Finite Group Databasebeta

← C3⋊D3×#872 #13175388

Subgroups — C3⋊D3×#872 #13175388

Order 1728.

2284 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic
2#11723913C211yesmaximal, characteristic, complemented
3C3×C42×C3⋊D3C211yesmaximal, characteristic, complemented
4C32×#872C211yesmaximal, characteristic, complemented
5C3⋊D3×(C2×D3)⋊C4C212yesmaximal
6#11723469C212yesmaximal, complemented
7C3⋊D3×Dic3⋊C4C212yesmaximal, complemented
8C4×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
9#11723459C212yesmaximal, complemented
10#11723452C211yesmaximal, characteristic, complemented
11#11723423C211yesmaximal, characteristic, complemented
12C3⋊D3×#117—31nomaximal
13D3×#872—34nomaximal
14C22×D3×C3⋊D3C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C32×C4×D3C2211yescharacteristic
19C2×#6289C2211yescharacteristic
20C2×Dic3×C3⋊D3C2212yes—
21C2×Dic3×C3⋊D3C2211yescharacteristic
22C2×D3×C3⋊Dic3C2211yescharacteristic
23C4×D3×C3⋊D3C418yescomplemented
24C33×C42C2211yescharacteristic, complemented
25#88859C2212yes—
26#88857C2212yescomplemented
27C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
28C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
29C3×C4×C3⋊Dic3C2212yescomplemented
30C32×(C2×D3)⋊C4C2212yes—
31C32×Dic3⋊C4C2212yescomplemented
32C32×C4×Dic3C2211yescharacteristic, complemented
33#88806C2212yescomplemented
34Dic3⋊(C3⋊Dic3)C2212yescomplemented
35#88804C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37Dic3×C3⋊Dic3C2212yescomplemented
38C22×C4×C3⋊D3—31no—
39C2×C4×D32—34no—
40C32×#117—31no—
41#64135—31no—
42#64101—32no—
43C4⋊C4×C3⋊D3—32no—
44C22⋊C4×C3⋊D3—32no—
45#64086—32no—
46#64081—31no—
47#64081—31no—
48#64081—34no—
49#64081—64no—
50C42×C3⋊D3—31no—
51C42×C3⋊D3D311yescharacteristic, complemented
52C3×#872D314yescomplemented
53C3×C42×D3—34no—
54D3×(C2×D3)⋊C4—38no—
55#63877—38no—
56D3×Dic3⋊C4—38no—
57C4×D3×Dic3—34no—
58#63874—34no—
59#63855—34no—
60#63836—38no—
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C22×C3×C3⋊D3C2311yescharacteristic
63C22×C32×D3C2311yescharacteristic
64C2×D3×C3⋊D3—24no—
65C2×D3×C3⋊D3C2×C414yes—
66C2×C33×C4C2312yes—
67C2×C33×C4C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C4×C3⋊(C3⋊D3)C2×C414yescomplemented
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C3×C4×C3⋊D3—24no—
74C3×C4×C3⋊D3C2×C414yes—
75C2×C32×Dic3C2312yes—
76C2×C32×Dic3C2311yescharacteristic
77C32×C4×D3C2×C414yescomplemented
78#6289C2×C414yescomplemented
79Dic3×C3⋊D3—24no—
80Dic3×C3⋊D3C2×C414yes—
81D3×C3⋊Dic3C2×C414yescomplemented
82D3×#117—94no—
83C2×#872—91no—
84C23×C3⋊D3—31no—
85C22×D32—34no—
