Finite Group Databasebeta

← #13177398

Subgroups — #13177398

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
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C22×#6289C211yesmaximal, characteristic, complemented
3#11723978C211yesmaximal, characteristic, complemented
4C32×#935C211yesmaximal, characteristic, complemented
5#11723639C211yesmaximal, characteristic, complemented
6#11723623C212yesmaximal, complemented
7#11723622C212yesmaximal, complemented
8#11723605C211yesmaximal, characteristic, complemented
9#11723595C211yesmaximal, characteristic, complemented
10#11723512C212yesmaximal, complemented
11#11723489C212yesmaximal, complemented
12#10596127—31nomaximal
13#10595159—34nomaximal
14C23×C3⋊(C3⋊D3)C2211yescharacteristic
15C2×C33×D4C2211yescharacteristic
16C2×#6312C2212yes—
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
24C22×C32×Dic3C2211yescharacteristic
25C2×#6292C2212yes—
26C2×#6289—24no—
27C2×#6289C2212yes—
28C2×#6289C2212yes—
29#88864C2211yescharacteristic
30#88859C2212yes—
31C32×#217C2211yescharacteristic
32C32×#217C2212yescomplemented
33C32×Dic3⋊C4C2212yescomplemented
34#88807C2212yescomplemented
35#88806C2212yescomplemented
36#88804C2212yes—
37#88803C2212yescomplemented
38Dic3×C3⋊Dic3C2212yescomplemented
39C2×#3830D311yescharacteristic, complemented
40C22×C4×C3⋊D3—31no—
41C2×C3×#237—34no—
42C22×#624—34no—
43C32×#123—31no—
44#64146—31no—
45#64146—34no—
46#64146—64no—
47#64138—31no—
48#64105—32no—
49#64102—32no—
50#64093—32no—
51#64091—31no—
52#64090—32no—
53#64086—31no—
54C3×#935D314yescomplemented
55#63981—34no—
56#63961—34no—
57#63958—38no—
58#63955—34no—
59#63952—38no—
60#63904—38no—
61#63888—38no—
62C23×C33C2311yescharacteristic
63C23×C33C2311yescharacteristic
64C22×C3⋊(C3⋊D3)—22no—
65C22×C3⋊(C3⋊D3)—24no—
66C22×C3⋊(C3⋊D3)C2312yes—
67C22×C3×C3⋊D3C2311yescharacteristic
68C33×D4—22no—
69C2×C33×C4C2311yescharacteristic
70#6312—24no—
71C2×C3⋊(C3⋊Dic3)C2312yes—
72C3×#638—22no—
73C3×#638—22no—
74C2×C3×C3⋊Dic3—22no—
75C2×C3×C3⋊Dic3D414yescomplemented
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C3×C4×C3⋊D3—22no—
80C3×#634—22no—
81C2×C32×Dic3—22no—
82C2×C32×Dic3C2312yes—
83C2×C32×Dic3C2312yes—
84#6292—24no—
85#6289—24no—
86#6289—24no—
87#5828—91no—
88#5656—94no—
89C23×C3⋊D3—31no—
90C23×C3⋊D3—34no—
91C23×C3⋊D3—64no—
92C2×C32×D4—24no—
93C2×C32×D4C2×D314yes—
94C2×C32×D4C2×D311yescharacteristic
95C22×C32×C4—31no—
96C2×#638—32no—
97C2×#638—38no—
98C2×#638—68no—
99C2×#638C2×D311yescharacteristic
100C2×#638C2×D311yescharacteristic, complemented
101C22×C3⋊Dic3C2×D311yescharacteristic
