Finite Group Databasebeta

← #13177540

Subgroups — #13177540

Order 1728.

2388 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C22×#6290C211yesmaximal, characteristic, complemented
3#11723976C211yesmaximal, characteristic, complemented
4C32×#937C211yesmaximal, characteristic, complemented
5#11723634C212yesmaximal, complemented
6#11723631C212yesmaximal, complemented
7#11723629C211yesmaximal, characteristic, complemented
8C3⋊D3×#217C211yesmaximal, characteristic, complemented
9#11723609C211yesmaximal, characteristic, complemented
10#11723519C212yesmaximal, complemented
11#11723518C212yesmaximal, complemented
12#10596060—31nomaximal
13#10595230—34nomaximal
14C23×C3×C3⋊D3C2211yescharacteristic
15C23×C3×C3⋊D3C2211yescharacteristic, complemented
16C23×C32×D3C2211yescharacteristic, complemented
17C2×C33×D4C2211yescharacteristic
18C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
19C2×C3×#638C2211yescharacteristic
20C2×C3×#638C2211yescharacteristic, complemented
21C3×D4×C3⋊D3—24no—
22C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
23C2×C3×C4×C3⋊D3C2211yescharacteristic, complemented
24C2×C32×#67C2212yescomplemented
25C2×#6290—24no—
26C2×#6290C2212yes—
27C2×#6290C2212yescomplemented
28C2×#6290C2212yescomplemented
29C2×Dic3×C3⋊D3C2212yescomplemented
30#88864C2211yescharacteristic, complemented
31#88864C2212yescomplemented
32#88857C2212yescomplemented
33C32×#217C2211yescharacteristic, complemented
34C32×(C2×D3)⋊C4C2212yescomplemented
35Dic3⋊(C3⋊Dic3)C2212yescomplemented
36#88803C2212yescomplemented
37#88803C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C22×#638—31no—
40C2×D4×C3⋊D3D311yescharacteristic, complemented
41C2×C3×D3×D4—34no—
42C22×#625—34no—
43C32×#120—31no—
44#64162—31no—
45#64147—32no—
46#64146—32no—
47#64144—31no—
48#64144—34no—
49#64144—64no—
50#64095—31no—
51#64093—32no—
52#64092—32no—
53C22⋊C4×C3⋊D3—31no—
54C3×#937D314yescomplemented
55#63976—38no—
56#63976—34no—
57#63970—38no—
58D3×#217—34no—
59#63962—34no—
60#63905—38no—
61#63885—38no—
62C23×C33C2311yescharacteristic
63C23×C33C2311yescharacteristic
64C22×C3×C3⋊D3—22no—
65C22×C3×C3⋊D3—42no—
66C22×C3×C3⋊D3—42no—
67C22×C3×C3⋊D3D414yescomplemented
68C22×C3×C3⋊D3C2311yescharacteristic
69C22×C3×C3⋊D3C2311yescharacteristic
70C22×C3×C3⋊D3C2311yescharacteristic
71C22×C32×D3—22no—
72C22×C32×D3—42no—
73C22×C32×D3C2312yes—
74C33×D4—22no—
75C2×C33×C4C2311yescharacteristic
76C2×C3⋊(C3⋊Dic3)—22no—
