Finite Group Databasebeta

← C3×#10589777 #13153089

Subgroups — C3×#10589777 #13153089

Order 1728.

886 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×D9C211yesmaximal, characteristic, complemented
2C2×C3×#3699C211yesmaximal, characteristic
3C3×C9×#124C211yesmaximal, characteristic, complemented
4C3×#63506C211yesmaximal, characteristic, complemented
5C3×#63498C211yesmaximal, characteristic, complemented
6C3×#63445C214yesmaximal, complemented
7C3×#63440C214yesmaximal, complemented
8C3×#63434C211yesmaximal, characteristic, complemented
9C3×#63431C211yesmaximal, characteristic, complemented
10C3×#5675—31nomaximal
11#10589777C311yesmaximal, characteristic, direct product factor
12C2×C3×Q8×C9C2211yescharacteristic
13C2×C3×D4×C9C2211yescharacteristic
14C2×C3×#611C2212yes—
15C22×C3×Dic9C2212yes—
16C3×Q8×D9—24no—
17C3×#3699—24no—
18C2×C3×C4×D9C2212yes—
19C2×C3×C4×D9C2211yescharacteristic
20C2×C3×Dic18C2211yescharacteristic
21C2×C3×Dic18C2211yescharacteristic
22C2×C3×Dic18C2212yes—
23C3×C9×C22⋊C4C2214yes—
24C3×C42×C9C2211yescharacteristic
25C3×#3676C2214yes—
26C3×(C2×D9)⋊C4C2212yes—
27C3×C4⋊Dic9C2214yes—
28C3×Dic9⋊C4C2212yes—
29C3×Dic9⋊C4C2214yes—
30C3×C4×Dic9C2211yescharacteristic
31C2×C3×D3×Q8—31no—
32C2×C3×#237—31no—
33C32×#124D311yescharacteristic
34C3×#944—31no—
35C3×#936—31no—
36C3×#883—34no—
37C3×#878—34no—
38C3×#872—31no—
39C3×#869—31no—
40C2×Q8×D9C2×C311yescharacteristic, complemented
41C2×#3699C2×C311yescharacteristic
42C9×#124—21no—
43C9×#124C2×C311yescharacteristic, complemented
44#63506C2×C311yescharacteristic, complemented
45#63498C2×C311yescharacteristic, complemented
46#63445C2×C314yescomplemented
47#63440C2×C314yescomplemented
48#63434C2×C311yescharacteristic, complemented
49#63431C2×C311yescharacteristic, complemented
50C23×C3×C9C2312yes—
51C22×C3×D9C2311yescharacteristic
52C3×Q8×C9—22no—
53C3×D4×C9—22no—
54C2×C3×C4×C9C2312yes—
55C2×C3×C4×C9C2312yes—
56C2×C3×C4×C9C2311yescharacteristic
57C3×#611—24no—
58C2×C3×Dic9—24no—
59C2×C3×Dic9C2312yes—
60C2×C3×Dic9C2312yes—
61C2×C3×Dic9C2312yes—
62C2×C3×Dic9C2311yescharacteristic
63C3×C4×D9—24no—
64C3×C4×D9D414yes—
65C3×Dic18—22no—
66C3×Dic18—22no—
67C3×Dic18—24no—
68C3×#536—91no—
69#5675—31no—
70C2×C32×Q8C2×D311yescharacteristic
71C2×C32×D4C2×D311yescharacteristic
72C2×C3×#67—32no—
73C22×C3×Dic3—32no—
74C3×D3×Q8—64no—
75C3×#237—64no—
76C2×C3×C4×D3—31no—
77C2×C3×C4×D3—32no—
78C2×C3×Dic6—32no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C32×C22⋊C4C2×D314yes—
82C32×C42C2×D311yescharacteristic
83C3×#217—34no—
84C3×(C2×D3)⋊C4—32no—
