Finite Group Databasebeta

← #13177335

Subgroups — #13177335

Order 1728.

2996 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×#89039C211yesmaximal, characteristic
3C3×#64138C211yesmaximal, characteristic, complemented
4C32×#929C211yesmaximal, characteristic, complemented
5#11723627C211yesmaximal, characteristic, complemented
6#11723609C211yesmaximal, characteristic, complemented
7#11723603C211yesmaximal, characteristic, complemented
8#11723539C212yesmaximal, complemented
9#11723470C212yesmaximal, complemented
10#11723421C212yesmaximal, complemented
11#11723420C212yesmaximal, complemented
12C3⋊D3×#123—31nomaximal
13#10595079—34nomaximal
14C23×C3⋊(C3⋊D3)C2211yescharacteristic
15C22×D3×C3⋊D3C2211yescharacteristic
16C22×C33×C4C2211yescharacteristic, complemented
17C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
18C2×C4×C3⋊(C3⋊D3)—24no—
19C2×C4×C3⋊(C3⋊D3)C2212yes—
20C2×C4×C3⋊(C3⋊D3)C2212yescomplemented
21C2×C3×#638C2211yescharacteristic
22C2×C32×#67C2211yescharacteristic
23#89039—24no—
24C2×#6292C2211yescharacteristic
25C2×#6291C2211yescharacteristic
26C2×#6289C2212yes—
27C2×#6289C2211yescharacteristic
28C3×#3757C2211yescharacteristic, complemented
29C3×#3752C2212yescomplemented
30C3×#3750C2212yescomplemented
31C32×#217C2211yescharacteristic, complemented
32C32×(C2×D3)⋊C4C2212yescomplemented
33C32×Dic3⋊C4C2212yescomplemented
34#88807C2212yescomplemented
35Dic3⋊(C3⋊Dic3)C2212yescomplemented
36#88804C2212yescomplemented
37#88803C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C2×D4×C3⋊D3—31no—
40C22×C4×C3⋊D3—31no—
41C22×C4×C3⋊D3—34no—
42C22×C4×C3⋊D3—64no—
43C2×#3811—34no—
44C32×#123—31no—
45#64144—31no—
46#64138D311yescharacteristic, complemented
47#64103—32no—
48C4⋊C4×C3⋊D3—32no—
49#64095—31no—
50#64092—32no—
51C22⋊C4×C3⋊D3—31no—
52C22⋊C4×C3⋊D3—32no—
53C3×#929—34no—
54C3×#929D314yescomplemented
55#63974—34no—
56#63962—34no—
57#63962—34no—
58#63921—38no—
59#63883—38no—
60#63853—38no—
61#63853—38no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)—24no—
64C22×C3⋊(C3⋊D3)D414yes—
65C22×C3⋊(C3⋊D3)C2311yescharacteristic
66C22×C3⋊(C3⋊D3)C2311yescharacteristic
67C22×C3×C3⋊D3C2311yescharacteristic
68C22×C32×D3C2311yescharacteristic
69C2×D3×C3⋊D3—22no—
70C2×D3×C3⋊D3—24no—
71C2×C33×C4—22no—
72C2×C33×C4C2312yes—
73C2×C3⋊(C3⋊Dic3)—22no—
74C2×C3⋊(C3⋊Dic3)C2312yes—
75C4×C3⋊(C3⋊D3)—24no—
