Finite Group Databasebeta

← C3×#10595994 #13179277

Subgroups — C3×#10595994 #13179277

Order 1728.

2068 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C33×#117C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4C3×#64101C212yesmaximal, complemented
5C3×C4⋊C4×C3⋊D3C212yesmaximal, complemented
6C3×#64090C214yesmaximal, complemented
7C3×C4×C4⋊(C3⋊D3)C212yesmaximal, complemented
8C3×C4×#634C212yesmaximal, complemented
9#10595994C311yesmaximal, characteristic, direct product factor
10C3×#5523—34nomaximal
11C22×C33×C4C2211yescharacteristic
12C2×C3×#638C2212yes—
13C22×C3×C3⋊Dic3C2212yes—
14C3×#3828C418yescomplemented
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2214yes—
17C2×C3×C4×C3⋊D3C2212yes—
18C2×C3×#634C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20C33×C4⋊C4C2211yescharacteristic
21C33×C22⋊C4C2212yes—
22C33×C42C2212yes—
23C3×#3752C2214yes—
24C3×C4⋊(C3⋊Dic3)C2212yes—
25C3×C4⋊(C3⋊Dic3)C2212yes—
26C3×#3750C2214yes—
27C3×C4×C3⋊Dic3C2214yes—
28C2×#3828C2×C311yescharacteristic
29C2×C3×#235—34no—
30C32×#117—24no—
31C32×#117D314yescomplemented
32C32×#117C2×C311yescharacteristic, complemented
33C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
34#64101C2×C312yescomplemented
35C4⋊C4×C3⋊D3C2×C312yescomplemented
36#64090C2×C314yescomplemented
37C4×C4⋊(C3⋊D3)C2×C312yescomplemented
38C4×#634C2×C312yescomplemented
39C2×C3×C4⋊Dic3—34no—
40C3×#892—38no—
41C3×D3×C4⋊C4—38no—
42C3×#67⋊C4—316no—
43C3×C4×D12—38no—
44C3×C4×Dic6—38no—
45C23×C33C2311yescharacteristic
46C22×C3×C3⋊D3C2312yes—
47C2×C33×C4C2×C414yescomplemented
48C2×C33×C4C2312yes—
49C2×C33×C4C2312yes—
50C2×C33×C4C2311yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C3×#638C2×C418yescomplemented
53C2×C3×C3⋊Dic3—42no—
54C2×C3×C3⋊Dic3C2314yes—
55C2×C3×C3⋊Dic3C2312yes—
56C3×C4⋊(C3⋊D3)C2×C414yescomplemented
57C3×C4×C3⋊D3—44no—
58C3×C4×C3⋊D3C2×C418yescomplemented
59C3×#634C2×C414yescomplemented
60C3×#520—91no—
61#5523—34no—
62C22×C32×C4—24no—
63C22×C32×C4C2×D314yes—
64C22×C32×C4C22×C311yescharacteristic
65C2×#638C22×C312yes—
66C22×C3⋊Dic3C22×C312yes—
67#3828C3×C418yescomplemented
68C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
69C2×C4×C3⋊D3C22×C314yes—
70C2×C4×C3⋊D3C22×C312yes—
71C2×#634C22×C311yescharacteristic
72C2×C3×#67—38no—
73C22×C3×Dic3—38no—
74C3×#235—332no—
75C2×C3×D12—34no—
76C2×C3×C4×D3—38no—
77C2×C3×C4×D3—316no—
78C2×C3×Dic6—34no—
79C32×C4⋊C4—24no—
80C32×C4⋊C4—24no—
81C32×C4⋊C4C2×D314yes—
82C32×C4⋊C4C2×D314yes—
83C32×C4⋊C4C22×C311yescharacteristic
84C32×C4⋊C4C22×C311yescharacteristic
85C32×C22⋊C4—28no—
86C32×C22⋊C4C2×D318yes—
87C32×C22⋊C4C22×C312yes—
88C32×C42—28no—
89C32×C42C2×D318yes—
90C32×C42C22×C312yes—
91#3752C22×C314yes—
92C4⋊(C3⋊Dic3)C22×C312yes—
93C4⋊(C3⋊Dic3)C22×C312yes—
94#3750C22×C314yes—
95C4×C3⋊Dic3C22×C314yes—
96C3×(C2×D3)⋊C4—316no—
