Finite Group Databasebeta

← #13172076

Subgroups — #13172076

Order 1728.

1766 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#3805C211yesmaximal, characteristic, complemented
3#11723715C211yesmaximal, characteristic, complemented
4#11723603C212yesmaximal, complemented
5C3×#63976C212yesmaximal, complemented
6#11722976C212yesmaximal, complemented
7#11722953C212yesmaximal, complemented
8#11722924C214yesmaximal, complemented
9#10595501—31nomaximal
10#10595154—32nomaximal
11C23×C32×D3C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C2×C32×#67C2212yes—
15C2×D3×C3⋊Dic3—24no—
16C2×D3×C3⋊Dic3C2212yes—
17C2×D3×C3⋊Dic3C2212yes—
18C2×C3×#627C2211yescharacteristic, complemented
19C2×C3×#625C2214yes—
20C2×C3×#625C2211yescharacteristic, complemented
21C3×#3805—24no—
22C2×C3×D3×Dic3C2212yescomplemented
23C3×#3757C2212yes—
24#88815C2212yescomplemented
25#88813C2212yescomplemented
26#88807C2214yescomplemented
27(C2×D3)⋊(C3⋊Dic3)C2214yescomplemented
28C3×(C2×D3)⋊Dic3C2214yes—
29C23×C3⋊Dic3—31no—
30C2×C3×#237—32no—
31C2×#3805D311yescharacteristic, complemented
32C22×D3×Dic3—32no—
33C22×D3×Dic3—61no—
34#64144—32no—
35C3×#953—32no—
36C3×#937—32no—
37#63982—32no—
38#63980—32no—
39#63976D312yescomplemented
40#67⋊Dic3—34no—
41Dic3×#67—34no—
42#63970—34no—
43#63962—32no—
44#63962—32no—
45#63962—62no—
46#63957—31no—
47#63955—32no—
48#63887—34no—
49#63879—34no—
50C23×C33C2311yescharacteristic
51C22×C32×D3—22no—
52C22×C32×D3—24no—
53C22×C32×D3C2312yes—
54C22×C32×D3C2312yes—
55C2×C3⋊(C3⋊Dic3)—22no—
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C2×C3×C3⋊Dic3D414yescomplemented
58C2×C3×C3⋊Dic3C2314yes—
59C2×C3×C3⋊Dic3C2311yescharacteristic
60C2×C3×C3⋊Dic3C2311yescharacteristic
61C32×#67—24no—
62C2×C32×Dic3C2312yes—
63D3×C3⋊Dic3—24no—
64D3×C3⋊Dic3—24no—
65C3×#627—22no—
66C3×#625—28no—
67C3×#625—22no—
68C3×D3×Dic3—24no—
69#5828—91no—
70#5586—92no—
71C24×C32—31no—
72C23×C3×D3—21no—
73C23×C3×D3C2×D312yes—
74C2×C32×D4—32no—
75C22×C3⋊Dic3—32no—
76C22×C3⋊Dic3—31no—
77C22×C3⋊Dic3—32no—
78C22×C3⋊Dic3—32no—
79C22×C3⋊Dic3—61no—
80C22×C3⋊Dic3—64no—
81C22×C3⋊Dic3—62no—
82C22×C3⋊Dic3—121no—
83C22×C3⋊Dic3C2×D311yescharacteristic
84C2×C3×#67—22no—
85C2×C3×#67—34no—
86C2×C3×#67—34no—
87C2×C3×#67—32no—
88C2×C3×#67C2×D312yes—
89C2×C3×#67C2×D312yes—
90C22×C3×Dic3—32no—
91C22×C3×Dic3—32no—
92C22×C3×Dic3—61no—
93C3×#237—68no—
94C2×C3×C4×D3—32no—
95C2×C3×Dic6—32no—
96C2×#627C2×D311yescharacteristic, complemented
97C2×#625C2×D314yes—
