Finite Group Databasebeta

← #13148835

Subgroups — #13148835

Order 1728.

910 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×Dic18C211yesmaximal, characteristic, complemented
2C2×D9×Dic6C211yesmaximal, characteristic, complemented
3#11721743C211yesmaximal, characteristic, complemented
4C9×#897C211yesmaximal, characteristic, complemented
5C3×#63459C211yesmaximal, characteristic, complemented
6#11721028C212yesmaximal
7C4×#6194C211yesmaximal, characteristic
8#11721026C212yesmaximal, complemented
9#11720999C212yesmaximal, complemented
10#11720971C212yesmaximal, complemented
11#11720930C211yesmaximal, characteristic, complemented
12#10594526—31nomaximal
13#10589734—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C4×D3×C9C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×C3×Dic18C2211yescharacteristic
19C2×#6194C2211yescharacteristic
20C2×D3×Dic9C2212yes—
21C2×Dic3×D9C2212yes—
22C2×#6189C2212yes—
23C2×#6189C2212yes—
24D3×Dic18—24no—
25D9×Dic6—24no—
26C3×C9×C4⋊C4C2211yescharacteristic
27#88534C2212yes—
28C4×C3⋊Dic9C2212yes—
29C4×C3⋊Dic9C2211yescharacteristic
30C9×(C2×D3)⋊C4C2212yes—
31C3×(C2×D9)⋊C4C2212yes—
32C9×C4⋊Dic3C2212yes—
33C3×C4⋊Dic9C2212yes—
34(C2×D3)⋊Dic9C2211yescharacteristic
35(C2×D9)⋊Dic3C2211yescharacteristic
36#88442C2211yescharacteristic
37#88442C2212yes—
38#88442C2212yes—
39C2×D3×Dic6—31no—
40C2×D3×Dic6—31no—
41#64096—31no—
42C3×#897—31no—
43C3×#897D311yescharacteristic
44#63919—32no—
45#63902—32no—
46C4×#625—31no—
47#63899—32no—
48#63899—32no—
49#63873—31no—
50C2×Q8×D9—31no—
51C22×Dic18—31no—
52C9×C22⋊Q8—31no—
53#63506—31no—
54C4×#611—31no—
55#63459D311yescharacteristic, complemented
56#63458—32no—
57#63450—32no—
58#63449—31no—
59#63449—61no—
60#611⋊C4—32no—
61#63440—32no—
62C22×D3×C9C2311yescharacteristic
63C22×C3×D9C2311yescharacteristic
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2311yescharacteristic
69#6226D414yescomplemented
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C4×D3×C9—22no—
75C3×C4×D9—22no—
76C9×Dic6—22no—
77C3×Dic18—22no—
78#6194Q814yescomplemented
79D3×Dic9—24no—
80Dic3×D9—24no—
81#6189—24no—
82#6189—24no—
83#5632—91no—
84#5632—91no—
85C2×#634—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3C2×D311yescharacteristic
88C2×C3×Dic6—31no—
89C2×C3×Dic6C2×D311yescharacteristic
90C2×#627—32no—
91C2×#627—32no—
92C2×#625—31no—
93C2×D3×Dic3—32no—
94C2×D3×Dic3—32no—
95D3×Dic6—64no—
96D3×Dic6—64no—
97C32×C4⋊C4C2×D311yescharacteristic
98#3750—32no—
99C4×C3⋊Dic3—31no—
100C4×C3⋊Dic3—32no—
101C3×(C2×D3)⋊C4—32no—
102C3×(C2×D3)⋊C4C2×D312yes—
103C3×C4⋊Dic3—32no—
104C3×C4⋊Dic3C2×D312yes—
105#3724—32no—
106#3724—32no—
107#3724—31no—
108(C2×D3)⋊Dic3—31no—
109(C2×D3)⋊Dic3—31no—
110C2×Q8×C9—31no—
111C22×C4×C9—31no—
112C2×#611—31no—
113C22×Dic9—32no—
114Q8×D9—64no—
115C2×C4×D9—32no—
116C2×C4×D9C2×D311yescharacteristic
117C2×Dic18—31no—