97C3×C4⋊Dic3—38no—
98C3×C4⋊Dic3—38no—
99C3×Dic3⋊C4—316no—
100C3×C4×Dic3—316no—
101C22×C33—21no—
102C22×C33—21no—
103C22×C33C22×C412yes—
104C22×C33C2411yescharacteristic
105C2×C3×C3⋊D3—22no—
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3C22×C414yes—
108C33×C4—22no—
109C33×C4—22no—
110C33×C4C22×C412yes—
111C33×C4C22×C412yes—
112C3×C3⋊Dic3—24no—
113C3×C3⋊Dic3C22×C414yes—
114C2×C3×#41—91no—
115C2×#235—34no—
116C3×C4×Q8—92no—
117C3×C4×D4—94no—
118C3×C4×D4—92no—
119C3×#117—24no—
120C3×#117—92no—
121C3×#117C3×D314yescomplemented
122C3×#117C3⋊D311yescharacteristic, complemented
123C2×C3×C4⋊C4—91no—
124C2×C3×C4⋊C4—92no—
125C2×C4⋊Dic3—34no—
126#892—38no—
127D3×C4⋊C4—38no—
128#67⋊C4—316no—
129C4×D12—38no—
130C4×Dic6—38no—
131C23×C32—24no—
132C23×C32C22×D314yes—
133C23×C32C23×C311yescharacteristic
134C22×C3⋊D3C23×C312yes—
135C22×C3×D3—38no—
136C2×C32×C4—216no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—28no—
140C2×C32×C4—28no—
141C2×C32×C4C4×D3116yescomplemented
142C2×C32×C4C2×C3×C414yescomplemented
143C2×C32×C4C22×D318yes—
144C2×C32×C4C22×D318yes—
145C2×C32×C4C22×D314yes—
146C2×C32×C4C22×D314yes—
147C2×C32×C4C23×C312yes—
148C2×C32×C4C23×C312yes—
149C2×C32×C4C23×C311yescharacteristic
150C2×C32×C4C23×C311yescharacteristic
151#638C2×C3×C418yescomplemented
152C2×C3⋊Dic3—42no—
153C2×C3⋊Dic3C23×C314yes—
154C2×C3⋊Dic3C23×C312yes—
155C4⋊(C3⋊D3)C2×C3×C414yescomplemented
156C4×C3⋊D3—44no—
157C4×C3⋊D3C2×C3×C418yescomplemented
158#634C2×C3×C414yescomplemented
159C3×#67—332no—
160C2×C3×Dic3—38no—
161C2×C3×Dic3—316no—
162C2×C3×Dic3—128no—
163C3×D12—316no—
164C3×C4×D3—332no—
165C3×C4×D3—1216no—
166C3×Dic6—316no—
167#520—91noSylow
168C2×C33—22no—
169C2×C33C23×C411yescharacteristic
170C2×C33H(5,𝔽2)11yescharacteristic
171C2×C332−1+411yescharacteristic
172C3×C3⋊D3—24no—
173C3×#41—98no—
174C2×C3×Q8—91no—
175C2×C3×D4—92no—
176C2×C3×D4—91no—
177C22×C3×C4—24no—
178C22×C3×C4—92no—
179C22×C3×C4—92no—
180C22×C3×C4—94no—
181C22×C3×C4C2×C3×D314yes—
182C22×C3×C4C2×C3⋊D311yescharacteristic
183C2×#67—38no—
184C22×Dic3—38no—
185#235—332no—
186C2×D12—34no—
187C2×C4×D3—38no—
188C2×C4×D3—316no—
189C2×Dic6—34no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—94no—
193C3×C4⋊C4—92no—
194C3×C4⋊C4—92no—
195C3×C4⋊C4C2×C3×D314yes—
196C3×C4⋊C4C2×C3×D314yes—
197C3×C4⋊C4C2×C3⋊D311yescharacteristic
198C3×C4⋊C4C2×C3⋊D311yescharacteristic
199C3×C22⋊C4—28no—
200C3×C22⋊C4—94no—
201C3×C22⋊C4C2×C3×D318yes—
202C3×C22⋊C4C2×C3⋊D312yes—
203C3×C42—28no—
204C3×C42—94no—
205C3×C42C2×C3×D318yes—
206C3×C42C2×C3⋊D312yes—
207(C2×D3)⋊C4—316no—
208C4⋊Dic3—38no—
209C4⋊Dic3—38no—
210Dic3⋊C4—316no—
211C4×Dic3—316no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—24no—
215C22×C32—28no—
216C22×C32—24no—
217C22×C32—24no—
218C22×C32—44no—
219C22×C32—44no—
220C22×C32C2×C4×D318yes—
221C22×C32C22×C3×C412yes—
