Finite Group Databasebeta

← D9×#888 #13148812

Subgroups — D9×#888 #13148812

Order 1728.

938 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D9×Dic6C212yesmaximal
2#11721744C211yesmaximal, characteristic, complemented
3C3×C4⋊C4×D9C211yesmaximal, characteristic
4C9×#888C211yesmaximal, characteristic, complemented
5D9×C4⋊Dic3C211yesmaximal, characteristic
6D9×Dic3⋊C4C212yesmaximal
7C4×Dic3×D9C211yesmaximal, characteristic
8#11720971C211yesmaximal, characteristic, complemented
9#11720970C211yesmaximal, characteristic, complemented
10#11720905C212yesmaximal, complemented
11#11720903C212yesmaximal, complemented
12D3×#888—31nomaximal
13D9×C4⋊Q8—31nomaximal
14C2×#6226C2212yes—
15C2×C3×C4×D9C2212yes—
16C2×C3×C4×D9C2211yescharacteristic
17C2×C9×Dic6C2212yes—
18C2×Dic3×D9C2212yes—
19C2×Dic3×D9C2211yescharacteristic
20C2×Dic3×D9C2211yescharacteristic
21C2×#6189C2212yes—
22C2×#6189C2212yes—
23D9×Dic6—28no—
24C3×C9×C4⋊C4C2211yescharacteristic
25C4⋊(C3⋊Dic9)C2211yescharacteristic
26#88534C2212yes—
27C4×C3⋊Dic9C2211yescharacteristic
28C9×C4⋊Dic3C2211yescharacteristic
29C9×Dic3⋊C4C2212yes—
30C4×C9×Dic3C2211yescharacteristic
31C3×C4⋊Dic9C2211yescharacteristic
32C3×Dic9⋊C4C2212yes—
33Dic3⋊Dic9C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2211yescharacteristic
37Dic3×Dic9C2211yescharacteristic
38C2×D3×Dic6—32no—
39#64097—31no—
40C3×D3×C4⋊C4—31no—
41C3×#888D311yescharacteristic
42#63919—31no—
43#63918—31no—
44D3×C4⋊Dic3—31no—
45D3×Dic3⋊C4—32no—
46C4×D3×Dic3—31no—
47#63840—32no—
48#63839—32no—
49C2×Q8×D9—32no—
50C9×C4⋊Q8—31no—
51#63506—32no—
52C4⋊C4×D9—32no—
53C4⋊C4×D9—31no—
54C4⋊C4×D9D311yescharacteristic
55#63450—31no—
56#63450—31no—
57#63450—32no—
58#63450—61no—
59C42×D9—31no—
60#63431—31no—
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67#6226—24no—
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C3×C4×D9—24no—
74C3×C4×D9Q814yescomplemented
75C9×Dic6—24no—
76Dic3×D9—24no—
77Dic3×D9D414yes—
78Dic3×D9Q814yescomplemented
79#6189—24no—
80#6189—24no—
81D3×C4⋊Q8—91no—
82C2×#888—91no—
83C2×#634—32no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—32no—
86C2×C3×Dic6C2×D312yes—
87C2×#627—32no—
88C2×#627—32no—
89C2×D3×Dic3—31no—
90C2×D3×Dic3—31no—
91C2×D3×Dic3—32no—
92D3×Dic6—68no—
93C32×C4⋊C4C2×D311yescharacteristic
94C4⋊(C3⋊Dic3)—31no—
95#3750—32no—
96C4×C3⋊Dic3—31no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4C2×D312yes—
101C3×C4×Dic3C2×D311yescharacteristic
102#3724—32no—
103Dic3⋊Dic3—32no—
104Dic3⋊Dic3—32no—
105Dic32—31no—
106Dic32—31no—
107C2×Q8×C9—32no—
108Q8×D9—68no—
109C2×C4×D9—31no—
110C2×C4×D9—31no—
111C2×C4×D9—32no—
112C2×C4×D9C2×D312yes—
113C2×C4×D9C2×D311yescharacteristic
114C2×Dic18—32no—
115C2×Dic18—32no—
116C2×Dic18—32no—
117C2×Dic18—62no—
118C9×C4⋊C4—21no—
