Finite Group Databasebeta

← #13178764

Subgroups — #13178764

Order 1728.

1264 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64156C212yesmaximal, complemented
2C3×#64155C211yesmaximal, characteristic, complemented
3#11723701C212yesmaximal, complemented
4#11723700C211yesmaximal, characteristic, complemented
5#11723695C211yesmaximal, characteristic, complemented
6#11723690C212yesmaximal, complemented
7#11723687C212yesmaximal, complemented
8#11723685C212yesmaximal, complemented
9#11723676C212yesmaximal, complemented
10#10595438—31nomaximal
11#10595435—32nomaximal
12C22×#1113C2211yescharacteristic
13C2×C33×Q8C2211yescharacteristic, complemented
14C2×C3×C4⋊(C3⋊D3)C2212yes—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×#6295C2212yes—
17C2×#6295C2212yes—
18C2×#6295C2211yescharacteristic, complemented
19C2×#6294C2211yescharacteristic
20C2×#6294C2212yescomplemented
21C3×#3752C2212yes—
22C3×#3752C2212yes—
23C3×#3752C2212yes—
24C3×#3752C2212yescomplemented
25C3×#3752C2212yescomplemented
26C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
27C3×#3750C2212yescomplemented
28C3×C4×C3⋊Dic3C2212yescomplemented
29#88815C2211yescharacteristic, complemented
30#88815C2212yescomplemented
31#88814C2212yes—
32#88814C2212yes—
33#64156D312yescomplemented
34#64155D311yescharacteristic, complemented
35C3×#947—32no—
36C3×#947—32no—
37C3×#947—62no—
38C3×#946—32no—
39C3×#946—61no—
40#63923—32no—
41#63922—32no—
42#63922—32no—
43#63921—31no—
44D3×(C2×D3)⋊C4—32no—
45#63912—31no—
46#63910—32no—
47#63906—32no—
48#63902—32no—
49#63891—32no—
50#63890—32no—
51#63886—32no—
52#63885—32no—
53#63880—32no—
54#63880—32no—
55#63868—32no—
56#63862—32no—
57#63859—32no—
58#63854—32no—
59C22×C3×C3⋊D3C2312yes—
60C22×C3×C3⋊D3C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C2×#1113—22no—
63C2×#1113—41no—
64C2×#1113—41no—
65C33×Q8—41no—
66C2×C33×C4C2312yes—
67C2×C33×C4C2311yescharacteristic
68C2×C3×C3⋊Dic3C2312yes—
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2312yes—
72C3×C4⋊(C3⋊D3)—42no—
73C3×C4×C3⋊D3—41no—
74#6295—24no—
75#6295—42no—
76#6295—41no—
77#6294—41no—
78#6294—42no—
79#5678—92no—
80#5644—91no—
81C22×D32—32no—
82C22×D32—31no—
83C2×C32×Q8—21no—
84C2×C32×Q8—22no—
85C2×C32×Q8—41no—
86C2×C32×Q8C2×D312yescomplemented
87C2×C32×Q8C2×D311yescharacteristic, complemented
88C2×C4⋊(C3⋊D3)C2×D312yes—
89C2×C4×C3⋊D3C2×D311yescharacteristic
90C2×C3×D12—32no—
91C2×C3×D12—32no—
