Finite Group Databasebeta

← #13175942

Subgroups — #13175942

Order 1728.

2604 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89039C211yesmaximal, characteristic, complemented
2C22×#6291C211yesmaximal, characteristic, complemented
3C22⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C3×#64095C211yesmaximal, characteristic, complemented
5C32×#882C211yesmaximal, characteristic, complemented
6#11723629C211yesmaximal, characteristic, complemented
7#11723602C211yesmaximal, characteristic, complemented
8#11723540C212yesmaximal, complemented
9#11723531C212yesmaximal, complemented
10#11723527C212yesmaximal, complemented
11#11723422C212yesmaximal, complemented
12#10596060—31nomaximal
13#10594296—34nomaximal
14C23×C3⋊(C3⋊D3)C2211yescharacteristic
15C23×C32×D3C2211yescharacteristic, complemented
16C22×D3×C3⋊D3C2211yescharacteristic, complemented
17C2×C4×C3⋊(C3⋊D3)C2212yescomplemented
18C2×C3×#638C2211yescharacteristic
19C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
20C2×C32×#67C2211yescharacteristic
21C2×C32×D12C2212yescomplemented
22#89039—24no—
23C2×#6292C2211yescharacteristic, complemented
24C2×#6291—24no—
25C2×#6291C2212yes—
26C2×#6291C2212yescomplemented
27C2×#6291C2211yescharacteristic, complemented
28C2×#6290C2212yescomplemented
29C2×#6289C2211yescharacteristic, complemented
30C33×C22⋊C4C2211yescharacteristic, complemented
31#88864C2211yescharacteristic, complemented
32#88859C2212yescomplemented
33C3×#3752C2212yescomplemented
34C3×C4⋊(C3⋊Dic3)C2212yescomplemented
35C32×(C2×D3)⋊C4C2212yescomplemented
36#88806C2212yescomplemented
37(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C22×#638—31no—
40C2×D4×C3⋊D3—31no—
41C2×#3811—34no—
42C22×#626—34no—
43C32×#120—31no—
44#64162—31no—
45#64147—32no—
46#64146—32no—
47#64144—31no—
48#64095D311yescharacteristic, complemented
49#64093—32no—
50#64092—32no—
51C22⋊C4×C3⋊D3—31no—
52C22⋊C4×C3⋊D3—34no—
53C22⋊C4×C3⋊D3—64no—
54C3×#886—34no—
55C3×#882D314yescomplemented
56#63976—34no—
57#63961—34no—
58#63922—38no—
59#63916—38no—
60#63912—38no—
61#63854—38no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)—42no—
64C22×C3⋊(C3⋊D3)D414yescomplemented
65C22×C3⋊(C3⋊D3)C2311yescharacteristic
66C22×C3⋊(C3⋊D3)C2311yescharacteristic
67C22×C3×C3⋊D3C2311yescharacteristic
68C22×C32×D3—22no—
69C22×C32×D3—42no—
70C22×C32×D3C2311yescharacteristic
71C22×C32×D3C2312yes—
72C2×D3×C3⋊D3—22no—
73C2×D3×C3⋊D3—42no—
74C2×C33×C4C2312yes—
75C2×C3⋊(C3⋊Dic3)C2312yes—