102C22×C3⋊Dic3C2×D311yescharacteristic, complemented
103#3830—24no—
104C2×C4⋊(C3⋊D3)—32no—
105C2×C4×C3⋊D3—32no—
106C2×C4×C3⋊D3—32no—
107C2×C4×C3⋊D3—64no—
108C2×C4×C3⋊D3C2×D311yescharacteristic, complemented
109C2×#634C2×D311yescharacteristic, complemented
110C2×C3×#67—34no—
111C2×C3×#67—34no—
112C22×C3×Dic3—34no—
113C22×C3×Dic3—34no—
114C22×C3×Dic3C2×D314yes—
115C3×#237—616no—
116C2×C3×C4×D3—34no—
117C2×C3×Dic6—34no—
118C2×#626—38no—
119C2×#624—38no—
120C2×#624—38no—
121C2×#624—616no—
122C32×C4⋊C4—32no—
123C32×C22⋊C4—31no—
124C32×C22⋊C4—32no—
125#3757—31no—
126#3757—34no—
127#3757—64no—
128#3752—32no—
129#3752—32no—
130#3752—38no—
131#3752—32no—
132#3752—68no—
133#3750—32no—
134#3750—32no—
135C4×C3⋊Dic3—32no—
136C3×#217C2×D314yes—
137C3×#217C2×D318yescomplemented
138C3×Dic3⋊C4C2×D318yescomplemented
139#3724—38no—
140Dic3⋊Dic3—38no—
141#3722—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)—28no—
150C2×C3⋊(C3⋊D3)—22no—
151C2×C3⋊(C3⋊D3)—22no—
152C2×C3⋊(C3⋊D3)#4114yescomplemented
153C2×C3×C3⋊D3—21no—
154C2×C3×C3⋊D3—22no—
155C33×C4—21no—
156C3⋊(C3⋊Dic3)—22no—
157C3×C3⋊Dic3—21no—
158C3×C3⋊Dic3C2×D412yes—
159C3×C3⋊Dic3C2×D412yes—
160C32×Dic3—22no—
161C32×Dic3—22no—
162C2×C3×#41—91no—
163C2×#237—34no—
164C22×C4×D3—94no—
165C22×C4×D3—91no—
166C3×#123—34no—
167#946—92no—
168#938—92no—
169#937—94no—
170#937—91no—
171#937—91no—
172#937—184no—
173#935—91no—
174#935C3⋊D311yescharacteristic, complemented
175#929—94no—
176C4×#67—92no—
177C4×#67—92no—
178#926—91no—
179#896—98no—
180#893—98no—
181#884—98no—
182#882—94no—
183#67⋊C4—98no—
184#877—94no—
185C23×C32—24no—
186C23×C32—24no—
187C23×C32C22×D314yes—
188C23×C32C22×D314yes—
189C23×C32C22×D311yescharacteristic
190C23×C32C22×D311yescharacteristic
191C22×C3⋊D3—32no—
192C22×C3⋊D3—38no—
193C22×C3⋊D3—68no—
194C22×C3⋊D3—64no—
195C22×C3⋊D3—62no—
196C22×C3⋊D3—616no—
197C22×C3⋊D3—68no—
198C22×C3⋊D3—1216no—
199C22×C3⋊D3—128no—
200C22×C3⋊D3C22×D311yescharacteristic
201C22×C3×D3—34no—
202C32×D4—22no—
203C32×D4—28no—
204C32×D4—48no—
205C2×C32×C4—24no—
206C2×C32×C4—32no—
207C2×C32×C4—32no—
208C2×C32×C4—62no—
209C2×C32×C4C22×D314yes—
210C2×C32×C4C22×D311yescharacteristic
211#638—22no—
212#638—22no—
213#638—64no—
214#638—616no—
215#638—1216no—
216C2×C3⋊Dic3—22no—
217C2×C3⋊Dic3—32no—
218C2×C3⋊Dic3—38no—
219C2×C3⋊Dic3—68no—
220C2×C3⋊Dic3#6714yescomplemented
221C2×C3⋊Dic3C22×D311yescharacteristic
222C2×C3⋊Dic3C22×D311yescharacteristic
223C2×C3⋊Dic3C22×D311yescharacteristic
224C4⋊(C3⋊D3)—64no—
225C4×C3⋊D3—22no—
226C4×C3⋊D3—64no—
227C4×C3⋊D3—64no—
228#634—22no—
229C3×#67—68no—
230C3×#67—68no—