92C2×C3×D12—62no—
93C2×C3×C4×D3—32no—
94C2×C3×C4×D3—61no—
95C2×#626—32no—
96C2×#626—32no—
97C2×#626—32no—
98C2×#626—32no—
99C2×#626—32no—
100C2×#625—32no—
101C2×#625—31no—
102C2×#625—32no—
103C2×#624—32no—
104C2×#624—31no—
105C2×D3×Dic3—32no—
106C2×D3×Dic3—32no—
107C2×D3×Dic3—32no—
108#3752C2×D312yes—
109#3752C2×D312yes—
110#3752C2×D312yes—
111#3752C2×D312yescomplemented
112#3752C2×D312yescomplemented
113C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
114#3750C2×D312yescomplemented
115C4×C3⋊Dic3C2×D312yescomplemented
116C3×(C2×D3)⋊C4—32no—
117C3×(C2×D3)⋊C4—32no—
118C3×(C2×D3)⋊C4—32no—
119C3×(C2×D3)⋊C4—32no—
120C3×(C2×D3)⋊C4—32no—
121C3×(C2×D3)⋊C4—32no—
122C3×(C2×D3)⋊C4—32no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—32no—
125C3×(C2×D3)⋊C4—32no—
126C3×(C2×D3)⋊C4—62no—
127C3×(C2×D3)⋊C4—62no—
128C3×(C2×D3)⋊C4—62no—
129C3×(C2×D3)⋊C4—62no—
130C3×(C2×D3)⋊C4—62no—
131C3×C4⋊Dic3—32no—
132C3×C4⋊Dic3—61no—
133C3×Dic3⋊C4—32no—
134C3×Dic3⋊C4—32no—
135C3×Dic3⋊C4—62no—
136C3×C4×Dic3—32no—
137C3×C4×Dic3—32no—
138C3×C4×Dic3—62no—
139#3724—32no—
140#3724—31no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—32no—
144#3722—32no—
145#3722—32no—
146(C2×D3)⋊Dic3—32no—
147(C2×D3)⋊Dic3—32no—
148(C2×D3)⋊Dic3—32no—
149(C2×D3)⋊Dic3—32no—
150C22×C33C2411yescharacteristic
151C2×C3×C3⋊D3—21no—
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3—22no—
154C2×C3×C3⋊D3—22no—
155C2×C3×C3⋊D3—22no—
156C2×C3×C3⋊D3—22no—
157C2×C3×C3⋊D3—22no—
158C2×C3×C3⋊D3—22no—
159C2×C3×C3⋊D3#4112yes—
160#1113—44no—
161C33×C4—21no—
162C33×C4—22no—
163C3×C3⋊Dic3—22no—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3—21no—
167C3×#124—92no—
168C3×C22⋊Q8—91no—
169#947—32no—
170#947—32no—
171#947—62no—
172#946—32no—
173#946—61no—
174#939—92no—
175#937—92no—
176#930—91no—
177C2×(C2×D3)⋊C4—91no—
178#898—92no—
179#895—92no—
180#894—92no—
181#886—92no—
182#884—92no—
183#884—92no—
184#883—92no—
185#882—92no—
186#882—92no—
187#67⋊C4—92no—
188#67⋊C4—92no—
189D3×C22⋊C4—92no—
190#879—92no—
191#876—92no—
192C4×D12—92no—
193C22×C3⋊D3C22×D312yes—
194C22×C3⋊D3C22×D312yes—
195C22×C3⋊D3C22×D311yescharacteristic
196C22×C3×D3—32no—
197C22×C3×D3—32no—
198C22×C3×D3—32no—
199C22×C3×D3—32no—
200C22×C3×D3—32no—
201C22×C3×D3—61no—
202C22×C3×D3—62no—
203C22×C3×D3—62no—
204C2×D32—64no—
205C2×D32—62no—
206C2×D32—122no—
207C2×D32—122no—
208C2×D32—121no—
209C2×D32—121no—
210C32×Q8—41no—
