Finite Group Databasebeta

← #13170800

Subgroups — #13170800

Order 1728.

2010 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3802C211yesmaximal, characteristic
2C2×D32×Dic3C212yesmaximal, complemented
3C4×#6294C211yesmaximal, characteristic, complemented
4C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
5D12×C3⋊Dic3C212yesmaximal, complemented
6C3×#63898C212yesmaximal, complemented
7#11722924C214yesmaximal, complemented
8#11722902C212yesmaximal, complemented
9D3×D4×Dic3—32nomaximal
10#10594605—31nomaximal
11C22×C3×D32C2212yes—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2212yes—
14C2×#6294C2211yescharacteristic
15C2×#6294C2211yescharacteristic
16C2×Dic3×C3⋊D3C2212yes—
17C2×D3×C3⋊Dic3C2214yescomplemented
18C2×C3×#625C2212yes—
19C2×C3×D3×Dic3C2214yescomplemented
20C3×#3802C418yescomplemented
21D32×Dic3—28no—
22C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
23C3×C4×C3⋊Dic3C2212yescomplemented
24C32×C4⋊Dic3C2211yescharacteristic, complemented
25#88813C2212yescomplemented
26#88807C2212yescomplemented
27(C2×D3)⋊(C3⋊Dic3)C2214yescomplemented
28C3×(C2×D3)⋊Dic3C2214yescomplemented
29C2×C3×D3×D4—32no—
30C22×D3×Dic3—34no—
31C2×#3802D311yescharacteristic
32C2×C4×D32—32no—
33D4×C3⋊Dic3—32no—
34C4⋊C4×C3⋊D3—31no—
35C3×D4×Dic3—32no—
36C3×#893—32no—
37#67⋊Dic3—34no—
38Dic3×#67—34no—
39D3×#217—34no—
40#63921—32no—
41#63898D312yescomplemented
42Dic3×D12—32no—
43Dic3×D12—32no—
44Dic3×D12—62no—
45D3×C4⋊Dic3—32no—
46D3×C4⋊Dic3—61no—
47C4×#624—31no—
48#63879—34no—
49C4×D3×Dic3—32no—
50C22×C3×C3⋊D3C2311yescharacteristic
51C22×C32×D3C2314yes—
52C2×C3×D32—24no—
53C2×C3×D32—24no—
54C2×C3×D32C2×C418yescomplemented
55C2×C33×C4C2311yescharacteristic
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C2×C3×C3⋊Dic3C2312yes—
58C2×C3×C3⋊Dic3C2311yescharacteristic
59C2×C3×C3⋊Dic3C2312yes—
60C3×C4×C3⋊D3C2×C414yescomplemented
61C2×C32×Dic3C2312yes—
62C32×D12C2×C418yescomplemented
63#6294—22no—
64#6294D414yescomplemented
65Dic3×C3⋊D3—24no—
66D3×C3⋊Dic3—28no—
67C3×#625C2×C418yescomplemented
68C3×D3×Dic3—28no—
69C2×D4×Dic3—91no—
70C4×D3×D4—92no—
71C23×C3×D3—34no—
72C22×D32C2×D312yes—
73C2×C32×D4—32no—
74C22×C3⋊Dic3—34no—
75C2×C4×C3⋊D3—32no—
76C2×C4×C3⋊D3C2×D311yescharacteristic
77C2×C3×#67—34no—
78C22×C3×Dic3—34no—
79C3×D3×D4—316no—
80C2×C3×D12—22no—
81C2×C3×D12C2×D312yes—
82C2×C3×D12C2×D312yes—
83C2×C3×C4×D3—32no—
84C2×C3×C4×D3—34no—
85C2×C3×C4×D3—61no—
86C2×#625C2×D312yes—
87C2×#624—31no—
88C2×#624—31no—
89C2×D3×Dic3—32no—
90C2×D3×Dic3—34no—
91C2×D3×Dic3—34no—
92C2×D3×Dic3—32no—
93C2×D3×Dic3—34no—
94C2×D3×Dic3—616no—
95C2×D3×Dic3—62no—
96C2×D3×Dic3—64no—
97C2×D3×Dic3C2×D314yescomplemented
98#3802Dic318yescomplemented
99C4×D32—68no—
