Finite Group Databasebeta

← C3×#10588854 #13143352

Subgroups — C3×#10588854 #13143352

Order 1728.

428 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×C3×C9×SD16C211yesmaximal, characteristic, complemented
2C3×#63508C211yesmaximal, characteristic, complemented
3C3×#63500C211yesmaximal, characteristic
4C2×C3×#3663C211yesmaximal, characteristic, complemented
5C3×#63395C211yesmaximal, characteristic, complemented
6C3×#63392C211yesmaximal, characteristic, complemented
7C3×#63378C211yesmaximal, characteristic, complemented
8C3×#5174—31nomaximal
9#10588854C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×D4×C9C2211yescharacteristic
12C2×C3×#611—21no—
13C2×C3×C4×D9C2211yescharacteristic
14C3×C9×SD16—22no—
15C2×C3×C8×C9C2211yescharacteristic
16C3×#3676—21no—
17C3×(C2×D9)⋊C4—21no—
18C3×C4⋊Dic9C2211yescharacteristic
19C3×C4⋊Dic9C2211yescharacteristic
20C3×Dic9⋊C4—21no—
21C2×C3×C9⋊C8C2211yescharacteristic
22C3×#3663—22no—
23C2×C32×SD16D311yescharacteristic
24C3×#946—31no—
25C3×#938—31no—
26C2×C3×ΓL(1,𝔽25)—31no—
27C3×#835—31no—
28C3×#832—31no—
29C3×#818—31no—
30C2×C9×SD16—21no—
31C2×C9×SD16C2×C311yescharacteristic, complemented
32#63508C2×C311yescharacteristic, complemented
33#63500C2×C311yescharacteristic
34C2×#3663C2×C311yescharacteristic, complemented
35#63395C2×C311yescharacteristic, complemented
36#63392C2×C311yescharacteristic, complemented
37#63378C2×C311yescharacteristic, complemented
38C23×C3×C9—21no—
39C22×C3×D9D411yescharacteristic
40C3×Q8×C9—21no—
41C3×Q8×C9—21no—
42C3×D4×C9—21no—
43C3×D4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2311yescharacteristic
46C3×#611—42no—
47C2×C3×Dic9—21no—
48C2×C3×Dic9—21no—
49C2×C3×Dic9D411yescharacteristic
50C3×C4×D9—21no—
51C3×C4×D9—21no—
52C3×C8×C9D412yescomplemented
53C3×C9⋊C8—21no—
54C3×#468—91no—
55#5174—31no—
56C2×C32×Q8C2×D311yescharacteristic
57C2×C32×D4C2×D311yescharacteristic
58C2×C3×#67—61no—
59C2×C3×C4×D3—31no—
60C32×SD16—22no—
61C2×C32×C8C2×D311yescharacteristic
62C3×#217—61no—
63C3×(C2×D3)⋊C4—61no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×Dic3⋊C4—61no—
67C2×C3×C3⋊C8—31no—
68C3×ΓL(1,𝔽25)—62no—
69C2×Q8×C9—21no—
70C2×Q8×C9C22×C311yescharacteristic
71C2×D4×C9—21no—
72C2×D4×C9C22×C311yescharacteristic
73C2×#611—21no—
74C2×C4×D9C22×C311yescharacteristic
75C9×SD16—24no—
76C9×SD16—22no—
77C2×C8×C9—21no—
78C2×C8×C9C22×C311yescharacteristic
79#3676—21no—
80(C2×D9)⋊C4—21no—
81C4⋊Dic9C22×C311yescharacteristic
82C4⋊Dic9C22×C311yescharacteristic
83Dic9⋊C4—21no—
84C2×C9⋊C8C22×C311yescharacteristic
85#3663—22no—
86C22×C3×C9—41no—
87C22×C3×C9—41no—
88C22×C3×C9—41no—
89C22×C3×C9C2×D411yescharacteristic
90C2×C3×D9—21no—
91C2×C3×D9—21no—
92C2×C3×D9—21no—
93C3×C4×C9—41no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C3×Dic9—21no—
97C3×Dic9—41no—
98C3×Dic9—41no—
99C2×C3×SD16—21no—
100C2×C3×SD16D911yescharacteristic, complemented
101C2×C3×SD16C3×D311yescharacteristic
102C2×C3×M16—91no—
103C3×C22⋊Q8—91no—
104C3×#121—91no—
105#946—31no—
106#938—31no—
107C2×ΓL(1,𝔽25)—31no—
108C3×#107—91no—
109C3×#103—91no—
110C3×#102—91no—
111#835—31no—
112#832—31no—
113#818—31no—
114C23×C32—21no—
115C22×C3×D3—31no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×D4—21no—
