Finite Group Databasebeta

← C3×#10592861 #13163200

Subgroups — C3×#10592861 #13163200

Order 1728.

992 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×SD16C211yesmaximal, characteristic
2C3×#64153C211yesmaximal, characteristic, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic
4C2×C3×#3754C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63643C211yesmaximal, characteristic, complemented
7C3×#63641C211yesmaximal, characteristic, complemented
8#10592861C311yesmaximal, characteristic, direct product factor
9C3×#5162—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C22×C3×C3⋊Dic3—21no—
13C2×C3×#634C2211yescharacteristic
14C33×SD16—22no—
15C2×C33×C8C2211yescharacteristic
16C3×#3757—21no—
17C3×#3754—22no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×#3750—21no—
20C3×#3750—21no—
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C2×C32×SD16—24no—
24C2×C32×SD16D314yes—
25C2×C32×SD16C2×C311yescharacteristic
26#64153C2×C311yescharacteristic, complemented
27D4×C3⋊Dic3C2×C311yescharacteristic
28C2×#3754C2×C311yescharacteristic
29C3×#944—34no—
30C3×D4×Dic3—34no—
31C2×C3×#214—34no—
32#63662C2×C311yescharacteristic, complemented
33#63643C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35C3×#835—34no—
36C3×#816—34no—
37C3×Dic3⋊C8—34no—
38C23×C33—21no—
39C33×Q8—21no—
40C33×Q8—21no—
41C33×D4—21no—
42C33×D4D412yes—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—42no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×#634—21no—
51C3×#634—21no—
52C33×C8—21no—
53C3×C3⋊(C3⋊C8)—21no—
54C3×#461—91no—
55#5162—34no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C22×C311yescharacteristic
59C2×C32×D4—24no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C22×C311yescharacteristic
62C22×C3⋊Dic3—21no—
63C2×#634C22×C311yescharacteristic
64C22×C3×Dic3—64no—
65C2×C3×Dic6—34no—
66C32×SD16—216no—
67C32×SD16—28no—
68C32×SD16—22no—
69C2×C32×C8—24no—
70C2×C32×C8C2×D314yes—
71C2×C32×C8C22×C311yescharacteristic
72#3757—21no—
73#3754—22no—
74C4⋊(C3⋊Dic3)C22×C311yescharacteristic
75#3750—21no—
76#3750—21no—
77C4×C3⋊Dic3C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C3×#217—64no—
80C3×#214—68no—
81C3×C4⋊Dic3—34no—
82C3×Dic3⋊C4—64no—
83C3×Dic3⋊C4—64no—
84C3×C4×Dic3—34no—
85C2×C3×C3⋊C8—34no—
86C22×C33—22no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C33×C4—41no—
91C33×C4C2×D411yescharacteristic
92C33×C4#4111yescharacteristic
93C3×C3⋊Dic3—21no—
94C3×C3⋊Dic3—41no—
95C3×C3⋊Dic3—41no—
96C3×C3⋊Dic3SD1612yescomplemented
97C2×C3×SD16—24no—
98C2×C3×SD16—91no—
99C2×C3×SD16C3×D314yes—
100C2×C3×SD16C3⋊D311yescharacteristic
101C3×C4⋊Q8—91no—
102C3×C4×D4—91no—
103#944—34no—
104D4×Dic3—34no—
105C2×#214—34no—
106C3×C4⋊C8—91no—
107C3×#103—91no—
108C3×#103—91no—
109#835—34no—
110#816—34no—
111Dic3⋊C8—34no—
112C23×C32—21no—
113C23×C32—24no—
114C23×C32—44no—
115C32×Q8—28no—
116C32×Q8—24no—
