Finite Group Databasebeta

← #13161983

Subgroups — #13161983

Order 1728.

940 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C211yesmaximal, characteristic, complemented
2C3×#64147C211yesmaximal, characteristic, complemented
3C2×C3×#3754C211yesmaximal, characteristic, complemented
4C2×C3×#3753C211yesmaximal, characteristic, complemented
5#11722076C211yesmaximal, characteristic, complemented
6#11722075C211yesmaximal, characteristic, complemented
7#11722072C211yesmaximal, characteristic, complemented
8#10591521—31nomaximal
9#10591520—31nomaximal
10#10591519—31nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634C2211yescharacteristic
16C2×#6295—21no—
17C2×#6294—21no—
18#89041—22no—
19#89041—22no—
20C3×#3757—21no—
21C3×#3754—22no—
22C3×#3753—22no—
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
25C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
26C2×#3799—31no—
27C2×#3796—31no—
28C2×#3795—31no—
29#64147D311yescharacteristic, complemented
30C2×#3754D311yescharacteristic, complemented
31C2×#3753D311yescharacteristic, complemented
32C3×#938—31no—
33C3×#938—31no—
34C3×#938—61no—
35C2×C3×#214—31no—
36C2×C3×#214—31no—
37C2×C3×#214—61no—
38C2×C3×#213—31no—
39C2×C3×#213—31no—
40C2×C3×#213—61no—
41#63567—31no—
42#63566—31no—
43#63565—31no—
44#63564—31no—
45#63563—31no—
46#63562—31no—
47#63557—31no—
48(C2×D3)⋊(C3⋊C8)—31no—
49(C2×D3)⋊(C3⋊C8)—31no—
50C23×C33—21no—
51C22×C3×C3⋊D3—21no—
52C22×C3×C3⋊D3D411yescharacteristic
53C33×D4—21no—
54C33×D4—21no—
55C2×C33×C4C2311yescharacteristic
56C3×#638—42no—
57C2×C3×C3⋊Dic3—21no—
58C2×C3×C3⋊Dic3—21no—
59C2×C3×C3⋊Dic3D411yescharacteristic
60C3×C4⋊(C3⋊D3)—21no—
61C3×C4⋊(C3⋊D3)D412yescomplemented
62C3×C4×C3⋊D3—21no—
63C3×C4×C3⋊D3—21no—
64C3×#634—21no—
65C3×#634D412yescomplemented
66#6295—24no—
67#6294—24no—
68C3×C3⋊(C3⋊C8)—21no—
69C3×C3⋊(C3⋊C8)—21no—
70#5172—91no—
71#5158—91no—
72#5037—91no—
73C2×C32×D4—21no—
74C2×C32×D4—21no—
75C2×C32×D4—21no—
76C2×C32×D4—41no—
77C2×C32×D4C2×D311yescharacteristic
78C2×C32×D4C2×D311yescharacteristic
79C2×C32×D4C2×D311yescharacteristic
80C2×#638—21no—
81C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
82C2×C4×C3⋊D3C2×D311yescharacteristic
83C2×#634C2×D311yescharacteristic
84C2×C3×#67—61no—
85C2×C3×#67—61no—
86C2×C3×#67—62no—
87C2×C3×D12—31no—
88C2×C3×D12—31no—
89C2×C3×D12—61no—
90C2×C3×C4×D3—31no—
91C2×C3×C4×D3—31no—
92C2×C3×C4×D3—61no—
93C2×C3×Dic6—31no—
94C2×C3×Dic6—31no—
95C2×C3×Dic6—61no—
96C2×#626—61no—
97C2×#626—61no—
98C2×#625—61no—
99C2×#624—61no—
