Finite Group Databasebeta

← C3×#10592508 #13162845

Subgroups — C3×#10592508 #13162845

Order 1728.

1056 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3C2×C3×#3746C211yesmaximal, characteristic, complemented
4C3×#64106C211yesmaximal, characteristic
5C33×#103C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic
7C3×#63637C211yesmaximal, characteristic, complemented
8#10592508C311yesmaximal, characteristic, direct product factor
9C3×#4809—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×Q8×C3⋊D3—22no—
12C3×Q8×C3⋊D3—22no—
13C2×C3×C4×C3⋊D3—21no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634—21no—
16C2×C3×#634C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C2×C33×C8C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20C3×#3756—22no—
21C3×#3752—21no—
22C3×C4⋊(C3⋊Dic3)—21no—
23C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
24C3×#3746—22no—
25C2×Q8×C3⋊D3C2×C311yescharacteristic
26C2×C3×D3×Q8—34no—
27C2×#3756C2×C311yescharacteristic, complemented
28C2×#3746C2×C311yescharacteristic, complemented
29#64106C2×C311yescharacteristic
30C2×C3×#216—34no—
31C2×C3×Dic12—34no—
32C3×#897—34no—
33C32×#103—24no—
34C32×#103D314yescomplemented
35C32×#103C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic
37#63637C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—34no—
39C3×#810—34no—
40C22×C3×C3⋊D3D411yescharacteristic
41C33×Q8—21no—
42C33×Q8D412yes—
43C2×C33×C4—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4×C3⋊D3—24no—
50C3×C4×C3⋊D3—21no—
51C3×C4×C3⋊D3—21no—
52C3×#634—24no—
53C3×#634—21no—
54C3×#634—21no—
55C3×#634—21no—
56C3×#634D412yes—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×#451—91no—
60#4809—34no—
61C2×C32×Q8—24no—
62C2×C32×Q8C2×D314yes—
63C2×C32×Q8C22×C311yescharacteristic
64Q8×C3⋊D3—22no—
65Q8×C3⋊D3—22no—
66C2×C4×C3⋊D3—21no—
67C2×C4×C3⋊D3C22×C311yescharacteristic
68C2×#634—21no—
69C2×#634C22×C311yescharacteristic
70C2×#634C22×C311yescharacteristic
71C3×D3×Q8—68no—
72C3×D3×Q8—68no—
73C2×C3×C4×D3—34no—
74C2×C3×C4×D3—64no—
75C2×C3×Dic6—34no—
76C2×C3×Dic6—34no—
77C2×C3×Dic6—64no—
78C2×C32×C8—24no—
79C2×C32×C8C2×D314yes—
80C2×C32×C8C22×C311yescharacteristic
81C32×C4⋊C4—24no—
82C32×C4⋊C4C2×D314yes—
83C32×C4⋊C4C22×C311yescharacteristic
84#3756—22no—
85#3752—21no—
86C4⋊(C3⋊Dic3)—21no—
87C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
88#3746—22no—
89C3×#216—68no—
90C3×(C2×D3)⋊C4—64no—
91C3×C4⋊Dic3—64no—
92C2×C3×C3⋊C8—34no—
93C3×Dic12—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—21no—
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3Q1612yescomplemented
98C33×C4—22no—
99C33×C4—41no—
100C33×C4C2×D411yescharacteristic
101C33×C4C2×D411yescharacteristic
102C3×C3⋊Dic3—21no—
103C3×C3⋊Dic3—22no—
104C3×C3⋊Dic3—41no—
105C22×C3×Q8—91no—
106C2×D3×Q8—34no—
107C2×C3×Q16—91no—
108C2×C3×Q16—91no—
109C3×C22⋊Q8—91no—
110C2×#216—34no—
111C2×Dic12—34no—
112#897—34no—
113C3×#103—24no—
114C3×#103—91no—
115C3×#103C3×D314yescomplemented
116C3×#103C3⋊D311yescharacteristic, complemented
117C3×C22⋊C8—91no—
118(C2×D3)⋊C8—34no—
119#810—34no—
120C22×C3⋊D3C3×D411yescharacteristic
