Finite Group Databasebeta

← #13177302

Subgroups — #13177302

Order 1728.

2380 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic, complemented
2C2×C3⋊D3×D12C211yesmaximal, characteristic, complemented
3#11723971C211yesmaximal, characteristic, complemented
4C32×#930C211yesmaximal, characteristic, complemented
5#11723643C212yesmaximal, complemented
6#11723633C212yesmaximal, complemented
7#11723541C212yesmaximal, complemented
8#11723526C211yesmaximal, characteristic, complemented
9#11723522C211yesmaximal, characteristic, complemented
10#11723521C212yesmaximal, complemented
11#11723404C211yesmaximal, characteristic, complemented
12#10596078—31nomaximal
13#10595064—34nomaximal
14C22×D3×C3⋊D3C2212yes—
15C22×C33×C4C2211yescharacteristic, complemented
16C2×#6312C2212yescomplemented
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18C2×C3×#638C2211yescharacteristic
19C2×C3×#638C2211yescharacteristic, complemented
20C3×#3828—24no—
21C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
22C2×C3×C4×C3⋊D3C2211yescharacteristic
23C2×C3×C4×C3⋊D3C2211yescharacteristic, complemented
24C2×C3×#634C2211yescharacteristic, complemented
25C2×C32×#67C2212yescomplemented
26C2×C32×D12C2211yescharacteristic
27C2×#6292C2212yescomplemented
28C2×#6291C2212yes—
29C2×#6290C2212yescomplemented
30C3⋊D3×D12—24no—
31#88859C2212yescomplemented
32C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
33C32×(C2×D3)⋊C4C2212yescomplemented
34C32×C4⋊Dic3C2211yescharacteristic, complemented
35#88804C2212yescomplemented
36#88803C2212yes—
37(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
38Dic3×C3⋊Dic3C2212yescomplemented
39C2×D4×C3⋊D3—31no—
40C2×#3828D311yescharacteristic, complemented
41C2×C3×#235—34no—
42C2×D3×D12—34no—
43C32×#121—31no—
44#64149—32no—
45#64147—31no—
46#64146—32no—
47#64139—31no—
48#64139—34no—
49#64139—64no—
50#64104—31no—
51#64101—31no—
52#64094—32no—
53#64091—32no—
54C3×#930D314yescomplemented
55#63983—38no—
56#63979—38no—
57#63922—38no—
58#63909—34no—
59#63908—34no—
60#63907—38no—
61#63841—34no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)C2312yes—
64C22×C3×C3⋊D3C2311yescharacteristic
65C22×C3×C3⋊D3C2311yescharacteristic
66C22×C32×D3C2312yes—
67C2×D3×C3⋊D3—24no—
68C2×D3×C3⋊D3—42no—
69C2×D3×C3⋊D3—42no—
70C2×C33×C4—22no—
71C2×C33×C4C2311yescharacteristic
72C2×C33×C4C2311yescharacteristic
73#6312—42no—
74C2×C3⋊(C3⋊Dic3)C2312yes—
75C4⋊(C3⋊(C3⋊D3))—22no—
76C3×#638—22no—
77C3×#638—22no—
