Finite Group Databasebeta

← #67×Dic18 #13150370

Subgroups — #67×Dic18 #13150370

Order 1728.

914 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Dic18C211yesmaximal, characteristic, complemented
2C2×D3×Dic18C211yesmaximal, characteristic, complemented
3#11721813C211yesmaximal, characteristic, complemented
4C4×C9×#67C211yesmaximal, characteristic
5#67×Dic9C212yesmaximal
6#11721253C212yesmaximal, complemented
7#11721018C212yesmaximal, complemented
8#11720960C211yesmaximal, characteristic, complemented
9#11720932C211yesmaximal, characteristic, complemented
10Dic3×Dic18C211yesmaximal, characteristic, complemented
11#11720904C212yesmaximal, complemented
12Dic6×#67—31nomaximal
13D4×Dic18—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C9×#67C2211yescharacteristic
17C22×C3×Dic9C2212yes—
18C2×C4×D3×C9C2211yescharacteristic
19C2×C3×Dic18—24no—
20C2×C3×Dic18C2211yescharacteristic
21C2×C3×Dic18C2212yes—
22C2×C3×Dic18C2211yescharacteristic
23C2×D3×Dic9C2212yes—
24C2×#6189C2212yes—
25D3×Dic18—24no—
26#88551C2212yes—
27C4⋊(C3⋊Dic9)C2211yescharacteristic
28#88534C2212yes—
29C9×#217C2211yescharacteristic
30C9×(C2×D3)⋊C4C2211yescharacteristic
31C9×Dic3⋊C4C2211yescharacteristic
32C4×C9×Dic3C2211yescharacteristic
33(C2×D3)⋊Dic9C2212yes—
34Dic3⋊Dic9C2212yes—
35Dic3⋊Dic9C2212yes—
36#88442C2212yes—
37Dic9⋊Dic3C2212yes—
38Dic3×Dic9C2212yes—
39C22×C3×Dic6—31no—
40C2×D3×Dic6—31no—
41#64133—31no—
42C3×C4×#67D311yescharacteristic
43Dic3×#67—32no—
44#63964—32no—
45#63887—32no—
46#63866—31no—
47#63856—31no—
48Dic3×Dic6—31no—
49#63839—32no—
50C22×Dic18—31no—
51C22×Dic18D311yescharacteristic, complemented
52C4×D4×C9—31no—
53D4×Dic9—32no—
54#63486—31no—
55#63486—31no—
56#63486—61no—
57#63450—32no—
58#63440—32no—
59#63440—32no—
60#63431—31no—
61C4×Dic18—31no—
62C23×C3×C9C2311yescharacteristic
63C22×D3×C9C2311yescharacteristic
64C2×C3×C4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2312yes—
69#6226—22no—
70C9×#67Q814yescomplemented
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C9×Dic3C2311yescharacteristic
74C2×C3×Dic9—24no—
75C2×C3×Dic9C2312yes—
76C2×C3×Dic9C2312yes—
77C4×D3×C9—22no—
78C3×Dic18—22no—
79C3×Dic18—24no—
80C3×Dic18D414yescomplemented
81D3×Dic9—24no—
82#6189—24no—
83Q8×#67—91no—
84D4×Dic6—91no—
85C22×C32×C4C2×D311yescharacteristic
86C2×#634—31no—
87C2×C3×#67C2×D311yescharacteristic
88C22×C3×Dic3—32no—
89C2×C3×C4×D3C2×D311yescharacteristic
90C2×C3×Dic6—31no—
91C2×C3×Dic6—32no—
92C2×C3×Dic6—31no—
93C2×C3×Dic6—64no—
94C2×#627—32no—
95C2×D3×Dic3—32no—
96D3×Dic6—64no—
97#3757—32no—
98C4⋊(C3⋊Dic3)—31no—
99#3750—32no—
100C3×#217C2×D311yescharacteristic
101C3×(C2×D3)⋊C4C2×D311yescharacteristic
102C3×Dic3⋊C4C2×D311yescharacteristic
103C3×C4×Dic3C2×D311yescharacteristic
104#3724—32no—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108(C2×D3)⋊Dic3—32no—
109Dic32—32no—
110C2×D4×C9—31no—
111C22×C4×C9—21no—
112C22×C4×C9—31no—
113C22×C4×C9C2×D311yescharacteristic
114C22×Dic9—32no—
115C22×Dic9C2×D312yes—
116C2×Dic18—24no—
117C2×Dic18—31no—
118C2×Dic18—32no—
