Finite Group Databasebeta

← #13177305

Subgroups — #13177305

Order 1728.

2380 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic, complemented
2C2×D3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3#11723971C211yesmaximal, characteristic, complemented
4C32×C4×#67C211yesmaximal, characteristic, complemented
5#11723643C212yesmaximal, complemented
6#11723624C212yesmaximal, complemented
7C3⋊D3×(C2×D3)⋊C4C212yesmaximal, complemented
8Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
9#11723441C211yesmaximal, characteristic, complemented
10#11723425C211yesmaximal, characteristic, complemented
11#11723417C212yesmaximal, complemented
12#10596021—31nomaximal
13#10595067—34nomaximal
14C22×D3×C3⋊D3C2212yes—
15C22×C33×C4C2211yescharacteristic, complemented
16C2×#6312C2212yescomplemented
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18C2×C3×#638C2212yescomplemented
19C3×#3828—24no—
20C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C3×C4×C3⋊D3C2212yescomplemented
22C2×C3×#634C2211yescharacteristic, complemented
23C2×C32×#67C2211yescharacteristic
24C2×C32×C4×D3C2211yescharacteristic
25C2×#6292C2212yes—
26C2×Dic3×C3⋊D3C2212yescomplemented
27D3×C4⋊(C3⋊D3)—24no—
28#88859C2212yescomplemented
29C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
30C32×#217C2211yescharacteristic, complemented
31C32×(C2×D3)⋊C4C2211yescharacteristic
32C32×Dic3⋊C4C2211yescharacteristic, complemented
33C32×C4×Dic3C2211yescharacteristic, complemented
34#88806C2212yescomplemented
35#88804C2212yescomplemented
36#88804C2212yescomplemented
37#88803C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yes—
39C2×#3828D311yescharacteristic, complemented
40C22×C4⋊(C3⋊D3)—31no—
41C2×C3×#235—34no—
42C2×D3×D12—34no—
43C32×C4×D4—31no—
44#64146—32no—
45#64139—31no—
46#64139—34no—
47#64139—64no—
48#64133—31no—
49#64103—32no—
50#64093—32no—
51C22⋊C4×C3⋊D3—32no—
52#64084—31no—
53C4×C4⋊(C3⋊D3)—31no—
54C3×C4×#67D314yescomplemented
55#63983—38no—
56#63971—38no—
57D3×(C2×D3)⋊C4—38no—
58Dic3×D12—34no—
59#63864—34no—
60#63856—34no—
61#63850—38no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)C2312yes—
64C22×C3×C3⋊D3C2312yes—
65C22×C32×D3C2311yescharacteristic
66C2×D3×C3⋊D3—24no—
67C2×D3×C3⋊D3—24no—
68C2×D3×C3⋊D3—42no—
69C2×C33×C4—22no—
70C2×C33×C4C2311yescharacteristic
71C2×C33×C4C2311yescharacteristic
72#6312—42no—
73C2×C3⋊(C3⋊Dic3)C2312yes—
74C4⋊(C3⋊(C3⋊D3))—22no—
75C3×#638—24no—
76C2×C3×C3⋊Dic3C2312yes—
77C3×C4⋊(C3⋊D3)D414yescomplemented
78C3×C4×C3⋊D3—24no—
79C3×#634—22no—
80C32×#67—22no—
81C2×C32×Dic3C2311yescharacteristic
82C2×C32×Dic3C2311yescharacteristic
83C2×C32×Dic3C2311yescharacteristic
84C32×C4×D3—22no—
85#6292—24no—
86Dic3×C3⋊D3—42no—
87#5829—91no—
88#5550—94no—
89C23×C3⋊D3—32no—
90C22×D32—38no—
91C2×C32×D4—31no—
92C22×C32×C4—24no—
93C22×C32×C4—31no—
94C22×C32×C4C2×D314yescomplemented
95C22×C32×C4C2×D311yescharacteristic, complemented
96C2×#638—32no—
97C2×#638—38no—