119C9×C4⋊C4—32no—
120C9×C4⋊C4—31no—
121C9×C4⋊C4C2×D311yescharacteristic
122C42×C9—31no—
123C4⋊Dic9—31no—
124C4⋊Dic9—32no—
125C4⋊Dic9—61no—
126C4⋊Dic9C2×D311yescharacteristic
127Dic9⋊C4—32no—
128Dic9⋊C4—32no—
129Dic9⋊C4—32no—
130Dic9⋊C4—62no—
131Dic9⋊C4C2×D312yes—
132C4×Dic9—31no—
133C4×Dic9—31no—
134C4×Dic9—31no—
135C4×Dic9—61no—
136C22×C3×C9C2411yescharacteristic
137C2×C3×D9C2×D412yes—
138C2×C3×D9C2×Q812yes—
139C2×C3×D9C2×Q812yes—
140C3×C4×C9—22no—
141C3×C4×C9C2×Q812yescomplemented
142C3⋊Dic9—22no—
143C3⋊Dic9C2×D412yes—
144C3⋊Dic9C2×Q812yescomplemented
145C9×Dic3—22no—
146C9×Dic3C2×D412yes—
147C9×Dic3C2×Q812yescomplemented
148C3×Dic9—22no—
149C3×Dic9C2×Q812yescomplemented
150C2×D3×Q8—92no—
151C22×Dic6—92no—
152C3×C4⋊Q8—31no—
153C2×C3×C4⋊C4—91no—
154#944—92no—
155C2×C4⋊Dic3—91no—
156C2×Dic3⋊C4—92no—
157C2×C4×Dic3—91no—
158D3×C4⋊C4—31no—
159D3×C4⋊C4—92no—
160D3×C4⋊C4—91no—
161#888—92no—
162#888—91no—
163#888—92no—
164#888—91no—
165#888—91no—
166#888—91no—
167#888—92no—
168#888—91no—
169#888—181no—
170#888D911yescharacteristic, direct product factor
171C42×D3—91no—
172#869—91no—
173C22×C3×D3—31no—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—32no—
179#634—64no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3C22×D312yes—
183C2×C3×Dic3C22×D311yescharacteristic
184C2×C3×Dic3C22×D311yescharacteristic
185C3×C4×D3—34no—
186C3×C4×D3—64no—
187C3×Dic6—24no—
188#627—64no—
189#627—64no—
190D3×Dic3—34no—
191D3×Dic3—34no—
192D3×Dic3—64no—
193C22×D9C22×D311yescharacteristic
194Q8×C9—64no—
195C2×C4×C9—21no—
196C2×C4×C9—22no—
197C2×C4×C9—31no—
198C2×C4×C9—31no—
199C2×C4×C9—32no—
200C2×C4×C9C22×D312yes—
201C2×C4×C9C22×D311yescharacteristic
202C2×Dic9—31no—
203C2×Dic9—31no—
204C2×Dic9—32no—
205C2×Dic9—61no—
206C2×Dic9—61no—
207C2×Dic9—62no—
208C2×Dic9C22×D312yes—
209C2×Dic9C22×D311yescharacteristic
210C4×D9—24no—
211C4×D9—34no—
212C4×D9—34no—
213C4×D9—64no—
214C4×D9Dic614yescomplemented
215Dic18—64no—
216Dic18—64no—
217Dic18—64no—
218Dic18—124no—
219C2×C4⋊Q8—271noSylow
220C2×C3×C9C22×D411yescharacteristic
221C2×C3×C9C22×Q811yescharacteristic
222C2×C3×C9C22×Q811yescharacteristic
223C3×D9C4⋊Q814yescomplemented
224C2×C3×Q8—32no—
225C22×C3×C4—91no—
226C22×C3×C4—92no—
227C22×Dic3—91no—
228C22×Dic3—91no—
229C22×Dic3—92no—
230D3×Q8—188no—
231C2×C4×D3—31no—
232C2×C4×D3—32no—
233C2×C4×D3—91no—
234C2×C4×D3—91no—
235C2×C4×D3—92no—
236C2×Dic6—92no—
237C2×Dic6—92no—
238C2×Dic6—92no—
239C2×Dic6—92no—
240C2×Dic6—92no—
241C2×Dic6—92no—
242C2×Dic6—182no—
243C2×Dic6—188no—
244C2×Dic6C2×D912yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—32no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4—91no—
250C3×C4⋊C4C2×D911yescharacteristic
251C3×C4⋊C4D3211yescharacteristic
252C3×C42—31no—
253C4⋊Dic3—31no—
254C4⋊Dic3—92no—
