Finite Group Databasebeta

← C3×#10588569 #13142197

Subgroups — C3×#10588569 #13142197

Order 1728.

396 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3664C211yesmaximal, characteristic
2C3×#63455C211yesmaximal, characteristic
3C3×#63449C211yesmaximal, characteristic, complemented
4C3×C9×#107C211yesmaximal, characteristic, complemented
5C3×#63375C211yesmaximal, characteristic, complemented
6C3×#63369C211yesmaximal, characteristic
7C3×#63368C211yesmaximal, characteristic, complemented
8C3×#4889—31nomaximal
9#10588569C311yesmaximal, characteristic, direct product factor
10C2×C3×D36C2211yescharacteristic
11C2×C3×C4×D9—21no—
12C2×C3×Dic18C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16C3×(C2×D9)⋊C4—21no—
17C3×Dic9⋊C4—21no—
18C3×C4×Dic9—21no—
19C3×C4×Dic9C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21C3×#3664C414yescomplemented
22C2×C3×#204—31no—
23C3×#893—31no—
24C3×#887—31no—
25C32×#107D311yescharacteristic
26C3×#815—31no—
27C3×#810—31no—
28C3×#809—31no—
29C2×#3664C2×C311yescharacteristic
30#63455C2×C311yescharacteristic
31#63449C2×C311yescharacteristic, complemented
32C9×#107—21no—
33C9×#107C2×C311yescharacteristic, complemented
34#63375C2×C311yescharacteristic, complemented
35#63369C2×C311yescharacteristic
36#63368C2×C311yescharacteristic, complemented
37C22×C3×D9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3×Dic9—21no—
42C2×C3×Dic9D411yescharacteristic
43C2×C3×Dic9D411yescharacteristic
44C3×D36—21no—
45C3×D36C2×C412yes—
46C3×C4×D9—42no—
47C3×Dic18—21no—
48C3×Dic18C2×C412yescomplemented
49C3×C8×C9C2×C412yescomplemented
50C3×C9⋊C8—21no—
51C3×#440—91no—
52#4889—31no—
53C2×C3×D12—31no—
54C2×C3×C4×D3—61no—
55C2×C3×Dic6—31no—
56C2×C32×C8C2×D311yescharacteristic
57C32×C4⋊C4C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59C3×(C2×D3)⋊C4—61no—
60C3×Dic3⋊C4—61no—
61C3×C4×Dic3—31no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—31no—
64C3×#204—34no—
65C2×D36C22×C311yescharacteristic
66C2×C4×D9—21no—
67C2×Dic18C22×C311yescharacteristic
68C2×C8×C9—21no—
69C2×C8×C9C22×C311yescharacteristic
70C9×C4⋊C4—21no—
71C9×C4⋊C4—21no—
72C9×C4⋊C4C22×C311yescharacteristic
73C9×C4⋊C4C22×C311yescharacteristic
74(C2×D9)⋊C4—21no—
75Dic9⋊C4—21no—
76C4×Dic9—21no—
77C4×Dic9C22×C311yescharacteristic
78C2×C9⋊C8C22×C311yescharacteristic
79#3664C3×C414yescomplemented
80C22×C3×C9C2×D411yescharacteristic
81C2×C3×D9—22no—
82C2×C3×D9—41no—
83C2×C3×D9—41no—
84C3×C4×C9—41no—
85C3×C4×C9—41no—
86C3×C4×C9C22×C411yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C3×Dic9—22no—
89C3×Dic9—21no—
90C3×Dic9—21no—
91C2×C3×SD16—91no—
92C3×C4×Q8—91no—
93C3×C4×D4—91no—
94C2×#204—31no—
95#893—31no—
96#887—31no—
97C3×#107—21no—
98C3×#107D911yescharacteristic, complemented
99C3×#107C3×D311yescharacteristic
100C3×#103—91no—
101C3×#102—91no—
102C3×#97—91no—
103#815—31no—
104#810—31no—
105#809—31no—
