Finite Group Databasebeta

← C3×#10588731 #13142858

Subgroups — C3×#10588731 #13142858

Order 1728.

446 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#3675C211yesmaximal, characteristic, complemented
3C3×#63492C211yesmaximal, characteristic
4C22×C3×C9⋊C8C211yesmaximal, characteristic, complemented
5C3×#63395C211yesmaximal, characteristic, complemented
6C3×#63369C211yesmaximal, characteristic, complemented
7C3×#63367C211yesmaximal, characteristic, complemented
8C3×#5048—31nomaximal
9#10588731C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C2×C3×#611—21no—
13C2×C3×D36C2211yescharacteristic
14C3×C9×C4⋊C4—21no—
15C3×C9×C4⋊C4C2211yescharacteristic
16C3×C9×C22⋊C4—21no—
17C3×#3675—22no—
18C3×(C2×D9)⋊C4—21no—
19C3×C4⋊Dic9C2211yescharacteristic
20C2×C3×C9⋊C8—22no—
21C2×C3×C9⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23C32×C22⋊Q8D311yescharacteristic
24C2×C3×#215—31no—
25C3×#930—31no—
26C22×C3×C3⋊C8—31no—
27C3×#835—31no—
28C3×#809—31no—
29C3×#808—31no—
30C9×C22⋊Q8—21no—
31C9×C22⋊Q8C2×C311yescharacteristic, complemented
32C2×#3675C2×C311yescharacteristic, complemented
33#63492C2×C311yescharacteristic
34C22×C9⋊C8C2×C311yescharacteristic, complemented
35#63395C2×C311yescharacteristic, complemented
36#63369C2×C311yescharacteristic, complemented
37#63367C2×C311yescharacteristic, complemented
38C23×C3×C9D411yescharacteristic
39C22×C3×D9—21no—
40C3×Q8×C9—21no—
41C3×Q8×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9D411yescharacteristic
47C2×C3×C4×C9C2311yescharacteristic
48C3×#611—42no—
49C2×C3×Dic9—21no—
50C3×D36—21no—
51C3×D36—21no—
52C3×C9⋊C8—21no—
53C3×C9⋊C8D412yescomplemented
54C3×#465—91no—
55#5048—31no—
56C2×C32×Q8C2×D311yescharacteristic
57C22×C32×C4C2×D311yescharacteristic
58C2×C3×#67—61no—
59C2×C3×D12—31no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C22⋊C4—21no—
63C3×#215—62no—
64C3×(C2×D3)⋊C4—61no—
65C3×C4⋊Dic3—31no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—62no—
69C2×Q8×C9—21no—
70C2×Q8×C9C22×C311yescharacteristic
71C22×C4×C9—21no—
72C22×C4×C9C22×C311yescharacteristic
73C2×#611—21no—
74C2×D36C22×C311yescharacteristic
75C9×C4⋊C4—22no—
76C9×C4⋊C4—21no—
77C9×C4⋊C4—21no—
78C9×C4⋊C4C22×C311yescharacteristic
79C9×C22⋊C4—22no—
80C9×C22⋊C4—21no—
81#3675—22no—
82(C2×D9)⋊C4—21no—
83C4⋊Dic9C22×C311yescharacteristic
84C2×C9⋊C8—22no—
85C2×C9⋊C8C22×C311yescharacteristic
86C2×C9⋊C8C22×C311yescharacteristic
87C22×C3×C9—21no—
88C22×C3×C9—21no—
89C22×C3×C9SD1612yescomplemented
90C22×C3×C9C2×D411yescharacteristic
91C2×C3×D9—41no—
92C2×C3×D9—41no—
93C2×C3×D9—41no—
94C3×C4×C9—21no—
95C3×C4×C9—41no—
96C3×C4×C9—41no—
97C3×C4×C9C2×D411yescharacteristic
98C3×C4×C9#4111yescharacteristic
99C3×Dic9—41no—
100C2×C3×SD16—91no—
101C22×C3×C8—91no—
102C3×C22⋊Q8—21no—
103C3×C22⋊Q8D911yescharacteristic, complemented
104C3×C22⋊Q8C3×D311yescharacteristic
105C3×#121—91no—
106C2×#215—31no—
107#930—31no—
108C22×C3⋊C8—31no—
109C3×#106—91no—
110C3×#103—91no—
111C3×#102—91no—
112#835—31no—
113#809—31no—
114#808—31no—
115C23×C32#6711yescharacteristic
116C22×C3×D3—61no—
117C32×Q8—21no—
118C32×Q8—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4#6711yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125C3×#67—122no—
