Finite Group Databasebeta

← C3×#10589837 #13153304

Subgroups — C3×#10589837 #13153304

Order 1728.

656 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C4⋊Q8C211yesmaximal, characteristic, complemented
2C3×#63509C211yesmaximal, characteristic
3C3×#63508C212yesmaximal, complemented
4C3×#63506C211yesmaximal, characteristic, complemented
5C3×#63460C212yesmaximal, complemented
6C3×#63458C212yesmaximal, complemented
7C3×#63456C212yesmaximal
8C3×#63451C212yesmaximal, complemented
9C3×#63438C212yesmaximal, complemented
10C3×#5735—31nomaximal
11#10589837C311yesmaximal, characteristic, direct product factor
12C2×C3×Q8×C9C2211yescharacteristic
13C2×C3×Q8×C9C2211yescharacteristic
14C2×C3×D36C2211yescharacteristic
15C2×C3×C4×D9C2212yes—
16C2×C3×Dic18C2211yescharacteristic
17C3×C9×C4⋊C4C2212yes—
18C3×C9×C4⋊C4C2212yes—
19C3×C42×C9C2211yescharacteristic
20C3×(C2×D9)⋊C4C2212yes—
21C3×(C2×D9)⋊C4C2212yes—
22C3×(C2×D9)⋊C4C2212yes—
23C3×(C2×D9)⋊C4C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C3×C4⋊Dic9C2212yes—
26C3×Dic9⋊C4C2212yes—
27C3×Dic9⋊C4C2212yes—
28C3×Dic9⋊C4C2212yes—
29C3×Dic9⋊C4C2212yes—
30C3×Dic9⋊C4C2212yes—
31C3×C4×Dic9C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33C32×C4⋊Q8D311yescharacteristic
34C3×#947—31no—
35C3×#946—32no—
36C3×#944—31no—
37C3×#898—32no—
38C3×#896—32no—
39C3×#894—32no—
40C3×#889—32no—
41C3×#876—32no—
42C9×C4⋊Q8—21no—
43C9×C4⋊Q8C2×C311yescharacteristic, complemented
44#63509C2×C311yescharacteristic
45#63508C2×C312yescomplemented
46#63506C2×C311yescharacteristic, complemented
47#63460C2×C312yescomplemented
48#63458C2×C312yescomplemented
49#63456C2×C312yes—
50#63451C2×C312yescomplemented
51#63438C2×C312yescomplemented
52C22×C3×D9C2312yes—
53C3×Q8×C9—41no—
54C3×Q8×C9—41no—
55C2×C3×C4×C9C2311yescharacteristic
56C2×C3×C4×C9C2312yes—
57C2×C3×C4×C9C2311yescharacteristic
58C2×C3×C4×C9C2311yescharacteristic
59C2×C3×C4×C9C2312yes—
60C2×C3×Dic9C2312yes—
61C2×C3×Dic9C2312yes—
62C2×C3×Dic9C2312yes—
63C3×D36—41no—
64C3×C4×D9—42no—
65C3×Dic18—41no—
66C3×#563—91no—
67#5735—31no—
68C2×C32×Q8C2×D311yescharacteristic
69C2×C32×Q8C2×D311yescharacteristic
70C2×C3×D12—31no—
71C2×C3×C4×D3—32no—
72C2×C3×Dic6—31no—
73C32×C4⋊C4C2×D312yes—
74C32×C4⋊C4C2×D312yes—
75C32×C42C2×D311yescharacteristic
76C3×(C2×D3)⋊C4—32no—
77C3×(C2×D3)⋊C4—32no—
78C3×(C2×D3)⋊C4—32no—
79C3×(C2×D3)⋊C4—32no—
80C3×(C2×D3)⋊C4—32no—
81C3×C4⋊Dic3—32no—
82C3×Dic3⋊C4—32no—
83C3×Dic3⋊C4—32no—
84C3×Dic3⋊C4—32no—
85C3×Dic3⋊C4—32no—
