Finite Group Databasebeta

← #13177393

Subgroups — #13177393

Order 1728.

2292 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C22×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3#11723976C211yesmaximal, characteristic, complemented
4C32×#937C211yesmaximal, characteristic, complemented
5#11723629C211yesmaximal, characteristic, complemented
6#67×C3⋊Dic3C212yesmaximal, complemented
7#11723615C212yesmaximal, complemented
8#11723609C211yesmaximal, characteristic, complemented
9#11723595C211yesmaximal, characteristic, complemented
10#11723507C212yesmaximal, complemented
11#11723487C212yesmaximal, complemented
12#10596057—31nomaximal
13#10595154—34nomaximal
14C23×C32×D3C2211yescharacteristic
15C2×C33×D4C2211yescharacteristic
16C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×#638C2211yescharacteristic
18C2×C3×#638C2211yescharacteristic, complemented
19C22×C3×C3⋊Dic3C2211yescharacteristic
20C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
21C3×#3830—24no—
22C2×C3×C4×C3⋊D3C2211yescharacteristic, complemented
23C2×C3×#634C2211yescharacteristic, complemented
24C2×C32×#67C2212yes—
25C2×#6290C2212yes—
26C2×D3×C3⋊Dic3—24no—
27C2×D3×C3⋊Dic3C2212yes—
28C2×D3×C3⋊Dic3C2212yes—
29#88864C2211yescharacteristic
30#88864C2212yescomplemented
31#88857C2212yescomplemented
32C32×#217C2211yescharacteristic
33C32×(C2×D3)⋊C4C2212yes—
34#88807C2212yescomplemented
35Dic3⋊(C3⋊Dic3)C2212yescomplemented
36#88803C2212yescomplemented
37(C2×D3)⋊(C3⋊Dic3)C2212yes—
38Dic3×C3⋊Dic3C2212yescomplemented
39C23×C3⋊Dic3—31no—
40C2×#3830D311yescharacteristic, complemented
41C2×C3×#237—34no—
42C22×D3×Dic3—34no—
43C32×#120—31no—
44#64162—31no—
45#64148—32no—
46#64144—31no—
47#64144—34no—
48#64144—64no—
49D4×C3⋊Dic3—32no—
50#64095—31no—
51#64090—32no—
52#64087—32no—
53#64086—31no—
54C3×#937D314yescomplemented
55#63976—34no—
56Dic3×#67—38no—
57#63970—38no—
58#63962—34no—
59#63955—34no—
60#63887—38no—
61#63879—38no—
62C23×C33C2311yescharacteristic
63C23×C33C2311yescharacteristic
64C22×C3×C3⋊D3C2311yescharacteristic
65C22×C32×D3—22no—
66C22×C32×D3—24no—
67C22×C32×D3C2312yes—
68C33×D4—22no—
69C2×C33×C4C2311yescharacteristic
70C2×C3⋊(C3⋊Dic3)—22no—
71C2×C3⋊(C3⋊Dic3)C2312yes—
72C2×C3⋊(C3⋊Dic3)C2312yes—
73C3×#638—22no—
74C3×#638—22no—
75C2×C3×C3⋊Dic3—22no—
76C2×C3×C3⋊Dic3D414yescomplemented
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C2×C3×C3⋊Dic3C2311yescharacteristic
80C3×C4×C3⋊D3—22no—
81C3×#634—22no—
82C32×#67—24no—
83C2×C32×Dic3C2312yes—
84#6290—24no—
85D3×C3⋊Dic3—24no—
86D3×C3⋊Dic3—24no—
87#5828—91no—
88#5586—94no—
89C24×C32—31no—
90C23×C3×D3C2×D314yes—
91C2×C32×D4—24no—