76C4×C3⋊(C3⋊D3)—42no—
77C3×#638—22no—
78C2×C3×C3⋊Dic3—22no—
79C2×C3×C3⋊Dic3C2311yescharacteristic
80C2×C3×C3⋊Dic3C2312yes—
81C32×#67—22no—
82C2×C32×Dic3C2311yescharacteristic
83C32×D12—42no—
84#6292—22no—
85#6291—24no—
86#6291—24no—
87#6291—22no—
88#6290—42no—
89#6289—22no—
90#5653—91no—
91#5589—94no—
92C24×C32—31no—
93C23×C3⋊D3—31no—
94C23×C3⋊D3—34no—
95C23×C3⋊D3—31no—
96C23×C3⋊D3—64no—
97C23×C3×D3C2×D314yescomplemented
98C22×D32—34no—
99C2×C32×D4—31no—
100C2×C32×D4—32no—
101C2×#638—32no—
102C2×#638—32no—
103C2×#638—32no—
104C2×#638—31no—
105C2×#638—64no—
106C2×#638C2×D311yescharacteristic
107C22×C3⋊Dic3C2×D311yescharacteristic, complemented
108D4×C3⋊D3—64no—
109C2×C4⋊(C3⋊D3)—31no—
110C2×C4×C3⋊D3—31no—
111C2×C4×C3⋊D3—32no—
112C2×C4×C3⋊D3—38no—
113C2×C4×C3⋊D3—68no—
114C2×C3×#67—34no—
115C2×C3×#67C2×D314yes—
116C22×C3×Dic3—34no—
117C2×C3×D12C2×D318yescomplemented
118#3811—616no—
119C2×#626—38no—
120C2×#626—38no—
121C2×#626—34no—
122C2×#626—34no—
123C2×#626—616no—
124C2×#625—38no—
125C2×#624—34no—
126C32×C22⋊C4—24no—
127C32×C22⋊C4—32no—
128C32×C22⋊C4C2×D314yescomplemented
129C32×C22⋊C4C2×D311yescharacteristic, complemented
130#3757—31no—
131#3757—34no—
132#3757—32no—
133#3757—32no—
134#3757—64no—
135#3752—32no—
136#3752—38no—
137#3752—68no—
138#3752C2×D312yescomplemented
139C4⋊(C3⋊Dic3)C2×D312yescomplemented
140#3750—32no—
141C3×(C2×D3)⋊C4—38no—
142C3×(C2×D3)⋊C4C2×D318yescomplemented
143C3×C4⋊Dic3—38no—
144#3724—38no—
145(C2×D3)⋊Dic3—38no—
146(C2×D3)⋊Dic3—38no—
147C22×C33—22no—
148C22×C33C2×D412yes—
149C22×C33C2411yescharacteristic
150C2×C3⋊(C3⋊D3)—22no—
151C2×C3⋊(C3⋊D3)—22no—
152C2×C3⋊(C3⋊D3)—44no—
153C2×C3⋊(C3⋊D3)C2×D412yes—
154C2×C3⋊(C3⋊D3)C2×D412yes—
155C2×C3×C3⋊D3—21no—
156C2×C3×C3⋊D3—22no—
157C2×C32×D3—22no—
158C2×C32×D3—21no—
159C2×C32×D3—22no—
160C2×C32×D3—22no—
161C2×C32×D3—22no—
162C2×C32×D3—44no—
163D3×C3⋊D3—44no—
164C33×C4—22no—
165C3⋊(C3⋊Dic3)—22no—
166C3×C3⋊Dic3—21no—
167C3×C3⋊Dic3—22no—
168C32×Dic3—21no—
169C22×#67—94no—
170C2×D3×D4—94no—
171C2×D3×D4—91no—
172C22×D12—91no—
173C3×#123—91no—
174C3×#120—34no—
175#953—94no—
176#938—98no—
177#937—98no—
178#937—91no—
179#935—94no—
180#929—91no—
181#895—92no—
182#894—92no—
183#886—34no—
184#884—98no—
185#884—92no—
186#883—98no—
187#882—92no—
188#882C3⋊D311yescharacteristic, complemented
189D3×C22⋊C4—94no—
190D3×C22⋊C4—91no—
191D3×C22⋊C4—184no—
192C23×C32—24no—
193C23×C32—31no—
194C23×C32—32no—
195C23×C32—62no—
