Finite Group Databasebeta

← C3×#10588852 #13143353

Subgroups — C3×#10588852 #13143353

Order 1728.

554 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3701C211yesmaximal, characteristic, complemented
2C2×C3×C9×SD16C211yesmaximal, characteristic, complemented
3C3×#63500C211yesmaximal, characteristic, complemented
4C2×C3×#3673C211yesmaximal, characteristic, complemented
5C3×#63395C211yesmaximal, characteristic, complemented
6C3×#63380C211yesmaximal, characteristic, complemented
7C3×(C2×D9)⋊C8C211yesmaximal, characteristic, complemented
8C3×#5172—31nomaximal
9#10588852C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×D4×C9C2211yescharacteristic
12C2×C3×#611—21no—
13C3×#3701—22no—
14C3×#3701—22no—
15C2×C3×D36—21no—
16C2×C3×D36C2211yescharacteristic
17C2×C3×C4×D9—21no—
18C2×C3×C4×D9C2211yescharacteristic
19C3×C9×SD16—22no—
20C2×C3×C8×C9C2211yescharacteristic
21C3×#3676—21no—
22C3×#3673—22no—
23C3×C4⋊Dic9C2211yescharacteristic
24C2×C3×C9⋊C8C2211yescharacteristic
25C2×C3×#239—31no—
26C2×C32×SD16D311yescharacteristic
27C3×#938—31no—
28C2×C3×#213—31no—
29C2×#3701C2×C311yescharacteristic, complemented
30C3×#835—31no—
31C3×#821—31no—
32C3×(C2×D3)⋊C8—31no—
33C2×C9×SD16—21no—
34C2×C9×SD16C2×C311yescharacteristic, complemented
35#63500C2×C311yescharacteristic, complemented
36C2×#3673C2×C311yescharacteristic, complemented
37#63395C2×C311yescharacteristic, complemented
38#63380C2×C311yescharacteristic, complemented
39(C2×D9)⋊C8C2×C311yescharacteristic, complemented
40C23×C3×C9—21no—
41C22×C3×D9—21no—
42C22×C3×D9D411yescharacteristic
43C3×Q8×C9—21no—
44C3×Q8×C9D412yescomplemented
45C3×D4×C9—21no—
46C3×D4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9C2311yescharacteristic
49C3×#611—42no—
50C2×C3×Dic9—21no—
51C2×C3×Dic9D411yescharacteristic
52C3×D36—24no—
53C3×D36—21no—
54C3×D36D412yescomplemented
55C3×C4×D9—24no—
56C3×C4×D9—21no—
57C3×C4×D9—21no—
58C3×C8×C9—21no—
59C3×C9⋊C8—21no—
60C3×#449—91no—
61#5172—31no—
62C2×C32×Q8C2×D311yescharacteristic
63C2×C32×D4C2×D311yescharacteristic
64C2×C3×#67—61no—
65C3×#239—62no—
66C3×#239—62no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×C4×D3—31no—
70C2×C3×C4×D3—61no—
71C32×SD16—22no—
72C2×C32×C8C2×D311yescharacteristic
73C3×#217—61no—
74C3×#213—62no—
75C3×C4⋊Dic3—31no—
76C2×C3×C3⋊C8—31no—
77C2×Q8×C9—21no—
78C2×Q8×C9C22×C311yescharacteristic
79C2×D4×C9—21no—
80C2×D4×C9C22×C311yescharacteristic
81C2×#611—21no—
82#3701—22no—
83#3701—22no—
84C2×D36—21no—
85C2×D36C22×C311yescharacteristic
86C2×C4×D9—21no—
87C2×C4×D9C22×C311yescharacteristic
88C9×SD16—24no—
89C9×SD16—22no—
90C2×C8×C9—21no—
91C2×C8×C9C22×C311yescharacteristic
92#3676—21no—
93#3673—22no—
94C4⋊Dic9C22×C311yescharacteristic
95C2×C9⋊C8C22×C311yescharacteristic
96C22×C3×C9—41no—
97C22×C3×C9—41no—
98C22×C3×C9—41no—
99C22×C3×C9C2×D411yescharacteristic
100C2×C3×D9—21no—
101C2×C3×D9—22no—
102C2×C3×D9—21no—
103C2×C3×D9—21no—
104C2×C3×D9—41no—
105C2×C3×D9—41no—
106C3×C4×C9—22no—
107C3×C4×C9C2×D411yescharacteristic
108C3×C4×C9C2×D411yescharacteristic
109C3×Dic9—21no—
110C3×Dic9—41no—
111C2×C3×#41—91no—
112C2×#239—31no—
113C2×C3×SD16—21no—
114C2×C3×SD16D911yescharacteristic, complemented
115C2×C3×SD16C3×D311yescharacteristic
116C2×C3×D8—91no—
117C3×#121—91no—
118#938—31no—
119C2×#213—31no—
120C3×#103—91no—
121C3×#102—91no—
122C3×C22⋊C8—91no—
123#835—31no—
124#821—31no—
125(C2×D3)⋊C8—31no—
126C23×C32—21no—
127C22×C3×D3—31no—
128C22×C3×D3—61no—
129C32×Q8—21no—
130C32×Q8#6712yes—
131C32×D4—21no—
132C32×D4—21no—
133C2×C32×C4—21no—