211C32×Q8—42no—
212C32×Q8—42no—
213C32×Q8—44no—
214C32×Q8—44no—
215C2×C32×C4—22no—
216C2×C32×C4—21no—
217C2×C32×C4—22no—
218C2×C32×C4—22no—
219C2×C32×C4—22no—
220C2×C32×C4—42no—
221C2×C32×C4—41no—
222C2×C32×C4C22×D312yes—
223C2×C32×C4C22×D312yes—
224C2×C32×C4C22×D312yes—
225C2×C32×C4C22×D312yes—
226C2×C32×C4C22×D311yescharacteristic
227C2×C3⋊Dic3C22×D311yescharacteristic
228C2×C3⋊Dic3C22×D312yes—
229C2×C3⋊Dic3C22×D312yes—
230C2×C3⋊Dic3C22×D312yes—
231C4⋊(C3⋊D3)—42no—
232C4×C3⋊D3—41no—
233C2×C3×Dic3—32no—
234C2×C3×Dic3—32no—
235C2×C3×Dic3—32no—
236C2×C3×Dic3—32no—
237C2×C3×Dic3—32no—
238C2×C3×Dic3—32no—
239C2×C3×Dic3—32no—
240C2×C3×Dic3—62no—
241C2×C3×Dic3—62no—
242C2×C3×Dic3—61no—
243C2×C3×Dic3—62no—
244C3×D12—122no—
245C3×D12—122no—
246C3×D12—124no—
247C3×C4×D3—122no—
248C3×C4×D3—122no—
249#626—64no—
250#626—64no—
251#626—122no—
252#626—122no—
253#626—122no—
254#625—64no—
255#625—121no—
256#625—122no—
257#624—121no—
258#624—122no—
259D3×Dic3—122no—
260D3×Dic3—122no—
261D3×Dic3—122no—
262#538—271noSylow
263C2×C33C2×#4111yescharacteristic
264C2×C33H(5,𝔽2)11yescharacteristic
265C2×C33H(5,𝔽2)11yescharacteristic
266C3×C3⋊D3—22no—
267C3×C3⋊D3—42no—
268C3×C3⋊D3—42no—
269C23×D3—92no—
270C23×D3—91no—
271C2×C3×Q8—21no—
272C2×C3×Q8—22no—
273C2×C3×Q8—41no—
274C2×C3×Q8D3212yescomplemented
275C2×C3×Q8D3211yescharacteristic, complemented
276C2×C3×D4—92no—
277C22×C3×C4—91no—
278C2×#67—92no—
279C2×#67—92no—
280C2×#67—92no—
281C2×#67—92no—
282C2×#67—92no—
283C22×Dic3—91no—
284C22×Dic3—92no—
285C2×D12—32no—
286C2×D12—32no—
287C2×D12—62no—
288C2×D12—92no—
289C2×D12—92no—
290C2×D12—91no—
291C2×C4×D3—32no—
292C2×C4×D3—61no—
293C2×C4×D3—92no—
294C2×C4×D3—92no—
295C2×C4×D3—92no—
296C3×C4⋊C4—91no—
297C3×C4⋊C4—92no—
298C3×C22⋊C4—92no—
299C3×C22⋊C4—92no—
300C3×C22⋊C4—92no—
301C3×C22⋊C4—92no—
302C3×C22⋊C4—92no—
303C3×C42—92no—
304#217—92no—
305#217—92no—
306(C2×D3)⋊C4—32no—
307(C2×D3)⋊C4—32no—
308(C2×D3)⋊C4—32no—
309(C2×D3)⋊C4—32no—
310(C2×D3)⋊C4—32no—
311(C2×D3)⋊C4—32no—
312(C2×D3)⋊C4—32no—
313(C2×D3)⋊C4—32no—
314(C2×D3)⋊C4—32no—
315(C2×D3)⋊C4—32no—
316(C2×D3)⋊C4—62no—
317(C2×D3)⋊C4—62no—
318(C2×D3)⋊C4—62no—
319(C2×D3)⋊C4—62no—
320(C2×D3)⋊C4—62no—
321(C2×D3)⋊C4—92no—
322(C2×D3)⋊C4—92no—
323(C2×D3)⋊C4—92no—
324(C2×D3)⋊C4—92no—
325C4⋊Dic3—32no—
326C4⋊Dic3—61no—
327C4⋊Dic3—92no—
328C4⋊Dic3—91no—
329Dic3⋊C4—32no—
330Dic3⋊C4—32no—
331Dic3⋊C4—62no—
332Dic3⋊C4—92no—
333Dic3⋊C4—92no—