86C3×Dic3⋊C4—32no—
87C3×C4×Dic3—31no—
88C3×C4×Dic3—31no—
89C2×Q8×C9—21no—
90C2×Q8×C9—21no—
91C2×Q8×C9C22×C311yescharacteristic
92C2×Q8×C9C22×C311yescharacteristic
93C2×D36C22×C311yescharacteristic
94C2×C4×D9C22×C312yes—
95C2×Dic18C22×C311yescharacteristic
96C9×C4⋊C4—22no—
97C9×C4⋊C4—22no—
98C9×C4⋊C4C22×C312yes—
99C9×C4⋊C4C22×C312yes—
100C42×C9—21no—
101C42×C9C22×C311yescharacteristic
102(C2×D9)⋊C4C22×C312yes—
103(C2×D9)⋊C4C22×C312yes—
104(C2×D9)⋊C4C22×C312yes—
105(C2×D9)⋊C4C22×C312yes—
106(C2×D9)⋊C4C22×C312yes—
107C4⋊Dic9C22×C312yes—
108Dic9⋊C4C22×C312yes—
109Dic9⋊C4C22×C312yes—
110Dic9⋊C4C22×C312yes—
111Dic9⋊C4C22×C312yes—
112Dic9⋊C4C22×C312yes—
113C4×Dic9C22×C311yescharacteristic
114C4×Dic9C22×C311yescharacteristic
115C22×C3×C9C2411yescharacteristic
116C2×C3×D9—22no—
117C2×C3×D9—24no—
118C3×C4×C9—21no—
119C3×C4×C9—21no—
120C3×C4×C9—22no—
121C3×C4×C9—21no—
122C3×C4×C9—22no—
123C3×Dic9—22no—
124C3×Dic9—22no—
125C3×Dic9—22no—
126C3×C4⋊Q8—21no—
127C3×C4⋊Q8—91no—
128C3×C4⋊Q8D911yescharacteristic, complemented
129C3×C4⋊Q8C3×D311yescharacteristic
130C3×#126—92no—
131C3×#126—92no—
132C3×#125—92no—
133C3×#124—91no—
134C3×#123—92no—
135C3×C22⋊Q8—92no—
136C3×C22⋊Q8—92no—
137#947—31no—
138#946—32no—
139#944—31no—
140#898—32no—
141#896—32no—
142#894—32no—
143#889—32no—
144#876—32no—
145C22×C3×D3—32no—
146C32×Q8—41no—
147C32×Q8—41no—
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D312yes—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D312yes—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—32no—
156C3×D12—121no—
157C3×C4×D3—122no—
158C3×Dic6—121no—
159C22×D9C23×C312yes—
160Q8×C9—41no—
161Q8×C9—41no—
162Q8×C9—42no—
163Q8×C9—42no—
164C2×C4×C9—22no—
165C2×C4×C9—21no—
166C2×C4×C9—21no—
167C2×C4×C9—21no—
168C2×C4×C9—22no—
169C2×C4×C9C23×C311yescharacteristic
170C2×C4×C9C23×C311yescharacteristic
171C2×C4×C9C23×C312yes—
172C2×C4×C9C23×C311yescharacteristic
173C2×C4×C9C23×C312yes—
174C2×Dic9C23×C312yes—
175C2×Dic9C23×C312yes—
176C2×Dic9C23×C312yes—
177D36—41no—
178C4×D9—42no—
179Dic18—41no—
180#563—91noSylow
181C2×C3×C9H(5,𝔽2)11yescharacteristic
182C2×C3×C92−1+412yes—
183C3×D9—42no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—91no—
187C2×C3×Q8C2×D911yescharacteristic
188C2×C3×Q8C2×D911yescharacteristic
189C2×C3×Q8C2×C3×D311yescharacteristic
190C2×C3×Q8C2×C3×D311yescharacteristic
191C2×C3×D4—91no—
192C22×C3×C4—92no—
193C2×D12—31no—
194C2×C4×D3—32no—
195C2×Dic6—31no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—92no—