76C4×C3⋊(C3⋊D3)—24no—
77C3×#638—22no—
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C2×C3×C3⋊Dic3C2312yes—
80C32×#67—22no—
81C2×C32×Dic3C2312yes—
82C2×C32×Dic3C2311yescharacteristic
83#6292—22no—
84#6291—22no—
85#6289—22no—
86#6289—24no—
87D3×#123—94no—
88D3×#123—91no—
89C23×C3⋊D3—31no—
90C23×C3⋊D3—34no—
91C23×C3⋊D3—31no—
92C23×C3⋊D3—64no—
93C22×D32—34no—
94C2×C32×D4—31no—
95C22×C32×C4—24no—
96C22×C32×C4C2×D314yescomplemented
97C22×C32×C4C2×D311yescharacteristic, complemented
98C2×#638—31no—
99C2×#638C2×D311yescharacteristic
100C22×C3⋊Dic3—31no—
101C22×C3⋊Dic3—34no—
102C22×C3⋊Dic3—64no—
103D4×C3⋊D3—64no—
104C2×C4⋊(C3⋊D3)—31no—
105C2×C4×C3⋊D3—32no—
106C2×C4×C3⋊D3—32no—
107C2×C4×C3⋊D3—38no—
108C2×C4×C3⋊D3—38no—
109C2×C4×C3⋊D3—32no—
110C2×C4×C3⋊D3—31no—
111C2×C4×C3⋊D3—64no—
112C2×C4×C3⋊D3—616no—
113C2×C4×C3⋊D3—68no—
114C2×C4×C3⋊D3—68no—
115C2×C4×C3⋊D3—632no—
116C2×C3×#67—34no—
117C2×C3×#67C2×D314yes—
118#3811—616no—
119C2×#626—34no—
120C2×#626—34no—
121C2×#624—34no—
122C2×#624—38no—
123C32×C4⋊C4—32no—
124C32×C22⋊C4—31no—
125C32×C22⋊C4—32no—
126#3757—32no—
127#3757C2×D311yescharacteristic, complemented
128#3752—32no—
129#3752—32no—
130#3752C2×D312yescomplemented
131C4⋊(C3⋊Dic3)—32no—
132#3750—32no—
133#3750C2×D312yescomplemented
134C3×#217—34no—
135C3×#217C2×D314yescomplemented
136C3×(C2×D3)⋊C4—38no—
137C3×(C2×D3)⋊C4C2×D318yescomplemented
138C3×Dic3⋊C4—38no—
139C3×Dic3⋊C4C2×D318yescomplemented
140Dic3⋊Dic3—38no—
141Dic3⋊Dic3—38no—
142#3722—38no—
143(C2×D3)⋊Dic3—38no—
144(C2×D3)⋊Dic3—38no—
145C22×C33—22no—
146C22×C33C2×D412yes—
147C22×C33C2411yescharacteristic
148C2×C3⋊(C3⋊D3)—22no—
149C2×C3⋊(C3⋊D3)—28no—
150C2×C3⋊(C3⋊D3)C2×D412yes—
151C2×C3⋊(C3⋊D3)C2×D412yes—
152C2×C3⋊(C3⋊D3)#4114yes—
153C2×C3×C3⋊D3—21no—
154C2×C3×C3⋊D3—22no—
155C2×C32×D3—21no—
156C2×C32×D3—22no—
157D3×C3⋊D3—44no—
158C33×C4—22no—
159C3⋊(C3⋊Dic3)—22no—
160C3×C3⋊Dic3—21no—
161C3×C3⋊Dic3—22no—
162C32×Dic3—21no—
163C32×Dic3—22no—
164C2×D3×D4—94no—
165C2×D3×D4—91no—
166C22×C4×D3—94no—
167C22×C4×D3—91no—
168C22×C4×D3—184no—
169C3×#123—34no—
170C3×#123—91no—
171#935—94no—
172#935—91no—
173#929—34no—
174#929C3⋊D311yescharacteristic, complemented
175#894—98no—
176#894—92no—
177D3×C4⋊C4—98no—
178D3×C4⋊C4—92no—
179#886—94no—
180#886—91no—
181#883—98no—
182#883—92no—
183D3×C22⋊C4—98no—
184D3×C22⋊C4—94no—
