Finite Group Databasebeta

← C3×#10592833 #13163171

Subgroups — C3×#10592833 #13163171

Order 1728.

992 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
1C2×C33×M16C211yesmaximal, characteristic, complemented
2C3×#64139C211yesmaximal, characteristic
3C3×#64133C211yesmaximal, characteristic, complemented
4C2×C3×#3744C211yesmaximal, characteristic, complemented
5C3×#63648C211yesmaximal, characteristic, complemented
6C3×#63645C211yesmaximal, characteristic, complemented
7C3×#63643C211yesmaximal, characteristic, complemented
8#10592833C311yesmaximal, characteristic, direct product factor
9C3×#5134—34nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×C3×#638—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C33×M16—22no—
15C2×C33×C8C2211yescharacteristic
16C2×C33×C8C2211yescharacteristic
17C3×#3757—21no—
18C3×#3752—21no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×#3750—21no—
22C3×#3744—22no—
23C2×C32×M16—24no—
24C2×C32×M16D314yescomplemented
25C2×C32×M16C2×C311yescharacteristic, complemented
26#64139C2×C311yescharacteristic
27#64133C2×C311yescharacteristic, complemented
28C2×#3744C2×C311yescharacteristic, complemented
29C3×#930—34no—
30C3×#924—34no—
31C2×C3×#204—34no—
32#63648C2×C311yescharacteristic, complemented
33#63645C2×C311yescharacteristic, complemented
34#63643C2×C311yescharacteristic, complemented
35C3×#821—34no—
36C3×#818—34no—
37C3×#816—34no—
38C23×C33D411yescharacteristic
39C22×C3×C3⋊D3—21no—
40C2×C33×C4—21no—
41C2×C33×C4—21no—
42C2×C33×C4D411yescharacteristic
43C2×C33×C4C2311yescharacteristic
44C3×#638—42no—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×#634—21no—
51C3×#634—21no—
52C33×C8—21no—
53C33×C8D412yescomplemented
54C3×#468—91no—
55#5134—34no—
56C22×C32×C4—24no—
57C22×C32×C4C2×D314yes—
58C22×C32×C4C22×C311yescharacteristic
59C2×#638—21no—
60C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
61C2×#634C22×C311yescharacteristic
62C2×C3×#67—64no—
63C2×C3×D12—34no—
64C2×C3×Dic6—34no—
65C32×M16—216no—
66C32×M16—28no—
67C32×M16—22no—
68C2×C32×C8—24no—
69C2×C32×C8—24no—
70C2×C32×C8C2×D314yes—
71C2×C32×C8C2×D314yes—
72C2×C32×C8C22×C311yescharacteristic
73C2×C32×C8C22×C311yescharacteristic
74#3757—21no—
75#3752—21no—
76C4⋊(C3⋊Dic3)C22×C311yescharacteristic
77C4⋊(C3⋊Dic3)C22×C311yescharacteristic
78#3750—21no—
79#3744—22no—
80C3×#217—64no—
81C3×(C2×D3)⋊C4—64no—
82C3×C4⋊Dic3—34no—
83C3×C4⋊Dic3—34no—
84C3×Dic3⋊C4—64no—
85C3×#204—68no—
86C22×C33—21no—
87C22×C33—21no—
88C22×C33—21no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C33×C4—21no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—41no—
97C3×C3⋊Dic3—41no—
98C3×C3⋊Dic3—41no—
99C2×C3×SD16—91no—
100C2×C3×M16—24no—
101C2×C3×M16C3×D314yescomplemented
102C2×C3×M16C3⋊D311yescharacteristic, complemented
103C3×C22⋊Q8—91no—
104C3×#121—91no—
105#930—34no—
106#924—34no—
107C2×#204—34no—
108C3×#107—91no—
109C3×#103—91no—
110C3×#102—91no—
111#821—34no—
112#818—34no—
113#816—34no—
114C23×C32—24no—
115C23×C32D1214yes—
116C23×C32C3×D411yescharacteristic
117C22×C3⋊D3—21no—
118C22×C3×D3—64no—
119C2×C32×C4—28no—
120C2×C32×C4—28no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4—24no—
127C2×C32×C4D1214yes—
128C2×C32×C4C3×D411yescharacteristic
129C2×C32×C4C22×D314yes—
130C2×C32×C4C23×C311yescharacteristic
131#638—42no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—21no—
137#634—21no—
138#634—21no—
139C3×#67—128no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3—64no—
142C2×C3×Dic3—64no—
143C3×D12—64no—
144C3×D12—64no—
145C3×Dic6—64no—
146C3×Dic6—64no—
147C32×C8—28no—
148C32×C8—24no—
149C32×C8—21no—
150C32×C8—28no—
151C32×C8#6718yescomplemented
152C32×C8C3×D412yescomplemented
153#468—91noSylow
154C2×C33—41no—
155C2×C33#12111yescharacteristic
156C2×C33AGL(1,ℤ/8)11yescharacteristic
157C2×C33#13711yescharacteristic
158C3×C3⋊D3—81no—
159C2×C3×Q8—91no—
160C2×C3×D4—91no—
161C2×C3×D4—181no—
162C22×C3×C4—24no—
163C22×C3×C4C2×C3×D314yes—
164C22×C3×C4C2×C3⋊D311yescharacteristic
165C2×#67—64no—
166C2×D12—34no—
167C2×Dic6—34no—
168C3×SD16—182no—
169C3×M16—216no—
170C3×M16—28no—
171C3×M16—22no—
172C2×C3×C8—24no—
173C2×C3×C8—24no—
174C2×C3×C8C2×C3×D314yes—
175C2×C3×C8C2×C3×D314yes—