86C22×C32×C4—31no—
87C22×C3⋊Dic3—31no—
88C2×C4×C3⋊D3—31no—
89C2×C4×C3⋊D3—31no—
90C2×C4×C3⋊D3—34no—
91C2×C4×C3⋊D3—31no—
92C2×C4×C3⋊D3—38no—
93C2×C4×C3⋊D3—32no—
94C2×C4×C3⋊D3—64no—
95C2×C4×C3⋊D3C2×D312yes—
96C2×C4×C3⋊D3C2×D311yescharacteristic
97C2×C3×C4×D3—34no—
98C2×C3×C4×D3—38no—
99C2×C3×C4×D3C2×D314yes—
100C2×#624—34no—
101C2×D3×Dic3—34no—
102C2×D3×Dic3—34no—
103C2×D3×Dic3—38no—
104C4×D32—332no—
105C32×C4⋊C4—32no—
106C32×C22⋊C4—32no—
107C32×C42—24no—
108C32×C42—31no—
109C32×C42C2×D314yescomplemented
110C32×C42C2×D311yescharacteristic, complemented
111#3757—32no—
112#3752—32no—
113#3752—38no—
114#3752—32no—
115#3752—68no—
116C4⋊(C3⋊Dic3)—32no—
117#3750—32no—
118#3750—38no—
119#3750—32no—
120#3750—68no—
121C4×C3⋊Dic3—31no—
122C4×C3⋊Dic3—34no—
123C4×C3⋊Dic3—32no—
124C4×C3⋊Dic3—64no—
125C4×C3⋊Dic3C2×D311yescharacteristic, complemented
126C4×C3⋊Dic3C2×D312yescomplemented
127C3×(C2×D3)⋊C4C2×D318yes—
128C3×Dic3⋊C4C2×D318yescomplemented
129C3×C4×Dic3—38no—
130C3×C4×Dic3—34no—
131C3×C4×Dic3C2×D314yescomplemented
132#3724—38no—
133Dic3⋊Dic3—38no—
134#3722—38no—
135(C2×D3)⋊Dic3—38no—
136Dic32—38no—
137C22×C33C2411yescharacteristic
138C2×C3⋊(C3⋊D3)—22no—
139C2×C3⋊(C3⋊D3)C22×C412yes—
140C2×C3×C3⋊D3C22×C412yes—
141C2×C3×C3⋊D3#4114yes—
142C2×C32×D3—22no—
143C2×C32×D3C22×C412yes—
144D3×C3⋊D3—28no—
145C33×C4—22no—
146C33×C4C22×C412yes—
147C3⋊(C3⋊Dic3)—22no—
148C3⋊(C3⋊Dic3)C22×C412yes—
149C3×C3⋊Dic3—22no—
150C3×C3⋊Dic3C22×C412yes—
151C32×Dic3—22no—
152C32×Dic3C22×C412yes—
153C22×C4×D3—94no—
154C22×C4×D3—91no—
155C3×#117—34no—
156C2×C3×C42—91no—
157C2×(C2×D3)⋊C4—92no—
158#926—94no—
159C2×Dic3⋊C4—92no—
160C2×C4×Dic3—91no—
161#892—98no—
162D3×C4⋊C4—98no—
163D3×C22⋊C4—98no—
164#877—98no—
165#872—94no—
166#872—91no—
167#872—94no—
168#872—91no—
169#872—92no—
170#872—92no—
171#872—91no—
172#872—184no—
173#872C3⋊D311yescharacteristic, direct product factor
174C42×D3—34no—
175C42×D3—94no—
176C23×C32—31no—
177C22×C3⋊D3—31no—
178C22×C3⋊D3—34no—
179C22×C3⋊D3—34no—
180C22×C3⋊D3—64no—
181C22×C3⋊D3—64no—
182C22×C3⋊D3C22×D311yescharacteristic
183C22×C3×D3—34no—
184C22×C3×D3C22×D314yes—
185C2×D32—316no—
186C2×D32—616no—
187C2×C32×C4—24no—
188C2×C32×C4—28no—
189C2×C32×C4—31no—
190C2×C32×C4—34no—
191C2×C32×C4—32no—
192C2×C32×C4C22×D318yes—
193C2×C32×C4C22×D314yes—
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D311yescharacteristic
196C2×C3⋊Dic3—31no—
197C2×C3⋊Dic3—34no—
198C2×C3⋊Dic3—32no—
199C2×C3⋊Dic3—34no—
200C2×C3⋊Dic3—38no—
201C2×C3⋊Dic3—64no—
202C2×C3⋊Dic3—68no—
203C2×C3⋊Dic3C22×D312yes—
204C2×C3⋊Dic3C22×D311yescharacteristic
205C4×C3⋊D3—24no—
