Finite Group Databasebeta

← C3×#10588795 #13143098

Subgroups — C3×#10588795 #13143098

Order 1728.

448 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C8×C9C211yesmaximal, characteristic, complemented
2C3×#63492C211yesmaximal, characteristic
3C3×#63486C211yesmaximal, characteristic, complemented
4C2×C3×#3664C211yesmaximal, characteristic, complemented
5C3×#63380C211yesmaximal, characteristic, complemented
6C3×#63377C211yesmaximal, characteristic, complemented
7C3×#63376C211yesmaximal, characteristic, complemented
8C3×#5114—31nomaximal
9#10588795C311yesmaximal, characteristic, direct product factor
10C22×C3×C4×C9C2211yescharacteristic
11C2×C3×#611—21no—
12C2×C3×D36C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×C3×C8×C9—22no—
15C2×C3×C8×C9C2211yescharacteristic
16C2×C3×C8×C9C2211yescharacteristic
17C3×#3676—21no—
18C3×(C2×D9)⋊C4—21no—
19C3×C4⋊Dic9C2211yescharacteristic
20C3×C4⋊Dic9C2211yescharacteristic
21C3×Dic9⋊C4—21no—
22C3×#3664—22no—
23C22×C32×C8D311yescharacteristic
24C3×#930—31no—
25C3×#924—31no—
26C2×C3×#204—31no—
27C3×#821—31no—
28C3×#817—31no—
29C3×#816—31no—
30C22×C8×C9—21no—
31C22×C8×C9C2×C311yescharacteristic, complemented
32#63492C2×C311yescharacteristic
33#63486C2×C311yescharacteristic, complemented
34C2×#3664C2×C311yescharacteristic, complemented
35#63380C2×C311yescharacteristic, complemented
36#63377C2×C311yescharacteristic, complemented
37#63376C2×C311yescharacteristic, complemented
38C23×C3×C9D411yescharacteristic
39C22×C3×D9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9D411yescharacteristic
43C2×C3×C4×C9C2311yescharacteristic
44C3×#611—42no—
45C2×C3×Dic9—21no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9—21no—
48C3×D36—21no—
49C3×D36—21no—
50C3×Dic18—21no—
51C3×Dic18—21no—
52C3×C8×C9—21no—
53C3×C8×C9D412yescomplemented
54C3×#465—91no—
55#5114—31no—
56C22×C32×C4C2×D311yescharacteristic
57C2×C3×#67—61no—
58C2×C3×D12—31no—
59C2×C3×Dic6—31no—
60C2×C32×C8—22no—
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C2×D311yescharacteristic
63C3×#217—61no—
64C3×(C2×D3)⋊C4—61no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×Dic3⋊C4—61no—
68C3×#204—62no—
69C22×C4×C9—21no—
70C22×C4×C9C22×C311yescharacteristic
71C2×#611—21no—
72C2×D36C22×C311yescharacteristic
73C2×Dic18C22×C311yescharacteristic
74C2×C8×C9—24no—
75C2×C8×C9—21no—
76C2×C8×C9—21no—
77C2×C8×C9—22no—
78C2×C8×C9C22×C311yescharacteristic
79C2×C8×C9C22×C311yescharacteristic
80#3676—21no—
81(C2×D9)⋊C4—21no—
82C4⋊Dic9C22×C311yescharacteristic
83C4⋊Dic9C22×C311yescharacteristic
84Dic9⋊C4—21no—
85#3664—22no—
86C22×C3×C9—21no—
87C22×C3×C9—21no—
88C22×C3×C9SD1612yescomplemented
89C22×C3×C9C2×D411yescharacteristic
90C2×C3×D9—41no—
91C2×C3×D9—41no—
92C2×C3×D9—41no—
93C3×C4×C9—21no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C3×Dic9—41no—
97C3×Dic9—41no—
98C3×Dic9—41no—
99C2×C3×SD16—91no—
100C22×C3×C8—21no—
101C22×C3×C8D911yescharacteristic, complemented
102C22×C3×C8C3×D311yescharacteristic
103C3×C22⋊Q8—91no—
104C3×#121—91no—
105#930—31no—
106#924—31no—
107C2×#204—31no—
108C3×#106—91no—
109C3×#103—91no—
110C3×#102—91no—
111#821—31no—
112#817—31no—
113#816—31no—
114C23×C32D1211yescharacteristic
