Finite Group Databasebeta

← #13161427

Subgroups — #13161427

Order 1728.

2492 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6312C211yesmaximal, characteristic
2C2×C3×#3757C211yesmaximal, characteristic, complemented
3C2×C32×#217C211yesmaximal, characteristic, complemented
4C2×#88806C211yesmaximal, characteristic, complemented
5C2×#88804C211yesmaximal, characteristic, complemented
6C2×#88804C211yesmaximal, characteristic, complemented
7C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592656—31nomaximal
9#10591704—34nomaximal
10C24×C33C2211yescharacteristic
11C23×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×#6312—22no—
13C2×#6312—22no—
14C2×#6312C414yescomplemented
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C22×C32×Dic3C2211yescharacteristic
20C3×#3757—22no—
21C32×#217—22no—
22#88806—22no—
23#88804—22no—
24#88804—22no—
25Dic3×C3⋊Dic3—22no—
26C22×#638—31no—
27C22×#638—34no—
28C22×#638—64no—
29C2×C32×C22⋊C4—31no—
30C2×#3757D311yescharacteristic, complemented
31C2×#3752—31no—
32C2×#3752—31no—
33C2×#3750—31no—
34C2×C4×C3⋊Dic3—31no—
35C2×C3×#217—34no—
36C2×C3×#217D314yescomplemented
37C2×#3724—34no—
38C2×#3722—34no—
39C2×#3722—34no—
40C2×Dic32—34no—
41C23×C33—21no—
42C23×C33—21no—
43C23×C33—21no—
44C23×C33—21no—
45C23×C33—21no—
46C23×C33—21no—
47C23×C33C2×C412yes—
48C23×C33C2311yescharacteristic
49C22×C3⋊(C3⋊D3)—21no—
50C22×C3⋊(C3⋊D3)—21no—
51C22×C3⋊(C3⋊D3)—21no—
52C22×C3⋊(C3⋊D3)—21no—
53C22×C3⋊(C3⋊D3)—21no—
54C22×C3⋊(C3⋊D3)—21no—
55C22×C3⋊(C3⋊D3)C2×C412yes—
56#6312—28no—
57C2×C3⋊(C3⋊Dic3)C2×C412yes—
58C2×C3⋊(C3⋊Dic3)D412yes—
59C2×C3⋊(C3⋊Dic3)D412yes—
60C2×C3×C3⋊Dic3—21no—
61C2×C3×C3⋊Dic3—21no—
62C2×C3×C3⋊Dic3—21no—
63C2×C3×C3⋊Dic3—21no—
64C2×C3×C3⋊Dic3—21no—
65C2×C3×C3⋊Dic3D412yescomplemented
66C2×C32×Dic3—21no—
67C2×C32×Dic3—21no—
68C2×C32×Dic3—21no—
69C2×C32×Dic3—21no—
70C2×C32×Dic3—21no—
71C2×C32×Dic3D412yescomplemented
72#4957—94no—
73#4957—91no—
74C24×C32—24no—
75C24×C32C2×D314yes—
76C24×C32C2×D311yescharacteristic
77C23×C3⋊D3—31no—
78C23×C3⋊D3—34no—
79C23×C3⋊D3—64no—
80C22×C32×C4—31no—
81C22×C32×C4—31no—
82C2×#638—34no—
83C2×#638—316no—
84C2×#638—62no—
85C2×#638—62no—
86C2×#638—68no—
87C2×#638—68no—
88C2×#638—616no—
89C2×#638—616no—
90C2×#638—616no—
91C22×C3⋊Dic3—31no—
92C22×C3⋊Dic3—34no—
93C22×C3⋊Dic3—64no—
94C22×C3⋊Dic3C2×D311yescharacteristic
95C22×C3⋊Dic3C2×D311yescharacteristic
96C22×C3×Dic3—34no—
97C22×C3×Dic3—34no—
98C22×C3×Dic3C2×D314yes—
99C22×C3×Dic3C2×D314yes—
100C32×C22⋊C4—62no—
101#3757—22no—
102#3752—62no—
103#3752—62no—
104#3750—62no—
105C4×C3⋊Dic3—62no—
106C3×#217—28no—
107C3×#217—68no—
108#3724—68no—
109#3722—68no—
110#3722—68no—
111Dic32—68no—
112C22×C33—22no—
113C22×C33—22no—
114C22×C33—22no—
115C22×C33—41no—
116C22×C33—41no—
117C22×C33—41no—
118C22×C33—41no—
119C22×C33C22×C411yescharacteristic
120C22×C33C2×D411yescharacteristic
