Finite Group Databasebeta

← #13156767

Subgroups — #13156767

Order 1728.

1586 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6290C211yesmaximal, characteristic
2C22×C3×#624C211yesmaximal, characteristic, complemented
3C2×#88814C211yesmaximal, characteristic, complemented
4C2×C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
5#11722061C212yesmaximal, complemented
6#10591722—31nomaximal
7#10591381—32nomaximal
8C23×C3×C3⋊D3C2211yescharacteristic
9C23×C32×D3C2211yescharacteristic
10C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C2×#6290—22no—
14C2×#6290—22no—
15C2×#6290C414yescomplemented
16C2×C3×#624—22no—
17C2×C3×#624—24no—
18#88814—22no—
19C3×(C2×D3)⋊Dic3—24no—
20C22×#638—31no—
21C22×C3×C4×D3—32no—
22C22×#625—32no—
23C22×#625—61no—
24C22×#624D311yescharacteristic, complemented
25C2×C3×#217—32no—
26C2×C3×(C2×D3)⋊C4—32no—
27C2×#3722—31no—
28C2×(C2×D3)⋊Dic3—32no—
29C2×(C2×D3)⋊Dic3D312yescomplemented
30#63658—32no—
31#63579—32no—
32#63579—32no—
33#63579—62no—
34C23×C33C2311yescharacteristic
35C22×C3×C3⋊D3—22no—
36C22×C3×C3⋊D3—22no—
37C22×C3×C3⋊D3C2×C412yes—
38C22×C3×C3⋊D3D412yes—
39C22×C3×C3⋊D3D412yes—
40C22×C32×D3—21no—
41C22×C32×D3—21no—
42C22×C32×D3—22no—
43C22×C32×D3—22no—
44C22×C32×D3C2×C412yescomplemented
45C2×C3⋊(C3⋊Dic3)—21no—
46C2×C3⋊(C3⋊Dic3)—21no—
47C2×C3⋊(C3⋊Dic3)C2×C412yescomplemented
48C2×C3×C3⋊Dic3—22no—
49C2×C3×C3⋊Dic3—22no—
50C2×C3×C3⋊Dic3—22no—
51C2×C32×Dic3—22no—
52C2×C32×Dic3—22no—
53C2×C32×Dic3D414yescomplemented
54#6290—28no—
55C3×#624—28no—
56#5213—91no—
57#4956—92no—
58C24×C32—31no—
59C23×C3⋊D3C2×D311yescharacteristic
60C23×C3×D3—21no—
61C23×C3×D3—32no—
62C23×C3×D3—61no—
63C23×C3×D3C2×D312yes—
64C22×C32×C4—32no—
65C2×#638—34no—
66C2×#638—62no—
67C2×#638—62no—
68C22×C3⋊Dic3—32no—
69C22×C3⋊Dic3—31no—
70C22×C3⋊Dic3—61no—
71C22×C3⋊Dic3—62no—
72C22×C3⋊Dic3—121no—
73C22×C3⋊Dic3C2×D312yes—
74C22×C3×Dic3—22no—
75C22×C3×Dic3—32no—
76C22×C3×Dic3—32no—
77C22×C3×Dic3—62no—
78C22×C3×Dic3C2×D312yes—
79C22×C3×Dic3C2×D312yes—
80C2×C3×C4×D3—64no—
81C2×C3×C4×D3—64no—
82C2×C3×C4×D3—64no—
83C2×#625—38no—
84C2×#625—64no—
85C2×#625—64no—
86C2×#625—64no—
87C2×#625—64no—
88C2×#625—64no—
89C2×#624—22no—
90C2×#624—24no—
91C3×#217—64no—
92C3×(C2×D3)⋊C4—64no—
93#3722—62no—
94(C2×D3)⋊Dic3—24no—
95(C2×D3)⋊Dic3—64no—
96C22×C33C22×C411yescharacteristic
97C22×C33C2×D411yescharacteristic
98C22×C33C2×D411yescharacteristic
99C22×C33C2×D412yes—
100C22×C33#4112yes—
101C2×C3×C3⋊D3—24no—
102C2×C3×C3⋊D3—22no—
103C2×C3×C3⋊D3—24no—
104C2×C3×C3⋊D3—22no—
105C2×C3×C3⋊D3C22⋊C414yescomplemented
106C2×C32×D3—22no—
107C2×C32×D3—24no—
108C2×C32×D3—41no—
109C2×C32×D3—42no—
110C2×C32×D3—41no—
111C3⋊(C3⋊Dic3)—22no—
112C3×C3⋊Dic3—42no—