117C32×Q8—21no—
118C32×Q8—21no—
119C32×Q8—24no—
120C32×Q8—44no—
121C32×D4—21no—
122C32×D4—28no—
123C32×D4—24no—
124C32×D4—44no—
125C32×D4#6718yes—
126C32×D4C3×D412yes—
127C2×C32×C4—21no—
128C2×C32×C4—24no—
129C2×C32×C4—24no—
130C2×C32×C4—44no—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C23×C311yescharacteristic
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—42no—
136C2×C3⋊Dic3C3×D411yescharacteristic
137C2×C3⋊Dic3C3×D411yescharacteristic
138#634—21no—
139#634—21no—
140C2×C3×Dic3—34no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—128no—
145C3×Dic6—64no—
146C3×Dic6—64no—
147C32×C8—28no—
148C32×C8—24no—
149C32×C8—21no—
150C3⋊(C3⋊C8)—21no—
151C3×C3⋊C8—64no—
152#461—91noSylow
153C2×C33—42no—
154C2×C33#12111yescharacteristic
155C2×C33C2×SD1611yescharacteristic
156C2×C33#13711yescharacteristic
157C2×C3×Q8—24no—
158C2×C3×Q8—91no—
159C2×C3×Q8C2×C3×D314yes—
160C2×C3×Q8C2×C3⋊D311yescharacteristic
161C2×C3×D4—24no—
162C2×C3×D4C2×C3×D314yes—
163C2×C3×D4C2×C3⋊D311yescharacteristic
164C22×C3×C4—181no—
165C22×Dic3—64no—
166C2×Dic6—34no—
167C3×SD16—216no—
168C3×SD16—28no—
169C3×SD16—22no—
170C3×SD16—182no—
171C2×C3×C8—24no—
172C2×C3×C8—91no—
173C2×C3×C8C2×C3×D314yes—
174C2×C3×C8C2×C3⋊D311yescharacteristic
175C3×C4⋊C4—91no—
176C3×C4⋊C4—181no—
177C3×C4⋊C4—181no—
178C3×C22⋊C4—181no—
179C3×C42—91no—
180#217—64no—
181#214—68no—
182C4⋊Dic3—34no—
183Dic3⋊C4—64no—
184Dic3⋊C4—64no—
185C4×Dic3—34no—
186C2×C3⋊C8—34no—
187C22×C32—22no—
188C22×C32—24no—
189C22×C32—28no—
190C22×C32—48no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32—44no—
194C22×C32—44no—
195C22×C32—84no—
196C22×C32—84no—
197C22×C32D3×D414yes—
198C22×C32C2×C3×D411yescharacteristic
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—48no—
202C32×C4—44no—
203C32×C4—41no—
204C32×C4#23714yes—
205C32×C4C2×#6714yes—
206C32×C4C2×C3×D411yescharacteristic
207C32×C4C3×#4111yescharacteristic
208C3⋊Dic3—21no—
209C3⋊Dic3—41no—
210C3⋊Dic3—41no—
211C3⋊Dic3C3×SD1612yescomplemented
212C3×Dic3—38no—
213C3×Dic3—64no—
214C3×Dic3—124no—
215C3×Dic3—124no—
216C2×SD16—91no—
217C2×SD16C3×C3⋊D311yescharacteristic
218C4⋊Q8—91no—
219C4×D4—91no—
220C4⋊C8—91no—
221#103—91no—
222#103—91no—
223C33#46111yescharacteristic, complemented, Sylow
224C23×C3—24no—
225C23×C3—21no—
226C23×C3—44no—
227C3×Q8—28no—
228C3×Q8—24no—
229C3×Q8—24no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—44no—
233C3×Q8—181no—
234C3×Q8—181no—
235C3×D4—24no—
236C3×D4—21no—
237C3×D4—28no—
238C3×D4—44no—
239C3×D4C3×#6718yes—
240C3×D4#63812yes—
241C2×C3×C4—24no—
242C2×C3×C4—21no—
243C2×C3×C4—24no—
244C2×C3×C4—44no—
245C2×C3×C4—91no—
246C2×C3×C4—91no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4—362no—
250C2×C3×C4C22×C3×D314yes—
251C2×C3×C4C22×C3⋊D311yescharacteristic
252C2×Dic3—34no—
253C2×Dic3—34no—
254C2×Dic3—64no—
255C2×Dic3—64no—
256C2×Dic3—128no—
257Dic6—64no—