121C22×C33C2×D411yescharacteristic
122C22×C33C2×D411yescharacteristic
123C22×C33C2×D411yescharacteristic
124C22×C33#4111yescharacteristic
125C22×C33#4111yescharacteristic
126C2×C3⋊(C3⋊D3)—22no—
127C2×C3⋊(C3⋊D3)—22no—
128C2×C3⋊(C3⋊D3)—22no—
129C2×C3⋊(C3⋊D3)—41no—
130C2×C3⋊(C3⋊D3)—41no—
131C2×C3⋊(C3⋊D3)—41no—
132C2×C3⋊(C3⋊D3)—41no—
133C3⋊(C3⋊Dic3)C22⋊C414yescomplemented
134C3×C3⋊Dic3—22no—
135C3×C3⋊Dic3—41no—
136C32×Dic3—22no—
137C32×Dic3—41no—
138C22×#67—94no—
139C22×#67—91no—
140C22×#67—184no—
141C2×C3×C22⋊C4—34no—
142C2×C3×C22⋊C4—91no—
143C2×#217—34no—
144C2×#217C3⋊D311yescharacteristic, complemented
145C2×(C2×D3)⋊C4—94no—
146C2×(C2×D3)⋊C4—94no—
147C2×(C2×D3)⋊C4—91no—
148C2×(C2×D3)⋊C4—91no—
149C2×Dic3⋊C4—94no—
150C2×Dic3⋊C4—91no—
151C2×C4×Dic3—94no—
152C2×C4×Dic3—91no—
153C23×C32—28no—
154C23×C32—28no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—24no—
158C23×C32—24no—
159C23×C32—21no—
160C23×C32—21no—
161C23×C32—21no—
162C23×C32—21no—
163C23×C32—28no—
164C23×C32—24no—
165C23×C32—24no—
166C23×C32—24no—
167C23×C32—24no—
168C23×C32—24no—
169C23×C32—44no—
170C23×C32—44no—
171C23×C32—44no—
172C23×C32—44no—
173C23×C32C4×D318yes—
174C23×C32C4×D312yes—
175C23×C32C22×D314yes—
176C23×C32C22×D311yescharacteristic
177C22×C3⋊D3—32no—
178C22×C3⋊D3—38no—
179C22×C3⋊D3—68no—
180C22×C3⋊D3—68no—
181C22×C3⋊D3—68no—
182C22×C3⋊D3—61no—
183C22×C3⋊D3—61no—
184C22×C3⋊D3—64no—
185C22×C3⋊D3—64no—
186C22×C3⋊D3—61no—
187C22×C3⋊D3—61no—
188C22×C3⋊D3—61no—
189C22×C3⋊D3—61no—
190C22×C3⋊D3—64no—
191C22×C3⋊D3—64no—
192C22×C3⋊D3—64no—
193C22×C3⋊D3—64no—
194C22×C3⋊D3—124no—
195C22×C3⋊D3—124no—
196C22×C3⋊D3—124no—
197C22×C3⋊D3—124no—
198C2×C32×C4—32no—
199C2×C32×C4—61no—
200C2×C32×C4—61no—
201C2×C32×C4—61no—
202C2×C32×C4—61no—
203C2×C32×C4—61no—
204#638—68no—
205#638—632no—
206#638—664no—
207C2×C3⋊Dic3—21no—
208C2×C3⋊Dic3—21no—
209C2×C3⋊Dic3—21no—
210C2×C3⋊Dic3—21no—
211C2×C3⋊Dic3—21no—
212C2×C3⋊Dic3—32no—
213C2×C3⋊Dic3—32no—
214C2×C3⋊Dic3—32no—
215C2×C3⋊Dic3—38no—
216C2×C3⋊Dic3—38no—
217C2×C3⋊Dic3—38no—
218C2×C3⋊Dic3—68no—
219C2×C3⋊Dic3—68no—
220C2×C3⋊Dic3—68no—
221C2×C3⋊Dic3#6712yescomplemented
222C2×C3×Dic3—24no—
223C2×C3×Dic3—24no—
224C2×C3×Dic3—24no—
225C2×C3×Dic3—24no—
226C2×C3×Dic3—24no—
227C2×C3×Dic3—38no—
228C2×C3×Dic3—64no—
229C2×C3×Dic3—64no—
230C2×C3×Dic3—64no—
231C2×C3×Dic3—64no—
232C2×C3×Dic3—64no—
233C2×C3×Dic3#6718yescomplemented
234#390—271noSylow
235C2×C33—42no—
236C2×C33C2×C22⋊C411yescharacteristic
237C2×C33C4×D411yescharacteristic
238C2×C33C4×D411yescharacteristic
239C2×C33#12111yescharacteristic
240C2×C33#12111yescharacteristic
241C2×C33#12411yescharacteristic
242C2×C33#12711yescharacteristic
243C3⋊(C3⋊D3)—42no—
244C24×C3—24no—
245C24×C3D3214yes—
246C24×C3C2×C3⋊D311yescharacteristic
247C23×D3—94no—
248C23×D3—91no—
249C23×D3—184no—
250C22×C3×C4—34no—
251C22×C3×C4—34no—
252C22×C3×C4—91no—
253C22×C3×C4—91no—
254C2×#67—916no—