98C2×#625C2×D311yescharacteristic, complemented
99#3805—24no—
100C2×D3×Dic3—34no—
101C2×D3×Dic3—34no—
102C2×D3×Dic3—68no—
103C2×D3×Dic3—62no—
104C2×D3×Dic3—62no—
105C2×D3×Dic3—68no—
106C2×D3×Dic3C2×D312yescomplemented
107#3757—34no—
108#3757C2×D312yes—
109#3750—34no—
110C3×#217—32no—
111C3×#217—32no—
112C3×#217—34no—
113C3×#217—62no—
114C3×(C2×D3)⋊C4—34no—
115#3724—32no—
116Dic3⋊Dic3—34no—
117Dic3⋊Dic3—34no—
118Dic3⋊Dic3—34no—
119Dic3⋊Dic3—64no—
120(C2×D3)⋊Dic3—34no—
121(C2×D3)⋊Dic3—34no—
122(C2×D3)⋊Dic3—64no—
123(C2×D3)⋊Dic3C2×D314yes—
124Dic32—34no—
125Dic32—32no—
126C22×C33—22no—
127C22×C33C2×D412yes—
128C22×C33C2411yescharacteristic
129C2×C32×D3—28no—
130C2×C32×D3—22no—
131C2×C32×D3—22no—
132C2×C32×D3—22no—
133C2×C32×D3—24no—
134C2×C32×D3#4114yescomplemented
135C3⋊(C3⋊Dic3)—22no—
136C3×C3⋊Dic3—24no—
137C3×C3⋊Dic3C2×D412yes—
138C3×C3⋊Dic3C2×D412yes—
139C32×Dic3—22no—
140C2×C3×#41—91no—
141C23×Dic3—92no—
142C23×Dic3—181no—
143C2×#237—32no—
144C22×C4×D3—91no—
145C3×#120—92no—
146#953—32no—
147#946—92no—
148#939—94no—
149#938—92no—
150#937—32no—
151#935—92no—
152#935—92no—
153#935—182no—
154D4×Dic3—94no—
155C4×#67—94no—
156#926—91no—
157#886—92no—
158#67⋊C4—94no—
159#878—94no—
160#877—92no—
161C23×C32—21no—
162C23×C32—22no—
163C23×C32—32no—
164C23×C32—32no—
165C23×C32—41no—
166C23×C32—64no—
167C23×C32—62no—
168C23×C32C22×D312yes—
169C23×C32C22×D311yescharacteristic
170C22×C3×D3—28no—
171C22×C3×D3—24no—
172C22×C3×D3—24no—
173C22×C3×D3—28no—
174C22×C3×D3—22no—
175C22×C3×D3—22no—
176C22×C3×D3C22×D312yes—
177C22×C3×D3C22×D312yes—
178C22×C3×D3C22×D314yes—
179C32×D4—64no—
180C2×C32×C4—32no—
181C2×C3⋊Dic3—32no—
182C2×C3⋊Dic3—34no—
183C2×C3⋊Dic3—62no—
184C2×C3⋊Dic3—64no—
185C2×C3⋊Dic3—62no—
186C2×C3⋊Dic3—64no—
187C2×C3⋊Dic3—64no—
188C2×C3⋊Dic3—64no—
189C2×C3⋊Dic3—64no—
190C2×C3⋊Dic3—122no—
191C2×C3⋊Dic3—124no—
192C2×C3⋊Dic3—124no—
193C2×C3⋊Dic3#6714yescomplemented
194C2×C3⋊Dic3C22×D314yes—
195C2×C3⋊Dic3C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D311yescharacteristic
197C3×#67—24no—
198C3×#67—24no—
199C3×#67—44no—
200C3×#67—64no—
201C3×#67—68no—
202C3×#67—68no—
203C2×C3×Dic3—22no—
204C2×C3×Dic3—38no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—34no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3—64no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—61no—
212C2×C3×Dic3—64no—
213C2×C3×Dic3—64no—
214C2×C3×Dic3C22×D312yes—
215C2×C3×Dic3C22×D312yes—
216C3×C4×D3—64no—
217C3×Dic6—64no—
218#627—22no—