85C3×C4⋊Dic3—34no—
86C3×Dic3⋊C4—32no—
87C3×Dic3⋊C4—34no—
88C3×C4×Dic3—31no—
89C2×Q8×C9—21no—
90C2×Q8×C9C22×C311yescharacteristic
91C2×D4×C9—21no—
92C2×D4×C9C22×C311yescharacteristic
93C2×#611C22×C312yes—
94C22×Dic9C22×C312yes—
95Q8×D9—24no—
96#3699—24no—
97C2×C4×D9C22×C312yes—
98C2×C4×D9C22×C311yescharacteristic
99C2×Dic18C22×C311yescharacteristic
100C2×Dic18C22×C311yescharacteristic
101C2×Dic18C22×C312yes—
102C9×C22⋊C4—24no—
103C9×C22⋊C4C22×C314yes—
104C42×C9—21no—
105C42×C9C22×C311yescharacteristic
106#3676C22×C314yes—
107(C2×D9)⋊C4C22×C312yes—
108C4⋊Dic9C22×C314yes—
109Dic9⋊C4C22×C314yes—
110Dic9⋊C4C22×C312yes—
111C4×Dic9C22×C311yescharacteristic
112C22×C3×C9—22no—
113C22×C3×C9—24no—
114C22×C3×C9C2411yescharacteristic
115C2×C3×D9—22no—
116C2×C3×D9C2×D412yes—
117C3×C4×C9—22no—
118C3×C4×C9—22no—
119C3×C4×C9C2×D412yes—
120C3×Dic9—22no—
121C3×Dic9—22no—
122C3×Dic9—22no—
123C3×Dic9C2×D412yes—
124C2×C3×#41—91no—
125C22×C3×Q8—91no—
126C2×D3×Q8—31no—
127C2×#237—31no—
128C3×C4⋊Q8—91no—
129C3×C4⋊Q8—91no—
130C3×#124—21no—
131C3×#124—91no—
132C3×#124D911yescharacteristic, complemented
133C3×#124C3×D311yescharacteristic
134C3×#123—94no—
135C3×C22⋊Q8—94no—
136C3×#117—91no—
137#944—31no—
138#936—31no—
139#883—34no—
140#878—34no—
141#872—31no—
142#869—31no—
143C23×C32C22×D312yes—
144C22×C3×D3—31no—
145C32×Q8—22no—
146C32×D4—22no—
147C2×C32×C4C22×D312yes—
148C2×C32×C4C22×D312yes—
149C2×C32×C4C22×D311yescharacteristic
150C3×#67—64no—
151C2×C3×Dic3—31no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—64no—
156C3×C4×D3—34no—
157C3×C4×D3—64no—
158C3×Dic6—64no—
159C3×Dic6—62no—
160C3×Dic6—62no—
161C23×C9—22no—
162C23×C9C23×C312yes—
163C22×D9C23×C311yescharacteristic
164Q8×C9—22no—
165Q8×C9—42no—
166D4×C9—22no—
167D4×C9—42no—
168C2×C4×C9—21no—
169C2×C4×C9—22no—
170C2×C4×C9—22no—
171C2×C4×C9C23×C312yes—
172C2×C4×C9C23×C312yes—
173C2×C4×C9C23×C311yescharacteristic
174#611—24no—
175C2×Dic9—24no—
176C2×Dic9C23×C312yes—
177C2×Dic9C23×C312yes—
178C2×Dic9C23×C312yes—
179C2×Dic9C23×C311yescharacteristic
180C4×D9—24no—
181C4×D9C3×D414yes—
182Dic18—22no—
183Dic18—22no—
184Dic18—24no—
185#536—91noSylow
186C2×C3×C9—42no—
187C2×C3×C9C22×D411yescharacteristic
188C2×C3×C92−1+412yes—
189C3×D9—22no—
190C3×#41—184no—
191C2×C3×Q8—21no—
192C2×C3×Q8—92no—
193C2×C3×Q8—91no—
194C2×C3×Q8—91no—
195C2×C3×Q8—184no—
196C2×C3×Q8C2×D911yescharacteristic
197C2×C3×Q8C2×C3×D311yescharacteristic
198C2×C3×D4—21no—
199C2×C3×D4—92no—
200C2×C3×D4C2×D911yescharacteristic
201C2×C3×D4C2×C3×D311yescharacteristic
202C22×C3×C4—91no—
203C22×C3×C4—92no—
