Finite Group Databasebeta

← C3×#10592865 #13163206

Subgroups — C3×#10592865 #13163206

Order 1728.

944 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×SD16C211yesmaximal, characteristic
2C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#3756C211yesmaximal, characteristic, complemented
4C3×#64145C211yesmaximal, characteristic
5C3×#63659C211yesmaximal, characteristic
6C3×#63643C211yesmaximal, characteristic, complemented
7C3×#63641C211yesmaximal, characteristic, complemented
8#10592865C311yesmaximal, characteristic, direct product factor
9C3×#5166—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C33×SD16—22no—
14C2×C33×C8C2211yescharacteristic
15C3×#3757—21no—
16C3×#3757—21no—
17C3×#3756—22no—
18C3×C4⋊(C3⋊Dic3)—21no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×C4×C3⋊Dic3—21no—
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C2×C32×SD16—24no—
24C2×C32×SD16D314yes—
25C2×C32×SD16C2×C311yescharacteristic
26Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
27C2×#3756C2×C311yescharacteristic, complemented
28#64145C2×C311yescharacteristic
29C3×Q8×Dic3—34no—
30C2×C3×#216—34no—
31C3×#936—34no—
32#63659C2×C311yescharacteristic
33#63643C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35C3×#832—34no—
36C3×#816—34no—
37C3×Dic3⋊C8—34no—
38C23×C33—21no—
39C33×Q8—21no—
40C33×Q8D412yes—
41C33×D4—21no—
42C33×D4—21no—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×#634—21no—
50C3×#634—21no—
51C33×C8—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C3×#464—91no—
54#5166—34no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C22×C311yescharacteristic
58C2×C32×D4—24no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C22×C311yescharacteristic
61C2×#634C22×C311yescharacteristic
62C2×C3×Dic6—34no—
63C32×SD16—216no—
64C32×SD16—28no—
65C32×SD16—22no—
66C2×C32×C8—24no—
67C2×C32×C8C2×D314yes—
68C2×C32×C8C22×C311yescharacteristic
69#3757—21no—
70#3757—21no—
71#3756—22no—
72C4⋊(C3⋊Dic3)—21no—
73C4⋊(C3⋊Dic3)C22×C311yescharacteristic
74C4×C3⋊Dic3—21no—
75C4×C3⋊Dic3C22×C311yescharacteristic
76C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
77C3×#217—64no—
78C3×#217—64no—
79C3×#216—68no—
80C3×C4⋊Dic3—34no—
81C3×C4⋊Dic3—64no—
82C3×C4×Dic3—34no—
83C3×C4×Dic3—64no—
84C2×C3×C3⋊C8—34no—
85C22×C33—41no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33C2×D411yescharacteristic
89C33×C4—22no—
90C33×C4C2×D411yescharacteristic
91C33×C4#4111yescharacteristic
92C3×C3⋊Dic3—21no—
93C3×C3⋊Dic3—21no—
94C3×C3⋊Dic3—41no—
95C3×C3⋊Dic3—41no—
96C2×C3×Q16—91no—
97C2×C3×SD16—24no—
98C2×C3×SD16C3×D314yes—
99C2×C3×SD16C3⋊D311yescharacteristic
100C3×#124—91no—
101C3×C4×Q8—91no—
102Q8×Dic3—34no—
103C2×#216—34no—
104#936—34no—
105C3×C4⋊C8—91no—
106C3×#103—91no—
107C3×#102—91no—
108#832—34no—
109#816—34no—
110Dic3⋊C8—34no—
111C23×C32—21no—
