Finite Group Databasebeta

← C3×#10588711 #13142780

Subgroups — C3×#10588711 #13142780

Order 1728.

406 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×Q8×C9C211yesmaximal, characteristic, complemented
2C2×C3×#3675C211yesmaximal, characteristic
3C2×C3×#3674C211yesmaximal, characteristic, complemented
4C3×#63437C211yesmaximal, characteristic
5C3×#63369C211yesmaximal, characteristic
6C3×#63368C211yesmaximal, characteristic, complemented
7C3×C4⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5028—31nomaximal
9#10588711C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C3×C9×C4⋊C4—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×C42×C9—21no—
16C3×C42×C9C2211yescharacteristic
17C3×#3675—22no—
18C3×#3674—22no—
19C3×(C2×D9)⋊C4—21no—
20C3×(C2×D9)⋊C4—21no—
21C2×C3×C9⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23C32×C4×Q8D311yescharacteristic
24C2×C3×#216—31no—
25C2×C3×#215—31no—
26C3×#875—31no—
27C3×#810—31no—
28C3×#809—31no—
29C3×C4⋊(C3⋊C8)—31no—
30C4×Q8×C9—21no—
31C4×Q8×C9C2×C311yescharacteristic, complemented
32C2×#3675C2×C311yescharacteristic
33C2×#3674C2×C311yescharacteristic, complemented
34#63437C2×C311yescharacteristic
35#63369C2×C311yescharacteristic
36#63368C2×C311yescharacteristic, complemented
37C4⋊(C9⋊C8)C2×C311yescharacteristic, complemented
38C22×C3×D9—21no—
39C3×Q8×C9—21no—
40C3×Q8×C9D412yes—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9D411yescharacteristic
44C2×C3×C4×C9D411yescharacteristic
45C2×C3×C4×C9C2311yescharacteristic
46C2×C3×Dic9—21no—
47C3×D36—21no—
48C3×D36—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51C3×C9⋊C8—21no—
52C3×C9⋊C8—21no—
53C3×#464—91no—
54#5028—31no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C3×D12—31no—
57C2×C3×Dic6—31no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4C2×D311yescharacteristic
60C32×C42—21no—
61C32×C42C2×D311yescharacteristic
62C3×#216—62no—
63C3×#215—62no—
64C3×(C2×D3)⋊C4—61no—
65C3×(C2×D3)⋊C4—61no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×Q8×C9—21no—
69C2×Q8×C9C22×C311yescharacteristic
70C2×D36C22×C311yescharacteristic
71C2×Dic18C22×C311yescharacteristic
72C9×C4⋊C4—22no—
73C9×C4⋊C4—21no—
74C9×C4⋊C4—21no—
75C9×C4⋊C4C22×C311yescharacteristic
76C42×C9—22no—
77C42×C9—21no—
78C42×C9—21no—
79C42×C9C22×C311yescharacteristic
80#3675—22no—
81#3674—22no—
82(C2×D9)⋊C4—21no—
83(C2×D9)⋊C4—21no—
84C2×C9⋊C8C22×C311yescharacteristic
85C2×C9⋊C8C22×C311yescharacteristic
86C22×C3×C9C2×D411yescharacteristic
87C2×C3×D9—41no—
88C2×C3×D9—41no—
89C2×C3×D9—41no—
90C3×C4×C9—22no—
91C3×C4×C9—21no—
92C3×C4×C9—21no—
93C3×C4×C9—41no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C3×Dic9—41no—
97C2×C3×Q16—91no—
98C2×C3×SD16—91no—
99C3×#124—91no—
100C3×C4×Q8—21no—
101C3×C4×Q8D911yescharacteristic, complemented
102C3×C4×Q8C3×D311yescharacteristic
103C2×#216—31no—
104C2×#215—31no—
105#875—31no—
106C3×C4⋊C8—91no—
107C3×#103—91no—
108C3×#102—91no—
109#810—31no—
110#809—31no—