118C2×Dic18—32no—
119C2×Dic18—32no—
120C2×Dic18—64no—
121C2×Dic18—61no—
122C2×Dic18C2×D311yescharacteristic
123C9×C4⋊C4—21no—
124C9×C4⋊C4—32no—
125C9×C4⋊C4C2×D311yescharacteristic
126C9×C22⋊C4—32no—
127#3676—31no—
128(C2×D9)⋊C4—31no—
129(C2×D9)⋊C4C2×D312yes—
130C4⋊Dic9C2×D312yes—
131Dic9⋊C4—32no—
132Dic9⋊C4—31no—
133Dic9⋊C4—32no—
134Dic9⋊C4—32no—
135Dic9⋊C4—62no—
136C4×Dic9—31no—
137C4×Dic9—32no—
138C4×Dic9—62no—
139C4×Dic9—61no—
140C22×C3×C9C2411yescharacteristic
141C2×D3×C9—21no—
142C2×D3×C9—21no—
143C2×D3×C9C2×Q812yescomplemented
144C2×C3×D9—21no—
145C2×C3×D9—21no—
146C2×C3×D9C2×Q812yescomplemented
147C3×C4×C9—22no—
148C3×C4×C9C2×D412yes—
149C3⋊Dic9—21no—
150C3⋊Dic9C2×D414yes—
151C3⋊Dic9C2×Q812yescomplemented
152C9×Dic3—21no—
153C9×Dic3—22no—
154C3×Dic9—21no—
155C3×Dic9—22no—
156C2×D3×Q8—91no—
157C2×D3×Q8—91no—
158C22×Dic6—91no—
159C22×Dic6—91no—
160C3×C22⋊Q8—31no—
161C3×C22⋊Q8—91no—
162#944—91no—
163#944—91no—
164C4×#67—91no—
165C4×#67—91no—
166#897—31no—
167#897D911yescharacteristic, complemented
168#896—92no—
169#896—92no—
170#888—92no—
171#888—92no—
172#887—91no—
173#887—91no—
174#887—181no—
175#67⋊C4—92no—
176#67⋊C4—92no—
177#878—92no—
178#878—92no—
179C22×C3×D3—31no—
180C22×C3×D3C22×D311yescharacteristic
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D311yescharacteristic
183C2×C3⋊Dic3—31no—
184C2×C3⋊Dic3—32no—
185C2×C3⋊Dic3—31no—
186#634—34no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3C22×D311yescharacteristic
190C2×C3×Dic3C22×D312yes—
191C3×C4×D3—22no—
192C3×C4×D3—62no—
193C3×Dic6—22no—
194C3×Dic6—62no—
195#627—64no—
196#627—64no—
197#625—34no—
198D3×Dic3—64no—
199D3×Dic3—64no—
200C23×C9—31no—
201C22×D9C22×D311yescharacteristic
202Q8×C9—62no—
203C2×C4×C9—21no—
204C2×C4×C9—22no—
205C2×C4×C9—32no—
206C2×C4×C9—31no—
207C2×C4×C9—62no—
208C2×C4×C9C22×D312yes—
209C2×C4×C9C22×D311yescharacteristic
210#611—34no—
211C2×Dic9—31no—
212C2×Dic9—32no—
213C2×Dic9—31no—
214C2×Dic9—61no—
215C2×Dic9—62no—
216C2×Dic9—61no—
217C2×Dic9—64no—
218C2×Dic9C22×D312yes—
219C2×Dic9C22×D311yescharacteristic
220C4×D9—22no—
221C4×D9—64no—
222Dic18—22no—
223Dic18—34no—
224Dic18—64no—
225Dic18—64no—
226Dic18—64no—
227D4×Q8—271noSylow
228C2×C3×C9C22×D411yescharacteristic
229C2×C3×C9C22×Q811yescharacteristic
230C2×C3×C92−1+411yescharacteristic
231D3×C9—22no—
232C3×D9—22no—
233C2×C3×Q8—31no—
234C2×C3×Q8—91no—
235C22×C3×C4—31no—
236C22×C3×C4—91no—
237C2×#67—91no—
238C2×#67—91no—
239C22×Dic3—92no—
240C22×Dic3—92no—
241D3×Q8—184no—
242D3×Q8—184no—
243C2×C4×D3—31no—
244C2×C4×D3—92no—
245C2×C4×D3—92no—
246C2×C4×D3C2×D911yescharacteristic
247C2×Dic6—31no—
248C2×Dic6—91no—
249C2×Dic6—91no—
250C2×Dic6—92no—
251C2×Dic6—92no—
252C2×Dic6—92no—
253C2×Dic6—92no—
254C2×Dic6—181no—
255C2×Dic6—184no—
256C2×Dic6—184no—
257C2×Dic6C2×D911yescharacteristic
258C3×C4⋊C4—21no—
259C3×C4⋊C4—32no—
260C3×C4⋊C4—92no—