77C2×C3⋊(C3⋊Dic3)C2312yes—
78C2×C3⋊(C3⋊Dic3)C2312yes—
79C3×#638—22no—
80C3×#638—22no—
81C2×C3×C3⋊Dic3C2311yescharacteristic
82C3×C4⋊(C3⋊D3)—22no—
83C3×C4×C3⋊D3—22no—
84C32×#67—42no—
85C2×C32×Dic3C2312yes—
86#6290—24no—
87#6290—24no—
88#6290—42no—
89Dic3×C3⋊D3—42no—
90#5804—91no—
91#5589—94no—
92C24×C32—31no—
93C23×C3⋊D3C2×D311yescharacteristic
94C23×C3⋊D3C2×D311yescharacteristic, complemented
95C23×C3×D3—34no—
96C23×C3×D3—34no—
97C23×C3×D3C2×D314yescomplemented
98C2×C32×D4—24no—
99C2×C32×D4—32no—
100C2×C32×D4C2×D314yes—
101C2×C32×D4C2×D311yescharacteristic
102C2×#638—32no—
103C2×#638—32no—
104C2×#638—32no—
105C2×#638—64no—
106C2×#638C2×D311yescharacteristic
107C2×#638C2×D311yescharacteristic, complemented
108C22×C3⋊Dic3—31no—
109C22×C3⋊Dic3—34no—
110C22×C3⋊Dic3—64no—
111D4×C3⋊D3—24no—
112C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
113C2×C4×C3⋊D3—32no—
114C2×C4×C3⋊D3C2×D311yescharacteristic, complemented
115C2×C3×#67—34no—
116C2×C3×#67—34no—
117C2×C3×#67C2×D318yescomplemented
118C3×D3×D4—616no—
119C2×C3×D12—34no—
120C2×C3×C4×D3—34no—
121C2×#625—38no—
122C2×#625—38no—
123C2×#625—38no—
124C2×#625—616no—
125C2×D3×Dic3—38no—
126C32×C22⋊C4—31no—
127C32×C22⋊C4—32no—
128#3757—31no—
129#3757—32no—
130#3757—34no—
131#3757—38no—
132#3757—32no—
133#3757—64no—
134#3757—68no—
135#3752—32no—
136#3752—32no—
137C4⋊(C3⋊Dic3)—32no—
138#3750—32no—
139#3750—38no—
140#3750—68no—
141C3×#217C2×D314yescomplemented
142C3×(C2×D3)⋊C4C2×D318yescomplemented
143Dic3⋊Dic3—38no—
144(C2×D3)⋊Dic3—38no—
145(C2×D3)⋊Dic3—38no—
146(C2×D3)⋊Dic3—38no—
147C22×C33—22no—
148C22×C33—21no—
149C22×C33—22no—
150C22×C33C2×D412yes—
151C22×C33C2411yescharacteristic
152C2×C3×C3⋊D3—22no—
153C2×C3×C3⋊D3—22no—
154C2×C3×C3⋊D3—21no—
155C2×C3×C3⋊D3—22no—
156C2×C3×C3⋊D3—44no—
157C2×C3×C3⋊D3—44no—
158C2×C3×C3⋊D3C2×D412yes—
159C2×C3×C3⋊D3C2×D412yes—
160C2×C32×D3—22no—
161C2×C32×D3—22no—
162C2×C32×D3—22no—
163C2×C32×D3—44no—
164C33×C4—21no—
165C3⋊(C3⋊Dic3)—22no—
166C3⋊(C3⋊Dic3)—22no—
167C3×C3⋊Dic3—21no—
168C32×Dic3—22no—
169C22×C3×D4—91no—
170C22×#67—94no—
171C22×#67—91no—
172C2×D3×D4—34no—
173C3×#120—34no—
174#953—94no—
175#953—92no—
176C2×#217—91no—
177#939—92no—
178#939—92no—
179#938—98no—
180#937—98no—
181#937—91no—
182#937C3⋊D311yescharacteristic, complemented
183#935—94no—
184#935—91no—
185#935—91no—
186#935—184no—
187#929—92no—
188#886—94no—
189#884—98no—
190#883—98no—
191D3×C22⋊C4—94no—
192C23×C32—24no—
