Finite Group Databasebeta

← D3×#64106 #13176546

Subgroups — D3×#64106 #13176546

Order 1728.

2228 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic
2C2×D3×#634C211yesmaximal, characteristic, complemented
3#11723938C211yesmaximal, characteristic, complemented
4C3×#64106C211yesmaximal, characteristic, complemented
5C32×D3×C4⋊C4C211yesmaximal, characteristic, complemented
6D3×#3752C212yesmaximal
7#11723485C212yesmaximal, complemented
8#11723483C212yesmaximal, complemented
9D3×C4⋊(C3⋊Dic3)C212yesmaximal, complemented
10#11723449C211yesmaximal, characteristic, complemented
11#11723431C211yesmaximal, characteristic, complemented
12C2×#64106—31nomaximal
13D3×#897—34nomaximal
14C22×D3×C3⋊D3C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×#6308C2211yescharacteristic, complemented
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic, complemented
19C2×C32×C4×D3C2212yes—
20C2×C32×C4×D3C2211yescharacteristic
21C2×#6293C2212yescomplemented
22C2×#6289C2211yescharacteristic
23C2×Dic3×C3⋊D3C2211yescharacteristic
24C2×D3×C3⋊Dic3C2212yes—
25C2×D3×C3⋊Dic3C2211yescharacteristic
26C4×D3×C3⋊D3—24no—
27D3×#634—24no—
28C33×C4⋊C4C2211yescharacteristic, complemented
29#88859C2212yes—
30C4⋊(C3⋊(C3⋊Dic3))C2212yescomplemented
31C3×#3752C2212yes—
32C3×C4⋊(C3⋊Dic3)C2212yescomplemented
33C32×C4⋊Dic3C2211yescharacteristic, complemented
34C32×Dic3⋊C4C2212yescomplemented
35Dic3⋊(C3⋊Dic3)C2212yescomplemented
36Dic3⋊(C3⋊Dic3)C2212yescomplemented
37#88804C2212yes—
38#88803C2212yes—
39C22×C4×C3⋊D3—31no—
40C22×#634—31no—
41C2×C4×D32—34no—
42C2×D3×Dic6—34no—
43C2×C32×C4⋊C4—31no—
44C2×#3752—32no—
45C2×C4⋊(C3⋊Dic3)—32no—
46#64106—31no—
47#64106—31no—
48#64106—31no—
49#64106—32no—
50#64106—32no—
51#64106—34no—
52#64106—64no—
53#64106D311yescharacteristic, direct product factor
54C3×#897—34no—
55C3×D3×C4⋊C4D314yescomplemented
56D3×(C2×D3)⋊C4—38no—
57#63895—38no—
58#63893—38no—
59D3×C4⋊Dic3—38no—
60#63871—34no—
61#63857—34no—
62C22×C3⋊(C3⋊D3)C2311yescharacteristic
63C22×C3×C3⋊D3C2311yescharacteristic
64C22×C32×D3C2311yescharacteristic
65C2×D3×C3⋊D3—22no—
66C2×D3×C3⋊D3—22no—
67C2×D3×C3⋊D3Q814yescomplemented
68C2×C33×C4C2312yes—
69C2×C33×C4C2311yescharacteristic
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)C2312yes—
72C4×C3⋊(C3⋊D3)—22no—
73#6308—22no—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2312yes—
