Finite Group Databasebeta

← C3×D9×#99 #13141650

Subgroups — C3×D9×#99 #13141650

Order 1728.

682 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×D9C211yesmaximal, characteristic
2C3×C22⋊C4×D9C212yesmaximal, complemented
3C3×C9×#99C211yesmaximal, characteristic, complemented
4C3×#63393C211yesmaximal, characteristic, complemented
5C3×#63365C212yesmaximal, complemented
6C3×D3×#99—31nomaximal
7D9×#99C311yesmaximal, characteristic, direct product factor
8C23×C3×D9C411yescharacteristic, complemented
9C23×C3×D9C2211yescharacteristic
10C2×C3×D4×C9C2211yescharacteristic
11C2×C3×#611C411yescharacteristic, complemented
12C2×C3×#611C2211yescharacteristic
13C3×D4×D9—22no—
14C3×D4×D9—22no—
15C2×C3×D36C411yescharacteristic, complemented
16C2×C3×C4×D9—22no—
17C2×C3×C4×D9C411yescharacteristic, complemented
18C3×C9×C22⋊C4C2212yescomplemented
19C3×#3676C2212yescomplemented
20C3×(C2×D9)⋊C4—22no—
21C2×C3×D3×D4—31no—
22C3×D3×C22⋊C4—32no—
23C2×D4×D9C2×C311yescharacteristic
24C32×#99D311yescharacteristic
25C3×#834—31no—
26C3×#806—32no—
27C22⋊C4×D9C2×C312yescomplemented
28C9×#99—21no—
29C9×#99C2×C311yescharacteristic, complemented
30#63393C2×C311yescharacteristic, complemented
31#63365C2×C312yescomplemented
32C23×C3×C9C2×C411yescharacteristic, complemented
33C23×C3×C9C2311yescharacteristic
34C22×C3×D9—21no—
35C22×C3×D9—21no—
36C22×C3×D9—42no—
37C22×C3×D9—42no—
38C22×C3×D9C2×C411yescharacteristic
39C22×C3×D9C2×C411yescharacteristic
40C22×C3×D9C2×C411yescharacteristic, complemented
41C22×C3×D9D412yes—
42C22×C3×D9D412yes—
43C3×D4×C9—22no—
44C2×C3×C4×C9—22no—
45C2×C3×C4×C9C2×C411yescharacteristic, complemented
46C3×#611—22no—
47C3×#611—22no—
48C2×C3×Dic9—22no—
49C2×C3×Dic9C2×C411yescharacteristic, complemented
50C3×D36—22no—
51C3×C4×D9—21no—
52C3×C4×D9—21no—
53C3×C4×D9—44no—
54C2×C3×#99—91no—
55D3×#99—31no—
56C23×C3×D3—31no—
57C23×C3×D3—31no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C3×#67—31no—
60C2×C3×#67—31no—
61C3×D3×D4—62no—
62C3×D3×D4—62no—
63C2×C3×D12—31no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—62no—
66C23×D9C3×C411yescharacteristic, complemented
67C23×D9C22×C311yescharacteristic
68C32×C22⋊C4C2×D312yes—
69C3×#217—32no—
70C3×(C2×D3)⋊C4—62no—
71C2×D4×C9—21no—
72C2×D4×C9C22×C311yescharacteristic
73C2×#611C3×C411yescharacteristic, complemented
74C2×#611C22×C311yescharacteristic
75D4×D9—22no—
76D4×D9—22no—
77C2×D36C3×C411yescharacteristic, complemented
78C2×C4×D9—22no—
79C2×C4×D9C3×C411yescharacteristic, complemented
80C9×C22⋊C4—22no—
81C9×C22⋊C4C22×C312yescomplemented
82#3676C22×C312yescomplemented
83(C2×D9)⋊C4—22no—
84C22×C3×C9—21no—
85C22×C3×C9—41no—
86C22×C3×C9—41no—
87C22×C3×C9C22×C411yescharacteristic
88C22×C3×C9C2×D412yes—
89C2×C3×D9—24no—
90C2×C3×D9—22no—
91C2×C3×D9—21no—
92C2×C3×D9—41no—
93C2×C3×D9—42no—
94C2×C3×D9—42no—
95C2×C3×D9—42no—
96C2×C3×D9—41no—