255C4⋊Dic3—91no—
256C4⋊Dic3—91no—
257C4⋊Dic3—92no—
258C4⋊Dic3—181no—
259C4⋊Dic3C2×D911yescharacteristic
260Dic3⋊C4—32no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267Dic3⋊C4—182no—
268Dic3⋊C4C2×D912yes—
269C4×Dic3—91no—
270C4×Dic3—91no—
271C4×Dic3—91no—
272C4×Dic3—91no—
273C4×Dic3—91no—
274C4×Dic3—91no—
275C4×Dic3—181no—
276C4×Dic3C2×D911yescharacteristic
277C22×C32C23×D311yescharacteristic
278C2×C3×D3—32no—
279C2×C3×D3—32no—
280C2×C3×D3—32no—
281C32×C4—22no—
282C32×C4D3×Q812yes—
283C3⋊Dic3—32no—
284C3⋊Dic3—32no—
285C3⋊Dic3—62no—
286C3×Dic3—22no—
287C3×Dic3—32no—
288C3×Dic3—62no—
289C3×Dic3D3×D412yes—
290C3×Dic3D3×Q812yes—
291C22×C9—21no—
292C22×C9C23×D311yescharacteristic
293C2×D9C2×Dic612yes—
294C2×D9D3×D412yes—
295C2×D9D3×Q812yes—
296C4×C9—22no—
297C4×C9—22no—
298C4×C9—32no—
299C4×C9—32no—
300C4×C9—42no—
301C4×C9—62no—
302C4×C9C2×Dic612yescomplemented
303Dic9—22no—
304Dic9—32no—
305Dic9—32no—
306Dic9—62no—
307Dic9—62no—
308Dic9—62no—
309Dic9—122no—
310Dic9C2×Dic612yescomplemented
311C22×Q8—272no—
312C4⋊Q8—31no—
313C4⋊Q8—271no—
314C4⋊Q8—272no—
315C4⋊Q8—271no—
316C4⋊Q8—272no—
317C4⋊Q8—271no—
318C2×C4⋊C4—91no—
319C2×C4⋊C4—272no—
320C2×C4⋊C4—271no—
321C2×C42—271no—
322C3×C9C2×C4⋊Q811yescharacteristic, complemented, Sylow
323C23×C3—91no—
324C22×D3—31no—
325C3×Q8—64no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—31no—
329C2×C3×C4—31no—
330C2×C3×C4—32no—
331C2×C3×C4—91no—
332C2×C3×C4—94no—
333C2×C3×C4—92no—
334C2×C3×C4—184no—
335C2×C3×C4C22×D912yes—
336C2×C3×C4C22×D911yescharacteristic
337C2×C3×C4C2×D3212yes—
338C2×C3×C4C2×D3211yescharacteristic
339C2×Dic3—31no—
340C2×Dic3—32no—
341C2×Dic3—91no—
342C2×Dic3—91no—
343C2×Dic3—91no—
344C2×Dic3—91no—
345C2×Dic3—94no—
346C2×Dic3—94no—
347C2×Dic3—92no—
348C2×Dic3—92no—
349C2×Dic3—181no—
350C2×Dic3—181no—
351C2×Dic3—182no—
352C2×Dic3—184no—
353C2×Dic3C22×D912yes—
354C2×Dic3C22×D911yescharacteristic
355C2×Dic3C22×D911yescharacteristic
356C4×D3—34no—
357C4×D3—64no—
358C4×D3—94no—
359C4×D3—94no—
360C4×D3—184no—
361Dic6—24no—
362Dic6—184no—
363Dic6—184no—
364Dic6—184no—
365Dic6—184no—
366Dic6—184no—
367Dic6—184no—
368Dic6—364no—
369C2×C32C2×D3×D411yescharacteristic
370C2×C32C2×D3×Q811yescharacteristic
371C2×C32C2×D3×Q811yescharacteristic
372C3×D3—34no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9C22×Dic611yescharacteristic
377C2×C9C2×D3×D411yescharacteristic
378C2×C9C2×D3×Q811yescharacteristic
379D9#88814yesdirect product factor
380C2×Q8—32no—
381C2×Q8—272no—
382C2×Q8—272no—
383C2×Q8—272no—
384C2×Q8—548no—
385C22×C4—91no—
386C22×C4—92no—
387C22×C4—271no—
388C22×C4—271no—
389C22×C4—272no—
390C4⋊C4—32no—
391C4⋊C4—31no—
392C4⋊C4—92no—
393C4⋊C4—91no—
394C4⋊C4—272no—
395C4⋊C4—272no—
396C4⋊C4—272no—
397C4⋊C4—272no—
398C4⋊C4—271no—
399C4⋊C4D3×D911yescharacteristic
