Finite Group Databasebeta

← C2×C3×#63919 #13174638

Subgroups — C2×C3×#63919 #13174638

Order 1728.

1506 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#627C212yesmaximal
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3C2×C32×C4⋊Dic3C212yesmaximal
4C2×C3×#3724C212yesmaximal
5C3×#63919C218yesmaximal, direct product factor
6C2×C3×#888—32nomaximal
7C2×#63919C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2214yes—
12C2×C3×#627—28no—
13C2×C3×#627C2218yes—
14C3×C4×C3⋊Dic3C2214yes—
15C32×C4⋊Dic3C2218yes—
16C3×#3724C2218yes—
17C22×C3×Dic6—34no—
18C22×#627C2×C312yes—
19C2×C32×C4⋊C4—32no—
20C2×C4×C3⋊Dic3C2×C311yescharacteristic
21C2×C3×C4⋊Dic3—22no—
22C2×C3×C4⋊Dic3D312yes—
23C2×C3×C4⋊Dic3C2×C312yes—
24C2×C3×Dic3⋊C4—34no—
25C2×C3×C4×Dic3—32no—
26C2×C3×C4×Dic3—61no—
27C3×#888—316no—
28C2×#3724C2×C312yes—
29#63919C2×C318yesdirect product factor
30C23×C33C2311yescharacteristic
31C2×C33×C4Q814yescomplemented
32C2×C33×C4C2312yes—
33C2×C3×C3⋊Dic3D414yes—
34C2×C3×C3⋊Dic3Q814yescomplemented
35C2×C3×C3⋊Dic3C2312yes—
36C2×C3×C3⋊Dic3C2312yes—
37C2×C32×Dic3—28no—
38C2×C32×Dic3C2318yes—
39C3×#627—216no—
40C2×C3×C4⋊Q8—91no—
41C2×#888—32no—
42C22×C32×C4—21no—
43C22×C32×C4—22no—
44C22×C32×C4—34no—
45C22×C32×C4—41no—
46C22×C32×C4C2×D312yes—
47C22×C32×C4C22×C311yescharacteristic
48C22×C3⋊Dic3C22×C311yescharacteristic
49C22×C3⋊Dic3C22×C311yescharacteristic
50C22×C3×Dic3—24no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—61no—
54C22×C3×Dic3—61no—
55C22×C3×Dic3C2×D314yes—
56C22×C3×Dic3C22×C314yes—
57C2×C3×Dic6—316no—
58C2×C3×Dic6—616no—
59C2×#627—28no—
60C2×#627C22×C318yes—
61C32×C4⋊C4—38no—
62C4×C3⋊Dic3C22×C314yes—
63C3×C4⋊Dic3—28no—
64C3×C4⋊Dic3C2×D318yes—
65C3×C4⋊Dic3C22×C318yes—
66C3×Dic3⋊C4—316no—
67C3×C4×Dic3—38no—
68C3×C4×Dic3—64no—
69#3724C22×C318yes—
70C22×C33C2×D412yes—
71C22×C33C2×Q812yes—
72C22×C33C2×Q812yes—
73C22×C33C2411yescharacteristic
74C33×C4C2×Q814yescomplemented
75C3×C3⋊Dic3C2×D414yes—
76C3×C3⋊Dic3C2×Q814yescomplemented
77C32×Dic3—28no—
78C22×C3×Q8—92no—
79C22×Dic6—34no—
80C3×C4⋊Q8—98no—
81C2×C3×C4⋊C4—32no—
82C2×C3×C4⋊C4—32no—
83C2×C3×C4⋊C4—62no—
84C2×C3×C4⋊C4—92no—
85C2×C3×C42—91no—
86C2×C4⋊Dic3C3×D312yes—
87C2×Dic3⋊C4—34no—
88C2×C4×Dic3—32no—
89C2×C4×Dic3—61no—
90#888—316no—
91C23×C32—21no—
92C23×C32—22no—
93C23×C32—41no—
94C23×C32C22×D312yes—
95C23×C32C23×C311yescharacteristic
96C2×C32×C4—24no—
97C2×C32×C4—28no—
98C2×C32×C4—22no—
99C2×C32×C4—24no—
100C2×C32×C4—38no—
101C2×C32×C4—44no—
102C2×C32×C4—42no—
103C2×C32×C4—68no—
104C2×C32×C4Dic618yescomplemented
105C2×C32×C4C3×Q814yescomplemented
106C2×C32×C4C22×D314yes—
107C2×C32×C4C23×C312yes—
108C2×C3⋊Dic3C3×D414yes—
109C2×C3⋊Dic3C3×Q814yescomplemented
110C2×C3⋊Dic3C23×C312yes—
111C2×C3⋊Dic3C23×C312yes—
112C2×C3×Dic3—28no—
113C2×C3×Dic3—28no—
114C2×C3×Dic3—28no—
115C2×C3×Dic3—38no—
116C2×C3×Dic3—38no—
117C2×C3×Dic3—34no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3—48no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3C22×D318yes—
