Finite Group Databasebeta

← C3×#10588727 #13142856

Subgroups — C3×#10588727 #13142856

Order 1728.

416 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#3674C211yesmaximal, characteristic, complemented
3C3×#63486C211yesmaximal, characteristic
4C22×C3×C9⋊C8C211yesmaximal, characteristic
5C3×#63395C211yesmaximal, characteristic, complemented
6C3×#63368C211yesmaximal, characteristic, complemented
7C3×#63366C211yesmaximal, characteristic, complemented
8C3×#5051—31nomaximal
9#10588727C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C3×C9×C4⋊C4—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×C9×C22⋊C4—21no—
16C3×#3676—21no—
17C3×#3674—22no—
18C3×C4⋊Dic9C2211yescharacteristic
19C3×Dic9⋊C4—21no—
20C2×C3×C9⋊C8—22no—
21C2×C3×C9⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23C32×C22⋊Q8D311yescharacteristic
24C2×C3×#216—31no—
25C3×#924—31no—
26C22×C3×C3⋊C8—31no—
27C3×#835—31no—
28C3×#810—31no—
29C3×#807—31no—
30C9×C22⋊Q8—21no—
31C9×C22⋊Q8C2×C311yescharacteristic
32C2×#3674C2×C311yescharacteristic, complemented
33#63486C2×C311yescharacteristic
34C22×C9⋊C8C2×C311yescharacteristic
35#63395C2×C311yescharacteristic, complemented
36#63368C2×C311yescharacteristic, complemented
37#63366C2×C311yescharacteristic, complemented
38C23×C3×C9D411yescharacteristic
39C3×Q8×C9—21no—
40C3×Q8×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9D411yescharacteristic
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3×Dic9—21no—
48C2×C3×Dic9—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51C3×C9⋊C8—21no—
52C3×C9⋊C8D412yes—
53C3×#467—91no—
54#5051—31no—
55C2×C32×Q8C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C2×C3×Dic6—31no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4C2×D311yescharacteristic
60C32×C22⋊C4—21no—
61C3×#217—61no—
62C3×#216—62no—
63C3×C4⋊Dic3—31no—
64C3×Dic3⋊C4—61no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—62no—
68C2×Q8×C9—21no—
69C2×Q8×C9C22×C311yescharacteristic
70C22×C4×C9—21no—
71C22×C4×C9C22×C311yescharacteristic
72C2×Dic18C22×C311yescharacteristic
73C9×C4⋊C4—22no—
74C9×C4⋊C4—21no—
75C9×C4⋊C4—21no—
76C9×C4⋊C4C22×C311yescharacteristic
77C9×C22⋊C4—22no—
78C9×C22⋊C4—21no—
79#3676—21no—
80#3674—22no—
81C4⋊Dic9C22×C311yescharacteristic
82Dic9⋊C4—21no—
83C2×C9⋊C8—22no—
84C2×C9⋊C8C22×C311yescharacteristic
85C2×C9⋊C8C22×C311yescharacteristic
86C22×C3×C9—21no—
87C22×C3×C9—21no—
88C22×C3×C9Q1612yescomplemented
89C22×C3×C9C2×D411yescharacteristic
90C3×C4×C9—21no—
91C3×C4×C9—41no—
92C3×C4×C9—41no—
93C3×C4×C9C2×D411yescharacteristic
94C3×C4×C9#4111yescharacteristic
95C3×Dic9—41no—
96C3×Dic9—41no—
97C2×C3×Q16—91no—
98C22×C3×C8—91no—
99C3×C22⋊Q8—21no—
100C3×C22⋊Q8—91no—
101C3×C22⋊Q8D911yescharacteristic
102C3×C22⋊Q8C3×D311yescharacteristic
103C2×#216—31no—
104#924—31no—
105C22×C3⋊C8—31no—
106C3×#107—91no—
107C3×#103—91no—
108C3×#103—91no—
109#835—31no—
110#810—31no—
111#807—31no—
112C23×C32#6711yescharacteristic
113C32×Q8—21no—
114C32×Q8—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4#6711yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C3×Dic6—61no—
124C3×Dic6—61no—
125C23×C9—21no—
126C23×C9C3×D411yescharacteristic
127C3×C3⋊C8—32no—
128C3×C3⋊C8—61no—
129Q8×C9—21no—
130Q8×C9—21no—
131Q8×C9—41no—
132Q8×C9—41no—
133C2×C4×C9—21no—
134C2×C4×C9—21no—
135C2×C4×C9—22no—
136C2×C4×C9—22no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9—41no—
142C2×C4×C9—41no—
143C2×C4×C9C3×D411yescharacteristic
144C2×C4×C9C23×C311yescharacteristic
145C2×Dic9—21no—
146C2×Dic9—21no—
147Dic18—21no—
148Dic18—21no—
149C9⋊C8—21no—
150C9⋊C8C3×D412yes—
151#467—91noSylow
152C2×C3×C9—22no—
153C2×C3×C9#12111yescharacteristic
154C2×C3×C9C2×Q1611yescharacteristic
155C2×C3×C9#13511yescharacteristic
156C2×C3×Q8—21no—
157C2×C3×Q8—91no—
158C2×C3×Q8C2×D911yescharacteristic
159C2×C3×Q8C2×C3×D311yescharacteristic
160C22×C3×C4—21no—
161C22×C3×C4C2×D911yescharacteristic
162C22×C3×C4C2×C3×D311yescharacteristic
163C2×Dic6—31no—
164C3×Q16—182no—
165C2×C3×C8—91no—
166C2×C3×C8—91no—
167C2×C3×C8—182no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—91no—
173C3×C4⋊C4—181no—
174C3×C4⋊C4C2×D911yescharacteristic
175C3×C4⋊C4C2×C3×D311yescharacteristic
176C3×C22⋊C4—22no—
177C3×C22⋊C4—21no—
178C3×C22⋊C4—21no—
179C3×C22⋊C4—181no—
180#217—61no—
181#216—62no—
182C4⋊Dic3—31no—