196C23×C32—122no—
197C23×C32C22×D314yes—
198C23×C32C22×D311yescharacteristic
199C22×C3⋊D3—31no—
200C22×C3⋊D3—31no—
201C22×C3⋊D3—34no—
202C22×C3⋊D3—34no—
203C22×C3⋊D3—34no—
204C22×C3⋊D3—316no—
205C22×C3⋊D3—64no—
206C22×C3⋊D3—64no—
207C22×C3⋊D3—616no—
208C22×C3⋊D3—62no—
209C22×C3⋊D3—122no—
210C22×C3⋊D3—122no—
211C22×C3⋊D3—128no—
212C22×C3⋊D3—1216no—
213C22×C3⋊D3C22×D311yescharacteristic
214C22×C3×D3—28no—
215C22×C3×D3—34no—
216C22×C3×D3—48no—
217C22×C3×D3C22×D314yes—
218C22×C3×D3C22×D318yes—
219C2×D32—68no—
220C2×D32—128no—
221C32×D4—62no—
222C32×D4—122no—
223C2×C32×C4—28no—
224C2×C32×C4—31no—
225C2×C32×C4C22×D318yes—
226C2×C32×C4C22×D312yes—
227#638—22no—
228#638—62no—
229#638—64no—
230#638—64no—
231#638—122no—
232C2×C3⋊Dic3—22no—
233C2×C3⋊Dic3—32no—
234C2×C3⋊Dic3—38no—
235C2×C3⋊Dic3—68no—
236C2×C3⋊Dic3C22×D312yes—
237C2×C3⋊Dic3C22×D311yescharacteristic
238C4⋊(C3⋊D3)—62no—
239C4×C3⋊D3—62no—
240C4×C3⋊D3—122no—
241C4×C3⋊D3—128no—
242C4×C3⋊D3—1216no—
243C3×#67—28no—
244C3×#67—68no—
245C2×C3×Dic3—34no—
246C2×C3×Dic3—38no—
247C2×C3×Dic3—68no—
248C2×C3×Dic3C22×D314yes—
249C3×D12—48no—
250#626—68no—
251#626—616no—
252#626—616no—
253#626—68no—
254#625—128no—
255#624—68no—
256#534—271noSylow
257C2×C33—22no—
258C2×C33C22×D411yescharacteristic
259C2×C33H(5,𝔽2)12yes—
260C3⋊(C3⋊D3)—22no—
261C3⋊(C3⋊D3)—22no—
262C3×C3⋊D3—41no—
263C32×D3—41no—
264C32×D3—42no—
265C24×C3—34no—
266C23×D3—94no—
267C23×D3—94no—
268C23×D3—91no—
269C23×D3—91no—
270C23×D3—184no—
271C23×D3C2×C3⋊D311yescharacteristic, complemented
272C2×C3×D4—34no—
273C2×C3×D4—38no—
274C2×C3×D4—91no—
275C22×C3×C4—91no—
276C2×#67—34no—
277C2×#67—98no—
278C2×#67—98no—
279C2×#67—94no—
280C2×#67—98no—
281C2×#67—92no—
282C2×#67—91no—
283C2×#67—1816no—
284C2×#67C2×C3⋊D311yescharacteristic
285C22×Dic3—34no—
286D3×D4—184no—
287D3×D4—1816no—
288C2×D12—94no—
289C2×D12—92no—
290C2×D12—92no—
291C2×D12—91no—
292C2×D12—184no—
293C2×D12C2×C3⋊D312yescomplemented
294C2×C4×D3—94no—
295C2×C4×D3—91no—
296C2×C4×D3—98no—
297C2×C4×D3—92no—
298C2×C4×D3—188no—
299C3×C4⋊C4—92no—
300C3×C22⋊C4—24no—
301C3×C22⋊C4—38no—
302C3×C22⋊C4—92no—
303C3×C22⋊C4D3214yescomplemented
304C3×C22⋊C4C2×C3⋊D311yescharacteristic, complemented
305#217—98no—
306#217—98no—
307#217—94no—
308#217—91no—
309#217—184no—
310(C2×D3)⋊C4—38no—
311(C2×D3)⋊C4—92no—
312(C2×D3)⋊C4—92no—
313(C2×D3)⋊C4—98no—
314(C2×D3)⋊C4—92no—
315(C2×D3)⋊C4—188no—
316(C2×D3)⋊C4C2×C3⋊D312yescomplemented
317C4⋊Dic3—38no—
318Dic3⋊C4—98no—
