Finite Group Databasebeta

← C2×D3×#3710 #13170486

Subgroups — C2×D3×#3710 #13170486

Order 1728.

1794 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic
2C2×#88808C211yesmaximal, characteristic, complemented
3C2×#88785C212yesmaximal, complemented
4C2×C3×#3710C211yesmaximal, characteristic, complemented
5C2×C3×D3×C3⋊C8C212yesmaximal, complemented
6D3×#3710C218yesmaximal, direct product factor
7C22×#3710—31nomaximal
8C2×D32×C8—32nomaximal
9C22×D3×C3⋊D3C411yescharacteristic
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6289C411yescharacteristic
14C2×Dic3×C3⋊D3C411yescharacteristic
15C2×D3×C3⋊Dic3C411yescharacteristic
16C4×D3×C3⋊D3C414yes—
17C4×D3×C3⋊D3C2214yes—
18C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
19C2×C32×C3⋊C8C2212yescomplemented
20#88808C2214yescomplemented
21#88785C2218yescomplemented
22C3×#3710C2214yescomplemented
23C3×D3×C3⋊C8C2218yescomplemented
24C22×C4×C3⋊D3—31no—
25C2×C4×D32—32no—
26C2×C4×D32—61no—
27C2×C8×C3⋊D3—32no—
28C22×C3×C3⋊C8—32no—
29C2×C3×D3×C8—32no—
30C2×C3×D3×C8—32no—
31C2×#3710—31no—
32C2×#3710—32no—
33C2×#3710—32no—
34C2×#3710—38no—
35C2×#3710—62no—
36C2×#3710D311yescharacteristic, direct product factor
37C2×D3×C3⋊C8—32no—
38C2×D3×C3⋊C8D312yescomplemented
39D32×C8—316no—
40C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
41C22×C3×C3⋊D3C2×C411yescharacteristic
42C22×C32×D3C2×C411yescharacteristic
43C2×D3×C3⋊D3C818yescomplemented
44C2×D3×C3⋊D3C2×C414yes—
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
47C4×C3⋊(C3⋊D3)C2×C412yes—
48C4×C3⋊(C3⋊D3)C2312yes—
49C2×C3×C3⋊Dic3C2×C411yescharacteristic
50C3×C4×C3⋊D3C2×C412yes—
51C3×C4×C3⋊D3C2312yes—
52C2×C32×Dic3C2×C411yescharacteristic
53C32×C4×D3C2×C412yes—
54C32×C4×D3C2312yes—
55#6289C2×C414yes—
56Dic3×C3⋊D3C2×C414yes—
57D3×C3⋊Dic3C2×C414yes—
58C3×C3⋊(C3⋊C8)C2314yescomplemented
59C32×C3⋊C8C2314yescomplemented
60C22×D3×C8—91no—
61C22×D3×C8—92no—
62C23×C3⋊D3—31no—
63C22×D32—32no—
64C22×D32—61no—
65C22×C32×C4—31no—
66C22×C3⋊Dic3—31no—
67C2×C4×C3⋊D3—31no—
68C2×C4×C3⋊D3—32no—
69C2×C4×C3⋊D3—31no—
70C2×C4×C3⋊D3—31no—
71C2×C4×C3⋊D3—34no—
72C2×C4×C3⋊D3—34no—
73C2×C4×C3⋊D3—61no—
74C2×C4×C3⋊D3—62no—
75C2×C4×C3⋊D3—121no—
76C2×C4×C3⋊D3C2×D311yescharacteristic
77C2×C3×C4×D3—21no—
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—61no—
80C2×C3×C4×D3C2×D312yes—
81C2×#624—32no—
82C2×#624—61no—
83C2×D3×Dic3—32no—
84C2×D3×Dic3—32no—
85C2×D3×Dic3—61no—
86C2×D3×Dic3—61no—
87C4×D32—38no—
88C4×D32—38no—
89C4×D32—64no—
90C4×D32—64no—
91C2×C32×C8—32no—
92C2×C3⋊(C3⋊C8)C2×D312yescomplemented
93C8×C3⋊D3—38no—
94C2×C3×C3⋊C8—22no—
95C2×C3×C3⋊C8—32no—
96C2×C3×C3⋊C8—32no—
97C2×C3×C3⋊C8—38no—
98C2×C3×C3⋊C8—62no—