100C32×C4⋊C4—31no—
101#3757—34no—
102C4⋊(C3⋊Dic3)—31no—
103C4⋊(C3⋊Dic3)—32no—
104C4⋊(C3⋊Dic3)—61no—
105C4⋊(C3⋊Dic3)—62no—
106C4⋊(C3⋊Dic3)—121no—
107#3750—32no—
108C4×C3⋊Dic3C2×D312yescomplemented
109C3×#217—34no—
110C3×(C2×D3)⋊C4—34no—
111C3×C4⋊Dic3—21no—
112C3×C4⋊Dic3C2×D312yescomplemented
113C3×C4×Dic3—32no—
114C3×C4×Dic3—32no—
115C3×C4×Dic3—62no—
116Dic3⋊Dic3—34no—
117Dic3⋊Dic3—62no—
118(C2×D3)⋊Dic3—34no—
119(C2×D3)⋊Dic3—34no—
120(C2×D3)⋊Dic3—64no—
121(C2×D3)⋊Dic3C2×D314yescomplemented
122Dic32—32no—
123Dic32—32no—
124Dic32—32no—
125C22×C33C2411yescharacteristic
126C2×C3×C3⋊D3C22×C412yes—
127C2×C3×C3⋊D3C2×D412yes—
128C2×C3×C3⋊D3#4112yes—
129C2×C32×D3—24no—
130C2×C32×D3—24no—
131C2×C32×D3C22×C418yescomplemented
132C3×D32—216no—
133C33×C4C22×C412yescomplemented
134C3⋊(C3⋊Dic3)—22no—
135C3×C3⋊Dic3—22no—
136C3×C3⋊Dic3C22×C412yescomplemented
137C3×C3⋊Dic3C2×D414yescomplemented
138C32×Dic3—22no—
139C22×C3×D4—91no—
140C23×Dic3—92no—
141C2×D3×D4—32no—
142C22×C4×D3—94no—
143C3×C4×D4—92no—
144C2×#217—92no—
145D4×Dic3—32no—
146D4×Dic3—92no—
147D4×Dic3—92no—
148D4×Dic3—94no—
149D4×Dic3—182no—
150C4×#67—94no—
151C2×C4⋊Dic3—91no—
152C2×C4×Dic3—91no—
153#893—32no—
154D3×C4⋊C4—92no—
155D3×C4⋊C4—181no—
156#67⋊C4—94no—
157D3×C22⋊C4—94no—
158C4×D12—92no—
159C42×D3—92no—
160C23×C32—34no—
161C22×C3⋊D3C22×D311yescharacteristic
162C22×C3×D3—24no—
163C22×C3×D3—32no—
164C22×C3×D3—316no—
165C22×C3×D3—61no—
166C22×C3×D3—68no—
167C22×C3×D3—68no—
168C22×C3×D3C22×D314yes—
169C22×C3×D3C22×D314yes—
170C2×D32—24no—
171C2×D32—24no—
172C2×D32C2×Dic318yescomplemented
173C32×D4—38no—
174C2×C32×C4—22no—
175C2×C32×C4—21no—
176C2×C32×C4—32no—
177C2×C32×C4—41no—
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D311yescharacteristic
180C2×C3⋊Dic3—32no—
181C2×C3⋊Dic3—34no—
182C2×C3⋊Dic3—64no—
183C2×C3⋊Dic3—62no—
184C2×C3⋊Dic3—68no—
185C2×C3⋊Dic3—122no—
186C2×C3⋊Dic3C22×D312yes—
187C2×C3⋊Dic3C22×D312yes—
188C2×C3⋊Dic3C22×D311yescharacteristic
189C4×C3⋊D3—64no—
190C4×C3⋊D3C2×Dic314yescomplemented
191C3×#67—316no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—68no—
202C2×C3×Dic3C22×D314yes—
203C3×D12—28no—
204C3×D12C4×D318yescomplemented
205C3×D12C2×Dic318yescomplemented
206C3×C4×D3—38no—
207C3×C4×D3—64no—
208C3×C4×D3—68no—
209#625C2×Dic318yescomplemented
210#624—34no—
211#624—62no—
212D3×Dic3—28no—
213D3×Dic3—38no—
214D3×Dic3—68no—
215D3×Dic3—68no—
216D3×Dic3—68no—
217D3×Dic3—64no—
218D3×Dic3—128no—
219D3×Dic3—124no—
220C2×C4×D4—271noSylow
221C2×C33C23×C411yescharacteristic
222C2×C33C22×D411yescharacteristic
223C2×C33C2×#4111yescharacteristic
224C3×C3⋊D3C4×D414yescomplemented
225C32×D3—28no—
226C24×C3—92no—
227C23×D3—34no—
228C2×C3×D4—32no—
229C2×C3×D4—32no—