219#625—28no—
220#625—22no—
221D3×Dic3—24no—
222D3×Dic3—68no—
223D3×Dic3—68no—
224D3×Dic3—68no—
225D3×Dic3—68no—
226#525—271noSylow
227C2×C33—22no—
228C2×C33C22×D411yescharacteristic
229C2×C33C2×#4112yes—
230C32×D3—24no—
231C32×D3—42no—
232C24×C3—32no—
233C24×C3—61no—
234C23×D3D3211yescharacteristic
235C3×#41—184no—
236C2×C3×Q8—91no—
237C2×C3×D4—32no—
238C2×C3×D4—32no—
239C2×C3×D4—62no—
240C2×C3×D4—94no—
241C2×C3×D4—91no—
242C22×C3×C4—92no—
243C22×C3×C4—91no—
244C2×#67—34no—
245C2×#67—34no—
246C2×#67—32no—
247C2×#67D3212yes—
248C22×Dic3—32no—
249C22×Dic3—32no—
250C22×Dic3—61no—
251C22×Dic3—94no—
252C22×Dic3—94no—
253C22×Dic3—92no—
254C22×Dic3—91no—
255C22×Dic3—182no—
256C22×Dic3—182no—
257C22×Dic3—181no—
258C22×Dic3—182no—
259C22×Dic3—188no—
260C22×Dic3—188no—
261C22×Dic3—361no—
262#237—68no—
263C2×C4×D3—32no—
264C2×C4×D3—92no—
265C2×C4×D3—92no—
266C2×C4×D3—184no—
267C2×Dic6—32no—
268C3×C22⋊C4—94no—
269C3×C22⋊C4—92no—
270#217—32no—
271#217—32no—
272#217—34no—
273#217—62no—
274#217—94no—
275#217—94no—
276#217—184no—
277(C2×D3)⋊C4—34no—
278(C2×D3)⋊C4—94no—
279C4⋊Dic3—92no—
280C4⋊Dic3—94no—
281Dic3⋊C4—94no—
282Dic3⋊C4—94no—
283Dic3⋊C4—94no—
284Dic3⋊C4—184no—
285C4×Dic3—94no—
286C4×Dic3—92no—
287C22×C32—22no—
288C22×C32—24no—
289C22×C32—24no—
290C22×C32—22no—
291C22×C32—21no—
292C22×C32—24no—
293C22×C32—22no—
294C22×C32—34no—
295C22×C32—42no—
296C22×C32—41no—
297C22×C32—44no—
298C22×C32—44no—
299C22×C32—68no—
300C22×C32—62no—
301C22×C32—62no—
302C22×C32—62no—
303C22×C32—64no—
304C22×C32D3×D414yes—
305C22×C32C2×#6712yes—
306C22×C32C23×D312yes—
307C22×C32C23×D311yescharacteristic
308C2×C3×D3—24no—
309C2×C3×D3—216no—
310C2×C3×D3—216no—
311C2×C3×D3—24no—
312C2×C3×D3—24no—
313C2×C3×D3—24no—
314C2×C3×D3—22no—
315C2×C3×D3—24no—
316C2×C3×D3—24no—
317C2×C3×D3—22no—
318C2×C3×D3—24no—
319C2×C3×D3—42no—
320C2×C3×D3—44no—
321C2×C3×D3#23718yescomplemented
322C32×C4—62no—
323C3⋊Dic3—24no—
324C3⋊Dic3—62no—
325C3⋊Dic3—64no—
326C3⋊Dic3—64no—
327C3⋊Dic3—124no—
328C3⋊Dic3—124no—
329C3⋊Dic3C2×#6712yes—
330C3⋊Dic3C2×#6712yes—
331C3×Dic3—22no—
332C3×Dic3—22no—
333C3×Dic3—34no—
334C3×Dic3—34no—
335C3×Dic3—42no—
336C3×Dic3—62no—
337C3×Dic3—62no—
338C3×Dic3—64no—
339C3×Dic3—64no—
340C3×Dic3—124no—
341C2×#41—91no—
342C23×C4—271no—
343#123—272no—
344C22⋊Q8—272no—
345#121—272no—
346#120—92no—
347C4×D4—274no—
348#117—271no—
349C33#52511yescharacteristic, complemented, Sylow
350C23×C3—21no—
351C23×C3—22no—
352C23×C3—34no—
353C23×C3—32no—