193C23×C32—24no—
194C23×C32—32no—
195C23×C32—62no—
196C23×C32—122no—
197C23×C32C22×D314yes—
198C23×C32C22×D314yes—
199C23×C32C22×D311yescharacteristic
200C23×C32C22×D311yescharacteristic
201C22×C3⋊D3—22no—
202C22×C3⋊D3—42no—
203C22×C3⋊D3—42no—
204C22×C3⋊D3#6714yescomplemented
205C22×C3⋊D3C22×D311yescharacteristic
206C22×C3⋊D3C22×D311yescharacteristic
207C22×C3⋊D3C22×D311yescharacteristic
208C22×C3×D3—28no—
209C22×C3×D3—34no—
210C22×C3×D3—34no—
211C22×C3×D3—316no—
212C22×C3×D3—34no—
213C22×C3×D3—48no—
214C22×C3×D3—68no—
215C22×C3×D3—128no—
216C22×C3×D3—128no—
217C22×C3×D3C22×D318yes—
218C32×D4—22no—
219C32×D4—28no—
220C32×D4—48no—
221C32×D4—122no—
222C2×C32×C4—24no—
223C2×C32×C4—32no—
224C2×C32×C4C22×D314yes—
225C2×C32×C4C22×D311yescharacteristic
226#638—22no—
227#638—22no—
228#638—64no—
229#638—64no—
230#638—122no—
231C2×C3⋊Dic3—32no—
232C2×C3⋊Dic3—32no—
233C2×C3⋊Dic3—38no—
234C2×C3⋊Dic3—38no—
235C2×C3⋊Dic3—68no—
236C2×C3⋊Dic3—68no—
237C2×C3⋊Dic3—62no—
238C2×C3⋊Dic3—68no—
239C2×C3⋊Dic3—128no—
240C2×C3⋊Dic3C22×D311yescharacteristic
241C4⋊(C3⋊D3)—22no—
242C4×C3⋊D3—22no—
243C4×C3⋊D3—122no—
244C3×#67—48no—
245C3×#67—68no—
246C3×#67—68no—
247C2×C3×Dic3—34no—
248C2×C3×Dic3C22×D318yes—
249C3×D12—68no—
250C3×C4×D3—68no—
251#625—616no—
252#625—616no—
253#625—128no—
254D3×Dic3—128no—
255#534—271noSylow
256C2×C33—22no—
257C2×C33—41no—
258C2×C33C22×D411yescharacteristic
259C2×C33H(5,𝔽2)12yes—
260C3×C3⋊D3—22no—
261C3×C3⋊D3—22no—
262C3×C3⋊D3—41no—
263C32×D3—42no—
264C24×C3—34no—
265C24×C3—91no—
266C24×C3—91no—
267C23×D3—34no—
268C23×D3—34no—
269C23×D3C2×C3⋊D311yescharacteristic, complemented
270C2×C3×D4—24no—
271C2×C3×D4—38no—
272C2×C3×D4—91no—
273C2×C3×D4—91no—
274C2×C3×D4—91no—
275C2×C3×D4—184no—
276C2×C3×D4D3214yes—
277C2×C3×D4C2×C3⋊D311yescharacteristic
278C22×C3×C4—91no—
279C2×#67—34no—
280C2×#67—34no—
281C2×#67—98no—
282C2×#67—98no—
283C2×#67—92no—
284C2×#67—92no—
285C2×#67—98no—
286C2×#67—92no—
287C2×#67—184no—
288C2×#67—1816no—
289C2×#67C2×C3⋊D312yescomplemented
290C22×Dic3—94no—
291C22×Dic3—91no—
292C22×Dic3—92no—
293C22×Dic3—184no—
294D3×D4—616no—
295C2×D12—34no—
296C2×C4×D3—34no—
297C2×C4×D3—98no—
298C3×C22⋊C4—34no—
299C3×C22⋊C4—38no—
300#217—98no—
301#217—94no—
302#217—98no—
303#217—91no—
304#217—92no—
305#217—92no—
306#217—92no—
307#217—184no—
308#217—188no—
309#217C2×C3⋊D311yescharacteristic, complemented
310(C2×D3)⋊C4—98no—
311(C2×D3)⋊C4—98no—
312(C2×D3)⋊C4—92no—