112C23×C32—24no—
113C23×C32—44no—
114C32×Q8—21no—
115C32×Q8—28no—
116C32×Q8—24no—
117C32×Q8—44no—
118C32×Q8#6718yes—
119C32×Q8C3×D412yes—
120C32×D4—28no—
121C32×D4—24no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—24no—
125C32×D4—44no—
126C2×C32×C4—21no—
127C2×C32×C4—24no—
128C2×C32×C4—24no—
129C2×C32×C4—44no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C23×C311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3C3×D411yescharacteristic
135C2×C3⋊Dic3C3×D411yescharacteristic
136#634—21no—
137#634—21no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3—64no—
142C3×Dic6—64no—
143C3×Dic6—64no—
144C32×C8—28no—
145C32×C8—24no—
146C32×C8—21no—
147C3⋊(C3⋊C8)—21no—
148C3×C3⋊C8—64no—
149#464—91noSylow
150C2×C33—81no—
151C2×C33#12111yescharacteristic
152C2×C33#13511yescharacteristic
153C2×C33#13711yescharacteristic
154C2×C3×Q8—24no—
155C2×C3×Q8—91no—
156C2×C3×Q8C2×C3×D314yes—
157C2×C3×Q8C2×C3⋊D311yescharacteristic
158C2×C3×D4—24no—
159C2×C3×D4C2×C3×D314yes—
160C2×C3×D4C2×C3⋊D311yescharacteristic
161C2×Dic6—34no—
162C3×Q16—182no—
163C3×SD16—216no—
164C3×SD16—28no—
165C3×SD16—22no—
166C2×C3×C8—24no—
167C2×C3×C8—91no—
168C2×C3×C8C2×C3×D314yes—
169C2×C3×C8C2×C3⋊D311yescharacteristic
170C3×C4⋊C4—91no—
171C3×C4⋊C4—181no—
172C3×C22⋊C4—181no—
173C3×C22⋊C4—181no—
174C3×C42—91no—
175C3×C42—181no—
176#217—64no—
177#217—64no—
178#216—68no—
179C4⋊Dic3—34no—
180C4⋊Dic3—64no—
181C4×Dic3—34no—
182C4×Dic3—64no—
183C2×C3⋊C8—34no—
184C22×C32—24no—
185C22×C32—48no—
186C22×C32—44no—
187C22×C32—41no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—44no—
191C22×C32—44no—
192C22×C32—84no—
193C22×C32—84no—
194C22×C32D3×D414yes—
195C22×C32C2×C3×D411yescharacteristic
196C32×C4—22no—
197C32×C4—24no—
198C32×C4—24no—
199C32×C4—28no—
200C32×C4—48no—
201C32×C4#23714yes—
202C32×C4C2×#6714yes—
203C32×C4C2×C3×D411yescharacteristic
204C32×C4C3×#4111yescharacteristic
205C3⋊Dic3—21no—
206C3⋊Dic3—21no—
207C3⋊Dic3—41no—
208C3⋊Dic3—41no—
209C3×Dic3—64no—
210C3×Dic3—64no—
211C3×Dic3—124no—
212C3×Dic3—124no—
213C2×Q16—91no—
214C2×SD16C3×C3⋊D311yescharacteristic
215#124—91no—
216C4×Q8—91no—
217C4⋊C8—91no—
218#103—91no—
219#102—91no—
220C33#46411yescharacteristic, complemented, Sylow
221C23×C3—24no—
222C23×C3—21no—
223C23×C3—44no—
224C3×Q8—24no—
225C3×Q8—21no—
226C3×Q8—28no—
227C3×Q8—44no—
228C3×Q8—181no—
229C3×Q8—181no—
230C3×Q8C3×#6718yes—
231C3×Q8#63812yes—
232C3×D4—28no—
233C3×D4—24no—
234C3×D4—24no—
235C3×D4—21no—
236C3×D4—21no—
237C3×D4—44no—
238C2×C3×C4—24no—
239C2×C3×C4—21no—
240C2×C3×C4—24no—
241C2×C3×C4—44no—
242C2×C3×C4—91no—
243C2×C3×C4—91no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4C22×C3×D314yes—
247C2×C3×C4C22×C3⋊D311yescharacteristic
248C2×Dic3—34no—
249C2×Dic3—34no—
250C2×Dic3—64no—
251C2×Dic3—64no—
252Dic6—64no—
253Dic6—64no—
254C3×C8—28no—
255C3×C8—24no—