185D3×C22⋊C4—91no—
186D3×C22⋊C4—92no—
187C23×C32—24no—
188C23×C32—31no—
189C23×C32C22×D314yes—
190C23×C32C22×D311yescharacteristic
191C22×C3⋊D3—31no—
192C22×C3⋊D3—34no—
193C22×C3⋊D3—31no—
194C22×C3⋊D3—34no—
195C22×C3⋊D3—316no—
196C22×C3⋊D3—34no—
197C22×C3⋊D3—616no—
198C22×C3⋊D3—64no—
199C22×C3⋊D3—64no—
200C22×C3⋊D3—64no—
201C22×C3⋊D3—62no—
202C22×C3⋊D3—64no—
203C22×C3⋊D3—616no—
204C22×C3⋊D3—632no—
205C22×C3⋊D3C22×D311yescharacteristic
206C22×C3×D3—34no—
207C22×C3×D3C22×D314yes—
208C2×D32—616no—
209C2×D32—68no—
210C32×D4—62no—
211C2×C32×C4—216no—
212C2×C32×C4—28no—
213C2×C32×C4—22no—
214C2×C32×C4—28no—
215C2×C32×C4—31no—
216C2×C32×C4—32no—
217C2×C32×C4C22×D318yes—
218C2×C32×C4C22×D312yes—
219#638—22no—
220#638—62no—
221C2×C3⋊Dic3—32no—
222C2×C3⋊Dic3—38no—
223C2×C3⋊Dic3—68no—
224C2×C3⋊Dic3—616no—
225C2×C3⋊Dic3—68no—
226C2×C3⋊Dic3—62no—
227C2×C3⋊Dic3C22×D312yes—
228C2×C3⋊Dic3C22×D311yescharacteristic
229C4⋊(C3⋊D3)—62no—
230C4×C3⋊D3—64no—
231C4×C3⋊D3—62no—
232C4×C3⋊D3—64no—
233C4×C3⋊D3—64no—
234C4×C3⋊D3—616no—
235C4×C3⋊D3—616no—
236C4×C3⋊D3—632no—
237C4×C3⋊D3—632no—
238C3×#67—28no—
239C3×#67—68no—
240C2×C3×Dic3—38no—
241C2×C3×Dic3—34no—
242C2×C3×Dic3C22×D318yes—
243C2×C3×Dic3C22×D314yes—
244#626—68no—
245#626—68no—
246#624—616no—
247#624—68no—
248C2×#123—271noSylow
249C2×C33—22no—
250C2×C33C22×D411yescharacteristic
251C2×C33C2×#4112yes—
252C3⋊(C3⋊D3)—22no—
253C3⋊(C3⋊D3)#12314yescomplemented
254C3×C3⋊D3—41no—
255C32×D3—41no—
256C23×D3—94no—
257C23×D3—94no—
258C23×D3—91no—
259C23×D3—91no—
260C23×D3—184no—
261C2×C3×D4—34no—
262C2×C3×D4—91no—
263C22×C3×C4—24no—
264C22×C3×C4D3214yescomplemented
265C22×C3×C4C2×C3⋊D311yescharacteristic, complemented
266C2×#67—34no—
267C2×#67—94no—
268C2×#67—91no—
269C2×#67C2×C3⋊D311yescharacteristic
270C22×Dic3—94no—
271C22×Dic3—91no—
272C22×Dic3—184no—
273D3×D4—184no—
274D3×D4—1816no—
275C2×D12—94no—
276C2×D12—91no—
277C2×C4×D3—98no—
278C2×C4×D3—94no—
279C2×C4×D3—98no—
280C2×C4×D3—98no—
281C2×C4×D3—92no—
282C2×C4×D3—92no—
283C2×C4×D3—92no—
284C2×C4×D3—91no—
285C2×C4×D3—188no—
286C2×C4×D3—188no—
287C2×C4×D3—1816no—
288C2×C4×D3—184no—
289C2×C4×D3—1832no—
290C3×C4⋊C4—38no—
291C3×C4⋊C4—92no—
292C3×C22⋊C4—34no—
293C3×C22⋊C4—38no—
294C3×C22⋊C4—91no—
295C3×C22⋊C4—92no—
296#217—34no—
297#217—98no—
298#217—92no—
299#217C2×C3⋊D311yescharacteristic, complemented
300(C2×D3)⋊C4—38no—