125C2×C3×Dic3C23×C318yes—
126C3×Dic6—632no—
127#627—216no—
128C2×C4⋊Q8—91noSylow
129C2×C33C4⋊Q814yescomplemented
130C2×C33C22×D411yescharacteristic
131C2×C33C22×Q811yescharacteristic
132C2×C33C22×Q811yescharacteristic
133C2×C3×Q8—98no—
134C2×C3×Q8—188no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—34no—
138C22×C3×C4—34no—
139C22×C3×C4—41no—
140C22×C3×C4—64no—
141C22×C3×C4—91no—
142C22×C3×C4—91no—
143C22×C3×C4D3211yescharacteristic
144C22×C3×C4C2×C3×D312yes—
145C22×Dic3—32no—
146C22×Dic3—32no—
147C22×Dic3—61no—
148C22×Dic3—61no—
149C22×Dic3C2×C3×D314yes—
150C2×Dic6—316no—
151C2×Dic6—616no—
152C3×C4⋊C4—38no—
153C3×C4⋊C4—38no—
154C3×C4⋊C4—68no—
155C3×C4⋊C4—98no—
156C3×C42—94no—
157C4⋊Dic3C2×C3×D318yes—
158Dic3⋊C4—316no—
159C4×Dic3—38no—
160C4×Dic3—64no—
161C22×C32—21no—
162C22×C32—22no—
163C22×C32—22no—
164C22×C32—22no—
165C22×C32—22no—
166C22×C32—24no—
167C22×C32—24no—
168C22×C32—24no—
169C22×C32—42no—
170C22×C32—42no—
171C22×C32—42no—
172C22×C32—41no—
173C22×C32C2×Dic614yes—
174C22×C32D3×D414yes—
175C22×C32D3×Q814yes—
176C22×C32C2×C3×D412yes—
177C22×C32C2×C3×Q812yes—
178C22×C32C2×C3×Q812yes—
179C22×C32C23×D312yes—
180C22×C32C24×C311yescharacteristic
181C32×C4—24no—
182C32×C4—28no—
183C32×C4—44no—
184C32×C4—68no—
185C32×C4C2×Dic618yescomplemented
186C32×C4C2×C3×Q814yescomplemented
187C3⋊Dic3C2×C3×D414yes—
188C3⋊Dic3C2×C3×Q814yescomplemented
189C3×Dic3—28no—
190C3×Dic3—28no—
191C3×Dic3—38no—
192C3×Dic3—38no—
193C3×Dic3—48no—
194C3×Dic3—64no—
195C3×Dic3—64no—
196C22×Q8—92no—
197C4⋊Q8—98no—
198C2×C4⋊C4—32no—
199C2×C4⋊C4—92no—
200C2×C42—91no—
201C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
202C23×C3—22no—
203C23×C3—21no—
204C23×C3—41no—
205C23×C3C2×D3211yescharacteristic
206C23×C3C22×C3×D312yes—
207C3×Q8—1816no—
208C2×C3×C4—28no—
209C2×C3×C4—24no—
210C2×C3×C4—24no—
211C2×C3×C4—22no—
212C2×C3×C4—38no—
213C2×C3×C4—38no—
214C2×C3×C4—44no—
215C2×C3×C4—42no—
216C2×C3×C4—68no—
217C2×C3×C4—68no—
218C2×C3×C4—68no—
219C2×C3×C4—92no—
220C2×C3×C4—92no—
221C2×C3×C4—94no—
222C2×C3×C4—94no—
223C2×C3×C4—128no—
224C2×C3×C4#62714yescomplemented
225C2×C3×C4C3×Dic618yescomplemented
226C2×C3×C4C2×D3212yes—
227C2×C3×C4C22×C3×D314yes—
228C2×Dic3—28no—
229C2×Dic3—34no—
230C2×Dic3—34no—
231C2×Dic3—38no—
232C2×Dic3—38no—
233C2×Dic3—64no—
234C2×Dic3—64no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3C22×C3×D318yes—
238Dic6—632no—
239C2×C32—28no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—22no—
244C2×C32—22no—
245C2×C32—22no—
246C2×C32—24no—
247C2×C32—44no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32#88818yescomplemented
252C2×C32C3×C4⋊Q814yescomplemented
253C2×C32C22×Dic612yes—
254C2×C32C2×D3×D412yes—
255C2×C32C2×D3×Q812yes—
256C2×C32C22×C3×D411yescharacteristic
257C2×C32C22×C3×Q811yescharacteristic
258C2×C32C22×C3×Q811yescharacteristic
259C2×Q8—98no—
260C2×Q8—188no—
261C22×C4—34no—
262C22×C4—91no—
263C22×C4—91no—
264C22×C4C3×D3211yescharacteristic
265C4⋊C4—38no—
266C4⋊C4—98no—
267C42—94no—
268C22×C3—21no—