261C3×C4⋊C4C2×D911yescharacteristic
262C3×C4⋊C4D3211yescharacteristic
263C3×C22⋊C4—32no—
264C3×C22⋊C4—92no—
265#217—91no—
266#217—91no—
267(C2×D3)⋊C4—32no—
268(C2×D3)⋊C4—91no—
269(C2×D3)⋊C4—91no—
270(C2×D3)⋊C4C2×D912yes—
271C4⋊Dic3—32no—
272C4⋊Dic3C2×D912yes—
273Dic3⋊C4—92no—
274Dic3⋊C4—92no—
275Dic3⋊C4—91no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—91no—
281Dic3⋊C4—182no—
282C4×Dic3—92no—
283C4×Dic3—91no—
284C4×Dic3—92no—
285C4×Dic3—91no—
286C4×Dic3—182no—
287C4×Dic3—181no—
288C22×C32C23×D311yescharacteristic
289C2×C3×D3—21no—
290C2×C3×D3—21no—
291C2×C3×D3—32no—
292C2×C3×D3—61no—
293C2×C3×D3—61no—
294C2×C3×D3D3×Q812yes—
295C32×C4—22no—
296C32×C4D3×D412yes—
297C3⋊Dic3—34no—
298C3⋊Dic3—32no—
299C3⋊Dic3—61no—
300C3×Dic3—22no—
301C3×Dic3—21no—
302C3×Dic3—62no—
303C3×Dic3—61no—
304C22×C9—21no—
305C22×C9—32no—
306C22×C9—61no—
307C22×C9—61no—
308C22×C9C23×D311yescharacteristic
309C2×D9—21no—
310C2×D9—21no—
311C2×D9D3×Q812yescomplemented
312C4×C9—22no—
313C4×C9—22no—
314C4×C9—42no—
315C4×C9—61no—
316C4×C9—62no—
317C4×C9D3×D412yes—
318Dic9—22no—
319Dic9—21no—
320Dic9—34no—
321Dic9—32no—
322Dic9—64no—
323Dic9—62no—
324Dic9—61no—
325Dic9—121no—
326C22×Q8—271no—
327C22×Q8—271no—
328C4⋊Q8—272no—
329C4⋊Q8—271no—
330C22⋊Q8—31no—
331C22⋊Q8—91no—
332C22⋊Q8—272no—
333C22⋊Q8—272no—
334C4×Q8—271no—
335C4×D4—272no—
336C4×D4—271no—
337C3×C9D4×Q811yescharacteristic, complemented, Sylow
338C23×C3—31no—
339C23×C3—91no—
340C22×D3—31no—
341C22×D3C22×D911yescharacteristic
342C3×Q8—62no—
343C3×Q8—182no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—32no—
347C2×C3×C4—31no—
348C2×C3×C4—62no—
349C2×C3×C4—92no—
350C2×C3×C4—91no—
351C2×C3×C4—182no—
352C2×C3×C4C22×D912yes—
353C2×C3×C4C22×D911yescharacteristic
354C2×C3×C4C2×D3212yes—
355C2×C3×C4C2×D3211yescharacteristic
356#67—94no—
357#67—94no—
358C2×Dic3—32no—
359C2×Dic3—31no—
360C2×Dic3—91no—
361C2×Dic3—92no—
362C2×Dic3—91no—
363C2×Dic3—92no—
364C2×Dic3—91no—
365C2×Dic3—91no—
366C2×Dic3—181no—
367C2×Dic3—182no—
368C2×Dic3—181no—
369C2×Dic3—184no—
370C2×Dic3—184no—
371C2×Dic3C22×D912yes—
372C2×Dic3C22×D911yescharacteristic
373C4×D3—22no—
374C4×D3—62no—
375C4×D3—184no—
376C4×D3—184no—
377Dic6—22no—
378Dic6—62no—
379Dic6—94no—
380Dic6—94no—
381Dic6—184no—
382Dic6—184no—
383Dic6—184no—
384Dic6—184no—
385Dic6—184no—
386C2×C32C2×D3×D411yescharacteristic
387C2×C32C2×D3×Q811yescharacteristic
388C2×C32#101011yescharacteristic
389C3×D3—22no—
390C3×D3—62no—
391C2×C9—21no—
392C2×C9—21no—
393C2×C9—21no—
394C2×C9—62no—
395C2×C9C2×D3×D411yescharacteristic
396C2×C9C2×D3×Q811yescharacteristic
397C2×C9#101011yescharacteristic
398D9—22no—
399C2×Q8—31no—
400C2×Q8—91no—
401C2×Q8—272no—
402C2×Q8—272no—
403C2×Q8—271no—
404C2×Q8—544no—
405C2×Q8—544no—
406C2×D4—271no—
407C22×C4—31no—
408C22×C4—91no—
409C22×C4—272no—
410C22×C4—272no—
411C4⋊C4—32no—
412C4⋊C4—92no—
413C4⋊C4—272no—
414C4⋊C4—271no—