97C2×C3×D9C22⋊C411yescharacteristic
98C2×C3×D9C22⋊C412yes—
99C2×C3×D9C22⋊C411yescharacteristic
100C3×C4×C9—21no—
101C3×C4×C9—42no—
102C3×Dic9—21no—
103C3×Dic9—42no—
104C22×C3×D4—91no—
105C2×D3×D4—31no—
106C2×C3×C22⋊C4—92no—
107D3×C22⋊C4—32no—
108C3×#99—21no—
109C3×#99—91no—
110C3×#99—92no—
111C3×#99D911yescharacteristic, direct product factor
112C3×#99C3×D311yescharacteristic
113#834—31no—
114#806—32no—
115C23×C32C4×D311yescharacteristic
116C23×C32C22×D311yescharacteristic
117C22×C3×D3—31no—
118C22×C3×D3—32no—
119C22×C3×D3—31no—
120C22×C3×D3—32no—
121C22×C3×D3—31no—
122C22×C3×D3—61no—
123C22×C3×D3—61no—
124C22×C3×D3—122no—
125C22×C3×D3—122no—
126C32×D4—22no—
127C2×C32×C4—22no—
128C2×C32×C4C4×D311yescharacteristic
129C3×#67—62no—
130C3×#67—62no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—62no—
133C3×D12—62no—
134C3×C4×D3—61no—
135C3×C4×D3—61no—
136C3×C4×D3—124no—
137C23×C9—21no—
138C23×C9—21no—
139C23×C9C2×C3×C411yescharacteristic, complemented
140C23×C9C23×C311yescharacteristic
141C22×D9—21no—
142C22×D9—21no—
143C22×D9—42no—
144C22×D9—42no—
145C22×D9C2×C3×C411yescharacteristic
146C22×D9C2×C3×C411yescharacteristic
147C22×D9C2×C3×C411yescharacteristic, complemented
148C22×D9C3×D412yes—
149C22×D9C3×D412yes—
150D4×C9—22no—
151D4×C9—42no—
152C2×C4×C9—22no—
153C2×C4×C9—21no—
154C2×C4×C9—42no—
155C2×C4×C9C2×C3×C411yescharacteristic, complemented
156#611—22no—
157#611—22no—
158C2×Dic9—22no—
159C2×Dic9C2×C3×C411yescharacteristic, complemented
160D36—22no—
161C4×D9—21no—
162C4×D9—21no—
163C4×D9—44no—
164C2×#99—91noSylow
165C2×C3×C9—22no—
166C2×C3×C9—21no—
167C2×C3×C9—41no—
168C2×C3×C9C2×C22⋊C411yescharacteristic
169C3×D9—21no—
170C3×D9—22no—
171C3×D9—41no—
172C3×D9#9912yesdirect product factor
173C24×C3—91no—
174C24×C3—91no—
175C23×D3—31no—
176C23×D3—31no—
177C2×C3×D4—21no—
178C2×C3×D4—91no—
179C2×C3×D4—91no—
180C2×C3×D4—91no—
181C2×C3×D4—182no—
182C2×C3×D4—182no—
183C2×C3×D4C2×D911yescharacteristic
184C2×C3×D4C2×C3×D311yescharacteristic
185C22×C3×C4—91no—
186C22×C3×C4—182no—
187C2×#67—31no—
188C2×#67—31no—
189D3×D4—62no—
190D3×D4—62no—
191C2×D12—31no—
192C2×C4×D3—31no—
193C2×C4×D3—62no—
194C3×C22⋊C4—22no—
195C3×C22⋊C4—92no—
196C3×C22⋊C4—182no—
197C3×C22⋊C4C2×D912yescomplemented
198C3×C22⋊C4C2×C3×D312yes—
199#217—32no—
200(C2×D3)⋊C4—62no—
201C22×C32—21no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32C2×C4×D311yescharacteristic
205C22×C32D3×D412yes—
206C2×C3×D3—31no—
207C2×C3×D3—31no—
208C2×C3×D3—32no—
209C2×C3×D3—64no—
210C2×C3×D3—62no—
211C2×C3×D3—61no—
212C2×C3×D3—122no—
213C2×C3×D3—121no—
214C2×C3×D3—122no—
215C2×C3×D3—122no—
216C2×C3×D3—121no—
217C32×C4—21no—
218C32×C4—42no—
219C3×Dic3—61no—
220C3×Dic3—122no—
221C22×C9—21no—
222C22×C9—21no—
223C22×C9—22no—
224C22×C9—41no—
225C22×C9—41no—
226C22×C9—41no—
227C22×C9—81no—
228C22×C9—81no—
229C22×C9C22×C3×C411yescharacteristic
230C22×C9C2×C3×D412yes—
