Finite Group Databasebeta

← C3×#10588846 #13143335

Subgroups — C3×#10588846 #13143335

Order 1728.

404 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×SD16C211yesmaximal, characteristic
2C3×Q8×Dic9C211yesmaximal, characteristic, complemented
3C2×C3×#3674C211yesmaximal, characteristic, complemented
4C3×#63498C211yesmaximal, characteristic
5C3×#63392C211yesmaximal, characteristic
6C3×#63376C211yesmaximal, characteristic, complemented
7C3×Dic9⋊C8C211yesmaximal, characteristic, complemented
8C3×#5166—31nomaximal
9#10588846C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×D4×C9C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C3×C9×SD16—22no—
14C2×C3×C8×C9C2211yescharacteristic
15C3×#3676—21no—
16C3×#3676—21no—
17C3×#3674—22no—
18C3×C4⋊Dic9—21no—
19C3×C4⋊Dic9C2211yescharacteristic
20C3×C4×Dic9—21no—
21C3×C4×Dic9C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23C2×C32×SD16D311yescharacteristic
24C3×Q8×Dic3—31no—
25C2×C3×#216—31no—
26C3×#936—31no—
27C3×#832—31no—
28C3×#816—31no—
29C3×Dic3⋊C8—31no—
30C2×C9×SD16—21no—
31C2×C9×SD16C2×C311yescharacteristic
32Q8×Dic9C2×C311yescharacteristic, complemented
33C2×#3674C2×C311yescharacteristic, complemented
34#63498C2×C311yescharacteristic
35#63392C2×C311yescharacteristic
36#63376C2×C311yescharacteristic, complemented
37Dic9⋊C8C2×C311yescharacteristic, complemented
38C23×C3×C9—21no—
39C3×Q8×C9—21no—
40C3×Q8×C9D412yes—
41C3×D4×C9—21no—
42C3×D4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3×Dic9—21no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9D411yescharacteristic
48C2×C3×Dic9D411yescharacteristic
49C3×Dic18—21no—
50C3×Dic18—21no—
51C3×C8×C9—21no—
52C3×C9⋊C8—21no—
53C3×#464—91no—
54#5166—31no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C32×D4C2×D311yescharacteristic
57C2×C3×Dic6—31no—
58C32×SD16—22no—
59C2×C32×C8C2×D311yescharacteristic
60C3×#217—61no—
61C3×#217—61no—
62C3×#216—62no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—61no—
65C3×C4×Dic3—31no—
66C3×C4×Dic3—61no—
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×SD16—24no—
74C9×SD16—22no—
75C2×C8×C9—21no—
76C2×C8×C9C22×C311yescharacteristic
77#3676—21no—
78#3676—21no—
79#3674—22no—
80C4⋊Dic9—21no—
81C4⋊Dic9C22×C311yescharacteristic
82C4×Dic9—21no—
83C4×Dic9C22×C311yescharacteristic
84C2×C9⋊C8C22×C311yescharacteristic
85C22×C3×C9—41no—
86C22×C3×C9—41no—
87C22×C3×C9—41no—
88C22×C3×C9C2×D411yescharacteristic
89C3×C4×C9—22no—
90C3×C4×C9C2×D411yescharacteristic
91C3×C4×C9#4111yescharacteristic
92C3×Dic9—21no—
93C3×Dic9—21no—
94C3×Dic9—41no—
95C3×Dic9—41no—
96C2×C3×Q16—91no—
97C2×C3×SD16—21no—
98C2×C3×SD16D911yescharacteristic
99C2×C3×SD16C3×D311yescharacteristic
100C3×#124—91no—
101C3×C4×Q8—91no—
102Q8×Dic3—31no—
103C2×#216—31no—
104#936—31no—
105C3×C4⋊C8—91no—
106C3×#103—91no—
107C3×#102—91no—
108#832—31no—
109#816—31no—
110Dic3⋊C8—31no—
111C23×C32—21no—