92C2×C32×D4—32no—
93C2×C32×D4C2×D314yes—
94C2×C32×D4C2×D311yescharacteristic
95C2×#638—32no—
96C2×#638C2×D311yescharacteristic
97C2×#638C2×D311yescharacteristic, complemented
98C22×C3⋊Dic3—31no—
99C22×C3⋊Dic3—34no—
100C22×C3⋊Dic3—32no—
101C22×C3⋊Dic3—32no—
102C22×C3⋊Dic3—64no—
103C22×C3⋊Dic3—64no—
104C22×C3⋊Dic3C2×D311yescharacteristic
105C22×C3⋊Dic3C2×D311yescharacteristic, complemented
106#3830—24no—
107C2×C4×C3⋊D3C2×D311yescharacteristic, complemented
108C2×#634C2×D311yescharacteristic, complemented
109C2×C3×#67—34no—
110C2×C3×#67—34no—
111C2×C3×#67C2×D318yes—
112C22×C3×Dic3—34no—
113C22×C3×Dic3—34no—
114C3×#237—616no—
115C2×C3×C4×D3—34no—
116C2×C3×Dic6—34no—
117C2×#625—38no—
118C2×D3×Dic3—38no—
119C2×D3×Dic3—38no—
120C2×D3×Dic3—616no—
121C32×C22⋊C4—31no—
122C32×C22⋊C4—32no—
123#3757—31no—
124#3757—32no—
125#3757—34no—
126#3757—38no—
127#3757—32no—
128#3757—64no—
129#3757—68no—
130#3752—32no—
131C4⋊(C3⋊Dic3)—32no—
132#3750—32no—
133#3750—38no—
134#3750—32no—
135#3750—68no—
136C4×C3⋊Dic3—32no—
137C3×#217C2×D314yes—
138C3×(C2×D3)⋊C4C2×D318yes—
139Dic3⋊Dic3—38no—
140Dic3⋊Dic3—38no—
141(C2×D3)⋊Dic3—38no—
142(C2×D3)⋊Dic3—38no—
143Dic32—38no—
144C22×C33—22no—
145C22×C33—21no—
146C22×C33—22no—
147C22×C33C2×D412yes—
148C22×C33C2411yescharacteristic
149C2×C3×C3⋊D3—21no—
150C2×C3×C3⋊D3—22no—
151C2×C32×D3—28no—
152C2×C32×D3—22no—
153C2×C32×D3—22no—
154C2×C32×D3#4114yescomplemented
155C33×C4—21no—
156C3⋊(C3⋊Dic3)—22no—
157C3⋊(C3⋊Dic3)—22no—
158C3×C3⋊Dic3—21no—
159C3×C3⋊Dic3C2×D412yes—
160C3×C3⋊Dic3C2×D412yes—
161C32×Dic3—22no—
162C2×C3×#41—91no—
163C23×Dic3—94no—
164C2×#237—34no—
165C22×C4×D3—91no—
166C3×#120—34no—
167#953—94no—
168#946—92no—
169#939—98no—
170#938—92no—
171#937—91no—
172#937C3⋊D311yescharacteristic, complemented
173#935—94no—
174#935—91no—
175#935—91no—
176#935—184no—
177D4×Dic3—98no—
178C4×#67—92no—
179C4×#67—92no—
180#926—91no—
181#886—94no—
182#67⋊C4—98no—
183#878—98no—
184#877—94no—
185C23×C32—24no—
186C23×C32—24no—
187C23×C32—32no—
188C23×C32—64no—
189C23×C32—62no—
190C23×C32C22×D314yes—
191C23×C32C22×D314yes—
192C23×C32C22×D311yescharacteristic
193C23×C32C22×D311yescharacteristic
194C22×C3⋊D3C22×D311yescharacteristic
195C22×C3×D3—28no—
196C22×C3×D3—216no—
197C22×C3×D3—34no—
198C22×C3×D3C22×D318yes—
199C32×D4—22no—
200C32×D4—28no—
201C32×D4—48no—
202C32×D4—64no—
203C2×C32×C4—24no—
204C2×C32×C4—32no—
205C2×C32×C4C22×D314yes—
206C2×C32×C4C22×D311yescharacteristic
207#638—22no—
208#638—22no—
209#638—64no—
210C2×C3⋊Dic3—22no—
211C2×C3⋊Dic3—32no—
212C2×C3⋊Dic3—32no—
213C2×C3⋊Dic3—38no—
214C2×C3⋊Dic3—38no—
215C2×C3⋊Dic3—68no—
216C2×C3⋊Dic3—68no—