319Dic3⋊C4—92no—
320C22×C32—22no—
321C22×C32—24no—
322C22×C32—28no—
323C22×C32—28no—
324C22×C32—48no—
325C22×C32—62no—
326C22×C32—61no—
327C22×C32—62no—
328C22×C32—62no—
329C22×C32—62no—
330C22×C32—124no—
331C22×C32D3×D418yes—
332C22×C32D3×D412yes—
333C22×C32C23×D314yes—
334C22×C32C23×D311yescharacteristic
335C2×C3⋊D3—21no—
336C2×C3⋊D3—22no—
337C2×C3⋊D3—32no—
338C2×C3⋊D3—32no—
339C2×C3⋊D3—38no—
340C2×C3⋊D3—38no—
341C2×C3⋊D3—68no—
342C2×C3⋊D3—68no—
343C2×C3⋊D3—616no—
344C2×C3⋊D3—68no—
345C2×C3⋊D3—62no—
346C2×C3⋊D3—62no—
347C2×C3⋊D3—68no—
348C2×C3⋊D3—128no—
349C2×C3⋊D3—124no—
350C2×C3⋊D3—124no—
351C2×C3⋊D3—1216no—
352C2×C3⋊D3—1232no—
353C2×C3×D3—28no—
354C2×C3×D3—24no—
355C2×C3×D3—28no—
356C2×C3×D3—28no—
357C2×C3×D3—28no—
358C2×C3×D3—416no—
359C2×C3×D3—64no—
360C2×C3×D3—68no—
361D32—1216no—
362C32×C4—22no—
363C32×C4—28no—
364C32×C4—48no—
365C32×C4—61no—
366C3⋊Dic3—21no—
367C3⋊Dic3—22no—
368C3⋊Dic3—62no—
369C3⋊Dic3—68no—
370C3⋊Dic3—128no—
371C3×Dic3—24no—
372C3×Dic3—64no—
373C3×Dic3—68no—
374C22×D4—271no—
375C22×D4—271no—
376#123—91no—
377#123—272no—
378#123—271no—
379#121—272no—
380#121—272no—
381#120—31no—
382#120—272no—
383#120—271no—
384C2×C22⋊C4—271no—
385C33#53411yescharacteristic, complemented, Sylow
386C23×C3—24no—
387C23×C3—34no—
388C23×C3—38no—
389C23×C3—68no—
390C23×C3—91no—
391C23×C3—128no—
392C23×C3C2×D3214yes—
393C23×C3C22×C3⋊D311yescharacteristic
394C22×D3—22no—
395C22×D3—34no—
396C22×D3—42no—
397C22×D3—94no—
398C22×D3—94no—
399C22×D3—91no—
400C22×D3—91no—
401C22×D3—916no—
402C22×D3—94no—
403C22×D3—184no—
404C22×D3—184no—
405C22×D3—1816no—
406C22×D3—182no—
407C22×D3—188no—
408C22×D3—362no—
409C22×D3—368no—
410C22×D3—368no—
411C22×D3—362no—
412C22×D3—3616no—
413C22×D3C22×C3⋊D311yescharacteristic
414C22×D3C22×C3⋊D312yes—
415C3×D4—68no—
416C3×D4—128no—
417C3×D4—182no—
418C2×C3×C4—28no—
419C2×C3×C4—34no—
420C2×C3×C4—92no—
421C2×C3×C4—91no—
422C2×C3×C4—182no—
423C2×C3×C4C2×D3218yes—
424C2×C3×C4C22×C3⋊D312yes—
425#67—22no—
426#67—68no—
427#67—182no—
428#67—188no—
429#67—1816no—
430#67—1816no—
431#67—362no—
432#67—368no—
433C2×Dic3—38no—
434C2×Dic3—34no—
435C2×Dic3—68no—
436C2×Dic3—98no—
437C2×Dic3—92no—
438C2×Dic3—188no—
439C2×Dic3C22×C3⋊D311yescharacteristic
440D12—42no—
441D12—188no—
442D12—184no—
443D12—184no—
444D12—182no—
445C4×D3—182no—
446C4×D3—188no—
447C4×D3—368no—
448C4×D3—362no—
449C4×D3—3616no—
450C2×C32—22no—
451C2×C32—24no—
452C2×C32—28no—
453C2×C32—28no—
454C2×C32—48no—