112C32×Q8—21no—
113C32×Q8#6712yes—
114C32×D4—21no—
115C32×D4—21no—
116C2×C32×C4—21no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3×Dic3—31no—
119C2×C3×Dic3—31no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C3×Dic6—61no—
123C3×Dic6—61no—
124C23×C9—21no—
125C23×C9—41no—
126C32×C8—21no—
127C3×C3⋊C8—61no—
128Q8×C9—21no—
129Q8×C9—22no—
130Q8×C9—41no—
131Q8×C9C3×D412yes—
132D4×C9—22no—
133D4×C9—21no—
134D4×C9—21no—
135D4×C9—41no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—41no—
139C2×C4×C9C23×C311yescharacteristic
140C2×Dic9—21no—
141C2×Dic9—21no—
142C2×Dic9C3×D411yescharacteristic
143C2×Dic9C3×D411yescharacteristic
144Dic18—21no—
145Dic18—21no—
146C8×C9—22no—
147C8×C9—21no—
148C9⋊C8—21no—
149#464—91noSylow
150C2×C3×C9—81no—
151C2×C3×C9#12111yescharacteristic
152C2×C3×C9#13511yescharacteristic
153C2×C3×C9#13711yescharacteristic
154C2×C3×Q8—21no—
155C2×C3×Q8—91no—
156C2×C3×Q8C2×D911yescharacteristic
157C2×C3×Q8C2×C3×D311yescharacteristic
158C2×C3×D4—21no—
159C2×C3×D4C2×D911yescharacteristic
160C2×C3×D4C2×C3×D311yescharacteristic
161C2×Dic6—31no—
162C3×Q16—182no—
163C3×SD16—24no—
164C3×SD16—22no—
165C3×SD16—22no—
166C2×C3×C8—21no—
167C2×C3×C8—91no—
168C2×C3×C8C2×D911yescharacteristic
169C2×C3×C8C2×C3×D311yescharacteristic
170C3×C4⋊C4—91no—
171C3×C4⋊C4—181no—
172C3×C22⋊C4—181no—
173C3×C22⋊C4—181no—
174C3×C42—91no—
175C3×C42—181no—
176#217—61no—
177#217—61no—
178#216—62no—
179C4⋊Dic3—31no—
180C4⋊Dic3—61no—
181C4×Dic3—31no—
182C4×Dic3—61no—
183C2×C3⋊C8—31no—
184C22×C32—41no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32D3×D411yescharacteristic
188C32×C4—22no—
189C32×C4#23711yescharacteristic
190C32×C4C2×#6711yescharacteristic
191C3×Dic3—61no—
192C3×Dic3—61no—
193C3×Dic3—121no—
194C3×Dic3—121no—
195C22×C9—21no—
196C22×C9—42no—
197C22×C9—41no—
198C22×C9—41no—
199C22×C9—41no—
200C22×C9—81no—
201C22×C9—81no—
202C22×C9C2×C3×D411yescharacteristic
203C4×C9—22no—
204C4×C9—21no—
205C4×C9—21no—
206C4×C9—42no—
207C4×C9C2×C3×D411yescharacteristic
208C4×C9C3×#4111yescharacteristic
209Dic9—21no—
210Dic9—21no—
211Dic9—41no—
212Dic9—41no—
213C2×Q16—91no—
214C2×SD16C3×D911yescharacteristic
215#124—91no—
216C4×Q8—91no—
217C4⋊C8—91no—
218#103—91no—
219#102—91no—
220C3×C9#46411yescharacteristic, complemented, Sylow
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—41no—
224C3×Q8—21no—
225C3×Q8—21no—
226C3×Q8—22no—
227C3×Q8—41no—
228C3×Q8—181no—
229C3×Q8—181no—
230C3×Q8#61112yes—
231C3×Q8C3×#6712yes—
232C3×D4—22no—
233C3×D4—21no—
234C3×D4—21no—
235C3×D4—21no—
236C3×D4—21no—
237C3×D4—41no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4—91no—
243C2×C3×C4—91no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4C22×D911yescharacteristic
247C2×C3×C4C22×C3×D311yescharacteristic
248C2×Dic3—31no—
249C2×Dic3—31no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252Dic6—61no—
253Dic6—61no—
254C3×C8—22no—