98C2×#638—68no—
99C2×#638C2×D312yescomplemented
100#3828—24no—
101C2×C4⋊(C3⋊D3)—32no—
102C2×C4⋊(C3⋊D3)—31no—
103C2×C4⋊(C3⋊D3)—34no—
104C2×C4⋊(C3⋊D3)—64no—
105C2×C4⋊(C3⋊D3)—64no—
106C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
107C2×C4×C3⋊D3—32no—
108C2×C4×C3⋊D3C2×D312yescomplemented
109C2×#634C2×D311yescharacteristic, complemented
110C2×C3×#67—38no—
111C2×C3×#67C2×D314yes—
112C3×#235—616no—
113C2×C3×D12—34no—
114C2×C3×C4×D3—38no—
115C2×C3×C4×D3C2×D314yes—
116C2×C3×Dic6—34no—
117C2×#626—38no—
118C2×D3×Dic3—38no—
119D3×D12—616no—
120C32×C4⋊C4—31no—
121C32×C22⋊C4—31no—
122C32×C22⋊C4—31no—
123C32×C42—31no—
124#3757—32no—
125#3752—32no—
126#3752—32no—
127#3752—38no—
128#3752—32no—
129#3752—32no—
130#3752—68no—
131C4⋊(C3⋊Dic3)—31no—
132C4⋊(C3⋊Dic3)—34no—
133C4⋊(C3⋊Dic3)—64no—
134#3750—32no—
135C3×#217C2×D314yescomplemented
136C3×(C2×D3)⋊C4C2×D314yes—
137C3×Dic3⋊C4C2×D314yescomplemented
138C3×C4×Dic3C2×D314yescomplemented
139#3724—38no—
140#3722—38no—
141#3722—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—38no—
144C22×C33—21no—
145C22×C33—21no—
146C22×C33—21no—
147C22×C33C2411yescharacteristic
148C2×C3⋊(C3⋊D3)—22no—
149C2×C3⋊(C3⋊D3)—22no—
150C2×C3⋊(C3⋊D3)—22no—
151C2×C3×C3⋊D3—22no—
152C2×C3×C3⋊D3—22no—
153C2×C3×C3⋊D3C2×D414yes—
154C2×C32×D3—21no—
155C2×C32×D3—21no—
156C2×C32×D3#4112yescomplemented
157D3×C3⋊D3—48no—
158C33×C4—21no—
159C33×C4C2×D412yes—
160C3⋊(C3⋊Dic3)—22no—
161C3×C3⋊Dic3—22no—
162C32×Dic3—21no—
163C32×Dic3—21no—
164C32×Dic3—21no—
165C2×C3×#41—91no—
166C2×D3×D4—91no—
167C2×#235—34no—
168C22×D12—94no—
169C3×C4×D4—34no—
170#947—91no—
171#946—91no—
172#939—92no—
173#937—98no—
174#937—92no—
175D4×Dic3—91no—
176#930—94no—
177#930—91no—
178#930—184no—
179#929—92no—
180C4×#67C3⋊D311yescharacteristic, complemented
181C2×(C2×D3)⋊C4—92no—
182#924—94no—
183#894—98no—
184#884—98no—
185D3×C22⋊C4—98no—
186#875—94no—
187C4×D12—94no—
188C23×C32—24no—
189C23×C32—31no—
190C23×C32C22×D314yes—
191C23×C32C22×D311yescharacteristic
192C22×C3⋊D3—32no—
193C22×C3⋊D3—38no—
194C22×C3⋊D3—68no—
195C22×C3⋊D3—64no—
196C22×C3⋊D3—64no—
197C22×C3⋊D3—122no—
198C22×C3⋊D3C22×D312yes—
199C22×C3×D3—38no—
200C22×C3×D3C22×D314yes—
201C2×D32—616no—
202C2×D32—616no—
203C2×D32—128no—
204C32×D4—62no—
205C2×C32×C4—22no—
206C2×C32×C4—24no—
207C2×C32×C4—24no—
208C2×C32×C4—28no—
209C2×C32×C4—31no—
210C2×C32×C4—31no—
211C2×C32×C4—31no—
212C2×C32×C4—48no—
213C2×C32×C4—62no—
214C2×C32×C4C22×D314yes—
215C2×C32×C4C22×D314yes—
216C2×C32×C4C22×D311yescharacteristic
217C2×C32×C4C22×D311yescharacteristic
218#638—24no—
219#638—122no—
220#638—128no—
221#638—1216no—
222C2×C3⋊Dic3—32no—
223C2×C3⋊Dic3—38no—
224C2×C3⋊Dic3—68no—
225C2×C3⋊Dic3C22×D312yes—
226C4⋊(C3⋊D3)—64no—
227C4⋊(C3⋊D3)—62no—
228C4⋊(C3⋊D3)—68no—
229C4⋊(C3⋊D3)—128no—
230C4⋊(C3⋊D3)#6714yescomplemented
231C4×C3⋊D3—24no—
232C4×C3⋊D3—122no—
233#634—22no—
234C3×#67—28no—
235C3×#67—616no—
236C2×C3×Dic3—38no—
237C2×C3×Dic3C22×D314yes—