313(C2×D3)⋊C4C2×C3⋊D312yescomplemented
314C4⋊Dic3—98no—
315Dic3⋊C4—98no—
316Dic3⋊C4—92no—
317Dic3⋊C4—92no—
318Dic3⋊C4—188no—
319C22×C32—22no—
320C22×C32—22no—
321C22×C32—21no—
322C22×C32—24no—
323C22×C32—28no—
324C22×C32—28no—
325C22×C32—24no—
326C22×C32—28no—
327C22×C32—48no—
328C22×C32—44no—
329C22×C32—48no—
330C22×C32—62no—
331C22×C32—62no—
332C22×C32—62no—
333C22×C32—124no—
334C22×C32D3×D418yes—
335C22×C32C2×#6712yes—
336C22×C32C23×D314yes—
337C22×C32C23×D311yescharacteristic
338C2×C3⋊D3—22no—
339C2×C3⋊D3—22no—
340C2×C3⋊D3—22no—
341C2×C3⋊D3—21no—
342C2×C3⋊D3—44no—
343C2×C3⋊D3—44no—
344C2×C3⋊D3C2×#6712yes—
345C2×C3⋊D3C2×#6712yes—
346C2×C3×D3—28no—
347C2×C3×D3—28no—
348C2×C3×D3—28no—
349C2×C3×D3—38no—
350C2×C3×D3—38no—
351C2×C3×D3—416no—
352C2×C3×D3—68no—
353C2×C3×D3—68no—
354C2×C3×D3—64no—
355C2×C3×D3—68no—
356C2×C3×D3—1216no—
357C2×C3×D3—1216no—
358C32×C4—21no—
359C32×C4—24no—
360C32×C4—44no—
361C32×C4—62no—
362C3⋊Dic3—21no—
363C3⋊Dic3—62no—
364C3⋊Dic3—62no—
365C3⋊Dic3—68no—
366C3⋊Dic3—68no—
367C3⋊Dic3—128no—
368C3⋊Dic3—128no—
369C3×Dic3—28no—
370C3×Dic3—64no—
371C22×D4—91no—
372C22×D4—271no—
373#123—272no—
374#123—271no—
375#123—271no—
376#121—272no—
377#121—272no—
378#120—31no—
379#120—272no—
380#120—271no—
381C2×C22⋊C4—271no—
382C33#53411yescharacteristic, complemented, Sylow
383C23×C3—24no—
384C23×C3—24no—
385C23×C3—38no—
386C23×C3—68no—
387C23×C3—91no—
388C23×C3—91no—
389C23×C3—94no—
390C23×C3—91no—
391C23×C3—128no—
392C23×C3—182no—
393C23×C3—362no—
394C23×C3—362no—
395C23×C3C2×D3214yes—
396C23×C3C2×D3214yes—
397C23×C3C22×C3⋊D311yescharacteristic
398C23×C3C22×C3⋊D311yescharacteristic
399C22×D3—22no—
400C22×D3—34no—
401C22×D3—34no—
402C22×D3—316no—
403C22×D3—34no—
404C22×D3—42no—
405C22×D3—68no—
406C22×D3—128no—
407C22×D3—128no—
408C22×D3C22×C3⋊D312yes—
409C3×D4—28no—
410C3×D4—22no—
411C3×D4—48no—
412C3×D4—128no—
413C3×D4—182no—
414C3×D4—182no—
415C3×D4—182no—
416C2×C3×C4—24no—
417C2×C3×C4—38no—
418C2×C3×C4—91no—
419C2×C3×C4—182no—
420C2×C3×C4C2×D3214yes—
421C2×C3×C4C22×C3⋊D311yescharacteristic
422#67—42no—
423#67—68no—
424#67—68no—
425#67—184no—
426#67—184no—
427#67—1816no—
428#67—1816no—
429#67—362no—
430#67—368no—
431C2×Dic3—34no—
432C2×Dic3—98no—
433C2×Dic3—98no—
434C2×Dic3—92no—
435C2×Dic3—92no—
436C2×Dic3—188no—
437C2×Dic3—188no—
438C2×Dic3—188no—
439C2×Dic3—182no—
440C2×Dic3—368no—
441C2×Dic3—362no—
442C2×Dic3C22×C3⋊D312yes—
443D12—68no—
