C2212yescomplemented
21C3×#3788C2214yescomplemented
22C3×#3775C2218yescomplemented
23C3×#3774C2218yescomplemented
24C2×C3×D3×D4—61no—
25C2×C3×#235—61no—
26C2×#3802C2×C311yescharacteristic, complemented
27C2×#3798C2×C311yescharacteristic, complemented
28C2×#3788C2×C311yescharacteristic, complemented
29C2×#3775C2×C312yescomplemented
30C2×#3774C2×C312yescomplemented
31#64224C2×C318yesdirect product factor
32C22×C3×C3⋊D3D411yescharacteristic
33C22×C32×D3—22no—
34C22×C32×D3—21no—
35C2×C3×D32—24no—
36C2×C3×D32—44no—
37C2×C33×C4D411yescharacteristic
38C2×C3×C3⋊Dic3C2311yescharacteristic
39C3×C4×C3⋊D3D412yes—
40C3×C4×C3⋊D3C2312yes—
41C2×C32×Dic3—21no—
42C32×D12—24no—
43C32×C4×D3—24no—
44C3×#627—21no—
45C3×#627C2312yescomplemented
46C3×#626—24no—
47C3×#625—21no—
48C3×#625—22no—
49C3×#625C2314yes—
50C3×#625C2312yescomplemented
51C3×#622C2314yescomplemented
52C2×C3×AGL(1,ℤ/8)—91nomaximal
53C23×C3×D3—62no—
54C22×D32—21no—
55C2×C32×D4—61no—
56C22×C32×C4—61no—
57C2×C4×C3⋊D3C22×C311yescharacteristic
58C2×C3×#67—61no—
59C2×C3×#67—61no—
60C2×C3×#67—62no—
61C3×D3×D4—64no—
62C3×D3×D4—64no—
63C3×#235—64no—
64C3×#235—64no—
65C2×C3×D12—21no—
66C2×C3×D12—21no—
67C2×C3×D12—41no—
68C2×C3×D12—61no—
69C2×C3×C4×D3—21no—
70C2×C3×C4×D3—21no—
71C2×C3×C4×D3—41no—
72C2×C3×C4×D3—61no—
73C2×C3×C4×D3—61no—
74C2×C3×Dic6—61no—
75C2×#627C22×C311yescharacteristic, complemented
76C2×#626—21no—
77C2×#625C22×C312yes—
78C2×#625C22×C311yescharacteristic, complemented
79#3802—22no—
80#3802C22×C314yescomplemented
81#3798—22no—
82#3798C22×C314yescomplemented
83C2×#622C22×C312yescomplemented
84#3788C22×C314yescomplemented
85#3775C22×C318yescomplemented
86#3774C22×C318yescomplemented
87C22×C33C2×D411yescharacteristic
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3C2×D412yes—
90C2×C32×D3—24no—
91C2×C32×D3—22no—
92C2×C32×D3—42no—
93C2×C32×D3—44no—
94C3×D32—48no—
95C33×C4C2×D412yes—
96C3×C3⋊Dic3C2×D411yescharacteristic
97C3×C3⋊Dic3C2411yescharacteristic
98C32×Dic3—22no—
99C2×C3×#41—91no—
100C22×C3×D4—91no—
101C2×D3×D4—61no—
102C2×#235—61no—
103C3×AGL(1,ℤ/8)—98no—
104C2×C3×SD16—92no—
105C2×C3×D8—92no—
106C2×C3×M16—91no—
107C23×C32—62no—
108C23×C32—61no—
109C22×C3⋊D3C3×D411yescharacteristic
110C22×C3×D3—22no—
111C22×C3×D3—21no—
112C22×C3×D3—22no—
113C22×C3×D3—21no—
114C22×C3×D3—41no—
115C22×C3×D3—42no—
116C22×C3×D3—61no—
117C22×C3×D3—61no—
118C22×C3×D3—68no—
119C22×C3×D3—128no—
120C2×D32—24no—
121C2×D32—44no—
122C32×D4—64no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—41no—
126C2×C32×C4—41no—
127C2×C32×C4—64no—
128C2×C32×C4—61no—
129C2×C32×C4C3×D411yescharacteristic
130C2×C3⋊Dic3C23×C311yescharacteristic
131C4×C3⋊D3C3×D412yes—
132C4×C3⋊D3C23×C312yes—
133C3×#67—64no—
134C3×#67—62no—
135C3×#67—64no—
136C3×#67—62no—
137C3×#67—64no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—41no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C3×D12—24no—
144C3×D12—24no—
145C3×D12—44no—
146C3×D12—64no—
147C3×C4×D3—24no—
148C3×C4×D3—24no—
149C3×C4×D3—44no—
150C3×C4×D3—62no—
151C3×C4×D3—62no—
152C3×C4×D3—62no—
153C3×C4×D3—62no—
154C3×Dic6—62no—
155C3×Dic6—62no—
156#627—21no—
157#627C23×C312yescomplemented
158#626—24no—
159#625—21no—
160#625—22no—
161#625C23×C314yes—
162#625C23×C312yescomplemented
163#622C23×C314yescomplemented
