Finite Group Databasebeta

← #13150384

Subgroups — #13150384

Order 1728.

996 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6226C211yesmaximal, characteristic, complemented
2C2×D3×Dic18C211yesmaximal, characteristic, complemented
3C4×C9×#67C211yesmaximal, characteristic
4C3×#63486C211yesmaximal, characteristic, complemented
5#67⋊Dic9C212yesmaximal
6#11721267C212yesmaximal, complemented
7#11721024C212yesmaximal, complemented
8#11720961C211yesmaximal, characteristic, complemented
9#11720932C211yesmaximal, characteristic, complemented
10#11720912C211yesmaximal, characteristic, complemented
11#11720905C212yesmaximal, complemented
12#10594915—31nomaximal
13D4×Dic18—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2212yes—
16C2×#6226—24no—
17C2×#6226C2211yescharacteristic
18C2×#6226C2212yes—
19C2×#6226C2211yescharacteristic
20C2×C9×#67C2211yescharacteristic
21C2×C4×D3×C9C2211yescharacteristic
22C2×C3×Dic18C2211yescharacteristic
23C2×D3×Dic9C2212yes—
24C2×#6189C2212yes—
25D3×Dic18—24no—
26C9×#217C2211yescharacteristic
27C3×#3676C2212yes—
28C9×(C2×D3)⋊C4C2211yescharacteristic
29C9×Dic3⋊C4C2211yescharacteristic
30C4×C9×Dic3C2211yescharacteristic
31C3×C4⋊Dic9C2211yescharacteristic
32C3×Dic9⋊C4C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34Dic3⋊Dic9C2212yes—
35Dic3⋊Dic9C2212yes—
36#88442C2212yes—
37Dic9⋊Dic3C2212yes—
38Dic3×Dic9C2212yes—
39C22×#634—31no—
40C2×D3×Dic6—31no—
41C3×C4×#67D311yescharacteristic
42C3×#924—31no—
43#63982—32no—
44#67⋊Dic3—32no—
45#63896—32no—
46#63866—31no—
47#63857—31no—
48#63844—31no—
49#63840—32no—
50C22×Dic18—31no—
51C22×Dic18—31no—
52C22×Dic18—61no—
53C4×D4×C9—31no—
54D4×Dic9—32no—
55#63486—31no—
56#63486D311yescharacteristic, complemented
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⋊Dic9—24no—
68C2×C3⋊Dic9C2312yes—
69C2×C3⋊Dic9C2312yes—
70#6226—22no—
71#6226—24no—
72#6226D414yescomplemented
73C9×#67Q814yescomplemented
74C2×C9×Dic3C2311yescharacteristic
75C2×C9×Dic3C2311yescharacteristic
76C2×C9×Dic3C2311yescharacteristic
77C2×C3×Dic9C2312yes—
78C2×C3×Dic9C2312yes—
79C4×D3×C9—22no—
80C3×Dic18—22no—
81D3×Dic9—24no—
82#6189—24no—
83#5632—91no—
84D4×Dic6—91no—
85C22×C32×C4C2×D311yescharacteristic
86C22×C3⋊Dic3—32no—
87C2×#634—31no—
88C2×#634—32no—
89C2×#634—31no—
90C2×#634—64no—
91C2×C3×#67C2×D311yescharacteristic
92C2×C3×C4×D3C2×D311yescharacteristic
93C2×C3×Dic6—31no—
94C2×#627—32no—
95C2×D3×Dic3—32no—
96D3×Dic6—64no—
97C3×#217—32no—
98C3×#217C2×D311yescharacteristic
99C3×(C2×D3)⋊C4C2×D311yescharacteristic
100C3×C4⋊Dic3—31no—
101C3×Dic3⋊C4—32no—
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×Dic9—32no—
116C22×Dic9—62no—
117C2×Dic18—31no—
118C2×Dic18—31no—
119C2×Dic18—32no—
120C2×Dic18—32no—
121C2×Dic18—64no—
122C2×Dic18—64no—
123C2×Dic18—61no—
124C2×Dic18—61no—
125C2×Dic18—62no—
126C2×Dic18—68no—
127C2×Dic18C2×D311yescharacteristic
128C9×C4⋊C4—31no—
129C9×C22⋊C4—31no—
130C9×C22⋊C4—31no—
131C42×C9—31no—