111C4⋊(C3⋊C8)—31no—
112C22×C3×D3—61no—
113C32×Q8—21no—
114C32×Q8D1212yes—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4#6711yescharacteristic
118C2×C32×C4#6711yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C3×Dic3—61no—
121C3×D12—61no—
122C3×D12—61no—
123C3×Dic6—61no—
124C3×Dic6—61no—
125C22×D9—21no—
126C3×C3⋊C8—61no—
127C3×C3⋊C8—61no—
128Q8×C9—22no—
129Q8×C9—21no—
130Q8×C9—22no—
131Q8×C9C3×D412yes—
132C2×C4×C9—22no—
133C2×C4×C9—21no—
134C2×C4×C9—22no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9C3×D411yescharacteristic
140C2×C4×C9C3×D411yescharacteristic
141C2×C4×C9C23×C311yescharacteristic
142C2×Dic9—21no—
143D36—21no—
144D36—21no—
145Dic18—21no—
146Dic18—21no—
147C9⋊C8—21no—
148C9⋊C8—21no—
149#464—91noSylow
150C2×C3×C9#12111yescharacteristic
151C2×C3×C9#13511yescharacteristic
152C2×C3×C9#13711yescharacteristic
153C3×D9—81no—
154C2×C3×Q8—21no—
155C2×C3×Q8—91no—
156C2×C3×Q8C2×D911yescharacteristic
157C2×C3×Q8C2×C3×D311yescharacteristic
158C2×C3×D4—91no—
159C2×D12—31no—
160C2×Dic6—31no—
161C3×Q16—182no—
162C3×SD16—182no—
163C2×C3×C8—91no—
164C2×C3×C8—91no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4C2×D911yescharacteristic
170C3×C4⋊C4C2×C3×D311yescharacteristic
171C3×C22⋊C4—181no—
172C3×C22⋊C4—181no—
173C3×C42—22no—
174C3×C42—21no—
175C3×C42—21no—
176C3×C42—21no—
177C3×C42C2×D911yescharacteristic
178C3×C42C2×C3×D311yescharacteristic
179#216—62no—
180#215—62no—
181(C2×D3)⋊C4—61no—
182(C2×D3)⋊C4—61no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—31no—
185C22×C32C2×#6711yescharacteristic
186C2×C3×D3—121no—
187C2×C3×D3—121no—
188C2×C3×D3—121no—
189C32×C4—22no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—41no—
193C32×C4C2×D1211yescharacteristic
194C32×C4#23511yescharacteristic
195C3×Dic3—121no—
196C22×C9—21no—
197C22×C9C2×C3×D411yescharacteristic
198C2×D9—41no—
199C2×D9—41no—
200C2×D9—41no—
201C4×C9—22no—
202C4×C9—24no—
203C4×C9—22no—
204C4×C9—21no—
205C4×C9—21no—
206C4×C9—21no—
207C4×C9—21no—
208C4×C9—41no—
209C4×C9—42no—
210C4×C9—41no—
211C4×C9C2×C3×D411yescharacteristic
212C4×C9C3×#4111yescharacteristic
213Dic9—41no—
214C2×Q16—91no—
215C2×SD16—91no—
216#124—91no—
217C4×Q8C3×D911yescharacteristic, complemented
218C4⋊C8—91no—
219#103—91no—
220#102—91no—
221C3×C9#46411yescharacteristic, complemented, Sylow
222C23×C3—181no—
223C22×D3—61no—
224C3×Q8—22no—
225C3×Q8—21no—
226C3×Q8—21no—
227C3×Q8—22no—
228C3×Q8—181no—
229C3×Q8—181no—
230C3×Q8D3612yes—
231C3×Q8C3×D1212yes—
232C3×D4—181no—
233C3×D4—181no—
234C2×C3×C4—22no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—181no—
244C2×C3×C4#61111yescharacteristic
245C2×C3×C4#61111yescharacteristic
246C2×C3×C4C22×D911yescharacteristic
247C2×C3×C4C3×#6711yescharacteristic
248C2×C3×C4C3×#6711yescharacteristic
249C2×C3×C4C22×C3×D311yescharacteristic
250C2×Dic3—61no—
251D12—61no—
252D12—61no—
253Dic6—61no—
254Dic6—61no—
255C3×C8—181no—
256C3×C8—181no—
257C3⋊C8—61no—
258C3⋊C8—61no—
259C2×C32#93011yescharacteristic
