Finite Group Databasebeta

← #13178051

Subgroups — #13178051

Order 1728.

2500 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×Dic3C211yesmaximal, characteristic, complemented
2C2×#89031C211yesmaximal, characteristic, complemented
3#11723994C211yesmaximal, characteristic, complemented
4C3×#64162C211yesmaximal, characteristic, complemented
5#11723640C212yesmaximal, complemented
6#11723632C212yesmaximal, complemented
7#11723618C212yesmaximal, complemented
8Dic3×#638C212yesmaximal, complemented
9#11723612C211yesmaximal, characteristic, complemented
10#11723605C211yesmaximal, characteristic, complemented
11#11723594C211yesmaximal, characteristic, complemented
12#10596262—31nomaximal
13#10595498—34nomaximal
14C24×C33C2211yescharacteristic
15C2×#6312C2211yescharacteristic
16C2×#6312C2212yes—
17C2×C3×#638C2211yescharacteristic
18C2×C3×#638C2212yes—
19C22×C32×Dic3—24no—
20C22×C32×Dic3C2211yescharacteristic
21C22×C32×Dic3C2212yes—
22C22×C32×Dic3C2212yes—
23C22×C32×Dic3C2211yescharacteristic
24C2×#6293C2211yescharacteristic, complemented
25C2×#6292C2211yescharacteristic, complemented
26C2×#6289C2211yescharacteristic, complemented
27#89031—24no—
28C2×Dic3×C3⋊D3C2211yescharacteristic, complemented
29#88864C2211yescharacteristic
30#88864C2212yes—
31C3×#3757C2211yescharacteristic
32C3×#3757C2212yes—
33#88807C2212yescomplemented
34#88806C2212yescomplemented
35Dic3⋊(C3⋊Dic3)C2212yescomplemented
36#88804C2212yescomplemented
37#88803C2212yescomplemented
38Dic3×C3⋊Dic3C2212yescomplemented
39C23×C32×C4—31no—
40C2×#3828—31no—
41C23×C3×Dic3D314yescomplemented
42C2×#3804—34no—
43#64162—31no—
44#64162—34no—
45#64162—64no—
46#64162D311yescharacteristic, complemented
47#64139—32no—
48#64138—31no—
49#64138—31no—
50C4×#638—32no—
51C4×#638—32no—
52#64135—31no—
53#64133—32no—
54C3×#953—34no—
55#63978—38no—
56Dic3×#67—38no—
57#63966—34no—
58#63964—38no—
59#63961—34no—
60#63955—34no—
61#63952—38no—
62C23×C33—22no—
63C23×C33—24no—
64C23×C33C2311yescharacteristic
65C23×C33C2312yes—
66C22×C3⋊(C3⋊D3)C2311yescharacteristic
67C22×C3×C3⋊D3C2311yescharacteristic
68#6312—22no—
69#6312—24no—
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)C2312yes—
72C3×#638—22no—
73C3×#638—24no—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2312yes—
76C2×C32×Dic3—22no—
77C2×C32×Dic3—24no—
78C2×C32×Dic3—24no—
79C2×C32×Dic3D414yescomplemented
80C2×C32×Dic3C2312yes—
81C2×C32×Dic3C2311yescharacteristic
82C2×C32×Dic3C2311yescharacteristic
83#6293—22no—
84#6292—22no—
85#6289—22no—
86Dic3×C3⋊D3—22no—
87#5791—94no—
88#5586—91no—
89C24×C32—24no—
90C24×C32C2×D314yes—
91C24×C32C2×D311yescharacteristic
92C22×C32×C4—31no—