113C32×Dic3—24no—
114C23×C3×C4—91no—
115C22×#67—92no—
116C22×#67—91no—
117C22×#67—181no—
118C22×C4×D3—32no—
119C2×C3×C22⋊C4—92no—
120C2×#217—32no—
121C2×#217—91no—
122C2×(C2×D3)⋊C4—32no—
123C2×(C2×D3)⋊C4—92no—
124C3⋊#95—92no—
125C3⋊#95—92no—
126C3⋊#95—92no—
127C3⋊#95—182no—
128C23×C32—21no—
129C23×C32—22no—
130C23×C32—32no—
131C23×C32—41no—
132C23×C32—61no—
133C23×C32—61no—
134C23×C32—62no—
135C23×C32—62no—
136C23×C32C22×D311yescharacteristic
137C23×C32C22×D312yes—
138C22×C3⋊D3—22no—
139C22×C3⋊D3—22no—
140C22×C3⋊D3C4×D312yes—
141C22×C3⋊D3D1212yes—
142C22×C3⋊D3#6712yes—
143C22×C3×D3—22no—
144C22×C3×D3—22no—
145C22×C3×D3—22no—
146C22×C3×D3—22no—
147C22×C3×D3—22no—
148C22×C3×D3—22no—
149C22×C3×D3—22no—
150C22×C3×D3—22no—
151C22×C3×D3—22no—
152C22×C3×D3—34no—
153C22×C3×D3—34no—
154C22×C3×D3—34no—
155C22×C3×D3—42no—
156C22×C3×D3—42no—
157C22×C3×D3—62no—
158C22×C3×D3—62no—
159C22×C3×D3—62no—
160C22×C3×D3—62no—
161C22×C3×D3—62no—
162C22×C3×D3—62no—
163C22×C3×D3—62no—
164C22×C3×D3—122no—
165C22×C3×D3—122no—
166C22×C3×D3C4×D314yescomplemented
167C2×C32×C4—34no—
168C2×C32×C4—62no—
169C2×C32×C4—62no—
170#638—68no—
171C2×C3⋊Dic3—22no—
172C2×C3⋊Dic3—22no—
173C2×C3⋊Dic3—22no—
174C2×C3⋊Dic3—34no—
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—62no—
177C2×C3⋊Dic3—64no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—61no—
180C2×C3⋊Dic3—62no—
181C2×C3⋊Dic3—62no—
182C2×C3⋊Dic3—62no—
183C2×C3⋊Dic3—62no—
184C2×C3⋊Dic3—122no—
185C2×C3⋊Dic3—122no—
186C2×C3⋊Dic3—122no—
187C2×C3⋊Dic3—122no—
188C2×C3⋊Dic3—122no—
189C2×C3×Dic3—22no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—24no—
194C2×C3×Dic3—42no—
195C2×C3×Dic3—42no—
196C2×C3×Dic3—64no—
197C2×C3×Dic3—62no—
198C2×C3×Dic3—62no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—62no—
202C2×C3×Dic3—62no—
203C2×C3×Dic3—122no—
204C2×C3×Dic3—122no—
205C2×C3×Dic3#6714yescomplemented
206C2×C3×Dic3#6714yescomplemented
207C3×C4×D3—616no—
208#625—616no—
209#625—616no—
210#624—28no—
211#386—271noSylow
212C2×C33C2×C22⋊C411yescharacteristic
213C2×C33C4×D412yes—
214C2×C33#12011yescharacteristic
215C2×C33#12112yes—
216C2×C33#12311yescharacteristic
217C3×C3⋊D3—24no—
218C32×D3—42no—
219C24×C3—32no—
220C24×C3—61no—
221C24×C3—91no—
222C23×D3—32no—
223C23×D3—61no—
224C23×D3D3211yescharacteristic
225C22×C3×C4—32no—
226C22×C3×C4—32no—
227C22×C3×C4—62no—
228C22×C3×C4—92no—
229C22×C3×C4—184no—
230C22×C3×C4—182no—
231C2×#67—98no—
232C2×#67—94no—
233C2×#67—184no—
234C2×#67—182no—
235C2×#67—182no—
236C2×#67—184no—
237C2×#67—184no—
238C2×#67—184no—
239C2×#67—184no—
240C22×Dic3—32no—
241C22×Dic3—32no—
242C22×Dic3—62no—
243C22×Dic3—92no—
244C22×Dic3—91no—
245C22×Dic3—181no—
246C22×Dic3—182no—
247C22×Dic3—361no—