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C2×C3×C3⋊Dic3C2311yescharacteristic
80C3×C4⋊(C3⋊D3)—22no—
81C3×C4×C3⋊D3—22no—
82C3×C4×C3⋊D3D414yescomplemented
83C3×#634—22no—
84C32×#67—42no—
85C2×C32×Dic3C2312yes—
86C32×D12—22no—
87#6292—42no—
88#6291—24no—
89#6290—42no—
90#5830—91no—
91#5607—94no—
92C23×C3⋊D3—32no—
93C22×D32—38no—
94C2×C32×D4—32no—
95C2×C32×D4—31no—
96C22×C32×C4—24no—
97C22×C32×C4C2×D314yescomplemented
98C22×C32×C4C2×D311yescharacteristic, complemented
99C2×#638—32no—
100C2×#638—38no—
101C2×#638—32no—
102C2×#638—32no—
103C2×#638—68no—
104C2×#638C2×D311yescharacteristic
105C2×#638C2×D311yescharacteristic, complemented
106D4×C3⋊D3—64no—
107#3828—24no—
108C2×C4⋊(C3⋊D3)—31no—
109C2×C4⋊(C3⋊D3)—34no—
110C2×C4⋊(C3⋊D3)—32no—
111C2×C4⋊(C3⋊D3)—64no—
112C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
113C2×C4×C3⋊D3C2×D311yescharacteristic
114C2×C4×C3⋊D3C2×D311yescharacteristic, complemented
115C2×#634C2×D311yescharacteristic, complemented
116C2×C3×#67—34no—
117C2×C3×#67—34no—
118C2×C3×#67C2×D318yescomplemented
119C3×#235—616no—
120C2×C3×D12—34no—
121C2×C3×D12C2×D314yes—
122C2×C3×C4×D3—34no—
123C2×C3×C4×D3—34no—
124C2×C3×Dic6—34no—
125C2×#626—38no—
126C2×#626—38no—
127C2×#625—38no—
128D3×D12—616no—
129C32×C4⋊C4—31no—
130C32×C22⋊C4—32no—
131#3757—32no—
132#3752—32no—
133#3752—32no—
134#3752—38no—
135#3752—32no—
136#3752—68no—
137C4⋊(C3⋊Dic3)—31no—
138C4⋊(C3⋊Dic3)—34no—
139C4⋊(C3⋊Dic3)—64no—
140C4×C3⋊Dic3—32no—
141C3×(C2×D3)⋊C4C2×D318yescomplemented
142C3×C4⋊Dic3C2×D314yescomplemented
143#3722—38no—
144(C2×D3)⋊Dic3—38no—
145(C2×D3)⋊Dic3—38no—
146Dic32—38no—
147C22×C33—21no—
148C22×C33—21no—
149C22×C33—21no—
150C22×C33C2411yescharacteristic
151C2×C3⋊(C3⋊D3)—22no—
152C2×C3⋊(C3⋊D3)—22no—
153C2×C3⋊(C3⋊D3)—22no—
154C2×C3×C3⋊D3—21no—
155C2×C3×C3⋊D3—21no—
156C2×C3×C3⋊D3—21no—
157C2×C3×C3⋊D3—21no—
158C2×C3×C3⋊D3—21no—
159C2×C3×C3⋊D3C2×D412yes—
160C2×C32×D3—22no—
161C2×C32×D3—22no—
162C2×C32×D3—22no—
163D3×C3⋊D3—48no—
164C33×C4—21no—
165C33×C4C2×D412yes—
166C3⋊(C3⋊Dic3)—22no—
167C3×C3⋊Dic3—21no—
168C3×C3⋊Dic3C2×D412yes—
169C32×Dic3—22no—
170C2×C3×#41—91no—
171C2×D3×D4—94no—
172C2×#235—34no—
173C22×D12—91no—
174C3×#121—34no—
175#947—92no—
176#940—98no—
177#940—92no—
178#938—94no—
179#937—98no—
180#937—92no—
181#937—92no—
182#930—94no—
183#930—91no—
184#930—91no—
185#930—91no—
186#930—184no—
187#930C3⋊D311yescharacteristic, complemented
188#926—91no—
189#895—94no—
190#892—94no—
191#885—98no—
192#882—98no—
193C23×C32—24no—
194C23×C32—32no—
195C23×C32C22×D314yes—