256C3×C8—21no—
257C3×C8—181no—
258C3⋊C8—64no—
259C2×C32—24no—
260C2×C32—24no—
261C2×C32—24no—
262C2×C32—88no—
263C2×C32—84no—
264C2×C32—81no—
265C2×C32#91514yes—
266C2×C32#91614yes—
267C2×C32#93914yes—
268C2×C32C3×#12111yescharacteristic
269C2×C32C3×#13511yescharacteristic
270C2×C32C3×#13711yescharacteristic
271C2×Q8—91no—
272C2×Q8C2×C3×C3⋊D311yescharacteristic
273C2×D4C2×C3×C3⋊D311yescharacteristic
274Q16—182no—
275SD16—22no—
276C2×C8—91no—
277C2×C8C2×C3×C3⋊D311yescharacteristic
278C4⋊C4—91no—
279C4⋊C4—181no—
280C22⋊C4—181no—
281C22⋊C4—181no—
282C42—91no—
283C42—181no—
284C22×C3—24no—
285C22×C3—48no—
286C22×C3—44no—
287C22×C3—44no—
288C22×C3—44no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—84no—
293C22×C3—84no—
294C22×C3C3×D3×D414yes—
295C22×C3D4×C3⋊D311yescharacteristic
296C3×C4—28no—
297C3×C4—22no—
298C3×C4—24no—
299C3×C4—24no—
300C3×C4—48no—
301C3×C4—181no—
302C3×C4—181no—
303C3×C4—361no—
304C3×C4—361no—
305C3×C4C3×#23714yes—
306C3×C4C2×C3×#6714yes—
307C3×C4#383011yescharacteristic
308C3×C4C2×#63811yescharacteristic
309Dic3—64no—
310Dic3—64no—
311Dic3—124no—
312Dic3—124no—
313C32—24no—
314C32#516614yescomplemented
315C32C3×#46411yescharacteristic, complemented
316C23—21no—
317Q8—21no—
318Q8—181no—
319Q8—181no—
320Q8C3×#63812yes—
321D4—21no—
322D4—21no—
323C2×C4—21no—
324C2×C4—91no—
325C2×C4—91no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4C22×C3×C3⋊D311yescharacteristic
329C8—21no—
330C8—181no—
331C2×C3—24no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—88no—
335C2×C3—84no—
336C2×C3—81no—
337C2×C3C3×#91514yes—
338C2×C3C3×#91614yes—
339C2×C3C3×#93914yes—
340C2×C3#6412411yescharacteristic
341C2×C3#6412511yescharacteristic
342C2×C3#6414811yescharacteristic
343C22—41no—
344C22—41no—
345C22—41no—
346C22C3×D4×C3⋊D311yescharacteristic
347C4—22no—
348C4—181no—
349C4—181no—
350C4—361no—
351C4—361no—
352C4C3×#383011yescharacteristic
353C4C2×C3×#63811yescharacteristic
354C3—24no—
355C3C3×#516614yescomplemented
356C3#1059286511yescharacteristic, direct product factor
357C2—81no—
358C2C3×#6412411yescharacteristic
359C2C3×#6412511yescharacteristic
360C2C3×#6414811yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC2×C33×D4C22
hypercenterC2×C3×SD16C3⋊D3
hypocenterC32C3×#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
2. C3×Q8×C3⋊Dic3C2semidirect
3. C2×C3×#3756C2semidirect
6. C3×#63643C2semidirect
7. C3×#63641C2semidirect
8. #10592865C3direct (N × H)
26. Q8×C3⋊Dic3C2×C3semidirect
27. C2×#3756C2×C3semidirect
33. #63643C2×C3semidirect
34. #63641C2×C3semidirect
220. C33#464semidirect
314. C32#5166semidirect
315. C32C3×#464semidirect
355. C3C3×#5166semidirect
356. C3#10592865direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×SD162
C3×Q8×C3⋊Dic32
C2×C3×#37562
C3×#641452
C3×#636592
C3×#636432
C3×#636412
#105928653
C3×#51663