93C22×C32×C4—32no—
94C22×C32×C4—32no—
95C22×C32×C4—31no—
96C22×C32×C4—64no—
97C2×#638—31no—
98C2×#638—32no—
99C2×#638—34no—
100C2×#638—38no—
101C2×#638—68no—
102C2×#638—64no—
103C2×#638C2×D311yescharacteristic
104C2×#638C2×D312yes—
105#3828—64no—
106C2×C4⋊(C3⋊D3)—31no—
107C2×C4×C3⋊D3—31no—
108C2×C4×C3⋊D3—31no—
109C2×#634—31no—
110C2×C3×#67—34no—
111C2×C3×#67—38no—
112C22×C3×Dic3—216no—
113C22×C3×Dic3C2×D318yes—
114C22×C3×Dic3C2×D314yes—
115C22×C3×Dic3C2×D318yes—
116C22×C3×Dic3C2×D314yes—
117C2×#627—34no—
118C2×#626—34no—
119C2×#624—34no—
120#3804—616no—
121C2×D3×Dic3—34no—
122#3757—32no—
123#3757—31no—
124#3757—34no—
125#3757—38no—
126#3757—68no—
127#3757—64no—
128#3757C2×D312yes—
129#3757C2×D311yescharacteristic
130#3752—32no—
131#3752—32no—
132C4⋊(C3⋊Dic3)—32no—
133#3750—32no—
134#3750—32no—
135C4×C3⋊Dic3—32no—
136C3×#217—38no—
137C3×#217—34no—
138#3724—38no—
139Dic3⋊Dic3—38no—
140Dic3⋊Dic3—38no—
141#3722—38no—
142(C2×D3)⋊Dic3—38no—
143Dic32—38no—
144C22×C33—28no—
145C22×C33—22no—
146C22×C33—22no—
147C22×C33—22no—
148C22×C33C2×D412yes—
149C22×C33#4114yescomplemented
150C22×C33C2411yescharacteristic
151C2×C3⋊(C3⋊D3)—21no—
152C2×C3⋊(C3⋊D3)—22no—
153C2×C3×C3⋊D3—21no—
154C2×C3×C3⋊D3—22no—
155C3⋊(C3⋊Dic3)—21no—
156C3⋊(C3⋊Dic3)—22no—
157C3×C3⋊Dic3—21no—
158C3×C3⋊Dic3—22no—
159C32×Dic3—22no—
160C32×Dic3C2×D412yes—
161C32×Dic3C2×D412yes—
162C23×C3×C4—34no—
163C23×Dic3C3⋊D311yescharacteristic, complemented
164C2×#237—91no—
165C2×#235—94no—
166C3×#120—91no—
167#953—34no—
168#953—94no—
169#953—91no—
170#953—184no—
171#939—92no—
172#935—91no—
173D4×Dic3—92no—
174#930—98no—
175#929—94no—
176#929—94no—
177C4×#67—98no—
178C4×#67—98no—
179#926—94no—
180#924—98no—
181#886—91no—
182#67⋊C4—92no—
183#878—92no—
184#877—91no—
185C23×C32—22no—
186C23×C32—24no—
187C23×C32—28no—
188C23×C32—24no—
189C23×C32—216no—
190C23×C32—28no—
191C23×C32—416no—
192C23×C32—48no—
193C23×C32C22×D318yes—
194C23×C32C22×D314yes—
195C23×C32C22×D311yescharacteristic
196C23×C32C22×D312yes—
197C22×C3⋊D3—31no—
198C22×C3⋊D3—34no—
199C22×C3⋊D3—64no—
200C22×C3⋊D3C22×D311yescharacteristic
201C22×C3×D3—34no—
202C2×C32×C4—34no—
203C2×C32×C4—31no—
204C2×C32×C4—31no—
205C2×C32×C4—32no—
206C2×C32×C4—64no—
207C2×C32×C4—64no—
208C2×C32×C4—62no—
209#638—22no—
210#638—24no—
211#638—62no—
212#638—64no—
213#638—616no—
214#638—68no—
215#638—1216no—
216#638—128no—
217C2×C3⋊Dic3—32no—
218C2×C3⋊Dic3—31no—
219C2×C3⋊Dic3—38no—
220C2×C3⋊Dic3—34no—
221C2×C3⋊Dic3—68no—
222C2×C3⋊Dic3—64no—
223C2×C3⋊Dic3C22×D312yes—
224C2×C3⋊Dic3C22×D311yescharacteristic
225C4⋊(C3⋊D3)—62no—
226C4×C3⋊D3—62no—
227C4×C3⋊D3—62no—
228#634—62no—
229C3×#67—616no—
