Finite Group Databasebeta

← C3×#10588451 #13141736

Subgroups — C3×#10588451 #13141736

Order 1728.

536 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C211yesmaximal, characteristic
2C2×C3×#3663C211yesmaximal, characteristic, complemented
3C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
4C3×C9×#102C211yesmaximal, characteristic, complemented
5C3×#63392C211yesmaximal, characteristic, complemented
6C3×#63376C211yesmaximal, characteristic, complemented
7C3×#63368C211yesmaximal, characteristic, complemented
8C3×#4771—31nomaximal
9#10588451C311yesmaximal, characteristic, direct product factor
10C2×C3×D4×C9C2211yescharacteristic
11C2×C3×#611—21no—
12C22×C3×Dic9—21no—
13C3×#3699—22no—
14C3×#3699C414yescomplemented
15C2×C3×C4×D9—21no—
16C2×C3×C4×D9C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C2×C3×C8×C9C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20C3×C4⋊Dic9C2211yescharacteristic
21C3×Dic9⋊C4—21no—
22C2×C3×C9⋊C8C2211yescharacteristic
23C3×#3663—22no—
24C2×C3×#237—31no—
25C2×C3×ΓL(1,𝔽25)—31no—
26C3×D3×C4⋊C4—31no—
27C2×#3699C2×C311yescharacteristic
28C32×#102D311yescharacteristic
29C3×#832—31no—
30C3×#816—31no—
31C3×#810—31no—
32C2×#3663C2×C311yescharacteristic, complemented
33C4⋊C4×D9C2×C311yescharacteristic, complemented
34C9×#102—21no—
35C9×#102C2×C311yescharacteristic, complemented
36#63392C2×C311yescharacteristic, complemented
37#63376C2×C311yescharacteristic, complemented
38#63368C2×C311yescharacteristic, complemented
39C23×C3×C9—21no—
40C22×C3×D9D411yescharacteristic
41C3×D4×C9—21no—
42C3×D4×C9C2×C412yescomplemented
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C3×#611—24no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9—24no—
48C2×C3×Dic9—21no—
49C2×C3×Dic9D411yescharacteristic
50C3×C4×D9—42no—
51C3×C4×D9C2×C412yes—
52C3×C4×D9D412yes—
53C3×Dic18—21no—
54C3×Dic18C2×C412yescomplemented
55C3×C8×C9—21no—
56C3×C9⋊C8—21no—
57C3×#417—91no—
58#4771—31no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C3×#67—61no—
61C22×C3×Dic3—61no—
62C3×#237—34no—
63C3×#237—62no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—61no—
66C2×C3×Dic6—31no—
67C2×C32×C8C2×D311yescharacteristic
68C32×C4⋊C4C2×D311yescharacteristic
69C3×C4⋊Dic3—31no—
70C3×Dic3⋊C4—61no—
71C2×C3×C3⋊C8—31no—
72C3×ΓL(1,𝔽25)—62no—
73C2×D4×C9—21no—
74C2×D4×C9C22×C311yescharacteristic
75C2×#611—21no—
76C22×Dic9—21no—
77#3699—22no—
78#3699C3×C414yescomplemented
79C2×C4×D9—21no—
80C2×C4×D9C22×C311yescharacteristic
81C2×Dic18C22×C311yescharacteristic
82C2×C8×C9—21no—
83C2×C8×C9C22×C311yescharacteristic
84C9×C4⋊C4—21no—
85C9×C4⋊C4C22×C311yescharacteristic
86C4⋊Dic9C22×C311yescharacteristic
87Dic9⋊C4—21no—
88C2×C9⋊C8C22×C311yescharacteristic
89#3663—22no—
90C22×C3×C9—22no—
91C22×C3×C9—41no—
92C22×C3×C9—41no—
93C22×C3×C9C2×D411yescharacteristic
94C2×C3×D9—21no—
95C2×C3×D9—21no—
96C2×C3×D9C22⋊C412yes—
97C3×C4×C9—41no—
98C3×C4×C9C22×C411yescharacteristic
99C3×C4×C9C2×D411yescharacteristic
100C3×Dic9—22no—
101C3×Dic9—41no—
102C3×Dic9C22⋊C412yes—
103C2×C3×#41—91no—
104C2×#237—31no—
105C2×C3×M16—91no—
106C2×C3×C4⋊C4—91no—
107C2×ΓL(1,𝔽25)—31no—
108D3×C4⋊C4—31no—
109C3×#103—91no—
110C3×#103—91no—
111C3×#102—21no—
112C3×#102—91no—
113C3×#102D911yescharacteristic, complemented
114C3×#102C3×D311yescharacteristic
115#832—31no—
116#816—31no—
117#810—31no—
118C23×C32—21no—
119C22×C3×D3—31no—
120C32×D4—21no—
