Finite Group Databasebeta

← C3×#10588697 #13142728

Subgroups — C3×#10588697 #13142728

Order 1728.

490 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×D4×C9C211yesmaximal, characteristic, complemented
2C2×C3×#3673C211yesmaximal, characteristic
3C3×C4×D36C211yesmaximal, characteristic, complemented
4C3×#63392C211yesmaximal, characteristic, complemented
5C3×#63369C211yesmaximal, characteristic, complemented
6C3×#63367C211yesmaximal, characteristic, complemented
7C3×#63362C211yesmaximal, characteristic, complemented
8C3×#5014—31nomaximal
9#10588697C311yesmaximal, characteristic, direct product factor
10C2×C3×D4×C9C2211yescharacteristic
11C22×C3×C4×C9—21no—
12C2×C3×D36C2211yescharacteristic
13C2×C3×C4×D9—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×C9×C22⋊C4—21no—
16C3×C42×C9C2211yescharacteristic
17C3×#3673C414yescomplemented
18C3×(C2×D9)⋊C4—21no—
19C3×C4⋊Dic9C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22C32×C4×D4D311yescharacteristic
23C2×C3×#213—31no—
24C3×C4×D12—31no—
25C3×#832—31no—
26C3×#809—31no—
27C3×#808—31no—
28C3×#803—31no—
29C4×D4×C9—21no—
30C4×D4×C9C2×C311yescharacteristic, complemented
31C2×#3673C2×C311yescharacteristic
32C4×D36C2×C311yescharacteristic, complemented
33#63392C2×C311yescharacteristic, complemented
34#63369C2×C311yescharacteristic, complemented
35#63367C2×C311yescharacteristic, complemented
36#63362C2×C311yescharacteristic, complemented
37C23×C3×C9—21no—
38C22×C3×D9—21no—
39C3×D4×C9—21no—
40C3×D4×C9C2×C412yescomplemented
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9—42no—
43C2×C3×C4×C9D411yescharacteristic
44C2×C3×C4×C9D411yescharacteristic
45C2×C3×C4×C9C2311yescharacteristic
46C2×C3×Dic9—21no—
47C3×D36—21no—
48C3×D36C2×C412yescomplemented
49C3×C4×D9—42no—
50C3×C9⋊C8—21no—
51C3×C9⋊C8C2×C412yescomplemented
52C3×#442—91no—
53#5014—31no—
54C2×C32×D4C2×D311yescharacteristic
55C22×C32×C4—21no—
56C2×C3×D12—31no—
57C2×C3×C4×D3—61no—
58C32×C4⋊C4C2×D311yescharacteristic
59C32×C22⋊C4—21no—
60C32×C42C2×D311yescharacteristic
61C3×#213—34no—
62C3×(C2×D3)⋊C4—61no—
63C3×C4⋊Dic3—31no—
64C2×C3×C3⋊C8—31no—
65C2×C3×C3⋊C8—31no—
66C2×D4×C9—21no—
67C2×D4×C9C22×C311yescharacteristic
68C22×C4×C9—22no—
69C22×C4×C9—21no—
70C2×D36C22×C311yescharacteristic
71C2×C4×D9—21no—
72C9×C4⋊C4—21no—
73C9×C4⋊C4C22×C311yescharacteristic
74C9×C22⋊C4—22no—
75C9×C22⋊C4—21no—
76C42×C9—21no—
77C42×C9C22×C311yescharacteristic
78#3673C3×C414yescomplemented
79(C2×D9)⋊C4—21no—
80C4⋊Dic9C22×C311yescharacteristic
81C2×C9⋊C8C22×C311yescharacteristic
82C2×C9⋊C8C22×C311yescharacteristic
83C22×C3×C9—22no—
84C22×C3×C9—41no—
85C22×C3×C9—41no—
86C22×C3×C9C2×D411yescharacteristic
87C2×C3×D9—22no—
88C2×C3×D9—41no—
89C2×C3×D9—41no—
90C3×C4×C9—21no—
91C3×C4×C9—21no—
92C3×C4×C9—41no—
93C3×C4×C9C22×C411yescharacteristic
94C3×C4×C9#4111yescharacteristic
95C3×Dic9—41no—
96C2×C3×D8—91no—
97C3×C4×D4—21no—
98C3×C4×D4—91no—
99C3×C4×D4D911yescharacteristic, complemented
100C3×C4×D4C3×D311yescharacteristic
101C2×#213—31no—
102C4×D12—31no—
103C3×#106—91no—
104C3×#102—91no—
105C3×#102—91no—
106C3×#97—91no—
107#832—31no—
108#809—31no—
109#808—31no—
110#803—31no—
111C23×C32—21no—
112C22×C3×D3—61no—
113C32×D4—21no—
114C32×D4C4×D312yes—
115C2×C32×C4—21no—
116C2×C32×C4—42no—
117C2×C32×C4#6711yescharacteristic
118C2×C32×C4#6711yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C3×Dic3—61no—