119C32×D4—21no—
120C2×C32×C4—21no—
121C2×C32×C4C22×D311yescharacteristic
122C3×#67—122no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C3×C4×D3—61no—
127C3×C4×D3—61no—
128C23×C9—21no—
129C23×C9—41no—
130C22×D9C3×D411yescharacteristic
131C32×C8#6712yes—
132C3×C3⋊C8—61no—
133Q8×C9—22no—
134Q8×C9—21no—
135Q8×C9—21no—
136Q8×C9—41no—
137D4×C9—22no—
138D4×C9—21no—
139D4×C9—21no—
140D4×C9—41no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—41no—
144C2×C4×C9C23×C311yescharacteristic
145#611—42no—
146C2×Dic9—21no—
147C2×Dic9—21no—
148C2×Dic9C3×D411yescharacteristic
149C4×D9—21no—
150C4×D9—21no—
151C8×C9—22no—
152C8×C9C3×D412yescomplemented
153C9⋊C8—21no—
154#468—91noSylow
155C2×C3×C9—81no—
156C2×C3×C9#12111yescharacteristic
157C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
158C2×C3×C9#13711yescharacteristic
159C3×D9—41no—
160C2×C3×Q8—21no—
161C2×C3×Q8C2×D911yescharacteristic
162C2×C3×Q8C2×C3×D311yescharacteristic
163C2×C3×D4—21no—
164C2×C3×D4—181no—
165C2×C3×D4C2×D911yescharacteristic
166C2×C3×D4C2×C3×D311yescharacteristic
167C22×C3×C4—91no—
168C2×#67—61no—
169C2×C4×D3—31no—
170C3×SD16—24no—
171C3×SD16—22no—
172C3×SD16—22no—
173C3×M16—182no—
174C2×C3×C8—21no—
175C2×C3×C8—91no—
176C2×C3×C8C2×D911yescharacteristic
177C2×C3×C8C2×C3×D311yescharacteristic
178C3×C4⋊C4—91no—
179C3×C4⋊C4—91no—
180C3×C4⋊C4—181no—
181C3×C22⋊C4—181no—
182C3×C22⋊C4—181no—
183#217—61no—
184(C2×D3)⋊C4—61no—
185C4⋊Dic3—31no—
186C4⋊Dic3—31no—
187Dic3⋊C4—61no—
188C2×C3⋊C8—31no—
189ΓL(1,𝔽25)—62no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32D3×D411yescharacteristic
194C2×C3×D3—61no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C32×C4—41no—
198C32×C4#23711yescharacteristic
199C32×C4C2×#6711yescharacteristic
200C3×Dic3—61no—
201C3×Dic3—121no—
202C3×Dic3—121no—
203C22×C9—21no—
204C22×C9—42no—
205C22×C9—41no—
206C22×C9—41no—
207C22×C9—41no—
208C22×C9—81no—
209C22×C9—81no—
210C22×C9C2×C3×D411yescharacteristic
211C2×D9—21no—
212C2×D9—21no—
213C2×D9—21no—
214C4×C9—21no—
215C4×C9—21no—
216C4×C9—42no—
217C4×C9—41no—
218C4×C9C2×C3×D411yescharacteristic
219C4×C9C3×#4111yescharacteristic
220Dic9—21no—
221Dic9—41no—
222Dic9—41no—
223C2×SD16C3×D911yescharacteristic, complemented
224C2×M16—91no—
225C22⋊Q8—91no—
226#121—91no—
227#107—91no—
228#103—91no—
229#102—91no—
230C3×C9#46811yescharacteristic, complemented, Sylow
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—41no—
234C23×C3—91no—
235C22×D3—31no—
236C3×Q8—22no—
237C3×Q8—21no—
238C3×Q8—21no—
239C3×Q8—21no—
240C3×Q8—21no—
241C3×Q8—41no—
242C3×D4—22no—
243C3×D4—21no—
244C3×D4—21no—
245C3×D4—21no—
246C3×D4—21no—
247C3×D4—41no—
248C3×D4—362no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—41no—
253C2×C3×C4—91no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4—181no—
257C2×C3×C4—181no—
258C2×C3×C4C22×D911yescharacteristic
259C2×C3×C4C22×C3×D311yescharacteristic
260#67—122no—
261C2×Dic3—31no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C4×D3—61no—
265C4×D3—61no—
266C3×C8—22no—
267C3×C8—181no—
268C3×C8#61112yescomplemented
269C3×C8C3×#6712yes—
270C3⋊C8—61no—
271C2×C32—81no—
272C2×C32#91411yescharacteristic
273C2×C32#91511yescharacteristic
274C2×C32#93811yescharacteristic