230C3×#67—68no—
231C2×C3×Dic3—28no—
232C2×C3×Dic3—216no—
233C2×C3×Dic3—216no—
234C2×C3×Dic3—38no—
235C2×C3×Dic3—34no—
236C2×C3×Dic3#67116yescomplemented
237C2×C3×Dic3C22×D318yes—
238C2×C3×Dic3C22×D314yes—
239C2×C3×Dic3C22×D314yes—
240#627—68no—
241#626—68no—
242#624—68no—
243D3×Dic3—68no—
244#525—271noSylow
245C2×C33—22no—
246C2×C33—24no—
247C2×C33C22×D411yescharacteristic
248C2×C33C2×#4112yes—
249C3⋊(C3⋊D3)—41no—
250C3×C3⋊D3—41no—
251C24×C3—24no—
252C24×C3D3214yes—
253C24×C3C2×C3⋊D311yescharacteristic
254C2×C3×D4—92no—
255C2×C3×D4—91no—
256C22×C3×C4—34no—
257C22×C3×C4—38no—
258C22×C3×C4—38no—
259C22×C3×C4—34no—
260C22×C3×C4—616no—
261C2×#67—38no—
262C2×#67—34no—
263C2×#67—98no—
264C2×#67—94no—
265C2×#67—92no—
266C2×#67—91no—
267C2×#67—91no—
268C2×#67—188no—
269C2×#67—184no—
270C22×Dic3—24no—
271C22×Dic3—91no—
272C22×Dic3C2×C3⋊D312yes—
273C22×Dic3C2×C3⋊D312yes—
274C22×Dic3C2×C3⋊D311yescharacteristic
275C22×Dic3C2×C3⋊D311yescharacteristic
276#237—184no—
277#235—1816no—
278C2×D12—94no—
279C2×C4×D3—94no—
280C2×C4×D3—94no—
281C2×C4×D3—91no—
282C2×Dic6—94no—
283C2×Dic6—91no—
284C3×C22⋊C4—91no—
285C3×C22⋊C4—92no—
286#217—38no—
287#217—34no—
288#217—98no—
289#217—94no—
290#217—91no—
291#217—92no—
292#217—92no—
293#217—184no—
294#217—188no—
295(C2×D3)⋊C4—98no—
296(C2×D3)⋊C4—98no—
297(C2×D3)⋊C4—92no—
298C4⋊Dic3—98no—
299C4⋊Dic3—92no—
300Dic3⋊C4—98no—
301Dic3⋊C4—98no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304C4×Dic3—98no—
305C4×Dic3—92no—
306C22×C32—28no—
307C22×C32—22no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—24no—
311C22×C32—216no—
312C22×C32—28no—
313C22×C32—232no—
314C22×C32—28no—
315C22×C32—28no—
316C22×C32—28no—
317C22×C32—48no—
318C22×C32—48no—
319C22×C32—48no—
320C22×C32—432no—
321C22×C32#235116yescomplemented
322C22×C32D3×D412yes—
323C22×C32#23714yescomplemented
324C22×C32C2×#6718yes—
325C22×C32C23×D314yes—
326C22×C32C23×D311yescharacteristic
327C2×C3⋊D3—21no—
328C2×C3⋊D3—22no—
329C2×C3⋊D3—61no—
330C2×C3⋊D3—62no—
331C2×C3⋊D3—68no—
332C2×C3⋊D3—64no—
333C2×C3⋊D3—128no—
334C2×C3⋊D3—124no—
335C2×C3×D3—68no—
336C2×C3×D3—64no—
337C32×C4—32no—
338C32×C4—32no—
339C32×C4—62no—
340C3⋊Dic3—21no—
341C3⋊Dic3—22no—
342C3⋊Dic3—61no—
343C3⋊Dic3—62no—
344C3⋊Dic3—68no—
345C3⋊Dic3—64no—
346C3⋊Dic3—128no—
347C3⋊Dic3—124no—
348C3×Dic3—28no—
349C3×Dic3—68no—
350C3×Dic3—64no—
351C3×Dic3C2×#6718yes—
352C3×Dic3C2×#6718yes—
353C2×#41—271no—
354C23×C4—31no—
355#123—271no—
356#123—271no—
357C22⋊Q8—272no—
358#121—272no—
359#120—91no—
360#120—271no—
361C4×D4—272no—
362C4×D4—272no—
363#117—271no—
364C33#52511yescharacteristic, complemented, Sylow
365C23×C3—216no—