248C22×Dic3D3212yes—
249C2×C4×D3—64no—
250C2×C4×D3—64no—
251C2×C4×D3—64no—
252C3×C22⋊C4—184no—
253#217—64no—
254#217—182no—
255(C2×D3)⋊C4—64no—
256(C2×D3)⋊C4—184no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—21no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—22no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—41no—
270C22×C32—41no—
271C22×C32—41no—
272C22×C32—42no—
273C22×C32—42no—
274C22×C32—62no—
275C22×C32—64no—
276C22×C32—121no—
277C22×C32—122no—
278C22×C32—121no—
279C22×C32C2×C4×D311yescharacteristic
280C22×C32C2×C4×D312yes—
281C22×C32C2×D1211yescharacteristic
282C22×C32#23512yes—
283C22×C32#23512yes—
284C22×C32D3×D412yes—
285C22×C32D3×D412yes—
286C22×C32D3×D412yes—
287C22×C32#23712yes—
288C22×C32C2×#6712yes—
289C22×C32C2×#6712yes—
290C22×C32C2×#6711yescharacteristic
291C2×C3⋊D3—24no—
292C2×C3⋊D3—22no—
293C2×C3⋊D3—24no—
294C2×C3⋊D3—22no—
295C2×C3⋊D3(C2×D3)⋊C414yescomplemented
296C2×C3×D3—24no—
297C2×C3×D3—24no—
298C2×C3×D3—24no—
299C2×C3×D3—24no—
300C2×C3×D3—38no—
301C2×C3×D3—44no—
302C2×C3×D3—44no—
303C2×C3×D3—42no—
304C2×C3×D3—42no—
305C2×C3×D3—42no—
306C2×C3×D3—42no—
307C2×C3×D3—42no—
308C2×C3×D3—42no—
309C2×C3×D3—64no—
310C2×C3×D3—64no—
311C2×C3×D3—64no—
312C2×C3×D3—64no—
313C2×C3×D3—64no—
314C2×C3×D3—64no—
315C2×C3×D3—64no—
316C2×C3×D3—124no—
317C2×C3×D3—124no—
318C2×C3×D3—122no—
319C2×C3×D3—122no—
320C32×C4—64no—
321C3⋊Dic3—42no—
322C3⋊Dic3—62no—
323C3⋊Dic3—64no—
324C3⋊Dic3—64no—
325C3⋊Dic3—124no—
326C3⋊Dic3—124no—
327C3×Dic3—24no—
328C3×Dic3—24no—
329C3×Dic3—44no—
330C3×Dic3—124no—
331C3×Dic3—122no—
332C3×Dic3—122no—
333C22×D4—271no—
334C23×C4—91no—
335C2×C22⋊C4—92no—
336C2×C22⋊C4—271no—
337#95—272no—
338C33#38611yescharacteristic, complemented, Sylow
339C23×C3—22no—
340C23×C3—21no—
341C23×C3—34no—
342C23×C3—41no—
343C23×C3—62no—
344C23×C3—62no—
345C23×C3—62no—
346C23×C3—62no—
347C23×C3—62no—
348C23×C3—62no—
349C23×C3—62no—
350C23×C3—62no—
351C23×C3—62no—
352C23×C3—92no—
353C23×C3—92no—
354C23×C3—92no—
355C23×C3—122no—
356C23×C3—122no—
357C23×C3—182no—
358C23×C3—182no—
359C23×C3C2×D3212yes—
360C23×C3C2×D3211yescharacteristic
361C22×D3—22no—
362C22×D3—22no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—34no—
366C22×D3—34no—
367C22×D3—34no—
368C22×D3—62no—
369C22×D3—62no—
370C22×D3—62no—
371C22×D3—62no—
372C22×D3—62no—
373C22×D3—62no—
374C22×D3—62no—
375C22×D3—122no—
376C22×D3—122no—
377C22×D3#62412yescomplemented
378C2×C3×C4—34no—
379C2×C3×C4—34no—
380C2×C3×C4—64no—
381C2×C3×C4—62no—
382C2×C3×C4—62no—
383C2×C3×C4—62no—
384C2×C3×C4—62no—
385C2×C3×C4—122no—
386C2×C3×C4—122no—
387C2×C3×C4—188no—
388C2×C3×C4—182no—
389C2×C3×C4—182no—
390C2×C3×C4—182no—
391#67—188no—
392#67—1816no—
