Finite Group Databasebeta

← C3×#10588736 #13142910

Subgroups — C3×#10588736 #13142910

Order 1728.

412 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×#124C211yesmaximal, characteristic
2C2×C3×#3674C211yesmaximal, characteristic, complemented
3C2×C3×#3672C211yesmaximal, characteristic
4C3×C4×Dic18C211yesmaximal, characteristic, complemented
5C3×#63395C211yesmaximal, characteristic, complemented
6C3×#63392C211yesmaximal, characteristic
7C3×C4⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5054—31nomaximal
9#10588736C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×D4×C9C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C3×C9×C22⋊C4—21no—
14C3×C9×C22⋊C4—21no—
15C3×C42×C9C2211yescharacteristic
16C3×#3674—22no—
17C3×#3672—22no—
18C3×C4⋊Dic9C2211yescharacteristic
19C3×Dic9⋊C4—21no—
20C3×C4×Dic9—21no—
21C2×C3×C9⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23C32×#124D311yescharacteristic
24C2×C3×#216—31no—
25C2×C3×#214—31no—
26C3×C4×Dic6—31no—
27C3×#835—31no—
28C3×#832—31no—
29C3×C4⋊(C3⋊C8)—31no—
30C9×#124—21no—
31C9×#124C2×C311yescharacteristic
32C2×#3674C2×C311yescharacteristic, complemented
33C2×#3672C2×C311yescharacteristic
34C4×Dic18C2×C311yescharacteristic, complemented
35#63395C2×C311yescharacteristic, complemented
36#63392C2×C311yescharacteristic
37C4⋊(C9⋊C8)C2×C311yescharacteristic, complemented
38C23×C3×C9—21no—
39C3×Q8×C9—21no—
40C3×Q8×C9—21no—
41C3×D4×C9—21no—
42C3×D4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9D411yescharacteristic
45C2×C3×C4×C9D411yescharacteristic
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3×Dic9—21no—
48C2×C3×Dic9—21no—
49C3×Dic18—21no—
50C3×Dic18D412yes—
51C3×C9⋊C8—21no—
52C3×C9⋊C8—21no—
53C3×#464—91no—
54#5054—31no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C32×D4C2×D311yescharacteristic
57C2×C3×Dic6—31no—
58C32×C22⋊C4—21no—
59C32×C22⋊C4—21no—
60C32×C42C2×D311yescharacteristic
61C3×#216—62no—
62C3×#214—62no—
63C3×C4⋊Dic3—31no—
64C3×Dic3⋊C4—61no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×Q8×C9—21no—
69C2×Q8×C9C22×C311yescharacteristic
70C2×D4×C9—21no—
71C2×D4×C9C22×C311yescharacteristic
72C2×Dic18C22×C311yescharacteristic
73C9×C22⋊C4—22no—
74C9×C22⋊C4—22no—
75C9×C22⋊C4—21no—
76C9×C22⋊C4—21no—
77C42×C9—21no—
78C42×C9C22×C311yescharacteristic
79#3674—22no—
80#3672—22no—
81C4⋊Dic9C22×C311yescharacteristic
82Dic9⋊C4—21no—
83C4×Dic9—21no—
84C2×C9⋊C8C22×C311yescharacteristic
85C2×C9⋊C8C22×C311yescharacteristic
86C22×C3×C9—41no—
87C22×C3×C9—41no—
88C22×C3×C9—41no—
89C22×C3×C9C2×D411yescharacteristic
90C3×C4×C9—21no—
91C3×C4×C9—21no—
92C3×C4×C9—41no—
93C3×C4×C9C2×D411yescharacteristic
94C3×C4×C9#4111yescharacteristic
95C3×Dic9—22no—
96C3×Dic9—41no—
97C2×C3×Q16—91no—
98C2×C3×SD16—91no—
99C3×#124—21no—
100C3×#124D911yescharacteristic
101C3×#124C3×D311yescharacteristic
102C3×C4×Q8—91no—
103C2×#216—31no—
104C2×#214—31no—
105C4×Dic6—31no—
106C3×C4⋊C8—91no—
107C3×#103—91no—
108C3×#102—91no—
