Finite Group Databasebeta

← #13140771

Subgroups — #13140771

Order 1728.

516 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#239C211yesmaximal, characteristic
2#11720906C211yesmaximal, characteristic, complemented
3C2×D3×C9⋊C8C211yesmaximal, characteristic, complemented
4#11720582C211yesmaximal, characteristic, complemented
5C3×#63395C211yesmaximal, characteristic, complemented
6#11720295C211yesmaximal, characteristic, complemented
7#11720294C211yesmaximal, characteristic, complemented
8#10591556—31nomaximal
9#10588917—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C9×#239—22no—
12C9×#239C414yescomplemented
13C2×C9×D12—21no—
14C2×C9×D12C2211yescharacteristic
15C2×C4×D3×C9—21no—
16C2×C4×D3×C9C2211yescharacteristic
17C4⋊(C3⋊Dic9)C2211yescharacteristic
18C2×C3⋊(C9⋊C8)C2211yescharacteristic
19C3×C4⋊Dic9C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21(C2×D3)⋊Dic9—21no—
22Dic3×Dic9—21no—
23D3×C9⋊C8—22no—
24C2×C3×#239D311yescharacteristic
25#63841—31no—
26C2×D3×C3⋊C8—31no—
27C2×C9×#41—31no—
28#63662—31no—
29C3×#835—31no—
30#63563—31no—
31#63562—31no—
32#63488—31no—
33C22×C9⋊C8—31no—
34#63395—31no—
35#63395—61no—
36#63395D311yescharacteristic, complemented
37#63392—31no—
38#63392—31no—
39C22×D3×C9—21no—
40C22×D3×C9D411yescharacteristic
41C3×Q8×C9—21no—
42C3×Q8×C9C2×C412yes—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46C2×C9×Dic3D411yescharacteristic
47C2×C3×Dic9—21no—
48C9×D12—24no—
49C9×D12—21no—
50C9×D12C2×C412yes—
51C4×D3×C9—24no—
52C4×D3×C9C2×C412yes—
53C4×D3×C9D412yes—
54C3⋊(C9⋊C8)—21no—
55C3×C9⋊C8—21no—
56#5233—91no—
57#4804—91no—
58C2×C32×Q8C2×D311yescharacteristic
59C3×#239—22no—
60C3×#239Dic314yes—
61C2×C3×D12—21no—
62C2×C3×D12C2×D311yescharacteristic
63C2×C3×C4×D3—21no—
64C2×C3×C4×D3C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)—31no—
66C2×C3⋊(C3⋊C8)—31no—
67C3×C4⋊Dic3—31no—
68C2×C3×C3⋊C8—31no—
69(C2×D3)⋊Dic3—61no—
70Dic32—61no—
71D3×C3⋊C8—62no—
72C9×#41—34no—
73C9×#41—62no—
74C2×Q8×C9—21no—
75C2×Q8×C9C2×D311yescharacteristic
76C2×D4×C9—31no—
77C2×D4×C9—61no—
78C22×C4×C9—31no—
79C22×C4×C9—61no—
80#3676—61no—
81C4⋊Dic9—31no—
82C4⋊Dic9—61no—
83C4⋊Dic9C2×D311yescharacteristic
84C4×Dic9—61no—
85C2×C9⋊C8—31no—
86C2×C9⋊C8—62no—
87C2×C9⋊C8—61no—
88C2×C9⋊C8C2×D311yescharacteristic
89C22×C3×C9C2×D411yescharacteristic
90C2×D3×C9—21no—
91C2×D3×C9—21no—
92C2×D3×C9—22no—
93C2×D3×C9—41no—
94C2×D3×C9—41no—
95C2×D3×C9C22⋊C412yes—
96C3×C4×C9—22no—
97C3×C4×C9C22×C411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C3⋊Dic9—41no—
100C9×Dic3C22⋊C412yes—
101C3×Dic9—41no—
102C2×C3×#41—31no—
103C2×#239D911yescharacteristic
104#926—91no—
105C22×C3⋊C8—91no—
106C2×D3×C8—91no—
107#892—91no—
108C3×#103—91no—
109#835—31no—
110#835—91no—
111#835—91no—
112#835—181no—
113#832—91no—
114#832—91no—
115#821—91no—
116#809—91no—
117C22×C3×D3—21no—
118C22×C3×D3#6711yescharacteristic
119C32×Q8—21no—
120C32×Q8C2×Dic312yes—
121C2×C32×C4—21no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3#6711yescharacteristic