258Dic6—64no—
259C3×C8—28no—
260C3×C8—24no—
261C3×C8—21no—
262C3×C8—181no—
263C3⋊C8—64no—
264C2×C32—24no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—42no—
268C2×C32—48no—
269C2×C32—88no—
270C2×C32D3×SD1614yes—
271C2×C32#91514yes—
272C2×C32#93914yes—
273C2×C32C3×#12111yescharacteristic
274C2×C32C2×C3×SD1611yescharacteristic
275C2×C32C3×#13711yescharacteristic
276C2×Q8—91no—
277C2×Q8C2×C3×C3⋊D311yescharacteristic
278C2×D4C2×C3×C3⋊D311yescharacteristic
279C22×C4—181no—
280SD16—22no—
281SD16—182no—
282C2×C8—91no—
283C2×C8C2×C3×C3⋊D311yescharacteristic
284C4⋊C4—91no—
285C4⋊C4—181no—
286C4⋊C4—181no—
287C22⋊C4—181no—
288C42—91no—
289C22×C3—28no—
290C22×C3—22no—
291C22×C3—24no—
292C22×C3—48no—
293C22×C3—44no—
294C22×C3—44no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—84no—
298C22×C3—84no—
299C22×C3C3×D3×D414yes—
300C22×C3D4×C3⋊D311yescharacteristic
301C3×C4—24no—
302C3×C4—24no—
303C3×C4—48no—
304C3×C4—44no—
305C3×C4—41no—
306C3×C4—92no—
307C3×C4—181no—
308C3×C4—361no—
309C3×C4—361no—
310C3×C4C3×#23714yes—
311C3×C4C2×C3×#6714yes—
312C3×C4#383011yescharacteristic
313C3×C4C2×#63811yescharacteristic
314Dic3—38no—
315Dic3—64no—
316Dic3—124no—
317Dic3—124no—
318C32—24no—
319C32#516214yescomplemented
320C32C3×#46111yescharacteristic, complemented
321C23—21no—
322Q8—21no—
323Q8—21no—
324Q8—181no—
325Q8—181no—
326D4—21no—
327D4C3×#63812yes—
328C2×C4—21no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—362no—
334C2×C4C22×C3×C3⋊D311yescharacteristic
335C8—21no—
336C8—181no—
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—48no—
341C2×C3—42no—
342C2×C3—88no—
343C2×C3C3×D3×SD1614yes—
344C2×C3C3×#91514yes—
345C2×C3C3×#93914yes—
346C2×C3SD16×C3⋊D311yescharacteristic
347C2×C3#6412411yescharacteristic
348C2×C3#6414811yescharacteristic
349C22—22no—
350C22—41no—
351C22—41no—
352C22C3×D4×C3⋊D311yescharacteristic
353C4—41no—
354C4—92no—
355C4—181no—
356C4—361no—
357C4—361no—
358C4C3×#383011yescharacteristic
359C4C2×C3×#63811yescharacteristic
360C3—24no—
361C3C3×#516214yescomplemented
362C3#1059286111yescharacteristic, direct product factor
363C2—42no—
364C2C3×SD16×C3⋊D311yescharacteristic
365C2C3×#6412411yescharacteristic
366C2C3×#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×#461
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×#64153C2semidirect
5. C3×#63662C2semidirect
6. C3×#63643C2semidirect
7. C3×#63641C2semidirect
8. #10592861C3direct (N × H)
26. #64153C2×C3semidirect
32. #63662C2×C3semidirect
33. #63643C2×C3semidirect
34. #63641C2×C3semidirect
96. C3×C3⋊Dic3SD16semidirect
211. C3⋊Dic3C3×SD16semidirect
223. C33#461semidirect
319. C32#5162semidirect
320. C32C3×#461semidirect
361. C3C3×#5162semidirect
362. C3#10592861direct (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×#641532
C3×D4×C3⋊Dic32
C2×C3×#37542
C3×#636622
C3×#636432
C3×#636412
#105928613
C3×#51623