334Dic3⋊C4—92no—
335C4×Dic3—32no—
336C4×Dic3—32no—
337C4×Dic3—62no—
338C22×C32—21no—
339C22×C32—22no—
340C22×C32—41no—
341C22×C32C23×D311yescharacteristic
342C22×C32C23×D312yes—
343C2×C3⋊D3—21no—
344C2×C3⋊D3—21no—
345C2×C3⋊D3—22no—
346C2×C3⋊D3—22no—
347C2×C3⋊D3—22no—
348C2×C3⋊D3—22no—
349C2×C3⋊D3—22no—
350C2×C3⋊D3—22no—
351C2×C3⋊D3#23912yes—
352C2×C3×D3—34no—
353C2×C3×D3—62no—
354C2×C3×D3—62no—
355C2×C3×D3—62no—
356C2×C3×D3—62no—
357C2×C3×D3—62no—
358C2×C3×D3—62no—
359C2×C3×D3—62no—
360C2×C3×D3—62no—
361C2×C3×D3—62no—
362C2×C3×D3—62no—
363C2×C3×D3—62no—
364C2×C3×D3—62no—
365C2×C3×D3—62no—
366C2×C3×D3—62no—
367C2×C3×D3—62no—
368C2×C3×D3—121no—
369C2×C3×D3—121no—
370C2×C3×D3—122no—
371C2×C3×D3—122no—
372C2×C3×D3—122no—
373C2×C3×D3—122no—
374C2×C3×D3—122no—
375C2×C3×D3—122no—
376D32—128no—
377D32—124no—
378C32×C4—21no—
379C32×C4—22no—
380C32×C4—22no—
381C32×C4—24no—
382C32×C4—22no—
383C32×C4—22no—
384C32×C4—22no—
385C32×C4—42no—
386C32×C4—42no—
387C32×C4—42no—
388C32×C4—42no—
389C32×C4—44no—
390C3⋊Dic3—21no—
391C3⋊Dic3—22no—
392C3⋊Dic3—22no—
393C3⋊Dic3—22no—
394C3×Dic3—62no—
395C3×Dic3—62no—
396C3×Dic3—62no—
397C3×Dic3—62no—
398C3×Dic3—62no—
399C3×Dic3—62no—
400C3×Dic3—62no—
401C3×Dic3—121no—
402C3×Dic3—122no—
403C3×Dic3—122no—
404C3×Dic3—122no—
405#126—272no—
406#124—92no—
407#123—272no—
408C22⋊Q8—91no—
409#121—271no—
410#121—272no—
411#120—272no—
412C4×D4—272no—
413C2×C22⋊C4—271no—
414C33#53811yescharacteristic, complemented, Sylow
415C23×C3—91no—
416C23×C3—92no—
417C23×C3—92no—
418C22×D3—32no—
419C22×D3—32no—
420C22×D3—32no—
421C22×D3—32no—
422C22×D3—32no—
423C22×D3—61no—
424C22×D3—62no—
425C22×D3—62no—
426C22×D3—184no—
427C22×D3—182no—
428C22×D3—362no—
429C22×D3—362no—
430C22×D3—361no—
431C22×D3—361no—
432C3×Q8—42no—
433C3×Q8—41no—
434C3×Q8—42no—
435C3×Q8—44no—
436C3×Q8—44no—
437C3×D4—362no—
438C2×C3×C4—22no—
439C2×C3×C4—21no—
440C2×C3×C4—22no—
441C2×C3×C4—22no—
442C2×C3×C4—22no—
443C2×C3×C4—42no—
444C2×C3×C4—41no—
445C2×C3×C4—92no—
446C2×C3×C4—92no—
447C2×C3×C4—92no—
448C2×C3×C4—91no—
449C2×C3×C4—361no—
450C2×C3×C4C2×D3212yes—
451C2×C3×C4C2×D3212yes—
452C2×C3×C4C2×D3212yes—
453C2×C3×C4C2×D3212yes—
454C2×C3×C4C2×D3211yescharacteristic
455#67—184no—
456#67—184no—
457#67—362no—
458#67—362no—
459#67—362no—
460C2×Dic3—32no—
461C2×Dic3—32no—
462C2×Dic3—32no—
463C2×Dic3—32no—
464C2×Dic3—32no—
465C2×Dic3—32no—
466C2×Dic3—32no—
467C2×Dic3—62no—