126C3×D12—24no—
127C3×D12—21no—
128C3×D12C2×Dic312yes—
129C3×C4×D3—24no—
130C3×C4×D3C2×Dic312yes—
131C3×C4×D3#6712yes—
132C23×C9—31no—
133C23×C9—61no—
134C3⋊(C3⋊C8)—61no—
135C3×C3⋊C8—61no—
136Q8×C9—21no—
137Q8×C9—22no—
138Q8×C9—41no—
139Q8×C9C4×D312yes—
140D4×C9—32no—
141D4×C9—64no—
142D4×C9—61no—
143C2×C4×C9—21no—
144C2×C4×C9—21no—
145C2×C4×C9—31no—
146C2×C4×C9—32no—
147C2×C4×C9—32no—
148C2×C4×C9—41no—
149C2×C4×C9—64no—
150C2×C4×C9C22×D311yescharacteristic
151C2×Dic9—21no—
152C2×Dic9—61no—
153C2×Dic9—121no—
154C9⋊C8—21no—
155C9⋊C8—61no—
156C9⋊C8—121no—
157#416—271noSylow
158C2×C3×C9C2×C22⋊C411yescharacteristic
159C2×C3×C9#13511yescharacteristic
160C2×C3×C9#13511yescharacteristic
161D3×C9—22no—
162D3×C9—42no—
163C3×#41—34no—
164C3×#41—62no—
165C2×C3×Q8—21no—
166C2×C3×Q8C2×D911yescharacteristic
167C2×C3×Q8D3211yescharacteristic
168C2×C3×D4—31no—
169C2×C3×D4—61no—
170C22×C3×C4—31no—
171C22×C3×C4—61no—
172#239—22no—
173#239Dic914yescomplemented
174C2×D12—21no—
175C2×D12C2×D911yescharacteristic
176C2×C4×D3—21no—
177C2×C4×D3C2×D911yescharacteristic
178C2×C3×C8—91no—
179C3×C4⋊C4—91no—
180#217—181no—
181(C2×D3)⋊C4—181no—
182C4⋊Dic3—31no—
183C4⋊Dic3—91no—
184C4⋊Dic3—91no—
185C4⋊Dic3—181no—
186C4×Dic3—181no—
187C4×Dic3—181no—
188C2×C3⋊C8—31no—
189C2×C3⋊C8—91no—
190C2×C3⋊C8—91no—
191C2×C3⋊C8—181no—
192C2×C3⋊C8—182no—
193D3×C8—182no—
194C22×C32C2×#6711yescharacteristic
195C2×C3×D3—21no—
196C2×C3×D3—21no—
197C2×C3×D3—22no—
198C2×C3×D3—41no—
199C2×C3×D3—41no—
200C2×C3×D3#21712yes—
201C32×C4—22no—
202C32×C4C22×Dic311yescharacteristic
203C32×C4C2×#6711yescharacteristic
204C3⋊Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3#21712yes—
207C22×C9—21no—
208C22×C9—32no—
209C22×C9—61no—
210C22×C9—61no—
211C22×C9—62no—
212C22×C9—121no—
213C22×C9—121no—
214C22×C9D3×D411yescharacteristic
215C4×C9—22no—
216C4×C9—21no—
217C4×C9—21no—
218C4×C9—32no—
219C4×C9—42no—
220C4×C9C2×C4×D311yescharacteristic
221C4×C9D3×D411yescharacteristic
222Dic9—41no—
223Dic9—121no—
224Dic9—241no—
225C2×#41—31no—
226C22×C8—271no—
227#117—271no—
228#103—91no—
229#103—271no—
230#102—271no—
231#102—271no—
232C3×C9#41611yescharacteristic, complemented, Sylow
233C23×C3—31no—
234C23×C3—61no—
235C22×D3—21no—
236C22×D3#61111yescharacteristic
237C3×Q8—21no—
238C3×Q8—21no—
239C3×Q8—22no—
240C3×Q8—41no—
241C3×Q8C2×Dic912yes—
242C3×Q8D3×Dic312yes—
243C3×D4—32no—
244C3×D4—64no—
245C3×D4—61no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—31no—
250C2×C3×C4—32no—
251C2×C3×C4—32no—
252C2×C3×C4—41no—
253C2×C3×C4—64no—
254C2×C3×C4—181no—
255C2×C3×C4C22×D911yescharacteristic
256C2×C3×C4C2×D3211yescharacteristic
257C2×Dic3—61no—
258C2×Dic3—181no—
259C2×Dic3—181no—
260C2×Dic3—361no—
261C2×Dic3#61111yescharacteristic
262D12—24no—
263D12—21no—
264D12C2×Dic912yes—
265C4×D3—24no—
266C4×D3C2×Dic912yes—
267C4×D3#61112yes—
268C3×C8—181no—
269C3⋊C8—61no—
270C3⋊C8—181no—
271C3⋊C8—181no—
272C3⋊C8—361no—