301(C2×D3)⋊C4—98no—
302(C2×D3)⋊C4—98no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—92no—
305(C2×D3)⋊C4C2×C3⋊D312yescomplemented
306C4⋊Dic3—98no—
307C4⋊Dic3—92no—
308Dic3⋊C4—38no—
309Dic3⋊C4—98no—
310Dic3⋊C4—92no—
311Dic3⋊C4C2×C3⋊D312yescomplemented
312C22×C32—216no—
313C22×C32—28no—
314C22×C32—22no—
315C22×C32—24no—
316C22×C32—28no—
317C22×C32—61no—
318C22×C32—62no—
319C22×C32D3×D418yes—
320C22×C32D3×D412yes—
321C22×C32C23×D314yes—
322C22×C32C23×D311yescharacteristic
323C2×C3⋊D3—21no—
324C2×C3⋊D3—22no—
325C2×C3⋊D3—32no—
326C2×C3⋊D3—34no—
327C2×C3⋊D3—32no—
328C2×C3⋊D3—38no—
329C2×C3⋊D3—316no—
330C2×C3⋊D3—38no—
331C2×C3⋊D3—68no—
332C2×C3⋊D3—68no—
333C2×C3⋊D3—616no—
334C2×C3⋊D3—68no—
335C2×C3⋊D3—632no—
336C2×C3⋊D3—664no—
337C2×C3⋊D3—616no—
338C2×C3⋊D3—68no—
339C2×C3⋊D3—62no—
340C2×C3⋊D3—124no—
341C2×C3×D3—24no—
342C2×C3×D3—28no—
343C2×C3×D3—64no—
344C2×C3×D3—68no—
345D32—1216no—
346C32×C4—216no—
347C32×C4—28no—
348C32×C4—22no—
349C32×C4—61no—
350C32×C4—62no—
351C3⋊Dic3—21no—
352C3⋊Dic3—22no—
353C3⋊Dic3—616no—
354C3⋊Dic3—68no—
355C3⋊Dic3—62no—
356C3×Dic3—24no—
357C3×Dic3—28no—
358C3×Dic3—64no—
359C3×Dic3—68no—
360C22×D4—271no—
361C23×C4—271no—
362#123—31no—
363#123—91no—
364#123—272no—
365#123—271no—
366#123—272no—
367#123—271no—
368C2×C4⋊C4—272no—
369C2×C22⋊C4—271no—
370C2×C22⋊C4—272no—
371C33C2×#12311yescharacteristic, complemented, Sylow
372C23×C3—24no—
373C23×C3—34no—
374C23×C3—91no—
375C23×C3C2×D3214yes—
376C23×C3C22×C3⋊D311yescharacteristic
377C22×D3—34no—
378C22×D3—94no—
379C22×D3—91no—
380C22×D3—94no—
381C22×D3—91no—
382C22×D3—916no—
383C22×D3—94no—
384C22×D3—184no—
385C22×D3—184no—
386C22×D3—1816no—
387C22×D3—182no—
388C22×D3—184no—
389C22×D3—188no—
390C22×D3—1816no—
391C22×D3—1816no—
392C22×D3—184no—
393C22×D3—1832no—
394C22×D3C22×C3⋊D311yescharacteristic
395C3×D4—68no—
396C3×D4—182no—
397C2×C3×C4—216no—
398C2×C3×C4—28no—
399C2×C3×C4—22no—
400C2×C3×C4—28no—
401C2×C3×C4—34no—
402C2×C3×C4—38no—
403C2×C3×C4—91no—
404C2×C3×C4—92no—
405C2×C3×C4C2×D3218yes—
406C2×C3×C4C22×C3⋊D312yes—
407#67—22no—
408#67—68no—
409#67—182no—
410#67—188no—
411C2×Dic3—34no—
412C2×Dic3—38no—
413C2×Dic3—98no—
414C2×Dic3—92no—
415C2×Dic3—188no—
416C2×Dic3—1816no—
417C2×Dic3—188no—
418C2×Dic3—182no—
419C2×Dic3C22×C3⋊D312yes—
420C2×Dic3C22×C3⋊D311yescharacteristic
421D12—188no—
422D12—182no—