119C2×Dic18—64no—
120C2×Dic18—61no—
121C2×Dic18C2×D311yescharacteristic
122C2×Dic18C2×D312yes—
123C2×Dic18C2×D311yescharacteristic
124C9×C4⋊C4—31no—
125C9×C22⋊C4—31no—
126C9×C22⋊C4—31no—
127C42×C9—31no—
128#3676—32no—
129#3676—32no—
130#3676—62no—
131C4⋊Dic9—31no—
132C4⋊Dic9—32no—
133C4⋊Dic9—32no—
134C4⋊Dic9—61no—
135Dic9⋊C4—32no—
136Dic9⋊C4—32no—
137Dic9⋊C4—32no—
138Dic9⋊C4—62no—
139C4×Dic9—32no—
140C22×C3×C9—21no—
141C22×C3×C9—21no—
142C22×C3×C9C2×Q812yescomplemented
143C22×C3×C9C2411yescharacteristic
144C2×D3×C9—21no—
145C2×D3×C9—21no—
146C2×D3×C9C2×Q812yescomplemented
147C3×C4×C9—21no—
148C3×C4×C9C2×D412yes—
149C3⋊Dic9—22no—
150C3⋊Dic9—22no—
151C9×Dic3—21no—
152C9×Dic3—21no—
153C9×Dic3C2×Q812yescomplemented
154C3×Dic9—22no—
155C3×Dic9C2×D414yes—
156C22×C3×Q8—91no—
157C2×D3×Q8—91no—
158C22×Dic6—31no—
159C22×Dic6—91no—
160C3×C4×D4—31no—
161#946—91no—
162#946—92no—
163Q8×Dic3—91no—
164#944—92no—
165#944—91no—
166D4×Dic3—92no—
167C4×#67—92no—
168C4×#67D911yescharacteristic
169#924—91no—
170#924—91no—
171#924—91no—
172#924—92no—
173#924—181no—
174#888—92no—
175#878—92no—
176#878—92no—
177#869—91no—
178C4×Dic6—91no—
179C23×C32C22×D311yescharacteristic
180C22×C3×D3C22×D311yescharacteristic
181C2×C32×C4—22no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C3⋊Dic3—32no—
185C2×C3⋊Dic3—32no—
186#634—62no—
187C3×#67Dic614yes—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—64no—
191C2×C3×Dic3C22×D311yescharacteristic
192C2×C3×Dic3C22×D311yescharacteristic
193C2×C3×Dic3C22×D311yescharacteristic
194C3×C4×D3—22no—
195C3×Dic6—34no—
196C3×Dic6—64no—
197C3×Dic6—62no—
198#627—64no—
199D3×Dic3—64no—
200C23×C9—21no—
201C23×C9—31no—
202C23×C9C22×D311yescharacteristic
203D4×C9—34no—
204C2×C4×C9—22no—
205C2×C4×C9—21no—
206C2×C4×C9—21no—
207C2×C4×C9—31no—
208C2×C4×C9—31no—
209C2×C4×C9—31no—
210C2×C4×C9—42no—
211C2×C4×C9—62no—
212C2×C4×C9C22×D311yescharacteristic
213C2×C4×C9C22×D311yescharacteristic
214C2×Dic9—24no—
215C2×Dic9—32no—
216C2×Dic9—32no—
217C2×Dic9—62no—
218C2×Dic9—62no—
219C2×Dic9—64no—
220C2×Dic9C22×D312yes—
221C2×Dic9C22×D312yes—
222Dic18—22no—
223Dic18—24no—
224Dic18—62no—
225Dic18—64no—
226Dic18—122no—
227Dic18#6714yesdirect product factor
228D4×Q8—271noSylow
229C2×C3×C9—22no—
230C2×C3×C9C22×D411yescharacteristic
231C2×C3×C9C22×Q811yescharacteristic
232C2×C3×C92−1+411yescharacteristic
233D3×C9—22no—
234C2×C3×Q8—91no—
235C2×C3×Q8—92no—
236C2×C3×Q8—91no—
237C2×C3×Q8—184no—
238C2×C3×D4—31no—
239C22×C3×C4—21no—
240C22×C3×C4—31no—
241C22×C3×C4—92no—
242C22×C3×C4C2×D911yescharacteristic
243C22×C3×C4D3211yescharacteristic
244C2×#67C2×D911yescharacteristic
245C22×Dic3—32no—
246C22×Dic3—92no—
247D3×Q8—184no—
248C2×C4×D3—92no—
249C2×C4×D3C2×D911yescharacteristic
250C2×Dic6—31no—
251C2×Dic6—31no—
252C2×Dic6—32no—
253C2×Dic6—64no—
254C2×Dic6—92no—
255C2×Dic6—91no—
256C2×Dic6—91no—
257C2×Dic6—92no—
258C2×Dic6—181no—
259C2×Dic6—184no—
260C3×C4⋊C4—31no—
261C3×C22⋊C4—31no—
262C3×C22⋊C4—31no—
263C3×C42—31no—
264#217—92no—
265#217—92no—
266#217—92no—