100C2×D3×Dic3—61no—
101C2×D3×Dic3—61no—
102#3799—62no—
103#3799—62no—
104#3796—62no—
105#3796—62no—
106#3795—62no—
107#3795—62no—
108#3757—21no—
109#3754—22no—
110#3753—22no—
111C4⋊(C3⋊Dic3)C2×D311yescharacteristic
112C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
113C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
114C3×#217—61no—
115C3×#217—61no—
116C3×#217—62no—
117C3×#214—62no—
118C3×#214—62no—
119C3×#214—64no—
120C3×#213—62no—
121C3×#213—62no—
122C3×#213—64no—
123C3×C4⋊Dic3—31no—
124C3×C4⋊Dic3—31no—
125C3×C4⋊Dic3—61no—
126C2×C3×C3⋊C8—31no—
127C2×C3×C3⋊C8—31no—
128C2×C3×C3⋊C8—31no—
129C2×C3×C3⋊C8—31no—
130C2×C3×C3⋊C8—61no—
131C2×C3×C3⋊C8—61no—
132C22×C33—41no—
133C22×C33—41no—
134C22×C33—41no—
135C22×C33C2×D411yescharacteristic
136C2×C3×C3⋊D3—22no—
137C2×C3×C3⋊D3—21no—
138C2×C3×C3⋊D3—21no—
139C2×C3×C3⋊D3—21no—
140C2×C3×C3⋊D3—41no—
141C2×C3×C3⋊D3—41no—
142C33×C4C2×D411yescharacteristic
143C33×C4C2×D411yescharacteristic
144C3×C3⋊Dic3—21no—
145C3×C3⋊Dic3—22no—
146C3×C3⋊Dic3—41no—
147C2×#239—91no—
148C2×#237—91no—
149C2×#235—91no—
150C2×C3×SD16—91no—
151C2×C3×D8—91no—
152C3×#121—91no—
153#938—31no—
154#938—31no—
155#938—61no—
156C2×#214—31no—
157C2×#214—31no—
158C2×#214—61no—
159C2×#213—31no—
160C2×#213—31no—
161C2×#213—61no—
162#835—91no—
163#832—91no—
164#830—91no—
165#821—91no—
166(C2×D3)⋊C8—91no—
167(C2×D3)⋊C8—91no—
168#816—91no—
169#810—91no—
170#809—91no—
171C23×C32—21no—
172C23×C32—22no—
173C23×C32—21no—
174C23×C32—21no—
175C23×C32—41no—
176C23×C32—41no—
177C23×C32—42no—
178C22×C3⋊D3—21no—
179C22×C3⋊D3#6711yescharacteristic
180C22×C3×D3—31no—
181C22×C3×D3—31no—
182C22×C3×D3—61no—
183C22×C3×D3—61no—
184C22×C3×D3—61no—
185C22×C3×D3—121no—
186C32×D4—21no—
187C32×D4—21no—
188C32×D4—22no—
189C32×D4—22no—
190C32×D4—21no—
191C32×D4—21no—
192C32×D4—21no—
193C32×D4—21no—
194C32×D4—41no—
195C32×D4—41no—
196C32×D4—41no—
197C32×D4—41no—
198C32×D4—42no—
199C32×D4—42no—
200C2×C32×C4—21no—
201C2×C32×C4—21no—
202C2×C32×C4—21no—
203C2×C32×C4—41no—
204C2×C32×C4C22×D311yescharacteristic
205C2×C32×C4C22×D311yescharacteristic
206C2×C32×C4C22×D311yescharacteristic
207#638—42no—
208C2×C3⋊Dic3—21no—
209C2×C3⋊Dic3—21no—
210C2×C3⋊Dic3#6711yescharacteristic
211C4⋊(C3⋊D3)—21no—
212C4⋊(C3⋊D3)#6712yescomplemented
213C4×C3⋊D3—21no—
214C4×C3⋊D3—21no—
215#634—21no—
216#634#6712yescomplemented
217C3×#67—124no—
218C3×#67—122no—
219C3×#67—122no—
220C2×C3×Dic3—31no—
221C2×C3×Dic3—31no—
222C2×C3×Dic3—61no—
223C2×C3×Dic3—61no—
224C2×C3×Dic3—61no—
225C2×C3×Dic3—62no—
