Finite Group Databasebeta

← #13178705

Subgroups — #13178705

Order 1728.

1726 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×C4×#1113C211yesmaximal, characteristic, complemented
3C3×#64147C212yesmaximal, complemented
4#11723729C212yesmaximal, complemented
5#11723726C214yesmaximal, complemented
6#11723724C212yesmaximal, complemented
7#11723674C212yesmaximal, complemented
8#11723667C211yesmaximal, characteristic, complemented
9#10595306—31nomaximal
10#10595301—32nomaximal
11C22×#1113C2211yescharacteristic
12C2×C33×D4C2211yescharacteristic, complemented
13C2×C3×#638C2214yes—
14C2×C3×#638C2212yes—
15C22×C3×C3⋊Dic3C2212yescomplemented
16C3×#3830—24no—
17C2×C3×C4×C3⋊D3C2212yes—
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C3×#634C2211yescharacteristic, complemented
20C2×#6294C2211yescharacteristic
21C2×#6294C2212yes—
22C4×#1113—24no—
23C3×#3757C2214yescomplemented
24C3×C4⋊(C3⋊Dic3)C2212yescomplemented
25#88815C2214yescomplemented
26#88814C2212yes—
27#88814C2214yes—
28#88813C2212yescomplemented
29C2×#3830D311yescharacteristic, complemented
30C2×C3×#237—32no—
31C2×C3×#237—61no—
32C2×C4×D32—32no—
33C2×C4×D32—31no—
34#64147D312yescomplemented
35C3×#938—32no—
36C3×#938—32no—
37C3×#938—62no—
38#63983—34no—
39#63983—32no—
40#63975—32no—
41#67⋊Dic3—34no—
42Dic3×#67—34no—
43#63969—34no—
44#63952—34no—
45#63871—32no—
46#63857—32no—
47#63856—32no—
48#63849—31no—
49#63841—32no—
50C23×C33C2312yes—
51C22×C3×C3⋊D3C2311yescharacteristic
52C22×C3×C3⋊D3C2312yes—
53C2×#1113—22no—
54C2×#1113—24no—
55C33×D4—22no—
56C2×C33×C4C2311yescharacteristic
57C3×#638—24no—
58C3×#638—28no—
59C2×C3×C3⋊Dic3—24no—
60C2×C3×C3⋊Dic3C2314yes—
61C2×C3×C3⋊Dic3C2312yes—
62C2×C3×C3⋊Dic3C2312yes—
63C2×C3×C3⋊Dic3C2311yescharacteristic
64C3×C4×C3⋊D3—24no—
65C3×C4×C3⋊D3D414yescomplemented
66C3×#634—22no—
67#6294—22no—
68#6294—24no—
69#5828—91no—
70#5606—92no—
71C22×D32—31no—
72C22×D32—32no—
73C2×C32×D4—21no—
74C2×C32×D4—22no—
75C2×C32×D4—41no—
76C2×C32×D4C2×D311yescharacteristic, complemented
77C2×C32×D4C2×D312yescomplemented
78C2×#638C2×D312yes—
79C2×#638C2×D314yes—
80C22×C3⋊Dic3C2×D312yescomplemented
81#3830—24no—
82C2×C4×C3⋊D3C2×D312yes—
83C2×C4×C3⋊D3C2×D311yescharacteristic
84C2×#634C2×D311yescharacteristic, complemented
85C2×C3×#67—34no—
86C2×C3×#67—34no—
87C2×C3×#67—34no—
88C2×C3×#67—62no—
89C2×C3×#67—64no—
90C22×C3×Dic3—34no—
91C22×C3×Dic3—62no—
92C3×#237—68no—
93C3×#237—68no—
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3—32no—
97C2×C3×C4×D3—61no—