255C2×#67—94no—
256C2×#67—1816no—
257C2×#67—188no—
258C2×#67—188no—
259C2×#67—182no—
260C2×#67—182no—
261C2×#67—1816no—
262C2×#67—1816no—
263C22×Dic3—34no—
264C22×Dic3—34no—
265C22×Dic3—94no—
266C22×Dic3—91no—
267C22×Dic3—184no—
268C22×Dic3C2×C3⋊D311yescharacteristic
269C22×Dic3C2×C3⋊D311yescharacteristic
270C3×C22⋊C4—68no—
271C3×C22⋊C4—182no—
272#217—22no—
273#217—68no—
274(C2×D3)⋊C4—182no—
275(C2×D3)⋊C4—188no—
276(C2×D3)⋊C4—188no—
277(C2×D3)⋊C4—182no—
278Dic3⋊C4—182no—
279Dic3⋊C4—188no—
280C4×Dic3—182no—
281C4×Dic3—188no—
282C22×C32—216no—
283C22×C32—28no—
284C22×C32—22no—
285C22×C32—22no—
286C22×C32—22no—
287C22×C32—24no—
288C22×C32—24no—
289C22×C32—24no—
290C22×C32—24no—
291C22×C32—24no—
292C22×C32—24no—
293C22×C32—24no—
294C22×C32—28no—
295C22×C32—28no—
296C22×C32—48no—
297C22×C32—48no—
298C22×C32—48no—
299C22×C32—48no—
300C22×C32—48no—
301C22×C32—48no—
302C22×C32—44no—
303C22×C32—41no—
304C22×C32—44no—
305C22×C32—41no—
306C22×C32—44no—
307C22×C32—41no—
308C22×C32—44no—
309C22×C32—41no—
310C22×C32C2×C4×D314yes—
311C22×C32C2×C4×D311yescharacteristic
312C22×C32#23514yes—
313C22×C32#23511yescharacteristic
314C22×C32D3×D414yes—
315C22×C32D3×D414yes—
316C22×C32D3×D414yes—
317C22×C32D3×D411yescharacteristic
318C22×C32D3×D411yescharacteristic
319C22×C32D3×D411yescharacteristic
320C22×C32#23714yes—
321C22×C32#23711yescharacteristic
322C22×C32C2×#6714yes—
323C22×C32C2×#6711yescharacteristic
324C2×C3⋊D3—616no—
325C2×C3⋊D3—68no—
326C2×C3⋊D3—62no—
327C2×C3⋊D3—62no—
328C2×C3⋊D3—62no—
329C2×C3⋊D3—68no—
330C2×C3⋊D3—68no—
331C2×C3⋊D3—128no—
332C2×C3⋊D3—128no—
333C2×C3⋊D3—128no—
334C2×C3⋊D3—128no—
335C2×C3⋊D3—128no—
336C2×C3⋊D3—128no—
337C2×C3⋊D3—124no—
338C2×C3⋊D3—121no—
339C2×C3⋊D3—124no—
340C2×C3⋊D3—121no—
341C2×C3⋊D3—124no—
342C2×C3⋊D3—121no—
343C2×C3⋊D3—124no—
344C2×C3⋊D3—121no—
345C32×C4—62no—
346C32×C4—121no—
347C3⋊Dic3—22no—
348C3⋊Dic3—34no—
349C3⋊Dic3—316no—
350C3⋊Dic3—41no—
351C3⋊Dic3—616no—
352C3×Dic3—28no—
353C3×Dic3—44no—
354C3×Dic3—68no—
355C3×Dic3—124no—
356C22×D4—271no—
357C2×C4⋊C4—271no—
358C2×C22⋊C4—31no—
359C2×C22⋊C4—91no—
360C2×C22⋊C4—271no—
361C2×C22⋊C4—271no—
362C2×C42—271no—
363C33#39011yescharacteristic, complemented, Sylow
364C23×C3—28no—
365C23×C3—28no—
366C23×C3—24no—
367C23×C3—24no—
368C23×C3—24no—
369C23×C3—24no—
370C23×C3—24no—
371C23×C3—24no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—28no—
379C23×C3—24no—
380C23×C3—44no—
381C23×C3—44no—
382C23×C3—44no—
383C23×C3—44no—
384C23×C3#62418yes—
385C23×C3C4×C3⋊D312yes—
386C23×C3C2×D3214yes—
387C23×C3C22×C3⋊D311yescharacteristic
388C22×D3—98no—
389C22×D3—92no—
390C22×D3—188no—
391C22×D3—188no—
392C22×D3—188no—
393C22×D3—184no—
394C22×D3—184no—
395C22×D3—184no—
396C22×D3—184no—
397C22×D3—184no—
398C22×D3—184no—
399C22×D3—181no—
400C22×D3—181no—
401C22×D3—181no—
402C22×D3—181no—
403C22×D3—181no—
404C22×D3—181no—
405C22×D3—364no—