204C22×C3×C4—92no—
205C2×#67—32no—
206C22×Dic3—32no—
207D3×Q8—64no—
208#237—64no—
209C2×C4×D3—31no—
210C2×C4×D3—32no—
211C2×Dic6—31no—
212C2×Dic6—31no—
213C2×Dic6—32no—
214C3×C4⋊C4—92no—
215C3×C4⋊C4—94no—
216C3×C4⋊C4—94no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—94no—
219C3×C22⋊C4—92no—
220C3×C22⋊C4C2×D914yes—
221C3×C22⋊C4C2×C3×D314yes—
222C3×C42—21no—
223C3×C42—91no—
224C3×C42C2×D911yescharacteristic
225C3×C42C2×C3×D311yescharacteristic
226#217—34no—
227(C2×D3)⋊C4—32no—
228C4⋊Dic3—34no—
229Dic3⋊C4—32no—
230Dic3⋊C4—34no—
231C4×Dic3—31no—
232C22×C32—22no—
233C22×C32—24no—
234C22×C32C23×D311yescharacteristic
235C2×C3×D3—32no—
236C2×C3×D3—62no—
237C32×C4—22no—
238C32×C4—22no—
239C32×C4D3×D412yes—
240C3×Dic3—32no—
241C3×Dic3—62no—
242C3×Dic3—62no—
243C3×Dic3—62no—
244C22×C9—22no—
245C22×C9—24no—
246C22×C9—21no—
247C22×C9—42no—
248C22×C9—44no—
249C22×C9C24×C311yescharacteristic
250C2×D9—22no—
251C2×D9C2×C3×D412yes—
252C4×C9—22no—
253C4×C9—22no—
254C4×C9—22no—
255C4×C9—42no—
256C4×C9—42no—
257C4×C9C2×C3×D412yes—
258Dic9—22no—
259Dic9—22no—
260Dic9—22no—
261Dic9C2×C3×D412yes—
262C2×#41—91no—
263C22×Q8—91no—
264C4⋊Q8—91no—
265C4⋊Q8—91no—
266#124—91no—
267#124C3×D911yescharacteristic, complemented
268#123—94no—
269C22⋊Q8—94no—
270#117—91no—
271C3×C9#53611yescharacteristic, complemented, Sylow
272C23×C3—22no—
273C23×C3—91no—
274C23×C3C22×D912yes—
275C23×C3C22×C3×D312yes—
276C22×D3—31no—
277C3×Q8—22no—
278C3×Q8—22no—
279C3×Q8—42no—
280C3×Q8—184no—
281C3×Q8—182no—
282C3×Q8—182no—
283C3×D4—22no—
284C3×D4—22no—
285C3×D4—42no—
286C3×D4—184no—
287C2×C3×C4—21no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—91no—
291C2×C3×C4—94no—
292C2×C3×C4—92no—
293C2×C3×C4—92no—
294C2×C3×C4—92no—
295C2×C3×C4—184no—
296C2×C3×C4—184no—
297C2×C3×C4C22×D912yes—
298C2×C3×C4C22×D912yes—
299C2×C3×C4C22×D911yescharacteristic
300C2×C3×C4C22×C3×D312yes—
301C2×C3×C4C22×C3×D312yes—
302C2×C3×C4C22×C3×D311yescharacteristic
303#67—64no—
304C2×Dic3—31no—
305C2×Dic3—32no—
306C2×Dic3—32no—
307C2×Dic3—32no—
308C2×Dic3—64no—
309C4×D3—34no—
310C4×D3—64no—
311Dic6—64no—
312Dic6—62no—
313Dic6—62no—
314C2×C32—42no—
315C2×C32C2×D3×D411yescharacteristic
316C2×C32#101012yes—
317C3×D3—62no—
318C2×C9—21no—
319C2×C9—22no—
320C2×C9—42no—
321C2×C9—82no—
322C2×C9C22×C3×D411yescharacteristic
323C2×C9C3×2−1+412yes—
324D9—22no—
325#41—184no—
326C2×Q8—92no—
327C2×Q8—91no—
328C2×Q8—91no—
329C2×Q8—184no—
330C2×Q8C2×C3×D911yescharacteristic
331C2×D4—92no—
332C2×D4C2×C3×D911yescharacteristic
333C22×C4—91no—
334C22×C4—92no—
335C22×C4—92no—
336C4⋊C4—92no—