126C2×C3×Dic3—61no—
127C3×D12—61no—
128C3×D12—61no—
129C23×C9—21no—
130C23×C9C3×D411yescharacteristic
131C22×D9—21no—
132C3×C3⋊C8—32no—
133C3×C3⋊C8—61no—
134Q8×C9—21no—
135Q8×C9—21no—
136Q8×C9—41no—
137Q8×C9—41no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—22no—
141C2×C4×C9—22no—
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9—21no—
145C2×C4×C9—21no—
146C2×C4×C9—41no—
147C2×C4×C9—41no—
148C2×C4×C9C3×D411yescharacteristic
149C2×C4×C9C23×C311yescharacteristic
150#611—42no—
151C2×Dic9—21no—
152D36—21no—
153D36—21no—
154C9⋊C8—21no—
155C9⋊C8C3×D412yescomplemented
156#465—91noSylow
157C2×C3×C9—22no—
158C2×C3×C9#12111yescharacteristic
159C2×C3×C9C2×SD1611yescharacteristic
160C2×C3×C9#13511yescharacteristic
161C3×D9—81no—
162C2×C3×Q8—21no—
163C2×C3×Q8C2×D911yescharacteristic
164C2×C3×Q8C2×C3×D311yescharacteristic
165C2×C3×D4—91no—
166C2×C3×D4—181no—
167C22×C3×C4—21no—
168C22×C3×C4C2×D911yescharacteristic
169C22×C3×C4C2×C3×D311yescharacteristic
170C2×#67—61no—
171C2×D12—31no—
172C3×SD16—182no—
173C2×C3×C8—91no—
174C2×C3×C8—91no—
175C2×C3×C8—182no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—91no—
181C3×C4⋊C4C2×D911yescharacteristic
182C3×C4⋊C4C2×C3×D311yescharacteristic
183C3×C22⋊C4—22no—
184C3×C22⋊C4—21no—
185C3×C22⋊C4—21no—
186C3×C22⋊C4—181no—
187#215—62no—
188(C2×D3)⋊C4—61no—
189C4⋊Dic3—31no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—31no—
192C2×C3⋊C8—62no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32#21512yes—
196C22×C32C2×#6711yescharacteristic
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C32×C4—21no—
201C32×C4—41no—
202C32×C4—41no—
203C32×C4D3×D411yescharacteristic
204C32×C4#23711yescharacteristic
205C3×Dic3—121no—
206C22×C9—21no—
207C22×C9—21no—
208C22×C9—22no—
209C22×C9—21no—
210C22×C9—41no—
211C22×C9—41no—
212C22×C9C3×SD1612yescomplemented
213C22×C9C2×C3×D411yescharacteristic
214C2×D9—41no—
215C2×D9—41no—
216C2×D9—41no—
217C4×C9—21no—
218C4×C9—21no—
219C4×C9—21no—
220C4×C9—41no—
221C4×C9—42no—
222C4×C9—42no—
223C4×C9—41no—
224C4×C9—41no—
225C4×C9C2×C3×D411yescharacteristic
226C4×C9C3×#4111yescharacteristic
227Dic9—41no—
228C2×SD16—91no—
229C22×C8—91no—
230C22⋊Q8C3×D911yescharacteristic, complemented
231#121—91no—
232#106—91no—
233#103—91no—
234#102—91no—
235C3×C9#46511yescharacteristic, complemented, Sylow
236C23×C3—21no—
237C23×C3—181no—
238C23×C3#61111yescharacteristic
239C23×C3C3×#6711yescharacteristic
240C22×D3—61no—
241C3×Q8—21no—
242C3×Q8—21no—
243C3×Q8—21no—
244C3×Q8—21no—
245C3×Q8—41no—
246C3×Q8—41no—
247C3×D4—181no—
248C3×D4—181no—
249C3×D4—362no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—181no—
265C2×C3×C4#61111yescharacteristic
266C2×C3×C4C22×D911yescharacteristic
267C2×C3×C4C3×#6711yescharacteristic
268C2×C3×C4C22×C3×D311yescharacteristic
269#67—122no—
270C2×Dic3—61no—
271D12—61no—
272D12—61no—
273C3×C8—92no—
274C3×C8—181no—
275C3⋊C8—32no—
276C3⋊C8—61no—
277C2×C32—22no—
278C2×C32#93911yescharacteristic
279C2×C32C2×#21511yescharacteristic
280C2×C32#95011yescharacteristic
281C3×D3—241no—
282C2×C9—22no—
283C2×C9—21no—
284C2×C9—21no—
285C2×C9—21no—
286C2×C9—42no—
287C2×C9C3×#12111yescharacteristic