406C22×D3—364no—
407C22×D3—364no—
408C22×D3—364no—
409C2×C3×C4—38no—
410C2×C3×C4—64no—
411C2×C3×C4—64no—
412C2×C3×C4—64no—
413C2×C3×C4—64no—
414C2×C3×C4—64no—
415C2×C3×C4—92no—
416C2×C3×C4—181no—
417C2×C3×C4—181no—
418C2×C3×C4—181no—
419C2×C3×C4—181no—
420C2×C3×C4—181no—
421#67—188no—
422#67—1832no—
423#67—1864no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—38no—
430C2×Dic3—64no—
431C2×Dic3—64no—
432C2×Dic3—64no—
433C2×Dic3—64no—
434C2×Dic3—64no—
435C2×Dic3—98no—
436C2×Dic3—98no—
437C2×Dic3—98no—
438C2×Dic3—92no—
439C2×Dic3—92no—
440C2×Dic3—92no—
441C2×Dic3—188no—
442C2×Dic3—188no—
443C2×Dic3—188no—
444C2×Dic3#63812yescomplemented
445C2×C32—24no—
446C2×C32—24no—
447C2×C32—24no—
448C2×C32—24no—
449C2×C32—24no—
450C2×C32—24no—
451C2×C32—24no—
452C2×C32—416no—
453C2×C32—48no—
454C2×C32—42no—
455C2×C32D3×C22⋊C414yes—
456C2×C32D3×C22⋊C411yescharacteristic
457C2×C32#67⋊C414yes—
458C2×C32#67⋊C411yescharacteristic
459C2×C32#88414yes—
460C2×C32#88411yescharacteristic
461C2×C32#88514yes—
462C2×C32#88511yescharacteristic
463C2×C32C4×#6714yes—
464C2×C32C4×#6711yescharacteristic
465C2×C32#93914yes—
466C2×C32#93911yescharacteristic
467C2×C32#94014yes—
468C2×C32#94011yescharacteristic
469C3⋊D3—1216no—
470C3⋊D3—128no—
471C3⋊D3—122no—
472C24—271no—
473C24D3×C3⋊D311yescharacteristic
474C2×D4—274no—
475C2×D4—542no—
476C2×D4—542no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—91no—
480C22×C4—91no—
481C22×C4—271no—
482C4⋊C4—542no—
483C22⋊C4—62no—
484C22⋊C4—182no—
485C22⋊C4—542no—
486C22⋊C4—542no—
487C42—542no—
488C22×C3—216no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—28no—
493C22×C3—28no—
494C22×C3—28no—
495C22×C3—24no—
496C22×C3—24no—
497C22×C3—24no—
498C22×C3—24no—
499C22×C3—24no—
500C22×C3—24no—
501C22×C3—24no—
502C22×C3—48no—
503C22×C3—48no—
504C22×C3—48no—
505C22×C3—48no—
506C22×C3—48no—
507C22×C3—48no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—44no—
513C22×C3—44no—
514C22×C3—44no—
515C22×C3—44no—
516C22×C3#380414yes—
517C22×C3#380414yes—
518C22×C3C2×#62414yes—
519C22×C3D3×#6714yes—
520C22×C3D3×#6714yes—
521C22×C3#381114yes—
522C22×C3#381114yes—
523C22×C3C2×C4×C3⋊D311yescharacteristic
524C22×C3#382811yescharacteristic
525C22×C3D4×C3⋊D311yescharacteristic
526C22×C3D4×C3⋊D311yescharacteristic
527C22×C3D4×C3⋊D311yescharacteristic
528C22×C3#383011yescharacteristic
529C22×C3C2×#63811yescharacteristic
530C2×D3—1816no—
531C2×D3—182no—
532C2×D3—182no—
533C2×D3—182no—
534C2×D3—188no—
535C2×D3—188no—
536C2×D3—188no—
537C2×D3—368no—
538C2×D3—368no—
539C2×D3—368no—
540C2×D3—368no—
541C2×D3—368no—
542C2×D3—368no—
543C2×D3—361no—
544C2×D3—361no—
545C2×D3—361no—
546C2×D3—361no—
547C2×D3—364no—
548C2×D3—364no—
549C2×D3—364no—
550C2×D3—364no—
551C3×C4—68no—
552C3×C4—124no—
553C3×C4—182no—
554C3×C4—361no—
555Dic3—22no—
556Dic3—41no—
557Dic3—68no—
558Dic3—94no—
559Dic3—916no—
560Dic3—124no—