415C4⋊C4—272no—
416C4⋊C4—272no—
417C4⋊C4D3×D911yescharacteristic
418C22⋊C4—32no—
419C22⋊C4—92no—
420C22⋊C4—271no—
421C22⋊C4—271no—
422C42—272no—
423C42—271no—
424C22×C3—21no—
425C22×C3—32no—
426C22×C3—61no—
427C22×C3—61no—
428C22×C3—92no—
429C22×C3—181no—
430C22×C3—181no—
431C22×C3C23×D911yescharacteristic
432C22×C3C22×D3211yescharacteristic
433C2×D3—21no—
434C2×D3—21no—
435C2×D3—32no—
436C2×D3—61no—
437C2×D3—61no—
438C2×D3Q8×D912yescomplemented
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—42no—
443C3×C4—61no—
444C3×C4—62no—
445C3×C4—181no—
446C3×C4—182no—
447C3×C4D4×D912yes—
448C3×C4#380212yes—
449Dic3—22no—
450Dic3—21no—
451Dic3—62no—
452Dic3—61no—
453Dic3—94no—
454Dic3—92no—
455Dic3—94no—
456Dic3—92no—
457Dic3—184no—
458Dic3—182no—
459Dic3—181no—
460Dic3—181no—
461Dic3—361no—
462C32#563211yescharacteristic
463C9—21no—
464C9#563211yescharacteristic, complemented
465C23—31no—
466C23—91no—
467Q8—62no—
468Q8—182no—
469Q8—274no—
470Q8—544no—
471Q8—544no—
472D4—274no—
473C2×C4—32no—
474C2×C4—31no—
475C2×C4—62no—
476C2×C4—92no—
477C2×C4—91no—
478C2×C4—182no—
479C2×C4—271no—
480C2×C4—272no—
481C2×C4—271no—
482C2×C4—544no—
483C2×C4—544no—
484C2×C4C2×D3×D912yes—
485C2×C4C2×D3×D911yescharacteristic
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—62no—
490C2×C3—182no—
491C2×C3C2×D4×D911yescharacteristic
492C2×C3C2×Q8×D911yescharacteristic
493C2×C3#6371611yescharacteristic
494C2×C3C2×#380211yescharacteristic
495C2×C3#6430911yescharacteristic
496C2×C3D32×Q811yescharacteristic
497D3—22no—
498D3—62no—
499C22—32no—
500C22—61no—
501C22—61no—
502C22—92no—
503C22—181no—
504C22—181no—
505C22C22×D3×D911yescharacteristic
506C4—22no—
507C4—61no—
508C4—62no—
509C4—181no—
510C4—182no—
511C4—272no—
512C4—274no—
513C4—541no—
514C4#8864612yes—
515C3—21no—
516C3#1058973411yescharacteristic, complemented
517C3#1059452611yescharacteristic
518C2—62no—
519C2—182no—
520C2C2×#8864611yescharacteristic
521C2#1172249411yescharacteristic
522C2D3×Q8×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#6194C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9D4×Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×Dic18C2semidirect
2. C2×D9×Dic6C2semidirect
3. #11721743C2semidirect
4. C9×#897C2semidirect
5. C3×#63459C2semidirect
8. #11721026C2semidirect
9. #11720999C2semidirect
10. #11720971C2semidirect
11. #11720930C2semidirect
55. #63459D3semidirect
69. #6226D4semidirect
78. #6194Q8semidirect
143. C2×D3×C9C2×Q8semidirect
146. C2×C3×D9C2×Q8semidirect
151. C3⋊Dic9C2×Q8semidirect
167. #897D9semidirect
311. C2×D9D3×Q8semidirect
337. C3×C9D4×Q8semidirect
438. C2×D3Q8×D9semidirect
464. C9#5632semidirect
516. C3#10589734semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Dic182
C2×D9×Dic62
#117217432
C9×#8972
C3×#634592
#117210282
C4×#61942
#117210262
#117209992
#117209712
#117209302
#105945263
#105897343