Finite Group Databasebeta

← C3×#10588441 #13141711

Subgroups — C3×#10588441 #13141711

Order 1728.

374 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×Dic9C211yesmaximal, characteristic
2C3×#63450C211yesmaximal, characteristic, complemented
3C3×C9×#102C211yesmaximal, characteristic
4C3×#63392C211yesmaximal, characteristic
5C3×#63377C211yesmaximal, characteristic, complemented
6C3×Dic9⋊C8C211yesmaximal, characteristic, complemented
7C3×#63367C211yesmaximal, characteristic, complemented
8C3×#4761—31nomaximal
9#10588441C311yesmaximal, characteristic, direct product factor
10C2×C3×D4×C9C2211yescharacteristic
11C22×C3×Dic9—21no—
12C2×C3×Dic18—21no—
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×#3676—21no—
16C3×C4⋊Dic9C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C3×Dic9⋊C4—21no—
19C3×C4×Dic9C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21C3×D4×Dic3—31no—
22C3×#888—31no—
23C32×#102D311yescharacteristic
24C3×#832—31no—
25C3×#817—31no—
26C3×Dic3⋊C8—31no—
27C3×#808—31no—
28D4×Dic9C2×C311yescharacteristic
29#63450C2×C311yescharacteristic, complemented
30C9×#102—21no—
31C9×#102C2×C311yescharacteristic
32#63392C2×C311yescharacteristic
33#63377C2×C311yescharacteristic, complemented
34Dic9⋊C8C2×C311yescharacteristic, complemented
35#63367C2×C311yescharacteristic, complemented
36C23×C3×C9—21no—
37C3×D4×C9—21no—
38C3×D4×C9Q812yescomplemented
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3×Dic9—21no—
42C2×C3×Dic9—21no—
43C2×C3×Dic9—42no—
44C2×C3×Dic9D411yescharacteristic
45C2×C3×Dic9D411yescharacteristic
46C3×Dic18—42no—
47C3×C8×C9—21no—
48C3×C9⋊C8—21no—
49C3×#476—91no—
50#4761—31no—
51C2×C32×D4C2×D311yescharacteristic
52C22×C3×Dic3—61no—
53C2×C3×Dic6—61no—
54C2×C32×C8C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C3×#217—61no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×Dic3⋊C4—61no—
60C3×C4×Dic3—31no—
61C2×C3×C3⋊C8—31no—
62C2×D4×C9—21no—
63C2×D4×C9C22×C311yescharacteristic
64C22×Dic9—21no—
65C2×Dic18—21no—
66C2×C8×C9—21no—
67C2×C8×C9C22×C311yescharacteristic
68C9×C4⋊C4—21no—
69C9×C4⋊C4C22×C311yescharacteristic
70#3676—21no—
71C4⋊Dic9C22×C311yescharacteristic
72C4⋊Dic9C22×C311yescharacteristic
73Dic9⋊C4—21no—
74C4×Dic9C22×C311yescharacteristic
75C2×C9⋊C8C22×C311yescharacteristic
76C22×C3×C9—22no—
77C22×C3×C9—41no—
78C22×C3×C9—41no—
79C22×C3×C9C2×D411yescharacteristic
80C3×C4×C9—41no—
81C3×C4×C9C2×Q811yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C3×Dic9—21no—
84C3×Dic9—41no—
85C3×Dic9—41no—
86C3×Dic9SD1612yes—
87C3×C4⋊Q8—91no—
88C3×C4×D4—91no—
89D4×Dic3—31no—
90#888—31no—
91C3×#106—91no—
92C3×#106—91no—
93C3×C4⋊C8—91no—
94C3×#102—21no—
95C3×#102—91no—
96C3×#102D911yescharacteristic
97C3×#102C3×D311yescharacteristic
98#832—31no—
99#817—31no—
100Dic3⋊C8—31no—
101#808—31no—
102C23×C32—21no—
103C32×D4—21no—
104C32×D4Dic612yes—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3×Dic3—31no—
108C2×C3×Dic3—31no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—122no—
112C3×Dic6—122no—
113C23×C9—21no—
114C23×C9—41no—
115C32×C8—21no—
116C3×C3⋊C8—61no—
117D4×C9—21no—
118D4×C9—22no—
119D4×C9—41no—
120D4×C9C3×Q812yescomplemented
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9—41no—
124C2×C4×C9C23×C311yescharacteristic
125C2×Dic9—21no—
126C2×Dic9—21no—
127C2×Dic9—42no—
128C2×Dic9C3×D411yescharacteristic
129C2×Dic9C3×D411yescharacteristic
130Dic18—42no—
131C8×C9—21no—
132C8×C9—41no—
133C9⋊C8—21no—
134#476—91noSylow
135C2×C3×C9—42no—
136C2×C3×C9C22⋊Q811yescharacteristic
137C2×C3×C9C2×SD1611yescharacteristic
138C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
139C2×C3×Q8—181no—
140C2×C3×D4—21no—
141C2×C3×D4C2×D911yescharacteristic
142C2×C3×D4C2×C3×D311yescharacteristic
143C22×C3×C4—181no—
144C22×Dic3—61no—
145C2×Dic6—61no—
146C2×C3×C8—21no—
147C2×C3×C8—91no—
148C2×C3×C8C2×D911yescharacteristic
149C2×C3×C8C2×C3×D311yescharacteristic
150C3×C4⋊C4—21no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—91no—
153C3×C4⋊C4—181no—
154C3×C4⋊C4C2×D911yescharacteristic
155C3×C4⋊C4C2×C3×D311yescharacteristic
156C3×C22⋊C4—181no—
157C3×C42—91no—
158#217—61no—
159C4⋊Dic3—31no—
160C4⋊Dic3—31no—
161Dic3⋊C4—61no—
162C4×Dic3—31no—
163C2×C3⋊C8—31no—
164C22×C32—22no—
165C22×C32—41no—
166C22×C32—41no—