76C3×C4×C3⋊D3—22no—
77C3×#634—22no—
78C2×C32×Dic3C2312yes—
79C2×C32×Dic3C2311yescharacteristic
80C32×C4×D3—24no—
81C32×C4×D3D414yes—
82#6293—24no—
83#6289—22no—
84Dic3×C3⋊D3—22no—
85D3×C3⋊Dic3—24no—
86D3×C3⋊Dic3—22no—
87D3×C22⋊Q8—91no—
88C2×#897—94no—
89C23×C3⋊D3—31no—
90C22×D32—34no—
91C22×C32×C4—31no—
92C22×C32×C4—32no—
93C22×C3⋊Dic3—32no—
94C22×C3⋊Dic3—31no—
95C2×C4×C3⋊D3—31no—
96C2×C4×C3⋊D3—31no—
97C2×C4×C3⋊D3—34no—
98C2×C4×C3⋊D3—31no—
99C2×C4×C3⋊D3—64no—
100C2×C4×C3⋊D3—64no—
101C2×C4×C3⋊D3C2×D311yescharacteristic
102C2×#634—31no—
103C2×#634—34no—
104C2×#634—32no—
105C2×#634—64no—
106C2×#634—64no—
107C2×#634C2×D311yescharacteristic, complemented
108C2×C3×C4×D3—34no—
109C2×C3×C4×D3C2×D318yes—
110C2×C3×C4×D3C2×D314yes—
111C2×C3×Dic6—34no—
112C2×#627—38no—
113C2×#624—34no—
114C2×D3×Dic3—38no—
115C2×D3×Dic3—34no—
116C2×D3×Dic3—34no—
117C4×D32—616no—
118D3×Dic6—616no—
119C32×C4⋊C4—24no—
120C32×C4⋊C4—32no—
121C32×C4⋊C4—31no—
122C32×C4⋊C4C2×D314yescomplemented
123C32×C4⋊C4C2×D311yescharacteristic, complemented
124#3752—32no—
125#3752—32no—
126#3752—38no—
127#3752—32no—
128#3752—68no—
129#3752C2×D312yes—
130C4⋊(C3⋊Dic3)—32no—
131C4⋊(C3⋊Dic3)—38no—
132C4⋊(C3⋊Dic3)—32no—
133C4⋊(C3⋊Dic3)—32no—
134C4⋊(C3⋊Dic3)—68no—
135C4⋊(C3⋊Dic3)C2×D312yescomplemented
136C3×(C2×D3)⋊C4—38no—
137C3×C4⋊Dic3—38no—
138C3×C4⋊Dic3C2×D314yescomplemented
139C3×Dic3⋊C4C2×D318yescomplemented
140Dic3⋊Dic3—38no—
141Dic3⋊Dic3—38no—
142#3722—38no—
143(C2×D3)⋊Dic3—38no—
144C22×C33C2411yescharacteristic
145C2×C3⋊(C3⋊D3)—21no—
146C2×C3⋊(C3⋊D3)—21no—
147C2×C3⋊(C3⋊D3)C2×Q812yescomplemented
148C2×C3×C3⋊D3—21no—
149C2×C3×C3⋊D3—21no—
150C2×C3×C3⋊D3C2×Q812yescomplemented
151C2×C32×D3C2×D412yes—
152C2×C32×D3C2×Q812yes—
153C2×C32×D3#4112yes—
154D3×C3⋊D3—28no—
155C33×C4—22no—
156C33×C4C2×D412yes—
157C3⋊(C3⋊Dic3)—21no—
158C3⋊(C3⋊Dic3)—22no—
159C3×C3⋊Dic3—21no—
160C3×C3⋊Dic3—22no—
161C32×Dic3—22no—
162C32×Dic3C2×D412yes—
163C2×D3×Q8—91no—
164C22×C4×D3—94no—
165C22×C4×D3—91no—
166C22×Dic6—94no—
167C3×C22⋊Q8—91no—
168C2×C3×C4⋊C4—34no—
169#946—91no—
170C2×(C2×D3)⋊C4—98no—
171C2×C4⋊Dic3—98no—
172#924—91no—
173#897—34no—
174#897—98no—
175#897—98no—
176#897—94no—
177#897—94no—
178#897—94no—
179#897—91no—
180#897—184no—
181#896—92no—
182D3×C4⋊C4—92no—
183D3×C4⋊C4C3⋊D311yescharacteristic, complemented
184D3×C22⋊C4—92no—