366C23×C3—28no—
367C23×C3—24no—
368C23×C3—22no—
369C23×C3—28no—
370C23×C3—24no—
371C23×C3—416no—
372C23×C3—48no—
373C23×C3—91no—
374C23×C3C2×D3218yes—
375C23×C3C2×D3214yes—
376C23×C3C22×C3⋊D312yes—
377C23×C3C22×C3⋊D311yescharacteristic
378C22×D3—34no—
379C22×D3—94no—
380C22×D3—91no—
381C22×D3—184no—
382C3×D4—184no—
383C3×D4—182no—
384C2×C3×C4—34no—
385C2×C3×C4—34no—
386C2×C3×C4—316no—
387C2×C3×C4—38no—
388C2×C3×C4—616no—
389C2×C3×C4—616no—
390C2×C3×C4—68no—
391C2×C3×C4—92no—
392C2×C3×C4—91no—
393#67—616no—
394#67—68no—
395#67—1816no—
396#67—188no—
397#67—184no—
398#67—182no—
399#67—182no—
400#67—3616no—
401#67—368no—
402C2×Dic3—22no—
403C2×Dic3—24no—
404C2×Dic3—24no—
405C2×Dic3—38no—
406C2×Dic3—34no—
407C2×Dic3—98no—
408C2×Dic3—94no—
409C2×Dic3—92no—
410C2×Dic3—91no—
411C2×Dic3—188no—
412C2×Dic3—184no—
413C2×Dic3—182no—
414C2×Dic3#63814yescomplemented
415C2×Dic3C22×C3⋊D312yes—
416C2×Dic3C22×C3⋊D311yescharacteristic
417C2×Dic3C22×C3⋊D311yescharacteristic
418D12—188no—
419C4×D3—188no—
420C4×D3—188no—
421C4×D3—182no—
422Dic6—188no—
423Dic6—182no—
424C2×C32—22no—
425C2×C32—24no—
426C2×C32—28no—
427C2×C32—24no—
428C2×C32—216no—
429C2×C32—28no—
430C2×C32—416no—
431C2×C32—48no—
432C2×C32C2×#23518yes—
433C2×C32C2×D3×D411yescharacteristic
434C2×C32C2×#23712yes—
435C2×C32C22×#6714yes—
436C3⋊D3—41no—
437C3⋊D3—121no—
438C3⋊D3—124no—
439C3⋊D3—244no—
440C3×D3—124no—
441C24D3×C3⋊D311yescharacteristic
442#41—544no—
443C2×Q8—271no—
444C2×D4—92no—
445C2×D4—91no—
446C2×D4—271no—
447C2×D4—272no—
448C2×D4—271no—
449C22×C4—31no—
450C22×C4—32no—
451C22×C4—32no—
452C22×C4—31no—
453C22×C4—64no—
454C22×C4—271no—
455C22×C4—271no—
456C4⋊C4—272no—
457C4⋊C4—272no—
458C4⋊C4—272no—
459C22⋊C4—91no—
460C22⋊C4—92no—
461C22⋊C4—272no—
462C22⋊C4—272no—
463C22⋊C4—271no—
464C22⋊C4—272no—
465C42—272no—
466C22×C3—28no—
467C22×C3—232no—
468C22×C3—28no—
469C22×C3—28no—
470C22×C3—28no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—24no—
475C22×C3—216no—
476C22×C3—28no—
477C22×C3—48no—
478C22×C3—48no—
479C22×C3—48no—
480C22×C3—432no—
481C22×C3—182no—
482C22×C3—181no—
483C22×C3#3804116yescomplemented
484C22×C3D3×#6718yes—
485C22×C3#382814yescomplemented
486C22×C3C2×#63812yes—
487C22×C3C22×D3214yes—
488C22×C3C23×C3⋊D311yescharacteristic
489C2×D3—68no—
490C2×D3—64no—
491C2×D3—188no—
492C2×D3—184no—
493C2×D3—182no—
494C2×D3—181no—
495C2×D3—368no—
496C2×D3—364no—
497C3×C4—38no—
498C3×C4—38no—
499C3×C4—68no—
500C3×C4—182no—
501C3×C4—181no—
502Dic3—22no—
503Dic3—68no—
504Dic3—64no—
505Dic3—188no—
506Dic3—184no—
507Dic3—182no—
508Dic3—181no—
509Dic3—368no—
510Dic3—364no—
511Dic3C2×#63812yes—
512Dic3C2×#63812yes—