121C22×C3×D3—34no—
122C32×Q8—21no—
123C32×Q8—28no—
124C32×Q8—24no—
125C32×Q8—44no—
126C32×Q8D1218yes—
127C32×Q8C3×D412yes—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4—44no—
134C2×C32×C4—44no—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C23×C311yescharacteristic
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3C3×D411yescharacteristic
140C4×C3⋊D3—24no—
141C4×C3⋊D3—21no—
142C4×C3⋊D3—21no—
143#634—24no—
144#634—21no—
145#634—21no—
146#634—21no—
147#634C3×D412yes—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—64no—
150C2×C3×Dic3—64no—
151C3×C4×D3—616no—
152C3×C4×D3—64no—
153C3×C4×D3—64no—
154C3×Dic6—38no—
155C3×Dic6—616no—
156C3×Dic6—64no—
157C3×Dic6—64no—
158C3×Dic6—64no—
159C32×C8—21no—
160C32×C8—24no—
161C32×C8—44no—
162C3⋊(C3⋊C8)—21no—
163C3×C3⋊C8—64no—
164#451—91noSylow
165C2×C33#12011yescharacteristic
166C2×C33C2×Q1611yescharacteristic
167C2×C33#13711yescharacteristic
168C3×C3⋊D3—22no—
169C2×C3×Q8—24no—
170C2×C3×Q8—91no—
171C2×C3×Q8—91no—
172C2×C3×Q8—181no—
173C2×C3×Q8—182no—
174C2×C3×Q8—182no—
175C2×C3×Q8C2×C3×D314yes—
176C2×C3×Q8C2×C3⋊D311yescharacteristic
177C22×C3×C4—91no—
178C22×C3×C4—181no—
179D3×Q8—68no—
180D3×Q8—68no—
181C2×C4×D3—34no—
182C2×C4×D3—64no—
183C2×Dic6—34no—
184C2×Dic6—34no—
185C2×Dic6—64no—
186C3×Q16—182no—
187C3×Q16—182no—
188C2×C3×C8—24no—
189C2×C3×C8—91no—
190C2×C3×C8C2×C3×D314yes—
191C2×C3×C8C2×C3⋊D311yescharacteristic
192C3×C4⋊C4—24no—
193C3×C4⋊C4—181no—
194C3×C4⋊C4C2×C3×D314yes—
195C3×C4⋊C4C2×C3⋊D311yescharacteristic
196C3×C22⋊C4—181no—
197#216—68no—
198(C2×D3)⋊C4—64no—
199C4⋊Dic3—64no—
200C2×C3⋊C8—34no—
201Dic12—68no—
202C22×C32—24no—
203C22×C32D3×D414yes—
204C22×C32C2×C3×D411yescharacteristic
205C2×C3⋊D3—21no—
206C2×C3⋊D3—21no—
207C2×C3⋊D3C3×Q1612yescomplemented
208C2×C3×D3—38no—
209C2×C3×D3—64no—
210C2×C3×D3—64no—
211C32×C4—22no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—28no—
215C32×C4—48no—
216C32×C4—41no—
217C32×C4—44no—
218C32×C4—84no—
219C32×C4C2×D1214yes—
220C32×C4D3×D414yes—
221C32×C4C2×C3×D411yescharacteristic
222C32×C4C2×C3×D411yescharacteristic
223C3⋊Dic3—21no—
224C3⋊Dic3—22no—
225C3⋊Dic3—41no—
226C3×Dic3—68no—
227C3×Dic3—64no—
228C3×Dic3—124no—
229C22×Q8—91no—
230C2×Q16—91no—
231C2×Q16—91no—
232C22⋊Q8—91no—
233#103—91no—
234#103C3×C3⋊D311yescharacteristic, complemented
235C22⋊C8—91no—
236C33#45111yescharacteristic, complemented, Sylow
237C23×C3—91no—
238C22×D3—34no—
239C3×Q8—24no—
240C3×Q8—21no—
241C3×Q8—28no—
242C3×Q8—44no—
243C3×Q8—92no—
244C3×Q8—184no—
245C3×Q8—181no—
246C3×Q8—181no—
247C3×Q8—181no—
248C3×Q8C3×D1218yes—
249C3×Q8C4⋊(C3⋊D3)12yes—
250C2×C3×C4—24no—
251C2×C3×C4—21no—
252C2×C3×C4—24no—
253C2×C3×C4—21no—
254C2×C3×C4—24no—
255C2×C3×C4—44no—
256C2×C3×C4—44no—
257C2×C3×C4—91no—
258C2×C3×C4—184no—
259C2×C3×C4—181no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4C22×C3×D314yes—
264C2×C3×C4C22×C3⋊D311yescharacteristic
265C2×Dic3—34no—
266C2×Dic3—64no—
267C2×Dic3—64no—
268C4×D3—616no—
269C4×D3—64no—
270C4×D3—64no—
271Dic6—38no—
272Dic6—616no—
273Dic6—64no—
274Dic6—64no—
275Dic6—64no—