206C4×C3⋊D3—34no—
207C4×C3⋊D3—34no—
208C4×C3⋊D3—34no—
209C4×C3⋊D3—316no—
210C4×C3⋊D3—616no—
211C4×C3⋊D3—64no—
212C4×C3⋊D3C4×D314yes—
213C2×C3×Dic3—34no—
214C2×C3×Dic3—38no—
215C2×C3×Dic3C22×D318yes—
216C2×C3×Dic3C22×D314yes—
217C3×C4×D3—316no—
218C3×C4×D3—616no—
219C3×C4×D3C4×D3116yescomplemented
220#624—316no—
221D3×Dic3—316no—
222D3×Dic3—316no—
223D3×Dic3—616no—
224C2×#117—271noSylow
225C2×C33C23×C411yescharacteristic
226C2×C33C2×#4112yes—
227C3⋊(C3⋊D3)—22no—
228C3×C3⋊D3#11714yescomplemented
229C32×D3—22no—
230C23×D3—94no—
231C23×D3—91no—
232C22×C3×C4—34no—
233C22×C3×C4—91no—
234C22×C3×C4—92no—
235C22×Dic3—94no—
236C22×Dic3—91no—
237C22×Dic3—92no—
238C2×C4×D3—34no—
239C2×C4×D3—38no—
240C2×C4×D3—932no—
241C2×C4×D3—94no—
242C2×C4×D3—94no—
243C2×C4×D3—91no—
244C2×C4×D3—94no—
245C2×C4×D3—91no—
246C2×C4×D3—91no—
247C2×C4×D3—98no—
248C2×C4×D3—98no—
249C2×C4×D3—184no—
250C2×C4×D3C2×C3⋊D311yescharacteristic
251C3×C4⋊C4—38no—
252C3×C22⋊C4—38no—
253C3×C42—24no—
254C3×C42—34no—
255C3×C42—92no—
256C3×C42—91no—
257C3×C42D3214yescomplemented
258C3×C42C2×C3⋊D311yescharacteristic, complemented
259#217—98no—
260(C2×D3)⋊C4—98no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—98no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—188no—
266(C2×D3)⋊C4C2×C3⋊D312yes—
267C4⋊Dic3—98no—
268Dic3⋊C4—98no—
269Dic3⋊C4—98no—
270Dic3⋊C4—92no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4—188no—
274Dic3⋊C4C2×C3⋊D312yescomplemented
275C4×Dic3—34no—
276C4×Dic3—38no—
277C4×Dic3—98no—
278C4×Dic3—94no—
279C4×Dic3—91no—
280C4×Dic3—92no—
281C4×Dic3—184no—
282C4×Dic3C2×C3⋊D311yescharacteristic, complemented
283C22×C32—24no—
284C22×C32—32no—
285C22×C32—62no—
286C22×C32C23×D314yes—
287C22×C32C23×D311yescharacteristic
288C2×C3⋊D3—32no—
289C2×C3⋊D3—38no—
290C2×C3⋊D3—68no—
291C2×C3⋊D3—62no—
292C2×C3⋊D3—68no—
293C2×C3⋊D3—68no—
294C2×C3⋊D3—128no—
295C2×C3⋊D3C2×C4×D312yes—
296C2×C3⋊D3#23514yes—
297C2×C3×D3—28no—
298C2×C3×D3—38no—
299C2×C3×D3—316no—
300C2×C3×D3C2×C4×D318yes—
301D32—632no—
302C32×C4—22no—
303C32×C4—28no—
304C32×C4—28no—
305C32×C4—32no—
306C32×C4—48no—
307C32×C4—62no—
308C32×C4C2×C4×D318yes—
309C32×C4C2×C4×D312yes—
310C3⋊Dic3—22no—
311C3⋊Dic3—32no—
312C3⋊Dic3—38no—
313C3⋊Dic3—68no—
314C3⋊Dic3—62no—
315C3⋊Dic3—68no—
316C3⋊Dic3—128no—
317C3⋊Dic3C2×C4×D312yes—
318C3×Dic3—28no—
319C3×Dic3—38no—
320C3×Dic3—68no—
321C3×Dic3C2×C4×D318yes—
322C23×C4—271no—
323#117—31no—
324#117—271no—
325#117—272no—
326#117—272no—
327#117—271no—
328#117—271no—
329C2×C4⋊C4—272no—
330C2×C22⋊C4—272no—
331C2×C42—91no—
332C2×C42—271no—
333C33C2×#11711yescharacteristic, complemented, Sylow