121C32×D4C4×D312yes—
122C2×C32×C4—21no—
123C2×C32×C4C22×D311yescharacteristic
124C3×#67—64no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C3×C4×D3—32no—
130C3×C4×D3—32no—
131C3×C4×D3—122no—
132C3×Dic6—32no—
133C3×Dic6—61no—
134C23×C9—21no—
135C23×C9—41no—
136C22×D9C3×D411yescharacteristic
137C32×C8—21no—
138C3×C3⋊C8—61no—
139D4×C9—21no—
140D4×C9—22no—
141D4×C9—41no—
142D4×C9C2×C3×C412yescomplemented
143C2×C4×C9—21no—
144C2×C4×C9—21no—
145C2×C4×C9—41no—
146C2×C4×C9C23×C311yescharacteristic
147#611—24no—
148C2×Dic9—21no—
149C2×Dic9—24no—
150C2×Dic9—21no—
151C2×Dic9C3×D411yescharacteristic
152C4×D9—42no—
153C4×D9C2×C3×C412yes—
154C4×D9C3×D412yes—
155Dic18—21no—
156Dic18C2×C3×C412yescomplemented
157C8×C9—21no—
158C8×C9—41no—
159C9⋊C8—21no—
160#417—91noSylow
161C2×C3×C9—42no—
162C2×C3×C9C2×C22⋊C411yescharacteristic
163C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
164C2×C3×C9#13711yescharacteristic
165C3×D9—22no—
166C3×#41—94no—
167C3×#41—182no—
168C2×C3×Q8—91no—
169C2×C3×D4—21no—
170C2×C3×D4—181no—
171C2×C3×D4C2×D911yescharacteristic
172C2×C3×D4C2×C3×D311yescharacteristic
173C22×C3×C4—91no—
174C22×C3×C4—181no—
175C22×C3×C4—181no—
176C2×#67—61no—
177C22×Dic3—61no—
178#237—34no—
179#237—62no—
180C2×C4×D3—31no—
181C2×C4×D3—61no—
182C2×Dic6—31no—
183C3×M16—182no—
184C2×C3×C8—21no—
185C2×C3×C8—91no—
186C2×C3×C8C2×D911yescharacteristic
187C2×C3×C8C2×C3×D311yescharacteristic
188C3×C4⋊C4—21no—
189C3×C4⋊C4—91no—
190C3×C4⋊C4—181no—
191C3×C4⋊C4C2×D911yescharacteristic
192C3×C4⋊C4C2×C3×D311yescharacteristic
193C4⋊Dic3—31no—
194Dic3⋊C4—61no—
195C2×C3⋊C8—31no—
196ΓL(1,𝔽25)—62no—
197C22×C32—22no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32D3×D411yescharacteristic
201C2×C3×D3—32no—
202C2×C3×D3—61no—
203C2×C3×D3—61no—
204C32×C4—41no—
205C32×C4C2×C4×D311yescharacteristic
206C32×C4D3×D411yescharacteristic
207C3×Dic3—32no—
208C3×Dic3—62no—
209C3×Dic3—121no—
210C22×C9—22no—
211C22×C9—21no—
212C22×C9—42no—
213C22×C9—41no—
214C22×C9—41no—
215C22×C9—81no—
216C22×C9—81no—
217C22×C9C2×C3×D411yescharacteristic
218C2×D9—21no—
219C2×D9—21no—
220C2×D9C3×C22⋊C412yes—
221C4×C9—21no—
222C4×C9—21no—
223C4×C9—41no—
224C4×C9—81no—
225C4×C9C22×C3×C411yescharacteristic
226C4×C9C2×C3×D411yescharacteristic
227Dic9—22no—
228Dic9—41no—
229Dic9C3×C22⋊C412yes—
230C2×#41—91no—
231C2×M16—91no—
232C2×C4⋊C4—91no—
233#103—91no—
234#103—91no—
235#102—91no—
236#102C3×D911yescharacteristic, complemented
237C3×C9#41711yescharacteristic, complemented, Sylow
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—41no—
241C23×C3—91no—
242C22×D3—31no—
243C3×Q8—92no—
244C3×Q8—181no—
245C3×D4—21no—
246C3×D4—21no—
247C3×D4—22no—
248C3×D4—41no—
249C3×D4—184no—
250C3×D4C4×D912yescomplemented
251C3×D4C3×C4×D312yes—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—41no—
256C2×C3×C4—91no—
257C2×C3×C4—92no—
258C2×C3×C4—92no—
259C2×C3×C4—184no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4—362no—
263C2×C3×C4C22×D911yescharacteristic
264C2×C3×C4C22×C3×D311yescharacteristic
265#67—64no—
266C2×Dic3—31no—
267C2×Dic3—64no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C4×D3—32no—
271C4×D3—32no—
272C4×D3—122no—
273Dic6—32no—
274Dic6—61no—
275C3×C8—21no—
276C3×C8—21no—
277C3×C8—41no—
278C3×C8—181no—
279C3⋊C8—61no—
280C2×C32—42no—
281C2×C32D3×C22⋊C411yescharacteristic
282C2×C32#91111yescharacteristic