393#67—1816no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—64no—
397C2×Dic3—62no—
398C2×Dic3—62no—
399C2×Dic3—62no—
400C2×Dic3—62no—
401C2×Dic3—62no—
402C2×Dic3—62no—
403C2×Dic3—94no—
404C2×Dic3—92no—
405C2×Dic3—122no—
406C2×Dic3—122no—
407C2×Dic3—182no—
408C2×Dic3—184no—
409C2×Dic3—182no—
410C2×Dic3—182no—
411C2×Dic3—182no—
412C2×Dic3—182no—
413C2×Dic3—181no—
414C2×Dic3—181no—
415C2×Dic3—362no—
416C2×Dic3—362no—
417C2×Dic3—362no—
418C2×Dic3—362no—
419C2×Dic3—362no—
420C2×Dic3#62514yescomplemented
421C4×D3—616no—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—22no—
425C2×C32—22no—
426C2×C32—22no—
427C2×C32—22no—
428C2×C32—22no—
429C2×C32—22no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—22no—
433C2×C32—21no—
434C2×C32—41no—
435C2×C32—42no—
436C2×C32—41no—
437C2×C32—42no—
438C2×C32—41no—
439C2×C32—122no—
440C2×C32D3×C22⋊C412yes—
441C2×C32#67⋊C412yes—
442C2×C32#88312yes—
443C2×C32#88412yes—
444C2×C32C2×(C2×D3)⋊C411yescharacteristic
445C2×C32C4×#6712yes—
446C2×C32C4×#6712yes—
447C2×C32#92911yescharacteristic
448C2×C32#93012yes—
449C2×C32#93712yes—
450C2×C32#93812yes—
451C2×C32#95311yescharacteristic
452C3⋊D3—24no—
453C3×D3—44no—
454C3×D3—44no—
455C3×D3—68no—
456C3×D3—124no—
457C24—31no—
458C24—91no—
459C2×D4—274no—
460C2×D4—542no—
461C2×D4—542no—
462C22×C4—32no—
463C22×C4—92no—
464C22×C4—184no—
465C22×C4—182no—
466C22×C4—271no—
467C22⋊C4—184no—
468C22⋊C4—542no—
469C22×C3—21no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—21no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—21no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—41no—
482C22×C3—42no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—42no—
486C22×C3—64no—
487C22×C3—64no—
488C22×C3—64no—
489C22×C3—64no—
490C22×C3—94no—
491C22×C3—124no—
492C22×C3—122no—
493C22×C3—124no—
494C22×C3—122no—
495C22×C3—122no—
496C22×C3—122no—
497C22×C3—122no—
498C22×C3—122no—
499C22×C3—184no—
500C22×C3—184no—
501C22×C3—182no—
502C22×C3—182no—
503C22×C3#379512yes—
504C22×C3#379812yes—
505C22×C3C4×D3212yes—
506C22×C3D3×D1212yes—
507C22×C3#380412yes—
508C22×C3C2×#62411yescharacteristic
509C22×C3C2×#62512yes—
510C22×C3D3×#6712yes—
511C22×C3D3×#6712yes—
512C22×C3D3×#6712yes—
513C22×C3#381111yescharacteristic
514C22×C3#381111yescharacteristic
515C2×D3—24no—
516C2×D3—22no—
517C2×D3—38no—
518C2×D3—42no—
519C2×D3—41no—
520C2×D3—41no—
521C2×D3—64no—
522C2×D3—64no—
523C2×D3—64no—
524C2×D3—64no—
525C2×D3—64no—
526C2×D3—64no—
527C2×D3—64no—
528C2×D3—124no—
529C2×D3—122no—
530C2×D3—124no—
531C2×D3—122no—
532C3×C4—64no—
533C3×C4—64no—
534C3×C4—124no—
535C3×C4—362no—
536Dic3—24no—
537Dic3—124no—
538Dic3—122no—
539Dic3—122no—
540Dic3—184no—
541Dic3—184no—
542Dic3—182no—
543Dic3—364no—