134C2×C32×C4C22×D311yescharacteristic
135C3×#67—122no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—61no—
138C3×D12—32no—
139C3×D12—64no—
140C3×D12—61no—
141C3×C4×D3—64no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C23×C9—21no—
145C23×C9—41no—
146C22×D9—21no—
147C22×D9C3×D411yescharacteristic
148C32×C8—21no—
149C3×C3⋊C8—61no—
150Q8×C9—21no—
151Q8×C9—22no—
152Q8×C9—41no—
153Q8×C9C3×D412yescomplemented
154D4×C9—22no—
155D4×C9—21no—
156D4×C9—21no—
157D4×C9—41no—
158C2×C4×C9—21no—
159C2×C4×C9—21no—
160C2×C4×C9—41no—
161C2×C4×C9C23×C311yescharacteristic
162#611—42no—
163C2×Dic9—21no—
164C2×Dic9C3×D411yescharacteristic
165D36—24no—
166D36—21no—
167D36C3×D412yescomplemented
168C4×D9—24no—
169C4×D9—21no—
170C4×D9—21no—
171C8×C9—22no—
172C8×C9—21no—
173C9⋊C8—21no—
174#449—91noSylow
175C2×C3×C9—81no—
176C2×C3×C9#12011yescharacteristic
177C2×C3×C9#13511yescharacteristic
178C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
179C3×D9—41no—
180C3×D9—42no—
181C3×#41—182no—
182C3×#41—182no—
183C2×C3×Q8—21no—
184C2×C3×Q8C2×D911yescharacteristic
185C2×C3×Q8C2×C3×D311yescharacteristic
186C2×C3×D4—21no—
187C2×C3×D4—91no—
188C2×C3×D4—181no—
189C2×C3×D4—181no—
190C2×C3×D4C2×D911yescharacteristic
191C2×C3×D4C2×C3×D311yescharacteristic
192C22×C3×C4—91no—
193C22×C3×C4—181no—
194C2×#67—61no—
195#239—62no—
196#239—62no—
197C2×D12—31no—
198C2×D12—61no—
199C2×C4×D3—31no—
200C2×C4×D3—61no—
201C3×SD16—24no—
202C3×SD16—22no—
203C3×SD16—22no—
204C3×D8—182no—
205C2×C3×C8—21no—
206C2×C3×C8—91no—
207C2×C3×C8C2×D911yescharacteristic
208C2×C3×C8C2×C3×D311yescharacteristic
209C3×C4⋊C4—91no—
210C3×C22⋊C4—181no—
211#217—61no—
212#213—62no—
213C4⋊Dic3—31no—
214C2×C3⋊C8—31no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32D3×D411yescharacteristic
219C2×C3×D3—61no—
220C2×C3×D3—61no—
221C2×C3×D3—62no—
222C2×C3×D3—61no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C32×C4—22no—
226C32×C4D3×D411yescharacteristic
227C32×C4C2×#6711yescharacteristic
228C3×Dic3—61no—
229C3×Dic3—121no—
230C22×C9—21no—
231C22×C9—42no—
232C22×C9—41no—
233C22×C9—41no—
234C22×C9—41no—
235C22×C9—81no—
236C22×C9—81no—
237C22×C9C2×C3×D411yescharacteristic
238C2×D9—21no—
239C2×D9—22no—
240C2×D9—21no—
241C2×D9—21no—
242C2×D9—41no—
243C2×D9—41no—
244C4×C9—22no—
245C4×C9—21no—
246C4×C9—21no—
247C4×C9—42no—
248C4×C9C2×C3×D411yescharacteristic
249C4×C9C2×C3×D411yescharacteristic
250Dic9—21no—
251Dic9—41no—
252C2×#41—91no—
253C2×SD16C3×D911yescharacteristic, complemented
254C2×D8—91no—
255#121—91no—
256#103—91no—
257#102—91no—
258C22⋊C8—91no—
259C3×C9#44911yescharacteristic, complemented, Sylow
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—41no—
263C23×C3—91no—
264C23×C3—181no—
265C22×D3—31no—
266C22×D3—61no—
267C3×Q8—21no—
268C3×Q8—21no—
269C3×Q8—22no—
270C3×Q8—41no—
271C3×Q8#61112yescomplemented
272C3×Q8C3×#6712yes—
273C3×D4—22no—
274C3×D4—21no—
275C3×D4—21no—
276C3×D4—21no—
277C3×D4—21no—
278C3×D4—41no—
279C3×D4—92no—
280C3×D4—184no—
281C3×D4—181no—
282C3×D4—362no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—41no—
287C2×C3×C4—91no—
288C2×C3×C4—184no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4—181no—
292C2×C3×C4C22×D911yescharacteristic
293C2×C3×C4C22×C3×D311yescharacteristic
294#67—122no—
295C2×Dic3—31no—
296C2×Dic3—61no—
297D12—32no—
298D12—64no—
299D12—61no—
300C4×D3—64no—
301C4×D3—61no—
302C4×D3—61no—
303C3×C8—22no—
304C3×C8—21no—
305C3×C8—21no—
306C3×C8—181no—
307C3⋊C8—61no—
308C2×C32—81no—
309C2×C32#91411yescharacteristic
310C2×C32#91611yescharacteristic