115C22×C3×D3—61no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4D1211yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C3×#67—122no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C3×D12—61no—
125C3×D12—61no—
126C3×Dic6—61no—
127C3×Dic6—61no—
128C23×C9—21no—
129C23×C9C3×D411yescharacteristic
130C22×D9—21no—
131C32×C8—21no—
132C32×C8#6712yes—
133C2×C4×C9—22no—
134C2×C4×C9—22no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9C3×D411yescharacteristic
140C2×C4×C9C23×C311yescharacteristic
141#611—42no—
142C2×Dic9—21no—
143C2×Dic9—21no—
144C2×Dic9—21no—
145D36—21no—
146D36—21no—
147Dic18—21no—
148Dic18—21no—
149C8×C9—22no—
150C8×C9—21no—
151C8×C9—22no—
152C8×C9C3×D412yescomplemented
153#465—91noSylow
154C2×C3×C9—22no—
155C2×C3×C9#12111yescharacteristic
156C2×C3×C9C2×SD1611yescharacteristic
157C2×C3×C9#13511yescharacteristic
158C3×D9—81no—
159C2×C3×Q8—91no—
160C2×C3×D4—91no—
161C2×C3×D4—181no—
162C22×C3×C4—21no—
163C22×C3×C4C2×D911yescharacteristic
164C22×C3×C4C2×C3×D311yescharacteristic
165C2×#67—61no—
166C2×D12—31no—
167C2×Dic6—31no—
168C3×SD16—182no—
169C2×C3×C8—24no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8C2×D911yescharacteristic
175C2×C3×C8C2×D911yescharacteristic
176C2×C3×C8C2×C3×D311yescharacteristic
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—
188#204—62no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32#20412yes—
192C22×C32C2×D1211yescharacteristic
193C2×C3×D3—121no—
194C2×C3×D3—121no—
195C2×C3×D3—121no—
196C32×C4—21no—
197C32×C4#23511yescharacteristic
198C32×C4C2×#6711yescharacteristic
199C3×Dic3—121no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202C22×C9—22no—
203C22×C9—22no—
204C22×C9—21no—
205C22×C9—21no—
206C22×C9—22no—
207C22×C9—21no—
208C22×C9C3×SD1612yescomplemented
209C22×C9C2×C3×D411yescharacteristic
210C2×D9—41no—
211C2×D9—41no—
212C2×D9—41no—
213C4×C9—22no—
214C4×C9—21no—
215C4×C9—21no—
216C4×C9—21no—
217C4×C9C2×C3×D411yescharacteristic
218C4×C9C3×#4111yescharacteristic
219Dic9—41no—
220Dic9—41no—
221Dic9—41no—
222C2×SD16—91no—
223C22×C8C3×D911yescharacteristic, complemented
224C22⋊Q8—91no—
225#121—91no—
226#106—91no—
227#103—91no—
228#102—91no—
229C3×C9#46511yescharacteristic, complemented, Sylow
230C23×C3—21no—
231C23×C3—181no—
232C23×C3D3611yescharacteristic
233C23×C3C3×D1211yescharacteristic
234C22×D3—61no—
235C3×Q8—181no—
236C3×Q8—181no—
237C3×D4—181no—
238C3×D4—181no—
239C3×D4—362no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4D3611yescharacteristic
252C2×C3×C4C22×D911yescharacteristic
253C2×C3×C4C3×D1211yescharacteristic
254C2×C3×C4C22×C3×D311yescharacteristic
255#67—122no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259D12—61no—
260D12—61no—
261Dic6—61no—
262Dic6—61no—
263C3×C8—22no—
264C3×C8—21no—
265C3×C8—21no—
266C3×C8—22no—
267C3×C8#61112yescomplemented
268C3×C8C3×#6712yes—
269C2×C32—22no—
270C2×C32C2×#20411yescharacteristic
271C2×C32#90411yescharacteristic
272C2×C32#93011yescharacteristic
273C3×D3—241no—
274C2×C9—24no—
275C2×C9—22no—
276C2×C9—21no—
277C2×C9—21no—
278C2×C9—21no—
279C2×C9C3×#12111yescharacteristic
280C2×C9C2×C3×SD1611yescharacteristic
281C2×C9C3×#13511yescharacteristic
