Finite Group Databasebeta

← #13177548

Subgroups — #13177548

Order 1728.

2524 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×#3830C211yesmaximal, characteristic, complemented
2C22×#6291C211yesmaximal, characteristic, complemented
3#11723978C211yesmaximal, characteristic, complemented
4C32×#937C211yesmaximal, characteristic, complemented
5#11723639C211yesmaximal, characteristic, complemented
6#11723635C212yesmaximal, complemented
7#11723633C212yesmaximal, complemented
8#11723629C211yesmaximal, characteristic, complemented
9#11723595C211yesmaximal, characteristic, complemented
10#11723521C212yesmaximal, complemented
11#11723520C212yesmaximal, complemented
12#10596063—31nomaximal
13#10595238—34nomaximal
14C23×C3⋊(C3⋊D3)C2211yescharacteristic
15C23×C32×D3C2211yescharacteristic
16C2×C33×D4C2211yescharacteristic
17C2×#6312C2212yescomplemented
18C2×C3×#638C2211yescharacteristic
19C2×C3×#638C2211yescharacteristic, complemented
20C22×C3×C3⋊Dic3C2211yescharacteristic
21C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
22C3×#3830—24no—
23C2×C3×C4×C3⋊D3C2211yescharacteristic, complemented
24C2×C3×#634C2211yescharacteristic, complemented
25C2×C32×#67C2212yescomplemented
26C2×#6292C2212yescomplemented
27C2×#6291—24no—
28C2×#6291C2212yes—
29C2×#6291C2212yes—
30C2×#6290C2212yescomplemented
31#88864C2211yescharacteristic
32#88859C2212yescomplemented
33C32×#217C2211yescharacteristic
34C32×(C2×D3)⋊C4C2212yescomplemented
35#88807C2212yescomplemented
36#88804C2212yescomplemented
37(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
38Dic3×C3⋊Dic3C2212yescomplemented
39C22×#638—31no—
40C2×#3830D311yescharacteristic, complemented
41C2×C3×#237—34no—
42C22×#626—34no—
43C32×#120—31no—
44#64162—31no—
45#64149—32no—
46#64148—32no—
47#64146—31no—
48#64146—34no—
49#64146—64no—
50#64094—32no—
51#64093—32no—
52#64091—31no—
53#64086—31no—
54C3×#937D314yescomplemented
55#63981—34no—
56#63979—38no—
57#63978—38no—
58#63976—34no—
59#63955—34no—
60#63907—38no—
61#63906—38no—
62C23×C33C2311yescharacteristic
63C23×C33C2311yescharacteristic
64C22×C3⋊(C3⋊D3)—22no—
65C22×C3⋊(C3⋊D3)—42no—
66C22×C3⋊(C3⋊D3)C2312yes—
67C22×C3×C3⋊D3C2311yescharacteristic
68C22×C32×D3—22no—
69C22×C32×D3—42no—
70C22×C32×D3C2312yes—
71C33×D4—22no—
72C2×C33×C4C2311yescharacteristic
73#6312—42no—
74C2×C3⋊(C3⋊Dic3)C2312yes—
75C3×#638—22no—
76C3×#638—22no—
77C2×C3×C3⋊Dic3—22no—
78C2×C3×C3⋊Dic3D414yescomplemented
79C2×C3×C3⋊Dic3C2311yescharacteristic
80C2×C3×C3⋊Dic3C2311yescharacteristic
81C2×C3×C3⋊Dic3C2311yescharacteristic
82C3×C4×C3⋊D3—22no—
83C3×#634—22no—
84C32×#67—42no—
85C2×C32×Dic3C2312yes—
86#6292—42no—
87#6291—24no—
88#6291—24no—
89#6290—42no—
90#5830—91no—
91#5592—94no—
92C24×C32—31no—
93C23×C3⋊D3—31no—