275C3×D3—121no—
276C2×C9—21no—
277C2×C9—21no—
278C2×C9—21no—
279C2×C9—82no—
280C2×C9—81no—
281C2×C9C3×#12111yescharacteristic
282C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
283C2×C9C3×#13711yescharacteristic
284D9—41no—
285C2×Q8C2×C3×D911yescharacteristic
286C2×D4—181no—
287C2×D4C2×C3×D911yescharacteristic
288C22×C4—91no—
289SD16—22no—
290M16—182no—
291C2×C8—91no—
292C2×C8C2×C3×D911yescharacteristic
293C4⋊C4—91no—
294C4⋊C4—91no—
295C4⋊C4—181no—
296C22⋊C4—181no—
297C22⋊C4—181no—
298C22×C3—21no—
299C22×C3—42no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—181no—
309C22×C3—181no—
310C22×C3—181no—
311C22×C3D4×D911yescharacteristic
312C22×C3C3×D3×D411yescharacteristic
313C2×D3—61no—
314C2×D3—61no—
315C2×D3—61no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—42no—
319C3×C4—41no—
320C3×C4—41no—
321C3×C4—181no—
322C3×C4—361no—
323C3×C4—361no—
324C3×C4#369911yescharacteristic
325C3×C4C2×#61111yescharacteristic
326C3×C4C3×#23711yescharacteristic
327C3×C4C2×C3×#6711yescharacteristic
328Dic3—61no—
329Dic3—121no—
330Dic3—121no—
331C32#517411yescharacteristic
332C9—21no—
333C9C3×#46811yescharacteristic, complemented
334C23—21no—
335C23—91no—
336Q8—21no—
337Q8—21no—
338D4—21no—
339D4—21no—
340D4—362no—
341C2×C4—21no—
342C2×C4—91no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—181no—
346C2×C4—181no—
347C2×C4C22×C3×D911yescharacteristic
348C8—181no—
349C8C3×#61112yescomplemented
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—82no—
354C2×C3—81no—
355C2×C3—81no—
356C2×C3—361no—
357C2×C3#6347611yescharacteristic
358C2×C3#6347711yescharacteristic
359C2×C3#6350011yescharacteristic
360C2×C3C3×#91411yescharacteristic
361C2×C3C3×#91511yescharacteristic
362C2×C3C3×#93811yescharacteristic
363D3—121no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—181no—
368C22—181no—
369C22—181no—
370C22C3×D4×D911yescharacteristic
371C4—41no—
372C4—181no—
373C4—361no—
374C4—361no—
375C4C3×#369911yescharacteristic
376C4C2×C3×#61111yescharacteristic
377C3—21no—
378C3#1058885411yescharacteristic, direct product factor
379C3C3×#517411yescharacteristic
380C2—81no—
381C2—361no—
382C2C3×#6347611yescharacteristic
383C2C3×#6347711yescharacteristic
384C2C3×#6350011yescharacteristic
subgroupG/H
centerC22×C3D4×D9
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C3×C9×SD16C2
socleC22×C32D3×D4
prime radicalC3×#63500C2
hypercenterC2×C3×SD16D9
hypocenterC9C3×#468
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×C9×SD16C2semidirect
2. C3×#63508C2semidirect
4. C2×C3×#3663C2semidirect
5. C3×#63395C2semidirect
6. C3×#63392C2semidirect
7. C3×#63378C2semidirect
9. #10588854C3direct (N × H)
31. C2×C9×SD16C2×C3semidirect
32. #63508C2×C3semidirect
34. C2×#3663C2×C3semidirect
35. #63395C2×C3semidirect
36. #63392C2×C3semidirect
37. #63378C2×C3semidirect
52. C3×C8×C9D4semidirect
100. C2×C3×SD16D9semidirect
152. C8×C9C3×D4semidirect
223. C2×SD16C3×D9semidirect
230. C3×C9#468semidirect
268. C3×C8#611semidirect
333. C9C3×#468semidirect
349. C8C3×#611semidirect
378. C3#10588854direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C9×SD162
C3×#635082
C3×#635002
C2×C3×#36632
C3×#633952
C3×#633922
C3×#633782
C3×#51743
#105888543