106C22×C3×D3—61no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3×Dic3—31no—
111C2×C3×Dic3—31no—
112C2×C3×Dic3—61no—
113C3×D12—32no—
114C3×D12—61no—
115C3×C4×D3—122no—
116C3×Dic6—32no—
117C3×Dic6—61no—
118C22×D9—21no—
119C32×C8C4×D312yes—
120C3×C3⋊C8—61no—
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—41no—
125C2×C4×C9—41no—
126C2×C4×C9C23×C311yescharacteristic
127C2×Dic9—21no—
128C2×Dic9C3×D411yescharacteristic
129C2×Dic9C3×D411yescharacteristic
130D36—21no—
131D36C2×C3×C412yes—
132C4×D9—42no—
133Dic18—21no—
134Dic18C2×C3×C412yescomplemented
135C8×C9—22no—
136C8×C9C2×C3×C412yescomplemented
137C9⋊C8—21no—
138#440—91noSylow
139C2×C3×C9C4×D411yescharacteristic
140C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
141C2×C3×C9#13711yescharacteristic
142C3×D9—42no—
143C2×C3×Q8—91no—
144C2×C3×D4—91no—
145C22×C3×C4—181no—
146C2×D12—31no—
147C2×C4×D3—61no—
148C2×Dic6—31no—
149C3×SD16—94no—
150C2×C3×C8—21no—
151C2×C3×C8—91no—
152C2×C3×C8C2×D911yescharacteristic
153C2×C3×C8C2×C3×D311yescharacteristic
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—181no—
157C3×C4⋊C4C2×D911yescharacteristic
158C3×C4⋊C4C2×D911yescharacteristic
159C3×C4⋊C4C2×C3×D311yescharacteristic
160C3×C4⋊C4C2×C3×D311yescharacteristic
161C3×C22⋊C4—181no—
162C3×C42—91no—
163C3×C42—181no—
164(C2×D3)⋊C4—61no—
165Dic3⋊C4—61no—
166C4×Dic3—31no—
167C4×Dic3—61no—
168C2×C3⋊C8—31no—
169#204—34no—
170C22×C32D3×D411yescharacteristic
171C2×C3×D3—62no—
172C2×C3×D3—121no—
173C2×C3×D3—121no—
174C32×C4—41no—
175C32×C4—41no—
176C32×C4C2×C4×D311yescharacteristic
177C32×C4#23911yescharacteristic
178C3×Dic3—62no—
179C3×Dic3—61no—
180C3×Dic3—61no—
181C22×C9—21no—
182C22×C9C2×C3×D411yescharacteristic
183C2×D9—22no—
184C2×D9—41no—
185C2×D9—41no—
186C4×C9—21no—
187C4×C9—21no—
188C4×C9—41no—
189C4×C9—41no—
190C4×C9—81no—
191C4×C9—81no—
192C4×C9C22×C3×C411yescharacteristic
193C4×C9C3×#4111yescharacteristic
194Dic9—22no—
195Dic9—21no—
196Dic9—21no—
197C2×SD16—91no—
198C4×Q8—91no—
199C4×D4—91no—
200#107C3×D911yescharacteristic, complemented
201#103—91no—
202#102—91no—
203#97—91no—
204C3×C9#44011yescharacteristic, complemented, Sylow
205C23×C3—181no—
206C22×D3—61no—
207C3×Q8—92no—
208C3×Q8—181no—
209C3×D4—92no—
210C3×D4—181no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—91no—
219C2×C3×C4—91no—
220C2×C3×C4—181no—
221C2×C3×C4—362no—
222C2×C3×C4C22×D911yescharacteristic
223C2×C3×C4C22×C3×D311yescharacteristic
224C2×Dic3—31no—
225C2×Dic3—31no—
226C2×Dic3—61no—
227D12—32no—
228D12—61no—
229C4×D3—122no—
230Dic6—32no—
231Dic6—61no—
232C3×C8—22no—
233C3×C8—181no—
234C3×C8C4×D912yescomplemented
235C3×C8C3×C4×D312yes—
236C3⋊C8—61no—
237C2×C32#89311yescharacteristic
238C2×C32#91111yescharacteristic
239C2×C32#91811yescharacteristic
240C3×D3—122no—
241C2×C9—21no—
242C2×C9—21no—
243C2×C9—21no—
244C2×C9C3×C4×D411yescharacteristic
245C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
