Finite Group Databasebeta

← #13161809

Subgroups — #13161809

Order 1728.

1002 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C3×#64135C211yesmaximal, characteristic, complemented
3C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#11722076C212yesmaximal, complemented
5#11722075C212yesmaximal, complemented
6#10591105—31nomaximal
7#10591084—31nomaximal
8#10591083—31nomaximal
9C22×C33×C4C2211yescharacteristic
10C2×C3×#638—21no—
11C3×#3828—22no—
12C3×#3828C414yescomplemented
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3—21no—
15C2×C3×#634C2211yescharacteristic
16C3×#3757—21no—
17C3×C4⋊(C3⋊Dic3)C2212yes—
18C3×C4×C3⋊Dic3—21no—
19C2×C3×C3⋊(C3⋊C8)—22no—
20C2×C3×C3⋊(C3⋊C8)C2212yes—
21C2×#3828D311yescharacteristic
22C2×C3×#235—31no—
23C2×C3×#235—31no—
24C2×C3×#235—61no—
25#64135D311yescharacteristic, complemented
26C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
27C3×#926—31no—
28C3×#926—31no—
29C3×#926—61no—
30C22×C3×C3⋊C8—31no—
31C22×C3×C3⋊C8—31no—
32C22×C3×C3⋊C8—61no—
33#63567—32no—
34#63566—32no—
35#63565—32no—
36#63564—32no—
37#63563—32no—
38#63562—32no—
39C23×C33D411yescharacteristic
40C22×C3×C3⋊D3—21no—
41C2×C33×C4C2×C412yes—
42C2×C33×C4D411yescharacteristic
43C2×C33×C4D412yes—
44C2×C33×C4C2311yescharacteristic
45C3×#638—24no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—22no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)C2×C412yes—
50C3×C4×C3⋊D3—24no—
51C3×#634—21no—
52C3×#634C2×C412yes—
53C3×C3⋊(C3⋊C8)—22no—
54#5233—91no—
55#5113—91no—
56#5002—91no—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4—41no—
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C2×#638—21no—
65#3828—22no—
66#3828Dic314yescomplemented
67C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C2×#634C2×D311yescharacteristic
70C2×C3×#67—62no—
71C2×C3×#67—61no—
72C2×C3×#67—61no—
73C3×#235—34no—
74C3×#235—34no—
75C3×#235—64no—
76C3×#235—64no—
77C3×#235—62no—
78C3×#235—62no—
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—61no—
82C2×C3×C4×D3—62no—
83C2×C3×C4×D3—61no—
84C2×C3×C4×D3—61no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—61no—
88#3757—21no—
89C4⋊(C3⋊Dic3)C2×D312yes—
90C4×C3⋊Dic3—21no—
91C2×C3⋊(C3⋊C8)—22no—
92C2×C3⋊(C3⋊C8)C2×D312yes—
93C3×#217—62no—
94C3×#217—61no—
95C3×#217—61no—
96C3×C4⋊Dic3—32no—
97C3×C4⋊Dic3—32no—
98C3×C4⋊Dic3—62no—
99C3×C4×Dic3—62no—
100C3×C4×Dic3—61no—
101C3×C4×Dic3—61no—
102C2×C3×C3⋊C8—32no—
103C2×C3×C3⋊C8—32no—
104C2×C3×C3⋊C8—62no—
105C2×C3×C3⋊C8—64no—
106C2×C3×C3⋊C8—62no—
107C2×C3×C3⋊C8—62no—
108C22×C33—22no—
109C22×C33C22⋊C412yes—
110C22×C33C2×D411yescharacteristic
111C2×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3—42no—
113C33×C4C22⋊C412yes—
114C33×C4C22×C411yescharacteristic
115C33×C4C2×D411yescharacteristic
116C3×C3⋊Dic3—22no—
117C3×C3⋊Dic3—42no—
118C2×C3×#41—91no—
119C2×#235—31no—
120C2×#235—31no—
121C2×#235—61no—
122C22×C3×C8—91no—
123C3×#117—91no—