226C2×C3×Dic3—61no—
227C2×C3×Dic3—61no—
228C2×C3×Dic3—121no—
229C3×D12—32no—
230C3×D12—32no—
231C3×D12—62no—
232C3×D12—61no—
233C3×D12—61no—
234C3×D12—121no—
235C3×C4×D3—61no—
236C3×C4×D3—61no—
237C3×C4×D3—61no—
238C3×C4×D3—61no—
239C3×C4×D3—121no—
240C3×C4×D3—121no—
241C3×Dic6—32no—
242C3×Dic6—32no—
243C3×Dic6—62no—
244C3×Dic6—61no—
245C3×Dic6—61no—
246C3×Dic6—121no—
247#626—64no—
248#626—64no—
249#625—64no—
250#624—64no—
251D3×Dic3—64no—
252D3×Dic3—64no—
253C3⋊(C3⋊C8)—21no—
254C3⋊(C3⋊C8)—21no—
255C3×C3⋊C8—62no—
256C3×C3⋊C8—62no—
257C3×C3⋊C8—61no—
258C3×C3⋊C8—61no—
259C3×C3⋊C8—61no—
260C3×C3⋊C8—61no—
261#449—271noSylow
262C2×C33—81no—
263C2×C33#12011yescharacteristic
264C2×C33#13511yescharacteristic
265C2×C33AGL(1,ℤ/8)11yescharacteristic
266C3×C3⋊D3—41no—
267C3×C3⋊D3—42no—
268C2×C3×Q8—91no—
269C2×C3×D4—21no—
270C2×C3×D4—21no—
271C2×C3×D4—21no—
272C2×C3×D4—41no—
273C2×C3×D4—91no—
274C2×C3×D4—181no—
275C2×C3×D4D3211yescharacteristic
276C2×C3×D4D3211yescharacteristic
277C2×C3×D4D3211yescharacteristic
278C22×C3×C4—91no—
279C2×#67—61no—
280C2×#67—61no—
281C2×#67—62no—
282C2×#67—181no—
283C2×#67—181no—
284C22×Dic3—181no—
285#239—182no—
286#239—182no—
287#237—182no—
288#237—182no—
289#235—182no—
290#235—182no—
291C2×D12—31no—
292C2×D12—31no—
293C2×D12—61no—
294C2×D12—181no—
295C2×C4×D3—31no—
296C2×C4×D3—31no—
297C2×C4×D3—61no—
298C2×C4×D3—181no—
299C2×C4×D3—181no—
300C2×Dic6—31no—
301C2×Dic6—31no—
302C2×Dic6—61no—
303C3×SD16—182no—
304C3×D8—182no—
305C2×C3×C8—91no—
306C2×C3×C8—91no—
307C3×C4⋊C4—91no—
308C3×C22⋊C4—181no—
309#217—61no—
310#217—61no—
311#217—62no—
312#214—62no—
313#214—62no—
314#214—64no—
315#213—62no—
316#213—62no—
317#213—64no—
318C4⋊Dic3—31no—
319C4⋊Dic3—31no—
320C4⋊Dic3—61no—
321C2×C3⋊C8—31no—
322C2×C3⋊C8—31no—
323C2×C3⋊C8—31no—
324C2×C3⋊C8—31no—
325C2×C3⋊C8—61no—
326C2×C3⋊C8—61no—
327C22×C32—21no—
328C22×C32—21no—
329C22×C32—21no—
330C22×C32—41no—
331C22×C32—42no—
332C22×C32—42no—
333C22×C32—42no—
334C22×C32—44no—
335C22×C32—41no—
336C22×C32—41no—
337C22×C32—41no—
338C22×C32—41no—
339C22×C32—41no—
340C22×C32—42no—
341C22×C32—42no—
342C22×C32—41no—
343C22×C32—41no—
344C22×C32—41no—
345C22×C32—41no—
346C22×C32—81no—
347C22×C32—81no—
348C22×C32—81no—
349C22×C32—81no—
350C22×C32—82no—
351C22×C32—82no—
352C22×C32D3×D411yescharacteristic
353C22×C32D3×D411yescharacteristic
354C22×C32C2×#6711yescharacteristic
355C2×C3⋊D3—21no—
356C2×C3⋊D3—22no—
357C2×C3⋊D3—21no—
358C2×C3⋊D3—21no—
359C2×C3⋊D3—41no—
360C2×C3⋊D3—41no—
361C2×C3×D3—62no—