98C2×C3×C4×D3—62no—
99C2×C3×Dic6—32no—
100C2×C3×Dic6—61no—
101C2×#624—32no—
102C2×#624—31no—
103C2×D3×Dic3—32no—
104C2×D3×Dic3—32no—
105C2×D3×Dic3—32no—
106C4×D32—68no—
107C4×D32—64no—
108#3757C2×D314yescomplemented
109C4⋊(C3⋊Dic3)C2×D312yescomplemented
110C3×#217—34no—
111C3×#217—34no—
112C3×#217—64no—
113C3×C4⋊Dic3—32no—
114C3×C4⋊Dic3—32no—
115C3×C4⋊Dic3—62no—
116#3724—34no—
117Dic3⋊Dic3—34no—
118Dic3⋊Dic3—34no—
119#3722—32no—
120#3722—34no—
121(C2×D3)⋊Dic3—34no—
122(C2×D3)⋊Dic3—34no—
123(C2×D3)⋊Dic3—34no—
124Dic32—34no—
125Dic32—32no—
126C22×C33—22no—
127C22×C33—24no—
128C22×C33C2411yescharacteristic
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—22no—
132C2×C3×C3⋊D3C2×D412yes—
133C2×C3×C3⋊D3#4114yescomplemented
134#1113—28no—
135C33×C4C2×D412yes—
136C3×C3⋊Dic3—22no—
137C3×C3⋊Dic3—22no—
138C3×C3⋊Dic3—24no—
139C3×C3⋊Dic3C2×D412yes—
140C2×C3×#41—91no—
141C2×#237—32no—
142C2×#237—61no—
143C22×C4×D3—91no—
144C22×C4×D3—92no—
145C3×#121—92no—
146#946—92no—
147#938—32no—
148#938—32no—
149#938—62no—
150#937—92no—
151#937—94no—
152#935—92no—
153D4×Dic3—94no—
154C4×#67—94no—
155#926—91no—
156#924—92no—
157#897—92no—
158#892—92no—
159#883—94no—
160#67⋊C4—94no—
161C23×C32—22no—
162C23×C32—24no—
163C23×C32—42no—
164C23×C32C22×D312yes—
165C23×C32C22×D314yes—
166C22×C3⋊D3C22×D312yes—
167C22×C3⋊D3C22×D311yescharacteristic
168C22×C3×D3—32no—
169C22×C3×D3—32no—
170C22×C3×D3—32no—
171C22×C3×D3—61no—
172C22×C3×D3—62no—
173C2×D32—64no—
174C2×D32—64no—
175C2×D32—64no—
176C2×D32—64no—
177C2×D32—62no—
178C32×D4—22no—
179C32×D4—24no—
180C32×D4—24no—
181C32×D4—44no—
182C32×D4—44no—
183C2×C32×C4—22no—
184C2×C32×C4—21no—
185C2×C32×C4—41no—
186C2×C32×C4C22×D312yes—
187C2×C32×C4C22×D311yescharacteristic
188#638—24no—
189#638—28no—
190C2×C3⋊Dic3—24no—
191C2×C3⋊Dic3C22×D314yes—
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D311yescharacteristic
195C4×C3⋊D3—24no—
196C4×C3⋊D3#6714yescomplemented
197#634—22no—
198C3×#67—68no—
199C3×#67—68no—
200C3×#67—68no—
201C3×#67—68no—
202C3×#67—128no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—34no—
205C2×C3×Dic3—34no—
206C2×C3×Dic3—34no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—64no—
212C2×C3×Dic3—62no—
213C2×C3×Dic3—68no—
214C2×C3×Dic3—68no—
215C3×C4×D3—38no—
216C3×C4×D3—64no—
217C3×C4×D3—64no—
218C3×C4×D3—64no—
219C3×C4×D3—124no—
220C3×Dic6—64no—
221C3×Dic6—64no—
222#624—64no—
223#624—62no—
224D3×Dic3—64no—
225D3×Dic3—64no—
226D3×Dic3—64no—
227#525—271noSylow
228C2×C33—42no—
229C2×C33C22×D411yescharacteristic
