Finite Group Databasebeta

← C3×#10588504 #13141968

Subgroups — C3×#10588504 #13141968

Order 1728.

362 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63513C211yesmaximal, characteristic
2C3×#63510C211yesmaximal, characteristic, complemented
3C3×#63469C211yesmaximal, characteristic, complemented
4C3×C9×#104C211yesmaximal, characteristic, complemented
5C3×#63381C211yesmaximal, characteristic, complemented
6C3×#63364C211yesmaximal, characteristic, complemented
7C3×C4×C9⋊C8C211yesmaximal, characteristic, complemented
8C3×#4824—31nomaximal
9#10588504C311yesmaximal, characteristic, direct product factor
10C3×C9×#41C2211yescharacteristic
11C3×#3697C2211yescharacteristic
12C3×C9×M16C2211yescharacteristic
13C3×C42×C9C2211yescharacteristic
14C3×#3675C411yescharacteristic, complemented
15C3×#3674C411yescharacteristic, complemented
16C3×#3673C411yescharacteristic, complemented
17C3×#3672C411yescharacteristic, complemented
18C3×#3667—21no—
19C3×#3667C2211yescharacteristic
20C2×C3×C9⋊C8—21no—
21C2×C3×C9⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23C3×#3663—21no—
24C3×C8×D9—21no—
25C3×#950—31no—
26C3×#948—31no—
27C3×#907—31no—
28C32×#104D311yescharacteristic
29C3×#822—31no—
30C3×#805—31no—
31C3×C4×C3⋊C8—31no—
32#63513C2×C311yescharacteristic
33#63510C2×C311yescharacteristic, complemented
34#63469C2×C311yescharacteristic, complemented
35C9×#104—21no—
36C9×#104C2×C311yescharacteristic, complemented
37#63381C2×C311yescharacteristic, complemented
38#63364C2×C311yescharacteristic, complemented
39C4×C9⋊C8C2×C311yescharacteristic, complemented
40C3×Q8×C9C2×C411yescharacteristic
41C3×D4×C9—21no—
42C3×D4×C9C2×C411yescharacteristic
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2311yescharacteristic
46C3×#611—21no—
47C3×D36C2×C411yescharacteristic
48C3×C4×D9—21no—
49C3×Dic18C2×C411yescharacteristic
50C3×C8×C9—21no—
51C3×C9⋊C8—21no—
52C3×C9⋊C8C2×C411yescharacteristic
53C3×C9⋊C8C2×C411yescharacteristic
54C3×C9⋊C8D411yescharacteristic
55C3×C9⋊C8D411yescharacteristic
56C3×#443—91no—
57#4824—31no—
58C32×#41C2×D311yescharacteristic
59C3×#235—31no—
60C32×M16C2×D311yescharacteristic
61C32×C42C2×D311yescharacteristic
62C3×#216—31no—
63C3×#215—31no—
64C3×#214—31no—
65C3×#213—31no—
66C3×#208—31no—
67C3×#208—61no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—61no—
71C3×ΓL(1,𝔽25)—61no—
72C3×D3×C8—61no—
73C9×#41—21no—
74C9×#41C22×C311yescharacteristic
75#3697C22×C311yescharacteristic
76C9×M16—21no—
77C9×M16C22×C311yescharacteristic
78C42×C9—21no—
79C42×C9C22×C311yescharacteristic
80#3675C3×C411yescharacteristic, complemented
81#3674C3×C411yescharacteristic, complemented
82#3673C3×C411yescharacteristic, complemented
83#3672C3×C411yescharacteristic, complemented
84#3667—21no—
85#3667C22×C311yescharacteristic
86C2×C9⋊C8—21no—
87C2×C9⋊C8C22×C311yescharacteristic
88C2×C9⋊C8C22×C311yescharacteristic
89#3663—21no—
90C8×D9—21no—
91C22×C3×C9—21no—
92C22×C3×C9#4111yescharacteristic
93C2×C3×D9—21no—
94C3×C4×C9—22no—
95C3×C4×C9—21no—
96C3×C4×C9C22×C411yescharacteristic
97C3×C4×C9C2×D411yescharacteristic
98C3×Dic9—21no—
99C3×#135—91no—
