Finite Group Databasebeta

← C2×C3×#63840 #13174637

Subgroups — C2×C3×#63840 #13174637

Order 1728.

1554 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic
2C22×C3×#627C211yesmaximal, characteristic
3C2×C32×Dic3⋊C4C212yesmaximal
4C2×C3×Dic3⋊Dic3C212yesmaximal
5C2×C3×Dic32C211yesmaximal, characteristic
6C3×#63840C218yesmaximal, direct product factor
7C2×C3×#888—32nomaximal
8C2×#63840C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C2×C3×#634—24no—
13C2×C3×#634C2214yes—
14C22×C32×Dic3C2212yes—
15C22×C32×Dic3C2212yes—
16C2×C3×#627—24no—
17C2×C3×#627C2214yes—
18C32×Dic3⋊C4C2218yes—
19C3×Dic3⋊Dic3C2218yes—
20C3×Dic32C2214yes—
21C22×#634C2×C311yescharacteristic
22C22×C3×Dic6—32no—
23C22×C3×Dic6—32no—
24C22×C3×Dic6—61no—
25C22×#627C2×C311yescharacteristic
26C2×C32×C4⋊C4—32no—
27C2×C3×C4⋊Dic3—32no—
28C2×C3×Dic3⋊C4—22no—
29C2×C3×Dic3⋊C4—32no—
30C2×C3×Dic3⋊C4D312yes—
31C2×C3×Dic3⋊C4C2×C312yes—
32C2×C3×C4×Dic3—32no—
33C3×#888—316no—
34C2×Dic3⋊Dic3C2×C312yes—
35C2×Dic32C2×C311yescharacteristic
36#63840C2×C318yesdirect product factor
37C23×C33C2311yescharacteristic
38C2×C33×C4—22no—
39C2×C33×C4C2312yes—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3D414yes—
42C2×C3×C3⋊Dic3C2312yes—
43C2×C3×C3⋊Dic3C2312yes—
44C3×#634—28no—
45C2×C32×Dic3—24no—
46C2×C32×Dic3Q818yescomplemented
47C2×C32×Dic3C2314yes—
48C2×C32×Dic3C2314yes—
49C3×#627—28no—
50C2×C3×C4⋊Q8—91no—
51C2×#888—32no—
52C22×C32×C4—21no—
53C22×C32×C4—22no—
54C22×C32×C4—32no—
55C22×C32×C4—32no—
56C22×C32×C4—41no—
57C22×C32×C4C2×D312yes—
58C22×C32×C4C22×C311yescharacteristic
59C22×C3⋊Dic3C22×C311yescharacteristic
60C22×C3⋊Dic3C22×C311yescharacteristic
61C2×#634—24no—
62C2×#634C22×C314yes—
63C22×C3×Dic3—22no—
64C22×C3×Dic3—22no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—61no—
68C22×C3×Dic3—61no—
69C22×C3×Dic3C2×D312yes—
70C22×C3×Dic3C2×D312yes—
71C22×C3×Dic3C22×C312yes—
72C22×C3×Dic3C22×C312yes—
73C2×C3×Dic6—38no—
74C2×C3×Dic6—38no—
75C2×C3×Dic6—68no—
76C2×C3×Dic6—68no—
77C2×C3×Dic6—64no—
78C2×C3×Dic6—68no—
79C2×#627—24no—
80C2×#627C22×C314yes—
81C32×C4⋊C4—38no—
82C3×C4⋊Dic3—38no—
83C3×Dic3⋊C4—28no—
84C3×Dic3⋊C4—38no—
85C3×Dic3⋊C4C2×D318yes—
86C3×Dic3⋊C4C22×C318yes—
87C3×C4×Dic3—38no—
88Dic3⋊Dic3C22×C318yes—
89Dic32C22×C314yes—
90C22×C33C2×D412yes—
91C22×C33C2×Q814yes—
92C22×C33C2411yescharacteristic
93C33×C4—22no—
94C3×C3⋊Dic3—22no—
95C3×C3⋊Dic3C2×D414yes—
96C32×Dic3—24no—
97C32×Dic3C2×Q818yescomplemented
98C22×C3×Q8—91no—
99C22×C3×Q8—91no—
100C22×Dic6—32no—
101C22×Dic6—32no—
102C22×Dic6—61no—
103C3×C4⋊Q8—98no—
104C2×C3×C4⋊C4—32no—
105C2×C3×C4⋊C4—32no—
106C2×C3×C4⋊C4—62no—
107C2×C3×C4⋊C4—92no—
108C2×C3×C42—91no—
109C2×C4⋊Dic3—32no—
110C2×Dic3⋊C4—32no—
111C2×Dic3⋊C4C3×D312yes—
112C2×C4×Dic3—32no—
113#888—316no—
114C23×C32—21no—
115C23×C32—22no—
116C23×C32—41no—
117C23×C32C22×D312yes—
118C23×C32C23×C311yescharacteristic
119C2×C32×C4—22no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—22no—