196C23×C32C22×D311yescharacteristic
197C22×C3⋊D3—32no—
198C22×C3⋊D3—38no—
199C22×C3⋊D3—68no—
200C22×C3⋊D3—64no—
201C22×C3⋊D3—122no—
202C22×C3⋊D3—122no—
203C22×C3⋊D3C22×D311yescharacteristic
204C22×C3⋊D3C22×D311yescharacteristic
205C22×C3×D3—34no—
206C22×C3×D3—34no—
207C22×C3×D3C22×D318yes—
208C2×D32—616no—
209C2×D32—128no—
210C2×D32—128no—
211C32×D4—62no—
212C32×D4—122no—
213C2×C32×C4—22no—
214C2×C32×C4—24no—
215C2×C32×C4—24no—
216C2×C32×C4—28no—
217C2×C32×C4—32no—
218C2×C32×C4—48no—
219C2×C32×C4C22×D314yes—
220C2×C32×C4C22×D314yes—
221C2×C32×C4C22×D311yescharacteristic
222C2×C32×C4C22×D311yescharacteristic
223#638—22no—
224#638—22no—
225#638—64no—
226#638—122no—
227#638—122no—
228#638—128no—
229#638—1216no—
230C2×C3⋊Dic3—32no—
231C2×C3⋊Dic3—38no—
232C2×C3⋊Dic3—68no—
233C2×C3⋊Dic3C22×D311yescharacteristic
234C2×C3⋊Dic3C22×D311yescharacteristic
235C4⋊(C3⋊D3)—22no—
236C4⋊(C3⋊D3)—62no—
237C4⋊(C3⋊D3)—68no—
238C4⋊(C3⋊D3)—128no—
239C4⋊(C3⋊D3)—122no—
240C4×C3⋊D3—22no—
241C4×C3⋊D3#6714yescomplemented
242#634—22no—
243C3×#67—48no—
244C3×#67—68no—
245C3×#67—68no—
246C2×C3×Dic3—34no—
247C2×C3×Dic3—34no—
248C2×C3×Dic3C22×D318yes—
249C3×D12—28no—
250C3×D12—68no—
251C3×C4×D3—316no—
252C3×C4×D3—68no—
253C3×Dic6—68no—
254#626—616no—
255#626—128no—
256#625—128no—
257#535—271noSylow
258C2×C33—41no—
259C2×C33C22×D411yescharacteristic
260C2×C33H(5,𝔽2)11yescharacteristic
261C2×C33H(5,𝔽2)11yescharacteristic
262C3⋊(C3⋊D3)—42no—
263C3×C3⋊D3—22no—
264C3×C3⋊D3—41no—
265C32×D3—42no—
266C23×D3—98no—
267C23×D3—92no—
268C3×#41—184no—
269C2×C3×Q8—91no—
270C2×C3×D4—38no—
271C2×C3×D4—34no—
272C2×C3×D4—91no—
273C2×C3×D4—91no—
274C2×C3×D4—91no—
275C22×C3×C4—24no—
276C22×C3×C4—91no—
277C22×C3×C4—91no—
278C22×C3×C4D3214yescomplemented
279C22×C3×C4C2×C3⋊D311yescharacteristic, complemented
280C2×#67—34no—
281C2×#67—34no—
282C2×#67—98no—
283C2×#67—98no—
284C2×#67—98no—
285C2×#67—92no—
286C2×#67—92no—
287C2×#67—92no—
288C2×#67—188no—
289C2×#67C2×C3⋊D312yescomplemented
290D3×D4—1816no—
291#235—616no—
292C2×D12—34no—
293C2×D12—92no—
294C2×D12—94no—
295C2×D12—91no—
296C2×D12—98no—
297C2×D12—184no—
298C2×D12—184no—
299C2×D12C2×C3⋊D311yescharacteristic
300C2×C4×D3—34no—
301C2×C4×D3—34no—
302C2×Dic6—34no—
303C3×C4⋊C4—34no—
304C3×C22⋊C4—38no—
305#217—98no—
306#217—92no—
307(C2×D3)⋊C4—98no—
308(C2×D3)⋊C4—98no—
309(C2×D3)⋊C4—92no—
310(C2×D3)⋊C4—98no—
311(C2×D3)⋊C4—92no—
312(C2×D3)⋊C4—92no—
313(C2×D3)⋊C4—188no—
314(C2×D3)⋊C4C2×C3⋊D312yescomplemented
315C4⋊Dic3—94no—