255C3×C8—21no—
256C3×C8—21no—
257C3×C8—181no—
258C3⋊C8—61no—
259C2×C32—81no—
260C2×C32#91511yescharacteristic
261C2×C32#91611yescharacteristic
262C2×C32#93911yescharacteristic
263C2×C9—21no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9—82no—
267C2×C9—81no—
268C2×C9C3×#12111yescharacteristic
269C2×C9C3×#13511yescharacteristic
270C2×C9C3×#13711yescharacteristic
271C2×Q8—91no—
272C2×Q8C2×C3×D911yescharacteristic
273C2×D4C2×C3×D911yescharacteristic
274Q16—182no—
275SD16—22no—
276C2×C8—91no—
277C2×C8C2×C3×D911yescharacteristic
278C4⋊C4—91no—
279C4⋊C4—181no—
280C22⋊C4—181no—
281C22⋊C4—181no—
282C42—91no—
283C42—181no—
284C22×C3—21no—
285C22×C3—42no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3D4×D911yescharacteristic
295C22×C3C3×D3×D411yescharacteristic
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—42no—
301C3×C4—181no—
302C3×C4—181no—
303C3×C4—361no—
304C3×C4—361no—
305C3×C4#369911yescharacteristic
306C3×C4C2×#61111yescharacteristic
307C3×C4C3×#23711yescharacteristic
308C3×C4C2×C3×#6711yescharacteristic
309Dic3—61no—
310Dic3—61no—
311Dic3—121no—
312Dic3—121no—
313C32#516611yescharacteristic
314C9—21no—
315C9C3×#46411yescharacteristic, complemented
316C23—21no—
317Q8—21no—
318Q8—181no—
319Q8—181no—
320Q8C3×#61112yes—
321D4—21no—
322D4—21no—
323C2×C4—21no—
324C2×C4—91no—
325C2×C4—91no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4C22×C3×D911yescharacteristic
329C8—21no—
330C8—181no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—82no—
335C2×C3—81no—
336C2×C3—81no—
337C2×C3#6347711yescharacteristic
338C2×C3#6347811yescharacteristic
339C2×C3#6350111yescharacteristic
340C2×C3C3×#91511yescharacteristic
341C2×C3C3×#91611yescharacteristic
342C2×C3C3×#93911yescharacteristic
343C22—41no—
344C22—41no—
345C22—41no—
346C22C3×D4×D911yescharacteristic
347C4—22no—
348C4—181no—
349C4—181no—
350C4—361no—
351C4—361no—
352C4C3×#369911yescharacteristic
353C4C2×C3×#61111yescharacteristic
354C3—21no—
355C3#1058884611yescharacteristic, direct product factor
356C3C3×#516611yescharacteristic
357C2—81no—
358C2C3×#6347711yescharacteristic
359C2C3×#6347811yescharacteristic
360C2C3×#6350111yescharacteristic
subgroupG/H
centerC22×C3D4×D9
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C3×C9×SD16C2
socleC22×C32D3×D4
prime radicalC2×C32×D4C2×D3
hypercenterC2×C3×SD16D9
hypocenterC9C3×#464
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
2. C3×Q8×Dic9C2semidirect
3. C2×C3×#3674C2semidirect
6. C3×#63376C2semidirect
7. C3×Dic9⋊C8C2semidirect
9. #10588846C3direct (N × H)
32. Q8×Dic9C2×C3semidirect
33. C2×#3674C2×C3semidirect
36. #63376C2×C3semidirect
37. Dic9⋊C8C2×C3semidirect
220. C3×C9#464semidirect
315. C9C3×#464semidirect
355. C3#10588846direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C9×SD162
C3×Q8×Dic92
C2×C3×#36742
C3×#634982
C3×#633922
C3×#633762
C3×Dic9⋊C82
C3×#51663
#105888463