Finite Group Databasebeta

← C3×#10592470 #13162808

Subgroups — C3×#10592470 #13162808

Order 1728.

1172 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic
2C2×C3×#3743C211yesmaximal, characteristic, complemented
3C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
4C33×#102C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6C3×#63643C211yesmaximal, characteristic, complemented
7C3×#63637C211yesmaximal, characteristic, complemented
8#10592470C311yesmaximal, characteristic, direct product factor
9C3×#4771—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638—21no—
12C22×C3×C3⋊Dic3—21no—
13C3×#3830—22no—
14C3×#3830C414yescomplemented
15C2×C3×C4×C3⋊D3—21no—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C2×C33×C8C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×#3750—21no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C3×#3743—22no—
24C2×#3830C2×C311yescharacteristic
25C2×C3×#237—34no—
26C2×#3743C2×C311yescharacteristic, complemented
27C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
28C2×C3×ΓL(1,𝔽25)—34no—
29C3×D3×C4⋊C4—34no—
30C32×#102—24no—
31C32×#102D314yescomplemented
32C32×#102C2×C311yescharacteristic, complemented
33#63659C2×C311yescharacteristic, complemented
34#63643C2×C311yescharacteristic, complemented
35#63637C2×C311yescharacteristic, complemented
36C3×#832—34no—
37C3×#816—34no—
38C3×#810—34no—
39C23×C33—21no—
40C22×C3×C3⋊D3D411yescharacteristic
41C33×D4—21no—
42C33×D4C2×C412yescomplemented
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C3×#638—24no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—24no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4×C3⋊D3—42no—
51C3×C4×C3⋊D3C2×C412yes—
52C3×C4×C3⋊D3D412yes—
53C3×#634—21no—
54C3×#634C2×C412yescomplemented
55C33×C8—21no—
56C3×C3⋊(C3⋊C8)—21no—
57C3×#417—91no—
58#4771—34no—
59C2×C32×D4—24no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C22×C311yescharacteristic
62C2×#638—21no—
63C22×C3⋊Dic3—21no—
64#3830—22no—
65#3830C3×C414yescomplemented
66C2×C4×C3⋊D3—21no—
67C2×C4×C3⋊D3C22×C311yescharacteristic
68C2×#634C22×C311yescharacteristic
69C2×C3×#67—64no—
70C22×C3×Dic3—64no—
71C3×#237—316no—
72C3×#237—68no—
73C2×C3×C4×D3—34no—
74C2×C3×C4×D3—64no—
75C2×C3×Dic6—34no—
76C2×C32×C8—24no—
77C2×C32×C8C2×D314yes—
78C2×C32×C8C22×C311yescharacteristic
79C32×C4⋊C4—24no—
80C32×C4⋊C4C2×D314yes—
81C32×C4⋊C4C22×C311yescharacteristic
82C4⋊(C3⋊Dic3)C22×C311yescharacteristic
83#3750—21no—
84C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
85#3743—22no—
86C3×C4⋊Dic3—34no—
87C3×Dic3⋊C4—64no—
88C2×C3×C3⋊C8—34no—
89C3×ΓL(1,𝔽25)—68no—
90C22×C33—22no—
91C22×C33—41no—
92C22×C33—41no—
93C22×C33C2×D411yescharacteristic
94C2×C3×C3⋊D3—21no—
95C2×C3×C3⋊D3—21no—
96C2×C3×C3⋊D3C22⋊C412yes—
97C33×C4—41no—
98C33×C4C22×C411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C3×C3⋊Dic3—22no—
101C3×C3⋊Dic3—41no—
102C3×C3⋊Dic3C22⋊C412yes—
103C2×C3×#41—91no—
104C2×#237—34no—
105C2×C3×M16—91no—
106C2×C3×C4⋊C4—91no—
107C2×ΓL(1,𝔽25)—34no—
108D3×C4⋊C4—34no—
109C3×#103—91no—
110C3×#103—91no—
111C3×#102—24no—
112C3×#102—91no—
113C3×#102C3×D314yescomplemented
114C3×#102C3⋊D311yescharacteristic, complemented
115#832—34no—
116#816—34no—
117#810—34no—
118C23×C32—21no—
119C23×C32—24no—
120C23×C32—44no—
121C22×C3⋊D3C3×D411yescharacteristic
122C22×C3×D3—34no—
123C32×D4—21no—
124C32×D4—28no—
125C32×D4—24no—
126C32×D4—44no—
127C32×D4C4×D318yescomplemented