94C23×C3⋊D3—34no—
95C23×C3⋊D3—64no—
96C23×C3×D3C2×D314yes—
97C2×C32×D4—24no—
98C2×C32×D4—32no—
99C2×C32×D4C2×D314yes—
100C2×C32×D4C2×D311yescharacteristic
101C2×#638—32no—
102C2×#638—38no—
103C2×#638—32no—
104C2×#638—32no—
105C2×#638—32no—
106C2×#638—68no—
107C2×#638—64no—
108C2×#638C2×D311yescharacteristic
109C2×#638C2×D311yescharacteristic, complemented
110C22×C3⋊Dic3C2×D311yescharacteristic
111C22×C3⋊Dic3C2×D311yescharacteristic, complemented
112#3830—24no—
113C2×C4⋊(C3⋊D3)—32no—
114C2×C4×C3⋊D3C2×D311yescharacteristic, complemented
115C2×#634C2×D311yescharacteristic, complemented
116C2×C3×#67—34no—
117C2×C3×#67—34no—
118C2×C3×#67C2×D318yescomplemented
119C22×C3×Dic3—34no—
120C22×C3×Dic3—34no—
121C3×#237—616no—
122C2×C3×C4×D3—34no—
123C2×C3×Dic6—34no—
124C2×#626—38no—
125C2×#626—38no—
126C2×#626—38no—
127C2×#626—616no—
128C2×#625—38no—
129C32×C22⋊C4—31no—
130C32×C22⋊C4—32no—
131#3757—31no—
132#3757—34no—
133#3757—32no—
134#3757—64no—
135#3752—32no—
136#3752—38no—
137#3752—32no—
138#3752—68no—
139#3750—32no—
140C4×C3⋊Dic3—32no—
141C3×#217C2×D314yes—
142C3×(C2×D3)⋊C4C2×D318yescomplemented
143Dic3⋊Dic3—38no—
144#3722—38no—
145(C2×D3)⋊Dic3—38no—
146Dic32—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)—22no—
155C2×C3⋊(C3⋊D3)—44no—
156C2×C3×C3⋊D3—21no—
157C2×C3×C3⋊D3—22no—
158C2×C32×D3—22no—
159C2×C32×D3—22no—
160C2×C32×D3—22no—
161C2×C32×D3—44no—
162C33×C4—21no—
163C3⋊(C3⋊Dic3)—22no—
164C3×C3⋊Dic3—21no—
165C3×C3⋊Dic3C2×D412yes—
166C3×C3⋊Dic3C2×D412yes—
167C32×Dic3—22no—
168C2×C3×#41—91no—
169C22×#67—94no—
170C2×#237—34no—
171C22×D12—91no—
172C3×#120—34no—
173#953—94no—
174#947—92no—
175#940—98no—
176#940—92no—
177#939—98no—
178#937—91no—
179#937—94no—
180#937—91no—
181#937—91no—
182#937—184no—
183#937C3⋊D311yescharacteristic, complemented
184#930—92no—
185#930—92no—
186#926—91no—
187#885—98no—
188#884—98no—
189#882—94no—
190#877—94no—
191C23×C32—24no—
192C23×C32—24no—
193C23×C32—32no—
194C23×C32—62no—
195C23×C32—122no—
196C23×C32C22×D314yes—
197C23×C32C22×D314yes—
198C23×C32C22×D311yescharacteristic
199C23×C32C22×D311yescharacteristic
200C22×C3⋊D3—32no—
201C22×C3⋊D3—38no—
202C22×C3⋊D3—68no—
203C22×C3⋊D3—62no—
204C22×C3⋊D3—68no—
205C22×C3⋊D3—128no—
206C22×C3⋊D3—122no—
207C22×C3⋊D3—128no—
208C22×C3⋊D3—1216no—
209C22×C3⋊D3C22×D311yescharacteristic
210C22×C3×D3—28no—
211C22×C3×D3—34no—
212C22×C3×D3—48no—
213C22×C3×D3C22×D318yes—
214C32×D4—22no—
215C32×D4—28no—
216C32×D4—48no—
217C32×D4—122no—
218C2×C32×C4—24no—
219C2×C32×C4—32no—
220C2×C32×C4C22×D314yes—
221C2×C32×C4C22×D311yescharacteristic
222#638—22no—
223#638—22no—
224#638—64no—
225#638—64no—
226#638—122no—
227#638—122no—
228#638—128no—
229#638—1216no—
230C2×C3⋊Dic3—22no—
231C2×C3⋊Dic3—32no—
232C2×C3⋊Dic3—38no—
