Finite Group Databasebeta

← C3×#10588824 #13143249

Subgroups — C3×#10588824 #13143249

Order 1728.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#131C211yesmaximal, characteristic, complemented
2C3×#63513C211yesmaximal, characteristic
3C3×#63464C211yesmaximal, characteristic, complemented
4C3×#63396C211yesmaximal, characteristic, complemented
5C3×#63384C211yesmaximal, characteristic, complemented
6C3×#63381C211yesmaximal, characteristic, complemented
7C3×C8×Dic9C211yesmaximal, characteristic, complemented
8C3×#5144—31nomaximal
9#10588824C311yesmaximal, characteristic, direct product factor
10C3×C9×#41C2211yescharacteristic
11C3×#3697C2211yescharacteristic
12C3×C9×M16—21no—
13C3×C9×M16C2211yescharacteristic
14C2×C3×C8×C9—21no—
15C2×C3×C8×C9C2211yescharacteristic
16C3×#3675C411yescharacteristic, complemented
17C3×#3674C411yescharacteristic, complemented
18C3×#3673C411yescharacteristic, complemented
19C3×#3672C411yescharacteristic, complemented
20C3×C4×Dic9C2211yescharacteristic
21C3×#3667C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23C3×#3663—21no—
24C3×C8×D9—21no—
25C32×#131D311yescharacteristic
26C3×#950—31no—
27C3×#901—31no—
28C3×#837—31no—
29C3×#825—31no—
30C3×#822—31no—
31C3×C8×Dic3—31no—
32C9×#131—21no—
33C9×#131C2×C311yescharacteristic, complemented
34#63513C2×C311yescharacteristic
35#63464C2×C311yescharacteristic, complemented
36#63396C2×C311yescharacteristic, complemented
37#63384C2×C311yescharacteristic, complemented
38#63381C2×C311yescharacteristic, complemented
39C8×Dic9C2×C311yescharacteristic, complemented
40C3×Q8×C9C2×C411yescharacteristic
41C3×D4×C9—21no—
42C3×D4×C9C2×C411yescharacteristic
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C3×#611—21no—
46C2×C3×Dic9—21no—
47C3×D36C2×C411yescharacteristic
48C3×C4×D9—21no—
49C3×Dic18C2×C411yescharacteristic
50C3×C8×C9—21no—
51C3×C8×C9D411yescharacteristic
52C3×C8×C9D411yescharacteristic
53C3×C9⋊C8—21no—
54C3×C9⋊C8C2×C411yescharacteristic
55C3×C9⋊C8C2×C411yescharacteristic
56C3×#443—91no—
57#5144—31no—
58C32×#41C2×D311yescharacteristic
59C3×#235—31no—
60C32×M16—21no—
61C32×M16C2×D311yescharacteristic
62C2×C32×C8—21no—
63C2×C32×C8C2×D311yescharacteristic
64C3×#216—31no—
65C3×#215—31no—
66C3×#214—31no—
67C3×#213—31no—
68C3×C4×Dic3—31no—
69C3×#208—31no—
70C2×C3×C3⋊C8—31no—
71C3×ΓL(1,𝔽25)—61no—
72C3×D3×C8—61no—
73C9×#41—21no—
74C9×#41C22×C311yescharacteristic
75#3697C22×C311yescharacteristic
76C9×M16—22no—
77C9×M16—21no—
78C9×M16—21no—
79C9×M16C22×C311yescharacteristic
80C2×C8×C9—22no—
81C2×C8×C9—21no—
82C2×C8×C9—21no—
83C2×C8×C9C22×C311yescharacteristic
84#3675C3×C411yescharacteristic, complemented
85#3674C3×C411yescharacteristic, complemented
86#3673C3×C411yescharacteristic, complemented
87#3672C3×C411yescharacteristic, complemented
88C4×Dic9C22×C311yescharacteristic
89#3667C22×C311yescharacteristic
90C2×C9⋊C8C22×C311yescharacteristic
91#3663—21no—
92C8×D9—21no—
93C22×C3×C9—21no—
94C22×C3×C9#4111yescharacteristic
95C2×C3×D9—21no—
96C3×C4×C9—21no—
97C3×C4×C9C22×C411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C3×Dic9—22no—
100C3×Dic9—21no—
101C3×#135—91no—
102C3×#131—21no—
103C3×#131—91no—
104C3×#131D911yescharacteristic, complemented