185#878—92no—
186C23×C32—31no—
187C22×C3⋊D3—31no—
188C22×C3⋊D3—34no—
189C22×C3⋊D3—34no—
190C22×C3⋊D3—64no—
191C22×C3⋊D3—62no—
192C22×C3⋊D3—62no—
193C22×C3⋊D3C22×D311yescharacteristic
194C22×C3×D3—34no—
195C22×C3×D3C22×D314yes—
196C2×D32—316no—
197C2×D32—68no—
198C2×D32—68no—
199C2×C32×C4—24no—
200C2×C32×C4—28no—
201C2×C32×C4—31no—
202C2×C32×C4—34no—
203C2×C32×C4—32no—
204C2×C32×C4—64no—
205C2×C32×C4C22×D318yes—
206C2×C32×C4C22×D314yes—
207C2×C32×C4C22×D312yes—
208C2×C32×C4C22×D311yescharacteristic
209C2×C3⋊Dic3—31no—
210C2×C3⋊Dic3—32no—
211C2×C3⋊Dic3—38no—
212C2×C3⋊Dic3—34no—
213C2×C3⋊Dic3—64no—
214C2×C3⋊Dic3—68no—
215C2×C3⋊Dic3—64no—
216C2×C3⋊Dic3—62no—
217C2×C3⋊Dic3C22×D311yescharacteristic
218C2×C3⋊Dic3C22×D312yes—
219C4×C3⋊D3—22no—
220C4×C3⋊D3—62no—
221C4×C3⋊D3—62no—
222C4×C3⋊D3—62no—
223C4×C3⋊D3—68no—
224C4×C3⋊D3—128no—
225#634—22no—
226#634—64no—
227#634—62no—
228#634—68no—
229#634—128no—
230C2×C3×Dic3—38no—
231C2×C3×Dic3—34no—
232C2×C3×Dic3C22×D318yes—
233C2×C3×Dic3C22×D314yes—
234C3×C4×D3—216no—
235C3×C4×D3—68no—
236C3×C4×D3D12116yes—
237C3×Dic6—68no—
238#627—616no—
239#624—68no—
240D3×Dic3—616no—
241D3×Dic3—68no—
242D3×Dic3—68no—
243C2×C22⋊Q8—271noSylow
244C2×C33C22×D411yescharacteristic
245C2×C33C22×Q811yescharacteristic
246C2×C33C2×#4111yescharacteristic
247C3⋊(C3⋊D3)—22no—
248C3×C3⋊D3—22no—
249C32×D3C22⋊Q814yescomplemented
250C23×D3—94no—
251C23×D3—91no—
252C2×C3×Q8—91no—
253C22×C3×C4—34no—
254C22×C3×C4—38no—
255C22×C3×C4—91no—
256C22×Dic3—94no—
257C22×Dic3—98no—
258C22×Dic3—91no—
259D3×Q8—184no—
260C2×C4×D3—34no—
261C2×C4×D3—94no—
262C2×C4×D3—94no—
263C2×C4×D3—92no—
264C2×C4×D3—91no—
265C2×C4×D3—94no—
266C2×C4×D3—91no—
267C2×C4×D3—91no—
268C2×C4×D3—184no—
269C2×C4×D3—1816no—
270C2×C4×D3—184no—
271C2×C4×D3C2×C3⋊D312yes—
272C2×C4×D3C2×C3⋊D311yescharacteristic
273C2×Dic6—34no—
274C2×Dic6—98no—
275C2×Dic6—94no—
276C2×Dic6—91no—
277C2×Dic6—92no—
278C2×Dic6—184no—
279C2×Dic6—1816no—
280C3×C4⋊C4—24no—
281C3×C4⋊C4—38no—
282C3×C4⋊C4—34no—
283C3×C4⋊C4—92no—
284C3×C4⋊C4D3214yescomplemented
285C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
286C3×C22⋊C4—92no—
287#217—92no—
288(C2×D3)⋊C4—38no—
289(C2×D3)⋊C4—98no—
290(C2×D3)⋊C4—98no—
291(C2×D3)⋊C4—98no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—188no—
295C4⋊Dic3—38no—
296C4⋊Dic3—98no—
297C4⋊Dic3—98no—