231C2×D9—24no—
232C2×D9—22no—
233C2×D9—21no—
234C2×D9—41no—
235C2×D9—42no—
236C2×D9—42no—
237C2×D9—42no—
238C2×D9—41no—
239C2×D9C3×C22⋊C411yescharacteristic
240C2×D9C3×C22⋊C412yes—
241C2×D9C3×C22⋊C411yescharacteristic
242C4×C9—21no—
243C4×C9—41no—
244C4×C9—42no—
245C4×C9—82no—
246Dic9—21no—
247Dic9—42no—
248C22×D4—91no—
249C2×C22⋊C4—92no—
250#99—91no—
251#99—92no—
252#99C3×D911yescharacteristic, direct product factor
253C3×C9C2×#9911yescharacteristic, complemented, Sylow
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—91no—
257C23×C3—92no—
258C23×C3—91no—
259C23×C3—92no—
260C23×C3—91no—
261C23×C3—181no—
262C23×C3—181no—
263C23×C3—362no—
264C23×C3—362no—
265C23×C3C4×D911yescharacteristic, complemented
266C23×C3C22×D911yescharacteristic
267C23×C3C3×C4×D311yescharacteristic
268C23×C3C22×C3×D311yescharacteristic
269C22×D3—31no—
270C22×D3—32no—
271C22×D3—31no—
272C22×D3—32no—
273C22×D3—31no—
274C22×D3—61no—
275C22×D3—61no—
276C22×D3—122no—
277C22×D3—122no—
278C3×D4—22no—
279C3×D4—22no—
280C3×D4—42no—
281C3×D4—182no—
282C3×D4—182no—
283C3×D4—182no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—21no—
287C2×C3×C4—42no—
288C2×C3×C4—91no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4—182no—
292C2×C3×C4—364no—
293C2×C3×C4C4×D911yescharacteristic, complemented
294C2×C3×C4C3×C4×D311yescharacteristic
295#67—62no—
296#67—62no—
297C2×Dic3—31no—
298C2×Dic3—62no—
299D12—62no—
300C4×D3—61no—
301C4×D3—61no—
302C4×D3—124no—
303C2×C32—22no—
304C2×C32—21no—
305C2×C32—41no—
306C2×C32D3×C22⋊C411yescharacteristic
307C3×D3—32no—
308C3×D3—62no—
309C3×D3—61no—
310C3×D3—121no—
311C2×C9—22no—
312C2×C9—21no—
313C2×C9—21no—
314C2×C9—42no—
315C2×C9—41no—
316C2×C9—41no—
317C2×C9—81no—
318C2×C9C2×C3×C22⋊C411yescharacteristic
319D9—21no—
320D9—22no—
321D9—41no—
322D9C3×#9912yesdirect product factor
323C24—91no—
324C24—91no—
325C2×D4—91no—
326C2×D4—91no—
327C2×D4—91no—
328C2×D4—182no—
329C2×D4—182no—
330C2×D4C2×C3×D911yescharacteristic
331C22×C4—91no—
332C22×C4—182no—
333C22⋊C4—92no—
334C22⋊C4—182no—
335C22⋊C4C2×C3×D912yescomplemented
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—91no—
348C22×C3—91no—
349C22×C3—92no—
350C22×C3—184no—
351C22×C3—182no—
352C22×C3—181no—
353C22×C3—362no—
354C22×C3—361no—
355C22×C3—361no—
356C22×C3—362no—
357C22×C3—362no—
358C22×C3C2×C4×D911yescharacteristic
359C22×C3D4×D912yes—
360C22×C3C2×C3×C4×D311yescharacteristic
361C22×C3C3×D3×D412yes—
362C2×D3—31no—
363C2×D3—31no—
364C2×D3—32no—
365C2×D3—64no—
366C2×D3—62no—
367C2×D3—61no—
368C2×D3—122no—
369C2×D3—122no—
370C2×D3—121no—
371C2×D3—122no—
372C2×D3—121no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—41no—
376C3×C4—42no—
377C3×C4—42no—
378C3×C4—82no—
379C3×C4—181no—
380C3×C4—362no—
381Dic3—61no—
382Dic3—122no—
383C32D3×#9911yescharacteristic
384C9—21no—
385C9C2×C3×#9911yescharacteristic, complemented