222C22×C32C23×D314yes—
223C22×C32C24×C311yescharacteristic
224C2×C3⋊D3—22no—
225C2×C3⋊D3—22no—
226C2×C3⋊D3C22×C3×C414yes—
227C2×C3×D3—316no—
228C2×C3×D3—68no—
229C2×C3×D3—68no—
230C32×C4—22no—
231C32×C4—22no—
232C32×C4—28no—
233C32×C4—28no—
234C32×C4—28no—
235C32×C4—28no—
236C32×C4—48no—
237C32×C4—48no—
238C32×C4C2×C4×D318yes—
239C32×C4C2×C4×D318yes—
240C32×C4C22×C3×C412yes—
241C32×C4C22×C3×C412yes—
242C3⋊Dic3—24no—
243C3⋊Dic3C22×C3×C414yes—
244C3×Dic3—316no—
245C3×Dic3—616no—
246C2×#41—91no—
247C4×Q8—92no—
248C4×D4—94no—
249C4×D4—92no—
250#117—92no—
251#117C3×C3⋊D311yescharacteristic, complemented
252C2×C4⋊C4—91no—
253C2×C4⋊C4—92no—
254C33#52011yescharacteristic, complemented, Sylow
255C23×C3—24no—
256C23×C3—92no—
257C23×C3C22×C3×D314yes—
258C23×C3C22×C3⋊D311yescharacteristic
259C22×D3—38no—
260C3×Q8—94no—
261C3×D4—98no—
262C3×D4—94no—
263C2×C3×C4—216no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—28no—
267C2×C3×C4—28no—
268C2×C3×C4—92no—
269C2×C3×C4—98no—
270C2×C3×C4—94no—
271C2×C3×C4—364no—
272C2×C3×C4—362no—
273C2×C3×C4C3×C4×D3116yescomplemented
274C2×C3×C4C4×C3⋊D314yescomplemented
275C2×C3×C4C22×C3×D318yes—
276C2×C3×C4C22×C3×D318yes—
277C2×C3×C4C22×C3×D314yes—
278C2×C3×C4C22×C3×D314yes—
279C2×C3×C4C22×C3⋊D312yes—
280C2×C3×C4C22×C3⋊D312yes—
281C2×C3×C4C22×C3⋊D311yescharacteristic
282C2×C3×C4C22×C3⋊D311yescharacteristic
283#67—332no—
284C2×Dic3—38no—
285C2×Dic3—316no—
286C2×Dic3—128no—
287D12—316no—
288C4×D3—332no—
289C4×D3—1216no—
290Dic6—316no—
291C2×C32—22no—
292C2×C32—24no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—28no—
296C2×C32—48no—
297C2×C32C22×C4×D314yes—
298C2×C32#100914yes—
299C2×C32#101014yes—
300C2×C32C23×C3×C411yescharacteristic
301C2×C32C3×H(5,𝔽2)11yescharacteristic
302C2×C32C3×2−1+411yescharacteristic
303C3⋊D3—24no—
304C3×D3—616no—
305#41—98no—
306C2×Q8—91no—
307C2×D4—92no—
308C2×D4—91no—
309C22×C4—92no—
310C22×C4—92no—
311C22×C4—94no—
312C22×C4C2×C3×C3⋊D311yescharacteristic
313C4⋊C4—92no—
314C4⋊C4—94no—
315C4⋊C4—92no—
316C4⋊C4C2×C3×C3⋊D311yescharacteristic
317C4⋊C4C2×C3×C3⋊D311yescharacteristic
318C22⋊C4—94no—
319C22⋊C4C2×C3×C3⋊D312yes—
320C42—94no—
321C42C2×C3×C3⋊D312yes—
322C22×C3—24no—
323C22×C3—24no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—24no—
327C22×C3—28no—
328C22×C3—44no—
329C22×C3—44no—
330C22×C3—94no—
331C22×C3—182no—
332C22×C3—182no—
333C22×C3C2×C3×C4×D318yes—
334C22×C3C2×C4×C3⋊D312yes—
335C22×C3C23×C3×D314yes—
336C22×C3C23×C3⋊D311yescharacteristic
337C2×D3—316no—
338C2×D3—68no—
339C2×D3—68no—
340C3×C4—28no—
341C3×C4—28no—
342C3×C4—22no—
343C3×C4—22no—
344C3×C4—28no—
345C3×C4—28no—
346C3×C4—48no—
347C3×C4—48no—
348C3×C4—94no—
349C3×C4—184no—
350C3×C4C2×C3×C4×D318yes—
351C3×C4C2×C3×C4×D318yes—
352C3×C4C2×C4×C3⋊D312yes—
353C3×C4C2×C4×C3⋊D312yes—
354Dic3—316no—
355Dic3—616no—
356C32—24no—