105C3×#131C3×D311yescharacteristic
106#950—31no—
107#901—31no—
108C3×#108—91no—
109C3×#104—91no—
110C3×#104—91no—
111C3×C4×C8—91no—
112#837—31no—
113#825—31no—
114#822—31no—
115C8×Dic3—31no—
116C32×Q8C4×D311yescharacteristic
117C32×D4—21no—
118C32×D4C4×D311yescharacteristic
119C2×C32×C4—21no—
120C2×C32×C4C22×D311yescharacteristic
121C3×#67—61no—
122C2×C3×Dic3—61no—
123C3×D12—31no—
124C3×C4×D3—61no—
125C3×Dic6—31no—
126C32×C8—21no—
127C32×C8#6711yescharacteristic
128C32×C8#6711yescharacteristic
129C3×C3⋊C8—31no—
130C3×C3⋊C8—31no—
131C3×C3⋊C8—61no—
132Q8×C9—21no—
133Q8×C9C2×C3×C411yescharacteristic
134D4×C9—22no—
135D4×C9—21no—
136D4×C9—21no—
137D4×C9C2×C3×C411yescharacteristic
138C2×C4×C9—22no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9C23×C311yescharacteristic
142#611—21no—
143C2×Dic9—21no—
144D36C2×C3×C411yescharacteristic
145C4×D9—21no—
146Dic18C2×C3×C411yescharacteristic
147C8×C9—22no—
148C8×C9—21no—
149C8×C9—21no—
150C8×C9—21no—
151C8×C9C3×D411yescharacteristic
152C8×C9C3×D411yescharacteristic
153C9⋊C8—21no—
154C9⋊C8C2×C3×C411yescharacteristic
155C9⋊C8C2×C3×C411yescharacteristic
156#443—91noSylow
157C2×C3×C9—21no—
158C2×C3×C9—41no—
159C2×C3×C9C4×D411yescharacteristic
160C3×D9—41no—
161C3×#41—21no—
162C3×#41—91no—
163C3×#41C2×D911yescharacteristic
164C3×#41C2×C3×D311yescharacteristic
165#235—31no—
166C3×Q16—91no—
167C3×SD16—91no—
168C3×SD16—91no—
169C3×D8—91no—
170C3×M16—22no—
171C3×M16—21no—
172C3×M16—21no—
173C3×M16—21no—
174C3×M16—91no—
175C3×M16—181no—
176C3×M16C2×D911yescharacteristic
177C3×M16C2×C3×D311yescharacteristic
178C2×C3×C8—22no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—91no—
183C2×C3×C8—181no—
184C2×C3×C8C2×D911yescharacteristic
185C2×C3×C8C2×C3×D311yescharacteristic
186C3×C42—91no—
187#216—31no—
188#215—31no—
189#214—31no—
190#213—31no—
191C4×Dic3—31no—
192#208—31no—
193C2×C3⋊C8—31no—
194ΓL(1,𝔽25)—61no—
195D3×C8—61no—
196C22×C32—21no—
197C22×C32#23511yescharacteristic
198C2×C3×D3—61no—
199C32×C4—21no—
200C32×C4C2×C4×D311yescharacteristic
201C32×C4C2×#6711yescharacteristic
202C3×Dic3—62no—
203C3×Dic3—61no—
204C22×C9—22no—
205C22×C9—21no—
206C22×C9—21no—
207C22×C9C3×#4111yescharacteristic
208C2×D9—21no—
209C4×C9—22no—
210C4×C9—21no—
211C4×C9—21no—
212C4×C9—21no—
213C4×C9C22×C3×C411yescharacteristic
214C4×C9C2×C3×D411yescharacteristic
215Dic9—22no—
216Dic9—21no—
217#135—91no—
218#131—91no—
219#131C3×D911yescharacteristic, complemented
220#108—91no—
221#104—91no—
222#104—91no—
223C4×C8—91no—
224C3×C9#44311yescharacteristic, complemented, Sylow
225C3×Q8—21no—
226C3×Q8—91no—
227C3×Q8C4×D911yescharacteristic
228C3×Q8C3×C4×D311yescharacteristic
229C3×D4—22no—
230C3×D4—21no—
231C3×D4—21no—
232C3×D4—21no—
233C3×D4—91no—
234C3×D4—181no—
235C3×D4C4×D911yescharacteristic
236C3×D4C3×C4×D311yescharacteristic
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4C22×D911yescharacteristic
244C2×C3×C4C22×C3×D311yescharacteristic
245#67—61no—
246C2×Dic3—61no—
247D12—31no—
248C4×D3—61no—
249Dic6—31no—
250C3×C8—22no—
251C3×C8—21no—
252C3×C8—21no—
253C3×C8—21no—
254C3×C8—21no—
255C3×C8—91no—
256C3×C8—91no—
257C3×C8—181no—
258C3×C8#61111yescharacteristic
