Finite Group Databasebeta

← C3×#10592747 #13163086

Subgroups — C3×#10592747 #13163086

Order 1728.

1040 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3C3×#64139C211yesmaximal, characteristic
4C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic, complemented
7C3×#63635C211yesmaximal, characteristic, complemented
8#10592747C311yesmaximal, characteristic, direct product factor
9C3×#5048—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C2×C3×#638—21no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C33×C4⋊C4—21no—
15C33×C4⋊C4C2211yescharacteristic
16C33×C22⋊C4—21no—
17C3×#3755—22no—
18C3×#3752—21no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)—22no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C32×C22⋊Q8—24no—
24C32×C22⋊Q8D314yescomplemented
25C32×C22⋊Q8C2×C311yescharacteristic, complemented
26C2×#3755C2×C311yescharacteristic, complemented
27#64139C2×C311yescharacteristic
28C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
29C2×C3×#215—34no—
30C3×#930—34no—
31C22×C3×C3⋊C8—34no—
32#63662C2×C311yescharacteristic, complemented
33#63636C2×C311yescharacteristic, complemented
34#63635C2×C311yescharacteristic, complemented
35C3×#835—34no—
36C3×#809—34no—
37C3×#808—34no—
38C23×C33D411yescharacteristic
39C22×C3×C3⋊D3—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—21no—
43C2×C33×C4—21no—
44C2×C33×C4—21no—
45C2×C33×C4—21no—
46C2×C33×C4D411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C3×#638—42no—
49C2×C3×C3⋊Dic3—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C3×C3⋊(C3⋊C8)D412yescomplemented
54C3×#465—91no—
55#5048—34no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C22×C311yescharacteristic
59C22×C32×C4—24no—
60C22×C32×C4C2×D314yes—
61C22×C32×C4C22×C311yescharacteristic
62C2×#638—21no—
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C2×C3×#67—64no—
65C2×C3×D12—34no—
66C32×C4⋊C4—28no—
67C32×C4⋊C4—24no—
68C32×C4⋊C4—24no—
69C32×C4⋊C4—21no—
70C32×C4⋊C4C2×D314yes—
71C32×C4⋊C4C22×C311yescharacteristic
72C32×C22⋊C4—28no—
73C32×C22⋊C4—24no—
74C32×C22⋊C4—21no—
75#3755—22no—
76#3752—21no—
77C4⋊(C3⋊Dic3)C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)—22no—
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81C3×#215—68no—
82C3×(C2×D3)⋊C4—64no—
83C3×C4⋊Dic3—34no—
84C2×C3×C3⋊C8—34no—
85C2×C3×C3⋊C8—34no—
86C2×C3×C3⋊C8—68no—
87C22×C33—21no—
88C22×C33—21no—
89C22×C33SD1612yescomplemented
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—41no—
94C33×C4—21no—
95C33×C4—41no—
96C33×C4—41no—
97C33×C4C2×D411yescharacteristic
98C33×C4#4111yescharacteristic
99C3×C3⋊Dic3—41no—
100C2×C3×SD16—91no—
101C22×C3×C8—91no—
102C3×C22⋊Q8—24no—
103C3×C22⋊Q8C3×D314yescomplemented
104C3×C22⋊Q8C3⋊D311yescharacteristic, complemented
105C3×#121—91no—
106C2×#215—34no—
107#930—34no—
108C22×C3⋊C8—34no—
109C3×#106—91no—
110C3×#103—91no—
111C3×#102—91no—
112#835—34no—
113#809—34no—
114#808—34no—
115C23×C32—24no—
116C23×C32#6714yes—
117C23×C32C3×D411yescharacteristic
118C22×C3⋊D3—21no—
119C22×C3×D3—64no—
120C32×Q8—21no—
121C32×Q8—21no—
122C32×Q8—24no—
123C32×Q8—24no—
124C32×Q8—44no—
125C32×Q8—44no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—28no—
129C2×C32×C4—28no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—24no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—44no—