298C4⋊Dic3—98no—
299C4⋊Dic3—92no—
300C4⋊Dic3—188no—
301C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4C2×C3⋊D312yescomplemented
305C22×C32—24no—
306C22×C32—32no—
307C22×C32—32no—
308C22×C32—32no—
309C22×C32C23×D314yes—
310C22×C32C23×D311yescharacteristic
311C2×C3⋊D3—21no—
312C2×C3⋊D3—21no—
313C2×C3⋊D3—32no—
314C2×C3⋊D3—38no—
315C2×C3⋊D3—68no—
316C2×C3⋊D3—68no—
317C2×C3⋊D3—61no—
318C2×C3⋊D3—61no—
319C2×C3⋊D3—64no—
320C2×C3⋊D3—64no—
321C2×C3⋊D3—124no—
322C2×C3⋊D3—124no—
323C2×C3⋊D3D3×Q812yescomplemented
324C2×C3×D3—38no—
325C2×C3×D3—64no—
326C2×C3×D3—64no—
327C2×C3×D3C2×D1218yes—
328C2×C3×D3#23718yes—
329C2×C3×D3D3×Q818yes—
330D32—632no—
331C32×C4—22no—
332C32×C4—28no—
333C32×C4—28no—
334C32×C4—32no—
335C32×C4—48no—
336C32×C4—62no—
337C32×C4C2×D1218yes—
338C32×C4D3×D412yes—
339C3⋊Dic3—22no—
340C3⋊Dic3—21no—
341C3⋊Dic3—62no—
342C3⋊Dic3—61no—
343C3⋊Dic3—64no—
344C3⋊Dic3—68no—
345C3⋊Dic3—128no—
346C3⋊Dic3—124no—
347C3×Dic3—28no—
348C3×Dic3—64no—
349C3×Dic3—68no—
350C3×Dic3C2×D1218yes—
351C22×Q8—271no—
352C23×C4—271no—
353C22⋊Q8—91no—
354C22⋊Q8—272no—
355C22⋊Q8—272no—
356C22⋊Q8—271no—
357C22⋊Q8—271no—
358C22⋊Q8—271no—
359C2×C4⋊C4—31no—
360C2×C4⋊C4—272no—
361C2×C22⋊C4—272no—
362C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
363C23×C3—34no—
364C23×C3—91no—
365C22×D3—34no—
366C22×D3—916no—
367C22×D3—94no—
368C22×D3—91no—
369C22×D3—94no—
370C22×D3—184no—
371C22×D3—188no—
372C22×D3—188no—
373C22×D3—182no—
374C22×D3—182no—
375C22×D3C22×C3⋊D311yescharacteristic
376C3×Q8—182no—
377C2×C3×C4—24no—
378C2×C3×C4—28no—
379C2×C3×C4—34no—
380C2×C3×C4—316no—
381C2×C3×C4—38no—
382C2×C3×C4—616no—
383C2×C3×C4—91no—
384C2×C3×C4—92no—
385C2×C3×C4—182no—
386C2×C3×C4C2×D3218yes—
387C2×C3×C4C2×D3214yes—
388C2×C3×C4C22×C3⋊D312yes—
389C2×C3×C4C22×C3⋊D311yescharacteristic
390C2×Dic3—34no—
391C2×Dic3—38no—
392C2×Dic3—94no—
393C2×Dic3—98no—
394C2×Dic3—91no—
395C2×Dic3—92no—
396C2×Dic3—184no—
397C2×Dic3—188no—
398C2×Dic3—1816no—
399C2×Dic3—188no—
400C2×Dic3—182no—
401C2×Dic3C22×C3⋊D312yes—
402C2×Dic3C22×C3⋊D311yescharacteristic
403C4×D3—24no—
404C4×D3—68no—
405C4×D3—188no—
406C4×D3—188no—
407C4×D3—182no—
408C4×D3—184no—
409C4×D3—188no—
410C4×D3—182no—
411C4×D3—182no—
412C4×D3—368no—
413C4×D3C4⋊(C3⋊D3)14yes—
414Dic6—68no—
415Dic6—1816no—
416Dic6—188no—
417Dic6—182no—
418Dic6—184no—
419Dic6—368no—