99C2×C3×C3⋊C8C2×D312yescomplemented
100C2×C3×C3⋊C8C2×D312yescomplemented
101C3×D3×C8—38no—
102C3×D3×C8—38no—
103#3710—34no—
104#3710—38no—
105#3710—38no—
106#3710—68no—
107#3710C2×D314yesdirect product factor
108D3×C3⋊C8—38no—
109D3×C3⋊C8C2×D318yescomplemented
110C22×C33C22×C411yescharacteristic
111C2×C3⋊(C3⋊D3)C2×C814yescomplemented
112C2×C3⋊(C3⋊D3)C22×C412yes—
113C2×C3×C3⋊D3C2×C814yescomplemented
114C2×C3×C3⋊D3C22×C412yes—
115C2×C32×D3C2×C814yescomplemented
116C2×C32×D3C22×C412yes—
117D3×C3⋊D3C2×C8116yescomplemented
118C33×C4C22×C411yescharacteristic
119C33×C4C2411yescharacteristic
120C3⋊(C3⋊Dic3)C22×C412yes—
121C3×C3⋊Dic3C22×C412yes—
122C32×Dic3C22×C412yes—
123C22×C4×D3—92no—
124C22×C4×D3—91no—
125C22×C4×D3—181no—
126C22×C3×C8—91no—
127C22×C3×C8—92no—
128C22×C3⋊C8—32no—
129C22×C3⋊C8—91no—
130C2×D3×C8—32no—
131C2×D3×C8—32no—
132C2×D3×C8—98no—
133C2×D3×C8—916no—
134C2×D3×C8—92no—
135C2×D3×C8—92no—
136C2×D3×C8—92no—
137C2×D3×C8—92no—
138C2×D3×C8—182no—
139C23×C32—31no—
140C22×C3⋊D3—32no—
141C22×C3⋊D3—31no—
142C22×C3⋊D3—38no—
143C22×C3⋊D3—34no—
144C22×C3⋊D3—61no—
145C22×C3⋊D3—62no—
146C22×C3⋊D3—121no—
147C22×C3⋊D3C4×D311yescharacteristic
148C22×C3×D3—21no—
149C22×C3×D3—32no—
150C22×C3×D3—61no—
151C22×C3×D3C4×D312yes—
152C2×D32—316no—
153C2×D32—38no—
154C2×D32—68no—
155C2×D32—64no—
156C2×C32×C4—22no—
157C2×C32×C4—21no—
158C2×C32×C4—31no—
159C2×C32×C4—32no—
160C2×C32×C4—32no—
161C2×C32×C4—41no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D312yes—
164C2×C3⋊Dic3—32no—
165C2×C3⋊Dic3—31no—
166C2×C3⋊Dic3—34no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—62no—
169C2×C3⋊Dic3—121no—
170C2×C3⋊Dic3C4×D311yescharacteristic
171C4×C3⋊D3—34no—
172C4×C3⋊D3—34no—
173C4×C3⋊D3—32no—
174C4×C3⋊D3—32no—
175C4×C3⋊D3—34no—
176C4×C3⋊D3—34no—
177C4×C3⋊D3—64no—
178C4×C3⋊D3—62no—
179C4×C3⋊D3—64no—
180C4×C3⋊D3—62no—
181C4×C3⋊D3—122no—
182C4×C3⋊D3—122no—
183C4×C3⋊D3C4×D312yes—
184C4×C3⋊D3C22×D312yes—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3C4×D312yes—
189C3×C4×D3—22no—
190C3×C4×D3—22no—
191C3×C4×D3—34no—
192C3×C4×D3—34no—
193C3×C4×D3—62no—
194C3×C4×D3—62no—
195C3×C4×D3C4×D314yes—
196C3×C4×D3C22×D314yes—
197#624—38no—
198#624—64no—
199D3×Dic3—38no—
200D3×Dic3—38no—
201D3×Dic3—64no—
202D3×Dic3—64no—
203C32×C8—34no—
204C3⋊(C3⋊C8)C22×D314yescomplemented
205C3×C3⋊C8—24no—
206C3×C3⋊C8—34no—
207C3×C3⋊C8—34no—
208C3×C3⋊C8—64no—
209C3×C3⋊C8C22×D314yescomplemented
210C3×C3⋊C8C22×D314yescomplemented
211C23×C8—271noSylow
212C2×C33C22×C812yescomplemented
213C2×C33C23×C411yescharacteristic
214C3⋊(C3⋊D3)C22×C814yescomplemented
215C3×C3⋊D3C22×C814yescomplemented
216C32×D3C22×C814yescomplemented
217C23×D3—92no—
218C23×D3—91no—
219C23×D3—181no—