231C2×C3×Dic3—28no—
232C2×C3×Dic3—316no—
233C2×C3×Dic3—34no—
234C2×C3×Dic3—34no—
235C2×C3×Dic3—34no—
236C2×C3×Dic3—68no—
237C2×C3×Dic3C22×D318yes—
238C2×C3×Dic3C22×D318yes—
239C3×C4×D3—68no—
240C3×Dic6—68no—
241#626—616no—
242#624—616no—
243#624—616no—
244#525—271noSylow
245C2×C33—22no—
246C2×C33—41no—
247C2×C33C22×D411yescharacteristic
248C2×C33C2×#4112yes—
249C3⋊(C3⋊D3)—24no—
250C3×C3⋊D3—41no—
251C23×D3—94no—
252C23×D3—91no—
253C23×D3—184no—
254C3×#41—184no—
255C2×C3×Q8—91no—
256C2×C3×D4—24no—
257C2×C3×D4—91no—
258C2×C3×D4—91no—
259C2×C3×D4D3214yes—
260C2×C3×D4C2×C3⋊D311yescharacteristic
261C22×C3×C4—34no—
262C22×C3×C4—91no—
263C22×C3×C4—91no—
264C22×C3×C4—91no—
265C2×#67—34no—
266C2×#67—34no—
267C2×#67—98no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—188no—
271C22×Dic3—34no—
272C22×Dic3—34no—
273C22×Dic3C2×C3⋊D311yescharacteristic
274#237—616no—
275C2×D12—98no—
276C2×C4×D3—34no—
277C2×C4×D3—98no—
278C2×C4×D3—98no—
279C2×C4×D3—92no—
280C2×C4×D3—92no—
281C2×C4×D3—184no—
282C2×C4×D3—1816no—
283C2×Dic6—34no—
284C3×C4⋊C4—38no—
285C3×C22⋊C4—34no—
286C3×C22⋊C4—38no—
287#217—94no—
288#217—91no—
289#217—92no—
290#217—184no—
291#217C2×C3⋊D311yescharacteristic
292#217C2×C3⋊D312yescomplemented
293(C2×D3)⋊C4—98no—
294(C2×D3)⋊C4—98no—
295(C2×D3)⋊C4—98no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—188no—
299C4⋊Dic3—92no—
300Dic3⋊C4—98no—
301Dic3⋊C4—98no—
302Dic3⋊C4—92no—
303Dic3⋊C4C2×C3⋊D312yescomplemented
304C4×Dic3—98no—
305C4×Dic3—92no—
306C22×C32—22no—
307C22×C32—22no—
308C22×C32—21no—
309C22×C32—24no—
310C22×C32—28no—
311C22×C32—28no—
312C22×C32—24no—
313C22×C32—28no—
314C22×C32—48no—
315C22×C32—48no—
316C22×C32—44no—
317C22×C32D3×D418yes—
318C22×C32C2×#6712yes—
319C22×C32C23×D314yes—
320C22×C32C23×D311yescharacteristic
321C2×C3⋊D3—22no—
322C2×C3⋊D3—21no—
323C2×C3⋊D3—34no—
324C2×C3⋊D3—316no—
325C2×C3⋊D3—616no—
326C2×C3⋊D3—68no—
327C2×C3⋊D3—62no—
328C2×C3⋊D3—62no—
329C2×C3⋊D3—632no—
330C2×C3⋊D3—68no—
331C2×C3⋊D3—68no—
332C2×C3⋊D3—128no—
333C2×C3⋊D3—128no—
334C2×C3⋊D3—1232no—
335C2×C3×D3—64no—
336C2×C3×D3—68no—
337C32×C4—21no—
338C32×C4—24no—
339C32×C4—44no—
340C32×C4—62no—
341C32×C4—62no—
342C3⋊Dic3—21no—
343C3⋊Dic3—62no—
344C3⋊Dic3—68no—
345C3⋊Dic3—128no—
346C3⋊Dic3C2×#6712yes—
347C3⋊Dic3C2×#6712yes—
348C3×Dic3—28no—
349C3×Dic3—28no—
350C3×Dic3—38no—
351C3×Dic3—38no—
352C3×Dic3—64no—
353C2×#41—91no—
354C23×C4—271no—
355#123—31no—
356#123—271no—
357C22⋊Q8—272no—
358#121—272no—
359#120—271no—
360#120—271no—
361C4×D4—272no—
362C4×D4—272no—
363#117—271no—
364C33#52511yescharacteristic, complemented, Sylow
