Finite Group Databasebeta

← #13156852

Subgroups — #13156852

Order 1728.

1442 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#625C211yesmaximal, characteristic
2C2×#88813C211yesmaximal, characteristic, complemented
3C2×#88807C211yesmaximal, characteristic, complemented
4C2×(C2×D3)⋊(C3⋊Dic3)C212yesmaximal, complemented
5C2×C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
6#10591707—32nomaximal
7#10591386—31nomaximal
8C23×C32×D3C2212yes—
9C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2211yescharacteristic
13C2×C3×#625—22no—
14C2×C3×#625—22no—
15C2×C3×#625C414yescomplemented
16#88813—22no—
17#88807—22no—
18(C2×D3)⋊(C3⋊Dic3)—24no—
19C3×(C2×D3)⋊Dic3—24no—
20C22×C3×#67—32no—
21C22×#625D311yescharacteristic
22C2×#3757—32no—
23C2×#3750—31no—
24C2×C3×#217—32no—
25C2×C3×(C2×D3)⋊C4—32no—
26C2×Dic3⋊Dic3—32no—
27C2×Dic3⋊Dic3—61no—
28C2×(C2×D3)⋊Dic3—32no—
29C2×(C2×D3)⋊Dic3—32no—
30C2×(C2×D3)⋊Dic3—62no—
31C2×(C2×D3)⋊Dic3D312yescomplemented
32C2×Dic32—31no—
33C2×Dic32—32no—
34C23×C33C2311yescharacteristic
35C22×C32×D3—22no—
36C22×C32×D3—22no—
37C22×C32×D3—22no—
38C22×C32×D3—22no—
39C22×C32×D3—22no—
40C22×C32×D3—22no—
41C22×C32×D3C2×C414yes—
42C2×C3⋊(C3⋊Dic3)—22no—
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3×C3⋊Dic3—22no—
45C2×C3×C3⋊Dic3—22no—
46C2×C3×C3⋊Dic3C2×C412yes—
47C2×C3×C3⋊Dic3D412yes—
48C2×C3×C3⋊Dic3D412yes—
49C2×C3×C3⋊Dic3D414yescomplemented
50C2×C32×Dic3—21no—
51C2×C32×Dic3—22no—
52C3×#625—28no—
53#5221—91no—
54#4957—92no—
55C24×C32—32no—
56C23×C3×D3—22no—
57C23×C3×D3C2×D312yes—
58C23×C3×D3C2×D312yes—
59C22×C32×C4—31no—
60C22×C3⋊Dic3—31no—
61C22×C3⋊Dic3—32no—
62C22×C3⋊Dic3—62no—
63C22×C3⋊Dic3—61no—
64C22×C3⋊Dic3—121no—
65C22×C3⋊Dic3C2×D312yes—
66C22×C3⋊Dic3C2×D311yescharacteristic
67C2×C3×#67—38no—
68C2×C3×#67—64no—
69C2×C3×#67—64no—
70C22×C3×Dic3—21no—
71C22×C3×Dic3—32no—
72C22×C3×Dic3—32no—
73C22×C3×Dic3—32no—
74C22×C3×Dic3—61no—
75C22×C3×Dic3—62no—
76C22×C3×Dic3C2×D312yes—
77C2×#625—22no—
78C2×#625—22no—
79C2×#625Dic314yescomplemented
80#3757—64no—
81#3750—62no—
82C3×#217—64no—
83C3×(C2×D3)⋊C4—64no—
84Dic3⋊Dic3—64no—
85Dic3⋊Dic3—64no—
86(C2×D3)⋊Dic3—24no—
87(C2×D3)⋊Dic3—64no—
88(C2×D3)⋊Dic3—64no—
89(C2×D3)⋊Dic3—68no—
90Dic32—64no—
91Dic32—62no—
92C22×C33C22×C411yescharacteristic
93C22×C33C2×D411yescharacteristic
94C22×C33C2×D411yescharacteristic
95C22×C33C2×D412yes—
96C22×C33#4112yes—
97C2×C32×D3—24no—
98C2×C32×D3—24no—
99C2×C32×D3—24no—
100C2×C32×D3—42no—
101C2×C32×D3—42no—