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—64no—
184(C2×D3)⋊C4—64no—
185C4⋊Dic3—34no—
186C4⋊Dic3—34no—
187Dic3⋊C4—64no—
188#204—68no—
189C22×C32—28no—
190C22×C32—24no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—24no—
195C22×C32—24no—
196C22×C32—24no—
197C22×C32—44no—
198C22×C32—44no—
199C22×C32C2×D1214yes—
200C22×C32C2×C3×D411yescharacteristic
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3×D3—124no—
205C2×C3×D3—124no—
206C2×C3×D3—124no—
207C32×C4—28no—
208C32×C4—24no—
209C32×C4—21no—
210C32×C4—24no—
211C32×C4—24no—
212C32×C4#23514yes—
213C32×C4C2×#6714yes—
214C32×C4C2×C3×D411yescharacteristic
215C32×C4C3×#4111yescharacteristic
216C3⋊Dic3—41no—
217C3⋊Dic3—41no—
218C3⋊Dic3—41no—
219C3×Dic3—124no—
220C3×Dic3—124no—
221C3×Dic3—124no—
222C2×SD16—91no—
223C2×M16C3×C3⋊D311yescharacteristic, complemented
224C22⋊Q8—91no—
225#121—91no—
226#107—91no—
227#103—91no—
228#102—91no—
229C33#46811yescharacteristic, complemented, Sylow
230C23×C3—24no—
231C23×C3—181no—
232C23×C3C3×D1214yes—
233C23×C3C4⋊(C3⋊D3)11yescharacteristic
234C22×D3—64no—
235C3×Q8—181no—
236C3×Q8—181no—
237C3×D4—181no—
238C3×D4—181no—
239C3×D4—362no—
240C2×C3×C4—28no—
241C2×C3×C4—28no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C3×D1214yes—
252C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
253C2×C3×C4C22×C3×D314yes—
254C2×C3×C4C22×C3⋊D311yescharacteristic
255#67—128no—
256C2×Dic3—64no—
257C2×Dic3—64no—
258C2×Dic3—64no—
259D12—64no—
260D12—64no—
261Dic6—64no—
262Dic6—64no—
263C3×C8—28no—
264C3×C8—24no—
265C3×C8—21no—
266C3×C8—28no—
267C3×C8C3×#6718yescomplemented
268C3×C8#63812yescomplemented
269C2×C32—24no—
270C2×C32—24no—
271C2×C32—24no—
272C2×C32—48no—
273C2×C32—44no—
274C2×C32—41no—
275C2×C32#90814yes—
276C2×C32#90914yes—
277C2×C32#93014yes—
278C2×C32C3×#12111yescharacteristic
279C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
280C2×C32C3×#13711yescharacteristic
281C3⋊D3—81no—
282C3×D3—244no—
283C2×Q8—91no—
284C2×D4—91no—
285C2×D4—181no—
286C22×C4C2×C3×C3⋊D311yescharacteristic
287SD16—182no—
288M16—22no—
289C2×C8C2×C3×C3⋊D311yescharacteristic
290C2×C8C2×C3×C3⋊D311yescharacteristic
291C4⋊C4—91no—
292C4⋊C4—91no—
293C4⋊C4—181no—
294C22⋊C4—181no—
295C22⋊C4—181no—
296C22×C3—28no—
297C22×C3—24no—
298C22×C3—24no—
299C22×C3—24no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—24no—
304C22×C3—44no—
305C22×C3—44no—
306C22×C3—361no—
307C22×C3—361no—
308C22×C3—361no—
309C22×C3C2×C3×D1214yes—
310C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
311C2×D3—124no—
312C2×D3—124no—
313C2×D3—124no—
314C3×C4—28no—
315C3×C4—24no—
316C3×C4—21no—
317C3×C4—24no—
318C3×C4—24no—
319C3×C4—361no—
320C3×C4—361no—
321C3×C4—361no—
322C3×C4C3×#23514yes—
323C3×C4C2×C3×#6714yes—
324C3×C4#382811yescharacteristic
325C3×C4C2×#63811yescharacteristic
326Dic3—124no—
327Dic3—124no—
328Dic3—124no—
329C32—24no—
330C32#513414yescomplemented
331C32C3×#46811yescharacteristic, complemented
332C23—181no—
333C23C3×C4⋊(C3⋊D3)11yescharacteristic
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×C4⋊(C3⋊D3)11yescharacteristic
345C2×C4C22×C3×C3⋊D311yescharacteristic
346C8—21no—
347C8C3×#63812yescomplemented
348C2×C3—24no—
349C2×C3—24no—
350C2×C3—24no—
351C2×C3—48no—
352C2×C3—41no—
353C2×C3—44no—
354C2×C3—721no—
355C2×C3C3×#90814yes—
356C2×C3C3×#90914yes—
357C2×C3C3×#93014yes—
358C2×C3#6411711yescharacteristic
359C2×C3#6411811yescharacteristic
360C2×C3#6413911yescharacteristic
361D3—244no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—361no—
366C22—361no—
367C22—361no—
368C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
369C4—21no—
370C4—361no—
371C4—361no—
372C4—361no—
373C4C3×#382811yescharacteristic
374C4C2×C3×#63811yescharacteristic
375C3—24no—
376C3C3×#513414yescomplemented
377C3#1059283311yescharacteristic, direct product factor
378C2—41no—
379C2—721no—
380C2C3×#6411711yescharacteristic
381C2C3×#6411811yescharacteristic
382C2C3×#6413911yescharacteristic
subgroupG/H
centerC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×M16C2
socleC22×C33C2×D4
prime radicalC3×#64139C2
hypercenterC2×C3×M16C3⋊D3
hypocenterC32C3×#468
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×M16