357C32#552314yescomplemented
358C32C3×#52011yescharacteristic, complemented
359C23—92no—
360C23C22×C3×C3⋊D311yescharacteristic
361Q8—94no—
362D4—98no—
363D4—94no—
364C2×C4—92no—
365C2×C4—98no—
366C2×C4—94no—
367C2×C4—364no—
368C2×C4—362no—
369C2×C4C3×C4×C3⋊D314yescomplemented
370C2×C4C22×C3×C3⋊D312yes—
371C2×C4C22×C3×C3⋊D312yes—
372C2×C4C22×C3×C3⋊D311yescharacteristic
373C2×C4C22×C3×C3⋊D311yescharacteristic
374C2×C3—28no—
375C2×C3—22no—
376C2×C3—24no—
377C2×C3—24no—
378C2×C3—24no—
379C2×C3—48no—
380C2×C3—184no—
381C2×C3C22×C3×C4×D314yes—
382C2×C3C3×#100914yes—
383C2×C3C3×#101014yes—
384C2×C3C22×C4×C3⋊D311yescharacteristic
385C2×C3#6436611yescharacteristic
386C2×C3#6436711yescharacteristic
387D3—616no—
388C22—21no—
389C22—21no—
390C22—94no—
391C22—182no—
392C22—182no—
393C22C2×C3×C4×C3⋊D312yes—
394C22C23×C3×C3⋊D311yescharacteristic
395C4—22no—
396C4—22no—
397C4—94no—
398C4—184no—
399C4C2×C3×C4×C3⋊D312yes—
400C4C2×C3×C4×C3⋊D312yes—
401C3—24no—
402C3C3×#552314yescomplemented
403C3#1059599411yescharacteristic, direct product factor
404C2—22no—
405C2—184no—
406C2C22×C3×C4×C3⋊D311yescharacteristic
407C2C3×#6436611yescharacteristic
408C2C3×#6436711yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutatorC2×C32C23×C3×C4
FrattiniC22C23×C3×C3⋊D3
FittingC33×#117C2
socleC22×C33C24
prime radicalC2×C3×#3828C2
cosocleC22×C32C24×C3
hypercenterC3×#117C3⋊D3
hypocenterC32C3×#520
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
2. C33×#117C2semidirect
3. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
4. C3×#64101C2semidirect
5. C3×C4⋊C4×C3⋊D3C2semidirect
6. C3×#64090C2semidirect
7. C3×C4×C4⋊(C3⋊D3)C2semidirect
8. C3×C4×#634C2semidirect
9. #10595994C3direct (N × H)
14. C3×#3828C4semidirect
31. C32×#117D3semidirect
32. C32×#117C2×C3semidirect
33. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
34. #64101C2×C3semidirect
35. C4⋊C4×C3⋊D3C2×C3semidirect
36. #64090C2×C3semidirect
37. C4×C4⋊(C3⋊D3)C2×C3semidirect
38. C4×#634C2×C3semidirect
47. C2×C33×C4C2×C4semidirect
52. C3×#638C2×C4semidirect
56. C3×C4⋊(C3⋊D3)C2×C4semidirect
58. C3×C4×C3⋊D3C2×C4semidirect
59. C3×#634C2×C4semidirect
67. #3828C3×C4semidirect
121. C3×#117C3×D3semidirect
122. C3×#117C3⋊D3semidirect
141. C2×C32×C4C4×D3semidirect
142. C2×C32×C4C2×C3×C4semidirect
151. #638C2×C3×C4semidirect
155. C4⋊(C3⋊D3)C2×C3×C4semidirect
157. C4×C3⋊D3C2×C3×C4semidirect
158. #634C2×C3×C4semidirect
251. #117C3×C3⋊D3semidirect
254. C33#520semidirect
273. C2×C3×C4C3×C4×D3semidirect
274. C2×C3×C4C4×C3⋊D3semidirect
357. C32#5523semidirect
358. C32C3×#520semidirect
369. C2×C4C3×C4×C3⋊D3semidirect
402. C3C3×#5523semidirect
403. C3#10595994direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C33×#1172
C2×C3×C4⋊(C3⋊Dic3)2
C3×#641012
C3×C4⋊C4×C3⋊D32
C3×#640902
C3×C4×C4⋊(C3⋊D3)2
C3×C4×#6342
#105959943
C3×#55233