102C2×C32×D3—42no—
103C2×C32×D3—42no—
104C3⋊(C3⋊Dic3)—41no—
105C3×C3⋊Dic3—24no—
106C3×C3⋊Dic3C22⋊C414yescomplemented
107C32×Dic3—41no—
108C22×C3×D4—91no—
109C22×#67—32no—
110C2×C3×C22⋊C4—92no—
111C2×#217—32no—
112C2×#217—92no—
113C2×#217—92no—
114C2×#217—182no—
115C2×(C2×D3)⋊C4—32no—
116C2×(C2×D3)⋊C4—92no—
117C2×C4⋊Dic3—91no—
118C2×Dic3⋊C4—92no—
119C2×Dic3⋊C4—181no—
120C2×C4×Dic3—92no—
121C2×C4×Dic3—91no—
122C23×C32—22no—
123C23×C32—21no—
124C23×C32—34no—
125C23×C32—41no—
126C23×C32—62no—
127C23×C32—62no—
128C23×C32—62no—
129C23×C32—62no—
130C23×C32—62no—
131C23×C32—62no—
132C23×C32C22×D312yes—
133C23×C32C22×D311yescharacteristic
134C22×C3×D3—24no—
135C22×C3×D3—24no—
136C22×C3×D3—22no—
137C22×C3×D3—22no—
138C22×C3×D3—22no—
139C22×C3×D3—22no—
140C22×C3×D3—22no—
141C22×C3×D3—22no—
142C22×C3×D3—22no—
143C22×C3×D3—22no—
144C22×C3×D3—22no—
145C22×C3×D3—22no—
146C22×C3×D3—22no—
147C22×C3×D3—22no—
148C22×C3×D3—24no—
149C22×C3×D3—42no—
150C22×C3×D3—42no—
151C22×C3×D3—42no—
152C22×C3×D3—42no—
153C22×C3×D3C4×D314yes—
154C22×C3×D3C2×Dic314yes—
155C2×C32×C4—61no—
156C2×C32×C4—62no—
157C2×C3⋊Dic3—22no—
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—62no—
161C2×C3⋊Dic3—64no—
162C2×C3⋊Dic3—62no—
163C2×C3⋊Dic3—62no—
164C2×C3⋊Dic3—62no—
165C2×C3⋊Dic3—62no—
166C2×C3⋊Dic3—122no—
167C2×C3⋊Dic3—122no—
168C2×C3⋊Dic3—122no—
169C2×C3⋊Dic3—122no—
170C2×C3⋊Dic3—124no—
171C2×C3⋊Dic3C2×Dic312yes—
172C2×C3⋊Dic3#6712yes—
173C2×C3⋊Dic3#6712yes—
174C2×C3⋊Dic3#6714yescomplemented
175C3×#67—616no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—34no—
181C2×C3×Dic3—34no—
182C2×C3×Dic3—34no—
183C2×C3×Dic3—34no—
184C2×C3×Dic3—34no—
185C2×C3×Dic3—42no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—64no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—62no—
194C2×C3×Dic3—122no—
195C2×C3×Dic3—122no—
196#625—28no—
197#390—271noSylow
198C2×C33C2×C22⋊C411yescharacteristic
199C2×C33C4×D412yes—
200C2×C33#12112yes—
201C2×C33#12411yescharacteristic
202C2×C33#12711yescharacteristic
203C32×D3—44no—
204C24×C3—32no—
205C24×C3—32no—
206C24×C3—62no—
207C23×D3D3212yes—
208C2×C3×D4—94no—
209C2×C3×D4—182no—
210C2×C3×D4—182no—
211C22×C3×C4—32no—
212C22×C3×C4—61no—
213C22×C3×C4—91no—
214C22×C3×C4—92no—
215C2×#67—38no—
216C2×#67—64no—
217C2×#67—64no—
218C22×Dic3—32no—
219C22×Dic3—32no—
220C22×Dic3—32no—
221C22×Dic3—61no—
222C22×Dic3—62no—
223C22×Dic3—92no—
224C22×Dic3—91no—
225C22×Dic3—181no—
226C22×Dic3—182no—
227C22×Dic3—361no—