334C23×C3—34no—
335C23×C3—91no—
336C22×D3—34no—
337C22×D3—916no—
338C22×D3—94no—
339C22×D3—91no—
340C22×D3—94no—
341C22×D3—184no—
342C22×D3—184no—
343C22×D3—1816no—
344C22×D3C22×C3⋊D311yescharacteristic
345C2×C3×C4—28no—
346C2×C3×C4—24no—
347C2×C3×C4—34no—
348C2×C3×C4—316no—
349C2×C3×C4—38no—
350C2×C3×C4—91no—
351C2×C3×C4—94no—
352C2×C3×C4—92no—
353C2×C3×C4—184no—
354C2×C3×C4C2×D3218yes—
355C2×C3×C4C2×D3214yes—
356C2×C3×C4C22×C3⋊D312yes—
357C2×C3×C4C22×C3⋊D311yescharacteristic
358C2×Dic3—34no—
359C2×Dic3—38no—
360C2×Dic3—916no—
361C2×Dic3—94no—
362C2×Dic3—91no—
363C2×Dic3—94no—
364C2×Dic3—98no—
365C2×Dic3—92no—
366C2×Dic3—184no—
367C2×Dic3—188no—
368C2×Dic3—184no—
369C2×Dic3C22×C3⋊D312yes—
370C2×Dic3C22×C3⋊D311yescharacteristic
371C4×D3—316no—
372C4×D3—616no—
373C4×D3—916no—
374C4×D3—916no—
375C4×D3—94no—
376C4×D3—916no—
377C4×D3—94no—
378C4×D3—94no—
379C4×D3—1816no—
380C4×D3—1816no—
381C4×D3C4×C3⋊D314yescomplemented
382C2×C32—24no—
383C2×C32—28no—
384C2×C32—62no—
385C2×C32C22×C4×D314yes—
386C2×C32C22×C4×D311yescharacteristic
387C2×C32C2×#23512yes—
388C2×C32D3×#4118yes—
389C3⋊D3—62no—
390C3⋊D3—68no—
391C3⋊D3—128no—
392C3⋊D3#87214yesdirect product factor
393C3×D3—28no—
394C3×D3—316no—
395C24—271no—
396C22×C4—31no—
397C22×C4—91no—
398C22×C4—92no—
399C22×C4—271no—
400C22×C4—278no—
401C22×C4—271no—
402C22×C4—272no—
403C22×C4—271no—
404C22×C4—271no—
405C4⋊C4—32no—
406C4⋊C4—272no—
407C4⋊C4—272no—
408C4⋊C4—272no—
409C22⋊C4—32no—
410C22⋊C4—272no—
411C22⋊C4—272no—
412C22⋊C4—272no—
413C42—31no—
414C42—92no—
415C42—91no—
416C42—272no—
417C42—271no—
418C42D3×C3⋊D311yescharacteristic, complemented
419C22×C3—24no—
420C22×C3—38no—
421C22×C3—68no—
422C22×C3—92no—
423C22×C3—94no—
424C22×C3C22×D3214yes—
425C22×C3C23×C3⋊D311yescharacteristic
426C2×D3—22no—
427C2×D3—38no—
428C2×D3—316no—
429C2×D3—98no—
430C2×D3—92no—
431C2×D3—188no—
432C2×D3—188no—
433C2×D3—1832no—
434C2×D3—188no—
435C2×D3—182no—
436C2×D3—368no—
437C2×D3C2×C4×C3⋊D312yes—
438C3×C4—28no—
439C3×C4—22no—
440C3×C4—28no—
441C3×C4—38no—
442C3×C4—48no—
443C3×C4—68no—
444C3×C4—92no—
445C3×C4—182no—
446C3×C4C4×D3218yes—
447C3×C4C2×C4×C3⋊D312yes—
448Dic3—22no—
449Dic3—38no—
450Dic3—68no—
451Dic3—98no—
452Dic3—92no—
453Dic3—188no—
454Dic3—188no—
455Dic3—182no—
456Dic3—368no—
457Dic3C2×C4×C3⋊D312yes—
458C32—24no—
459C32C2×#87211yescharacteristic, complemented
460C32D3×#11714yescomplemented
461C23—31no—
462C23—91no—
463C23—274no—
464C23—271no—
465C23—544no—
466C2×C4—31no—
467C2×C4—34no—
468C2×C4—32no—
469C2×C4—91no—
470C2×C4—94no—
471C2×C4—92no—
472C2×C4—184no—
473C2×C4—274no—