167C22×C32D3×D411yescharacteristic
168C32×C4—41no—
169C32×C4C2×Dic611yescharacteristic
170C32×C4#23711yescharacteristic
171C3×Dic3—32no—
172C3×Dic3—61no—
173C3×Dic3—121no—
174C3×Dic3—121no—
175C22×C9—22no—
176C22×C9—21no—
177C22×C9—42no—
178C22×C9—41no—
179C22×C9—41no—
180C22×C9—81no—
181C22×C9—81no—
182C22×C9C2×C3×D411yescharacteristic
183C4×C9—21no—
184C4×C9—21no—
185C4×C9—41no—
186C4×C9—81no—
187C4×C9C2×C3×Q811yescharacteristic
188C4×C9C3×#4111yescharacteristic
189Dic9—21no—
190Dic9—41no—
191Dic9—41no—
192Dic9C3×SD1612yes—
193C4⋊Q8—91no—
194C4×D4—91no—
195#106—91no—
196#106—91no—
197C4⋊C8—91no—
198#102—91no—
199#102C3×D911yescharacteristic
200C3×C9#47611yescharacteristic, complemented, Sylow
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—41no—
204C3×Q8—362no—
205C3×D4—21no—
206C3×D4—21no—
207C3×D4—22no—
208C3×D4—41no—
209C3×D4Dic1812yescomplemented
210C3×D4C3×Dic612yes—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—41no—
215C2×C3×C4—91no—
216C2×C3×C4—91no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4—362no—
220C2×C3×C4C22×D911yescharacteristic
221C2×C3×C4C22×C3×D311yescharacteristic
222C2×Dic3—31no—
223C2×Dic3—31no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—122no—
227Dic6—122no—
228C3×C8—21no—
229C3×C8—21no—
230C3×C8—41no—
231C3×C8—181no—
232C3⋊C8—61no—
233C2×C32—42no—
234C2×C32#87811yescharacteristic
235C2×C32#91111yescharacteristic
236C2×C32D3×SD1611yescharacteristic
237C2×C9—21no—
238C2×C9—21no—
239C2×C9—21no—
240C2×C9—42no—
241C2×C9—82no—
242C2×C9C3×C22⋊Q811yescharacteristic
243C2×C9C2×C3×SD1611yescharacteristic
244C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
245C2×Q8—181no—
246C2×D4C2×C3×D911yescharacteristic
247C22×C4—181no—
248C2×C8—91no—
249C2×C8C2×C3×D911yescharacteristic
250C4⋊C4—91no—
251C4⋊C4—91no—
252C4⋊C4—181no—
253C4⋊C4C2×C3×D911yescharacteristic
254C22⋊C4—181no—
255C42—91no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—42no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—81no—
265C22×C3—81no—
266C22×C3D4×D911yescharacteristic
267C22×C3C3×D3×D411yescharacteristic
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—81no—
273C3×C4—92no—
274C3×C4—181no—
275C3×C4—361no—
276C3×C4—361no—
277C3×C4C2×Dic1811yescharacteristic
278C3×C4#369911yescharacteristic
279C3×C4C2×C3×Dic611yescharacteristic
280C3×C4C3×#23711yescharacteristic
281Dic3—32no—
282Dic3—61no—
283Dic3—121no—
284Dic3—121no—
285C32#476111yescharacteristic
286C9—21no—
287C9C3×#47611yescharacteristic, complemented
288C23—21no—
289Q8—362no—
290D4—21no—
291D4C3×Dic1812yescomplemented
292C2×C4—21no—
293C2×C4—91no—
294C2×C4—91no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4—362no—
298C2×C4C22×C3×D911yescharacteristic
299C8—21no—
300C8—181no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—42no—
305C2×C3—42no—
306C2×C3—82no—
307C2×C3#6344011yescharacteristic
308C2×C3#6347311yescharacteristic
309C2×C3SD16×D911yescharacteristic
310C2×C3C3×#87811yescharacteristic
311C2×C3C3×#91111yescharacteristic
312C2×C3C3×D3×SD1611yescharacteristic
313C22—22no—
314C22—41no—
315C22—41no—
316C22C3×D4×D911yescharacteristic
317C4—41no—
318C4—92no—
319C4—181no—
320C4—361no—
321C4—361no—
322C4C2×C3×Dic1811yescharacteristic
323C4C3×#369911yescharacteristic
324C3—21no—
325C3#1058844111yescharacteristic, direct product factor
326C3C3×#476111yescharacteristic
327C2—42no—
328C2C3×#6344011yescharacteristic
329C2C3×#6347311yescharacteristic
330C2C3×SD16×D911yescharacteristic
subgroupG/H
centerC22×C3D4×D9
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
prime radicalC2×C32×D4C2×D3
hypercenterC3×#102D9
hypocenterC9C3×#476
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. C3×#63450C2semidirect
5. C3×#63377C2semidirect
6. C3×Dic9⋊C8C2semidirect
7. C3×#63367C2semidirect
9. #10588441C3direct (N × H)
29. #63450C2×C3semidirect
33. #63377C2×C3semidirect
34. Dic9⋊C8C2×C3semidirect
35. #63367C2×C3semidirect
38. C3×D4×C9Q8semidirect
120. D4×C9C3×Q8semidirect
200. C3×C9#476semidirect
209. C3×D4Dic18semidirect
287. C9C3×#476semidirect
291. D4C3×Dic18semidirect
325. C3#10588441direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D4×Dic92
C3×#634502
C3×C9×#1022
C3×#633922
C3×#633772
C3×Dic9⋊C82
C3×#633672
C3×#47613
#105884413