354C23×C3—32no—
355C23×C3—41no—
356C23×C3—62no—
357C23×C3—62no—
358C23×C3—68no—
359C23×C3—64no—
360C23×C3—68no—
361C23×C3—64no—
362C23×C3C2×D3212yes—
363C23×C3C2×D3211yescharacteristic
364C22×D3—22no—
365C22×D3—24no—
366C22×D3C2×D3212yes—
367C22×D3C2×D3212yes—
368C3×Q8—182no—
369C3×D4—64no—
370C3×D4—64no—
371C3×D4—124no—
372C3×D4—182no—
373C3×D4—188no—
374C2×C3×C4—32no—
375C2×C3×C4—32no—
376C2×C3×C4—62no—
377C2×C3×C4—91no—
378C2×C3×C4—91no—
379C2×C3×C4—94no—
380C2×C3×C4—94no—
381C2×C3×C4—184no—
382#67—24no—
383#67—64no—
384#67—68no—
385#67—68no—
386C2×Dic3—32no—
387C2×Dic3—32no—
388C2×Dic3—38no—
389C2×Dic3—34no—
390C2×Dic3—34no—
391C2×Dic3—64no—
392C2×Dic3—61no—
393C2×Dic3—61no—
394C2×Dic3—64no—
395C2×Dic3—64no—
396C2×Dic3—92no—
397C2×Dic3—94no—
398C2×Dic3—182no—
399C2×Dic3—184no—
400C2×Dic3—188no—
401C2×Dic3—188no—
402C2×Dic3—188no—
403C2×Dic3—188no—
404C2×Dic3—184no—
405C2×Dic3—184no—
406C2×Dic3—182no—
407C2×Dic3—362no—
408C2×Dic3—364no—
409C2×Dic3—364no—
410C2×Dic3C2×D3212yes—
411C4×D3—64no—
412C4×D3—184no—
413C4×D3—184no—
414Dic6—64no—
415C2×C32—22no—
416C2×C32—24no—
417C2×C32—24no—
418C2×C32—24no—
419C2×C32—22no—
420C2×C32—22no—
421C2×C32—21no—
422C2×C32—41no—
423C2×C32—42no—
424C2×C32—44no—
425C2×C32—44no—
426C2×C32—64no—
427C2×C32—122no—
428C2×C32C2×D3×D412yes—
429C2×C32C2×#23714yes—
430C2×C32D3×#4112yes—
431C2×C32C22×#6711yescharacteristic
432C3×D3—28no—
433C3×D3—28no—
434C3×D3—42no—
435C3×D3—42no—
436C3×D3—82no—
437C24—31no—
438#41—184no—
439C2×Q8—91no—
440C2×D4—32no—
441C2×D4—94no—
442C2×D4—91no—
443C22×C4—92no—
444C22×C4—91no—
445C22×C4—272no—
446C22×C4—272no—
447C22×C4—271no—
448C22×C4—544no—
449C4⋊C4—274no—
450C4⋊C4—272no—
451C22⋊C4—94no—
452C22⋊C4—92no—
453C22⋊C4—274no—
454C42—272no—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—21no—
460C22×C3—22no—
461C22×C3—24no—
462C22×C3—38no—
463C22×C3—44no—
464C22×C3—44no—
465C22×C3—41no—
466C22×C3—42no—
467C22×C3—64no—
468C22×C3—64no—
469C22×C3—616no—
470C22×C3—616no—
471C22×C3—64no—
472C22×C3—64no—
473C22×C3—64no—
474C22×C3—64no—
475C22×C3—64no—
476C22×C3—62no—
477C22×C3—62no—
478C22×C3—122no—
479C22×C3—124no—
480C22×C3D3×#6714yes—
481C22×C3#381112yes—
482C22×C3C22×D3212yes—
483C22×C3C22×D3211yescharacteristic
484C2×D3—28no—
485C2×D3—22no—
486C2×D3—22no—
487C2×D3—24no—
488C2×D3—22no—
489C2×D3#380514yescomplemented
490C3×C4—62no—
491C3×C4—62no—
492C3×C4—92no—
493C3×C4—92no—
494C3×C4—122no—
495C3×C4—184no—
496Dic3—22no—
497Dic3—34no—
498Dic3—34no—