230C2×C33C2×#4112yes—
231C3×C3⋊D3—22no—
232C3×C3⋊D3—24no—
233C23×D3—92no—
234C23×D3—91no—
235C3×#41—184no—
236C2×C3×Q8—91no—
237C2×C3×D4—21no—
238C2×C3×D4—22no—
239C2×C3×D4—41no—
240C2×C3×D4—94no—
241C2×C3×D4—92no—
242C2×C3×D4D3211yescharacteristic, complemented
243C2×C3×D4D3212yescomplemented
244C22×C3×C4—91no—
245C22×C3×C4—92no—
246C22×C3×C4—92no—
247C2×#67—34no—
248C2×#67—34no—
249C2×#67—34no—
250C2×#67—62no—
251C2×#67—64no—
252C22×Dic3—34no—
253C22×Dic3—62no—
254C22×Dic3—92no—
255C22×Dic3—91no—
256#237—68no—
257#237—68no—
258C2×C4×D3—32no—
259C2×C4×D3—32no—
260C2×C4×D3—32no—
261C2×C4×D3—61no—
262C2×C4×D3—62no—
263C2×C4×D3—92no—
264C2×C4×D3—92no—
265C2×C4×D3—92no—
266C2×C4×D3—188no—
267C2×C4×D3—184no—
268C2×Dic6—32no—
269C2×Dic6—61no—
270C3×C4⋊C4—92no—
271C3×C22⋊C4—94no—
272#217—34no—
273#217—34no—
274#217—64no—
275#217—94no—
276#217—92no—
277(C2×D3)⋊C4—94no—
278(C2×D3)⋊C4—94no—
279(C2×D3)⋊C4—94no—
280C4⋊Dic3—32no—
281C4⋊Dic3—32no—
282C4⋊Dic3—62no—
283C4⋊Dic3—94no—
284Dic3⋊C4—94no—
285Dic3⋊C4—94no—
286C4×Dic3—94no—
287C4×Dic3—92no—
288C22×C32—22no—
289C22×C32—24no—
290C22×C32—24no—
291C22×C32—24no—
292C22×C32—24no—
293C22×C32—24no—
294C22×C32—21no—
295C22×C32—22no—
296C22×C32—41no—
297C22×C32—44no—
298C22×C32—44no—
299C22×C32—44no—
300C22×C32—44no—
301C22×C32—44no—
302C22×C32—84no—
303C22×C32C23×D312yes—
304C22×C32C23×D311yescharacteristic
305C2×C3⋊D3—22no—
306C2×C3⋊D3—22no—
307C2×C3⋊D3—22no—
308C2×C3⋊D3#23714yescomplemented
309C2×C3⋊D3C2×#6712yes—
310C2×C3×D3—34no—
311C2×C3×D3—34no—
312C2×C3×D3—34no—
313C2×C3×D3—62no—
314C2×C3×D3—64no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C2×C3×D3—62no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—122no—
322C2×C3×D3—122no—
323C2×C3×D3—122no—
324D32—68no—
325D32—616no—
326C32×C4—24no—
327C32×C4—22no—
328C32×C4—42no—
329C32×C4D3×D414yes—
330C32×C4C2×#6712yes—
331C3⋊Dic3—22no—
332C3⋊Dic3—22no—
333C3⋊Dic3—24no—
334C3⋊Dic3C2×#6712yes—
335C3×Dic3—34no—
336C3×Dic3—62no—
337C3×Dic3—64no—
338C3×Dic3—62no—
339C3×Dic3—64no—
340C3×Dic3—64no—
341C3×Dic3—64no—
342C3×Dic3—62no—
343C3×Dic3—122no—
344C3×Dic3—124no—
345C2×#41—91no—
346C23×C4—271no—
347#123—272no—
348C22⋊Q8—272no—
349#121—92no—
350#120—272no—
351C4×D4—274no—
352#117—271no—
353C33#52511yescharacteristic, complemented, Sylow
354C23×C3—22no—
355C23×C3—24no—
356C23×C3—42no—
357C23×C3—92no—
358C23×C3—91no—
359C23×C3C2×D3212yes—
360C23×C3C2×D3214yes—
361C22×D3—32no—
362C22×D3—32no—
363C22×D3—32no—
364C22×D3—61no—
365C22×D3—62no—