Complements

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

NH ≅ G/Ntype
1. C2×C33×M16C2semidirect
3. C3×#64133C2semidirect
4. C2×C3×#3744C2semidirect
5. C3×#63648C2semidirect
6. C3×#63645C2semidirect
7. C3×#63643C2semidirect
8. #10592833C3direct (N × H)
24. C2×C32×M16D3semidirect
25. C2×C32×M16C2×C3semidirect
27. #64133C2×C3semidirect
28. C2×#3744C2×C3semidirect
32. #63648C2×C3semidirect
33. #63645C2×C3semidirect
34. #63643C2×C3semidirect
53. C33×C8D4semidirect
101. C2×C3×M16C3×D3semidirect
102. C2×C3×M16C3⋊D3semidirect
151. C32×C8#67semidirect
152. C32×C8C3×D4semidirect
223. C2×M16C3×C3⋊D3semidirect
229. C33#468semidirect
267. C3×C8C3×#67semidirect
268. C3×C8#638semidirect
330. C32#5134semidirect
331. C32C3×#468semidirect
347. C8C3×#638semidirect
376. C3C3×#5134semidirect
377. C3#10592833direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×M162
C3×#641392
C3×#641332
C2×C3×#37442
C3×#636482
C3×#636452
C3×#636432
#105928333
C3×#51343