423C4×D3—184no—
424C4×D3—182no—
425C4×D3—1816no—
426C4×D3—188no—
427C4×D3—1816no—
428C4×D3—1816no—
429C4×D3—184no—
430C4×D3—184no—
431C4×D3—1832no—
432C4×D3—1832no—
433C2×C32—216no—
434C2×C32—28no—
435C2×C32—22no—
436C2×C32—28no—
437C2×C32—24no—
438C2×C32—121no—
439C2×C32C2×D3×D414yes—
440C2×C32C2×D3×D411yescharacteristic
441C2×C32D3×#4118yes—
442C2×C32D3×#4112yes—
443C3⋊D3—34no—
444C3⋊D3—316no—
445C3⋊D3—41no—
446C3⋊D3—616no—
447C3⋊D3—616no—
448C3⋊D3—68no—
449C3⋊D3—62no—
450C3×D3—44no—
451C3×D3—124no—
452C24—271no—
453C24—271no—
454C2×D4—31no—
455C2×D4—91no—
456C2×D4—271no—
457C2×D4—271no—
458C2×D4—544no—
459C22×C4—272no—
460C22×C4—271no—
461C22×C4—271no—
462C22×C4—272no—
463C22×C4—272no—
464C22×C4—544no—
465C22×C4D3×C3⋊D311yescharacteristic, complemented
466C4⋊C4—32no—
467C4⋊C4—92no—
468C4⋊C4—272no—
469C4⋊C4—272no—
470C22⋊C4—31no—
471C22⋊C4—32no—
472C22⋊C4—91no—
473C22⋊C4—92no—
474C22⋊C4—272no—
475C22⋊C4—272no—
476C22⋊C4—272no—
477C22×C3—216no—
478C22×C3—28no—
479C22×C3—22no—
480C22×C3—24no—
481C22×C3—28no—
482C22×C3—68no—
483C22×C3—64no—
484C22×C3—181no—
485C22×C3—182no—
486C22×C3#381118yes—
487C22×C3D4×C3⋊D312yes—
488C22×C3C22×D3214yes—
489C22×C3C23×C3⋊D311yescharacteristic
490C2×D3—21no—
491C2×D3—22no—
492C2×D3—68no—
493C2×D3—64no—
494C2×D3—916no—
495C2×D3—98no—
496C2×D3—92no—
497C2×D3—94no—
498C2×D3—98no—
499C2×D3—92no—
500C2×D3—188no—
501C2×D3—188no—
502C2×D3—1816no—
503C2×D3—1832no—
504C2×D3—188no—
505C2×D3—1864no—
506C2×D3—1816no—
507C2×D3—188no—
508C2×D3—182no—
509C2×D3—364no—
510C2×D3—3616no—
511C3×C4—216no—
512C3×C4—28no—
513C3×C4—22no—
514C3×C4—68no—
515C3×C4—64no—
516C3×C4—181no—
517C3×C4—182no—
518Dic3—21no—
519Dic3—22no—
520Dic3—68no—
521Dic3—64no—
522Dic3—1816no—
523Dic3—188no—
524Dic3—182no—
525C32—24no—
526C32D3×#12314yescomplemented
527C32D3×#12311yescharacteristic, complemented
528C23—31no—
529C23—91no—
530C23—271no—
531C23—271no—
532C23—274no—
533C23—544no—
534C23—542no—
535C23—544no—
536C23C2×D3×C3⋊D311yescharacteristic
537D4—62no—
538D4—182no—
539D4—542no—
540D4—542no—
541C2×C4—22no—
542C2×C4—32no—
543C2×C4—31no—
544C2×C4—92no—
545C2×C4—91no—
546C2×C4—272no—
547C2×C4—544no—
548C2×C4—544no—
549C2×C4—542no—
550C2×C4—544no—
551C2×C4—542no—
552C2×C4C2×D3×C3⋊D312yes—
553C2×C3—216no—
554C2×C3—28no—
555C2×C3—22no—
556C2×C3—28no—
557C2×C3—24no—
558C2×C3—124no—