288C2×C9C2×C3×SD1611yescharacteristic
289C2×C9C3×#13511yescharacteristic
290D9—81no—
291C2×Q8C2×C3×D911yescharacteristic
292C2×D4—91no—
293C2×D4—181no—
294C22×C4C2×C3×D911yescharacteristic
295SD16—182no—
296C2×C8—91no—
297C2×C8—91no—
298C2×C8—182no—
299C4⋊C4—21no—
300C4⋊C4—91no—
301C4⋊C4C2×C3×D911yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—181no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—22no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—361no—
313C22×C3—361no—
314C22×C3—361no—
315C22×C3#367512yescomplemented
316C22×C3C2×#61111yescharacteristic
317C22×C3C3×#21512yes—
318C22×C3C2×C3×#6711yescharacteristic
319C2×D3—121no—
320C2×D3—121no—
321C2×D3—121no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—41no—
327C3×C4—42no—
328C3×C4—42no—
329C3×C4—41no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—361no—
334C3×C4D4×D911yescharacteristic
335C3×C4#369911yescharacteristic
336C3×C4C3×D3×D411yescharacteristic
337C3×C4C3×#23711yescharacteristic
338Dic3—121no—
339C32#504811yescharacteristic
340C9—21no—
341C9C3×#46511yescharacteristic, complemented
342C23—181no—
343C23C3×#61111yescharacteristic
344Q8—21no—
345Q8—21no—
346D4—181no—
347D4—181no—
348D4—362no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—181no—
354C2×C4C3×#61111yescharacteristic
355C2×C4C22×C3×D911yescharacteristic
356C8—92no—
357C8—181no—
358C2×C3—22no—
359C2×C3—22no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—42no—
364C2×C3—721no—
365C2×C3#6350111yescharacteristic
366C2×C3C2×#367511yescharacteristic
367C2×C3#6351311yescharacteristic
368C2×C3C3×#93911yescharacteristic
369C2×C3C2×C3×#21511yescharacteristic
370C2×C3C3×#95011yescharacteristic
371D3—241no—
372C22—21no—
373C22—21no—
374C22—361no—
375C22—361no—
376C22—361no—
377C22C3×#367512yescomplemented
378C22C2×C3×#61111yescharacteristic
379C4—21no—
380C4—41no—
381C4—41no—
382C4—361no—
383C4C3×D4×D911yescharacteristic
384C4C3×#369911yescharacteristic
385C3—21no—
386C3#1058873111yescharacteristic, direct product factor
387C3C3×#504811yescharacteristic
388C2—22no—
389C2—721no—
390C2C3×#6350111yescharacteristic
391C2C2×C3×#367511yescharacteristic
392C2C3×#6351311yescharacteristic
subgroupG/H
centerC22×C3C2×#611
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C2×#67
prime radicalC3×#63492C2
hypercenterC3×C22⋊Q8D9
hypocenterC9C3×#465
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C9×C22⋊Q8C2semidirect
2. C2×C3×#3675C2semidirect
4. C22×C3×C9⋊C8C2semidirect
5. C3×#63395C2semidirect
6. C3×#63369C2semidirect
7. C3×#63367C2semidirect
9. #10588731C3direct (N × H)
31. C9×C22⋊Q8C2×C3semidirect
32. C2×#3675C2×C3semidirect
34. C22×C9⋊C8C2×C3semidirect
35. #63395C2×C3semidirect
36. #63369C2×C3semidirect
37. #63367C2×C3semidirect
53. C3×C9⋊C8D4semidirect
89. C22×C3×C9SD16semidirect
103. C3×C22⋊Q8D9semidirect
155. C9⋊C8C3×D4semidirect
212. C22×C9C3×SD16semidirect
230. C22⋊Q8C3×D9semidirect
235. C3×C9#465semidirect
315. C22×C3#3675semidirect
341. C9C3×#465semidirect
377. C22C3×#3675semidirect
386. C3#10588731direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×C22⋊Q82
C2×C3×#36752
C3×#634922
C22×C3×C9⋊C82
C3×#633952
C3×#633692
C3×#633672
C3×#50483
#105887313