217C2×C3⋊Dic3—62no—
218C2×C3⋊Dic3—64no—
219C2×C3⋊Dic3—64no—
220C2×C3⋊Dic3—68no—
221C2×C3⋊Dic3—128no—
222C2×C3⋊Dic3#6714yescomplemented
223C2×C3⋊Dic3C22×D311yescharacteristic
224C2×C3⋊Dic3C22×D311yescharacteristic
225C2×C3⋊Dic3C22×D311yescharacteristic
226C4×C3⋊D3—22no—
227#634—22no—
228C3×#67—216no—
229C3×#67—68no—
230C3×#67—68no—
231C2×C3×Dic3—316no—
232C2×C3×Dic3—34no—
233C2×C3×Dic3—34no—
234C2×C3×Dic3—34no—
235C2×C3×Dic3—68no—
236C2×C3×Dic3C22×D318yes—
237C3×C4×D3—68no—
238C3×Dic6—68no—
239#625—616no—
240D3×Dic3—616no—
241D3×Dic3—616no—
242#525—271noSylow
243C2×C33—22no—
244C2×C33—41no—
245C2×C33C22×D411yescharacteristic
246C2×C33C2×#4112yes—
247C3×C3⋊D3—41no—
248C32×D3—24no—
249C24×C3—34no—
250C23×D3C2×C3⋊D311yescharacteristic
251C3×#41—184no—
252C2×C3×Q8—91no—
253C2×C3×D4—24no—
254C2×C3×D4—38no—
255C2×C3×D4—91no—
256C2×C3×D4—91no—
257C2×C3×D4D3214yes—
258C2×C3×D4C2×C3⋊D311yescharacteristic
259C22×C3×C4—91no—
260C22×C3×C4—91no—
261C22×C3×C4—91no—
262C2×#67—34no—
263C2×#67—34no—
264C2×#67—98no—
265C2×#67—92no—
266C2×#67C2×C3⋊D312yes—
267C22×Dic3—34no—
268C22×Dic3—34no—
269C22×Dic3—98no—
270C22×Dic3—98no—
271C22×Dic3—94no—
272C22×Dic3—91no—
273C22×Dic3—184no—
274C22×Dic3—1816no—
275#237—616no—
276C2×C4×D3—34no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—184no—
280C2×Dic6—34no—
281C3×C22⋊C4—34no—
282C3×C22⋊C4—38no—
283#217—98no—
284#217—94no—
285#217—98no—
286#217—91no—
287#217—92no—
288#217—92no—
289#217—184no—
290#217—188no—
291#217C2×C3⋊D311yescharacteristic
292(C2×D3)⋊C4—98no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4C2×C3⋊D312yes—
295C4⋊Dic3—98no—
296C4⋊Dic3—92no—
297Dic3⋊C4—98no—
298Dic3⋊C4—98no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—188no—
302C4×Dic3—98no—
303C4×Dic3—92no—
304C22×C32—22no—
305C22×C32—22no—
306C22×C32—21no—
307C22×C32—24no—
308C22×C32—28no—
309C22×C32—28no—
310C22×C32—24no—
311C22×C32—28no—
312C22×C32—34no—
313C22×C32—48no—
314C22×C32—44no—
315C22×C32—48no—
316C22×C32—68no—
317C22×C32—62no—
318C22×C32—62no—
319C22×C32D3×D418yes—
320C22×C32C2×#6712yes—
321C22×C32C23×D314yes—
322C22×C32C23×D311yescharacteristic
323C2×C3⋊D3—22no—
324C2×C3⋊D3—21no—
325C2×C3×D3—232no—
326C2×C3×D3—28no—
327C2×C3×D3—28no—
328C2×C3×D3—64no—
329C2×C3×D3—68no—
330C2×C3×D3#237116yescomplemented
331C32×C4—21no—
332C32×C4—24no—
333C32×C4—44no—
334C32×C4—62no—
335C3⋊Dic3—21no—
336C3⋊Dic3—62no—
337C3⋊Dic3—62no—
338C3⋊Dic3—68no—
339C3⋊Dic3—68no—
340C3⋊Dic3—128no—
341C3⋊Dic3—128no—
342C3⋊Dic3C2×#6712yes—
343C3⋊Dic3C2×#6712yes—
344C3×Dic3—28no—
345C3×Dic3—38no—
346C3×Dic3—38no—
347C3×Dic3—64no—
348C2×#41—91no—
349C23×C4—271no—
