Finite Group Databasebeta

← C4×D32×Dic3 #13170700

Subgroups — C4×D32×Dic3 #13170700

Order 1728.

2346 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D32C211yesmaximal, characteristic
2C2×D32×Dic3C212yesmaximal
3C4×#6294C211yesmaximal, characteristic, complemented
4C4×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
5C4×D3×C3⋊Dic3C212yesmaximal, complemented
6C3×C4×D3×Dic3C212yesmaximal, complemented
7Dic3×#624C212yesmaximal, complemented
8D3×Dic32C214yesmaximal, complemented
9C2×C4×D3×Dic3—32nomaximal
10C42×D32—31nomaximal
11C22×C3×D32C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×C4×D3C2212yes—
14C2×#6294C2212yes—
15C2×Dic3×C3⋊D3C2212yes—
16C2×D3×C3⋊Dic3C2214yes—
17C2×C3×#624C2211yescharacteristic
18C2×C3×D3×Dic3C2214yes—
19C2×C3×D3×Dic3C2212yes—
20C3×C4×D32C418yescomplemented
21D32×Dic3C4116yesdirect product factor
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
23C3×C4×C3⋊Dic3C2212yescomplemented
24C32×C4×Dic3C2211yescharacteristic, complemented
25#88813C2212yescomplemented
26Dic3×C3⋊Dic3C2212yescomplemented
27Dic3×C3⋊Dic3C2214yescomplemented
28C3×Dic32C2214yescomplemented
29C22×C3×C4×D3—32no—
30C22×D3×Dic3—34no—
31C2×C4×D32—32no—
32C2×C4×D32D311yescharacteristic
33C2×C4×C3⋊Dic3—32no—
34C42×C3⋊D3—31no—
35C2×C3×C4×Dic3—32no—
36C3×C42×D3—32no—
37C2×Dic32—34no—
38C4×#624—31no—
39C4×#624—32no—
40C4×D3×Dic3—32no—
41C4×D3×Dic3—32no—
42C4×D3×Dic3—34no—
43C4×D3×Dic3—34no—
44C4×D3×Dic3—32no—
45C4×D3×Dic3—32no—
46C4×D3×Dic3—34no—
47C4×D3×Dic3—62no—
48C4×D3×Dic3—61no—
49C4×D3×Dic3D312yesdirect product factor
50C22×C3×C3⋊D3C2311yescharacteristic
51C22×C32×D3C2312yes—
52C2×C3×D32C2×C418yes—
53C2×C3×D32C2×C414yes—
54C2×C33×C4C2311yescharacteristic
55C2×C3⋊(C3⋊Dic3)C2312yes—
56C2×C3×C3⋊Dic3C2314yes—
57C2×C3×C3⋊Dic3C2311yescharacteristic
58C3×C4×C3⋊D3C2×C414yescomplemented
59C2×C32×Dic3C2312yes—
60C2×C32×Dic3C2312yes—
61C32×C4×D3C2×C418yescomplemented
62#6294C2×C418yescomplemented
63Dic3×C3⋊D3C2×C418yescomplemented
64D3×C3⋊Dic3C2×C4116yescomplemented
65C3×#624C2×C414yescomplemented
66C3×D3×Dic3C2×C4116yescomplemented
67C3×D3×Dic3C2×C418yescomplemented
68C22×C4×Dic3—91no—
69C2×C42×D3—92no—
70C23×C3×D3—32no—
71C22×D32C2×D311yescharacteristic
72C22×C32×C4—32no—
73C22×C3⋊Dic3—34no—
74C2×C4×C3⋊D3—32no—
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C22×C3×Dic3—34no—
77C22×C3×Dic3—32no—
78C2×C3×C4×D3—22no—
79C2×C3×C4×D3—34no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—32no—
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—316no—
84C2×C3×C4×D3—61no—
85C2×C3×C4×D3C2×D312yes—
86C2×C3×C4×D3C2×D312yes—
87C2×#624—32no—
88C2×#624C2×D311yescharacteristic
89C2×D3×Dic3—34no—
90C2×D3×Dic3—34no—
91C2×D3×Dic3—34no—
92C2×D3×Dic3—34no—
93C2×D3×Dic3—332no—
94C2×D3×Dic3—62no—
95C2×D3×Dic3—64no—