362C2×C3×D3—61no—
363C2×C3×D3—61no—
364C2×C3×D3—61no—
365C2×C3×D3—61no—
366C2×C3×D3—62no—
367C2×C3×D3—62no—
368C2×C3×D3—61no—
369C2×C3×D3—61no—
370C2×C3×D3—121no—
371C2×C3×D3—121no—
372C2×C3×D3—122no—
373C2×C3×D3—121no—
374C2×C3×D3—121no—
375C2×C3×D3—121no—
376C2×C3×D3—121no—
377C2×C3×D3—241no—
378C2×C3×D3—241no—
379C32×C4—21no—
380C32×C4—21no—
381C32×C4—21no—
382C32×C4—21no—
383C32×C4—21no—
384C32×C4—21no—
385C32×C4—41no—
386C32×C4—41no—
387C32×C4D3×D411yescharacteristic
388C32×C4D3×D411yescharacteristic
389C32×C4D3×D411yescharacteristic
390C32×C4D3×D411yescharacteristic
391C32×C4C2×#6711yescharacteristic
392C32×C4C2×#6711yescharacteristic
393C3⋊Dic3—21no—
394C3⋊Dic3—22no—
395C3⋊Dic3—41no—
396C3×Dic3—61no—
397C3×Dic3—62no—
398C3×Dic3—61no—
399C3×Dic3—62no—
400C3×Dic3—122no—
401C3×Dic3—121no—
402C3×Dic3—122no—
403C3×Dic3—121no—
404C3×Dic3—121no—
405C2×#41—271no—
406C2×SD16—91no—
407C2×D8—91no—
408#121—91no—
409#103—271no—
410#102—271no—
411C22⋊C8—271no—
412C33#44911yescharacteristic, complemented, Sylow
413C23×C3—22no—
414C23×C3—21no—
415C23×C3—21no—
416C23×C3—21no—
417C23×C3—41no—
418C23×C3—41no—
419C23×C3—42no—
420C23×C3—91no—
421C23×C3—181no—
422C22×D3—31no—
423C22×D3—31no—
424C22×D3—61no—
425C22×D3—61no—
426C22×D3—61no—
427C22×D3—121no—
428C3×Q8—92no—
429C3×Q8—181no—
430C3×D4—22no—
431C3×D4—22no—
432C3×D4—21no—
433C3×D4—21no—
434C3×D4—21no—
435C3×D4—21no—
436C3×D4—21no—
437C3×D4—21no—
438C3×D4—41no—
439C3×D4—41no—
440C3×D4—41no—
441C3×D4—41no—
442C3×D4—42no—
443C3×D4—42no—
444C3×D4—92no—
445C3×D4—181no—
446C3×D4—362no—
447C2×C3×C4—21no—
448C2×C3×C4—21no—
449C2×C3×C4—21no—
450C2×C3×C4—41no—
451C2×C3×C4—91no—
452C2×C3×C4—181no—
453C2×C3×C4—181no—
454C2×C3×C4—181no—
455C2×C3×C4—181no—
456C2×C3×C4C2×D3211yescharacteristic
457C2×C3×C4C2×D3211yescharacteristic
458C2×C3×C4C2×D3211yescharacteristic
459#67—124no—
460#67—122no—
461#67—122no—
462#67—184no—
463#67—184no—
464C2×Dic3—31no—
465C2×Dic3—31no—
466C2×Dic3—61no—
467C2×Dic3—61no—
468C2×Dic3—61no—
469C2×Dic3—62no—
470C2×Dic3—61no—
471C2×Dic3—61no—
472C2×Dic3—121no—
473C2×Dic3—184no—
474D12—32no—
475D12—32no—
476D12—62no—
477D12—61no—
478D12—61no—
479D12—121no—
480D12—184no—
481C4×D3—61no—
482C4×D3—61no—
483C4×D3—61no—
484C4×D3—61no—
485C4×D3—121no—
486C4×D3—121no—
487C4×D3—184no—
488C4×D3—184no—
489Dic6—32no—
490Dic6—32no—
491Dic6—62no—
492Dic6—61no—
493Dic6—61no—
494Dic6—121no—
495C3×C8—181no—
496C3×C8—181no—
497C3⋊C8—62no—
498C3⋊C8—62no—
499C3⋊C8—61no—
500C3⋊C8—61no—
501C3⋊C8—61no—
502C3⋊C8—61no—