220C22×C3×C4—32no—
221C22×C3×C4—61no—
222C22×C3×C4—91no—
223C22×Dic3—92no—
224C22×Dic3—91no—
225C22×Dic3—181no—
226C2×C4×D3—32no—
227C2×C4×D3—61no—
228C2×C4×D3—98no—
229C2×C4×D3—98no—
230C2×C4×D3—92no—
231C2×C4×D3—92no—
232C2×C4×D3—91no—
233C2×C4×D3—92no—
234C2×C4×D3—91no—
235C2×C4×D3—91no—
236C2×C4×D3—94no—
237C2×C4×D3—94no—
238C2×C4×D3—184no—
239C2×C4×D3—184no—
240C2×C4×D3—181no—
241C2×C4×D3—182no—
242C2×C4×D3—181no—
243C2×C4×D3—181no—
244C2×C4×D3—361no—
245C2×C4×D3D3211yescharacteristic
246C2×C3×C8—32no—
247C2×C3×C8—32no—
248C2×C3×C8—62no—
249C2×C3×C8—94no—
250C2×C3×C8—98no—
251C2×C3×C8—92no—
252C2×C3⋊C8—32no—
253C2×C3⋊C8—32no—
254C2×C3⋊C8—38no—
255C2×C3⋊C8—62no—
256C2×C3⋊C8—94no—
257C2×C3⋊C8D3212yescomplemented
258D3×C8—38no—
259D3×C8—38no—
260D3×C8—98no—
261D3×C8—98no—
262D3×C8—98no—
263D3×C8—98no—
264D3×C8—188no—
265C22×C32—22no—
266C22×C32—21no—
267C22×C32—34no—
268C22×C32—32no—
269C22×C32—41no—
270C22×C32C2×C4×D311yescharacteristic
271C22×C32C2×C4×D312yes—
272C2×C3⋊D3—38no—
273C2×C3⋊D3—34no—
274C2×C3⋊D3—34no—
275C2×C3⋊D3—32no—
276C2×C3⋊D3—316no—
277C2×C3⋊D3—64no—
278C2×C3⋊D3—68no—
279C2×C3⋊D3—62no—
280C2×C3⋊D3—64no—
281C2×C3⋊D3—124no—
282C2×C3⋊D3—122no—
283C2×C3⋊D3D3×C814yescomplemented
284C2×C3⋊D3C2×C4×D312yes—
285C2×C3×D3—24no—
286C2×C3×D3—22no—
287C2×C3×D3—38no—
288C2×C3×D3—34no—
289C2×C3×D3—64no—
290C2×C3×D3—62no—
291C2×C3×D3D3×C818yescomplemented
292C2×C3×D3C2×C4×D314yes—
293D32—332no—
294D32—616no—
295C32×C4—22no—
296C32×C4—21no—
297C32×C4—22no—
298C32×C4—21no—
299C32×C4—32no—
300C32×C4—41no—
301C32×C4—41no—
302C32×C4C2×C4×D311yescharacteristic
303C32×C4C2×C4×D312yes—
304C32×C4C23×D311yescharacteristic
305C32×C4C23×D312yes—
306C3⋊Dic3—34no—
307C3⋊Dic3—32no—
308C3⋊Dic3—62no—
309C3⋊Dic3—64no—
310C3⋊Dic3—122no—
311C3⋊Dic3C2×C4×D312yes—
312C3×Dic3—22no—
313C3×Dic3—34no—
314C3×Dic3—62no—
315C3×Dic3C2×C4×D314yes—
316C23×C4—271no—
317C22×C8—91no—
318C22×C8—92no—
319C22×C8—272no—
320C22×C8—278no—
321C22×C8—271no—
322C33C23×C811yescharacteristic, complemented, Sylow
323C23×C3—32no—
324C23×C3—61no—
325C23×C3—91no—
326C22×D3—32no—
327C22×D3—61no—
328C22×D3—98no—
329C22×D3—916no—
330C22×D3—92no—
331C22×D3—91no—
332C22×D3—98no—
333C22×D3—94no—
334C22×D3—184no—
335C22×D3—188no—
336C22×D3—182no—
337C22×D3—181no—
338C22×D3—361no—
339C22×D3#62411yescharacteristic
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—32no—
343C2×C3×C4—34no—
344C2×C3×C4—34no—
345C2×C3×C4—41no—
346C2×C3×C4—62no—
347C2×C3×C4—62no—
348C2×C3×C4—61no—
349C2×C3×C4—91no—
350C2×C3×C4—92no—
351C2×C3×C4—92no—
352C2×C3×C4C2×D3212yes—
353C2×C3×C4C2×D3211yescharacteristic
354C2×Dic3—32no—
355C2×Dic3—61no—
356C2×Dic3—98no—