96C2×D3×Dic3C2×D314yes—
97C2×D3×Dic3C2×D312yes—
98C4×D32—316no—
99C4×D32Dic318yesdirect product factor
100C32×C42—31no—
101C4×C3⋊Dic3—31no—
102C4×C3⋊Dic3—32no—
103C4×C3⋊Dic3—32no—
104C4×C3⋊Dic3—34no—
105C4×C3⋊Dic3—61no—
106C4×C3⋊Dic3—62no—
107C4×C3⋊Dic3—121no—
108C4×C3⋊Dic3C2×D312yescomplemented
109C3×C4×Dic3—21no—
110C3×C4×Dic3—32no—
111C3×C4×Dic3—32no—
112C3×C4×Dic3—34no—
113C3×C4×Dic3—34no—
114C3×C4×Dic3—62no—
115C3×C4×Dic3C2×D312yescomplemented
116Dic32—32no—
117Dic32—34no—
118Dic32—34no—
119Dic32—34no—
120Dic32—34no—
121Dic32—64no—
122Dic32—62no—
123Dic32C2×D314yescomplemented
124C22×C33C2411yescharacteristic
125C2×C3×C3⋊D3C22×C414yes—
126C2×C3×C3⋊D3C22×C412yes—
127C2×C32×D3C22×C418yes—
128C2×C32×D3C22×C414yes—
129C3×D32C42116yescomplemented
130C33×C4C22×C412yescomplemented
131C3⋊(C3⋊Dic3)C22×C414yescomplemented
132C3×C3⋊Dic3C22×C418yescomplemented
133C3×C3⋊Dic3C22×C412yescomplemented
134C32×Dic3C22×C414yescomplemented
135C32×Dic3C22×C414yescomplemented
136C23×C3×C4—91no—
137C23×Dic3—92no—
138C22×C4×D3—32no—
139C22×C4×D3—94no—
140C2×C3×C42—92no—
141C2×C4×Dic3—32no—
142C2×C4×Dic3—91no—
143C2×C4×Dic3—92no—
144C2×C4×Dic3—94no—
145C2×C4×Dic3—94no—
146C2×C4×Dic3—92no—
147C2×C4×Dic3—92no—
148C2×C4×Dic3—91no—
149C2×C4×Dic3—182no—
150C42×D3—32no—
151C42×D3—92no—
152C42×D3—94no—
153C42×D3—92no—
154C42×D3—94no—
155C42×D3—92no—
156C42×D3—181no—
157C23×C32—32no—
158C22×C3⋊D3C22×D311yescharacteristic
159C22×C3×D3—22no—
160C22×C3×D3—32no—
161C22×C3×D3—316no—
162C22×C3×D3—38no—
163C22×C3×D3—61no—
164C22×C3×D3C22×D312yes—
165C22×C3×D3C22×D312yes—
166C2×D32C4×D318yes—
167C2×D32C2×Dic314yes—
168C2×C32×C4—21no—
169C2×C32×C4—22no—
170C2×C32×C4—32no—
171C2×C32×C4—32no—
172C2×C32×C4—38no—
173C2×C32×C4—41no—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—32no—
177C2×C3⋊Dic3—34no—
178C2×C3⋊Dic3—316no—
179C2×C3⋊Dic3—64no—
180C2×C3⋊Dic3—62no—
181C2×C3⋊Dic3—122no—
182C2×C3⋊Dic3C22×D314yes—
183C2×C3⋊Dic3C22×D311yescharacteristic
184C4×C3⋊D3—38no—
185C4×C3⋊D3C2×Dic314yescomplemented
186C2×C3×Dic3—22no—
187C2×C3×Dic3—22no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—34no—
190C2×C3×Dic3—34no—
191C2×C3×Dic3—316no—
192C2×C3×Dic3—38no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3C22×D312yes—
196C2×C3×Dic3C22×D314yes—
197C2×C3×Dic3C22×D312yes—
198C3×C4×D3—28no—
199C3×C4×D3—316no—
200C3×C4×D3—38no—
201C3×C4×D3—38no—
202C3×C4×D3—38no—
203C3×C4×D3—64no—
204C3×C4×D3C4×D318yescomplemented
205C3×C4×D3C2×Dic318yescomplemented
206#624—38no—
207#624C2×Dic314yescomplemented
208D3×Dic3—316no—
209D3×Dic3—316no—
210D3×Dic3—316no—
211D3×Dic3—316no—
212D3×Dic3—616no—
213D3×Dic3—68no—
214D3×Dic3C4×D3116yesdirect product factor
215D3×Dic3C2×Dic318yescomplemented
216C22×C42—271noSylow
217C2×C33C23×C412yes—
218C2×C33C23×C411yescharacteristic