124#926—31no—
125#926—31no—
126#926—61no—
127C22×C3⋊C8—31no—
128C22×C3⋊C8—31no—
129C22×C3⋊C8—61no—
130#835—92no—
131#832—92no—
132#821—92no—
133#816—92no—
134#810—92no—
135#809—92no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32—21no—
139C23×C32—41no—
140C23×C32D1211yescharacteristic
141C23×C32#6711yescharacteristic
142C23×C32#6711yescharacteristic
143C22×C3⋊D3—21no—
144C22×C3×D3—62no—
145C22×C3×D3—61no—
146C22×C3×D3—61no—
147C2×C32×C4—22no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—21no—
152C2×C32×C4—22no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—42no—
160C2×C32×C4—42no—
161C2×C32×C4—41no—
162C2×C32×C4—41no—
163C2×C32×C4C4×D312yes—
164C2×C32×C4C4×D312yes—
165C2×C32×C4D1211yescharacteristic
166C2×C32×C4D1212yes—
167C2×C32×C4C2×Dic312yes—
168C2×C32×C4#6711yescharacteristic
169C2×C32×C4#6712yes—
170C2×C32×C4#6711yescharacteristic
171C2×C32×C4#6712yes—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175#638—24no—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3—22no—
178C4⋊(C3⋊D3)—21no—
179C4⋊(C3⋊D3)C2×Dic312yes—
180C4×C3⋊D3—24no—
181#634—21no—
182#634C2×Dic312yes—
183C3×#67—68no—
184C3×#67—64no—
185C3×#67—64no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—64no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—62no—
192C3×D12—32no—
193C3×D12—32no—
194C3×D12—62no—
195C3×D12—62no—
196C3×D12—61no—
197C3×D12—61no—
198C3×C4×D3—68no—
199C3×C4×D3—64no—
200C3×C4×D3—64no—
201C3×Dic6—32no—
202C3×Dic6—32no—
203C3×Dic6—62no—
204C3×Dic6—62no—
205C3×Dic6—61no—
206C3×Dic6—61no—
207C3⋊(C3⋊C8)—22no—
208C3×C3⋊C8—64no—
209C3×C3⋊C8—62no—
210C3×C3⋊C8—62no—
211#416—271noSylow
212C2×C33—22no—
213C2×C33C2×C22⋊C411yescharacteristic
214C2×C33#13512yes—
215C3×C3⋊D3—42no—
216C3×#41—94no—
217C3×#41—182no—
218C2×C3×Q8—91no—
219C2×C3×D4—91no—
220C2×C3×D4—181no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—41no—
225C22×C3×C4—181no—
226C22×C3×C4D3211yescharacteristic
227C22×C3×C4D3211yescharacteristic
228C22×C3×C4D3211yescharacteristic
229C2×#67—62no—
230C2×#67—61no—
231C2×#67—61no—
232#235—34no—
233#235—34no—
234#235—64no—
235#235—64no—
236#235—62no—
237#235—62no—
238C2×D12—31no—
239C2×D12—31no—
240C2×D12—61no—
241C2×C4×D3—62no—
242C2×C4×D3—61no—
243C2×C4×D3—61no—
244C2×Dic6—31no—
245C2×Dic6—31no—
246C2×Dic6—61no—
247C2×C3×C8—92no—
248C2×C3×C8—182no—
249C3×C4⋊C4—92no—
250C3×C22⋊C4—181no—
251C3×C42—181no—
252#217—62no—
253#217—61no—
254#217—61no—
255C4⋊Dic3—32no—
256C4⋊Dic3—32no—
257C4⋊Dic3—62no—
258C4×Dic3—62no—
259C4×Dic3—61no—
260C4×Dic3—61no—
261C2×C3⋊C8—32no—
262C2×C3⋊C8—32no—
263C2×C3⋊C8—62no—
264C2×C3⋊C8—64no—
265C2×C3⋊C8—62no—
266C2×C3⋊C8—62no—
267C22×C32—22no—
268C22×C32—24no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—42no—
278C22×C32—41no—
279C22×C32—42no—
280C22×C32—42no—
281C22×C32—44no—
282C22×C32(C2×D3)⋊C412yes—
283C22×C32(C2×D3)⋊C412yes—
284C22×C32#21712yes—
285C22×C32C2×D1211yescharacteristic