468C2×Dic3—62no—
469C2×Dic3—61no—
470C2×Dic3—62no—
471C2×Dic3—361no—
472C2×Dic3—362no—
473D12—122no—
474D12—122no—
475D12—124no—
476D12—184no—
477D12—362no—
478D12—361no—
479C4×D3—122no—
480C4×D3—122no—
481C4×D3—362no—
482C4×D3—362no—
483C4×D3—362no—
484C2×C32—22no—
485C2×C32—21no—
486C2×C32—21no—
487C2×C32—22no—
488C2×C32—21no—
489C2×C32—22no—
490C2×C32—41no—
491C2×C32—41no—
492C2×C32—41no—
493C2×C32#100411yescharacteristic
494C2×C32#100412yes—
495C2×C32C2×#23911yescharacteristic
496C2×C32D3×#4112yes—
497C2×C32#100911yescharacteristic
498C2×C32#100912yes—
499C3⋊D3—22no—
500C3⋊D3—42no—
501C3⋊D3—42no—
502C3×D3—64no—
503C3×D3—122no—
504C3×D3—122no—
505C3×D3—122no—
506C3×D3—122no—
507C3×D3—122no—
508C3×D3—242no—
509C3×D3—242no—
510C24—271no—
511C2×Q8#111311yescharacteristic, complemented
512C2×D4—92no—
513C2×D4—272no—
514C2×D4—272no—
515C2×D4—271no—
516C22×C4—91no—
517C22×C4—271no—
518C22×C4—272no—
519C4⋊C4—92no—
520C4⋊C4—91no—
521C4⋊C4—272no—
522C4⋊C4—271no—
523C22⋊C4—92no—
524C22⋊C4—92no—
525C22⋊C4—92no—
526C22⋊C4—92no—
527C22⋊C4—92no—
528C22⋊C4—272no—
529C22⋊C4—272no—
530C42—92no—
531C22×C3—21no—
532C22×C3—22no—
533C22×C3—41no—
534C22×C3—92no—
535C22×C3—181no—
536C22×C3—181no—
537C22×C3—182no—
538C22×C3—182no—
539C22×C3—182no—
540C22×C3—182no—
541C22×C3—182no—
542C22×C3—182no—
543C22×C3C22×D3212yes—
544C22×C3C22×D3211yescharacteristic
545C2×D3—34no—
546C2×D3—62no—
547C2×D3—62no—
548C2×D3—62no—
549C2×D3—62no—
550C2×D3—62no—
551C2×D3—62no—
552C2×D3—62no—
553C2×D3—62no—
554C2×D3—62no—
555C2×D3—62no—
556C2×D3—62no—
557C2×D3—62no—
558C2×D3—62no—
559C2×D3—62no—
560C2×D3—62no—
561C2×D3—121no—
562C2×D3—121no—
563C2×D3—122no—
564C2×D3—122no—
565C2×D3—122no—
566C2×D3—122no—
567C2×D3—122no—
568C2×D3—122no—
569C2×D3—368no—
570C2×D3—364no—
571C3×C4—22no—
572C3×C4—24no—
573C3×C4—22no—
574C3×C4—22no—
575C3×C4—22no—
576C3×C4—21no—
577C3×C4—22no—
578C3×C4—42no—
579C3×C4—42no—
580C3×C4—42no—
581C3×C4—42no—
582C3×C4—44no—
583C3×C4—182no—
584C3×C4—182no—
585C3×C4—182no—
586C3×C4—181no—
587Dic3—62no—
588Dic3—62no—
589Dic3—62no—
590Dic3—62no—
591Dic3—62no—
592Dic3—62no—
593Dic3—62no—
594Dic3—121no—
595Dic3—122no—
596Dic3—122no—
597Dic3—122no—
598C32—22no—
599C32—21no—
600C32—41no—
601C32#564411yescharacteristic, complemented
602C32#567812yescomplemented
603C23—91no—
604C23—92no—
605C23—92no—
606C23—542no—
607C23—1081no—
608C23—1081no—
609Q8—41no—
610D4—362no—
611D4—544no—