199C3×C4⋊C4—92no—
200C3×C4⋊C4—92no—
201C3×C4⋊C4—92no—
202C3×C4⋊C4—92no—
203C3×C4⋊C4—92no—
204C3×C4⋊C4C2×D912yes—
205C3×C4⋊C4C2×D912yes—
206C3×C4⋊C4C2×C3×D312yes—
207C3×C4⋊C4C2×C3×D312yes—
208C3×C22⋊C4—92no—
209C3×C22⋊C4—92no—
210C3×C22⋊C4—92no—
211C3×C22⋊C4—92no—
212C3×C22⋊C4—92no—
213C3×C42—21no—
214C3×C42—91no—
215C3×C42—91no—
216C3×C42C2×D911yescharacteristic
217C3×C42C2×C3×D311yescharacteristic
218(C2×D3)⋊C4—32no—
219(C2×D3)⋊C4—32no—
220(C2×D3)⋊C4—32no—
221(C2×D3)⋊C4—32no—
222(C2×D3)⋊C4—32no—
223C4⋊Dic3—32no—
224Dic3⋊C4—32no—
225Dic3⋊C4—32no—
226Dic3⋊C4—32no—
227Dic3⋊C4—32no—
228Dic3⋊C4—32no—
229C4×Dic3—31no—
230C4×Dic3—31no—
231C22×C32C23×D311yescharacteristic
232C2×C3×D3—62no—
233C2×C3×D3—64no—
234C32×C4—21no—
235C32×C4—22no—
236C32×C4—21no—
237C32×C4—22no—
238C32×C4—21no—
239C3×Dic3—62no—
240C3×Dic3—62no—
241C3×Dic3—62no—
242C22×C9—21no—
243C22×C9C24×C311yescharacteristic
244C2×D9—22no—
245C2×D9—24no—
246C4×C9—24no—
247C4×C9—22no—
248C4×C9—21no—
249C4×C9—22no—
250C4×C9—21no—
251C4×C9—21no—
252C4×C9—22no—
253C4×C9—41no—
254C4×C9—42no—
255C4×C9—41no—
256Dic9—22no—
257Dic9—22no—
258Dic9—22no—
259C4⋊Q8—91no—
260C4⋊Q8C3×D911yescharacteristic, complemented
261#126—92no—
262#126—92no—
263#125—92no—
264#124—91no—
265#123—92no—
266C22⋊Q8—92no—
267C22⋊Q8—92no—
268C3×C9#56311yescharacteristic, complemented, Sylow
269C23×C3—92no—
270C22×D3—32no—
271C3×Q8—41no—
272C3×Q8—41no—
273C3×Q8—41no—
274C3×Q8—41no—
275C3×Q8—42no—
276C3×Q8—42no—
277C3×Q8—361no—
278C3×D4—361no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—22no—
284C2×C3×C4—92no—
285C2×C3×C4—92no—
286C2×C3×C4—92no—
287C2×C3×C4—362no—
288C2×C3×C4C22×D911yescharacteristic
289C2×C3×C4C22×D911yescharacteristic
290C2×C3×C4C22×D912yes—
291C2×C3×C4C22×D911yescharacteristic
292C2×C3×C4C22×D912yes—
293C2×C3×C4C22×C3×D311yescharacteristic
294C2×C3×C4C22×C3×D311yescharacteristic
295C2×C3×C4C22×C3×D312yes—
296C2×C3×C4C22×C3×D311yescharacteristic
297C2×C3×C4C22×C3×D312yes—
298C2×Dic3—32no—
299C2×Dic3—32no—
300C2×Dic3—32no—
301D12—121no—
302C4×D3—122no—
303Dic6—121no—
304C2×C32#100411yescharacteristic
305C2×C32#100712yes—
306C3×D3—122no—
307C2×C9—21no—
308C2×C9—22no—
309C2×C9C3×H(5,𝔽2)11yescharacteristic
310C2×C9C3×2−1+412yes—
311D9—42no—
312C2×Q8—91no—
313C2×Q8C2×C3×D911yescharacteristic
314C2×Q8C2×C3×D911yescharacteristic
315C2×D4—91no—
316C22×C4—92no—
317C4⋊C4—92no—
318C4⋊C4—92no—
319C4⋊C4—92no—
320C4⋊C4—92no—