123C2×C32×C4—24no—
124C2×C32×C4—38no—
125C2×C32×C4—34no—
126C2×C32×C4—34no—
127C2×C32×C4—42no—
128C2×C32×C4—44no—
129C2×C32×C4—44no—
130C2×C32×C4—64no—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C23×C312yes—
133C2×C3⋊Dic3—22no—
134C2×C3⋊Dic3C3×D414yes—
135C2×C3⋊Dic3C23×C312yes—
136C2×C3⋊Dic3C23×C312yes—
137#634—28no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—28no—
141C2×C3×Dic3—24no—
142C2×C3×Dic3—24no—
143C2×C3×Dic3—38no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—34no—
146C2×C3×Dic3—44no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—64no—
151C2×C3×Dic3—64no—
152C2×C3×Dic3Dic618yescomplemented
153C2×C3×Dic3C3×Q818yescomplemented
154C2×C3×Dic3C22×D314yes—
155C2×C3×Dic3C22×D314yes—
156C2×C3×Dic3C23×C314yes—
157C2×C3×Dic3C23×C314yes—
158C3×Dic6—616no—
159C3×Dic6—616no—
160C3×Dic6—616no—
161#627—28no—
162C2×C4⋊Q8—91noSylow
163C2×C33C4⋊Q814yescomplemented
164C2×C33C22×D411yescharacteristic
165C2×C33C22×Q812yes—
166C2×C3×Q8—94no—
167C2×C3×Q8—94no—
168C2×C3×Q8—184no—
169C2×C3×Q8—184no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—32no—
173C22×C3×C4—32no—
174C22×C3×C4—32no—
175C22×C3×C4—32no—
176C22×C3×C4—41no—
177C22×C3×C4—62no—
178C22×C3×C4—62no—
179C22×C3×C4—91no—
180C22×C3×C4—91no—
181C22×C3×C4D3211yescharacteristic
182C22×C3×C4C2×C3×D312yes—
183C22×Dic3—32no—
184C22×Dic3—32no—
185C22×Dic3—61no—
186C22×Dic3—61no—
187C22×Dic3C2×C3×D312yes—
188C22×Dic3C2×C3×D312yes—
189C2×Dic6—38no—
190C2×Dic6—38no—
191C2×Dic6—64no—
192C2×Dic6—68no—
193C2×Dic6—68no—
194C2×Dic6—68no—
195C3×C4⋊C4—38no—
196C3×C4⋊C4—38no—
197C3×C4⋊C4—68no—
198C3×C4⋊C4—98no—
199C3×C42—94no—
200C4⋊Dic3—38no—
201Dic3⋊C4—38no—
202Dic3⋊C4C2×C3×D318yes—
203C4×Dic3—38no—
204C22×C32—21no—
205C22×C32—22no—
206C22×C32—24no—
207C22×C32—22no—
208C22×C32—24no—
209C22×C32—24no—
210C22×C32—24no—
211C22×C32—42no—
212C22×C32—44no—
213C22×C32—41no—
214C22×C32C2×Dic614yes—
215C22×C32D3×D414yes—
216C22×C32D3×Q814yes—
217C22×C32C2×C3×D412yes—
218C22×C32C2×C3×Q814yes—
219C22×C32C23×D312yes—
220C22×C32C24×C311yescharacteristic
221C32×C4—22no—
222C32×C4—24no—
223C32×C4—24no—
224C32×C4—38no—
225C32×C4—44no—
226C32×C4—44no—
227C32×C4—64no—
228C3⋊Dic3—22no—
229C3⋊Dic3C2×C3×D414yes—
230C3×Dic3—24no—
231C3×Dic3—24no—
232C3×Dic3—28no—
233C3×Dic3—38no—
234C3×Dic3—44no—
235C3×Dic3—64no—
236C3×Dic3—64no—
237C3×Dic3—64no—
238C3×Dic3C2×Dic618yescomplemented
239C3×Dic3C2×C3×Q818yescomplemented
240C22×Q8—91no—
241C22×Q8—91no—
242C4⋊Q8—98no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—92no—
245C2×C42—91no—
246C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
247C23×C3—22no—
248C23×C3—21no—
249C23×C3—41no—
250C23×C3C2×D3211yescharacteristic
251C23×C3C22×C3×D312yes—
252C3×Q8—188no—
253C3×Q8—188no—
254C2×C3×C4—24no—
255C2×C3×C4—22no—
256C2×C3×C4—24no—
257C2×C3×C4—22no—
258C2×C3×C4—24no—
259C2×C3×C4—34no—
260C2×C3×C4—34no—
261C2×C3×C4—38no—
262C2×C3×C4—38no—
263C2×C3×C4—34no—
264C2×C3×C4—34no—