286C22×C32C2×#6711yescharacteristic
287C22×C32C2×#6711yescharacteristic
288C2×C3⋊D3—22no—
289C2×C3⋊D3—42no—
290C2×C3×D3—64no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—124no—
294C2×C3×D3—122no—
295C2×C3×D3—122no—
296C32×C4—22no—
297C32×C4—22no—
298C32×C4—22no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—42no—
306C32×C4—41no—
307C32×C4—41no—
308C32×C4(C2×D3)⋊C412yes—
309C32×C4(C2×D3)⋊C412yes—
310C32×C4#21712yes—
311C32×C4C2×C4×D311yescharacteristic
312C32×C4C2×C4×D311yescharacteristic
313C32×C4C2×D1211yescharacteristic
314C32×C4C22×Dic311yescharacteristic
315C32×C4C2×#6711yescharacteristic
316C32×C4C2×#6711yescharacteristic
317C3⋊Dic3—22no—
318C3⋊Dic3—42no—
319C3×Dic3—64no—
320C3×Dic3—62no—
321C3×Dic3—62no—
322C3×Dic3—124no—
323C3×Dic3—122no—
324C3×Dic3—122no—
325C2×#41—91no—
326C22×C8—91no—
327#117—91no—
328#103—272no—
329#102—272no—
330C33#41611yescharacteristic, complemented, Sylow
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—41no—
335C23×C3—181no—
336C23×C3#62511yescharacteristic
337C23×C3#62611yescharacteristic
338C23×C3#62611yescharacteristic
339C22×D3—62no—
340C22×D3—61no—
341C22×D3—61no—
342C3×Q8—92no—
343C3×Q8—181no—
344C3×D4—92no—
345C3×D4—184no—
346C3×D4—181no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—181no—
364C2×C3×C4—184no—
365C2×C3×C4—182no—
366C2×C3×C4D3×Dic312yes—
367C2×C3×C4D3×Dic312yes—
368C2×C3×C4#62412yes—
369C2×C3×C4#62512yes—
370C2×C3×C4#62511yescharacteristic
371C2×C3×C4#62612yes—
372C2×C3×C4#62612yes—
373C2×C3×C4#62611yescharacteristic
374C2×C3×C4#62611yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377C2×C3×C4C2×D3211yescharacteristic
378#67—68no—
379#67—64no—
380#67—64no—
381C2×Dic3—62no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—64no—
385C2×Dic3—62no—
386C2×Dic3—62no—
387D12—32no—
388D12—32no—
389D12—62no—
390D12—62no—
391D12—61no—
392D12—61no—
393C4×D3—68no—
394C4×D3—64no—
395C4×D3—64no—
396Dic6—32no—
397Dic6—32no—
398Dic6—62no—
399Dic6—62no—
400Dic6—61no—
401Dic6—61no—
402C3×C8—182no—
403C3⋊C8—64no—
404C3⋊C8—62no—
405C3⋊C8—62no—
406C2×C32—22no—
407C2×C32—24no—
408C2×C32—22no—
409C2×C32—22no—
410C2×C32—22no—
411C2×C32—22no—
412C2×C32—22no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—41no—
417C2×C32—42no—
418C2×C32—42no—
419C2×C32—42no—
420C2×C32—44no—
421C2×C32#90412yes—
422C2×C32C2×(C2×D3)⋊C411yescharacteristic
423C2×C32C2×(C2×D3)⋊C411yescharacteristic
424C2×C32#95012yes—
425C2×C32#95012yes—
426C2×C32C2×#21711yescharacteristic
427C3⋊D3—42no—
428C3×D3—124no—
429C3×D3—122no—
430C3×D3—122no—
431#41—94no—
432#41—182no—
433C2×Q8—91no—
434C2×D4—91no—
435C2×D4—181no—
436C22×C4—181no—
437C22×C4#111311yescharacteristic
438C2×C8—92no—
439C2×C8—182no—
440C4⋊C4—92no—
441C22⋊C4—181no—
442C42—181no—
443C22×C3—24no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—44no—
456C22×C3—41no—
457C22×C3—42no—
458C22×C3—182no—
459C22×C3—362no—