350#123—271no—
351#123—271no—
352C22⋊Q8—272no—
353#121—272no—
354#120—31no—
355#120—271no—
356C4×D4—272no—
357C4×D4—272no—
358#117—271no—
359C33#52511yescharacteristic, complemented, Sylow
360C23×C3—24no—
361C23×C3—24no—
362C23×C3—38no—
363C23×C3—616no—
364C23×C3—68no—
365C23×C3—91no—
366C23×C3C2×D3214yes—
367C23×C3C2×D3214yes—
368C23×C3C22×C3⋊D311yescharacteristic
369C23×C3C22×C3⋊D311yescharacteristic
370C22×D3—22no—
371C22×D3—24no—
372C22×D3—34no—
373C22×D3C22×C3⋊D312yes—
374C3×Q8—182no—
375C3×D4—28no—
376C3×D4—22no—
377C3×D4—48no—
378C3×D4—616no—
379C3×D4—182no—
380C3×D4—182no—
381C2×C3×C4—24no—
382C2×C3×C4—38no—
383C2×C3×C4—91no—
384C2×C3×C4—91no—
385C2×C3×C4—94no—
386C2×C3×C4—91no—
387C2×C3×C4—182no—
388C2×C3×C4—182no—
389C2×C3×C4C2×D3214yes—
390C2×C3×C4C22×C3⋊D311yescharacteristic
391#67—24no—
392#67—68no—
393#67—68no—
394#67—1816no—
395#67—184no—
396C2×Dic3—34no—
397C2×Dic3—34no—
398C2×Dic3—316no—
399C2×Dic3—34no—
400C2×Dic3—68no—
401C2×Dic3—98no—
402C2×Dic3—98no—
403C2×Dic3—92no—
404C2×Dic3—92no—
405C2×Dic3—188no—
406C2×Dic3—188no—
407C2×Dic3—1816no—
408C2×Dic3—1816no—
409C2×Dic3—188no—
410C2×Dic3—182no—
411C2×Dic3—368no—
412C2×Dic3C22×C3⋊D312yes—
413C4×D3—68no—
414C4×D3—184no—
415C4×D3—184no—
416Dic6—68no—
417C2×C32—22no—
418C2×C32—28no—
419C2×C32—24no—
420C2×C32—28no—
421C2×C32—48no—
422C2×C32—41no—
423C2×C32—44no—
424C2×C32—64no—
425C2×C32—84no—
426C2×C32C2×D3×D414yes—
427C2×C32C2×#23718yes—
428C2×C32D3×#4112yes—
429C2×C32C22×#6711yescharacteristic
430C3⋊D3—41no—
431C3×D3—216no—
432C3×D3—124no—
433C24—31no—
434#41—184no—
435C2×Q8—91no—
436C2×D4—32no—
437C2×D4—91no—
438C2×D4—91no—
439C2×D4—272no—
440C2×D4D3×C3⋊D311yescharacteristic
441C22×C4—91no—
442C22×C4—91no—
443C22×C4—91no—
444C22×C4—272no—
445C22×C4—272no—
446C22×C4—271no—
447C22×C4—544no—
448C4⋊C4—272no—
449C4⋊C4—272no—
450C4⋊C4—272no—
451C22⋊C4—31no—
452C22⋊C4—32no—
453C22⋊C4—272no—
454C22⋊C4—272no—
455C22⋊C4—271no—
456C22⋊C4—272no—
457C42—272no—
458C22×C3—28no—
459C22×C3—22no—
460C22×C3—28no—
461C22×C3—24no—
462C22×C3—21no—
463C22×C3—22no—
464C22×C3—24no—
465C22×C3—28no—
466C22×C3—316no—
467C22×C3—48no—
468C22×C3—44no—
469C22×C3—48no—
470C22×C3—632no—
471C22×C3—68no—
472C22×C3—68no—
473C22×C3—181no—
474C22×C3—182no—
475C22×C3D3×#6718yes—
476C22×C3D4×C3⋊D312yes—
477C22×C3C22×D3214yes—
478C22×C3C23×C3⋊D311yescharacteristic
479C2×D3—28no—
480C2×D3—22no—
481C2×D3—22no—
482C2×D3—68no—
483C2×D3—64no—
484C2×D3#383014yescomplemented
485C3×C4—24no—
486C3×C4—21no—
487C3×C4—44no—
488C3×C4—68no—
489C3×C4—92no—
490C3×C4—92no—
491C3×C4—181no—
492Dic3—22no—
493Dic3—38no—
494Dic3—38no—
495Dic3—64no—