219C3×C3⋊D3C2×C4214yescomplemented
220C32×D3C2×C4218yescomplemented
221C24×C3—91no—
222C23×D3—32no—
223C22×C3×C4—32no—
224C22×C3×C4—32no—
225C22×C3×C4—62no—
226C22×C3×C4—91no—
227C22×C3×C4—98no—
228C22×C3×C4—92no—
229C22×C3×C4—94no—
230C22×C3×C4—91no—
231C22×Dic3—34no—
232C22×Dic3—32no—
233C22×Dic3—92no—
234C22×Dic3—916no—
235C22×Dic3—94no—
236C22×Dic3—94no—
237C22×Dic3—92no—
238C22×Dic3—184no—
239C2×C4×D3—34no—
240C2×C4×D3—32no—
241C2×C4×D3—32no—
242C2×C4×D3—32no—
243C2×C4×D3—316no—
244C2×C4×D3—61no—
245C2×C4×D3—932no—
246C2×C4×D3—94no—
247C2×C4×D3—94no—
248C2×C4×D3—94no—
249C2×C4×D3—182no—
250C2×C4×D3D3212yes—
251C3×C42—32no—
252C3×C42—61no—
253C3×C42—94no—
254C3×C42—92no—
255C4×Dic3—32no—
256C4×Dic3—32no—
257C4×Dic3—34no—
258C4×Dic3—34no—
259C4×Dic3—62no—
260C4×Dic3—94no—
261C4×Dic3—92no—
262C4×Dic3—94no—
263C4×Dic3—94no—
264C4×Dic3—94no—
265C4×Dic3—92no—
266C4×Dic3—91no—
267C4×Dic3—94no—
268C4×Dic3—92no—
269C4×Dic3—182no—
270C4×Dic3—184no—
271C4×Dic3—181no—
272C4×Dic3—182no—
273C4×Dic3—361no—
274C4×Dic3D3211yescharacteristic, direct product factor
275C22×C32—21no—
276C22×C32—22no—
277C22×C32—38no—
278C22×C32—34no—
279C22×C32—41no—
280C22×C32C23×D312yes—
281C22×C32C23×D311yescharacteristic
282C2×C3⋊D3C2×C4×D314yes—
283C2×C3⋊D3C22×Dic312yes—
284C2×C3×D3—28no—
285C2×C3×D3—24no—
286C2×C3×D3—38no—
287C2×C3×D3—34no—
288C2×C3×D3—332no—
289C2×C3×D3—64no—
290C2×C3×D3—62no—
291C2×C3×D3C2×C4×D318yes—
292C2×C3×D3C2×C4×D318yes—
293C2×C3×D3C2×C4×D314yes—
294C2×C3×D3C22×Dic314yes—
295D32C4×Dic3116yesdirect product factor
296C32×C4—22no—
297C32×C4—24no—
298C32×C4—34no—
299C32×C4—34no—
300C32×C4—42no—
301C32×C4C2×C4×D314yescomplemented
302C32×C4C22×Dic312yescomplemented
303C3⋊Dic3—34no—
304C3⋊Dic3—38no—
305C3⋊Dic3—68no—
306C3⋊Dic3—64no—
307C3⋊Dic3—124no—
308C3⋊Dic3C2×C4×D318yescomplemented
309C3⋊Dic3C22×Dic312yescomplemented
310C3×Dic3—24no—
311C3×Dic3—24no—
312C3×Dic3—38no—
313C3×Dic3—38no—
314C3×Dic3—34no—
315C3×Dic3—68no—
316C3×Dic3—62no—
317C3×Dic3C2×C4×D318yescomplemented
318C3×Dic3C2×C4×D314yescomplemented
319C3×Dic3C22×Dic314yescomplemented
320C23×C4—91no—
321C23×C4—272no—
322C2×C42—92no—
323C2×C42—271no—
324C2×C42—272no—
325C2×C42—271no—
326C2×C42—272no—
327C2×C42—274no—
328C33C22×C4211yescharacteristic, complemented, Sylow
329C23×C3—32no—
330C23×C3—32no—
331C23×C3—62no—
332C23×C3—98no—
333C23×C3—94no—
334C23×C3—91no—
335C22×D3—32no—
336C22×D3—316no—
337C22×D3—38no—
338C22×D3—61no—
339C22×D3C2×D3212yes—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—32no—
343C2×C3×C4—34no—
344C2×C3×C4—32no—
345C2×C3×C4—38no—
346C2×C3×C4—38no—
347C2×C3×C4—41no—
348C2×C3×C4—68no—
349C2×C3×C4—62no—
350C2×C3×C4—62no—
351C2×C3×C4—94no—
352C2×C3×C4—916no—
353C2×C3×C4—98no—
354C2×C3×C4—91no—
355C2×C3×C4—94no—
356C2×C3×C4—94no—