366C22×D3—184no—
367C22×D3—184no—
368C22×D3—184no—
369C22×D3—182no—
370C22×D3—184no—
371C3×Q8—182no—
372C3×D4—24no—
373C3×D4—24no—
374C3×D4—22no—
375C3×D4—44no—
376C3×D4—44no—
377C3×D4—184no—
378C3×D4—188no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—41no—
382C2×C3×C4—91no—
383C2×C3×C4—94no—
384C2×C3×C4—94no—
385C2×C3×C4—92no—
386C2×C3×C4—92no—
387C2×C3×C4—184no—
388C2×C3×C4—184no—
389C2×C3×C4C2×D3212yes—
390C2×C3×C4C2×D3211yescharacteristic
391#67—68no—
392#67—68no—
393#67—68no—
394#67—68no—
395#67—128no—
396C2×Dic3—32no—
397C2×Dic3—34no—
398C2×Dic3—34no—
399C2×Dic3—32no—
400C2×Dic3—34no—
401C2×Dic3—32no—
402C2×Dic3—62no—
403C2×Dic3—61no—
404C2×Dic3—64no—
405C2×Dic3—62no—
406C2×Dic3—68no—
407C2×Dic3—68no—
408C2×Dic3—184no—
409C2×Dic3—182no—
410C4×D3—38no—
411C4×D3—64no—
412C4×D3—64no—
413C4×D3—64no—
414C4×D3—124no—
415C4×D3—184no—
416C4×D3—184no—
417C4×D3—184no—
418Dic6—64no—
419Dic6—64no—
420C2×C32—22no—
421C2×C32—22no—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—21no—
425C2×C32—41no—
426C2×C32—42no—
427C2×C32—42no—
428C2×C32—44no—
429C2×C32—44no—
430C2×C32—84no—
431C2×C32—84no—
432C2×C32C2×D3×D412yes—
433C2×C32C2×#23712yes—
434C2×C32D3×#4112yes—
435C2×C32D3×#4112yes—
436C2×C32C22×#6711yescharacteristic
437C3⋊D3—22no—
438C3⋊D3—24no—
439C3×D3—64no—
440C3×D3—64no—
441C3×D3—64no—
442C3×D3—124no—
443C3×D3—122no—
444C24—271no—
445#41—184no—
446C2×Q8—91no—
447C2×D4—94no—
448C2×D4—92no—
449C2×D4#111311yescharacteristic, complemented
450C22×C4—91no—
451C22×C4—92no—
452C22×C4—92no—
453C22×C4—272no—
454C22×C4—271no—
455C22×C4—544no—
456C4⋊C4—92no—
457C4⋊C4—274no—
458C22⋊C4—94no—
459C22⋊C4—272no—
460C22⋊C4—274no—
461C42—272no—
462C22×C3—24no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—24no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—21no—
470C22×C3—44no—
471C22×C3—44no—
472C22×C3—44no—
473C22×C3—44no—
474C22×C3—44no—
475C22×C3—41no—
476C22×C3—84no—
477C22×C3—94no—
478C22×C3—92no—
479C22×C3—182no—
480C22×C3—182no—
481C22×C3—182no—
482C22×C3C22×D3211yescharacteristic
483C22×C3C22×D3212yes—
484C2×D3—34no—
485C2×D3—34no—
486C2×D3—34no—
487C2×D3—62no—
488C2×D3—64no—
489C2×D3—62no—
490C2×D3—62no—
491C2×D3—62no—
492C2×D3—62no—
493C2×D3—62no—
494C2×D3—62no—
495C2×D3—122no—
496C2×D3—122no—
497C2×D3—122no—
498C2×D3—1816no—
499C2×D3—188no—
500C3×C4—22no—
501C3×C4—24no—
502C3×C4—42no—
503C3×C4—92no—
504C3×C4—184no—
505C3×C4—182no—
506C3×C4—182no—
507C3×C4D3×#6714yes—
508C3×C4#381112yes—
509Dic3—34no—
510Dic3—62no—
511Dic3—64no—