164C2×AGL(1,ℤ/8)—91noSylow
165C2×C33AGL(1,ℤ/8)12yescomplemented
166C2×C33C22×D411yescharacteristic
167C3×C3⋊D3—22no—
168C32×D3—42no—
169C32×D3—44no—
170C24×C3—181no—
171C23×D3—62no—
172C3×#41—94no—
173C3×#41—182no—
174C2×C3×Q8—91no—
175C2×C3×D4—61no—
176C2×C3×D4—61no—
177C2×C3×D4—94no—
178C2×C3×D4—91no—
179C2×C3×D4—92no—
180C2×C3×D4—121no—
181C2×C3×D4—181no—
182C2×C3×D4—182no—
183C22×C3×C4—61no—
184C22×C3×C4—61no—
185C22×C3×C4—91no—
186C22×C3×C4—121no—
187C2×#67—62no—
188C2×#67—61no—
189C2×#67—61no—
190D3×D4—64no—
191D3×D4—64no—
192#235—64no—
193#235—64no—
194C2×D12—21no—
195C2×D12—61no—
196C2×C4×D3—21no—
197C2×C4×D3—61no—
198C2×C4×D3—61no—
199C2×Dic6—61no—
200C3×SD16—98no—
201C3×D8—98no—
202C3×M16—94no—
203C2×C3×C8—92no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—62no—
209C22×C32—64no—
210C22×C32—122no—
211C22×C32—124no—
212C22×C32C2×C3×D411yescharacteristic
213C2×C3⋊D3—22no—
214C2×C3⋊D3C2×C3×D412yes—
215C2×C3×D3—22no—
216C2×C3×D3—24no—
217C2×C3×D3—24no—
218C2×C3×D3—22no—
219C2×C3×D3—42no—
220C2×C3×D3—44no—
221C2×C3×D3—42no—
222C2×C3×D3—44no—
223C2×C3×D3—44no—
224C2×C3×D3—44no—
225C2×C3×D3—42no—
226C2×C3×D3—62no—
227C2×C3×D3—62no—
228C2×C3×D3—64no—
229C2×C3×D3—84no—
230C2×C3×D3—122no—
231C2×C3×D3—1216no—
232D32—48no—
233C32×C4—22no—
234C32×C4—22no—
235C32×C4—42no—
236C32×C4—42no—
237C32×C4—62no—
238C32×C4C2×C3×D412yes—
239C3⋊Dic3C2×C3×D411yescharacteristic
240C3⋊Dic3C24×C311yescharacteristic
241C3×Dic3—22no—
242C3×Dic3—22no—
243C3×Dic3—42no—
244C3×Dic3—61no—
245C3×Dic3—61no—
246C3×Dic3—61no—
247C3×Dic3—61no—
248C2×#41—91no—
249C22×D4—91no—
250AGL(1,ℤ/8)—98no—
251C2×SD16—92no—
252C2×D8—92no—
253C2×M16—91no—
254C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
255C23×C3—61no—
256C23×C3—62no—
257C23×C3—61no—
258C23×C3—62no—
259C23×C3—91no—
260C23×C3—121no—
261C23×C3—122no—
262C23×C3—184no—
263C23×C3—364no—
264C22×D3—21no—
265C22×D3—22no—
266C22×D3—61no—
267C22×D3—61no—
268C22×D3—68no—
269C22×D3—128no—
270C3×Q8—92no—
271C3×Q8—181no—
272C3×D4—64no—
273C3×D4—64no—
274C3×D4—92no—
275C3×D4—94no—
276C3×D4—124no—
277C3×D4—184no—
278C3×D4—182no—
279C3×D4—181no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—64no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—64no—
288C2×C3×C4—91no—
289C2×C3×C4—92no—
290C2×C3×C4—92no—
291C2×C3×C4—124no—
292C2×C3×C4—121no—
293C2×C3×C4GO+(4,𝔽2)11yescharacteristic
294#67—64no—
295#67—62no—
296#67—64no—
297#67—62no—
298#67—64no—
299C2×Dic3—21no—
300C2×Dic3—61no—
301C2×Dic3—61no—
302D12—24no—
303D12—64no—
304C4×D3—24no—
305C4×D3—62no—
306C4×D3—62no—
307C4×D3—62no—
308C4×D3—62no—
309Dic6—62no—
310Dic6—62no—
311C3×C8—94no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—42no—
317C2×C32—42no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—122no—
321C2×C32—124no—
322C2×C32C3×AGL(1,ℤ/8)12yescomplemented
323C2×C32C22×C3×D411yescharacteristic
324C3⋊D3—22no—
325C3×D3—42no—
326C3×D3—44no—
327C3×D3—44no—
328C3×D3—44no—
329C3×D3—42no—
330C3×D3—64no—
331C3×D3—84no—
332C3×D3—122no—
333C24—181no—
334#41—94no—
335#41—182no—
336C2×Q8—91no—
337C2×D4—61no—
338C2×D4—94no—
339C2×D4—91no—
340C2×D4—92no—
341C2×D4—181no—
342C2×D4—182no—
343C22×C4—61no—
344C22×C4—91no—
345SD16—98no—
346D8—98no—
347M16—94no—
348C2×C8—92no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—62no—