283C2×C32#91511yescharacteristic
284C3×D3—62no—
285C2×C9—21no—
286C2×C9—21no—
287C2×C9—21no—
288C2×C9—42no—
289C2×C9—82no—
290C2×C9C2×C3×C22⋊C411yescharacteristic
291C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
292C2×C9C3×#13711yescharacteristic
293D9—22no—
294#41—94no—
295#41—182no—
296C2×Q8—91no—
297C2×D4—181no—
298C2×D4C2×C3×D911yescharacteristic
299C22×C4—91no—
300C22×C4—181no—
301C22×C4—181no—
302M16—182no—
303C2×C8—91no—
304C2×C8C2×C3×D911yescharacteristic
305C4⋊C4—91no—
306C4⋊C4—181no—
307C4⋊C4C2×C3×D911yescharacteristic
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—21no—
311C22×C3—42no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—92no—
319C22×C3—181no—
320C22×C3—181no—
321C22×C3D4×D911yescharacteristic
322C22×C3C3×D3×D411yescharacteristic
323C2×D3—32no—
324C2×D3—61no—
325C2×D3—61no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—81no—
331C3×C4—92no—
332C3×C4—182no—
333C3×C4—361no—
334C3×C4C2×C4×D911yescharacteristic
335C3×C4D4×D911yescharacteristic
336C3×C4C2×C3×C4×D311yescharacteristic
337C3×C4C3×D3×D411yescharacteristic
338Dic3—32no—
339Dic3—62no—
340Dic3—121no—
341C32#477111yescharacteristic
342C9—21no—
343C9C3×#41711yescharacteristic, complemented
344C23—21no—
345C23—91no—
346Q8—92no—
347Q8—181no—
348D4—21no—
349D4—184no—
350D4C3×C4×D912yescomplemented
351C2×C4—21no—
352C2×C4—91no—
353C2×C4—92no—
354C2×C4—92no—
355C2×C4—184no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—362no—
359C2×C4C22×C3×D911yescharacteristic
360C8—21no—
361C8—181no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—42no—
366C2×C3—42no—
367C2×C3—82no—
368C2×C3—182no—
369C2×C3C22⋊C4×D911yescharacteristic
370C2×C3#6347311yescharacteristic
371C2×C3#6347711yescharacteristic
372C2×C3C3×D3×C22⋊C411yescharacteristic
373C2×C3C3×#91111yescharacteristic
374C2×C3C3×#91511yescharacteristic
375D3—62no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—92no—
380C22—181no—
381C22—181no—
382C22C3×D4×D911yescharacteristic
383C4—41no—
384C4—92no—
385C4—182no—
386C4—361no—
387C4C2×C3×C4×D911yescharacteristic
388C4C3×D4×D911yescharacteristic
389C3—21no—
390C3#1058845111yescharacteristic, direct product factor
391C3C3×#477111yescharacteristic
392C2—42no—
393C2—182no—
394C2C3×C22⋊C4×D911yescharacteristic
395C2C3×#6347311yescharacteristic
396C2C3×#6347711yescharacteristic
subgroupG/H
centerC22×C3D4×D9
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
prime radicalC2×C3×#3699C2
cosocleC2×C4×C9C23×C3
hypercenterC3×#102D9
hypocenterC9C3×#417
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3663C2semidirect
3. C3×C4⋊C4×D9C2semidirect
4. C3×C9×#102C2semidirect
5. C3×#63392C2semidirect
6. C3×#63376C2semidirect
7. C3×#63368C2semidirect
9. #10588451C3direct (N × H)
14. C3×#3699C4semidirect
32. C2×#3663C2×C3semidirect
33. C4⋊C4×D9C2×C3semidirect
35. C9×#102C2×C3semidirect
36. #63392C2×C3semidirect
37. #63376C2×C3semidirect
38. #63368C2×C3semidirect
42. C3×D4×C9C2×C4semidirect
54. C3×Dic18C2×C4semidirect
78. #3699C3×C4semidirect
113. C3×#102D9semidirect
142. D4×C9C2×C3×C4semidirect
156. Dic18C2×C3×C4semidirect
236. #102C3×D9semidirect
237. C3×C9#417semidirect
250. C3×D4C4×D9semidirect
343. C9C3×#417semidirect
350. D4C3×C4×D9semidirect
390. C3#10588451direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C2×C3×#36632
C3×C4⋊C4×D92
C3×C9×#1022
C3×#633922
C3×#633762
C3×#633682
C3×#47713
#105884513