260C2×C32#94211yescharacteristic
261C2×C32#95011yescharacteristic
262C3×D3—241no—
263C2×C9—21no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9C3×#12111yescharacteristic
267C2×C9C3×#13511yescharacteristic
268C2×C9C3×#13711yescharacteristic
269D9—81no—
270C2×Q8—91no—
271C2×Q8C2×C3×D911yescharacteristic
272C2×D4—91no—
273Q16—182no—
274SD16—182no—
275C2×C8—91no—
276C2×C8—91no—
277C4⋊C4—21no—
278C4⋊C4C2×C3×D911yescharacteristic
279C22⋊C4—181no—
280C22⋊C4—181no—
281C42—21no—
282C42C2×C3×D911yescharacteristic
283C22×C3—21no—
284C22×C3—361no—
285C22×C3—361no—
286C22×C3—361no—
287C22×C3C2×#61111yescharacteristic
288C22×C3C2×C3×#6711yescharacteristic
289C2×D3—121no—
290C2×D3—121no—
291C2×D3—121no—
292C3×C4—24no—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—22no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—41no—
303C3×C4—42no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—361no—
307C3×C4C2×D3611yescharacteristic
308C3×C4#369711yescharacteristic
309C3×C4C2×C3×D1211yescharacteristic
310C3×C4C3×#23511yescharacteristic
311Dic3—121no—
312C32#502811yescharacteristic
313C9—21no—
314C9C3×#46411yescharacteristic, complemented
315C23—181no—
316Q8—21no—
317Q8—181no—
318Q8—181no—
319Q8C3×D3612yes—
320D4—181no—
321D4—181no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—181no—
325C2×C4C3×#61111yescharacteristic
326C2×C4C3×#61111yescharacteristic
327C2×C4C22×C3×D911yescharacteristic
328C8—181no—
329C8—181no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—721no—
334C2×C3#6349211yescharacteristic
335C2×C3#6350511yescharacteristic
336C2×C3#6351311yescharacteristic
337C2×C3C3×#93011yescharacteristic
338C2×C3C3×#94211yescharacteristic
339C2×C3C3×#95011yescharacteristic
340D3—241no—
341C22—361no—
342C22—361no—
343C22—361no—
344C22C2×C3×#61111yescharacteristic
345C4—22no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—361no—
350C4C2×C3×D3611yescharacteristic
351C4C3×#369711yescharacteristic
352C3—21no—
353C3#1058871111yescharacteristic, direct product factor
354C3C3×#502811yescharacteristic
355C2—721no—
356C2C3×#6349211yescharacteristic
357C2C3×#6350511yescharacteristic
358C2C3×#6351311yescharacteristic
subgroupG/H
centerC22×C3C2×#611
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C4×Q8×C9C2
socleC22×C32C2×#67
prime radicalC2×C3×D36C22
hypercenterC3×C4×Q8D9
hypocenterC9C3×#464
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C3×C4×Q8×C9C2semidirect
3. C2×C3×#3674C2semidirect
6. C3×#63368C2semidirect
7. C3×C4⋊(C9⋊C8)C2semidirect
9. #10588711C3direct (N × H)
31. C4×Q8×C9C2×C3semidirect
33. C2×#3674C2×C3semidirect
36. #63368C2×C3semidirect
37. C4⋊(C9⋊C8)C2×C3semidirect
101. C3×C4×Q8D9semidirect
217. C4×Q8C3×D9semidirect
221. C3×C9#464semidirect
314. C9C3×#464semidirect
353. C3#10588711direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×Q8×C92
C2×C3×#36752
C2×C3×#36742
C3×#634372
C3×#633692
C3×#633682
C3×C4⋊(C9⋊C8)2
C3×#50283
#105887113