109#835—31no—
110#832—31no—
111C4⋊(C3⋊C8)—31no—
112C23×C32—21no—
113C32×Q8—21no—
114C32×Q8—21no—
115C32×D4—21no—
116C32×D4—21no—
117C2×C32×C4—21no—
118C2×C32×C4#6711yescharacteristic
119C2×C32×C4#6711yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C3×Dic6—32no—
124C3×Dic6—61no—
125C23×C9—22no—
126C23×C9—21no—
127C3×C3⋊C8—61no—
128C3×C3⋊C8—61no—
129Q8×C9—21no—
130Q8×C9—21no—
131Q8×C9—41no—
132Q8×C9—41no—
133D4×C9—21no—
134D4×C9—21no—
135D4×C9—41no—
136D4×C9—41no—
137C2×C4×C9—22no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9C3×D411yescharacteristic
143C2×C4×C9C3×D411yescharacteristic
144C2×C4×C9C23×C311yescharacteristic
145C2×Dic9—21no—
146C2×Dic9—21no—
147Dic18—21no—
148Dic18C3×D412yes—
149C9⋊C8—21no—
150C9⋊C8—21no—
151#464—91noSylow
152C2×C3×C9—81no—
153C2×C3×C9#12111yescharacteristic
154C2×C3×C9#13511yescharacteristic
155C2×C3×C9#13711yescharacteristic
156C2×C3×Q8—21no—
157C2×C3×Q8—91no—
158C2×C3×Q8C2×D911yescharacteristic
159C2×C3×Q8C2×C3×D311yescharacteristic
160C2×C3×D4—21no—
161C2×C3×D4C2×D911yescharacteristic
162C2×C3×D4C2×C3×D311yescharacteristic
163C2×Dic6—31no—
164C3×Q16—182no—
165C3×SD16—182no—
166C2×C3×C8—91no—
167C2×C3×C8—91no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—181no—
170C3×C22⋊C4—22no—
171C3×C22⋊C4—22no—
172C3×C22⋊C4—21no—
173C3×C22⋊C4—21no—
174C3×C22⋊C4—21no—
175C3×C22⋊C4—21no—
176C3×C42—21no—
177C3×C42—181no—
178C3×C42C2×D911yescharacteristic
179C3×C42C2×C3×D311yescharacteristic
180#216—62no—
181#214—62no—
182C4⋊Dic3—31no—
183Dic3⋊C4—61no—
184C4×Dic3—61no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—31no—
187C22×C32—41no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32C2×#6711yescharacteristic
191C32×C4—21no—
192C32×C4—21no—
193C32×C4—41no—
194C32×C4D3×D411yescharacteristic
195C32×C4#23711yescharacteristic
196C3×Dic3—62no—
197C3×Dic3—121no—
198C22×C9—21no—
199C22×C9—42no—
200C22×C9—42no—
201C22×C9—42no—
202C22×C9—41no—
203C22×C9—41no—
204C22×C9—41no—
205C22×C9C2×C3×D411yescharacteristic
206C4×C9—21no—
207C4×C9—21no—
208C4×C9—21no—
209C4×C9—21no—
210C4×C9—41no—
211C4×C9—41no—
212C4×C9—42no—
213C4×C9—41no—
214C4×C9C2×C3×D411yescharacteristic
215C4×C9C3×#4111yescharacteristic
216Dic9—22no—
217Dic9—41no—
218C2×Q16—91no—
219C2×SD16—91no—
220#124C3×D911yescharacteristic
221C4×Q8—91no—
222C4⋊C8—91no—
223#103—91no—
224#102—91no—
225C3×C9#46411yescharacteristic, complemented, Sylow
226C23×C3—22no—
227C23×C3—21no—
228C23×C3—21no—
229C3×Q8—21no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×Q8—92no—
236C3×Q8—181no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—21no—
240C3×D4—21no—
241C3×D4—41no—
242C3×D4—41no—
243C2×C3×C4—22no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4#61111yescharacteristic
252C2×C3×C4#61111yescharacteristic
253C2×C3×C4C22×D911yescharacteristic
254C2×C3×C4C3×#6711yescharacteristic
255C2×C3×C4C3×#6711yescharacteristic
256C2×C3×C4C22×C3×D311yescharacteristic
257C2×Dic3—61no—
258C2×Dic3—61no—