365C23×C3—24no—
366C23×C3—24no—
367C23×C3—91no—
368C23×C3C2×D3214yes—
369C23×C3C2×D3214yes—
370C23×C3C22×C3⋊D311yescharacteristic
371C23×C3C22×C3⋊D311yescharacteristic
372C22×D3—34no—
373C22×D3—98no—
374C22×D3—92no—
375C22×D3—188no—
376C22×D3—184no—
377C22×D3—1816no—
378C22×D3—188no—
379C22×D3—182no—
380C22×D3—3616no—
381C22×D3—368no—
382C3×Q8—182no—
383C3×D4—28no—
384C3×D4—22no—
385C3×D4—48no—
386C3×D4—182no—
387C3×D4—182no—
388C2×C3×C4—24no—
389C2×C3×C4—38no—
390C2×C3×C4—38no—
391C2×C3×C4—68no—
392C2×C3×C4—91no—
393C2×C3×C4—91no—
394C2×C3×C4—94no—
395C2×C3×C4—91no—
396C2×C3×C4—182no—
397C2×C3×C4—182no—
398C2×C3×C4C2×D3214yes—
399C2×C3×C4C22×C3⋊D311yescharacteristic
400#67—68no—
401#67—68no—
402#67—184no—
403#67—184no—
404#67—1816no—
405#67—3616no—
406C2×Dic3—22no—
407C2×Dic3—34no—
408C2×Dic3—34no—
409C2×Dic3—316no—
410C2×Dic3—34no—
411C2×Dic3—68no—
412C2×Dic3—98no—
413C2×Dic3—92no—
414C2×Dic3—188no—
415C2×Dic3C22×C3⋊D312yes—
416C2×Dic3C22×C3⋊D312yes—
417D12—1816no—
418C4×D3—68no—
419C4×D3—184no—
420C4×D3—184no—
421C4×D3—1816no—
422C4×D3—1816no—
423Dic6—68no—
424C2×C32—22no—
425C2×C32—28no—
426C2×C32—24no—
427C2×C32—28no—
428C2×C32—48no—
429C2×C32—41no—
430C2×C32—44no—
431C2×C32—84no—
432C2×C32C2×D3×D414yes—
433C2×C32D3×#4118yes—
434C2×C32D3×#4112yes—
435C2×C32C22×#6711yescharacteristic
436C3⋊D3—41no—
437C3⋊D3—64no—
438C3⋊D3—616no—
439C3⋊D3—1216no—
440C3×D3—124no—
441C24—271no—
442#41—184no—
443C2×Q8—91no—
444C2×D4—91no—
445C2×D4—91no—
446C2×D4—272no—
447C2×D4—272no—
448C2×D4D3×C3⋊D311yescharacteristic
449C22×C4—31no—
450C22×C4—91no—
451C22×C4—91no—
452C22×C4—91no—
453C22×C4—272no—
454C22×C4—272no—
455C22×C4—544no—
456C4⋊C4—32no—
457C4⋊C4—272no—
458C4⋊C4—272no—
459C22⋊C4—31no—
460C22⋊C4—32no—
461C22⋊C4—272no—
462C22⋊C4—272no—
463C22⋊C4—271no—
464C22⋊C4—272no—
465C42—272no—
466C22×C3—28no—
467C22×C3—22no—
468C22×C3—28no—
469C22×C3—24no—
470C22×C3—21no—
471C22×C3—22no—
472C22×C3—24no—
473C22×C3—28no—
474C22×C3—48no—
475C22×C3—48no—
476C22×C3—44no—
477C22×C3—181no—
478C22×C3—182no—
479C22×C3D3×#6718yes—
480C22×C3D4×C3⋊D312yes—
481C22×C3C22×D3214yes—
482C22×C3C23×C3⋊D311yescharacteristic
483C2×D3—68no—
484C2×D3—64no—
485C2×D3—916no—
486C2×D3—94no—
487C2×D3—1816no—
488C2×D3—188no—
489C2×D3—1832no—
490C2×D3—188no—
491C2×D3—188no—
492C2×D3—182no—
493C2×D3—182no—
494C2×D3—368no—
495C2×D3—368no—
496C2×D3—3632no—
497C3×C4—24no—
498C3×C4—21no—
499C3×C4—44no—
500C3×C4—68no—
501C3×C4—68no—
502C3×C4—92no—
503C3×C4—92no—
504C3×C4—181no—
505Dic3—22no—
506Dic3—22no—
507Dic3—38no—
508Dic3—38no—
509Dic3—64no—
510Dic3—188no—