357C2×C3×C4C2×D3211yescharacteristic
358C2×C3×C4C2×D3212yes—
359C2×Dic3—34no—
360C2×Dic3—34no—
361C2×Dic3—32no—
362C2×Dic3—316no—
363C2×Dic3—38no—
364C2×Dic3—64no—
365C2×Dic3—61no—
366C2×Dic3—98no—
367C2×Dic3—916no—
368C2×Dic3—916no—
369C2×Dic3—94no—
370C2×Dic3—92no—
371C2×Dic3—98no—
372C2×Dic3—1816no—
373C2×Dic3—182no—
374C2×Dic3—184no—
375C2×Dic3—362no—
376C2×Dic3C2×D3212yes—
377C2×Dic3C2×D3212yes—
378C4×D3—316no—
379C4×D3—38no—
380C4×D3—38no—
381C4×D3—38no—
382C4×D3—64no—
383C4×D3—916no—
384C4×D3—916no—
385C4×D3—916no—
386C4×D3—188no—
387C4×D3D3×Dic318yesdirect product factor
388C2×C32—22no—
389C2×C32—24no—
390C2×C32—21no—
391C2×C32—22no—
392C2×C32—38no—
393C2×C32—42no—
394C2×C32—41no—
395C2×C32C22×C4×D314yes—
396C2×C32C22×C4×D312yes—
397C2×C32C22×C4×D312yes—
398C2×C32C23×Dic311yescharacteristic
399C3⋊D3C2×C4×Dic314yescomplemented
400C3×D3—28no—
401C3×D3—38no—
402C3×D3—64no—
403C3×D3C42×D318yescomplemented
404C3×D3C2×C4×Dic318yescomplemented
405C24—91no—
406C22×C4—32no—
407C22×C4—91no—
408C22×C4—98no—
409C22×C4—94no—
410C22×C4—92no—
411C22×C4—91no—
412C22×C4—272no—
413C22×C4—272no—
414C22×C4—2716no—
415C22×C4—274no—
416C42—31no—
417C42—94no—
418C42—92no—
419C42—272no—
420C42—274no—
421C42—272no—
422C42—271no—
423C22×C3—22no—
424C22×C3—21no—
425C22×C3—38no—
426C22×C3—38no—
427C22×C3—34no—
428C22×C3—34no—
429C22×C3—41no—
430C22×C3—68no—
431C22×C3—64no—
432C22×C3—916no—
433C22×C3—94no—
434C22×C3—92no—
435C22×C3C22×D3211yescharacteristic
436C22×C3C22×D3212yes—
437C2×D3—38no—
438C2×D3—34no—
439C2×D3—332no—
440C2×D3—62no—
441C2×D3—64no—
442C2×D3C4×D3218yes—
443C2×D3C2×D3×Dic314yes—
444C3×C4—22no—
445C3×C4—24no—
446C3×C4—38no—
447C3×C4—34no—
448C3×C4—34no—
449C3×C4—42no—
450C3×C4—64no—
451C3×C4—64no—
452C3×C4—98no—
453C3×C4—92no—
454C3×C4C4×D3212yescomplemented
455C3×C4C2×D3×Dic314yescomplemented
456Dic3—38no—
457Dic3—38no—
458Dic3—34no—
459Dic3—62no—
460Dic3—68no—
461Dic3—98no—
462Dic3—94no—
463Dic3—184no—
464Dic3—188no—
465Dic3—364no—
466Dic3C4×D3214yesdirect product factor
467Dic3C2×D3×Dic314yescomplemented
468C32—22no—
469C32—21no—
470C32—41no—
471C32C2×C42×D312yescomplemented
472C32C22×C4×Dic311yescharacteristic, complemented
473C23—32no—
474C23—94no—
475C23—98no—
476C23—91no—
477C2×C4—32no—
478C2×C4—32no—
479C2×C4—38no—
480C2×C4—94no—
481C2×C4—98no—
482C2×C4—916no—
483C2×C4—94no—
484C2×C4—91no—
485C2×C4—94no—
486C2×C4—278no—
487C2×C4—278no—
488C2×C4—2716no—
489C2×C4—272no—
490C2×C4D3311yescharacteristic
491C2×C3—24no—
492C2×C3—22no—
493C2×C3—21no—
494C2×C3—22no—
495C2×C3—38no—
496C2×C3—38no—
497C2×C3—42no—
498C2×C3—41no—
499C2×C3—68no—
500C2×C3—94no—
501C2×C3C2×C4×D3214yes—
502C2×C3C2×C4×D3212yes—
503C2×C3C2×C4×D3211yescharacteristic
504C2×C3C22×D3×Dic312yes—
505D3—38no—
506D3—64no—