544Dic3—364no—
545C32—22no—
546C32—21no—
547C32—41no—
548C32#495612yescomplemented
549C32#521311yescharacteristic, complemented
550C23—32no—
551C23—62no—
552C23—61no—
553C23—61no—
554C23—62no—
555C23—92no—
556C23—92no—
557C23—92no—
558C23—182no—
559C23—182no—
560C23D3311yescharacteristic
561D4—548no—
562C2×C4—34no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—188no—
566C2×C4—182no—
567C2×C4—182no—
568C2×C4—182no—
569C2×C4—272no—
570C2×C4—541no—
571C2×C4—541no—
572C2×C3—21no—
573C2×C3—22no—
574C2×C3—22no—
575C2×C3—22no—
576C2×C3—22no—
577C2×C3—22no—
578C2×C3—21no—
579C2×C3—22no—
580C2×C3—22no—
581C2×C3—22no—
582C2×C3—21no—
583C2×C3—22no—
584C2×C3—41no—
585C2×C3—42no—
586C2×C3—41no—
587C2×C3—42no—
588C2×C3—41no—
589C2×C3—124no—
590C2×C3—124no—
591C2×C3—184no—
592C2×C3#6387912yes—
593C2×C3C4×#62512yes—
594C2×C3#6390512yes—
595C2×C3#6390612yes—
596C2×C3#6390812yes—
597C2×C3D3×(C2×D3)⋊C412yes—
598C2×C3#6395212yes—
599C2×C3#6395812yes—
600C2×C3#6397411yescharacteristic
601C2×C3#6397511yescharacteristic
602C2×C3#6397612yes—
603C2×C3#6398311yescharacteristic
604D3—42no—
605D3—68no—
606D3—124no—
607C22—62no—
608C22—64no—
609C22—94no—
610C22—121no—
611C22—122no—
612C22—121no—
613C22—184no—
614C22—182no—
615C22—184no—
616C22—182no—
617C22D3×#62411yescharacteristic
618C22D3×#62512yes—
619C22#8895211yescharacteristic
620C22#8895511yescharacteristic
621C22#8896012yes—
622C4—64no—
623C4—362no—
624C4—542no—
625C3—21no—
626C3—22no—
627C3—41no—
628C3#1059138112yescomplemented
629C3#1059172211yescharacteristic, complemented
630C2—122no—
631C2—184no—
632C2#1172290411yescharacteristic
633C2#1172293012yes—
634C2#1172294611yescharacteristic
635C2#1172295712yes—
636C2#1172299211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, cosocleC23×C33C23
prime radicalC22×#6290C2
hypocenterC33#386
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3×#624C2semidirect
3. C2×#88814C2semidirect
4. C2×C3×(C2×D3)⋊Dic3C2semidirect
5. #11722061C2semidirect
15. C2×#6290C4semidirect
24. C22×#624D3semidirect
29. C2×(C2×D3)⋊Dic3D3semidirect
44. C22×C32×D3C2×C4semidirect
47. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
53. C2×C32×Dic3D4semidirect
105. C2×C3×C3⋊D3C22⋊C4semidirect
166. C22×C3×D3C4×D3semidirect
205. C2×C3×Dic3#67semidirect
206. C2×C3×Dic3#67semidirect
295. C2×C3⋊D3(C2×D3)⋊C4semidirect
338. C33#386semidirect
377. C22×D3#624semidirect
420. C2×Dic3#625semidirect
548. C32#4956semidirect
549. C32#5213semidirect
628. C3#10591381semidirect
629. C3#10591722semidirect

Maximal subgroups

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

subgroupindex
C22×#62902
C22×C3×#6242
C2×#888142
C2×C3×(C2×D3)⋊Dic32
#117220612
#105917223
#105913813