460C22×C3(C2×D3)⋊Dic312yes—
461C22×C3(C2×D3)⋊Dic312yes—
462C22×C3#372212yes—
463C22×C3C2×#62511yescharacteristic
464C22×C3C2×#62611yescharacteristic
465C22×C3C2×#62611yescharacteristic
466C2×D3—64no—
467C2×D3—62no—
468C2×D3—62no—
469C2×D3—124no—
470C2×D3—122no—
471C2×D3—122no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—42no—
484C3×C4—182no—
485C3×C4—362no—
486C3×C4(C2×D3)⋊Dic312yes—
487C3×C4(C2×D3)⋊Dic312yes—
488C3×C4#372212yes—
489C3×C4C2×D3×Dic311yescharacteristic
490C3×C4C2×D3×Dic311yescharacteristic
491C3×C4C2×#62411yescharacteristic
492C3×C4C2×#62511yescharacteristic
493C3×C4C2×#62611yescharacteristic
494C3×C4C2×#62611yescharacteristic
495Dic3—64no—
496Dic3—62no—
497Dic3—62no—
498Dic3—124no—
499Dic3—122no—
500Dic3—122no—
501C32—21no—
502C32—21no—
503C32—21no—
504C32—41no—
505C32#500211yescharacteristic, complemented
506C32#511311yescharacteristic, complemented
507C32#523311yescharacteristic, complemented
508C23—181no—
509C23#629511yescharacteristic
510Q8—92no—
511Q8—181no—
512D4—92no—
513D4—184no—
514D4—181no—
515C2×C4—181no—
516C2×C4—184no—
517C2×C4—182no—
518C2×C4#629412yes—
519C2×C4#629512yes—
520C2×C4#629511yescharacteristic
521C2×C4C2×#111311yescharacteristic
522C8—182no—
523C2×C3—24no—
524C2×C3—22no—
525C2×C3—22no—
526C2×C3—22no—
527C2×C3—22no—
528C2×C3—22no—
529C2×C3—22no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—42no—
534C2×C3—42no—
535C2×C3—44no—
536C2×C3—42no—
537C2×C3—41no—
538C2×C3—362no—
539C2×C3#6382212yes—
540C2×C3#6382912yes—
541C2×C3#6382912yes—
542C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
543C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
544C2×C3C2×#372211yescharacteristic
545D3—124no—
546D3—122no—
547D3—122no—
548C22—22no—
549C22—182no—
550C22—362no—
551C22#8881412yes—
552C22C2×#629511yescharacteristic
553C4—182no—
554C4—362no—
555C4#8881412yes—
556C4C2×#629411yescharacteristic
557C4C2×#629511yescharacteristic
558C3—21no—
559C3—21no—
560C3—21no—
561C3—41no—
562C3#1059108311yescharacteristic, complemented
563C3#1059108411yescharacteristic, complemented
564C3#1059110511yescharacteristic, complemented
565C2—22no—
566C2—362no—
567C2#1172366012yes—
568C2C2×#8881411yescharacteristic
subgroupG/H
centerC2×C4#6295
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3828C2
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33#416
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64135C2semidirect
3. C22×C3×C3⋊(C3⋊C8)C2semidirect
4. #11722076C2semidirect
5. #11722075C2semidirect
12. C3×#3828C4semidirect
25. #64135D3semidirect
26. C22×C3⋊(C3⋊C8)D3semidirect
66. #3828Dic3semidirect
330. C33#416semidirect
505. C32#5002semidirect
506. C32#5113semidirect
507. C32#5233semidirect
562. C3#10591083semidirect
563. C3#10591084semidirect
564. C3#10591105semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C3×#641352
C22×C3×C3⋊(C3⋊C8)2
#117220762
#117220752
#105911053
#105910843
#105910833