183Dic3⋊C4—61no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—62no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32#21612yes—
190C22×C32C2×#6711yescharacteristic
191C32×C4—21no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4D3×D411yescharacteristic
195C32×C4#23711yescharacteristic
196C3×Dic3—121no—
197C3×Dic3—121no—
198C22×C9—21no—
199C22×C9—21no—
200C22×C9—22no—
201C22×C9—21no—
202C22×C9—41no—
203C22×C9—41no—
204C22×C9C3×Q1612yescomplemented
205C22×C9C2×C3×D411yescharacteristic
206C4×C9—21no—
207C4×C9—21no—
208C4×C9—21no—
209C4×C9—41no—
210C4×C9—42no—
211C4×C9—42no—
212C4×C9—41no—
213C4×C9—41no—
214C4×C9C2×C3×D411yescharacteristic
215C4×C9C3×#4111yescharacteristic
216Dic9—41no—
217Dic9—41no—
218C2×Q16—91no—
219C22×C8—91no—
220C22⋊Q8—91no—
221C22⋊Q8C3×D911yescharacteristic
222#107—91no—
223#103—91no—
224#103—91no—
225C3×C9#46711yescharacteristic, complemented, Sylow
226C23×C3—21no—
227C23×C3#61111yescharacteristic
228C23×C3C3×#6711yescharacteristic
229C3×Q8—21no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×Q8—181no—
236C3×Q8—181no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4#61111yescharacteristic
254C2×C3×C4C22×D911yescharacteristic
255C2×C3×C4C3×#6711yescharacteristic
256C2×C3×C4C22×C3×D311yescharacteristic
257C2×Dic3—61no—
258C2×Dic3—61no—
259Dic6—61no—
260Dic6—61no—
261C3×C8—92no—
262C3×C8—181no—
263C3⋊C8—32no—
264C3⋊C8—61no—
265C2×C32—22no—
266C2×C32#93911yescharacteristic
267C2×C32C2×#21611yescharacteristic
268C2×C32#95011yescharacteristic
269C2×C9—22no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9—42no—
274C2×C9C3×#12111yescharacteristic
275C2×C9C2×C3×Q1611yescharacteristic
276C2×C9C3×#13511yescharacteristic
277C2×Q8—91no—
278C2×Q8C2×C3×D911yescharacteristic
279C22×C4C2×C3×D911yescharacteristic
280Q16—182no—
281C2×C8—91no—
282C2×C8—91no—
283C2×C8—182no—
284C4⋊C4—21no—
285C4⋊C4—91no—
286C4⋊C4—181no—
287C4⋊C4C2×C3×D911yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—181no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—22no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3#367412yescomplemented
299C22×C3C2×#61111yescharacteristic
300C22×C3C3×#21612yes—
301C22×C3C2×C3×#6711yescharacteristic
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—41no—
307C3×C4—42no—
308C3×C4—42no—
309C3×C4—41no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—361no—
314C3×C4—361no—
315C3×C4D4×D911yescharacteristic
316C3×C4#369911yescharacteristic
317C3×C4C3×D3×D411yescharacteristic
318C3×C4C3×#23711yescharacteristic
319Dic3—121no—
320Dic3—121no—
321C32#505111yescharacteristic
322C9—21no—
323C9C3×#46711yescharacteristic, complemented
324C23C3×#61111yescharacteristic
325Q8—21no—
326Q8—21no—
327Q8—181no—
328Q8—181no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4C3×#61111yescharacteristic
336C2×C4C22×C3×D911yescharacteristic
337C8—92no—
338C8—181no—
339C2×C3—22no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—42no—
345C2×C3#6350111yescharacteristic
346C2×C3C2×#367411yescharacteristic
347C2×C3#6351311yescharacteristic
348C2×C3C3×#93911yescharacteristic
349C2×C3C2×C3×#21611yescharacteristic
350C2×C3C3×#95011yescharacteristic
351C22—21no—
352C22—21no—
353C22C3×#367412yescomplemented
354C22C2×C3×#61111yescharacteristic
355C4—21no—
356C4—41no—
357C4—41no—
358C4—361no—
359C4—361no—
360C4C3×D4×D911yescharacteristic
361C4C3×#369911yescharacteristic
362C3—21no—
363C3#1058872711yescharacteristic, direct product factor
364C3C3×#505111yescharacteristic
365C2—22no—
366C2C3×#6350111yescharacteristic
367C2C2×C3×#367411yescharacteristic
368C2C3×#6351311yescharacteristic
subgroupG/H
centerC22×C3C2×#611
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C2×#67
prime radicalC23×C32#67
hypercenterC3×C22⋊Q8D9
hypocenterC9C3×#467
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3674C2semidirect
5. C3×#63395C2semidirect
6. C3×#63368C2semidirect
7. C3×#63366C2semidirect
9. #10588727C3direct (N × H)
32. C2×#3674C2×C3semidirect
35. #63395C2×C3semidirect
36. #63368C2×C3semidirect
37. #63366C2×C3semidirect
88. C22×C3×C9Q16semidirect
204. C22×C9C3×Q16semidirect
225. C3×C9#467semidirect
298. C22×C3#3674semidirect
323. C9C3×#467semidirect
353. C22C3×#3674semidirect
363. C3#10588727direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×C22⋊Q82
C2×C3×#36742
C3×#634862
C22×C3×C9⋊C82
C3×#633952
C3×#633682
C3×#633662
C3×#50513
#105887273