100C3×#131—91no—
101C3×#131—91no—
102#950—31no—
103#948—31no—
104#907—31no—
105C3×#108—91no—
106C3×#104—21no—
107C3×#104—91no—
108C3×#104D911yescharacteristic, complemented
109C3×#104C3×D311yescharacteristic
110C3×C4×C8—91no—
111#822—31no—
112#805—31no—
113C4×C3⋊C8—31no—
114C32×Q8C4×D311yescharacteristic
115C32×D4—21no—
116C32×D4C4×D311yescharacteristic
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C3×#67—61no—
121C3×D12—31no—
122C3×C4×D3—61no—
123C3×Dic6—31no—
124C32×C8—21no—
125C3×C3⋊C8—31no—
126C3×C3⋊C8—31no—
127C3×C3⋊C8—31no—
128C3×C3⋊C8—31no—
129C3×C3⋊C8—61no—
130Q8×C9—21no—
131Q8×C9C2×C3×C411yescharacteristic
132D4×C9—21no—
133D4×C9—21no—
134D4×C9—41no—
135D4×C9C2×C3×C411yescharacteristic
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—41no—
140C2×C4×C9—41no—
141C2×C4×C9C23×C311yescharacteristic
142#611—21no—
143D36C2×C3×C411yescharacteristic
144C4×D9—21no—
145Dic18C2×C3×C411yescharacteristic
146C8×C9—21no—
147C8×C9—41no—
148C9⋊C8—21no—
149C9⋊C8C2×C3×C411yescharacteristic
150C9⋊C8C2×C3×C411yescharacteristic
151C9⋊C8C3×D411yescharacteristic
152C9⋊C8C3×D411yescharacteristic
153#443—91noSylow
154C2×C3×C9—21no—
155C2×C3×C9—41no—
156C2×C3×C9C4×D411yescharacteristic
157C3×D9—41no—
158C3×#41—21no—
159C3×#41—91no—
160C3×#41C2×D911yescharacteristic
161C3×#41C2×C3×D311yescharacteristic
162#235—31no—
163C3×Q16—91no—
164C3×SD16—91no—
165C3×SD16—91no—
166C3×D8—91no—
167C3×M16—21no—
168C3×M16—91no—
169C3×M16—181no—
170C3×M16—181no—
171C3×M16C2×D911yescharacteristic
172C3×M16C2×C3×D311yescharacteristic
173C2×C3×C8—91no—
174C2×C3×C8—91no—
175C2×C3×C8—181no—
176C2×C3×C8—181no—
177C3×C42—21no—
178C3×C42C2×D911yescharacteristic
179C3×C42C2×C3×D311yescharacteristic
180#216—31no—
181#215—31no—
182#214—31no—
183#213—31no—
184#208—31no—
185#208—61no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—61no—
189ΓL(1,𝔽25)—61no—
190D3×C8—61no—
191C22×C32—21no—
192C22×C32#23711yescharacteristic
193C2×C3×D3—61no—
194C32×C4—22no—
195C32×C4—21no—
196C32×C4C2×C4×D311yescharacteristic
197C32×C4D3×D411yescharacteristic
198C3×Dic3—61no—
199C22×C9—21no—
200C22×C9—21no—
201C22×C9—41no—
202C22×C9C3×#4111yescharacteristic
203C2×D9—21no—
204C4×C9—22no—
205C4×C9—21no—
206C4×C9—21no—
207C4×C9—21no—
208C4×C9—41no—
209C4×C9—42no—
210C4×C9C22×C3×C411yescharacteristic
211C4×C9C2×C3×D411yescharacteristic
212Dic9—21no—
213#135—91no—
214#131—91no—
215#131—91no—
216#108—91no—
217#104—91no—
218#104C3×D911yescharacteristic, complemented
219C4×C8—91no—
220C3×C9#44311yescharacteristic, complemented, Sylow
221C3×Q8—21no—
222C3×Q8—91no—
223C3×Q8C4×D911yescharacteristic
224C3×Q8C3×C4×D311yescharacteristic
225C3×D4—21no—
226C3×D4—21no—
227C3×D4—21no—
228C3×D4—41no—
229C3×D4—91no—
230C3×D4—181no—
231C3×D4C4×D911yescharacteristic
232C3×D4C3×C4×D311yescharacteristic
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—181no—
241C2×C3×C4C22×D911yescharacteristic
242C2×C3×C4C22×C3×D311yescharacteristic
243#67—61no—
244D12—31no—
245C4×D3—61no—
246Dic6—31no—
247C3×C8—21no—
248C3×C8—21no—
249C3×C8—41no—
250C3×C8—91no—
251C3×C8—91no—
252C3×C8—91no—
253C3×C8—91no—