121C3×D12—32no—
122C3×D12—61no—
123C3×C4×D3—122no—
124C23×C9—22no—
125C23×C9—21no—
126C22×D9—21no—
127C3×C3⋊C8—32no—
128C3×C3⋊C8—61no—
129D4×C9—22no—
130D4×C9—21no—
131D4×C9—22no—
132D4×C9C2×C3×C412yescomplemented
133C2×C4×C9—22no—
134C2×C4×C9—21no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—44no—
139C2×C4×C9—42no—
140C2×C4×C9C3×D411yescharacteristic
141C2×C4×C9C3×D411yescharacteristic
142C2×C4×C9C23×C311yescharacteristic
143C2×Dic9—21no—
144D36—21no—
145D36C2×C3×C412yescomplemented
146C4×D9—42no—
147C9⋊C8—21no—
148C9⋊C8C2×C3×C412yescomplemented
149#442—91noSylow
150C2×C3×C9—42no—
151C2×C3×C9C4×D411yescharacteristic
152C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
153C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
154C3×D9—42no—
155C2×C3×D4—21no—
156C2×C3×D4—91no—
157C2×C3×D4C2×D911yescharacteristic
158C2×C3×D4C2×C3×D311yescharacteristic
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—181no—
163C2×D12—31no—
164C2×C4×D3—61no—
165C3×D8—94no—
166C2×C3×C8—91no—
167C2×C3×C8—91no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4C2×D911yescharacteristic
171C3×C4⋊C4C2×C3×D311yescharacteristic
172C3×C22⋊C4—22no—
173C3×C22⋊C4—21no—
174C3×C22⋊C4—21no—
175C3×C22⋊C4—181no—
176C3×C42—21no—
177C3×C42C2×D911yescharacteristic
178C3×C42C2×C3×D311yescharacteristic
179#213—34no—
180(C2×D3)⋊C4—61no—
181C4⋊Dic3—31no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—31no—
184C22×C32—22no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32C2×#6711yescharacteristic
188C2×C3×D3—62no—
189C2×C3×D3—121no—
190C2×C3×D3—121no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4—41no—
194C32×C4C2×C4×D311yescharacteristic
195C32×C4#23511yescharacteristic
196C3×Dic3—121no—
197C22×C9—24no—
198C22×C9—22no—
199C22×C9—21no—
200C22×C9—42no—
201C22×C9—42no—
202C22×C9—41no—
203C22×C9—41no—
204C22×C9C2×C3×D411yescharacteristic
205C2×D9—22no—
206C2×D9—41no—
207C2×D9—41no—
208C4×C9—22no—
209C4×C9—21no—
210C4×C9—21no—
211C4×C9—21no—
212C4×C9—21no—
213C4×C9—41no—
214C4×C9—42no—
215C4×C9—41no—
216C4×C9C22×C3×C411yescharacteristic
217C4×C9C3×#4111yescharacteristic
218Dic9—41no—
219C2×D8—91no—
220C4×D4—91no—
221C4×D4C3×D911yescharacteristic, complemented
222#106—91no—
223#102—91no—
224#102—91no—
225#97—91no—
226C3×C9#44211yescharacteristic, complemented, Sylow
227C23×C3—22no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—181no—
231C22×D3—61no—
232C3×D4—22no—
233C3×D4—21no—
234C3×D4—21no—
235C3×D4—22no—
236C3×D4—92no—
237C3×D4—181no—
238C3×D4C4×D912yescomplemented
239C3×D4C3×C4×D312yes—
240C2×C3×C4—22no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—42no—
247C2×C3×C4—44no—
248C2×C3×C4—42no—
249C2×C3×C4—181no—
250C2×C3×C4—362no—
251C2×C3×C4#61111yescharacteristic
252C2×C3×C4#61111yescharacteristic
253C2×C3×C4C22×D911yescharacteristic
254C2×C3×C4C3×#6711yescharacteristic
255C2×C3×C4C3×#6711yescharacteristic
256C2×C3×C4C22×C3×D311yescharacteristic
257C2×Dic3—61no—
258D12—32no—
259D12—61no—
260C4×D3—122no—
261C3×C8—92no—
262C3×C8—181no—
263C3⋊C8—32no—
264C3⋊C8—61no—
265C2×C32—42no—
266C2×C32C4×#6711yescharacteristic
267C2×C32#93211yescharacteristic
268C2×C32#94911yescharacteristic
269C3×D3—122no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9—44no—
274C2×C9—42no—
275C2×C9C3×C4×D411yescharacteristic