316C4⋊Dic3—91no—
317C4⋊Dic3—184no—
318C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
319C4×Dic3—98no—
320C4×Dic3—92no—
321C22×C32—28no—
322C22×C32—24no—
323C22×C32—21no—
324C22×C32—21no—
325C22×C32—21no—
326C22×C32—24no—
327C22×C32—24no—
328C22×C32—24no—
329C22×C32—44no—
330C22×C32—44no—
331C22×C32—62no—
332C22×C32—62no—
333C22×C32—62no—
334C22×C32C23×D314yes—
335C22×C32C23×D311yescharacteristic
336C2×C3⋊D3—21no—
337C2×C3⋊D3—21no—
338C2×C3⋊D3—21no—
339C2×C3⋊D3—21no—
340C2×C3⋊D3—21no—
341C2×C3⋊D3—62no—
342C2×C3⋊D3—62no—
343C2×C3⋊D3—62no—
344C2×C3⋊D3—68no—
345C2×C3⋊D3—68no—
346C2×C3⋊D3—68no—
347C2×C3⋊D3—128no—
348C2×C3⋊D3—128no—
349C2×C3⋊D3—128no—
350C2×C3⋊D3—128no—
351C2×C3⋊D3C2×#6712yes—
352C2×C3×D3—28no—
353C2×C3×D3—28no—
354C2×C3×D3—28no—
355C2×C3×D3—38no—
356C2×C3×D3—64no—
357C2×C3×D3—64no—
358C2×C3×D3—64no—
359C2×C3×D3—64no—
360C2×C3×D3—64no—
361D32—1232no—
362C32×C4—21no—
363C32×C4—28no—
364C32×C4—24no—
365C32×C4—44no—
366C32×C4—62no—
367C32×C4D3×D418yes—
368C32×C4C2×#6712yes—
369C3⋊Dic3—21no—
370C3⋊Dic3—62no—
371C3⋊Dic3—68no—
372C3⋊Dic3—128no—
373C3⋊Dic3C2×#6712yes—
374C3×Dic3—28no—
375C3×Dic3—38no—
376C3×Dic3—64no—
377C2×#41—91no—
378C22×D4—271no—
379#127—272no—
380#124—272no—
381#121—31no—
382#121—271no—
383#121—271no—
384#121—271no—
385#120—272no—
386#120—272no—
387#117—271no—
388C33#53511yescharacteristic, complemented, Sylow
389C23×C3—24no—
390C23×C3—38no—
391C23×C3—91no—
392C23×C3—91no—
393C23×C3C2×D3214yes—
394C23×C3C22×C3⋊D311yescharacteristic
395C22×D3—34no—
396C22×D3—34no—
397C22×D3—98no—
398C22×D3—92no—
399C22×D3—188no—
400C22×D3—184no—
401C22×D3—1816no—
402C22×D3—362no—
403C22×D3—362no—
404C22×D3—368no—
405C22×D3—368no—
406C22×D3C22×C3⋊D312yes—
407C3×Q8—182no—
408C3×D4—68no—
409C3×D4—128no—
410C3×D4—182no—
411C3×D4—182no—
412C3×D4—182no—
413C2×C3×C4—28no—
414C2×C3×C4—22no—
415C2×C3×C4—24no—
416C2×C3×C4—24no—
417C2×C3×C4—38no—
418C2×C3×C4—48no—
419C2×C3×C4—91no—
420C2×C3×C4—94no—
421C2×C3×C4—91no—
422C2×C3×C4—182no—
423C2×C3×C4C2×D3214yes—
424C2×C3×C4C2×D3214yes—
425C2×C3×C4C22×C3⋊D311yescharacteristic
426C2×C3×C4C22×C3⋊D311yescharacteristic
427#67—42no—
428#67—68no—
429#67—68no—
430#67—1816no—
431#67—362no—
432#67—362no—
433#67—368no—
434#67—368no—
435#67—362no—
436#67—3616no—
437C2×Dic3—34no—
438C2×Dic3—34no—
439C2×Dic3—98no—
440C2×Dic3—92no—
441C2×Dic3—188no—
442C2×Dic3C22×C3⋊D312yes—
443D12—22no—
444D12—68no—
445D12—184no—
446D12—188no—
447D12—182no—
448D12—368no—
449D12—368no—
450C4×D3—316no—