474C2×C4—274no—
475C2×C4—274no—
476C2×C4—271no—
477C2×C4—274no—
478C2×C4—272no—
479C2×C4—544no—
480C2×C4C2×D3×C3⋊D312yes—
481C2×C4C2×D3×C3⋊D311yescharacteristic
482C2×C3—24no—
483C2×C3—28no—
484C2×C3—68no—
485C2×C3—94no—
486C2×C3C2×C4×D3214yes—
487C2×C3D3×#23518yes—
488C2×C3C22×C4×C3⋊D311yescharacteristic
489C2×C3#41×C3⋊D312yes—
490D3—22no—
491D3—316no—
492D3—182no—
493D3—188no—
494D3—368no—
495C22—32no—
496C22—62no—
497C22—92no—
498C22—94no—
499C22—272no—
500C22—548no—
501C22—542no—
502C22C22×D3×C3⋊D311yescharacteristic
503C4—22no—
504C4—32no—
505C4—62no—
506C4—92no—
507C4—182no—
508C4—272no—
509C4—542no—
510C4C4×D3×C3⋊D312yes—
511C3—24no—
512C3D3×#87214yescomplemented
513C3C3⋊D3×#11711yescharacteristic, complemented
514C2—62no—
515C2—94no—
516C2—542no—
517C2C2×C4×D3×C3⋊D311yescharacteristic
518C2C3⋊D3×#23512yes—
subgroupG/H
centerC2×C4C2×D3×C3⋊D3
commutatorC2×C33C23×C4
FrattiniC22C22×D3×C3⋊D3
FittingC33×C42C22
socle, cosocleC22×C33C24
prime radicalC22×D3×C3⋊D3C22
hypercenterC42D3×C3⋊D3
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11723913C2semidirect
3. C3×C42×C3⋊D3C2semidirect
4. C32×#872C2semidirect
6. #11723469C2semidirect
7. C3⋊D3×Dic3⋊C4C2semidirect
8. C4×Dic3×C3⋊D3C2semidirect
9. #11723459C2semidirect
10. #11723452C2semidirect
11. #11723423C2semidirect
23. C4×D3×C3⋊D3C4semidirect
24. C33×C42C22semidirect
26. #88857C22semidirect
27. C4×C3⋊(C3⋊Dic3)C22semidirect
28. C3×C4×C3⋊Dic3C22semidirect
29. C3×C4×C3⋊Dic3C22semidirect
31. C32×Dic3⋊C4C22semidirect
32. C32×C4×Dic3C22semidirect
33. #88806C22semidirect
34. Dic3⋊(C3⋊Dic3)C22semidirect
37. Dic3×C3⋊Dic3C22semidirect
51. C42×C3⋊D3D3semidirect
52. C3×#872D3semidirect
70. C4×C3⋊(C3⋊D3)C2×C4semidirect
77. C32×C4×D3C2×C4semidirect
78. #6289C2×C4semidirect
81. D3×C3⋊Dic3C2×C4semidirect
109. C32×C42C2×D3semidirect
110. C32×C42C2×D3semidirect
125. C4×C3⋊Dic3C2×D3semidirect
126. C4×C3⋊Dic3C2×D3semidirect
128. C3×Dic3⋊C4C2×D3semidirect
131. C3×C4×Dic3C2×D3semidirect
173. #872C3⋊D3direct (N × H)
219. C3×C4×D3C4×D3semidirect
228. C3×C3⋊D3#117semidirect
257. C3×C42D32semidirect
258. C3×C42C2×C3⋊D3semidirect
274. Dic3⋊C4C2×C3⋊D3semidirect
282. C4×Dic3C2×C3⋊D3semidirect
333. C33C2×#117semidirect
381. C4×D3C4×C3⋊D3semidirect
392. C3⋊D3#872direct (N × H)
418. C42D3×C3⋊D3semidirect
459. C32C2×#872semidirect
460. C32D3×#117semidirect
512. C3D3×#872semidirect
513. C3C3⋊D3×#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×C3⋊D32
#117239132
C3×C42×C3⋊D32
C32×#8722
C3⋊D3×(C2×D3)⋊C42
#117234692
C3⋊D3×Dic3⋊C42
C4×Dic3×C3⋊D32
#117234592
#117234522
#117234232
C3⋊D3×#1173
D3×#8723