267#217—182no—
268#217C2×D911yescharacteristic
269(C2×D3)⋊C4—92no—
270(C2×D3)⋊C4C2×D911yescharacteristic
271C4⋊Dic3—92no—
272C4⋊Dic3—92no—
273C4⋊Dic3—91no—
274C4⋊Dic3—91no—
275C4⋊Dic3—92no—
276C4⋊Dic3—181no—
277Dic3⋊C4—92no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—182no—
285Dic3⋊C4C2×D911yescharacteristic
286C4×Dic3—92no—
287C4×Dic3—92no—
288C4×Dic3C2×D911yescharacteristic
289C22×C32—21no—
290C22×C32—21no—
291C22×C32C2×Dic612yes—
292C22×C32C23×D311yescharacteristic
293C2×C3×D3—21no—
294C2×C3×D3—21no—
295C2×C3×D3C2×Dic612yes—
296C32×C4—21no—
297C32×C4D3×D412yes—
298C3⋊Dic3—62no—
299C3⋊Dic3—62no—
300C3×Dic3—21no—
301C3×Dic3—21no—
302C3×Dic3—34no—
303C3×Dic3—62no—
304C3×Dic3C2×Dic612yes—
305C22×C9—21no—
306C22×C9—21no—
307C22×C9—21no—
308C22×C9—22no—
309C22×C9—32no—
310C22×C9—41no—
311C22×C9—41no—
312C22×C9—61no—
313C22×C9—61no—
314C22×C9D3×Q812yescomplemented
315C22×C9C23×D311yescharacteristic
316C4×C9—21no—
317C4×C9—22no—
318C4×C9—32no—
319C4×C9—41no—
320C4×C9—61no—
321C4×C9—61no—
322C4×C9C2×#6712yes—
323Dic9—22no—
324Dic9—62no—
325Dic9—62no—
326Dic9—122no—
327Dic9—122no—
328Dic9C2×#6714yes—
329C22×Q8—91no—
330C22×Q8—271no—
331C4⋊Q8—271no—
332C4⋊Q8—272no—
333C22⋊Q8—272no—
334C22⋊Q8—272no—
335C22⋊Q8—271no—
336C22⋊Q8—271no—
337C4×Q8—271no—
338C4×D4—31no—
339C4×D4—272no—
340C3×C9D4×Q811yescharacteristic, complemented, Sylow
341C23×C3—21no—
342C23×C3—31no—
343C23×C3C22×D911yescharacteristic
344C23×C3C2×D3211yescharacteristic
345C22×D3C22×D911yescharacteristic
346C3×Q8—94no—
347C3×Q8—184no—
348C3×Q8—182no—
349C3×D4—34no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—42no—
358C2×C3×C4—62no—
359C2×C3×C4—92no—
360C2×C3×C4—92no—
361C2×C3×C4—184no—
362C2×C3×C4C22×D911yescharacteristic
363C2×C3×C4C22×D911yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365C2×C3×C4C2×D3211yescharacteristic
366#67Dic1814yesdirect product factor
367C2×Dic3—32no—
368C2×Dic3—32no—
369C2×Dic3—64no—
370C2×Dic3—92no—
371C2×Dic3—92no—
372C2×Dic3—92no—
373C2×Dic3—92no—
374C2×Dic3—182no—
375C2×Dic3—182no—
376C2×Dic3—184no—
377C2×Dic3C22×D911yescharacteristic
378C2×Dic3C22×D911yescharacteristic
379C2×Dic3C22×D911yescharacteristic
380C4×D3—22no—
381C4×D3—184no—
382Dic6—34no—
383Dic6—64no—
384Dic6—62no—
385Dic6—184no—
386Dic6—184no—
387Dic6—182no—
388Dic6—182no—
389Dic6—362no—
390C2×C32—22no—
391C2×C32C22×Dic611yescharacteristic
392C2×C32C2×D3×D411yescharacteristic
393C2×C32#101011yescharacteristic
394C3×D3—22no—
395C2×C9—22no—
396C2×C9—21no—
397C2×C9—21no—
398C2×C9—21no—
399C2×C9—42no—
400C2×C9—62no—
401C2×C9C2×D3×Q811yescharacteristic
402C2×C9#100711yescharacteristic
403C2×C9C22×#6711yescharacteristic
404C2×Q8—91no—
405C2×Q8—92no—
406C2×Q8—91no—
407C2×Q8—184no—
408C2×Q8—272no—
409C2×Q8—271no—
410C2×Q8—544no—
411C2×D4—31no—
412C22×C4—31no—
413C22×C4—92no—
414C22×C4—272no—
415C22×C4D3×D911yescharacteristic
416C4⋊C4—31no—
417C4⋊C4—272no—
418C4⋊C4—272no—
419C4⋊C4—272no—
420C4⋊C4—272no—
421C4⋊C4—271no—
422C4⋊C4—272no—
423C22⋊C4—31no—
424C22⋊C4—31no—
425C22⋊C4—272no—