254C3×C8—181no—
255C3⋊C8—31no—
256C3⋊C8—31no—
257C3⋊C8—31no—
258C3⋊C8—31no—
259C3⋊C8—61no—
260C2×C32—21no—
261C2×C32—41no—
262C2×C32#67⋊C411yescharacteristic
263C3×D3—121no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9—41no—
267C2×C9—41no—
268C2×C9—81no—
269C2×C9C3×C4×D411yescharacteristic
270D9—41no—
271#41—91no—
272#41C2×C3×D911yescharacteristic
273Q16—91no—
274SD16—91no—
275SD16—91no—
276D8—91no—
277M16—91no—
278M16—181no—
279M16—181no—
280M16C2×C3×D911yescharacteristic
281C2×C8—91no—
282C2×C8—91no—
283C2×C8—181no—
284C2×C8—181no—
285C42C2×C3×D911yescharacteristic
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—41no—
290C22×C3—181no—
291C22×C3#369911yescharacteristic
292C22×C3C3×#23711yescharacteristic
293C2×D3—61no—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—41no—
301C3×C4—42no—
302C3×C4—181no—
303C3×C4C2×C4×D911yescharacteristic
304C3×C4D4×D911yescharacteristic
305C3×C4C2×C3×C4×D311yescharacteristic
306C3×C4C3×D3×D411yescharacteristic
307Dic3—61no—
308C32#482411yescharacteristic
309C9—21no—
310C9C3×#44311yescharacteristic, complemented
311Q8—91no—
312Q8C3×C4×D911yescharacteristic
313D4—21no—
314D4—91no—
315D4—181no—
316D4C3×C4×D911yescharacteristic
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—181no—
320C2×C4C22×C3×D911yescharacteristic
321C8—21no—
322C8—91no—
323C8—91no—
324C8—91no—
325C8—91no—
326C8—181no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—41no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—81no—
334C2×C3—361no—
335C2×C3#611⋊C411yescharacteristic
336C2×C3C3×#67⋊C411yescharacteristic
337D3—121no—
338C22—21no—
339C22—181no—
340C22C3×#369911yescharacteristic
341C4—22no—
342C4—21no—
343C4—181no—
344C4C2×C3×C4×D911yescharacteristic
345C4C3×D4×D911yescharacteristic
346C3—21no—
347C3#1058850411yescharacteristic, direct product factor
348C3C3×#482411yescharacteristic
349C2—21no—
350C2—41no—
351C2—361no—
352C2C3×#611⋊C411yescharacteristic
subgroupG/H
centerC3×C4D4×D9
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#104C2
socleC2×C32#67⋊C4
prime radicalC3×#63513C2
cosocleC2×C4×C9C23×C3
hypercenterC3×#104D9
hypocenterC9C3×#443
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×#104

Complements

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

NH ≅ G/Ntype
2. C3×#63510C2semidirect
3. C3×#63469C2semidirect
4. C3×C9×#104C2semidirect
5. C3×#63381C2semidirect
6. C3×#63364C2semidirect
7. C3×C4×C9⋊C8C2semidirect
9. #10588504C3direct (N × H)
14. C3×#3675C4semidirect
15. C3×#3674C4semidirect
16. C3×#3673C4semidirect
17. C3×#3672C4semidirect
33. #63510C2×C3semidirect
34. #63469C2×C3semidirect
36. C9×#104C2×C3semidirect
37. #63381C2×C3semidirect
38. #63364C2×C3semidirect
39. C4×C9⋊C8C2×C3semidirect
80. #3675C3×C4semidirect
81. #3674C3×C4semidirect
82. #3673C3×C4semidirect
83. #3672C3×C4semidirect
108. C3×#104D9semidirect
218. #104C3×D9semidirect
220. C3×C9#443semidirect
310. C9C3×#443semidirect
347. C3#10588504direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#635132
C3×#635102
C3×#634692
C3×C9×#1042
C3×#633812
C3×#633642
C3×C4×C9⋊C82
C3×#48243
#105885043