233C2×C3⋊Dic3—68no—
234C2×C3⋊Dic3#6714yescomplemented
235C2×C3⋊Dic3C22×D311yescharacteristic
236C2×C3⋊Dic3C22×D311yescharacteristic
237C2×C3⋊Dic3C22×D311yescharacteristic
238C4⋊(C3⋊D3)—122no—
239C4×C3⋊D3—22no—
240#634—22no—
241C3×#67—48no—
242C3×#67—68no—
243C3×#67—68no—
244C2×C3×Dic3—316no—
245C2×C3×Dic3—34no—
246C2×C3×Dic3—34no—
247C2×C3×Dic3—34no—
248C2×C3×Dic3—68no—
249C2×C3×Dic3C22×D318yes—
250C3×C4×D3—68no—
251C3×Dic6—68no—
252#626—616no—
253#626—616no—
254#626—128no—
255#625—128no—
256#535—271noSylow
257C2×C33—22no—
258C2×C33—41no—
259C2×C33C22×D411yescharacteristic
260C2×C33H(5,𝔽2)12yes—
261C3⋊(C3⋊D3)—42no—
262C3×C3⋊D3—41no—
263C32×D3—42no—
264C24×C3—34no—
265C23×D3—94no—
266C23×D3—91no—
267C23×D3—184no—
268C23×D3C2×C3⋊D311yescharacteristic
269C3×#41—184no—
270C2×C3×Q8—91no—
271C2×C3×D4—24no—
272C2×C3×D4—38no—
273C2×C3×D4—91no—
274C2×C3×D4—91no—
275C2×C3×D4D3214yes—
276C2×C3×D4C2×C3⋊D311yescharacteristic
277C22×C3×C4—91no—
278C22×C3×C4—91no—
279C22×C3×C4—91no—
280C2×#67—34no—
281C2×#67—34no—
282C2×#67—98no—
283C2×#67—98no—
284C2×#67—98no—
285C2×#67—98no—
286C2×#67—92no—
287C2×#67—92no—
288C2×#67—92no—
289C2×#67—188no—
290C2×#67—1816no—
291C2×#67C2×C3⋊D312yescomplemented
292C22×Dic3—34no—
293C22×Dic3—34no—
294#237—616no—
295C2×D12—92no—
296C2×D12—92no—
297C2×D12—98no—
298C2×D12—184no—
299C2×C4×D3—34no—
300C2×Dic6—34no—
301C3×C22⋊C4—34no—
302C3×C22⋊C4—38no—
303#217—98no—
304#217—94no—
305#217—91no—
306#217—184no—
307#217C2×C3⋊D311yescharacteristic
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—92no—
316Dic3⋊C4—98no—
317C4×Dic3—98no—
318C4×Dic3—92no—
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—48no—
329C22×C32—44no—
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—21no—
340C2×C3⋊D3—616no—
341C2×C3⋊D3—68no—
342C2×C3⋊D3—62no—
343C2×C3⋊D3—62no—
344C2×C3⋊D3—62no—
345C2×C3⋊D3—68no—
346C2×C3⋊D3—68no—
347C2×C3⋊D3—128no—
348C2×C3⋊D3—128no—
349C2×C3⋊D3—124no—
350C2×C3⋊D3—1216no—
351C2×C3⋊D3—1232no—
352C2×C3×D3—28no—
353C2×C3×D3—28no—
354C2×C3×D3—28no—
355C2×C3×D3—416no—
356C2×C3×D3—64no—
357C2×C3×D3—68no—
358C32×C4—21no—
359C32×C4—24no—
360C32×C4—44no—
361C32×C4—62no—
362C3⋊Dic3—21no—
363C3⋊Dic3—62no—
364C3⋊Dic3—68no—
365C3⋊Dic3—128no—
366C3⋊Dic3C2×#6712yes—
367C3⋊Dic3C2×#6712yes—
368C3×Dic3—28no—
369C3×Dic3—38no—
370C3×Dic3—38no—
371C3×Dic3—64no—
372C2×#41—91no—
373C22×D4—271no—
374#127—272no—
375#124—272no—
376#121—272no—
377#121—272no—
378#120—31no—
379#120—271no—
380#120—271no—
381#120—271no—
382#117—271no—
383C33#53511yescharacteristic, complemented, Sylow
384C23×C3—24no—
385C23×C3—24no—
386C23×C3—38no—
387C23×C3—68no—
388C23×C3—91no—
389C23×C3—128no—
390C23×C3C2×D3214yes—
391C23×C3C2×D3214yes—
392C23×C3C22×C3⋊D311yescharacteristic