259C3×C8#61111yescharacteristic
260C3×C8C3×#6711yescharacteristic
261C3×C8C3×#6711yescharacteristic
262C3⋊C8—31no—
263C3⋊C8—31no—
264C3⋊C8—61no—
265C2×C32—21no—
266C2×C32—41no—
267C2×C32C4×#6711yescharacteristic
268C3×D3—121no—
269C2×C9—21no—
270C2×C9—21no—
271C2×C9—41no—
272C2×C9—42no—
273C2×C9—41no—
274C2×C9C3×C4×D411yescharacteristic
275D9—41no—
276#41—91no—
277#41C2×C3×D911yescharacteristic
278Q16—91no—
279SD16—91no—
280SD16—91no—
281D8—91no—
282M16—21no—
283M16—91no—
284M16—181no—
285M16C2×C3×D911yescharacteristic
286C2×C8—21no—
287C2×C8—91no—
288C2×C8—181no—
289C2×C8C2×C3×D911yescharacteristic
290C42—91no—
291C22×C3—22no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—181no—
296C22×C3#369711yescharacteristic
297C22×C3C3×#23511yescharacteristic
298C2×D3—61no—
299C3×C4—22no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—182no—
305C3×C4—181no—
306C3×C4C2×C4×D911yescharacteristic
307C3×C4C2×#61111yescharacteristic
308C3×C4C2×C3×C4×D311yescharacteristic
309C3×C4C2×C3×#6711yescharacteristic
310Dic3—62no—
311Dic3—61no—
312C32#514411yescharacteristic
313C9—21no—
314C9C3×#44311yescharacteristic, complemented
315Q8—91no—
316Q8C3×C4×D911yescharacteristic
317D4—21no—
318D4—91no—
319D4—181no—
320D4C3×C4×D911yescharacteristic
321C2×C4—21no—
322C2×C4—181no—
323C2×C4—181no—
324C2×C4C22×C3×D911yescharacteristic
325C8—21no—
326C8—91no—
327C8—91no—
328C8—181no—
329C8C3×#61111yescharacteristic
330C8C3×#61111yescharacteristic
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—41no—
335C2×C3—42no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—361no—
339C2×C3C4×#61111yescharacteristic
340C2×C3C3×C4×#6711yescharacteristic
341D3—121no—
342C22—21no—
343C22—181no—
344C22C3×#369711yescharacteristic
345C4—21no—
346C4—182no—
347C4—181no—
348C4C2×C3×C4×D911yescharacteristic
349C4C2×C3×#61111yescharacteristic
350C3—21no—
351C3#1058882411yescharacteristic, direct product factor
352C3C3×#514411yescharacteristic
353C2—21no—
354C2—41no—
355C2—361no—
356C2C3×C4×#61111yescharacteristic
subgroupG/H
centerC3×C8#611
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#131C2
socleC2×C32C4×#67
prime radicalC3×#63513C2
cosocleC2×C4×C9C23×C3
hypercenterC3×#131D9
hypocenterC9C3×#443
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C3×C8 ◁ C2×C3×C8 ◁ C3×#131

Complements

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

NH ≅ G/Ntype
1. C3×C9×#131C2semidirect
3. C3×#63464C2semidirect
4. C3×#63396C2semidirect
5. C3×#63384C2semidirect
6. C3×#63381C2semidirect
7. C3×C8×Dic9C2semidirect
9. #10588824C3direct (N × H)
16. C3×#3675C4semidirect
17. C3×#3674C4semidirect
18. C3×#3673C4semidirect
19. C3×#3672C4semidirect
33. C9×#131C2×C3semidirect
35. #63464C2×C3semidirect
36. #63396C2×C3semidirect
37. #63384C2×C3semidirect
38. #63381C2×C3semidirect
39. C8×Dic9C2×C3semidirect
84. #3675C3×C4semidirect
85. #3674C3×C4semidirect
86. #3673C3×C4semidirect
87. #3672C3×C4semidirect
104. C3×#131D9semidirect
219. #131C3×D9semidirect
224. C3×C9#443semidirect
314. C9C3×#443semidirect
351. C3#10588824direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1312
C3×#635132
C3×#634642
C3×#633962
C3×#633842
C3×#633812
C3×C8×Dic92
C3×#51443
#105888243