132#3676—32no—
133#3676C2×D312yes—
134C4⋊Dic9—32no—
135C4⋊Dic9—32no—
136C4⋊Dic9C2×D311yescharacteristic
137Dic9⋊C4—32no—
138Dic9⋊C4—32no—
139Dic9⋊C4C2×D312yes—
140C4×Dic9—32no—
141C22×C3×C9—21no—
142C22×C3×C9—21no—
143C22×C3×C9C2×Q812yescomplemented
144C22×C3×C9C2411yescharacteristic
145C2×D3×C9—21no—
146C2×D3×C9—21no—
147C2×D3×C9C2×Q812yescomplemented
148C3×C4×C9—21no—
149C3×C4×C9C2×D412yes—
150C3⋊Dic9—22no—
151C3⋊Dic9C2×D414yes—
152C9×Dic3—21no—
153C9×Dic3—21no—
154C9×Dic3C2×Q812yescomplemented
155C3×Dic9—22no—
156C3×Dic9—22no—
157C2×D3×Q8—91no—
158C22×Dic6—91no—
159C22×Dic6—91no—
160C22×Dic6—91no—
161C22×Dic6—181no—
162C3×C22⋊Q8—91no—
163C3×C4×D4—31no—
164#944—91no—
165D4×Dic3—92no—
166C4×#67D911yescharacteristic
167#924—31no—
168#924—91no—
169#897—91no—
170#896—92no—
171#888—92no—
172#888—92no—
173#887—91no—
174#67⋊C4—92no—
175#878—92no—
176#878—92no—
177#878—92no—
178#869—91no—
179C4×Dic6—91no—
180C23×C32C22×D311yescharacteristic
181C22×C3×D3C22×D311yescharacteristic
182C2×C32×C4—22no—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3—32no—
186C2×C3⋊Dic3—32no—
187C2×C3⋊Dic3—64no—
188#634—34no—
189#634—64no—
190#634—62no—
191C3×#67Dic614yes—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3C22×D311yescharacteristic
195C2×C3×Dic3C22×D311yescharacteristic
196C2×C3×Dic3C22×D311yescharacteristic
197C3×C4×D3—22no—
198C3×Dic6—62no—
199#627—64no—
200D3×Dic3—64no—
201C23×C9—21no—
202C23×C9—31no—
203C23×C9C22×D311yescharacteristic
204D4×C9—34no—
205C2×C4×C9—24no—
206C2×C4×C9—22no—
207C2×C4×C9—21no—
208C2×C4×C9—21no—
209C2×C4×C9—31no—
210C2×C4×C9—31no—
211C2×C4×C9—31no—
212C2×C4×C9—62no—
213C2×C4×C9C22×D311yescharacteristic
214C2×C4×C9C22×D311yescharacteristic
215C2×Dic9—32no—
216C2×Dic9—32no—
217C2×Dic9—62no—
218C2×Dic9—62no—
219C2×Dic9—64no—
220C2×Dic9—64no—
221C2×Dic9—68no—
222C2×Dic9C22×D312yes—
223C2×Dic9C22×D312yes—
224Dic18—22no—
225Dic18—34no—
226Dic18—64no—
227Dic18—64no—
228Dic18—64no—
229Dic18—68no—
230Dic18—64no—
231Dic18—62no—
232D4×Q8—271noSylow
233C2×C3×C9—22no—
234C2×C3×C9C22×D411yescharacteristic
235C2×C3×C9C22×Q811yescharacteristic
236C2×C3×C92−1+411yescharacteristic
237D3×C9—22no—
238C2×C3×Q8—91no—
239C2×C3×D4—31no—
240C22×C3×C4—21no—
241C22×C3×C4—31no—
242C22×C3×C4C2×D911yescharacteristic
243C22×C3×C4D3211yescharacteristic
244C2×#67C2×D911yescharacteristic
245C22×Dic3—92no—
246C22×Dic3—92no—
247C22×Dic3—92no—
248C22×Dic3—182no—
249D3×Q8—184no—
250C2×C4×D3—92no—
251C2×C4×D3C2×D911yescharacteristic
252C2×Dic6—31no—
253C2×Dic6—91no—
254C2×Dic6—91no—
255C2×Dic6—92no—
256C2×Dic6—92no—
257C2×Dic6—91no—
258C2×Dic6—91no—
259C2×Dic6—92no—
260C2×Dic6—92no—
261C2×Dic6—181no—
262C2×Dic6—182no—
263C2×Dic6—181no—
264C2×Dic6—184no—
265C2×Dic6—184no—
266C2×Dic6—184no—
267C2×Dic6—188no—
268C3×C4⋊C4—31no—
269C3×C4⋊C4—91no—
270C3×C4⋊C4—92no—
271C3×C22⋊C4—31no—
272C3×C22⋊C4—31no—
273C3×C22⋊C4—92no—
274C3×C42—31no—
275#217—32no—
276#217—92no—
277#217C2×D911yescharacteristic