451C4×D3—68no—
452Dic6—68no—
453C2×C32—24no—
454C2×C32—24no—
455C2×C32—24no—
456C2×C32—48no—
457C2×C32—41no—
458C2×C32—44no—
459C2×C32—122no—
460C2×C32C2×D3×D414yes—
461C2×C32#100414yes—
462C2×C32#100914yes—
463C2×C32#100911yescharacteristic
464C2×C32#100911yescharacteristic
465C2×C32C22×#6711yescharacteristic
466C3⋊D3—22no—
467C3⋊D3—41no—
468C3⋊D3—122no—
469C3⋊D3—128no—
470C3⋊D3—248no—
471C3×D3—48no—
472C3×D3—68no—
473C3×D3—124no—
474C24—272no—
475#41—184no—
476C2×Q8—91no—
477C2×D4—32no—
478C2×D4—31no—
479C2×D4—91no—
480C2×D4—91no—
481C2×D4—91no—
482C2×D4—272no—
483C2×D4—272no—
484C2×D4—272no—
485C2×D4—272no—
486C2×D4—271no—
487C2×D4—544no—
488C22×C4—91no—
489C22×C4—91no—
490C22×C4D3×C3⋊D311yescharacteristic, complemented
491C4⋊C4—31no—
492C4⋊C4—271no—
493C22⋊C4—32no—
494C22⋊C4—272no—
495C22⋊C4—272no—
496C22⋊C4—272no—
497C22⋊C4—272no—
498C42—272no—
499C22×C3—28no—
500C22×C3—24no—
501C22×C3—24no—
502C22×C3—24no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—24no—
507C22×C3—44no—
508C22×C3—44no—
509C22×C3—68no—
510C22×C3—68no—
511C22×C3—68no—
512C22×C3—92no—
513C22×C3—181no—
514C22×C3—181no—
515C22×C3—181no—
516C22×C3—181no—
517C22×C3—181no—
518C22×C3C22×D3214yes—
519C22×C3C23×C3⋊D311yescharacteristic
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—38no—
524C2×D3—64no—
525C2×D3—64no—
526C2×D3—64no—
527C2×D3—64no—
528C2×D3—64no—
529C2×D3—188no—
530C2×D3—188no—
531C2×D3—188no—
532C2×D3—182no—
533C2×D3—182no—
534C2×D3—182no—
535C2×D3—368no—
536C2×D3—368no—
537C2×D3—368no—
538C2×D3—368no—
539C2×D3—3632no—
540C3×C4—24no—
541C3×C4—21no—
542C3×C4—28no—
543C3×C4—44no—
544C3×C4—68no—
545C3×C4—92no—
546C3×C4—181no—
547C3×C4D3×#6718yes—
548C3×C4D4×C3⋊D312yes—
549Dic3—22no—
550Dic3—38no—
551Dic3—64no—
552Dic3—188no—
553Dic3—182no—
554Dic3—368no—
555C32—24no—
556C32#560714yescomplemented
557C32#583011yescharacteristic, complemented
558C23—32no—
559C23—91no—
560C23—91no—
561C23—272no—
562C23—544no—
563C23—1082no—
564C23—1082no—
565C23C2×D3×C3⋊D311yescharacteristic
566Q8—182no—
567D4—62no—
568D4—122no—
569D4—182no—
570D4—182no—
571D4—182no—
572D4—544no—
573D4—542no—
574D4—1082no—
575D4—1082no—
576D4—1082no—
577C2×C4—22no—
578C2×C4—32no—
579C2×C4—91no—
580C2×C4—94no—
581C2×C4—91no—
582C2×C4—182no—
583C2×C4—272no—
584C2×C4C2×D3×C3⋊D311yescharacteristic
585C2×C4C2×D3×C3⋊D311yescharacteristic
586C2×C3—24no—
587C2×C3—24no—
588C2×C3—24no—
589C2×C3—48no—
590C2×C3—41no—
591C2×C3—44no—
592C2×C3—128no—
593C2×C3—182no—
594C2×C3—361no—