139C2×C32×C4—44no—
140C2×C32×C4#6714yes—
141C2×C32×C4C3×D411yescharacteristic
142C2×C32×C4C22×D314yes—
143C2×C32×C4C23×C311yescharacteristic
144#638—42no—
145C2×C3⋊Dic3—21no—
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)—21no—
148C3×#67—128no—
149C2×C3×Dic3—64no—
150C3×D12—64no—
151C3×D12—64no—
152C3⋊(C3⋊C8)—21no—
153C3⋊(C3⋊C8)C3×D412yescomplemented
154C3×C3⋊C8—38no—
155C3×C3⋊C8—64no—
156#465—91noSylow
157C2×C33—22no—
158C2×C33#12111yescharacteristic
159C2×C33C2×SD1611yescharacteristic
160C2×C33#13511yescharacteristic
161C3×C3⋊D3—81no—
162C2×C3×Q8—24no—
163C2×C3×Q8C2×C3×D314yes—
164C2×C3×Q8C2×C3⋊D311yescharacteristic
165C2×C3×D4—91no—
166C2×C3×D4—181no—
167C22×C3×C4—24no—
168C22×C3×C4C2×C3×D314yes—
169C22×C3×C4C2×C3⋊D311yescharacteristic
170C2×#67—64no—
171C2×D12—34no—
172C3×SD16—182no—
173C2×C3×C8—91no—
174C2×C3×C8—91no—
175C2×C3×C8—182no—
176C3×C4⋊C4—28no—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—91no—
181C3×C4⋊C4C2×C3×D314yes—
182C3×C4⋊C4C2×C3⋊D311yescharacteristic
183C3×C22⋊C4—28no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—21no—
186C3×C22⋊C4—181no—
187#215—68no—
188(C2×D3)⋊C4—64no—
189C4⋊Dic3—34no—
190C2×C3⋊C8—34no—
191C2×C3⋊C8—34no—
192C2×C3⋊C8—68no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—28no—
196C22×C32—24no—
197C22×C32—24no—
198C22×C32—24no—
199C22×C32—44no—
200C22×C32—44no—
201C22×C32#21518yescomplemented
202C22×C32C3×SD1612yescomplemented
203C22×C32C2×#6714yes—
204C22×C32C2×C3×D411yescharacteristic
205C2×C3⋊D3—41no—
206C2×C3⋊D3—41no—
207C2×C3⋊D3—41no—
208C2×C3×D3—124no—
209C2×C3×D3—124no—
210C2×C3×D3—124no—
211C32×C4—21no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—24no—
215C32×C4—44no—
216C32×C4—48no—
217C32×C4—48no—
218C32×C4—44no—
219C32×C4—44no—
220C32×C4—41no—
221C32×C4—41no—
222C32×C4D3×D414yes—
223C32×C4#23714yes—
224C32×C4C2×C3×D411yescharacteristic
225C32×C4C3×#4111yescharacteristic
226C3⋊Dic3—41no—
227C3×Dic3—124no—
228C2×SD16—91no—
229C22×C8—91no—
230C22⋊Q8C3×C3⋊D311yescharacteristic, complemented
231#121—91no—
232#106—91no—
233#103—91no—
234#102—91no—
235C33#46511yescharacteristic, complemented, Sylow
236C23×C3—24no—
237C23×C3—181no—
238C23×C3C3×#6714yes—
239C23×C3#63811yescharacteristic
240C22×D3—64no—
241C3×Q8—24no—
242C3×Q8—24no—
243C3×Q8—21no—
244C3×Q8—21no—
245C3×Q8—44no—
246C3×Q8—44no—
247C3×D4—181no—
248C3×D4—181no—
249C3×D4—362no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—28no—
255C2×C3×C4—28no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—44no—
263C2×C3×C4—44no—
264C2×C3×C4—181no—
265C2×C3×C4C3×#6714yes—
266C2×C3×C4#63811yescharacteristic
267C2×C3×C4C22×C3×D314yes—
268C2×C3×C4C22×C3⋊D311yescharacteristic
269#67—128no—
270C2×Dic3—64no—
271D12—64no—
272D12—64no—
273C3×C8—92no—
274C3×C8—181no—
275C3⋊C8—38no—
276C3⋊C8—64no—
277C2×C32—22no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—24no—
281C2×C32—28no—
282C2×C32—48no—
283C2×C32#93914yes—
284C2×C32C2×#21514yes—
285C2×C32#95014yes—
286C2×C32C3×#12111yescharacteristic
287C2×C32C2×C3×SD1611yescharacteristic
288C2×C32C3×#13511yescharacteristic
289C3⋊D3—81no—
290C3×D3—244no—
291C2×Q8C2×C3×C3⋊D311yescharacteristic
292C2×D4—91no—