612D4—1082no—
613D4—1081no—
614C2×C4—92no—
615C2×C4—92no—
616C2×C4—92no—
617C2×C4—91no—
618C2×C4—361no—
619C2×C4—1081no—
620C2×C4—1082no—
621C2×C4C2×#111312yes—
622C2×C4C2×#111311yescharacteristic
623C2×C3—22no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—22no—
627C2×C3—21no—
628C2×C3—22no—
629C2×C3—41no—
630C2×C3—41no—
631C2×C3—41no—
632C2×C3—182no—
633C2×C3—362no—
634C2×C3—362no—
635C2×C3D3×#23912yes—
636C2×C3#6431911yescharacteristic
637C2×C3#6432012yes—
638C2×C3#6432011yescharacteristic
639C2×C3#6433012yes—
640C2×C3#6433011yescharacteristic
641D3—64no—
642D3—122no—
643D3—122no—
644D3—122no—
645D3—122no—
646D3—122no—
647D3—242no—
648D3—242no—
649C22—92no—
650C22—181no—
651C22—181no—
652C22—182no—
653C22—182no—
654C22—182no—
655C22—182no—
656C22—182no—
657C22—182no—
658C22—1084no—
659C22C22×#111311yescharacteristic
660C4—21no—
661C4—22no—
662C4—182no—
663C4—182no—
664C4—182no—
665C4—181no—
666C3—21no—
667C3—22no—
668C3—41no—
669C3#1059543512yescomplemented
670C3#1059543811yescharacteristic, complemented
671C2—182no—
672C2—362no—
673C2—362no—
674C2#1172462611yescharacteristic
675C2#1172462711yescharacteristic
676C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
hypercenterC2×Q8#1113
hypocenterC33#538
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64156C2semidirect
2. C3×#64155C2semidirect
3. #11723701C2semidirect
4. #11723700C2semidirect
5. #11723695C2semidirect
6. #11723690C2semidirect
7. #11723687C2semidirect
8. #11723685C2semidirect
9. #11723676C2semidirect
13. C2×C33×Q8C22semidirect
18. C2×#6295C22semidirect
20. C2×#6294C22semidirect
24. C3×#3752C22semidirect
25. C3×#3752C22semidirect
26. C3×C4⋊(C3⋊Dic3)C22semidirect
27. C3×#3750C22semidirect
28. C3×C4×C3⋊Dic3C22semidirect
29. #88815C22semidirect
30. #88815C22semidirect
33. #64156D3semidirect
34. #64155D3semidirect
86. C2×C32×Q8C2×D3semidirect
87. C2×C32×Q8C2×D3semidirect
111. #3752C2×D3semidirect
112. #3752C2×D3semidirect
113. C4⋊(C3⋊Dic3)C2×D3semidirect
114. #3750C2×D3semidirect
115. C4×C3⋊Dic3C2×D3semidirect
274. C2×C3×Q8D32semidirect
275. C2×C3×Q8D32semidirect
414. C33#538semidirect
511. C2×Q8#1113semidirect
601. C32#5644semidirect
602. C32#5678semidirect
669. C3#10595435semidirect
670. C3#10595438semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C3×#641552
#117237012
#117237002
#117236952
#117236902
#117236872
#117236852
#117236762
#105954383
#105954353