595C2×C3#6430714yes—
596C2×C3#6430814yes—
597C2×C3C2×D3×#6714yes—
598C2×C3C2×D4×C3⋊D311yescharacteristic
599C2×C3#6436111yescharacteristic
600C2×C3#6436611yescharacteristic
601D3—42no—
602D3—68no—
603D3—124no—
604D3—362no—
605D3—368no—
606D3—728no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—62no—
611C22—62no—
612C22—62no—
613C22—92no—
614C22—181no—
615C22—181no—
616C22—181no—
617C22—181no—
618C22—181no—
619C22—542no—
620C22—542no—
621C22—542no—
622C22—1088no—
623C22C22×D3×C3⋊D311yescharacteristic
624C4—21no—
625C4—62no—
626C4—92no—
627C4—181no—
628C4—542no—
629C4C3⋊D3×#6712yes—
630C3—24no—
631C3#1059506414yescomplemented
632C3#1059607811yescharacteristic, complemented
633C2—41no—
634C2—122no—
635C2—182no—
636C2—361no—
637C2—1082no—
638C2#1172457911yescharacteristic
639C2#1172458111yescharacteristic
640C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3828C2semidirect
2. C2×C3⋊D3×D12C2semidirect
3. #11723971C2semidirect
4. C32×#930C2semidirect
5. #11723643C2semidirect
6. #11723633C2semidirect
7. #11723541C2semidirect
8. #11723526C2semidirect
9. #11723522C2semidirect
10. #11723521C2semidirect
11. #11723404C2semidirect
15. C22×C33×C4C22semidirect
16. C2×#6312C22semidirect
19. C2×C3×#638C22semidirect
23. C2×C3×C4×C3⋊D3C22semidirect
24. C2×C3×#634C22semidirect
25. C2×C32×#67C22semidirect
27. C2×#6292C22semidirect
29. C2×#6290C22semidirect
31. #88859C22semidirect
32. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
33. C32×(C2×D3)⋊C4C22semidirect
34. C32×C4⋊Dic3C22semidirect
35. #88804C22semidirect
37. (C2×D3)⋊(C3⋊Dic3)C22semidirect
38. Dic3×C3⋊Dic3C22semidirect
40. C2×#3828D3semidirect
54. C3×#930D3semidirect
82. C3×C4×C3⋊D3D4semidirect
97. C22×C32×C4C2×D3semidirect
98. C22×C32×C4C2×D3semidirect
105. C2×#638C2×D3semidirect
114. C2×C4×C3⋊D3C2×D3semidirect
115. C2×#634C2×D3semidirect
118. C2×C3×#67C2×D3semidirect
141. C3×(C2×D3)⋊C4C2×D3semidirect
142. C3×C4⋊Dic3C2×D3semidirect
187. #930C3⋊D3semidirect
241. C4×C3⋊D3#67semidirect
278. C22×C3×C4D32semidirect
279. C22×C3×C4C2×C3⋊D3semidirect
289. C2×#67C2×C3⋊D3semidirect
314. (C2×D3)⋊C4C2×C3⋊D3semidirect
318. C4⋊Dic3C2×C3⋊D3semidirect
388. C33#535semidirect
490. C22×C4D3×C3⋊D3semidirect
556. C32#5607semidirect
557. C32#5830semidirect
631. C3#10595064semidirect
632. C3#10596078semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C3⋊D3×D122
#117239712
C32×#9302
#117236432
#117236332
#117235412
#117235262
#117235222
#117235212
#117234042
#105960783
#105950643