444C4×D3—68no—
445C4×D3—368no—
446C2×C32—22no—
447C2×C32—28no—
448C2×C32—24no—
449C2×C32—28no—
450C2×C32—48no—
451C2×C32—41no—
452C2×C32—44no—
453C2×C32—84no—
454C2×C32—122no—
455C2×C32C2×D3×D414yes—
456C2×C32#100418yes—
457C2×C32#100412yes—
458C2×C32C22×#6711yescharacteristic
459C3⋊D3—22no—
460C3⋊D3—22no—
461C3⋊D3—41no—
462C3×D3—48no—
463C3×D3—68no—
464C3×D3—68no—
465C3×D3—124no—
466C24—31no—
467C24—91no—
468C24—91no—
469C2×D4—32no—
470C2×D4—91no—
471C2×D4—91no—
472C2×D4—91no—
473C2×D4—184no—
474C2×D4—272no—
475C2×D4—272no—
476C2×D4—272no—
477C2×D4—544no—
478C2×D4D3×C3⋊D311yescharacteristic
479C22×C4—91no—
480C22×C4—272no—
481C22×C4—271no—
482C4⋊C4—272no—
483C4⋊C4—272no—
484C22⋊C4—31no—
485C22⋊C4—32no—
486C22⋊C4—272no—
487C22⋊C4—272no—
488C22⋊C4—272no—
489C22⋊C4—271no—
490C22⋊C4—272no—
491C22×C3—28no—
492C22×C3—22no—
493C22×C3—28no—
494C22×C3—24no—
495C22×C3—21no—
496C22×C3—22no—
497C22×C3—24no—
498C22×C3—28no—
499C22×C3—48no—
500C22×C3—44no—
501C22×C3—48no—
502C22×C3—68no—
503C22×C3—68no—
504C22×C3—68no—
505C22×C3—92no—
506C22×C3—92no—
507C22×C3—1216no—
508C22×C3—182no—
509C22×C3—182no—
510C22×C3—181no—
511C22×C3—182no—
512C22×C3—364no—
513C22×C3—364no—
514C22×C3D3×#6718yes—
515C22×C3D4×C3⋊D312yes—
516C22×C3C22×D3214yes—
517C22×C3C23×C3⋊D311yescharacteristic
518C2×D3—22no—
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—38no—
522C2×D3—38no—
523C2×D3—44no—
524C2×D3—68no—
525C2×D3—68no—
526C2×D3—68no—
527C2×D3—64no—
528C2×D3—1216no—
529C2×D3—1216no—
530C3×C4—24no—
531C3×C4—21no—
532C3×C4—44no—
533C3×C4—68no—
534C3×C4—181no—
535Dic3—22no—
536Dic3—64no—
537Dic3—188no—
538Dic3—188no—
539Dic3—182no—
540Dic3—182no—
541Dic3—368no—
542Dic3—368no—
543C32—24no—
544C32#558914yescomplemented
545C32#580411yescharacteristic, complemented
546C23—32no—
547C23—62no—
548C23—91no—
549C23—91no—
550C23—94no—
551C23—91no—
552C23—122no—
553C23—182no—
554C23—362no—
555C23—362no—
556C23C2×D3×C3⋊D311yescharacteristic
557C23C2×D3×C3⋊D311yescharacteristic
558D4—22no—
559D4—122no—
560D4—182no—
561D4—182no—
562D4—182no—
563D4—544no—
564D4—544no—
565D4—1082no—
566C2×C4—32no—
567C2×C4—91no—
568C2×C4—182no—
569C2×C4—272no—
570C2×C4—272no—
571C2×C4—542no—
572C2×C4—1082no—
573C2×C4C2×D3×C3⋊D311yescharacteristic
574C2×C3—28no—
575C2×C3—22no—
576C2×C3—28no—
577C2×C3—24no—
578C2×C3—48no—
579C2×C3—44no—
580C2×C3—41no—
581C2×C3—84no—
582C2×C3—128no—
583C2×C3—182no—
584C2×C3—182no—
585C2×C3—361no—