561Dic3—1816no—
562C32—24no—
563C32#495714yescomplemented
564C32#495711yescharacteristic, complemented
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—272no—
572C23—541no—
573C23—541no—
574C23—541no—
575C23—541no—
576C23—541no—
577C23—541no—
578C23#628912yes—
579C23C2×D3×C3⋊D311yescharacteristic
580D4—548no—
581C2×C4—32no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—92no—
588C2×C4—181no—
589C2×C4—181no—
590C2×C4—181no—
591C2×C4—181no—
592C2×C4—181no—
593C2×C4—272no—
594C2×C4—272no—
595C2×C4—272no—
596C2×C3—24no—
597C2×C3—24no—
598C2×C3—24no—
599C2×C3—24no—
600C2×C3—24no—
601C2×C3—24no—
602C2×C3—24no—
603C2×C3—416no—
604C2×C3—42no—
605C2×C3—48no—
606C2×C3#6395214yes—
607C2×C3#6395214yes—
608C2×C3#6395314yes—
609C2×C3#6397114yes—
610C2×C3#6397114yes—
611C2×C3#6397414yes—
612C2×C3#6397914yes—
613C2×C3C22⋊C4×C3⋊D311yescharacteristic
614C2×C3#6409011yescharacteristic
615C2×C3#6409311yescharacteristic
616C2×C3#6409411yescharacteristic
617C2×C3C4×#63811yescharacteristic
618C2×C3#6414811yescharacteristic
619C2×C3#6414911yescharacteristic
620D3—3616no—
621D3—362no—
622D3—368no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—542no—
631C22—542no—
632C22—542no—
633C22—1081no—
634C22—1081no—
635C22—1081no—
636C22—1081no—
637C22#8902811yescharacteristic
638C22#8903111yescharacteristic
639C22C2×#628911yescharacteristic
640C22D3×#63811yescharacteristic
641C22C3⋊D3×#6711yescharacteristic
642C22#8903911yescharacteristic
643C22#8903911yescharacteristic
644C4—62no—
645C4—121no—
646C4—182no—
647C4—274no—
648C4—361no—
649C3—24no—
650C3#1059170414yescomplemented
651C3#1059265611yescharacteristic, complemented
652C2—42no—
653C2—1082no—
654C2#1172359211yescharacteristic
655C2#1172361611yescharacteristic
656C2#1172361811yescharacteristic
657C2#1172362211yescharacteristic
658C2#1172362411yescharacteristic
659C2#1172362711yescharacteristic
660C2#1172363311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC22×C33C22×C4
FittingC24×C33C22
socle, cosocleC23×C33C23
prime radicalC22×#6312C2
hypercenterC24D3×C3⋊D3
hypocenterC33#390
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3757C2semidirect
3. C2×C32×#217C2semidirect
4. C2×#88806C2semidirect
5. C2×#88804C2semidirect
6. C2×#88804C2semidirect
7. C2×Dic3×C3⋊Dic3C2semidirect
14. C2×#6312C4semidirect
30. C2×#3757D3semidirect
36. C2×C3×#217D3semidirect
65. C2×C3×C3⋊Dic3D4semidirect
71. C2×C32×Dic3D4semidirect
133. C3⋊(C3⋊Dic3)C22⋊C4semidirect
144. C2×#217C3⋊D3semidirect
221. C2×C3⋊Dic3#67semidirect
233. C2×C3×Dic3#67semidirect
363. C33#390semidirect
444. C2×Dic3#638semidirect
563. C32#4957semidirect
564. C32#4957semidirect
650. C3#10591704semidirect
651. C3#10592656semidirect

Maximal subgroups

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

subgroupindex
C22×#63122
C2×C3×#37572
C2×C32×#2172
C2×#888062
C2×#888042
C2×Dic3×C3⋊Dic32
#105926563
#105917043