420C2×C32—24no—
421C2×C32—24no—
422C2×C32—24no—
423C2×C32—34no—
424C2×C32C22×D1214yes—
425C2×C32C2×D3×D411yescharacteristic
426C2×C32C2×#23714yes—
427C2×C32C2×D3×Q814yes—
428C2×C32C2×D3×Q811yescharacteristic
429C2×C32D3×#4111yescharacteristic
430C3⋊D3—22no—
431C3⋊D3—62no—
432C3⋊D3—68no—
433C3⋊D3—128no—
434C3×D3—68no—
435C3×D3#897116yescomplemented
436C24—271no—
437C2×Q8—91no—
438C2×Q8—272no—
439C2×Q8—271no—
440C2×Q8—544no—
441C22×C4—31no—
442C22×C4—32no—
443C22×C4—91no—
444C22×C4—271no—
445C22×C4—271no—
446C22×C4—272no—
447C22×C4—271no—
448C22×C4—271no—
449C22×C4—544no—
450C4⋊C4—32no—
451C4⋊C4—31no—
452C4⋊C4—92no—
453C4⋊C4—272no—
454C4⋊C4—272no—
455C4⋊C4—272no—
456C4⋊C4D3×C3⋊D311yescharacteristic, complemented
457C22⋊C4—92no—
458C22⋊C4—272no—
459C22⋊C4—272no—
460C22⋊C4—272no—
461C22×C3—24no—
462C22×C3—38no—
463C22×C3—38no—
464C22×C3—38no—
465C22×C3—92no—
466C22×C3—181no—
467C22×C3—181no—
468C22×C3C22×D3214yes—
469C22×C3C23×C3⋊D311yescharacteristic
470C2×D3—38no—
471C2×D3—64no—
472C2×D3—64no—
473C2×D3—98no—
474C2×D3—92no—
475C2×D3—188no—
476C2×D3—1832no—
477C2×D3—184no—
478C2×D3—184no—
479C2×D3—181no—
480C2×D3—181no—
481C2×D3—188no—
482C2×D3—364no—
483C2×D3—364no—
484C2×D3C2×C4⋊(C3⋊D3)12yes—
485C2×D3#383012yes—
486C2×D3Q8×C3⋊D312yes—
487C3×C4—28no—
488C3×C4—22no—
489C3×C4—28no—
490C3×C4—38no—
491C3×C4—48no—
492C3×C4—68no—
493C3×C4—182no—
494C3×C4—181no—
495C3×C4D3×D1218yes—
496C3×C4C2×C4⋊(C3⋊D3)12yes—
497Dic3—22no—
498Dic3—68no—
499Dic3—64no—
500Dic3—188no—
501Dic3—184no—
502Dic3—182no—
503Dic3—181no—
504Dic3—368no—
505Dic3—364no—
506Dic3C2×C4⋊(C3⋊D3)12yes—
507C32—24no—
508C32C2×#89714yescomplemented
509C32D3×C22⋊Q811yescharacteristic, complemented
510C23—31no—
511C23—91no—
512C23—274no—
513C23—271no—
514C23—542no—
515C23—542no—
516Q8—182no—
517Q8—544no—
518Q8—542no—
519C2×C4—31no—
520C2×C4—34no—
521C2×C4—32no—
522C2×C4—64no—
523C2×C4—92no—
524C2×C4—91no—
525C2×C4—182no—
526C2×C4—272no—
527C2×C4—271no—
528C2×C4—542no—
529C2×C4—542no—
530C2×C4—542no—
531C2×C4—544no—
532C2×C4—542no—
533C2×C4C2×D3×C3⋊D312yes—
534C2×C4C2×D3×C3⋊D311yescharacteristic
535C2×C3—24no—
536C2×C3—24no—
537C2×C3—24no—
538C2×C3—316no—
539C2×C3—182no—
540C2×C3C2×D3×D1214yes—
541C2×C3D3×#23714yes—
542C2×C3D32×Q814yes—
543C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
544C2×C3C2×#383011yescharacteristic