499Dic3—62no—
500Dic3—62no—
501Dic3—64no—
502Dic3—64no—
503Dic3—124no—
504Dic3—184no—
505Dic3—184no—
506Dic3—182no—
507Dic3—364no—
508Dic3—364no—
509C32—22no—
510C32—21no—
511C32—41no—
512C32#558612yescomplemented
513C32#582811yescharacteristic, complemented
514C23—32no—
515C23—32no—
516C23—64no—
517C23—62no—
518C23D3311yescharacteristic
519Q8—182no—
520D4—64no—
521D4—182no—
522D4—188no—
523C2×C4—32no—
524C2×C4—91no—
525C2×C4—91no—
526C2×C4—94no—
527C2×C4—94no—
528C2×C4—184no—
529C2×C4—272no—
530C2×C4—544no—
531C2×C4—544no—
532C2×C4—542no—
533C2×C3—24no—
534C2×C3—24no—
535C2×C3—22no—
536C2×C3—24no—
537C2×C3—22no—
538C2×C3—21no—
539C2×C3—22no—
540C2×C3—44no—
541C2×C3—44no—
542C2×C3—42no—
543C2×C3—41no—
544C2×C3—68no—
545C2×C3—68no—
546C2×C3—122no—
547C2×C3—122no—
548C2×C3—242no—
549C2×C3D3×#23714yes—
550C2×C3C2×#380512yes—
551C2×C3C2×D3×#6712yes—
552C2×C3C2×#381111yescharacteristic
553D3—24no—
554D3—42no—
555C22—22no—
556C22—34no—
557C22—68no—
558C22—62no—
559C22—62no—
560C22—64no—
561C22—62no—
562C22#8895212yes—
563C22C2×D3311yescharacteristic
564C4—62no—
565C4—92no—
566C4—92no—
567C4—184no—
568C4—542no—
569C3—21no—
570C3—22no—
571C3—41no—
572C3#1059515412yescomplemented
573C3#1059550111yescharacteristic, complemented
574C2—22no—
575C2—64no—
576C2—122no—
577C2D3×#380512yes—
578C2C2×#8895211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC23×C33C23
hypercenterC23D33
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×D3×C3⋊Dic3C2semidirect
2. C2×C3×#3805C2semidirect
3. #11723715C2semidirect
4. #11723603C2semidirect
5. C3×#63976C2semidirect
6. #11722976C2semidirect
7. #11722953C2semidirect
8. #11722924C2semidirect
18. C2×C3×#627C22semidirect
20. C2×C3×#625C22semidirect
22. C2×C3×D3×Dic3C22semidirect
24. #88815C22semidirect
25. #88813C22semidirect
26. #88807C22semidirect
27. (C2×D3)⋊(C3⋊Dic3)C22semidirect
31. C2×#3805D3semidirect
39. #63976D3semidirect
57. C2×C3×C3⋊Dic3D4semidirect
96. C2×#627C2×D3semidirect
98. C2×#625C2×D3semidirect
106. C2×D3×Dic3C2×D3semidirect
134. C2×C32×D3#41semidirect
193. C2×C3⋊Dic3#67semidirect
321. C2×C3×D3#237semidirect
349. C33#525semidirect
489. C2×D3#3805semidirect
512. C32#5586semidirect
513. C32#5828semidirect
572. C3#10595154semidirect
573. C3#10595501semidirect

Maximal subgroups

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

subgroupindex
C22×D3×C3⋊Dic32
C2×C3×#38052
#117237152
#117236032
C3×#639762
#117229762
#117229532
#117229242
#105955013
#105951543