321C4⋊C4—92no—
322C4⋊C4—92no—
323C4⋊C4C2×C3×D912yes—
324C4⋊C4C2×C3×D912yes—
325C22⋊C4—92no—
326C22⋊C4—92no—
327C22⋊C4—92no—
328C22⋊C4—92no—
329C22⋊C4—92no—
330C42—91no—
331C42—91no—
332C42C2×C3×D911yescharacteristic
333C22×C3—21no—
334C22×C3—182no—
335C22×C3—184no—
336C22×C3C23×D911yescharacteristic
337C22×C3C23×C3×D311yescharacteristic
338C2×D3—62no—
339C2×D3—64no—
340C3×C4—22no—
341C3×C4—24no—
342C3×C4—21no—
343C3×C4—22no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—22no—
347C3×C4—22no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—22no—
352C3×C4—42no—
353C3×C4—41no—
354C3×C4—41no—
355C3×C4—182no—
356C3×C4—182no—
357C3×C4—182no—
358Dic3—62no—
359Dic3—62no—
360Dic3—62no—
361C32#573511yescharacteristic
362C9—21no—
363C9C3×#56311yescharacteristic, complemented
364C23—92no—
365Q8—41no—
366Q8—41no—
367Q8—361no—
368D4—361no—
369C2×C4—92no—
370C2×C4—92no—
371C2×C4—92no—
372C2×C4—362no—
373C2×C4C22×C3×D911yescharacteristic
374C2×C4C22×C3×D911yescharacteristic
375C2×C4C22×C3×D912yes—
376C2×C4C22×C3×D911yescharacteristic
377C2×C4C22×C3×D912yes—
378C2×C3—21no—
379C2×C3—22no—
380C2×C3—362no—
381C2×C3#6371011yescharacteristic
382C2×C3#6371312yes—
383C2×C3C3×#100411yescharacteristic
384C2×C3C3×#100712yes—
385D3—122no—
386C22—182no—
387C22—184no—
388C22C23×C3×D911yescharacteristic
389C4—22no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—182no—
395C4—182no—
396C4—182no—
397C3—21no—
398C3#1058983711yescharacteristic, direct product factor
399C3C3×#573511yescharacteristic
400C2—362no—
401C2C3×#6371011yescharacteristic
402C2C3×#6371312yes—
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×C4⋊Q8C2
socleC22×C32C23×D3
prime radicalC2×C3×D36C22
hypercenterC3×C4⋊Q8D9
hypocenterC9C3×#563
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C9×C4⋊Q8C2semidirect
3. C3×#63508C2semidirect
4. C3×#63506C2semidirect
5. C3×#63460C2semidirect
6. C3×#63458C2semidirect
8. C3×#63451C2semidirect
9. C3×#63438C2semidirect
11. #10589837C3direct (N × H)
43. C9×C4⋊Q8C2×C3semidirect
45. #63508C2×C3semidirect
46. #63506C2×C3semidirect
47. #63460C2×C3semidirect
48. #63458C2×C3semidirect
50. #63451C2×C3semidirect
51. #63438C2×C3semidirect
128. C3×C4⋊Q8D9semidirect
260. C4⋊Q8C3×D9semidirect
268. C3×C9#563semidirect
363. C9C3×#563semidirect
398. C3#10589837direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×C4⋊Q82
C3×#635092
C3×#635082
C3×#635062
C3×#634602
C3×#634582
C3×#634562
C3×#634512
C3×#634382
C3×#57353
#105898373