586C2×C3#6431318yes—
587C2×C3C2×D3×#6714yes—
588C2×C3C2×D4×C3⋊D311yescharacteristic
589C2×C3#6436112yes—
590D3—42no—
591D3—68no—
592D3—68no—
593D3—124no—
594C22—22no—
595C22—22no—
596C22—21no—
597C22—62no—
598C22—62no—
599C22—62no—
600C22—92no—
601C22—92no—
602C22—124no—
603C22—182no—
604C22—182no—
605C22—182no—
606C22—181no—
607C22—364no—
608C22—364no—
609C22C3⋊D3×#6712yes—
610C22C22×D3×C3⋊D311yescharacteristic
611C4—21no—
612C4—62no—
613C4—181no—
614C4—542no—
615C4—542no—
616C3—24no—
617C3#1059523014yescomplemented
618C3#1059606011yescharacteristic, complemented
619C2—22no—
620C2—41no—
621C2—122no—
622C2—182no—
623C2—182no—
624C2—361no—
625C2#1172458712yes—
626C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#534
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×C3⋊D3C2semidirect
2. C22×#6290C2semidirect
3. #11723976C2semidirect
4. C32×#937C2semidirect
5. #11723634C2semidirect
6. #11723631C2semidirect
7. #11723629C2semidirect
8. C3⋊D3×#217C2semidirect
9. #11723609C2semidirect
10. #11723519C2semidirect
11. #11723518C2semidirect
15. C23×C3×C3⋊D3C22semidirect
16. C23×C32×D3C22semidirect
18. C22×C3⋊(C3⋊Dic3)C22semidirect
20. C2×C3×#638C22semidirect
22. C2×C3×C4⋊(C3⋊D3)C22semidirect
23. C2×C3×C4×C3⋊D3C22semidirect
24. C2×C32×#67C22semidirect
27. C2×#6290C22semidirect
28. C2×#6290C22semidirect
29. C2×Dic3×C3⋊D3C22semidirect
30. #88864C22semidirect
31. #88864C22semidirect
32. #88857C22semidirect
33. C32×#217C22semidirect
34. C32×(C2×D3)⋊C4C22semidirect
35. Dic3⋊(C3⋊Dic3)C22semidirect
36. #88803C22semidirect
37. #88803C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
40. C2×D4×C3⋊D3D3semidirect
54. C3×#937D3semidirect
67. C22×C3×C3⋊D3D4semidirect
94. C23×C3⋊D3C2×D3semidirect
97. C23×C3×D3C2×D3semidirect
107. C2×#638C2×D3semidirect
112. C2×C4⋊(C3⋊D3)C2×D3semidirect
114. C2×C4×C3⋊D3C2×D3semidirect
117. C2×C3×#67C2×D3semidirect
141. C3×#217C2×D3semidirect
142. C3×(C2×D3)⋊C4C2×D3semidirect
182. #937C3⋊D3semidirect
204. C22×C3⋊D3#67semidirect
269. C23×D3C2×C3⋊D3semidirect
289. C2×#67C2×C3⋊D3semidirect
309. #217C2×C3⋊D3semidirect
313. (C2×D3)⋊C4C2×C3⋊D3semidirect
382. C33#534semidirect
544. C32#5589semidirect
545. C32#5804semidirect
617. C3#10595230semidirect
618. C3#10596060semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C22×#62902
#117239762
C32×#9372
#117236342
#117236312
#117236292
C3⋊D3×#2172
#117236092
#117235192
#117235182
#105960603
#105952303