512Dic3—64no—
513Dic3—64no—
514Dic3—64no—
515Dic3—62no—
516Dic3—62no—
517Dic3—122no—
518Dic3—124no—
519C32—22no—
520C32—21no—
521C32—41no—
522C32#560612yescomplemented
523C32#582811yescharacteristic, complemented
524C23—92no—
525C23—91no—
526C23—542no—
527C23—544no—
528C23C2×#111312yes—
529Q8—182no—
530D4—22no—
531D4—184no—
532D4—188no—
533C2×C4—91no—
534C2×C4—94no—
535C2×C4—94no—
536C2×C4—92no—
537C2×C4—92no—
538C2×C4—184no—
539C2×C4—184no—
540C2×C4—544no—
541C2×C4—542no—
542C2×C4C2×#111311yescharacteristic
543C2×C3—22no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—21no—
547C2×C3—22no—
548C2×C3—44no—
549C2×C3—44no—
550C2×C3—42no—
551C2×C3—42no—
552C2×C3—41no—
553C2×C3—84no—
554C2×C3—84no—
555C2×C3—184no—
556C2×C3—182no—
557C2×C3D3×#23712yes—
558C2×C3D3×#23712yes—
559C2×C3#6431212yes—
560C2×C3C2×D3×#6712yes—
561C2×C3C2×#381111yescharacteristic
562D3—64no—
563D3—64no—
564D3—64no—
565D3—122no—
566D3—124no—
567C22—24no—
568C22—22no—
569C22—94no—
570C22—92no—
571C22—182no—
572C22—182no—
573C22—182no—
574C22—548no—
575C22C22×#111311yescharacteristic
576C4—92no—
577C4—184no—
578C4—182no—
579C4—182no—
580C4#8904912yes—
581C3—21no—
582C3—22no—
583C3—41no—
584C3#1059530112yescomplemented
585C3#1059530611yescharacteristic, complemented
586C2—42no—
587C2—184no—
588C2—182no—
589C2#1172462212yes—
590C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
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. C2×C4×#1113C2semidirect
3. C3×#64147C2semidirect
4. #11723729C2semidirect
5. #11723726C2semidirect
6. #11723724C2semidirect
7. #11723674C2semidirect
8. #11723667C2semidirect
12. C2×C33×D4C22semidirect
15. C22×C3×C3⋊Dic3C22semidirect
19. C2×C3×#634C22semidirect
23. C3×#3757C22semidirect
24. C3×C4⋊(C3⋊Dic3)C22semidirect
25. #88815C22semidirect
28. #88813C22semidirect
29. C2×#3830D3semidirect
34. #64147D3semidirect
65. C3×C4×C3⋊D3D4semidirect
76. C2×C32×D4C2×D3semidirect
77. C2×C32×D4C2×D3semidirect
80. C22×C3⋊Dic3C2×D3semidirect
84. C2×#634C2×D3semidirect
108. #3757C2×D3semidirect
109. C4⋊(C3⋊Dic3)C2×D3semidirect
133. C2×C3×C3⋊D3#41semidirect
196. C4×C3⋊D3#67semidirect
242. C2×C3×D4D32semidirect
243. C2×C3×D4D32semidirect
308. C2×C3⋊D3#237semidirect
353. C33#525semidirect
449. C2×D4#1113semidirect
522. C32#5606semidirect
523. C32#5828semidirect
584. C3#10595301semidirect
585. C3#10595306semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C4×#11132
C3×#641472
#117237292
#117237262
#117237242
#117236742
#117236672
#105953063
#105953013