238C2×C3×Dic3C22×D314yes—
239C2×C3×Dic3C22×D314yes—
240C3×D12—316no—
241C3×C4×D3—28no—
242C3×C4×D3—616no—
243C3×Dic6—68no—
244#626—616no—
245D3×Dic3—128no—
246#546—271noSylow
247C2×C33—41no—
248C2×C33C22×D411yescharacteristic
249C2×C33C2×#4111yescharacteristic
250C2×C33H(5,𝔽2)11yescharacteristic
251C3⋊(C3⋊D3)—42no—
252C3×C3⋊D3—24no—
253C32×D3—22no—
254C23×D3—98no—
255C23×D3—92no—
256C3×#41—184no—
257C2×C3×Q8—91no—
258C2×C3×D4—34no—
259C2×C3×D4—92no—
260C2×C3×D4—91no—
261C22×C3×C4—24no—
262C22×C3×C4—34no—
263C22×C3×C4—92no—
264C22×C3×C4D3214yescomplemented
265C22×C3×C4C2×C3⋊D311yescharacteristic, complemented
266C2×#67—38no—
267C2×#67—98no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—188no—
271C2×#67C2×C3⋊D311yescharacteristic
272C22×Dic3—92no—
273D3×D4—184no—
274#235—616no—
275C2×D12—34no—
276C2×D12—98no—
277C2×D12—94no—
278C2×D12—91no—
279C2×D12—184no—
280C2×D12—1816no—
281C2×C4×D3—38no—
282C2×C4×D3—98no—
283C2×C4×D3C2×C3⋊D311yescharacteristic
284C2×Dic6—34no—
285C3×C4⋊C4—34no—
286C3×C22⋊C4—34no—
287C3×C22⋊C4—34no—
288C3×C42—34no—
289#217—98no—
290#217—92no—
291#217C2×C3⋊D311yescharacteristic, complemented
292(C2×D3)⋊C4—98no—
293(C2×D3)⋊C4—98no—
294(C2×D3)⋊C4—98no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—98no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—188no—
301(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
302C4⋊Dic3—94no—
303C4⋊Dic3—91no—
304C4⋊Dic3—184no—
305Dic3⋊C4—98no—
306Dic3⋊C4—92no—
307Dic3⋊C4C2×C3⋊D311yescharacteristic, complemented
308C4×Dic3C2×C3⋊D311yescharacteristic, complemented
309C22×C32—28no—
310C22×C32—21no—
311C22×C32—24no—
312C22×C32—21no—
313C22×C32—21no—
314C22×C32—24no—
315C22×C32—24no—
316C22×C32—24no—
317C22×C32—32no—
318C22×C32—44no—
319C22×C32—44no—
320C22×C32—61no—
321C22×C32—61no—
322C22×C32C23×D314yes—
323C22×C32C23×D311yescharacteristic
324C2×C3⋊D3—22no—
325C2×C3⋊D3—22no—
326C2×C3⋊D3—62no—
327C2×C3⋊D3—62no—
328C2×C3⋊D3—62no—
329C2×C3⋊D3—68no—
330C2×C3⋊D3—68no—
331C2×C3⋊D3—68no—
332C2×C3⋊D3—128no—
333C2×C3⋊D3—128no—
334C2×C3⋊D3—128no—
335C2×C3⋊D3—128no—
336C2×C3⋊D3C2×#6714yes—
337C2×C3×D3—24no—
338C2×C3×D3—24no—
339C2×C3×D3—316no—
340C2×C3×D3—68no—
341C2×C3×D3—68no—
342C2×C3×D3#23518yescomplemented
343D32—1232no—
344C32×C4—21no—
345C32×C4—28no—
346C32×C4—24no—
347C32×C4—44no—
348C32×C4—61no—
349C32×C4—61no—
350C32×C4—61no—
351C32×C4D3×D418yes—
352C32×C4C2×#6712yes—
353C3⋊Dic3—22no—
354C3⋊Dic3—62no—
355C3⋊Dic3—68no—
356C3⋊Dic3—128no—
357C3×Dic3—24no—
358C3×Dic3—24no—
359C3×Dic3—24no—
360C3×Dic3—68no—
361C2×#41—91no—
362C22×D4—271no—
363#124—271no—
364#123—272no—
365C22⋊Q8—271no—
366#121—272no—
367#121—271no—
368#120—272no—
369C4×D4—31no—
370C4×D4—271no—
371C2×C22⋊C4—272no—
372C33#54611yescharacteristic, complemented, Sylow
373C23×C3—24no—
374C23×C3—34no—
375C23×C3—92no—
376C23×C3C2×D3214yes—
377C23×C3C22×C3⋊D311yescharacteristic
378C22×D3—38no—
379C22×D3—98no—
380C22×D3—92no—
381C22×D3—188no—