426C22⋊C4—272no—
427C42—31no—
428C42—272no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—22no—
434C22×C3—21no—
435C22×C3—32no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—61no—
439C22×C3—61no—
440C22×C3C2×Dic1812yescomplemented
441C22×C3C23×D911yescharacteristic
442C22×C3D3×Dic612yes—
443C22×C3C22×D3211yescharacteristic
444C2×D3—21no—
445C2×D3—21no—
446C2×D3C2×Dic1812yescomplemented
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—22no—
450C3×C4—32no—
451C3×C4—41no—
452C3×C4—61no—
453C3×C4—61no—
454C3×C4—94no—
455C3×C4—182no—
456C3×C4D4×D912yes—
457C3×C4D3×#6712yes—
458Dic3—21no—
459Dic3—21no—
460Dic3—34no—
461Dic3—62no—
462Dic3—182no—
463Dic3—182no—
464Dic3—182no—
465Dic3—182no—
466Dic3—362no—
467Dic3—362no—
468Dic3C2×Dic1812yescomplemented
469C32D4×Dic611yescharacteristic
470C9—21no—
471C9Q8×#6711yescharacteristic, complemented
472C23—31no—
473C23C2×D3×D911yescharacteristic
474Q8—94no—
475Q8—184no—
476Q8—182no—
477Q8—544no—
478Q8—542no—
479D4—34no—
480C2×C4—22no—
481C2×C4—31no—
482C2×C4—31no—
483C2×C4—31no—
484C2×C4—62no—
485C2×C4—92no—
486C2×C4—92no—
487C2×C4—184no—
488C2×C4—272no—
489C2×C4—272no—
490C2×C4—544no—
491C2×C4C2×D3×D911yescharacteristic
492C2×C4C2×D3×D911yescharacteristic
493C2×C3—22no—
494C2×C3—22no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—42no—
499C2×C3—62no—
500C2×C3C22×Dic1811yescharacteristic
501C2×C3C2×D4×D911yescharacteristic
502C2×C3#6371611yescharacteristic
503C2×C3C2×D3×Dic611yescharacteristic
504C2×C3#6429711yescharacteristic
505C2×C3C2×D3×#6711yescharacteristic
506D3—22no—
507C22—21no—
508C22—21no—
509C22—32no—
510C22—61no—
511C22—61no—
512C22D3×Dic1812yescomplemented
513C22C22×D3×D911yescharacteristic
514C4—21no—
515C4—32no—
516C4—61no—
517C4—61no—
518C4—94no—
519C4—182no—
520C4—542no—
521C4—542no—
522C4D9×#6712yes—
523C3—21no—
524C3D4×Dic1811yescharacteristic, complemented
525C3Dic6×#6711yescharacteristic
526C2—22no—
527C2—62no—
528C2C2×D3×Dic1811yescharacteristic
529C2#1172246411yescharacteristic
530C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×#67C2×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9D4×Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×Dic18C2semidirect
2. C2×D3×Dic18C2semidirect
3. #11721813C2semidirect
6. #11721253C2semidirect
7. #11721018C2semidirect
8. #11720960C2semidirect
9. #11720932C2semidirect
10. Dic3×Dic18C2semidirect
11. #11720904C2semidirect
51. C22×Dic18D3semidirect
70. C9×#67Q8semidirect
80. C3×Dic18D4semidirect
142. C22×C3×C9C2×Q8semidirect
146. C2×D3×C9C2×Q8semidirect
153. C9×Dic3C2×Q8semidirect
227. Dic18#67direct (N × H)
314. C22×C9D3×Q8semidirect
340. C3×C9D4×Q8semidirect
366. #67Dic18direct (N × H)
440. C22×C3C2×Dic18semidirect
446. C2×D3C2×Dic18semidirect
468. Dic3C2×Dic18semidirect
471. C9Q8×#67semidirect
512. C22D3×Dic18semidirect
524. C3D4×Dic18semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic182
C2×D3×Dic182
#117218132
C4×C9×#672
#67×Dic92
#117212532
#117210182
#117209602
#117209322
Dic3×Dic182
#117209042
Dic6×#673
D4×Dic183