230C2×C3×D4—62no—
231C2×C3×D4—98no—
232C2×C3×D4—92no—
233C22×C3×C4—94no—
234C22×C3×C4—91no—
235C2×#67—34no—
236C22×Dic3—34no—
237C22×Dic3—91no—
238C22×Dic3—91no—
239C22×Dic3—94no—
240C22×Dic3—94no—
241C22×Dic3—92no—
242C22×Dic3—184no—
243C22×Dic3—188no—
244D3×D4—316no—
245C2×D12D3212yes—
246C2×C4×D3—32no—
247C2×C4×D3—34no—
248C2×C4×D3—61no—
249C2×C4×D3—92no—
250C2×C4×D3—92no—
251C2×C4×D3—94no—
252C2×C4×D3—94no—
253C2×C4×D3—182no—
254C2×C4×D3—1816no—
255C3×C4⋊C4—32no—
256C3×C4⋊C4—61no—
257C3×C22⋊C4—94no—
258C3×C42—92no—
259#217—34no—
260#217—94no—
261#217—94no—
262#217—184no—
263(C2×D3)⋊C4—34no—
264(C2×D3)⋊C4—94no—
265C4⋊Dic3—92no—
266C4⋊Dic3—91no—
267C4⋊Dic3—92no—
268C4⋊Dic3—181no—
269C4⋊Dic3—182no—
270C4⋊Dic3—361no—
271C4⋊Dic3D3211yescharacteristic, complemented
272Dic3⋊C4—94no—
273Dic3⋊C4—182no—
274C4×Dic3—32no—
275C4×Dic3—32no—
276C4×Dic3—62no—
277C4×Dic3—92no—
278C4×Dic3—92no—
279C4×Dic3—92no—
280C22×C32—21no—
281C22×C32—22no—
282C22×C32—38no—
283C22×C32—41no—
284C22×C32—64no—
285C22×C32—64no—
286C22×C32C23×D312yes—
287C22×C32C23×D311yescharacteristic
288C2×C3⋊D3D3×D412yes—
289C2×C3⋊D3#23712yes—
290C2×C3⋊D3C22×Dic312yes—
291C2×C3×D3—24no—
292C2×C3×D3—24no—
293C2×C3×D3—24no—
294C2×C3×D3—24no—
295C2×C3×D3—28no—
296C2×C3×D3—34no—
297C2×C3×D3—34no—
298C2×C3×D3—34no—
299C2×C3×D3—44no—
300C2×C3×D3—44no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C2×C3×D3—632no—
305C2×C3×D3C2×C4×D318yescomplemented
306C2×C3×D3C22×Dic318yescomplemented
307D32—216no—
308C32×C4—22no—
309C32×C4—24no—
310C32×C4—42no—
311C32×C4—62no—
312C32×C4C2×C4×D314yescomplemented
313C32×C4C22×Dic312yescomplemented
314C3⋊Dic3—22no—
315C3⋊Dic3—62no—
316C3⋊Dic3—64no—
317C3⋊Dic3—124no—
318C3⋊Dic3—122no—
319C3⋊Dic3—242no—
320C3⋊Dic3D3×D414yescomplemented
321C3⋊Dic3C22×Dic312yescomplemented
322C3×Dic3—24no—
323C3×Dic3—34no—
324C3×Dic3—34no—
325C3×Dic3—34no—
326C3×Dic3—42no—
327C3×Dic3—62no—
328C3×Dic3—64no—
329C3×Dic3—62no—
330C3×Dic3—62no—
331C3×Dic3—122no—
332C22×D4—91no—
333C23×C4—272no—
334C4×D4—92no—
335C4×D4—272no—
336C4×D4—274no—
337C2×C4⋊C4—271no—
338C2×C22⋊C4—272no—
339C2×C42—271no—
340C33C2×C4×D411yescharacteristic, complemented, Sylow
341C23×C3—34no—
342C23×C3—34no—
343C23×C3—64no—
344C23×C3—98no—
345C23×C3—91no—
346C23×C3—184no—
347C23×C3—184no—
348C22×D3—32no—
349C22×D3—316no—
350C22×D3—61no—
351C22×D3—68no—
352C22×D3—68no—
353C22×D3C2×D3214yes—
354C3×D4—38no—
355C3×D4—38no—
356C3×D4—68no—
357C3×D4—98no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—34no—
361C2×C3×C4—41no—
362C2×C3×C4—62no—
363C2×C3×C4—91no—
364C2×C3×C4—92no—
365C2×C3×C4—94no—
366C2×C3×C4—92no—
367C2×C3×C4—188no—
368C2×C3×C4C2×D3211yescharacteristic
369C2×C3×C4C2×D3212yes—