259Dic6—32no—
260Dic6—61no—
261C3×C8—181no—
262C3×C8—181no—
263C3⋊C8—61no—
264C3⋊C8—61no—
265C2×C32—81no—
266C2×C32#93811yescharacteristic
267C2×C32#95011yescharacteristic
268C2×C32#95111yescharacteristic
269C2×C9—21no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—82no—
273C2×C9—81no—
274C2×C9C3×#12111yescharacteristic
275C2×C9C3×#13511yescharacteristic
276C2×C9C3×#13711yescharacteristic
277C2×Q8—91no—
278C2×Q8C2×C3×D911yescharacteristic
279C2×D4C2×C3×D911yescharacteristic
280Q16—182no—
281SD16—182no—
282C2×C8—91no—
283C2×C8—91no—
284C4⋊C4—91no—
285C4⋊C4—181no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C42—181no—
289C42C2×C3×D911yescharacteristic
290C22×C3—21no—
291C22×C3—42no—
292C22×C3—42no—
293C22×C3—42no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3C2×#61111yescharacteristic
301C22×C3C2×C3×#6711yescharacteristic
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—42no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—182no—
314C3×C4—361no—
315C3×C4D4×D911yescharacteristic
316C3×C4#369911yescharacteristic
317C3×C4C3×D3×D411yescharacteristic
318C3×C4C3×#23711yescharacteristic
319Dic3—62no—
320Dic3—121no—
321C32#505411yescharacteristic
322C9—21no—
323C9C3×#46411yescharacteristic, complemented
324C23—21no—
325Q8—21no—
326Q8—21no—
327Q8—92no—
328Q8—181no—
329D4—21no—
330D4—21no—
331C2×C4—21no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4C3×#61111yescharacteristic
335C2×C4C3×#61111yescharacteristic
336C2×C4C22×C3×D911yescharacteristic
337C8—181no—
338C8—181no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—82no—
343C2×C3—81no—
344C2×C3—81no—
345C2×C3#6350011yescharacteristic
346C2×C3#6351111yescharacteristic
347C2×C3#6351311yescharacteristic
348C2×C3C3×#93811yescharacteristic
349C2×C3C3×#95011yescharacteristic
350C2×C3C3×#95111yescharacteristic
351C22—41no—
352C22—41no—
353C22—41no—
354C22C2×C3×#61111yescharacteristic
355C4—21no—
356C4—21no—
357C4—41no—
358C4—182no—
359C4—361no—
360C4C3×D4×D911yescharacteristic
361C4C3×#369911yescharacteristic
362C3—21no—
363C3#1058873611yescharacteristic, direct product factor
364C3C3×#505411yescharacteristic
365C2—81no—
366C2C3×#6350011yescharacteristic
367C2C3×#6351111yescharacteristic
368C2C3×#6351311yescharacteristic
subgroupG/H
centerC22×C3C2×#611
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#124C2
socleC22×C32C2×#67
prime radicalC2×C32×D4C2×D3
hypercenterC3×#124D9
hypocenterC9C3×#464
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3674C2semidirect
4. C3×C4×Dic18C2semidirect
5. C3×#63395C2semidirect
7. C3×C4⋊(C9⋊C8)C2semidirect
9. #10588736C3direct (N × H)
32. C2×#3674C2×C3semidirect
34. C4×Dic18C2×C3semidirect
35. #63395C2×C3semidirect
37. C4⋊(C9⋊C8)C2×C3semidirect
225. C3×C9#464semidirect
323. C9C3×#464semidirect
363. C3#10588736direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1242
C2×C3×#36742
C2×C3×#36722
C3×C4×Dic182
C3×#633952
C3×#633922
C3×C4⋊(C9⋊C8)2
C3×#50543
#105887363