400C42—31no—
401C42—271no—
402C42—271no—
403C42—271no—
404C22×C3—21no—
405C22×C3—92no—
406C22×C3—92no—
407C22×C3—92no—
408C22×C3C23×D911yescharacteristic
409C22×C3C22×D3211yescharacteristic
410C2×D3—32no—
411C2×D3—32no—
412C2×D3—32no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—32no—
417C3×C4—32no—
418C3×C4—42no—
419C3×C4—62no—
420C3×C4—92no—
421C3×C4—182no—
422C3×C4Q8×D912yescomplemented
423C3×C4D3×Dic612yes—
424Dic3—22no—
425Dic3—32no—
426Dic3—62no—
427Dic3—92no—
428Dic3—92no—
429Dic3—92no—
430Dic3—92no—
431Dic3—182no—
432Dic3—182no—
433Dic3—182no—
434Dic3—182no—
435Dic3—362no—
436Dic3D4×D912yes—
437Dic3Q8×D912yescomplemented
438C32D3×C4⋊Q811yescharacteristic
439C9—21no—
440C9C2×#88811yescharacteristic, complemented
441C23—91no—
442Q8—64no—
443Q8—544no—
444Q8—544no—
445Q8—544no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—32no—
449C2×C4—91no—
450C2×C4—94no—
451C2×C4—92no—
452C2×C4—184no—
453C2×C4—274no—
454C2×C4—274no—
455C2×C4—271no—
456C2×C4—271no—
457C2×C4—272no—
458C2×C4—544no—
459C2×C4C2×D3×D912yes—
460C2×C4C2×D3×D911yescharacteristic
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—94no—
465C2×C3C2×D4×D911yescharacteristic
466C2×C3C2×Q8×D911yescharacteristic
467C2×C3C2×Q8×D911yescharacteristic
468C2×C3C2×D3×Dic611yescharacteristic
469C2×C3D32×D411yescharacteristic
470C2×C3D32×Q811yescharacteristic
471D3—34no—
472C22—92no—
473C22—92no—
474C22—92no—
475C22C22×D3×D911yescharacteristic
476C4—22no—
477C4—32no—
478C4—32no—
479C4—62no—
480C4—92no—
481C4—182no—
482C4—272no—
483C4—272no—
484C4—542no—
485C4D9×Dic612yescomplemented
486C3—21no—
487C3D9×C4⋊Q811yescharacteristic, complemented
488C3D3×#88811yescharacteristic
489C2—94no—
490C2C2×D9×Dic611yescharacteristic
491C2D3×D4×D911yescharacteristic
492C2D3×Q8×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3×D9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C2×C4⋊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
2. #11721744C2semidirect
4. C9×#888C2semidirect
8. #11720971C2semidirect
9. #11720970C2semidirect
10. #11720905C2semidirect
11. #11720903C2semidirect
74. C3×C4×D9Q8semidirect
78. Dic3×D9Q8semidirect
141. C3×C4×C9C2×Q8semidirect
144. C3⋊Dic9C2×Q8semidirect
147. C9×Dic3C2×Q8semidirect
149. C3×Dic9C2×Q8semidirect
170. #888D9direct (N × H)
214. C4×D9Dic6semidirect
223. C3×D9C4⋊Q8semidirect
302. C4×C9C2×Dic6semidirect
310. Dic9C2×Dic6semidirect
322. C3×C9C2×C4⋊Q8semidirect
379. D9#888direct (N × H)
422. C3×C4Q8×D9semidirect
437. Dic3Q8×D9semidirect
440. C9C2×#888semidirect
485. C4D9×Dic6semidirect
487. C3D9×C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×D9×Dic62
#117217442
C3×C4⋊C4×D92
C9×#8882
D9×C4⋊Dic32
D9×Dic3⋊C42
C4×Dic3×D92
#117209712
#117209702
#117209052
#117209032
D3×#8883
D9×C4⋊Q83