503C2×C32—21no—
504C2×C32—21no—
505C2×C32—21no—
506C2×C32—21no—
507C2×C32—21no—
508C2×C32—21no—
509C2×C32—21no—
510C2×C32—21no—
511C2×C32—21no—
512C2×C32—41no—
513C2×C32—41no—
514C2×C32—41no—
515C2×C32—82no—
516C2×C32—82no—
517C2×C32—82no—
518C2×C32—84no—
519C2×C32—81no—
520C2×C32—81no—
521C2×C32—81no—
522C2×C32#91111yescharacteristic
523C2×C32#91211yescharacteristic
524C2×C32#91411yescharacteristic
525C2×C32#91611yescharacteristic
526C2×C32#93211yescharacteristic
527C2×C32#93711yescharacteristic
528C2×C32#93711yescharacteristic
529C2×C32#93711yescharacteristic
530C2×C32#95011yescharacteristic
531C3⋊D3—41no—
532C3⋊D3—42no—
533C3×D3—122no—
534C3×D3—121no—
535C3×D3—121no—
536C3×D3—122no—
537C3×D3—122no—
538C3×D3—242no—
539#41—542no—
540#41—542no—
541C2×Q8—91no—
542C2×D4—91no—
543C2×D4—181no—
544C2×D4—541no—
545C2×D4#111311yescharacteristic
546C22×C4—91no—
547C22×C4—541no—
548SD16—182no—
549D8—182no—
550C2×C8—91no—
551C2×C8—91no—
552C4⋊C4—91no—
553C22⋊C4—181no—
554C22×C3—21no—
555C22×C3—21no—
556C22×C3—21no—
557C22×C3—42no—
558C22×C3—42no—
559C22×C3—42no—
560C22×C3—42no—
561C22×C3—42no—
562C22×C3—44no—
563C22×C3—41no—
564C22×C3—41no—
565C22×C3—41no—
566C22×C3—41no—
567C22×C3—41no—
568C22×C3—41no—
569C22×C3—41no—
570C22×C3—41no—
571C22×C3—41no—
572C22×C3—41no—
573C22×C3—81no—
574C22×C3—81no—
575C22×C3—81no—
576C22×C3—81no—
577C22×C3—82no—
578C22×C3—82no—
579C22×C3—181no—
580C22×C3—181no—
581C22×C3—182no—
582C22×C3—181no—
583C22×C3—361no—
584C22×C3—361no—
585C22×C3D3×#6711yescharacteristic
586C22×C3D3×#6711yescharacteristic
587C22×C3#381111yescharacteristic
588C2×D3—62no—
589C2×D3—61no—
590C2×D3—61no—
591C2×D3—61no—
592C2×D3—61no—
593C2×D3—61no—
594C2×D3—61no—
595C2×D3—62no—
596C2×D3—62no—
597C2×D3—121no—
598C2×D3—121no—
599C2×D3—122no—
600C2×D3—121no—
601C2×D3—121no—
602C2×D3—121no—
603C2×D3—121no—
604C2×D3—241no—
605C2×D3—241no—
606C3×C4—21no—
607C3×C4—21no—
608C3×C4—21no—
609C3×C4—21no—
610C3×C4—21no—
611C3×C4—21no—
612C3×C4—41no—
613C3×C4—41no—
614C3×C4—181no—
615C3×C4—182no—
616C3×C4—361no—
617C3×C4D3×#6711yescharacteristic
618C3×C4D3×#6711yescharacteristic
619C3×C4D3×#6711yescharacteristic
620C3×C4D3×#6711yescharacteristic
621C3×C4#381111yescharacteristic
622C3×C4#381111yescharacteristic
623Dic3—62no—
624Dic3—62no—
625Dic3—61no—
626Dic3—61no—
627Dic3—122no—
628Dic3—121no—
629Dic3—122no—
630Dic3—121no—
631Dic3—121no—
632C32—21no—
633C32—21no—
634C32—21no—
635C32—41no—
636C32#503711yescharacteristic, complemented
637C32#515811yescharacteristic, complemented