246C2×C9C3×#13711yescharacteristic
247D9—42no—
248C2×Q8—91no—
249C2×D4—91no—
250C22×C4—181no—
251SD16—94no—
252C2×C8—91no—
253C2×C8C2×C3×D911yescharacteristic
254C4⋊C4—181no—
255C4⋊C4C2×C3×D911yescharacteristic
256C4⋊C4C2×C3×D911yescharacteristic
257C22⋊C4—181no—
258C42—91no—
259C42—181no—
260C22×C3—21no—
261C22×C3—182no—
262C22×C3—361no—
263C22×C3—361no—
264C22×C3D4×D911yescharacteristic
265C22×C3C3×D3×D411yescharacteristic
266C2×D3—62no—
267C2×D3—121no—
268C2×D3—121no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—81no—
276C3×C4—81no—
277C3×C4—182no—
278C3×C4—181no—
279C3×C4—181no—
280C3×C4C2×C4×D911yescharacteristic
281C3×C4#370111yescharacteristic
282C3×C4C2×C3×C4×D311yescharacteristic
283C3×C4C3×#23911yescharacteristic
284Dic3—62no—
285Dic3—61no—
286Dic3—61no—
287C32#488911yescharacteristic
288C9—21no—
289C9C3×#44011yescharacteristic, complemented
290C23—181no—
291Q8—92no—
292Q8—181no—
293D4—92no—
294D4—181no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—91no—
298C2×C4—91no—
299C2×C4—181no—
300C2×C4—362no—
301C2×C4C22×C3×D911yescharacteristic
302C8—181no—
303C8C3×C4×D912yescomplemented
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—362no—
308C2×C3#6345511yescharacteristic
309C2×C3#6347311yescharacteristic
310C2×C3#6348011yescharacteristic
311C2×C3C3×#89311yescharacteristic
312C2×C3C3×#91111yescharacteristic
313C2×C3C3×#91811yescharacteristic
314D3—122no—
315C22—182no—
316C22—361no—
317C22—361no—
318C22C3×D4×D911yescharacteristic
319C4—41no—
320C4—41no—
321C4—181no—
322C4—181no—
323C4—182no—
324C4C2×C3×C4×D911yescharacteristic
325C4C3×#370111yescharacteristic
326C3—21no—
327C3#1058856911yescharacteristic, direct product factor
328C3C3×#488911yescharacteristic
329C2—362no—
330C2C3×#6345511yescharacteristic
331C2C3×#6347311yescharacteristic
332C2C3×#6348011yescharacteristic
subgroupG/H
centerC22×C3D4×D9
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#107C2
socleC22×C32D3×D4
prime radicalC2×C3×D36C22
cosocleC2×C4×C9C23×C3
hypercenterC3×#107D9
hypocenterC9C3×#440
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#107

Complements

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

NH ≅ G/Ntype
3. C3×#63449C2semidirect
4. C3×C9×#107C2semidirect
5. C3×#63375C2semidirect
7. C3×#63368C2semidirect
9. #10588569C3direct (N × H)
21. C3×#3664C4semidirect
31. #63449C2×C3semidirect
33. C9×#107C2×C3semidirect
34. #63375C2×C3semidirect
36. #63368C2×C3semidirect
48. C3×Dic18C2×C4semidirect
49. C3×C8×C9C2×C4semidirect
79. #3664C3×C4semidirect
98. C3×#107D9semidirect
134. Dic18C2×C3×C4semidirect
136. C8×C9C2×C3×C4semidirect
200. #107C3×D9semidirect
204. C3×C9#440semidirect
234. C3×C8C4×D9semidirect
289. C9C3×#440semidirect
303. C8C3×C4×D9semidirect
327. C3#10588569direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36642
C3×#634552
C3×#634492
C3×C9×#1072
C3×#633752
C3×#633692
C3×#633682
C3×#48893
#105885693