496Dic3—188no—
497Dic3—188no—
498Dic3—182no—
499Dic3—182no—
500Dic3—368no—
501Dic3—368no—
502C32—24no—
503C32#558614yescomplemented
504C32#582811yescharacteristic, complemented
505C23—32no—
506C23—64no—
507C23—62no—
508C23—91no—
509C23C2×D3×C3⋊D311yescharacteristic
510C23C2×D3×C3⋊D311yescharacteristic
511Q8—182no—
512D4—22no—
513D4—64no—
514D4—182no—
515D4—182no—
516D4—544no—
517C2×C4—32no—
518C2×C4—91no—
519C2×C4—91no—
520C2×C4—94no—
521C2×C4—91no—
522C2×C4—182no—
523C2×C4—182no—
524C2×C4—272no—
525C2×C4—272no—
526C2×C4—544no—
527C2×C4—544no—
528C2×C4—542no—
529C2×C4C2×D3×C3⋊D311yescharacteristic
530C2×C3—28no—
531C2×C3—22no—
532C2×C3—28no—
533C2×C3—24no—
534C2×C3—48no—
535C2×C3—44no—
536C2×C3—41no—
537C2×C3—616no—
538C2×C3—84no—
539C2×C3—361no—
540C2×C3D3×#23718yes—
541C2×C3C2×D3×#6714yes—
542C2×C3C2×D4×C3⋊D311yescharacteristic
543C2×C3C2×#383012yes—
544D3—24no—
545D3—124no—
546C22—22no—
547C22—22no—
548C22—21no—
549C22—34no—
550C22—68no—
551C22—62no—
552C22—62no—
553C22—182no—
554C22—181no—
555C22C3⋊D3×#6712yes—
556C22C22×D3×C3⋊D311yescharacteristic
557C4—21no—
558C4—62no—
559C4—92no—
560C4—92no—
561C4—181no—
562C4—542no—
563C4—542no—
564C3—24no—
565C3#1059515414yescomplemented
566C3#1059605711yescharacteristic, complemented
567C2—22no—
568C2—41no—
569C2—64no—
570C2—361no—
571C2D3×#383012yes—
572C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#525
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×D3×C3⋊Dic3C2semidirect
3. #11723976C2semidirect
4. C32×#937C2semidirect
5. #11723629C2semidirect
6. #67×C3⋊Dic3C2semidirect
7. #11723615C2semidirect
8. #11723609C2semidirect
9. #11723595C2semidirect
10. #11723507C2semidirect
11. #11723487C2semidirect
18. C2×C3×#638C22semidirect
20. C22×C3×C3⋊Dic3C22semidirect
22. C2×C3×C4×C3⋊D3C22semidirect
23. C2×C3×#634C22semidirect
30. #88864C22semidirect
31. #88857C22semidirect
34. #88807C22semidirect
35. Dic3⋊(C3⋊Dic3)C22semidirect
36. #88803C22semidirect
38. Dic3×C3⋊Dic3C22semidirect
40. C2×#3830D3semidirect
54. C3×#937D3semidirect
76. C2×C3×C3⋊Dic3D4semidirect
97. C2×#638C2×D3semidirect
105. C22×C3⋊Dic3C2×D3semidirect
107. C2×C4×C3⋊D3C2×D3semidirect
108. C2×#634C2×D3semidirect
154. C2×C32×D3#41semidirect
172. #937C3⋊D3semidirect
222. C2×C3⋊Dic3#67semidirect
330. C2×C3×D3#237semidirect
359. C33#525semidirect
484. C2×D3#3830semidirect
503. C32#5586semidirect
504. C32#5828semidirect
565. C3#10595154semidirect
566. C3#10596057semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C22×D3×C3⋊Dic32
#117239762
C32×#9372
#117236292
#67×C3⋊Dic32
#117236152
#117236092
#117235952
#117235072
#117234872
#105960573
#105951543