370#67—316no—
371C2×Dic3—32no—
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—32no—
376C2×Dic3—62no—
377C2×Dic3—61no—
378C2×Dic3—62no—
379C2×Dic3—68no—
380C2×Dic3—94no—
381C2×Dic3—94no—
382C2×Dic3—92no—
383C2×Dic3—182no—
384C2×Dic3—184no—
385C2×Dic3—184no—
386C2×Dic3—188no—
387C2×Dic3—188no—
388C2×Dic3—182no—
389C2×Dic3—362no—
390C2×Dic3—368no—
391C2×Dic3C2×D3212yes—
392D12D3×Dic318yescomplemented
393C4×D3—38no—
394C4×D3—64no—
395C4×D3—68no—
396C4×D3—98no—
397C4×D3—184no—
398C4×D3—188no—
399C4×D3—188no—
400C4×D3—364no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—22no—
404C2×C32—22no—
405C2×C32—21no—
406C2×C32—22no—
407C2×C32—41no—
408C2×C32—41no—
409C2×C32—41no—
410C2×C32—68no—
411C2×C32C22×C4×D312yes—
412C2×C32C2×D3×D411yescharacteristic
413C2×C32C2×D3×D412yes—
414C2×C32C2×#23711yescharacteristic
415C2×C32D3×#4112yes—
416C2×C32C23×Dic311yescharacteristic
417C3⋊D3D4×Dic314yescomplemented
418C3×D3—28no—
419C3×D3—28no—
420C3×D3—38no—
421C3×D3—48no—
422C3×D3—64no—
423C24—92no—
424C2×D4—32no—
425C2×D4—98no—
426C2×D4—92no—
427C22×C4—94no—
428C22×C4—91no—
429C22×C4—272no—
430C22×C4—271no—
431C22×C4—271no—
432C22×C4—274no—
433C22×C4—548no—
434C4⋊C4—31no—
435C4⋊C4—272no—
436C4⋊C4—271no—
437C22⋊C4—94no—
438C22⋊C4—274no—
439C42—92no—
440C42—272no—
441C22×C3—22no—
442C22×C3—21no—
443C22×C3—38no—
444C22×C3—38no—
445C22×C3—41no—
446C22×C3—68no—
447C22×C3—64no—
448C22×C3—64no—
449C22×C3—64no—
450C22×C3—64no—
451C22×C3—92no—
452C22×C3—92no—
453C22×C3—92no—
454C22×C3—124no—
455C22×C3—124no—
456C22×C3—1816no—
457C22×C3C22×D3211yescharacteristic
458C22×C3C22×D3212yes—
459C2×D3—24no—
460C2×D3—24no—
461C2×D3—34no—
462C2×D3—34no—
463C2×D3—34no—
464C2×D3—62no—
465C2×D3—62no—
466C2×D3—62no—
467C2×D3—632no—
468C2×D3C2×D3×Dic318yescomplemented
469C3×C4—22no—
470C3×C4—24no—
471C3×C4—42no—
472C3×C4—64no—
473C3×C4—92no—
474C3×C4—94no—
475C3×C4—122no—
476C3×C4—182no—
477C3×C4C4×D3212yescomplemented
478C3×C4C2×D3×Dic314yescomplemented
479Dic3—22no—
480Dic3—34no—
481Dic3—34no—
482Dic3—34no—
483Dic3—62no—
484Dic3—64no—
485Dic3—62no—
486Dic3—62no—
487Dic3—122no—
488Dic3—184no—
489Dic3—182no—
490Dic3—362no—
491Dic3—364no—
492Dic3—722no—
493C32—21no—
494C32—22no—
495C32—41no—
496C32C4×D3×D412yescomplemented
497C32C2×D4×Dic311yescharacteristic, complemented
498C23—34no—
499C23—98no—
500C23—91no—
501C23—184no—
502C23—184no—
503D4—38no—
504D4—98no—
505C2×C4—32no—
506C2×C4—92no—
507C2×C4—91no—
508C2×C4—94no—
509C2×C4—92no—
510C2×C4—188no—
511C2×C4—274no—
512C2×C4—272no—
513C2×C4—544no—
514C2×C4—548no—
515C2×C4—542no—
516C2×C4D3311yescharacteristic
517C2×C3—22no—
518C2×C3—21no—
519C2×C3—22no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—22no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—68no—