382C22×D3—184no—
383C22×D3—1816no—
384C22×D3—1816no—
385C22×D3—184no—
386C22×D3—362no—
387C22×D3—368no—
388C22×D3C22×C3⋊D311yescharacteristic
389C3×Q8—182no—
390C3×D4—68no—
391C3×D4—94no—
392C3×D4—184no—
393C2×C3×C4—28no—
394C2×C3×C4—22no—
395C2×C3×C4—24no—
396C2×C3×C4—24no—
397C2×C3×C4—34no—
398C2×C3×C4—34no—
399C2×C3×C4—34no—
400C2×C3×C4—48no—
401C2×C3×C4—68no—
402C2×C3×C4—92no—
403C2×C3×C4—184no—
404C2×C3×C4C2×D3214yes—
405C2×C3×C4C2×D3214yes—
406C2×C3×C4C22×C3⋊D311yescharacteristic
407C2×C3×C4C22×C3⋊D311yescharacteristic
408#67—22no—
409#67—616no—
410#67—184no—
411#67—368no—
412#67—362no—
413#67—3616no—
414C2×Dic3—38no—
415C2×Dic3—98no—
416C2×Dic3—92no—
417C2×Dic3—188no—
418C2×Dic3—362no—
419C2×Dic3C22×C3⋊D311yescharacteristic
420C2×Dic3C22×C3⋊D311yescharacteristic
421C2×Dic3C22×C3⋊D311yescharacteristic
422D12—316no—
423D12—1816no—
424D12—188no—
425D12—182no—
426D12—368no—
427C4×D3—22no—
428C4×D3—616no—
429C4×D3—368no—
430Dic6—68no—
431C2×C32—24no—
432C2×C32—24no—
433C2×C32—24no—
434C2×C32—48no—
435C2×C32—44no—
436C2×C32—41no—
437C2×C32—62no—
438C2×C32C2×#23514yes—
439C2×C32C2×D3×D414yes—
440C2×C32D3×#4111yescharacteristic
441C2×C32#100914yes—
442C2×C32#100911yescharacteristic
443C2×C32C22×#6711yescharacteristic
444C3⋊D3—24no—
445C3⋊D3—122no—
446C3⋊D3—128no—
447C3⋊D3—248no—
448C3×D3—28no—
449C3×D3—616no—
450C24—272no—
451#41—184no—
452C2×Q8—91no—
453C2×D4—31no—
454C2×D4—92no—
455C2×D4—91no—
456C2×D4—272no—
457C2×D4—272no—
458C2×D4—271no—
459C2×D4—544no—
460C22×C4—31no—
461C22×C4—92no—
462C22×C4—272no—
463C22×C4D3×C3⋊D311yescharacteristic, complemented
464C4⋊C4—31no—
465C4⋊C4—272no—
466C4⋊C4—271no—
467C22⋊C4—31no—
468C22⋊C4—31no—
469C22⋊C4—272no—
470C22⋊C4—272no—
471C22⋊C4—272no—
472C22⋊C4—272no—
473C22⋊C4—272no—
474C42—31no—
475C22×C3—28no—
476C22×C3—24no—
477C22×C3—24no—
478C22×C3—24no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—24no—
483C22×C3—38no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—64no—
487C22×C3—64no—
488C22×C3—94no—
489C22×C3—182no—
490C22×C3—182no—
491C22×C3C22×D3214yes—
492C22×C3C23×C3⋊D311yescharacteristic
493C2×D3—21no—
494C2×D3—21no—
495C2×D3—316no—
496C2×D3—68no—
497C2×D3—68no—
498C2×D3—188no—
499C2×D3—188no—
500C2×D3—188no—
501C2×D3—182no—
502C2×D3—182no—
503C2×D3—182no—
504C2×D3—368no—
505C2×D3—368no—
506C2×D3—368no—
507C2×D3—368no—
508C2×D3—3632no—
509C2×D3#382812yescomplemented
510C3×C4—24no—
511C3×C4—21no—
512C3×C4—28no—
513C3×C4—44no—
514C3×C4—64no—
515C3×C4—64no—
516C3×C4—64no—
517C3×C4—182no—
518C3×C4D3×#6718yes—
519C3×C4D4×C3⋊D312yes—
520Dic3—21no—
521Dic3—21no—
522Dic3—21no—
523Dic3—68no—
524Dic3—188no—
525Dic3—182no—
526Dic3—368no—
527C32—24no—
528C32#555014yescomplemented
529C32#582911yescharacteristic, complemented
530C23—31no—
531C23—92no—
532C23—272no—
533C23—544no—
534C23—544no—
535C23—1082no—
536C23C2×D3×C3⋊D311yescharacteristic
537Q8—182no—
538D4—62no—
539D4—94no—
540D4—184no—
541D4—544no—