337C4⋊C4—94no—
338C4⋊C4—94no—
339C22⋊C4—94no—
340C22⋊C4—92no—
341C22⋊C4C2×C3×D914yes—
342C42—91no—
343C42C2×C3×D911yescharacteristic
344C22×C3—22no—
345C22×C3—24no—
346C22×C3—22no—
347C22×C3—24no—
348C22×C3—21no—
349C22×C3—42no—
350C22×C3—44no—
351C22×C3—92no—
352C22×C3—182no—
353C22×C3C23×D911yescharacteristic
354C22×C3C23×C3×D311yescharacteristic
355C2×D3—32no—
356C2×D3—62no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—22no—
362C3×C4—42no—
363C3×C4—42no—
364C3×C4—92no—
365C3×C4—182no—
366C3×C4—182no—
367C3×C4—182no—
368C3×C4D4×D912yes—
369C3×C4C3×D3×D412yes—
370Dic3—32no—
371Dic3—62no—
372Dic3—62no—
373Dic3—62no—
374C32#567511yescharacteristic
375C9—21no—
376C9C3×#53611yescharacteristic, complemented
377C23—91no—
378C23C22×C3×D912yes—
379Q8—22no—
380Q8—184no—
381Q8—182no—
382Q8—182no—
383D4—22no—
384D4—184no—
385C2×C4—91no—
386C2×C4—94no—
387C2×C4—92no—
388C2×C4—92no—
389C2×C4—92no—
390C2×C4—184no—
391C2×C4—184no—
392C2×C4C22×C3×D912yes—
393C2×C4C22×C3×D912yes—
394C2×C4C22×C3×D911yescharacteristic
395C2×C3—21no—
396C2×C3—22no—
397C2×C3—42no—
398C2×C3—42no—
399C2×C3—82no—
400C2×C3—182no—
401C2×C3C2×D4×D911yescharacteristic
402C2×C3#6371612yes—
403C2×C3C2×C3×D3×D411yescharacteristic
404C2×C3C3×#101012yes—
405D3—62no—
406C22—22no—
407C22—24no—
408C22—92no—
409C22—182no—
410C22C23×C3×D911yescharacteristic
411C4—22no—
412C4—22no—
413C4—92no—
414C4—182no—
415C4—182no—
416C4—182no—
417C4C3×D4×D912yes—
418C3—21no—
419C3#1058977711yescharacteristic, direct product factor
420C3C3×#567511yescharacteristic
421C2—42no—
422C2—182no—
423C2C2×C3×D4×D911yescharacteristic
424C2C3×#6371612yes—
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#124C2
socleC22×C32C23×D3
prime radicalC2×C3×#3699C2
hypercenterC3×#124D9
hypocenterC9C3×#536
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×D9C2semidirect
3. C3×C9×#124C2semidirect
4. C3×#63506C2semidirect
5. C3×#63498C2semidirect
6. C3×#63445C2semidirect
7. C3×#63440C2semidirect
8. C3×#63434C2semidirect
9. C3×#63431C2semidirect
11. #10589777C3direct (N × H)
40. C2×Q8×D9C2×C3semidirect
43. C9×#124C2×C3semidirect
44. #63506C2×C3semidirect
45. #63498C2×C3semidirect
46. #63445C2×C3semidirect
47. #63440C2×C3semidirect
48. #63434C2×C3semidirect
49. #63431C2×C3semidirect
132. C3×#124D9semidirect
267. #124C3×D9semidirect
271. C3×C9#536semidirect
376. C9C3×#536semidirect
419. C3#10589777direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D92
C2×C3×#36992
C3×C9×#1242
C3×#635062
C3×#634982
C3×#634452
C3×#634402
C3×#634342
C3×#634312
C3×#56753
#105897773