559C2×C3—361no—
560C2×C3#6430618yes—
561C2×C3C2×#381114yes—
562C2×C3C2×D4×C3⋊D311yescharacteristic
563C2×C3#41×C3⋊D312yes—
564D3—41no—
565D3—916no—
566D3—94no—
567D3—124no—
568D3—1816no—
569D3—1816no—
570D3—188no—
571D3—182no—
572C22—22no—
573C22—62no—
574C22—61no—
575C22—182no—
576C22—181no—
577C22—274no—
578C22—272no—
579C22—272no—
580C22—548no—
581C22—542no—
582C22—1084no—
583C22#8903912yes—
584C22C22×D3×C3⋊D311yescharacteristic
585C4—22no—
586C4—62no—
587C4—61no—
588C4—182no—
589C4—181no—
590C4—542no—
591C3—24no—
592C3#1059507914yescomplemented
593C3C3⋊D3×#12311yescharacteristic, complemented
594C2—22no—
595C2—121no—
596C2—274no—
597C2—361no—
598C2—542no—
599C2#1172457812yes—
600C2C2×#8903911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
prime radicalC2×#89039C2
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×C3⋊(C3⋊D3)C2semidirect
3. C3×#64138C2semidirect
4. C32×#929C2semidirect
5. #11723627C2semidirect
6. #11723609C2semidirect
7. #11723603C2semidirect
8. #11723539C2semidirect
9. #11723470C2semidirect
10. #11723421C2semidirect
11. #11723420C2semidirect
16. C22×C33×C4C22semidirect
17. C22×C3⋊(C3⋊Dic3)C22semidirect
20. C2×C4×C3⋊(C3⋊D3)C22semidirect
28. C3×#3757C22semidirect
29. C3×#3752C22semidirect
30. C3×#3750C22semidirect
31. C32×#217C22semidirect
32. C32×(C2×D3)⋊C4C22semidirect
33. C32×Dic3⋊C4C22semidirect
34. #88807C22semidirect
35. Dic3⋊(C3⋊Dic3)C22semidirect
36. #88804C22semidirect
37. #88803C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
46. #64138D3semidirect
54. C3×#929D3semidirect
96. C22×C32×C4C2×D3semidirect
97. C22×C32×C4C2×D3semidirect
127. #3757C2×D3semidirect
130. #3752C2×D3semidirect
133. #3750C2×D3semidirect
135. C3×#217C2×D3semidirect
137. C3×(C2×D3)⋊C4C2×D3semidirect
139. C3×Dic3⋊C4C2×D3semidirect
174. #929C3⋊D3semidirect
253. C3⋊(C3⋊D3)#123semidirect
264. C22×C3×C4D32semidirect
265. C22×C3×C4C2×C3⋊D3semidirect
299. #217C2×C3⋊D3semidirect
305. (C2×D3)⋊C4C2×C3⋊D3semidirect
311. Dic3⋊C4C2×C3⋊D3semidirect
371. C33C2×#123semidirect
465. C22×C4D3×C3⋊D3semidirect
526. C32D3×#123semidirect
527. C32D3×#123semidirect
592. C3#10595079semidirect
593. C3C3⋊D3×#123semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊(C3⋊D3)2
C2×#890392
C3×#641382
C32×#9292
#117236272
#117236092
#117236032
#117235392
#117234702
#117234212
#117234202
C3⋊D3×#1233
#105950793