511Dic3—182no—
512Dic3—368no—
513C32—24no—
514C32#565614yescomplemented
515C32#582811yescharacteristic, complemented
516C23—91no—
517C23—272no—
518C23—544no—
519C23—542no—
520C23C2×D3×C3⋊D311yescharacteristic
521C23C2×D3×C3⋊D311yescharacteristic
522Q8—182no—
523D4—22no—
524D4—182no—
525D4—182no—
526D4—544no—
527D4—544no—
528C2×C4—32no—
529C2×C4—32no—
530C2×C4—62no—
531C2×C4—91no—
532C2×C4—91no—
533C2×C4—94no—
534C2×C4—91no—
535C2×C4—182no—
536C2×C4—182no—
537C2×C4—272no—
538C2×C4—544no—
539C2×C4—544no—
540C2×C4C2×D3×C3⋊D311yescharacteristic
541C2×C3—22no—
542C2×C3—28no—
543C2×C3—28no—
544C2×C3—24no—
545C2×C3—48no—
546C2×C3—41no—
547C2×C3—44no—
548C2×C3—84no—
549C2×C3—361no—
550C2×C3#6431218yes—
551C2×C3C2×D3×#6714yes—
552C2×C3C2×D4×C3⋊D311yescharacteristic
553C2×C3#41×C3⋊D312yes—
554D3—124no—
555D3—184no—
556D3—1816no—
557D3—3616no—
558C22—22no—
559C22—22no—
560C22—21no—
561C22—182no—
562C22—181no—
563C22—274no—
564C22—548no—
565C22—542no—
566C22—542no—
567C22C3⋊D3×#6712yes—
568C22C22×D3×C3⋊D311yescharacteristic
569C4—21no—
570C4—62no—
571C4—62no—
572C4—92no—
573C4—92no—
574C4—181no—
575C4—542no—
576C3—24no—
577C3#1059515914yescomplemented
578C3#1059612711yescharacteristic, complemented
579C2—22no—
580C2—41no—
581C2—361no—
582C2—544no—
583C2#1172458612yes—
584C2C2×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×#6289C2semidirect
3. #11723978C2semidirect
4. C32×#935C2semidirect
5. #11723639C2semidirect
6. #11723623C2semidirect
7. #11723622C2semidirect
8. #11723605C2semidirect
9. #11723595C2semidirect
10. #11723512C2semidirect
11. #11723489C2semidirect
18. C2×C3×#638C22semidirect
20. C22×C3×C3⋊Dic3C22semidirect
22. C2×C3×C4×C3⋊D3C22semidirect
23. C2×C3×#634C22semidirect
32. C32×#217C22semidirect
33. C32×Dic3⋊C4C22semidirect
34. #88807C22semidirect
35. #88806C22semidirect
37. #88803C22semidirect
38. Dic3×C3⋊Dic3C22semidirect
39. C2×#3830D3semidirect
54. C3×#935D3semidirect
75. C2×C3×C3⋊Dic3D4semidirect
100. C2×#638C2×D3semidirect
102. C22×C3⋊Dic3C2×D3semidirect
108. C2×C4×C3⋊D3C2×D3semidirect
109. C2×#634C2×D3semidirect
137. C3×#217C2×D3semidirect
138. C3×Dic3⋊C4C2×D3semidirect
152. C2×C3⋊(C3⋊D3)#41semidirect
174. #935C3⋊D3semidirect
220. C2×C3⋊Dic3#67semidirect
292. #217C2×C3⋊D3semidirect
303. Dic3⋊C4C2×C3⋊D3semidirect
364. C33#525semidirect
514. C32#5656semidirect
515. C32#5828semidirect
577. C3#10595159semidirect
578. C3#10596127semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C22×#62892
#117239782
C32×#9352
#117236392
#117236232
#117236222
#117236052
#117235952
#117235122
#117234892
#105961273
#105951593