638C32#517211yescharacteristic, complemented
639C23—21no—
640C23—91no—
641C23—181no—
642Q8—92no—
643Q8—181no—
644D4—21no—
645D4—21no—
646D4—92no—
647D4—181no—
648D4—362no—
649D4—544no—
650C2×C4—91no—
651C2×C4—181no—
652C2×C4—181no—
653C2×C4—181no—
654C2×C4—181no—
655C2×C4—544no—
656C2×C4C2×#111311yescharacteristic
657C8—181no—
658C8—181no—
659C2×C3—21no—
660C2×C3—21no—
661C2×C3—21no—
662C2×C3—21no—
663C2×C3—21no—
664C2×C3—21no—
665C2×C3—21no—
666C2×C3—21no—
667C2×C3—21no—
668C2×C3—41no—
669C2×C3—41no—
670C2×C3—41no—
671C2×C3—82no—
672C2×C3—82no—
673C2×C3—82no—
674C2×C3—84no—
675C2×C3—81no—
676C2×C3—81no—
677C2×C3—81no—
678C2×C3—361no—
679C2×C3—362no—
680C2×C3#6393211yescharacteristic
681C2×C3#6393311yescharacteristic
682C2×C3#6393411yescharacteristic
683C2×C3#6393511yescharacteristic
684C2×C3#6393611yescharacteristic
685C2×C3#6393711yescharacteristic
686C2×C3#6398311yescharacteristic
687C2×C3#6398311yescharacteristic
688C2×C3#6398311yescharacteristic
689D3—122no—
690D3—121no—
691D3—122no—
692D3—122no—
693D3—121no—
694D3—242no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22—181no—
699C22—181no—
700C22—181no—
701C22—182no—
702C22—361no—
703C22—361no—
704C22#8904911yescharacteristic
705C4—181no—
706C4—182no—
707C4—361no—
708C4#8904911yescharacteristic
709C4#8904911yescharacteristic
710C3—21no—
711C3—21no—
712C3—21no—
713C3—41no—
714C3#1059151911yescharacteristic, complemented
715C3#1059152011yescharacteristic, complemented
716C3#1059152111yescharacteristic, complemented
717C2—81no—
718C2—361no—
719C2—362no—
720C2#1172370611yescharacteristic
721C2#1172370711yescharacteristic
722C2#1172372911yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4#1113
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89041C2semidirect
2. C3×#64147C2semidirect
3. C2×C3×#3754C2semidirect
4. C2×C3×#3753C2semidirect
5. #11722076C2semidirect
6. #11722075C2semidirect
7. #11722072C2semidirect
29. #64147D3semidirect
30. C2×#3754D3semidirect
31. C2×#3753D3semidirect
61. C3×C4⋊(C3⋊D3)D4semidirect
65. C3×#634D4semidirect
212. C4⋊(C3⋊D3)#67semidirect
216. #634#67semidirect
412. C33#449semidirect
636. C32#5037semidirect
637. C32#5158semidirect
638. C32#5172semidirect
714. C3#10591519semidirect
715. C3#10591520semidirect
716. C3#10591521semidirect

Maximal subgroups

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

subgroupindex
C2×#890412
C3×#641472
C2×C3×#37542
C2×C3×#37532
#117220762
#117220752
#117220722
#105915213
#105915203
#105915193