545C2×C3C2×Q8×C3⋊D311yescharacteristic
546D3—68no—
547D3—188no—
548D3—182no—
549D3—368no—
550D3#6410614yesdirect product factor
551C22—32no—
552C22—32no—
553C22—32no—
554C22—92no—
555C22—181no—
556C22—181no—
557C22—272no—
558C22—548no—
559C22—541no—
560C22—541no—
561C22C22×D3×C3⋊D311yescharacteristic
562C4—22no—
563C4—32no—
564C4—62no—
565C4—181no—
566C4—182no—
567C4—541no—
568C4—542no—
569C4D3×C4⋊(C3⋊D3)12yes—
570C3—24no—
571C3D3×#89714yescomplemented
572C3C2×#6410611yescharacteristic, complemented
573C2—34no—
574C2—182no—
575C2—542no—
576C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
577C2D3×#383011yescharacteristic
578C2D3×Q8×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC22×D3×C3⋊D3C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C2×C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×D3×#634C2semidirect
3. #11723938C2semidirect
4. C3×#64106C2semidirect
5. C32×D3×C4⋊C4C2semidirect
7. #11723485C2semidirect
8. #11723483C2semidirect
9. D3×C4⋊(C3⋊Dic3)C2semidirect
10. #11723449C2semidirect
11. #11723431C2semidirect
16. C2×#6308C22semidirect
18. C2×C3×#634C22semidirect
21. C2×#6293C22semidirect
28. C33×C4⋊C4C22semidirect
30. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
32. C3×C4⋊(C3⋊Dic3)C22semidirect
33. C32×C4⋊Dic3C22semidirect
34. C32×Dic3⋊C4C22semidirect
35. Dic3⋊(C3⋊Dic3)C22semidirect
36. Dic3⋊(C3⋊Dic3)C22semidirect
53. #64106D3direct (N × H)
55. C3×D3×C4⋊C4D3semidirect
67. C2×D3×C3⋊D3Q8semidirect
107. C2×#634C2×D3semidirect
122. C32×C4⋊C4C2×D3semidirect
123. C32×C4⋊C4C2×D3semidirect
135. C4⋊(C3⋊Dic3)C2×D3semidirect
138. C3×C4⋊Dic3C2×D3semidirect
139. C3×Dic3⋊C4C2×D3semidirect
147. C2×C3⋊(C3⋊D3)C2×Q8semidirect
150. C2×C3×C3⋊D3C2×Q8semidirect
183. D3×C4⋊C4C3⋊D3semidirect
249. C32×D3C22⋊Q8semidirect
284. C3×C4⋊C4D32semidirect
285. C3×C4⋊C4C2×C3⋊D3semidirect
301. C4⋊Dic3C2×C3⋊D3semidirect
304. Dic3⋊C4C2×C3⋊D3semidirect
323. C2×C3⋊D3D3×Q8semidirect
362. C33C2×C22⋊Q8semidirect
435. C3×D3#897semidirect
456. C4⋊C4D3×C3⋊D3semidirect
508. C32C2×#897semidirect
509. C32D3×C22⋊Q8semidirect
550. D3#64106direct (N × H)
571. C3D3×#897semidirect
572. C3C2×#64106semidirect

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×D3×#6342
#117239382
C3×#641062
C32×D3×C4⋊C42
D3×#37522
#117234852
#117234832
D3×C4⋊(C3⋊Dic3)2
#117234492
#117234312
C2×#641063
D3×#8973