228C22×Dic3D3211yescharacteristic
229C3×C22⋊C4—184no—
230#217—64no—
231#217—184no—
232#217—184no—
233#217—188no—
234(C2×D3)⋊C4—64no—
235(C2×D3)⋊C4—184no—
236C4⋊Dic3—182no—
237Dic3⋊C4—184no—
238Dic3⋊C4—184no—
239C4×Dic3—184no—
240C4×Dic3—182no—
241C22×C32—22no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—21no—
252C22×C32—22no—
253C22×C32—41no—
254C22×C32—41no—
255C22×C32—42no—
256C22×C32—41no—
257C22×C32—42no—
258C22×C32—64no—
259C22×C32—64no—
260C22×C32—64no—
261C22×C32—122no—
262C22×C32—122no—
263C22×C32—122no—
264C22×C32—122no—
265C22×C32C2×C4×D312yes—
266C22×C32#23512yes—
267C22×C32D3×D412yes—
268C22×C32D3×D412yes—
269C22×C32D3×D412yes—
270C22×C32D3×D412yes—
271C22×C32#23712yes—
272C22×C32#23712yes—
273C22×C32C22×Dic311yescharacteristic
274C22×C32C2×#6712yes—
275C22×C32C2×#6711yescharacteristic
276C22×C32C2×#6711yescharacteristic
277C2×C3×D3—28no—
278C2×C3×D3—24no—
279C2×C3×D3—24no—
280C2×C3×D3—24no—
281C2×C3×D3—24no—
282C2×C3×D3—24no—
283C2×C3×D3—24no—
284C2×C3×D3—44no—
285C2×C3×D3—44no—
286C2×C3×D3—44no—
287C2×C3×D3—44no—
288C2×C3×D3—44no—
289C2×C3×D3—44no—
290C2×C3×D3—42no—
291C2×C3×D3—42no—
292C2×C3×D3—42no—
293C2×C3×D3—42no—
294C2×C3×D3—42no—
295C2×C3×D3—42no—
296C2×C3×D3—42no—
297C2×C3×D3—42no—
298C32×C4—121no—
299C3⋊Dic3—24no—
300C3⋊Dic3—122no—
301C3⋊Dic3—124no—
302C3⋊Dic3—124no—
303C3⋊Dic3—122no—
304C3⋊Dic3—121no—
305C3⋊Dic3#21714yescomplemented
306C3×Dic3—38no—
307C3×Dic3—42no—
308C3×Dic3—42no—
309C3×Dic3—64no—
310C3×Dic3—64no—
311C3×Dic3—64no—
312C3×Dic3—124no—
313C22×D4—91no—
314C2×C4⋊C4—271no—
315C2×C22⋊C4—92no—
316C2×C22⋊C4—272no—
317C2×C42—271no—
318C33#39011yescharacteristic, complemented, Sylow
319C23×C3—21no—
320C23×C3—22no—
321C23×C3—34no—
322C23×C3—34no—
323C23×C3—41no—
324C23×C3—64no—
325C23×C3—64no—
326C23×C3—64no—
327C23×C3—62no—
328C23×C3—62no—
329C23×C3—62no—
330C23×C3—62no—
331C23×C3—62no—
332C23×C3—62no—
333C23×C3—62no—
334C23×C3—62no—
335C23×C3—62no—
336C23×C3—62no—
337C23×C3—62no—
338C23×C3—62no—
339C23×C3—122no—
340C23×C3—122no—
341C23×C3—122no—
342C23×C3—122no—
343C23×C3C2×D3211yescharacteristic
344C23×C3C2×D3212yes—
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—22no—
348C22×D3—22no—
349C22×D3—22no—
350C22×D3—22no—
351C22×D3D3×Dic314yes—
352C3×D4—188no—
353C2×C3×C4—62no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—62no—
357C2×C3×C4—92no—
358C2×C3×C4—92no—
359C2×C3×C4—92no—
360C2×C3×C4—94no—
361C2×C3×C4—122no—
362C2×C3×C4—182no—
363C2×C3×C4—182no—