354C22×C3—64no—
355C22×C3—62no—
356C22×C3—64no—
357C22×C3—92no—
358C22×C3—122no—
359C22×C3—124no—
360C22×C3—124no—
361C22×C3—122no—
362C22×C3—124no—
363C22×C3—124no—
364C22×C3—122no—
365C22×C3—182no—
366C22×C3—244no—
367C22×C3—368no—
368C22×C3C2×GO+(4,𝔽2)11yescharacteristic
369C2×D3—22no—
370C2×D3—24no—
371C2×D3—44no—
372C2×D3—42no—
373C2×D3—62no—
374C2×D3—62no—
375C2×D3—64no—
376C2×D3—122no—
377C2×D3—1216no—
378C3×C4—22no—
379C3×C4—22no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—62no—
383C3×C4—62no—
384C3×C4—91no—
385C3×C4—91no—
386C3×C4—122no—
387C3×C4C2×GO+(4,𝔽2)12yes—
388Dic3—22no—
389Dic3—61no—
390Dic3—61no—
391Dic3—61no—
392Dic3—61no—
393C32—21no—
394C32—21no—
395C32—41no—
396C32—41no—
397C32C2×C3×AGL(1,ℤ/8)11yescharacteristic, complemented
398C23—61no—
399C23—62no—
400C23—91no—
401C23—184no—
402C23—364no—
403Q8—92no—
404Q8—181no—
405D4—64no—
406D4—92no—
407D4—94no—
408D4—184no—
409D4—182no—
410D4—181no—
411C2×C4—61no—
412C2×C4—64no—
413C2×C4—91no—
414C2×C4—92no—
415C2×C4—92no—
416C2×C4C3×GO+(4,𝔽2)11yescharacteristic
417C8—94no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—22no—
421C2×C3—22no—
422C2×C3—41no—
423C2×C3—41no—
424C2×C3—42no—
425C2×C3—42no—
426C2×C3—124no—
427C2×C3—122no—
428C2×C3—124no—
429C2×C3—124no—
430C2×C3—122no—
431C2×C3—182no—
432C2×C3—244no—
433C2×C3#6422412yesdirect product factor
434C2×C3C22×GO+(4,𝔽2)11yescharacteristic
435D3—44no—
436D3—42no—
437D3—64no—
438D3—122no—
439C22—62no—
440C22—64no—
441C22—92no—
442C22—124no—
443C22—122no—
444C22—182no—
445C22—368no—
446C22C2×C3×GO+(4,𝔽2)11yescharacteristic
447C4—62no—
448C4—91no—
449C4—91no—
450C4C2×C3×GO+(4,𝔽2)12yes—
451C3—21no—
452C3—21no—
453C3—41no—
454C3—41no—
455C3C2×#6422411yescharacteristic, direct product factor
456C2—124no—
457C2—122no—
458C2—182no—
459C2C3×#6422412yesdirect product factor
460C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×AGL(1,ℤ/8)
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3802C2semidirect
2. C2×C3×#3798C2semidirect
3. C2×C3×#3788C2semidirect
4. C2×C3×#3775C2semidirect
5. C2×C3×#3774C2semidirect
6. C3×#64224C2direct (N × H)
7. C2×#64224C3direct (N × H)
12. C2×C3×#627C22semidirect
15. C2×C3×#625C22semidirect
17. C3×#3802C22semidirect
19. C3×#3798C22semidirect
20. C2×C3×#622C22semidirect
21. C3×#3788C22semidirect
22. C3×#3775C22semidirect
23. C3×#3774C22semidirect
26. C2×#3802C2×C3semidirect
27. C2×#3798C2×C3semidirect
28. C2×#3788C2×C3semidirect
29. C2×#3775C2×C3semidirect
30. C2×#3774C2×C3semidirect
31. #64224C2×C3direct (N × H)
45. C3×#627C23semidirect
50. C3×#625C23semidirect
51. C3×#622C23semidirect
75. C2×#627C22×C3semidirect
78. C2×#625C22×C3semidirect
80. #3802C22×C3semidirect
82. #3798C22×C3semidirect
83. C2×#622C22×C3semidirect
84. #3788C22×C3semidirect
85. #3775C22×C3semidirect
86. #3774C22×C3semidirect
157. #627C23×C3semidirect
162. #625C23×C3semidirect
163. #622C23×C3semidirect
165. C2×C33AGL(1,ℤ/8)semidirect
254. C33C2×AGL(1,ℤ/8)semidirect
322. C2×C32C3×AGL(1,ℤ/8)semidirect
397. C32C2×C3×AGL(1,ℤ/8)semidirect
433. C2×C3#64224direct (N × H)
455. C3C2×#64224direct (N × H)
459. C2C3×#64224direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C2×C3×#37982
C2×C3×#37882
C2×C3×#37752
C2×C3×#37742
C3×#642242
C2×#642243
C2×C3×AGL(1,ℤ/8)9