393C23×C3C22×C3⋊D311yescharacteristic
394C22×D3—22no—
395C22×D3—34no—
396C22×D3—42no—
397C22×D3—98no—
398C22×D3—92no—
399C22×D3—188no—
400C22×D3—188no—
401C22×D3—182no—
402C22×D3—368no—
403C22×D3—368no—
404C22×D3—362no—
405C22×D3—3616no—
406C22×D3C22×C3⋊D312yes—
407C3×Q8—182no—
408C3×D4—28no—
409C3×D4—22no—
410C3×D4—48no—
411C3×D4—128no—
412C3×D4—182no—
413C3×D4—182no—
414C2×C3×C4—24no—
415C2×C3×C4—38no—
416C2×C3×C4—91no—
417C2×C3×C4—91no—
418C2×C3×C4—94no—
419C2×C3×C4—91no—
420C2×C3×C4—182no—
421C2×C3×C4—182no—
422C2×C3×C4C2×D3214yes—
423C2×C3×C4C22×C3⋊D311yescharacteristic
424#67—42no—
425#67—68no—
426#67—68no—
427#67—1816no—
428#67—1816no—
429#67—362no—
430#67—362no—
431#67—368no—
432#67—368no—
433#67—362no—
434#67—3616no—
435C2×Dic3—34no—
436C2×Dic3—34no—
437C2×Dic3—316no—
438C2×Dic3—34no—
439C2×Dic3—68no—
440C2×Dic3—98no—
441C2×Dic3—92no—
442C2×Dic3—188no—
443C2×Dic3C22×C3⋊D312yes—
444D12—184no—
445D12—184no—
446D12—368no—
447C4×D3—68no—
448Dic6—68no—
449C2×C32—22no—
450C2×C32—28no—
451C2×C32—24no—
452C2×C32—28no—
453C2×C32—48no—
454C2×C32—41no—
455C2×C32—44no—
456C2×C32—84no—
457C2×C32—122no—
458C2×C32C2×D3×D414yes—
459C2×C32#100418yes—
460C2×C32#100912yes—
461C2×C32C22×#6711yescharacteristic
462C3⋊D3—41no—
463C3⋊D3—1216no—
464C3⋊D3—122no—
465C3⋊D3—128no—
466C3×D3—48no—
467C3×D3—124no—
468C24—31no—
469C24—271no—
470#41—184no—
471C2×Q8—91no—
472C2×D4—32no—
473C2×D4—91no—
474C2×D4—91no—
475C2×D4—272no—
476C2×D4—272no—
477C2×D4—272no—
478C2×D4—272no—
479C2×D4—272no—
480C2×D4—544no—
481C2×D4D3×C3⋊D311yescharacteristic
482C22×C4—91no—
483C22×C4—91no—
484C22×C4—91no—
485C4⋊C4—272no—
486C22⋊C4—31no—
487C22⋊C4—32no—
488C22⋊C4—272no—
489C22⋊C4—272no—
490C22⋊C4—271no—
491C22⋊C4—272no—
492C42—272no—
493C22×C3—28no—
494C22×C3—22no—
495C22×C3—28no—
496C22×C3—24no—
497C22×C3—21no—
498C22×C3—22no—
499C22×C3—24no—
500C22×C3—28no—
501C22×C3—48no—
502C22×C3—48no—
503C22×C3—44no—
504C22×C3—68no—
505C22×C3—68no—
506C22×C3—68no—
507C22×C3—1216no—
508C22×C3—181no—
509C22×C3—182no—
510C22×C3D3×#6718yes—
511C22×C3D4×C3⋊D312yes—
512C22×C3C22×D3214yes—
513C22×C3C23×C3⋊D311yescharacteristic
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—22no—
517C2×D3—44no—
518C2×D3—68no—
519C2×D3—64no—
520C2×D3—1816no—
521C2×D3—188no—
522C2×D3—188no—
523C2×D3—188no—
524C2×D3—182no—
525C2×D3—182no—
526C2×D3—182no—
527C2×D3—368no—
528C2×D3—368no—
529C2×D3—3616no—
530C2×D3—364no—
531C2×D3—3632no—
532C3×C4—24no—
533C3×C4—21no—
534C3×C4—44no—
535C3×C4—68no—
536C3×C4—92no—
537C3×C4—92no—
538C3×C4—181no—
539Dic3—22no—
540Dic3—38no—
541Dic3—38no—
542Dic3—64no—
543Dic3—188no—
544Dic3—182no—
545Dic3—368no—
546C32—24no—
547C32#559214yescomplemented
548C32#583011yescharacteristic, complemented
549C23—32no—
550C23—62no—