527C2×C3—68no—
528C2×C3—94no—
529C2×C3—128no—
530C2×C3C2×C4×D3211yescharacteristic
531C2×C3D32×D411yescharacteristic
532C2×C3D32×D412yes—
533C2×C3D3×#23712yes—
534C2×C3#6431911yescharacteristic
535C2×C3C22×D3×Dic312yes—
536D3—28no—
537D3—38no—
538D3—64no—
539C22—38no—
540C22—64no—
541C22—64no—
542C22—92no—
543C22—92no—
544C22—92no—
545C22—1816no—
546C22C2×D3311yescharacteristic
547C4—62no—
548C4—94no—
549C4—92no—
550C4—182no—
551C4—542no—
552C4D32×Dic312yescomplemented
553C3—21no—
554C3—22no—
555C3—41no—
556C3#1059460511yescharacteristic, complemented
557C3D3×D4×Dic312yescomplemented
558C2—68no—
559C2—94no—
560C2#1172437411yescharacteristic
561C2#1172440311yescharacteristic
562C2C2×D32×Dic311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC2×C3×#3802C2
hypercenterC2×C4D33
hypocenterC33C2×C4×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×D32×Dic3C2semidirect
3. C4×#6294C2semidirect
4. C3⋊D3×C4⋊Dic3C2semidirect
5. D12×C3⋊Dic3C2semidirect
6. C3×#63898C2semidirect
7. #11722924C2semidirect
8. #11722902C2semidirect
17. C2×D3×C3⋊Dic3C22semidirect
19. C2×C3×D3×Dic3C22semidirect
20. C3×#3802C4semidirect
22. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
23. C3×C4×C3⋊Dic3C22semidirect
24. C32×C4⋊Dic3C22semidirect
25. #88813C22semidirect
26. #88807C22semidirect
27. (C2×D3)⋊(C3⋊Dic3)C22semidirect
28. C3×(C2×D3)⋊Dic3C22semidirect
41. #63898D3semidirect
54. C2×C3×D32C2×C4semidirect
60. C3×C4×C3⋊D3C2×C4semidirect
62. C32×D12C2×C4semidirect
64. #6294D4semidirect
67. C3×#625C2×C4semidirect
97. C2×D3×Dic3C2×D3semidirect
98. #3802Dic3semidirect
108. C4×C3⋊Dic3C2×D3semidirect
112. C3×C4⋊Dic3C2×D3semidirect
121. (C2×D3)⋊Dic3C2×D3semidirect
131. C2×C32×D3C22×C4semidirect
133. C33×C4C22×C4semidirect
136. C3×C3⋊Dic3C22×C4semidirect
137. C3×C3⋊Dic3C2×D4semidirect
172. C2×D32C2×Dic3semidirect
190. C4×C3⋊D3C2×Dic3semidirect
204. C3×D12C4×D3semidirect
205. C3×D12C2×Dic3semidirect
209. #625C2×Dic3semidirect
224. C3×C3⋊D3C4×D4semidirect
271. C4⋊Dic3D32semidirect
305. C2×C3×D3C2×C4×D3semidirect
306. C2×C3×D3C22×Dic3semidirect
312. C32×C4C2×C4×D3semidirect
313. C32×C4C22×Dic3semidirect
320. C3⋊Dic3D3×D4semidirect
321. C3⋊Dic3C22×Dic3semidirect
340. C33C2×C4×D4semidirect
392. D12D3×Dic3semidirect
417. C3⋊D3D4×Dic3semidirect
468. C2×D3C2×D3×Dic3semidirect
477. C3×C4C4×D32semidirect
478. C3×C4C2×D3×Dic3semidirect
496. C32C4×D3×D4semidirect
497. C32C2×D4×Dic3semidirect
552. C4D32×Dic3semidirect
556. C3#10594605semidirect
557. C3D3×D4×Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C2×D32×Dic32
C4×#62942
C3⋊D3×C4⋊Dic32
D12×C3⋊Dic32
C3×#638982
#117229242
#117229022
D3×D4×Dic33
#105946053