455C2×C32—121no—
456C2×C32—122no—
457C2×C32C2×D3×D414yes—
458C2×C32C2×D3×D411yescharacteristic
459C2×C32#100418yes—
460C2×C32#100912yes—
461C3⋊D3—41no—
462C3⋊D3—616no—
463C3⋊D3—62no—
464C3⋊D3—62no—
465C3⋊D3—68no—
466C3⋊D3—68no—
467C3⋊D3—128no—
468C3×D3—44no—
469C3×D3—48no—
470C3×D3—124no—
471C24—31no—
472C24—271no—
473C24—271no—
474C2×D4—31no—
475C2×D4—32no—
476C2×D4—91no—
477C2×D4—271no—
478C2×D4—272no—
479C2×D4—272no—
480C2×D4—271no—
481C2×D4—272no—
482C2×D4—544no—
483C2×D4—544no—
484C22×C4—91no—
485C22×C4—271no—
486C22×C4—272no—
487C4⋊C4—92no—
488C4⋊C4—272no—
489C22⋊C4—32no—
490C22⋊C4—92no—
491C22⋊C4—272no—
492C22⋊C4—272no—
493C22⋊C4—271no—
494C22⋊C4—272no—
495C22⋊C4D3×C3⋊D311yescharacteristic, complemented
496C22×C3—28no—
497C22×C3—22no—
498C22×C3—24no—
499C22×C3—28no—
500C22×C3—48no—
501C22×C3—68no—
502C22×C3—64no—
503C22×C3—68no—
504C22×C3—68no—
505C22×C3—68no—
506C22×C3—1216no—
507C22×C3—181no—
508C22×C3—182no—
509C22×C3#380218yes—
510C22×C3D4×C3⋊D312yes—
511C22×C3C22×D3214yes—
512C22×C3C23×C3⋊D311yescharacteristic
513C2×D3—22no—
514C2×D3—21no—
515C2×D3—22no—
516C2×D3—22no—
517C2×D3—22no—
518C2×D3—44no—
519C2×D3—64no—
520C2×D3—68no—
521C2×D3—98no—
522C2×D3—98no—
523C2×D3—92no—
524C2×D3—92no—
525C2×D3—188no—
526C2×D3—188no—
527C2×D3—1816no—
528C2×D3—188no—
529C2×D3—188no—
530C2×D3—182no—
531C2×D3—182no—
532C2×D3—368no—
533C2×D3—364no—
534C2×D3—3616no—
535C2×D3—364no—
536C2×D3—3616no—
537C2×D3—3632no—
538C3×C4—28no—
539C3×C4—22no—
540C3×C4—48no—
541C3×C4—64no—
542C3×C4—181no—
543C3×C4—182no—
544Dic3—21no—
545Dic3—64no—
546Dic3—68no—
547Dic3—188no—
548Dic3—182no—
549Dic3—368no—
550C32—24no—
551C32#558914yescomplemented
552C32#565311yescharacteristic, complemented
553C23—32no—
554C23—31no—
555C23—62no—
556C23—91no—
557C23—122no—
558C23—271no—
559C23—271no—
560C23—274no—
561C23—542no—
562C23—1082no—
563C23—1082no—
564C23C2×D3×C3⋊D311yescharacteristic
565D4—62no—
566D4—122no—
567D4—182no—
568D4—542no—
569D4—544no—
570D4—544no—
571D4—542no—
572D4—1082no—
573C2×C4—31no—
574C2×C4—92no—
575C2×C4—91no—
576C2×C4—182no—
577C2×C4—272no—
578C2×C4—542no—
579C2×C4—1082no—
580C2×C4C2×D3×C3⋊D312yes—
581C2×C3—22no—
582C2×C3—28no—
583C2×C3—24no—
584C2×C3—28no—
585C2×C3—48no—
586C2×C3—124no—
587C2×C3—128no—
588C2×C3—361no—
589C2×C3C2×#380214yes—
590C2×C3#6431418yes—
591C2×C3C2×D4×C3⋊D311yescharacteristic
592C2×C3#6436112yes—
593D3—42no—
594D3—41no—
595D3—124no—
596D3—1816no—
597D3—182no—