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4C2×D911yescharacteristic
280C4⋊Dic3—31no—
281C4⋊Dic3—92no—
282C4⋊Dic3—92no—
283C4⋊Dic3—92no—
284Dic3⋊C4—32no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4C2×D911yescharacteristic
291C4×Dic3—92no—
292C4×Dic3—92no—
293C4×Dic3C2×D911yescharacteristic
294C22×C32—21no—
295C22×C32—21no—
296C22×C32C2×Dic612yes—
297C22×C32C23×D311yescharacteristic
298C2×C3×D3—21no—
299C2×C3×D3—21no—
300C2×C3×D3C2×Dic612yes—
301C32×C4—21no—
302C32×C4D3×D412yes—
303C3⋊Dic3—34no—
304C3⋊Dic3—62no—
305C3×Dic3—21no—
306C3×Dic3—21no—
307C3×Dic3—62no—
308C3×Dic3—62no—
309C3×Dic3C2×Dic612yes—
310C22×C9—22no—
311C22×C9—22no—
312C22×C9—21no—
313C22×C9—21no—
314C22×C9—21no—
315C22×C9—22no—
316C22×C9—32no—
317C22×C9—61no—
318C22×C9—61no—
319C22×C9D3×Q812yescomplemented
320C22×C9C23×D311yescharacteristic
321C4×C9—22no—
322C4×C9—21no—
323C4×C9—22no—
324C4×C9—32no—
325C4×C9—61no—
326C4×C9—61no—
327C4×C9D3×D412yes—
328Dic9—22no—
329Dic9—22no—
330Dic9—34no—
331Dic9—64no—
332Dic9—64no—
333Dic9—62no—
334C22×Q8—271no—
335C22×Q8—271no—
336C4⋊Q8—271no—
337C4⋊Q8—272no—
338C22⋊Q8—91no—
339C22⋊Q8—272no—
340C22⋊Q8—272no—
341C22⋊Q8—271no—
342C4×Q8—271no—
343C4×D4—31no—
344C4×D4—272no—
345C3×C9D4×Q811yescharacteristic, complemented, Sylow
346C23×C3—21no—
347C23×C3—31no—
348C23×C3C22×D911yescharacteristic
349C23×C3C2×D3211yescharacteristic
350C22×D3C22×D911yescharacteristic
351C3×Q8—182no—
352C3×D4—34no—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—62no—
362C2×C3×C4—92no—
363C2×C3×C4—92no—
364C2×C3×C4C22×D911yescharacteristic
365C2×C3×C4C22×D911yescharacteristic
366C2×C3×C4C2×D3211yescharacteristic
367C2×C3×C4C2×D3211yescharacteristic
368#67Dic1814yescomplemented
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—92no—
372C2×Dic3—92no—
373C2×Dic3—92no—
374C2×Dic3—92no—
375C2×Dic3—182no—
376C2×Dic3—182no—
377C2×Dic3—184no—
378C2×Dic3—184no—
379C2×Dic3—184no—
380C2×Dic3—188no—
381C2×Dic3C22×D911yescharacteristic
382C2×Dic3C22×D911yescharacteristic
383C2×Dic3C22×D911yescharacteristic
384C4×D3—22no—
385C4×D3—184no—
386Dic6—62no—
387Dic6—94no—
388Dic6—94no—
389Dic6—184no—
390Dic6—184no—
391Dic6—184no—
392Dic6—184no—
393Dic6—184no—
394Dic6—188no—
395Dic6—184no—
396Dic6—182no—
397Dic6—182no—
398C2×C32—22no—
399C2×C32C22×Dic611yescharacteristic
400C2×C32C2×D3×D411yescharacteristic
401C2×C32#101011yescharacteristic
402C3×D3—22no—
403C2×C9—24no—
404C2×C9—22no—
405C2×C9—21no—
406C2×C9—21no—
407C2×C9—21no—
408C2×C9—62no—
409C2×C9C2×D3×D411yescharacteristic
410C2×C9C2×D3×Q811yescharacteristic
411C2×C9#101011yescharacteristic
412C2×Q8—91no—
413C2×Q8—272no—
414C2×Q8—271no—
415C2×Q8—272no—
416C2×Q8—271no—
417C2×Q8—544no—
418C2×Q8—544no—
419C2×D4—31no—
420C22×C4—31no—
421C22×C4—272no—
422C22×C4—272no—
423C22×C4D3×D911yescharacteristic
424C4⋊C4—31no—
425C4⋊C4—91no—
426C4⋊C4—92no—
427C4⋊C4—272no—
428C4⋊C4—272no—
429C4⋊C4—272no—
430C4⋊C4—272no—
431C22⋊C4—31no—