311C2×C32#93711yescharacteristic
312C3×D3—121no—
313C3×D3—122no—
314C2×C9—21no—
315C2×C9—21no—
316C2×C9—21no—
317C2×C9—82no—
318C2×C9—81no—
319C2×C9C3×#12011yescharacteristic
320C2×C9C3×#13511yescharacteristic
321C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
322D9—41no—
323D9—42no—
324#41—182no—
325#41—182no—
326C2×Q8C2×C3×D911yescharacteristic
327C2×D4—91no—
328C2×D4—181no—
329C2×D4—181no—
330C2×D4C2×C3×D911yescharacteristic
331C22×C4—91no—
332C22×C4—181no—
333SD16—22no—
334D8—182no—
335C2×C8—91no—
336C2×C8C2×C3×D911yescharacteristic
337C4⋊C4—91no—
338C22⋊C4—181no—
339C22×C3—21no—
340C22×C3—42no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3—182no—
350C22×C3—181no—
351C22×C3—181no—
352C22×C3—181no—
353C22×C3—361no—
354C22×C3—361no—
355C22×C3D4×D911yescharacteristic
356C22×C3C3×D3×D411yescharacteristic
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—62no—
360C2×D3—61no—
361C2×D3—121no—
362C2×D3—121no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—42no—
368C3×C4—181no—
369C3×C4—361no—
370C3×C4D4×D911yescharacteristic
371C3×C4C2×#61111yescharacteristic
372C3×C4C3×D3×D411yescharacteristic
373C3×C4C2×C3×#6711yescharacteristic
374Dic3—61no—
375Dic3—121no—
376C32#517211yescharacteristic
377C9—21no—
378C9C3×#44911yescharacteristic, complemented
379C23—21no—
380C23—91no—
381C23—181no—
382Q8—21no—
383Q8C3×#61112yescomplemented
384D4—21no—
385D4—21no—
386D4—92no—
387D4—184no—
388D4—181no—
389D4—362no—
390C2×C4—21no—
391C2×C4—91no—
392C2×C4—184no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4C22×C3×D911yescharacteristic
397C8—21no—
398C8—181no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—82no—
403C2×C3—81no—
404C2×C3—81no—
405C2×C3—361no—
406C2×C3—362no—
407C2×C3#6347611yescharacteristic
408C2×C3#6347811yescharacteristic
409C2×C3#6349911yescharacteristic
410C2×C3C3×#91411yescharacteristic
411C2×C3C3×#91611yescharacteristic
412C2×C3C3×#93711yescharacteristic
413D3—121no—
414D3—122no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—181no—
419C22—181no—
420C22—182no—
421C22—181no—
422C22—361no—
423C22—361no—
424C22C3×D4×D911yescharacteristic
425C4—22no—
426C4—181no—
427C4—361no—
428C4C3×D4×D911yescharacteristic
429C4C2×C3×#61111yescharacteristic
430C3—21no—
431C3#1058885211yescharacteristic, direct product factor
432C3C3×#517211yescharacteristic
433C2—81no—
434C2—361no—
435C2—362no—
436C2C3×#6347611yescharacteristic
437C2C3×#6347811yescharacteristic
438C2C3×#6349911yescharacteristic
subgroupG/H
centerC22×C3D4×D9
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C3×C9×SD16C2
socleC22×C32D3×D4
hypercenterC2×C3×SD16D9
hypocenterC9C3×#449
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3701C2semidirect
2. C2×C3×C9×SD16C2semidirect
3. C3×#63500C2semidirect
4. C2×C3×#3673C2semidirect
5. C3×#63395C2semidirect
6. C3×#63380C2semidirect
7. C3×(C2×D9)⋊C8C2semidirect
9. #10588852C3direct (N × H)
29. C2×#3701C2×C3semidirect
34. C2×C9×SD16C2×C3semidirect
35. #63500C2×C3semidirect
36. C2×#3673C2×C3semidirect
37. #63395C2×C3semidirect
38. #63380C2×C3semidirect
39. (C2×D9)⋊C8C2×C3semidirect
44. C3×Q8×C9D4semidirect
54. C3×D36D4semidirect
114. C2×C3×SD16D9semidirect
153. Q8×C9C3×D4semidirect
167. D36C3×D4semidirect
253. C2×SD16C3×D9semidirect
259. C3×C9#449semidirect
271. C3×Q8#611semidirect
378. C9C3×#449semidirect
383. Q8C3×#611semidirect
431. C3#10588852direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37012
C2×C3×C9×SD162
C3×#635002
C2×C3×#36732
C3×#633952
C3×#633802
C3×(C2×D9)⋊C82
C3×#51723
#105888523