357C2×Dic3—92no—
358C2×Dic3—91no—
359C2×Dic3—94no—
360C2×Dic3—184no—
361C2×Dic3—182no—
362C2×Dic3—181no—
363C2×Dic3—361no—
364C2×Dic3#62411yescharacteristic
365C4×D3—34no—
366C4×D3—34no—
367C4×D3—62no—
368C4×D3—62no—
369C4×D3—98no—
370C4×D3—98no—
371C4×D3—94no—
372C4×D3—94no—
373C4×D3—92no—
374C4×D3—94no—
375C4×D3—92no—
376C4×D3—94no—
377C4×D3—184no—
378C4×D3—184no—
379C4×D3—184no—
380C4×D3—182no—
381C4×D3—182no—
382C4×D3—184no—
383C4×D3—362no—
384C4×D3—362no—
385C4×D3#62412yes—
386C4×D3C2×D3212yes—
387C3×C8—34no—
388C3×C8—34no—
389C3×C8—64no—
390C3×C8—94no—
391C3⋊C8—34no—
392C3⋊C8—34no—
393C3⋊C8—64no—
394C3⋊C8C2×D3214yescomplemented
395C2×C32—24no—
396C2×C32—22no—
397C2×C32—21no—
398C2×C32—22no—
399C2×C32—34no—
400C2×C32—42no—
401C2×C32—41no—
402C2×C32C2×D3×C812yescomplemented
403C2×C32C2×D3×C814yescomplemented
404C2×C32C22×C4×D311yescharacteristic
405C2×C32C22×C4×D312yes—
406C3⋊D3—38no—
407C3⋊D3—34no—
408C3⋊D3—64no—
409C3⋊D3—68no—
410C3⋊D3—124no—
411C3⋊D3C2×D3×C814yescomplemented
412C3×D3—24no—
413C3×D3—38no—
414C3×D3—64no—
415C3×D3C2×D3×C818yescomplemented
416C24—271no—
417C22×C4—31no—
418C22×C4—91no—
419C22×C4—274no—
420C22×C4—271no—
421C22×C4—274no—
422C22×C4—271no—
423C22×C4—271no—
424C22×C4—271no—
425C2×C8—32no—
426C2×C8—94no—
427C2×C8—98no—
428C2×C8—92no—
429C2×C8—278no—
430C2×C8—274no—
431C22×C3—21no—
432C22×C3—22no—
433C22×C3—38no—
434C22×C3—34no—
435C22×C3—41no—
436C22×C3—62no—
437C22×C3—64no—
438C22×C3—92no—
439C22×C3—94no—
440C22×C3C4×D3212yes—
441C22×C3C2×#62411yescharacteristic
442C2×D3—38no—
443C2×D3—34no—
444C2×D3—62no—
445C2×D3—64no—
446C2×D3—932no—
447C2×D3—98no—
448C2×D3—94no—
449C2×D3—92no—
450C2×D3—94no—
451C2×D3—916no—
452C2×D3—1816no—
453C2×D3—182no—
454C2×D3—184no—
455C2×D3—184no—
456C2×D3—188no—
457C2×D3—362no—
458C2×D3—364no—
459C2×D3#371014yesdirect product factor
460C2×D3C2×#62412yes—
461C3×C4—22no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—22no—
465C3×C4—34no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—62no—
469C3×C4—92no—
470C3×C4C4×D3212yes—
471C3×C4C2×#62411yescharacteristic
472C3×C4C22×D3212yes—
473C3×C4C22×D3211yescharacteristic
474Dic3—34no—
475Dic3—62no—
476Dic3—94no—
477Dic3—92no—
478Dic3—182no—
479Dic3—184no—
480Dic3—362no—
481Dic3C2×#62412yes—
482C32—22no—
483C32—21no—
484C32—41no—
485C32C22×D3×C811yescharacteristic, complemented
486C32C22×D3×C812yescomplemented
487C23—31no—
488C23—91no—
489C23—274no—
490C23—278no—
491C23—271no—
492C2×C4—31no—
493C2×C4—32no—
494C2×C4—32no—
495C2×C4—91no—
496C2×C4—92no—
497C2×C4—92no—
498C2×C4—274no—
499C2×C4—274no—
500C2×C4—274no—
501C2×C4—271no—
502C2×C4—272no—
503C2×C4—272no—
504C2×C4D3311yescharacteristic
505C8—34no—