128C32×D4C2×C3×C412yescomplemented
129C2×C32×C4—21no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—44no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C23×C311yescharacteristic
135#638—24no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—24no—
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3C3×D411yescharacteristic
140C4×C3⋊D3—42no—
141C4×C3⋊D3C2×C3×C412yes—
142C4×C3⋊D3C3×D412yes—
143#634—21no—
144#634C2×C3×C412yescomplemented
145C3×#67—616no—
146C2×C3×Dic3—34no—
147C2×C3×Dic3—616no—
148C2×C3×Dic3—64no—
149C2×C3×Dic3—64no—
150C3×C4×D3—38no—
151C3×C4×D3—38no—
152C3×C4×D3—128no—
153C3×Dic6—38no—
154C3×Dic6—64no—
155C32×C8—21no—
156C32×C8—24no—
157C32×C8—44no—
158C3⋊(C3⋊C8)—21no—
159C3×C3⋊C8—64no—
160#417—91noSylow
161C2×C33—42no—
162C2×C33C2×C22⋊C411yescharacteristic
163C2×C33AGL(1,ℤ/8)11yescharacteristic
164C2×C33#13711yescharacteristic
165C3×C3⋊D3—22no—
166C3×#41—94no—
167C3×#41—182no—
168C2×C3×Q8—91no—
169C2×C3×D4—24no—
170C2×C3×D4—181no—
171C2×C3×D4C2×C3×D314yes—
172C2×C3×D4C2×C3⋊D311yescharacteristic
173C22×C3×C4—91no—
174C22×C3×C4—181no—
175C22×C3×C4—181no—
176C2×#67—64no—
177C22×Dic3—64no—
178#237—316no—
179#237—68no—
180C2×C4×D3—34no—
181C2×C4×D3—64no—
182C2×Dic6—34no—
183C3×M16—182no—
184C2×C3×C8—24no—
185C2×C3×C8—91no—
186C2×C3×C8C2×C3×D314yes—
187C2×C3×C8C2×C3⋊D311yescharacteristic
188C3×C4⋊C4—24no—
189C3×C4⋊C4—91no—
190C3×C4⋊C4—181no—
191C3×C4⋊C4C2×C3×D314yes—
192C3×C4⋊C4C2×C3⋊D311yescharacteristic
193C4⋊Dic3—34no—
194Dic3⋊C4—64no—
195C2×C3⋊C8—34no—
196ΓL(1,𝔽25)—68no—
197C22×C32—22no—
198C22×C32—24no—
199C22×C32—28no—
200C22×C32—48no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32—44no—
204C22×C32—44no—
205C22×C32—84no—
206C22×C32—84no—
207C22×C32D3×D414yes—
208C22×C32C2×C3×D411yescharacteristic
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3C3×C22⋊C412yes—
212C2×C3×D3—38no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—41no—
218C32×C4—44no—
219C32×C4—84no—
220C32×C4C2×C4×D314yes—
221C32×C4D3×D414yes—
222C32×C4C22×C3×C411yescharacteristic
223C32×C4C2×C3×D411yescharacteristic
224C3⋊Dic3—22no—
225C3⋊Dic3—41no—
226C3⋊Dic3C3×C22⋊C412yes—
227C3×Dic3—38no—
228C3×Dic3—68no—
229C3×Dic3—124no—
230C2×#41—91no—
231C2×M16—91no—
232C2×C4⋊C4—91no—
233#103—91no—
234#103—91no—
235#102—91no—
236#102C3×C3⋊D311yescharacteristic, complemented
237C33#41711yescharacteristic, complemented, Sylow
238C23×C3—24no—
239C23×C3—21no—
240C23×C3—44no—
241C23×C3—91no—
242C22×D3—34no—
243C3×Q8—92no—
244C3×Q8—181no—
245C3×D4—24no—
246C3×D4—21no—
247C3×D4—28no—
248C3×D4—44no—
249C3×D4—184no—
250C3×D4C3×C4×D318yescomplemented
251C3×D4C4×C3⋊D312yescomplemented
252C2×C3×C4—24no—
253C2×C3×C4—21no—
254C2×C3×C4—24no—
255C2×C3×C4—44no—
256C2×C3×C4—91no—
257C2×C3×C4—92no—
258C2×C3×C4—92no—
259C2×C3×C4—184no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4—362no—
263C2×C3×C4C22×C3×D314yes—
264C2×C3×C4C22×C3⋊D311yescharacteristic
265#67—616no—
266C2×Dic3—34no—
267C2×Dic3—616no—
268C2×Dic3—64no—
269C2×Dic3—64no—
270C4×D3—38no—
271C4×D3—38no—
272C4×D3—128no—
273Dic6—38no—
274Dic6—64no—
275C3×C8—24no—
276C3×C8—21no—
277C3×C8—44no—
278C3×C8—181no—
279C3⋊C8—64no—
280C2×C32—24no—
281C2×C32—24no—
282C2×C32—24no—
283C2×C32—42no—
284C2×C32—48no—
285C2×C32—88no—
286C2×C32D3×C22⋊C414yes—
287C2×C32#91114yes—
288C2×C32#91514yes—
289C2×C32C2×C3×C22⋊C411yescharacteristic