542D4—542no—
543D4—1082no—
544C2×C4—22no—
545C2×C4—31no—
546C2×C4—31no—
547C2×C4—31no—
548C2×C4—62no—
549C2×C4—92no—
550C2×C4—184no—
551C2×C4—272no—
552C2×C4—1082no—
553C2×C4C2×D3×C3⋊D311yescharacteristic
554C2×C4C2×D3×C3⋊D311yescharacteristic
555C2×C3—24no—
556C2×C3—24no—
557C2×C3—24no—
558C2×C3—48no—
559C2×C3—41no—
560C2×C3—44no—
561C2×C3—68no—
562C2×C3—184no—
563C2×C3D3×#23514yes—
564C2×C3#6430814yes—
565C2×C3C2×D3×#6714yes—
566C2×C3C2×#382811yescharacteristic
567C2×C3C2×D4×C3⋊D311yescharacteristic
568C2×C3#6436611yescharacteristic
569D3—22no—
570D3—616no—
571D3—362no—
572D3—368no—
573D3—728no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—32no—
578C22—61no—
579C22—61no—
580C22—94no—
581C22—182no—
582C22—182no—
583C22—542no—
584C22—542no—
585C22—542no—
586C22—1088no—
587C22C22×D3×C3⋊D311yescharacteristic
588C4—21no—
589C4—61no—
590C4—61no—
591C4—61no—
592C4—182no—
593C4—542no—
594C4C3⋊D3×#6712yes—
595C3—24no—
596C3#1059506714yescomplemented
597C3#1059602111yescharacteristic, complemented
598C2—41no—
599C2—62no—
600C2—184no—
601C2—1082no—
602C2D3×#382811yescharacteristic
603C2#1172458111yescharacteristic
604C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#546
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×D3×C4⋊(C3⋊D3)C2semidirect
3. #11723971C2semidirect
4. C32×C4×#67C2semidirect
5. #11723643C2semidirect
6. #11723624C2semidirect
7. C3⋊D3×(C2×D3)⋊C4C2semidirect
8. Dic3×C4⋊(C3⋊D3)C2semidirect
9. #11723441C2semidirect
10. #11723425C2semidirect
11. #11723417C2semidirect
15. C22×C33×C4C22semidirect
16. C2×#6312C22semidirect
18. C2×C3×#638C22semidirect
21. C2×C3×C4×C3⋊D3C22semidirect
22. C2×C3×#634C22semidirect
26. C2×Dic3×C3⋊D3C22semidirect
28. #88859C22semidirect
29. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
30. C32×#217C22semidirect
32. C32×Dic3⋊C4C22semidirect
33. C32×C4×Dic3C22semidirect
34. #88806C22semidirect
35. #88804C22semidirect
36. #88804C22semidirect
37. #88803C22semidirect
39. C2×#3828D3semidirect
54. C3×C4×#67D3semidirect
77. C3×C4⋊(C3⋊D3)D4semidirect
94. C22×C32×C4C2×D3semidirect
95. C22×C32×C4C2×D3semidirect
99. C2×#638C2×D3semidirect
108. C2×C4×C3⋊D3C2×D3semidirect
109. C2×#634C2×D3semidirect
135. C3×#217C2×D3semidirect
137. C3×Dic3⋊C4C2×D3semidirect
138. C3×C4×Dic3C2×D3semidirect
156. C2×C32×D3#41semidirect
180. C4×#67C3⋊D3semidirect
230. C4⋊(C3⋊D3)#67semidirect
264. C22×C3×C4D32semidirect
265. C22×C3×C4C2×C3⋊D3semidirect
291. #217C2×C3⋊D3semidirect
307. Dic3⋊C4C2×C3⋊D3semidirect
308. C4×Dic3C2×C3⋊D3semidirect
342. C2×C3×D3#235semidirect
372. C33#546semidirect
463. C22×C4D3×C3⋊D3semidirect
509. C2×D3#3828semidirect
528. C32#5550semidirect
529. C32#5829semidirect
596. C3#10595067semidirect
597. C3#10596021semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×D3×C4⋊(C3⋊D3)2
#117239712
C32×C4×#672
#117236432
#117236242
C3⋊D3×(C2×D3)⋊C42
Dic3×C4⋊(C3⋊D3)2
#117234412
#117234252
#117234172
#105960213
#105950673