282D9—81no—
283C2×Q8—91no—
284C2×D4—91no—
285C2×D4—181no—
286C22×C4C2×C3×D911yescharacteristic
287SD16—182no—
288C2×C8—22no—
289C2×C8C2×C3×D911yescharacteristic
290C2×C8C2×C3×D911yescharacteristic
291C4⋊C4—91no—
292C4⋊C4—91no—
293C4⋊C4—181no—
294C22⋊C4—181no—
295C22⋊C4—181no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—22no—
304C22×C3—361no—
305C22×C3—361no—
306C22×C3—361no—
307C22×C3#366412yescomplemented
308C22×C3C2×D3611yescharacteristic
309C22×C3C3×#20412yes—
310C22×C3C2×C3×D1211yescharacteristic
311C2×D3—121no—
312C2×D3—121no—
313C2×D3—121no—
314C3×C4—22no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—361no—
320C3×C4—361no—
321C3×C4—361no—
322C3×C4#369711yescharacteristic
323C3×C4C2×#61111yescharacteristic
324C3×C4C3×#23511yescharacteristic
325C3×C4C2×C3×#6711yescharacteristic
326Dic3—121no—
327Dic3—121no—
328Dic3—121no—
329C32#511411yescharacteristic
330C9—21no—
331C9C3×#46511yescharacteristic, complemented
332C23—181no—
333C23C3×D3611yescharacteristic
334Q8—181no—
335Q8—181no—
336D4—181no—
337D4—181no—
338D4—362no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—181no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4C3×D3611yescharacteristic
345C2×C4C22×C3×D911yescharacteristic
346C8—21no—
347C8C3×#61112yescomplemented
348C2×C3—24no—
349C2×C3—22no—
350C2×C3—22no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—721no—
355C2×C3C2×#366411yescharacteristic
356C2×C3#6346711yescharacteristic
357C2×C3#6349211yescharacteristic
358C2×C3C2×C3×#20411yescharacteristic
359C2×C3C3×#90411yescharacteristic
360C2×C3C3×#93011yescharacteristic
361D3—241no—
362C22—21no—
363C22—21no—
364C22—361no—
365C22—361no—
366C22—361no—
367C22C3×#366412yescomplemented
368C22C2×C3×D3611yescharacteristic
369C4—21no—
370C4—361no—
371C4—361no—
372C4—361no—
373C4C3×#369711yescharacteristic
374C4C2×C3×#61111yescharacteristic
375C3—21no—
376C3#1058879511yescharacteristic, direct product factor
377C3C3×#511411yescharacteristic
378C2—22no—
379C2—721no—
380C2C2×C3×#366411yescharacteristic
381C2C3×#6346711yescharacteristic
382C2C3×#6349211yescharacteristic
subgroupG/H
centerC22×C3C2×D36
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC22×C3×C8×C9C2
socleC22×C32C2×D12
prime radicalC3×#63492C2
hypercenterC22×C3×C8D9
hypocenterC9C3×#465
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C8×C9C2semidirect
3. C3×#63486C2semidirect
4. C2×C3×#3664C2semidirect
5. C3×#63380C2semidirect
6. C3×#63377C2semidirect
7. C3×#63376C2semidirect
9. #10588795C3direct (N × H)
31. C22×C8×C9C2×C3semidirect
33. #63486C2×C3semidirect
34. C2×#3664C2×C3semidirect
35. #63380C2×C3semidirect
36. #63377C2×C3semidirect
37. #63376C2×C3semidirect
53. C3×C8×C9D4semidirect
88. C22×C3×C9SD16semidirect
101. C22×C3×C8D9semidirect
152. C8×C9C3×D4semidirect
208. C22×C9C3×SD16semidirect
223. C22×C8C3×D9semidirect
229. C3×C9#465semidirect
267. C3×C8#611semidirect
307. C22×C3#3664semidirect
331. C9C3×#465semidirect
347. C8C3×#611semidirect
367. C22C3×#3664semidirect
376. C3#10588795direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C8×C92
C3×#634922
C3×#634862
C2×C3×#36642
C3×#633802
C3×#633772
C3×#633762
C3×#51143
#105887953