290C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
291C2×C32C3×#13711yescharacteristic
292C3⋊D3—22no—
293C3×D3—68no—
294#41—94no—
295#41—182no—
296C2×Q8—91no—
297C2×D4—181no—
298C2×D4C2×C3×C3⋊D311yescharacteristic
299C22×C4—91no—
300C22×C4—181no—
301C22×C4—181no—
302M16—182no—
303C2×C8—91no—
304C2×C8C2×C3×C3⋊D311yescharacteristic
305C4⋊C4—91no—
306C4⋊C4—181no—
307C4⋊C4C2×C3×C3⋊D311yescharacteristic
308C22×C3—28no—
309C22×C3—22no—
310C22×C3—24no—
311C22×C3—48no—
312C22×C3—44no—
313C22×C3—44no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—84no—
317C22×C3—84no—
318C22×C3—92no—
319C22×C3—181no—
320C22×C3—181no—
321C22×C3C3×D3×D414yes—
322C22×C3D4×C3⋊D311yescharacteristic
323C2×D3—38no—
324C2×D3—64no—
325C2×D3—64no—
326C3×C4—24no—
327C3×C4—24no—
328C3×C4—44no—
329C3×C4—41no—
330C3×C4—84no—
331C3×C4—92no—
332C3×C4—182no—
333C3×C4—361no—
334C3×C4C2×C3×C4×D314yes—
335C3×C4C3×D3×D414yes—
336C3×C4C2×C4×C3⋊D311yescharacteristic
337C3×C4D4×C3⋊D311yescharacteristic
338Dic3—38no—
339Dic3—68no—
340Dic3—124no—
341C32—24no—
342C32#477114yescomplemented
343C32C3×#41711yescharacteristic, complemented
344C23—21no—
345C23—91no—
346Q8—92no—
347Q8—181no—
348D4—21no—
349D4—184no—
350D4C3×C4×C3⋊D312yescomplemented
351C2×C4—21no—
352C2×C4—91no—
353C2×C4—92no—
354C2×C4—92no—
355C2×C4—184no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—362no—
359C2×C4C22×C3×C3⋊D311yescharacteristic
360C8—21no—
361C8—181no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—48no—
366C2×C3—42no—
367C2×C3—88no—
368C2×C3—182no—
369C2×C3C3×D3×C22⋊C414yes—
370C2×C3C3×#91114yes—
371C2×C3C3×#91514yes—
372C2×C3C22⋊C4×C3⋊D311yescharacteristic
373C2×C3#6412011yescharacteristic
374C2×C3#6412411yescharacteristic
375D3—68no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—92no—
380C22—181no—
381C22—181no—
382C22C3×D4×C3⋊D311yescharacteristic
383C4—41no—
384C4—92no—
385C4—182no—
386C4—361no—
387C4C2×C3×C4×C3⋊D311yescharacteristic
388C4C3×D4×C3⋊D311yescharacteristic
389C3—24no—
390C3C3×#477114yescomplemented
391C3#1059247011yescharacteristic, direct product factor
392C2—42no—
393C2—182no—
394C2C3×C22⋊C4×C3⋊D311yescharacteristic
395C2C3×#6412011yescharacteristic
396C2C3×#6412411yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#102C2
socleC22×C33C2×D4
prime radicalC2×C3×#3830C2
cosocleC2×C32×C4C23×C3
hypercenterC3×#102C3⋊D3
hypocenterC32C3×#417
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3743C2semidirect
3. C3×C4⋊C4×C3⋊D3C2semidirect
4. C33×#102C2semidirect
5. C3×#63659C2semidirect
6. C3×#63643C2semidirect
7. C3×#63637C2semidirect
8. #10592470C3direct (N × H)
14. C3×#3830C4semidirect
26. C2×#3743C2×C3semidirect
27. C4⋊C4×C3⋊D3C2×C3semidirect
31. C32×#102D3semidirect
32. C32×#102C2×C3semidirect
33. #63659C2×C3semidirect
34. #63643C2×C3semidirect
35. #63637C2×C3semidirect
42. C33×D4C2×C4semidirect
54. C3×#634C2×C4semidirect
65. #3830C3×C4semidirect
113. C3×#102C3×D3semidirect
114. C3×#102C3⋊D3semidirect
127. C32×D4C4×D3semidirect
128. C32×D4C2×C3×C4semidirect
144. #634C2×C3×C4semidirect
236. #102C3×C3⋊D3semidirect
237. C33#417semidirect
250. C3×D4C3×C4×D3semidirect
251. C3×D4C4×C3⋊D3semidirect
342. C32#4771semidirect
343. C32C3×#417semidirect
350. D4C3×C4×C3⋊D3semidirect
390. C3C3×#4771semidirect
391. C3#10592470direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C3×#37432
C3×C4⋊C4×C3⋊D32
C33×#1022
C3×#636592
C3×#636432
C3×#636372
#105924703
C3×#47713