551C23—91no—
552C23—122no—
553C23—272no—
554C23—542no—
555C23—1082no—
556C23C2×D3×C3⋊D311yescharacteristic
557C23C2×D3×C3⋊D311yescharacteristic
558Q8—182no—
559D4—22no—
560D4—122no—
561D4—182no—
562D4—182no—
563D4—544no—
564D4—544no—
565D4—1082no—
566D4—1082no—
567D4—1082no—
568C2×C4—32no—
569C2×C4—91no—
570C2×C4—91no—
571C2×C4—94no—
572C2×C4—91no—
573C2×C4—182no—
574C2×C4—182no—
575C2×C4—272no—
576C2×C4C2×D3×C3⋊D311yescharacteristic
577C2×C3—28no—
578C2×C3—22no—
579C2×C3—28no—
580C2×C3—24no—
581C2×C3—48no—
582C2×C3—44no—
583C2×C3—41no—
584C2×C3—84no—
585C2×C3—128no—
586C2×C3—361no—
587C2×C3#6431418yes—
588C2×C3C2×D3×#6714yes—
589C2×C3C2×D4×C3⋊D311yescharacteristic
590C2×C3#6436112yes—
591D3—42no—
592D3—124no—
593D3—3616no—
594D3—368no—
595D3—362no—
596C22—22no—
597C22—22no—
598C22—21no—
599C22—62no—
600C22—62no—
601C22—62no—
602C22—124no—
603C22—182no—
604C22—181no—
605C22—542no—
606C22—542no—
607C22—542no—
608C22—1084no—
609C22C3⋊D3×#6712yes—
610C22C22×D3×C3⋊D311yescharacteristic
611C4—21no—
612C4—62no—
613C4—92no—
614C4—92no—
615C4—181no—
616C4—542no—
617C3—24no—
618C3#1059523814yescomplemented
619C3#1059606311yescharacteristic, complemented
620C2—22no—
621C2—41no—
622C2—122no—
623C2—361no—
624C2—1082no—
625C2#1172458812yes—
626C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#535
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×#3830C2semidirect
2. C22×#6291C2semidirect
3. #11723978C2semidirect
4. C32×#937C2semidirect
5. #11723639C2semidirect
6. #11723635C2semidirect
7. #11723633C2semidirect
8. #11723629C2semidirect
9. #11723595C2semidirect
10. #11723521C2semidirect
11. #11723520C2semidirect
17. C2×#6312C22semidirect
19. C2×C3×#638C22semidirect
21. C22×C3×C3⋊Dic3C22semidirect
23. C2×C3×C4×C3⋊D3C22semidirect
24. C2×C3×#634C22semidirect
25. C2×C32×#67C22semidirect
26. C2×#6292C22semidirect
30. C2×#6290C22semidirect
32. #88859C22semidirect
34. C32×(C2×D3)⋊C4C22semidirect
35. #88807C22semidirect
36. #88804C22semidirect
37. (C2×D3)⋊(C3⋊Dic3)C22semidirect
38. Dic3×C3⋊Dic3C22semidirect
40. C2×#3830D3semidirect
54. C3×#937D3semidirect
78. C2×C3×C3⋊Dic3D4semidirect
109. C2×#638C2×D3semidirect
111. C22×C3⋊Dic3C2×D3semidirect
114. C2×C4×C3⋊D3C2×D3semidirect
115. C2×#634C2×D3semidirect
118. C2×C3×#67C2×D3semidirect
142. C3×(C2×D3)⋊C4C2×D3semidirect
183. #937C3⋊D3semidirect
234. C2×C3⋊Dic3#67semidirect
291. C2×#67C2×C3⋊D3semidirect
314. (C2×D3)⋊C4C2×C3⋊D3semidirect
383. C33#535semidirect
547. C32#5592semidirect
548. C32#5830semidirect
618. C3#10595238semidirect
619. C3#10596063semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C22×#62912
#117239782
C32×#9372
#117236392
#117236352
#117236332
#117236292
#117235952
#117235212
#117235202
#105960633
#105952383