432C22⋊C4—31no—
433C22⋊C4—92no—
434C22⋊C4—272no—
435C42—31no—
436C42—272no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—22no—
444C22×C3—21no—
445C22×C3—32no—
446C22×C3—61no—
447C22×C3—61no—
448C22×C3C2×Dic1812yescomplemented
449C22×C3C23×D911yescharacteristic
450C22×C3D3×Dic612yes—
451C22×C3C22×D3211yescharacteristic
452C2×D3—21no—
453C2×D3—21no—
454C2×D3C2×Dic1812yescomplemented
455C3×C4—22no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—22no—
459C3×C4—32no—
460C3×C4—61no—
461C3×C4—61no—
462C3×C4—182no—
463C3×C4—182no—
464C3×C4D4×D912yes—
465C3×C4#381112yes—
466Dic3—21no—
467Dic3—21no—
468Dic3—62no—
469Dic3—62no—
470Dic3—94no—
471Dic3—94no—
472Dic3—184no—
473Dic3—184no—
474Dic3—182no—
475Dic3—182no—
476Dic3C2×Dic1812yescomplemented
477C32D4×Dic611yescharacteristic
478C9—21no—
479C9#563211yescharacteristic, complemented
480C23—31no—
481C23C2×D3×D911yescharacteristic
482Q8—182no—
483Q8—274no—
484Q8—544no—
485Q8—544no—
486Q8—542no—
487D4—34no—
488C2×C4—22no—
489C2×C4—31no—
490C2×C4—31no—
491C2×C4—31no—
492C2×C4—62no—
493C2×C4—92no—
494C2×C4—92no—
495C2×C4—272no—
496C2×C4—272no—
497C2×C4—544no—
498C2×C4—544no—
499C2×C4C2×D3×D911yescharacteristic
500C2×C4C2×D3×D911yescharacteristic
501C2×C3—24no—
502C2×C3—22no—
503C2×C3—22no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—62no—
508C2×C3C22×Dic1811yescharacteristic
509C2×C3C2×D4×D911yescharacteristic
510C2×C3#6371611yescharacteristic
511C2×C3C2×D3×Dic611yescharacteristic
512C2×C3#6429811yescharacteristic
513C2×C3C2×#381111yescharacteristic
514D3—22no—
515C22—21no—
516C22—21no—
517C22—32no—
518C22—61no—
519C22—61no—
520C22D3×Dic1812yescomplemented
521C22C22×D3×D911yescharacteristic
522C4—21no—
523C4—32no—
524C4—61no—
525C4—61no—
526C4—182no—
527C4—182no—
528C4—274no—
529C4—542no—
530C4#8866812yes—
531C3—21no—
532C3D4×Dic1811yescharacteristic, complemented
533C3#1059491511yescharacteristic
534C2—22no—
535C2—62no—
536C2C2×D3×Dic1811yescharacteristic
537C2#1172246511yescharacteristic
538C2C2×#8866811yescharacteristic
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×#6226C2semidirect
2. C2×D3×Dic18C2semidirect
4. C3×#63486C2semidirect
6. #11721267C2semidirect
7. #11721024C2semidirect
8. #11720961C2semidirect
9. #11720932C2semidirect
10. #11720912C2semidirect
11. #11720905C2semidirect
56. #63486D3semidirect
72. #6226D4semidirect
73. C9×#67Q8semidirect
143. C22×C3×C9C2×Q8semidirect
147. C2×D3×C9C2×Q8semidirect
154. C9×Dic3C2×Q8semidirect
319. C22×C9D3×Q8semidirect
345. C3×C9D4×Q8semidirect
368. #67Dic18semidirect
448. C22×C3C2×Dic18semidirect
454. C2×D3C2×Dic18semidirect
476. Dic3C2×Dic18semidirect
479. C9#5632semidirect
520. C22D3×Dic18semidirect
532. C3D4×Dic18semidirect

Maximal subgroups

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

subgroupindex
C22×#62262
C2×D3×Dic182
C4×C9×#672
C3×#634862
#67⋊Dic92
#117212672
#117210242
#117209612
#117209322
#117209122
#117209052
#105949153
D4×Dic183