386C23—91no—
387C23—92no—
388C23—91no—
389C23—92no—
390C23—91no—
391C23—181no—
392C23—181no—
393C23—362no—
394C23—362no—
395C23C3×C4×D911yescharacteristic, complemented
396C23C22×C3×D911yescharacteristic
397D4—22no—
398D4—182no—
399D4—182no—
400D4—182no—
401C2×C4—22no—
402C2×C4—91no—
403C2×C4—181no—
404C2×C4—181no—
405C2×C4—182no—
406C2×C4—364no—
407C2×C4C3×C4×D911yescharacteristic, complemented
408C2×C3—22no—
409C2×C3—21no—
410C2×C3—22no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—42no—
414C2×C3—41no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—81no—
418C2×C3—92no—
419C2×C3—181no—
420C2×C3—182no—
421C2×C3—361no—
422C2×C3C22⋊C4×D911yescharacteristic
423C2×C3C3×D3×C22⋊C411yescharacteristic
424D3—32no—
425D3—61no—
426D3—62no—
427D3—121no—
428C22—21no—
429C22—41no—
430C22—41no—
431C22—91no—
432C22—91no—
433C22—92no—
434C22—184no—
435C22—182no—
436C22—181no—
437C22—362no—
438C22—361no—
439C22—361no—
440C22—362no—
441C22—362no—
442C22C2×C3×C4×D911yescharacteristic
443C22C3×D4×D912yes—
444C4—21no—
445C4—42no—
446C4—181no—
447C4—362no—
448C3—21no—
449C3D9×#9911yescharacteristic, direct product factor
450C3C3×D3×#9911yescharacteristic
451C2—22no—
452C2—21no—
453C2—41no—
454C2—92no—
455C2—181no—
456C2—182no—
457C2—361no—
458C2C3×C22⋊C4×D911yescharacteristic
subgroupG/H
centerC2×C3C22⋊C4×D9
commutatorC22×C9C22×C3×C4
FrattiniC23×C3C22×C3×D3
FittingC3×C9×#99C2
socleC2×C32D3×C22⋊C4
prime radicalC2×C3×D4×D9C2
cosocleC23×C9C23×C3
hypercenterC3×#99D9
hypocenterC9C2×C3×#99
LCSG ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
2. C3×C22⋊C4×D9C2semidirect
3. C3×C9×#99C2semidirect
4. C3×#63393C2semidirect
5. C3×#63365C2semidirect
7. D9×#99C3direct (N × H)
8. C23×C3×D9C4semidirect
11. C2×C3×#611C4semidirect
15. C2×C3×D36C4semidirect
17. C2×C3×C4×D9C4semidirect
18. C3×C9×C22⋊C4C22semidirect
19. C3×#3676C22semidirect
27. C22⋊C4×D9C2×C3semidirect
29. C9×#99C2×C3semidirect
30. #63393C2×C3semidirect
31. #63365C2×C3semidirect
32. C23×C3×C9C2×C4semidirect
40. C22×C3×D9C2×C4semidirect
45. C2×C3×C4×C9C2×C4semidirect
49. C2×C3×Dic9C2×C4semidirect
66. C23×D9C3×C4semidirect
73. C2×#611C3×C4semidirect
77. C2×D36C3×C4semidirect
79. C2×C4×D9C3×C4semidirect
81. C9×C22⋊C4C22×C3semidirect
82. #3676C22×C3semidirect
111. C3×#99D9direct (N × H)
139. C23×C9C2×C3×C4semidirect
147. C22×D9C2×C3×C4semidirect
155. C2×C4×C9C2×C3×C4semidirect
159. C2×Dic9C2×C3×C4semidirect
172. C3×D9#99direct (N × H)
197. C3×C22⋊C4C2×D9semidirect
252. #99C3×D9direct (N × H)
253. C3×C9C2×#99semidirect
265. C23×C3C4×D9semidirect
293. C2×C3×C4C4×D9semidirect
322. D9C3×#99direct (N × H)
335. C22⋊C4C2×C3×D9semidirect
385. C9C2×C3×#99semidirect
395. C23C3×C4×D9semidirect
407. C2×C4C3×C4×D9semidirect
449. C3D9×#99direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×D92
C3×C22⋊C4×D92
C3×C9×#992
C3×#633932
C3×#633652
C3×D3×#993
D9×#993