598D3—182no—
599D3—188no—
600D3—188no—
601D3—368no—
602C22—22no—
603C22—61no—
604C22—62no—
605C22—62no—
606C22—62no—
607C22—62no—
608C22—124no—
609C22—181no—
610C22—182no—
611C22—272no—
612C22—272no—
613C22—542no—
614C22—542no—
615C22—1084no—
616C22—1084no—
617C22#8902612yes—
618C22C22×D3×C3⋊D311yescharacteristic
619C4—22no—
620C4—61no—
621C4—182no—
622C4—181no—
623C4—542no—
624C3—24no—
625C3#1059429614yescomplemented
626C3#1059606011yescharacteristic, complemented
627C2—22no—
628C2—122no—
629C2—121no—
630C2—361no—
631C2—542no—
632C2—542no—
633C2C2×#8902611yescharacteristic
634C2#1172458812yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#534
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#89039C2semidirect
2. C22×#6291C2semidirect
3. C22⋊C4×C3⋊(C3⋊D3)C2semidirect
4. C3×#64095C2semidirect
5. C32×#882C2semidirect
6. #11723629C2semidirect
7. #11723602C2semidirect
8. #11723540C2semidirect
9. #11723531C2semidirect
10. #11723527C2semidirect
11. #11723422C2semidirect
15. C23×C32×D3C22semidirect
16. C22×D3×C3⋊D3C22semidirect
17. C2×C4×C3⋊(C3⋊D3)C22semidirect
19. C22×C3×C3⋊Dic3C22semidirect
21. C2×C32×D12C22semidirect
23. C2×#6292C22semidirect
26. C2×#6291C22semidirect
27. C2×#6291C22semidirect
28. C2×#6290C22semidirect
29. C2×#6289C22semidirect
30. C33×C22⋊C4C22semidirect
31. #88864C22semidirect
32. #88859C22semidirect
33. C3×#3752C22semidirect
34. C3×C4⋊(C3⋊Dic3)C22semidirect
35. C32×(C2×D3)⋊C4C22semidirect
36. #88806C22semidirect
37. (C2×D3)⋊(C3⋊Dic3)C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
48. #64095D3semidirect
55. C3×#882D3semidirect
64. C22×C3⋊(C3⋊D3)D4semidirect
97. C23×C3×D3C2×D3semidirect
107. C22×C3⋊Dic3C2×D3semidirect
117. C2×C3×D12C2×D3semidirect
128. C32×C22⋊C4C2×D3semidirect
129. C32×C22⋊C4C2×D3semidirect
138. #3752C2×D3semidirect
139. C4⋊(C3⋊Dic3)C2×D3semidirect
142. C3×(C2×D3)⋊C4C2×D3semidirect
188. #882C3⋊D3semidirect
271. C23×D3C2×C3⋊D3semidirect
293. C2×D12C2×C3⋊D3semidirect
303. C3×C22⋊C4D32semidirect
304. C3×C22⋊C4C2×C3⋊D3semidirect
316. (C2×D3)⋊C4C2×C3⋊D3semidirect
385. C33#534semidirect
495. C22⋊C4D3×C3⋊D3semidirect
551. C32#5589semidirect
552. C32#5653semidirect
625. C3#10594296semidirect
626. C3#10596060semidirect

Maximal subgroups

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

subgroupindex
C2×#890392
C22×#62912
C22⋊C4×C3⋊(C3⋊D3)2
C3×#640952
C32×#8822
#117236292
#117236022
#117235402
#117235312
#117235272
#117234222
#105960603
#105942963