513C32—24no—
514C32#558611yescharacteristic, complemented
515C32#579114yescomplemented
516C23—24no—
517C23—22no—
518C23—91no—
519C23—271no—
520C23C2×D3×C3⋊D312yes—
521C23C2×D3×C3⋊D311yescharacteristic
522Q8—542no—
523D4—184no—
524D4—182no—
525D4—542no—
526D4—544no—
527D4—542no—
528C2×C4—31no—
529C2×C4—31no—
530C2×C4—34no—
531C2×C4—32no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—62no—
535C2×C4—92no—
536C2×C4—91no—
537C2×C4—272no—
538C2×C4—271no—
539C2×C4—542no—
540C2×C4—542no—
541C2×C3—24no—
542C2×C3—22no—
543C2×C3—216no—
544C2×C3—28no—
545C2×C3—28no—
546C2×C3—24no—
547C2×C3—416no—
548C2×C3—48no—
549C2×C3—361no—
550C2×C3C2×#380418yes—
551C2×C3C2×D3×#6714yes—
552C2×C3C2×#382812yes—
553C2×C3C22×#63811yescharacteristic
554D3—124no—
555D3—361no—
556D3—364no—
557D3—724no—
558C22—28no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—182no—
563C22—181no—
564C22—542no—
565C22—541no—
566C22#8903114yescomplemented
567C22D3×#63812yes—
568C22C22×D3×C3⋊D311yescharacteristic
569C4—32no—
570C4—32no—
571C4—62no—
572C4—182no—
573C4—181no—
574C4—542no—
575C4—541no—
576C3—24no—
577C3#1059549814yescomplemented
578C3#1059626211yescharacteristic, complemented
579C2—24no—
580C2—22no—
581C2—361no—
582C2—1081no—
583C2C2×#8903112yes—
584C2C2×D3×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC24×C33C22
hypercenterC24D3×C3⋊D3
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C32×Dic3C2semidirect
2. C2×#89031C2semidirect
3. #11723994C2semidirect
4. C3×#64162C2semidirect
5. #11723640C2semidirect
6. #11723632C2semidirect
7. #11723618C2semidirect
8. Dic3×#638C2semidirect
9. #11723612C2semidirect
10. #11723605C2semidirect
11. #11723594C2semidirect
24. C2×#6293C22semidirect
25. C2×#6292C22semidirect
26. C2×#6289C22semidirect
28. C2×Dic3×C3⋊D3C22semidirect
33. #88807C22semidirect
34. #88806C22semidirect
35. Dic3⋊(C3⋊Dic3)C22semidirect
36. #88804C22semidirect
37. #88803C22semidirect
38. Dic3×C3⋊Dic3C22semidirect
41. C23×C3×Dic3D3semidirect
46. #64162D3semidirect
79. C2×C32×Dic3D4semidirect
149. C22×C33#41semidirect
163. C23×Dic3C3⋊D3semidirect
236. C2×C3×Dic3#67semidirect
321. C22×C32#235semidirect
323. C22×C32#237semidirect
364. C33#525semidirect
414. C2×Dic3#638semidirect
483. C22×C3#3804semidirect
485. C22×C3#3828semidirect
514. C32#5586semidirect
515. C32#5791semidirect
566. C22#89031semidirect
577. C3#10595498semidirect
578. C3#10596262semidirect

Maximal subgroups

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

subgroupindex
C23×C32×Dic32
C2×#890312
#117239942
C3×#641622
#117236402
#117236322
#117236182
Dic3×#6382
#117236122
#117236052
#117235942
#105962623
#105954983