364#67—616no—
365C2×Dic3—22no—
366C2×Dic3—21no—
367C2×Dic3—34no—
368C2×Dic3—34no—
369C2×Dic3—34no—
370C2×Dic3—34no—
371C2×Dic3—34no—
372C2×Dic3—62no—
373C2×Dic3—62no—
374C2×Dic3—62no—
375C2×Dic3—64no—
376C2×Dic3—62no—
377C2×Dic3—62no—
378C2×Dic3—62no—
379C2×Dic3—62no—
380C2×Dic3—122no—
381C2×Dic3—122no—
382C2×Dic3—182no—
383C2×Dic3—184no—
384C2×Dic3—182no—
385C2×Dic3—182no—
386C2×Dic3—182no—
387C2×Dic3—181no—
388C2×Dic3—182no—
389C2×Dic3—362no—
390C2×Dic3—362no—
391C2×Dic3—362no—
392C2×Dic3—364no—
393C2×Dic3—362no—
394C2×C32—22no—
395C2×C32—22no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—22no—
399C2×C32—22no—
400C2×C32—22no—
401C2×C32—22no—
402C2×C32—22no—
403C2×C32—22no—
404C2×C32—21no—
405C2×C32—22no—
406C2×C32—41no—
407C2×C32—42no—
408C2×C32—42no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—124no—
412C2×C32D3×C22⋊C412yes—
413C2×C32#67⋊C412yes—
414C2×C32#88412yes—
415C2×C32#88512yes—
416C2×C32C4×#6712yes—
417C2×C32D4×Dic312yes—
418C2×C32#93611yescharacteristic
419C2×C32#93812yes—
420C2×C32#93912yes—
421C2×C32#94012yes—
422C2×C32#94011yescharacteristic
423C2×C32C2×#21711yescharacteristic
424C3×D3—48no—
425C3×D3—44no—
426C3×D3—44no—
427C24—32no—
428C2×D4—94no—
429C2×D4—182no—
430C2×D4—182no—
431C22×C4—31no—
432C22×C4—91no—
433C22×C4—92no—
434C22×C4—271no—
435C4⋊C4—542no—
436C22⋊C4—184no—
437C22⋊C4—544no—
438C42—542no—
439C22×C3—21no—
440C22×C3—22no—
441C22×C3—21no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—21no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—42no—
454C22×C3—41no—
455C22×C3—42no—
456C22×C3—68no—
457C22×C3—64no—
458C22×C3—64no—
459C22×C3—64no—
460C22×C3—64no—
461C22×C3—64no—
462C22×C3—64no—
463C22×C3—124no—
464C22×C3—124no—
465C22×C3—124no—
466C22×C3—124no—
467C22×C3—124no—
468C22×C3—124no—
469C22×C3—122no—
470C22×C3—122no—
471C22×C3—122no—
472C22×C3—122no—
473C22×C3—122no—
474C22×C3—122no—
475C22×C3—122no—
476C22×C3—122no—
477C22×C3#379512yes—
478C22×C3C4×D3211yescharacteristic
479C22×C3#380211yescharacteristic
480C22×C3C2×D3×Dic312yes—
481C22×C3#380412yes—
482C22×C3#380512yes—
483C22×C3D3×#6712yes—
484C22×C3D3×#6712yes—
485C22×C3D3×#6712yes—
486C22×C3D3×#6712yes—
487C22×C3#381111yescharacteristic
488C22×C3#381112yes—
489C2×D3—24no—
490C2×D3—24no—
491C2×D3—24no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—42no—
496C3×C4—94no—
497C3×C4—122no—
498C3×C4—122no—
499C3×C4—184no—
500Dic3—38no—
501Dic3—41no—
502Dic3—64no—
503Dic3—64no—