506C8—94no—
507C2×C3—22no—
508C2×C3—24no—
509C2×C3—21no—
510C2×C3—22no—
511C2×C3—38no—
512C2×C3—42no—
513C2×C3—41no—
514C2×C3—64no—
515C2×C3—94no—
516C2×C3D32×C814yescomplemented
517C2×C3C2×#371012yescomplemented
518C2×C3C2×C4×D3212yes—
519C2×C3C22×#62411yescharacteristic
520D3—38no—
521D3—64no—
522D3—98no—
523D3—94no—
524D3—184no—
525D3—188no—
526D3—364no—
527D3C2×#371014yesdirect product factor
528C22—34no—
529C22—32no—
530C22—94no—
531C22—92no—
532C22—2716no—
533C22—274no—
534C22—272no—
535C22D3×#62411yescharacteristic
536C4—32no—
537C4—92no—
538C4—272no—
539C4D3×#62411yescharacteristic
540C4C2×D3311yescharacteristic
541C3—21no—
542C3—22no—
543C3—41no—
544C3C2×D32×C812yescomplemented
545C3C22×#371011yescharacteristic, complemented
546C2—34no—
547C2—94no—
548C2—274no—
549C2D3×#371012yesdirect product factor
550C2C2×D3×#62411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D33
commutator, hypocenterC33C23×C8
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC22×D3×C3⋊D3C4
cosocleC33×C4C24
LCSG ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88808C2semidirect
3. C2×#88785C2semidirect
4. C2×C3×#3710C2semidirect
5. C2×C3×D3×C3⋊C8C2semidirect
6. D3×#3710C2direct (N × H)
18. C2×C3×C3⋊(C3⋊C8)C22semidirect
19. C2×C32×C3⋊C8C22semidirect
20. #88808C22semidirect
21. #88785C22semidirect
22. C3×#3710C22semidirect
23. C3×D3×C3⋊C8C22semidirect
36. C2×#3710D3direct (N × H)
38. C2×D3×C3⋊C8D3semidirect
43. C2×D3×C3⋊D3C8semidirect
58. C3×C3⋊(C3⋊C8)C23semidirect
59. C32×C3⋊C8C23semidirect
92. C2×C3⋊(C3⋊C8)C2×D3semidirect
99. C2×C3×C3⋊C8C2×D3semidirect
100. C2×C3×C3⋊C8C2×D3semidirect
107. #3710C2×D3direct (N × H)
109. D3×C3⋊C8C2×D3semidirect
111. C2×C3⋊(C3⋊D3)C2×C8semidirect
113. C2×C3×C3⋊D3C2×C8semidirect
115. C2×C32×D3C2×C8semidirect
117. D3×C3⋊D3C2×C8semidirect
204. C3⋊(C3⋊C8)C22×D3semidirect
209. C3×C3⋊C8C22×D3semidirect
210. C3×C3⋊C8C22×D3semidirect
212. C2×C33C22×C8semidirect
214. C3⋊(C3⋊D3)C22×C8semidirect
215. C3×C3⋊D3C22×C8semidirect
216. C32×D3C22×C8semidirect
257. C2×C3⋊C8D32semidirect
283. C2×C3⋊D3D3×C8semidirect
291. C2×C3×D3D3×C8semidirect
322. C33C23×C8semidirect
394. C3⋊C8C2×D32semidirect
402. C2×C32C2×D3×C8semidirect
403. C2×C32C2×D3×C8semidirect
411. C3⋊D3C2×D3×C8semidirect
415. C3×D3C2×D3×C8semidirect
459. C2×D3#3710direct (N × H)
485. C32C22×D3×C8semidirect
486. C32C22×D3×C8semidirect
516. C2×C3D32×C8semidirect
517. C2×C3C2×#3710semidirect
527. D3C2×#3710direct (N × H)
544. C3C2×D32×C8semidirect
545. C3C22×#3710semidirect
549. C2D3×#3710direct (N × H)

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
C2×#888082
C2×#887852
C2×C3×#37102
C2×C3×D3×C3⋊C82
D3×#37102
C22×#37103
C2×D32×C83