276C3×C8—24no—
277C3×C8—21no—
278C3×C8—44no—
279C3×C8—181no—
280C3⋊C8—64no—
281C2×C32—24no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32#88214yes—
285C2×C32#91514yes—
286C2×C32D3×Q1614yes—
287C2×C32C3×#12011yescharacteristic
288C2×C32C2×C3×Q1611yescharacteristic
289C2×C32C3×#13711yescharacteristic
290C3⋊D3—22no—
291C3×D3—68no—
292C2×Q8—91no—
293C2×Q8—91no—
294C2×Q8—181no—
295C2×Q8—182no—
296C2×Q8—182no—
297C2×Q8C2×C3×C3⋊D311yescharacteristic
298C22×C4—91no—
299C22×C4—181no—
300Q16—182no—
301Q16—182no—
302C2×C8—91no—
303C2×C8C2×C3×C3⋊D311yescharacteristic
304C4⋊C4—181no—
305C4⋊C4C2×C3×C3⋊D311yescharacteristic
306C22⋊C4—181no—
307C22×C3—24no—
308C22×C3—92no—
309C22×C3—181no—
310C22×C3—181no—
311C22×C3C3×D3×D414yes—
312C22×C3D4×C3⋊D311yescharacteristic
313C2×D3—38no—
314C2×D3—64no—
315C2×D3—64no—
316C3×C4—28no—
317C3×C4—22no—
318C3×C4—24no—
319C3×C4—24no—
320C3×C4—48no—
321C3×C4—44no—
322C3×C4—41no—
323C3×C4—84no—
324C3×C4—181no—
325C3×C4—182no—
326C3×C4—361no—
327C3×C4C2×C3×D1214yes—
328C3×C4C3×D3×D414yes—
329C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
330C3×C4D4×C3⋊D311yescharacteristic
331Dic3—68no—
332Dic3—64no—
333Dic3—124no—
334C32—24no—
335C32#480914yescomplemented
336C32C3×#45111yescharacteristic, complemented
337C23—91no—
338Q8—21no—
339Q8—92no—
340Q8—184no—
341Q8—181no—
342Q8—181no—
343Q8—181no—
344Q8C3×C4⋊(C3⋊D3)12yes—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—91no—
348C2×C4—184no—
349C2×C4—181no—
350C2×C4—181no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4C22×C3×C3⋊D311yescharacteristic
354C8—21no—
355C8—181no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—182no—
360C2×C3C3×#88214yes—
361C2×C3C3×#91514yes—
362C2×C3C3×D3×Q1614yes—
363C2×C3#6409111yescharacteristic
364C2×C3#6412411yescharacteristic
365C2×C3Q16×C3⋊D311yescharacteristic
366D3—68no—
367C22—92no—
368C22—181no—
369C22—181no—
370C22C3×D4×C3⋊D311yescharacteristic
371C4—22no—
372C4—41no—
373C4—181no—
374C4—182no—
375C4—361no—
376C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
377C4C3×D4×C3⋊D311yescharacteristic
378C3—24no—
379C3C3×#480914yescomplemented
380C3#1059250811yescharacteristic, direct product factor
381C2—182no—
382C2C3×#6409111yescharacteristic
383C2C3×#6412411yescharacteristic
384C2C3×Q16×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC3×#103C3⋊D3
hypocenterC32C3×#451
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3756C2semidirect
3. C2×C3×#3746C2semidirect
5. C33×#103C2semidirect
7. C3×#63637C2semidirect
8. #10592508C3direct (N × H)
27. C2×#3756C2×C3semidirect
28. C2×#3746C2×C3semidirect
34. C32×#103D3semidirect
35. C32×#103C2×C3semidirect
37. #63637C2×C3semidirect
97. C2×C3×C3⋊D3Q16semidirect
115. C3×#103C3×D3semidirect
116. C3×#103C3⋊D3semidirect
207. C2×C3⋊D3C3×Q16semidirect
234. #103C3×C3⋊D3semidirect
236. C33#451semidirect
335. C32#4809semidirect
336. C32C3×#451semidirect
379. C3C3×#4809semidirect
380. C3#10592508direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×C3×#37562
C2×C3×#37462
C3×#641062
C33×#1032
C3×#636472
C3×#636372
#105925083
C3×#48093