276C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
277C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
278D9—42no—
279C2×D4—91no—
280C2×D4C2×C3×D911yescharacteristic
281C22×C4—21no—
282C22×C4—181no—
283D8—94no—
284C2×C8—91no—
285C2×C8—91no—
286C4⋊C4—91no—
287C4⋊C4C2×C3×D911yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—181no—
290C42C2×C3×D911yescharacteristic
291C22×C3—24no—
292C22×C3—22no—
293C22×C3—22no—
294C22×C3—21no—
295C22×C3—42no—
296C22×C3—42no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—182no—
302C22×C3—361no—
303C22×C3—361no—
304C22×C3C2×#61111yescharacteristic
305C22×C3C2×C3×#6711yescharacteristic
306C2×D3—62no—
307C2×D3—121no—
308C2×D3—121no—
309C3×C4—22no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—41no—
317C3×C4—42no—
318C3×C4—41no—
319C3×C4—41no—
320C3×C4—361no—
321C3×C4C2×C4×D911yescharacteristic
322C3×C4#369711yescharacteristic
323C3×C4C2×C3×C4×D311yescharacteristic
324C3×C4C3×#23511yescharacteristic
325Dic3—121no—
326C32#501411yescharacteristic
327C9—21no—
328C9C3×#44211yescharacteristic, complemented
329C23—21no—
330C23—181no—
331D4—21no—
332D4—92no—
333D4—181no—
334D4C3×C4×D912yescomplemented
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—181no—
338C2×C4—362no—
339C2×C4C3×#61111yescharacteristic
340C2×C4C3×#61111yescharacteristic
341C2×C4C22×C3×D911yescharacteristic
342C8—92no—
343C8—181no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—44no—
348C2×C3—42no—
349C2×C3—42no—
350C2×C3—362no—
351C2×C3C4×#61111yescharacteristic
352C2×C3#6349511yescharacteristic
353C2×C3#6351211yescharacteristic
354C2×C3C3×C4×#6711yescharacteristic
355C2×C3C3×#93211yescharacteristic
356C2×C3C3×#94911yescharacteristic
357D3—122no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—182no—
362C22—361no—
363C22—361no—
364C22C2×C3×#61111yescharacteristic
365C4—21no—
366C4—21no—
367C4—41no—
368C4—361no—
369C4C2×C3×C4×D911yescharacteristic
370C4C3×#369711yescharacteristic
371C3—21no—
372C3#1058869711yescharacteristic, direct product factor
373C3C3×#501411yescharacteristic
374C2—42no—
375C2—362no—
376C2C3×C4×#61111yescharacteristic
377C2C3×#6349511yescharacteristic
378C2C3×#6351211yescharacteristic
subgroupG/H
centerC22×C3C2×#611
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C4×D4×C9C2
socleC22×C32C2×#67
prime radicalC2×C3×#3673C2
cosocleC2×C4×C9C23×C3
hypercenterC3×C4×D4D9
hypocenterC9C3×#442
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C3×C4×D4×C9C2semidirect
3. C3×C4×D36C2semidirect
4. C3×#63392C2semidirect
5. C3×#63369C2semidirect
6. C3×#63367C2semidirect
7. C3×#63362C2semidirect
9. #10588697C3direct (N × H)
17. C3×#3673C4semidirect
30. C4×D4×C9C2×C3semidirect
32. C4×D36C2×C3semidirect
33. #63392C2×C3semidirect
34. #63369C2×C3semidirect
35. #63367C2×C3semidirect
36. #63362C2×C3semidirect
40. C3×D4×C9C2×C4semidirect
48. C3×D36C2×C4semidirect
51. C3×C9⋊C8C2×C4semidirect
78. #3673C3×C4semidirect
99. C3×C4×D4D9semidirect
132. D4×C9C2×C3×C4semidirect
145. D36C2×C3×C4semidirect
148. C9⋊C8C2×C3×C4semidirect
221. C4×D4C3×D9semidirect
226. C3×C9#442semidirect
238. C3×D4C4×D9semidirect
328. C9C3×#442semidirect
334. D4C3×C4×D9semidirect
372. C3#10588697direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×D4×C92
C2×C3×#36732
C3×C4×D362
C3×#633922
C3×#633692
C3×#633672
C3×#633622
C3×#50143
#105886973