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—24no—
272C22×C3—22no—
273C22×C3—24no—
274C22×C3—24no—
275C22×C3—22no—
276C22×C3—41no—
277C22×C3—42no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3#379712yes—
281C22×C3#380212yes—
282C22×C3C2×#62712yes—
283C22×C3C2×C3×Dic614yes—
284C22×C3C3×D3×D414yes—
285C22×C3C3×D3×Q814yes—
286C22×C3C22×D3211yescharacteristic
287C22×C3C23×C3×D312yes—
288C3×C4—28no—
289C3×C4—24no—
290C3×C4—44no—
291C3×C4—68no—
292C3×C4—68no—
293C3×C4—94no—
294C3×C4—94no—
295C3×C4—128no—
296C3×C4C2×#62714yescomplemented
297C3×C4C2×C3×Dic618yescomplemented
298Dic3—28no—
299Dic3—38no—
300Dic3—38no—
301Dic3—64no—
302Dic3—64no—
303C32—22no—
304C32—21no—
305C32—41no—
306C32C2×#88812yescomplemented
307C32C2×C3×C4⋊Q811yescharacteristic, complemented
308C23C2×C3×D3211yescharacteristic
309Q8—1816no—
310C2×C4—38no—
311C2×C4—68no—
312C2×C4—92no—
313C2×C4—92no—
314C2×C4—94no—
315C2×C4—94no—
316C2×C4C3×#62714yescomplemented
317C2×C4C2×C3×D3212yes—
318C2×C3—22no—
319C2×C3—21no—
320C2×C3—28no—
321C2×C3—24no—
322C2×C3—21no—
323C2×C3—22no—
324C2×C3—21no—
325C2×C3—22no—
326C2×C3—41no—
327C2×C3—44no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3#6391914yesdirect product factor
331C2×C3C3×#88818yescomplemented
332C2×C3C2×#379711yescharacteristic
333C2×C3C2×#380211yescharacteristic
334C2×C3C22×#62711yescharacteristic
335C2×C3C22×C3×Dic612yes—
336C2×C3C2×C3×D3×D412yes—
337C2×C3C2×C3×D3×Q812yes—
338C22C3×#379712yes—
339C22C3×#380212yes—
340C22C2×C3×#62712yes—
341C22C22×C3×D3211yescharacteristic
342C4—68no—
343C4—94no—
344C4—94no—
345C4C2×C3×#62714yescomplemented
346C3—22no—
347C3—21no—
348C3—41no—
349C3C2×#6391911yescharacteristic, direct product factor
350C3C2×C3×#88812yescomplemented
351C2C3×#6391914yesdirect product factor
352C2C2×C3×#379711yescharacteristic
353C2C2×C3×#380211yescharacteristic
354C2C22×C3×#62711yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×C4⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
5. C3×#63919C2direct (N × H)
7. C2×#63919C3direct (N × H)
29. #63919C2×C3direct (N × H)
31. C2×C33×C4Q8semidirect
34. C2×C3×C3⋊Dic3Q8semidirect
74. C33×C4C2×Q8semidirect
76. C3×C3⋊Dic3C2×Q8semidirect
104. C2×C32×C4Dic6semidirect
105. C2×C32×C4C3×Q8semidirect
109. C2×C3⋊Dic3C3×Q8semidirect
129. C2×C33C4⋊Q8semidirect
185. C32×C4C2×Dic6semidirect
186. C32×C4C2×C3×Q8semidirect
188. C3⋊Dic3C2×C3×Q8semidirect
201. C33C2×C4⋊Q8semidirect
224. C2×C3×C4#627semidirect
225. C2×C3×C4C3×Dic6semidirect
251. C2×C32#888semidirect
252. C2×C32C3×C4⋊Q8semidirect
296. C3×C4C2×#627semidirect
297. C3×C4C2×C3×Dic6semidirect
306. C32C2×#888semidirect
307. C32C2×C3×C4⋊Q8semidirect
316. C2×C4C3×#627semidirect
330. C2×C3#63919direct (N × H)
331. C2×C3C3×#888semidirect
345. C4C2×C3×#627semidirect
349. C3C2×#63919direct (N × H)
350. C3C2×C3×#888semidirect
351. C2C3×#63919direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6272
C2×C3×C4×C3⋊Dic32
C2×C32×C4⋊Dic32
C2×C3×#37242
C3×#639192
C2×C3×#8883
C2×#639193