507D3C4×D3×Dic318yesdirect product factor
508C22—38no—
509C22—34no—
510C22—916no—
511C22—94no—
512C22—92no—
513C22C2×D3311yescharacteristic
514C4—34no—
515C4—34no—
516C4—98no—
517C4—92no—
518C4—274no—
519C4D32×Dic312yesdirect product factor
520C3—21no—
521C3—22no—
522C3—41no—
523C3C42×D3211yescharacteristic, complemented
524C3C2×C4×D3×Dic312yescomplemented
525C2—38no—
526C2—94no—
527C2C4×D3312yes—
528C2C2×D32×Dic311yescharacteristic
subgroupG/H
center, hypercenterC2×C4D33
commutator, hypocenterC33C22×C42
FrattiniC22C2×D33
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC22×C3×D32C22
LCSG ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C4×#6294C2semidirect
4. C4×Dic3×C3⋊D3C2semidirect
5. C4×D3×C3⋊Dic3C2semidirect
6. C3×C4×D3×Dic3C2semidirect
7. Dic3×#624C2semidirect
8. D3×Dic32C2semidirect
20. C3×C4×D32C4semidirect
21. D32×Dic3C4direct (N × H)
22. C4×C3⋊(C3⋊Dic3)C22semidirect
23. C3×C4×C3⋊Dic3C22semidirect
24. C32×C4×Dic3C22semidirect
25. #88813C22semidirect
26. Dic3×C3⋊Dic3C22semidirect
27. Dic3×C3⋊Dic3C22semidirect
28. C3×Dic32C22semidirect
49. C4×D3×Dic3D3direct (N × H)
58. C3×C4×C3⋊D3C2×C4semidirect
61. C32×C4×D3C2×C4semidirect
62. #6294C2×C4semidirect
63. Dic3×C3⋊D3C2×C4semidirect
64. D3×C3⋊Dic3C2×C4semidirect
65. C3×#624C2×C4semidirect
66. C3×D3×Dic3C2×C4semidirect
67. C3×D3×Dic3C2×C4semidirect
99. C4×D32Dic3direct (N × H)
108. C4×C3⋊Dic3C2×D3semidirect
115. C3×C4×Dic3C2×D3semidirect
123. Dic32C2×D3semidirect
129. C3×D32C42semidirect
130. C33×C4C22×C4semidirect
131. C3⋊(C3⋊Dic3)C22×C4semidirect
132. C3×C3⋊Dic3C22×C4semidirect
133. C3×C3⋊Dic3C22×C4semidirect
134. C32×Dic3C22×C4semidirect
135. C32×Dic3C22×C4semidirect
185. C4×C3⋊D3C2×Dic3semidirect
204. C3×C4×D3C4×D3semidirect
205. C3×C4×D3C2×Dic3semidirect
207. #624C2×Dic3semidirect
214. D3×Dic3C4×D3direct (N × H)
215. D3×Dic3C2×Dic3semidirect
219. C3×C3⋊D3C2×C42semidirect
220. C32×D3C2×C42semidirect
274. C4×Dic3D32direct (N × H)
295. D32C4×Dic3direct (N × H)
301. C32×C4C2×C4×D3semidirect
302. C32×C4C22×Dic3semidirect
308. C3⋊Dic3C2×C4×D3semidirect
309. C3⋊Dic3C22×Dic3semidirect
317. C3×Dic3C2×C4×D3semidirect
318. C3×Dic3C2×C4×D3semidirect
319. C3×Dic3C22×Dic3semidirect
328. C33C22×C42semidirect
387. C4×D3D3×Dic3direct (N × H)
399. C3⋊D3C2×C4×Dic3semidirect
403. C3×D3C42×D3semidirect
404. C3×D3C2×C4×Dic3semidirect
454. C3×C4C4×D32semidirect
455. C3×C4C2×D3×Dic3semidirect
466. Dic3C4×D32direct (N × H)
467. Dic3C2×D3×Dic3semidirect
471. C32C2×C42×D3semidirect
472. C32C22×C4×Dic3semidirect
507. D3C4×D3×Dic3direct (N × H)
519. C4D32×Dic3direct (N × H)
523. C3C42×D32semidirect
524. C3C2×C4×D3×Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D322
C2×D32×Dic32
C4×#62942
C4×Dic3×C3⋊D32
C4×D3×C3⋊Dic32
C3×C4×D3×Dic32
Dic3×#6242
D3×Dic322
C2×C4×D3×Dic33
C42×D323