293C2×D4—181no—
294C22×C4C2×C3×C3⋊D311yescharacteristic
295SD16—182no—
296C2×C8—91no—
297C2×C8—91no—
298C2×C8—182no—
299C4⋊C4—21no—
300C4⋊C4—91no—
301C4⋊C4C2×C3×C3⋊D311yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—181no—
304C22×C3—24no—
305C22×C3—24no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—28no—
309C22×C3—24no—
310C22×C3—44no—
311C22×C3—44no—
312C22×C3—361no—
313C22×C3—361no—
314C22×C3—361no—
315C22×C3C3×#21518yescomplemented
316C22×C3#375512yescomplemented
317C22×C3C2×C3×#6714yes—
318C22×C3C2×#63811yescharacteristic
319C2×D3—124no—
320C2×D3—124no—
321C2×D3—124no—
322C3×C4—24no—
323C3×C4—21no—
324C3×C4—24no—
325C3×C4—24no—
326C3×C4—44no—
327C3×C4—48no—
328C3×C4—48no—
329C3×C4—44no—
330C3×C4—44no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—361no—
334C3×C4C3×D3×D414yes—
335C3×C4C3×#23714yes—
336C3×C4D4×C3⋊D311yescharacteristic
337C3×C4#383011yescharacteristic
338Dic3—124no—
339C32—24no—
340C32#504814yescomplemented
341C32C3×#46511yescharacteristic, complemented
342C23—181no—
343C23C3×#63811yescharacteristic
344Q8—21no—
345Q8—21no—
346D4—181no—
347D4—181no—
348D4—362no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—181no—
354C2×C4C3×#63811yescharacteristic
355C2×C4C22×C3×C3⋊D311yescharacteristic
356C8—92no—
357C8—181no—
358C2×C3—28no—
359C2×C3—22no—
360C2×C3—24no—
361C2×C3—24no—
362C2×C3—24no—
363C2×C3—48no—
364C2×C3—721no—
365C2×C3C3×#93914yes—
366C2×C3C2×C3×#21514yes—
367C2×C3C3×#95014yes—
368C2×C3#6414811yescharacteristic
369C2×C3C2×#375511yescharacteristic
370C2×C3#6415911yescharacteristic
371D3—244no—
372C22—21no—
373C22—21no—
374C22—361no—
375C22—361no—
376C22—361no—
377C22C3×#375512yescomplemented
378C22C2×C3×#63811yescharacteristic
379C4—21no—
380C4—41no—
381C4—41no—
382C4—361no—
383C4C3×D4×C3⋊D311yescharacteristic
384C4C3×#383011yescharacteristic
385C3—24no—
386C3C3×#504814yescomplemented
387C3#1059274711yescharacteristic, direct product factor
388C2—22no—
389C2—721no—
390C2C3×#6414811yescharacteristic
391C2C2×C3×#375511yescharacteristic
392C2C3×#6415911yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C22⋊Q8C2
socleC22×C33C2×D4
prime radicalC3×#64139C2
hypercenterC3×C22⋊Q8C3⋊D3
hypocenterC32C3×#465
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C33×C22⋊Q8C2semidirect
2. C2×C3×#3755C2semidirect
4. C22×C3×C3⋊(C3⋊C8)C2semidirect
5. C3×#63662C2semidirect
6. C3×#63636C2semidirect
7. C3×#63635C2semidirect
8. #10592747C3direct (N × H)
24. C32×C22⋊Q8D3semidirect
25. C32×C22⋊Q8C2×C3semidirect
26. C2×#3755C2×C3semidirect
28. C22×C3⋊(C3⋊C8)C2×C3semidirect
32. #63662C2×C3semidirect
33. #63636C2×C3semidirect
34. #63635C2×C3semidirect
53. C3×C3⋊(C3⋊C8)D4semidirect
89. C22×C33SD16semidirect
103. C3×C22⋊Q8C3×D3semidirect
104. C3×C22⋊Q8C3⋊D3semidirect
153. C3⋊(C3⋊C8)C3×D4semidirect
201. C22×C32#215semidirect
202. C22×C32C3×SD16semidirect
230. C22⋊Q8C3×C3⋊D3semidirect
235. C33#465semidirect
315. C22×C3C3×#215semidirect
316. C22×C3#3755semidirect
340. C32#5048semidirect
341. C32C3×#465semidirect
377. C22C3×#3755semidirect
386. C3C3×#5048semidirect
387. C3#10592747direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C22⋊Q82
C2×C3×#37552
C3×#641392
C22×C3×C3⋊(C3⋊C8)2
C3×#636622
C3×#636362
C3×#636352
#105927473
C3×#50483