504Dic3—64no—
505Dic3—124no—
506Dic3—362no—
507Dic3—364no—
508Dic3—364no—
509Dic3—361no—
510Dic3—362no—
511C32—21no—
512C32—22no—
513C32—41no—
514C32#495712yescomplemented
515C32#522111yescharacteristic, complemented
516C23—34no—
517C23—62no—
518C23—62no—
519C23—62no—
520C23—62no—
521C23—62no—
522C23—62no—
523C23D3311yescharacteristic
524D4—188no—
525C2×C4—62no—
526C2×C4—61no—
527C2×C4—92no—
528C2×C4—92no—
529C2×C4—92no—
530C2×C4—94no—
531C2×C4—182no—
532C2×C4—182no—
533C2×C4—542no—
534C2×C4—541no—
535C2×C3—21no—
536C2×C3—22no—
537C2×C3—21no—
538C2×C3—22no—
539C2×C3—22no—
540C2×C3—22no—
541C2×C3—22no—
542C2×C3—22no—
543C2×C3—22no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—21no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—42no—
550C2×C3—42no—
551C2×C3—41no—
552C2×C3—128no—
553C2×C3—124no—
554C2×C3—124no—
555C2×C3#6387912yes—
556C2×C3#6388512yes—
557C2×C3#6391311yescharacteristic
558C2×C3#6392111yescharacteristic
559C2×C3#6395812yes—
560C2×C3#6395912yes—
561C2×C3D3×#21712yes—
562C2×C3#6397012yes—
563C2×C3Dic3×#6712yes—
564C2×C3#67⋊Dic312yes—
565C2×C3#6397911yescharacteristic
566C2×C3#6397912yes—
567D3—44no—
568C22—64no—
569C22—64no—
570C22—64no—
571C22—122no—
572C22—122no—
573C22—122no—
574C22—122no—
575C22D32×Dic311yescharacteristic
576C22#8895211yescharacteristic
577C22#8895212yes—
578C22#8895311yescharacteristic
579C22#8896112yes—
580C4—94no—
581C4—121no—
582C4—184no—
583C4—1081no—
584C3—22no—
585C3—21no—
586C3—41no—
587C3#1059138611yescharacteristic, complemented
588C3#1059170712yescomplemented
589C2—124no—
590C2#1172290211yescharacteristic
591C2#1172292412yes—
592C2#1172296512yes—
593C2#1172300411yescharacteristic
594C2#1172303111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, cosocleC23×C33C23
prime radicalC22×C3×#625C2
hypocenterC33#390
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88813C2semidirect
3. C2×#88807C2semidirect
4. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
5. C2×C3×(C2×D3)⋊Dic3C2semidirect
15. C2×C3×#625C4semidirect
31. C2×(C2×D3)⋊Dic3D3semidirect
49. C2×C3×C3⋊Dic3D4semidirect
79. C2×#625Dic3semidirect
106. C3×C3⋊Dic3C22⋊C4semidirect
174. C2×C3⋊Dic3#67semidirect
305. C3⋊Dic3#217semidirect
318. C33#390semidirect
514. C32#4957semidirect
515. C32#5221semidirect
587. C3#10591386semidirect
588. C3#10591707semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6252
C2×#888132
C2×#888072
C2×(C2×D3)⋊(C3⋊Dic3)2
C2×C3×(C2×D3)⋊Dic32
#105917073
#105913863