273C2×C32#95011yescharacteristic
274C2×C32#95011yescharacteristic
275C2×C32C2×#21711yescharacteristic
276C3×D3—22no—
277C3×D3—42no—
278C2×C9—21no—
279C2×C9—21no—
280C2×C9—21no—
281C2×C9—62no—
282C2×C9—122no—
283C2×C9D3×C22⋊C411yescharacteristic
284C2×C9#91611yescharacteristic
285C2×C9#91911yescharacteristic
286#41—34no—
287#41—62no—
288C2×Q8D3×D911yescharacteristic
289C2×D4—31no—
290C2×D4—61no—
291C22×C4—31no—
292C22×C4—61no—
293C2×C8—91no—
294C2×C8—271no—
295C2×C8—542no—
296C4⋊C4—91no—
297C4⋊C4—271no—
298C22⋊C4—541no—
299C42—541no—
300C22×C3—21no—
301C22×C3—32no—
302C22×C3—61no—
303C22×C3—61no—
304C22×C3—62no—
305C22×C3—121no—
306C22×C3—121no—
307C22×C3C2×#61111yescharacteristic
308C22×C3D3×#6711yescharacteristic
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—22no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3#367612yes—
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—32no—
320C3×C4—42no—
321C3×C4—361no—
322C3×C4C22×Dic911yescharacteristic
323C3×C4C2×#61111yescharacteristic
324C3×C4C2×D3×Dic311yescharacteristic
325C3×C4D3×#6711yescharacteristic
326Dic3—121no—
327Dic3—361no—
328Dic3—361no—
329Dic3—721no—
330Dic3#367612yes—
331C32#523311yescharacteristic
332C9—21no—
333C9#480411yescharacteristic, complemented
334C23—31no—
335C23—61no—
336Q8—21no—
337Q8D3×Dic912yes—
338D4—32no—
339D4—64no—
340D4—61no—
341C2×C4—21no—
342C2×C4—31no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—64no—
346C2×C4—181no—
347C2×C4—541no—
348C2×C4C2×D3×D911yescharacteristic
349C8—181no—
350C8—541no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—62no—
355C2×C3—122no—
356C2×C3#6351311yescharacteristic
357C2×C3#6351311yescharacteristic
358C2×C3C2×#367611yescharacteristic
359C2×C3#6394711yescharacteristic
360C2×C3#6394911yescharacteristic
361C2×C3D3×#21711yescharacteristic
362D3—22no—
363D3—42no—
364C22—32no—
365C22—61no—
366C22—61no—
367C22—62no—
368C22—121no—
369C22—121no—
370C22D3×#61111yescharacteristic
371C4—22no—
372C4—32no—
373C4—361no—
374C4—1081no—
375C4C2×D3×Dic911yescharacteristic
376C4D3×#61111yescharacteristic
377C3—21no—
378C3#1058891711yescharacteristic, complemented
379C3#1059155611yescharacteristic
380C2—62no—
381C2—122no—
382C2#1172120711yescharacteristic
383C2#1172120911yescharacteristic
384C2D3×#367611yescharacteristic
subgroupG/H
centerC22D3×#611
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32C2×#67
prime radicalC2×C3×#239D3
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9#416
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11720906C2semidirect
3. C2×D3×C9⋊C8C2semidirect
4. #11720582C2semidirect
5. C3×#63395C2semidirect
6. #11720295C2semidirect
7. #11720294C2semidirect
12. C9×#239C4semidirect
36. #63395D3semidirect
173. #239Dic9semidirect
232. C3×C9#416semidirect
333. C9#4804semidirect
378. C3#10588917semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2392
#117209062
C2×D3×C9⋊C82
#117205822
C3×#633952
#117202952
#117202942
#105915563
#105889173