265C2×C3×C4—44no—
266C2×C3×C4—44no—
267C2×C3×C4—42no—
268C2×C3×C4—64no—
269C2×C3×C4—68no—
270C2×C3×C4—64no—
271C2×C3×C4—64no—
272C2×C3×C4—64no—
273C2×C3×C4—92no—
274C2×C3×C4—94no—
275C2×C3×C4—92no—
276C2×C3×C4—124no—
277C2×C3×C4—182no—
278C2×C3×C4C2×D3212yes—
279C2×C3×C4C22×C3×D314yes—
280C2×Dic3—24no—
281C2×Dic3—34no—
282C2×Dic3—38no—
283C2×Dic3—34no—
284C2×Dic3—64no—
285C2×Dic3—62no—
286C2×Dic3—62no—
287C2×Dic3—64no—
288C2×Dic3—64no—
289C2×Dic3C3×Dic618yescomplemented
290C2×Dic3C22×C3×D314yes—
291C2×Dic3C22×C3×D314yes—
292Dic6—616no—
293Dic6—616no—
294Dic6—616no—
295C2×C32—28no—
296C2×C32—22no—
297C2×C32—21no—
298C2×C32—22no—
299C2×C32—22no—
300C2×C32—22no—
301C2×C32—24no—
302C2×C32—44no—
303C2×C32—41no—
304C2×C32—42no—
305C2×C32#88818yescomplemented
306C2×C32C3×C4⋊Q814yescomplemented
307C2×C32C22×Dic612yes—
308C2×C32C2×D3×D412yes—
309C2×C32C2×D3×Q812yes—
310C2×C32C22×C3×D411yescharacteristic
311C2×C32C22×C3×Q812yes—
312C2×Q8—94no—
313C2×Q8—94no—
314C2×Q8—184no—
315C2×Q8—184no—
316C22×C4—32no—
317C22×C4—32no—
318C22×C4—91no—
319C22×C4—91no—
320C22×C4C3×D3211yescharacteristic
321C4⋊C4—38no—
322C4⋊C4—98no—
323C42—94no—
324C22×C3—21no—
325C22×C3—22no—
326C22×C3—24no—
327C22×C3—24no—
328C22×C3—24no—
329C22×C3—22no—
330C22×C3—24no—
331C22×C3—41no—
332C22×C3—44no—
333C22×C3—42no—
334C22×C3D3×Dic614yes—
335C22×C3#381112yes—
336C22×C3C2×C3×Dic614yes—
337C22×C3C3×D3×D414yes—
338C22×C3C3×D3×Q814yes—
339C22×C3C22×D3211yescharacteristic
340C22×C3C23×C3×D312yes—
341C3×C4—24no—
342C3×C4—22no—
343C3×C4—24no—
344C3×C4—38no—
345C3×C4—38no—
346C3×C4—44no—
347C3×C4—44no—
348C3×C4—68no—
349C3×C4—64no—
350C3×C4—64no—
351C3×C4—94no—
352C3×C4—124no—
353C3×C4—182no—
354Dic3—24no—
355Dic3—38no—
356Dic3—64no—
357Dic3—64no—
358Dic3—64no—
359Dic3C2×C3×Dic618yescomplemented
360C32—22no—
361C32—21no—
362C32—41no—
363C32C2×#88812yescomplemented
364C32C2×C3×C4⋊Q811yescharacteristic, complemented
365C23C2×C3×D3211yescharacteristic
366Q8—188no—
367Q8—188no—
368C2×C4—22no—
369C2×C4—34no—
370C2×C4—38no—
371C2×C4—34no—
372C2×C4—64no—
373C2×C4—92no—
374C2×C4—94no—
375C2×C4—92no—
376C2×C4—182no—
377C2×C4C2×C3×D3212yes—
378C2×C3—21no—
379C2×C3—22no—
380C2×C3—28no—
381C2×C3—24no—
382C2×C3—22no—
383C2×C3—22no—
384C2×C3—22no—
385C2×C3—41no—
386C2×C3—44no—
387C2×C3—42no—
388C2×C3#6384014yesdirect product factor
389C2×C3C3×#88818yescomplemented
390C2×C3C2×D3×Dic612yes—
391C2×C3C2×#381111yescharacteristic
392C2×C3C22×C3×Dic612yes—
393C2×C3C2×C3×D3×D412yes—
394C2×C3C2×C3×D3×Q812yes—
395C22C3×D3×Dic614yes—
396C22C3×#381112yes—
397C22C22×C3×D3211yescharacteristic
398C4—22no—
399C4—38no—
400C4—64no—
401C4—94no—
402C4—182no—
403C3—22no—
404C3—21no—
405C3—41no—
406C3C2×#6384011yescharacteristic, direct product factor
407C3C2×C3×#88812yescomplemented
408C2C3×#6384014yesdirect product factor
409C2C2×C3×D3×Dic612yes—
410C2C2×C3×#381111yescharacteristic
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
6. C3×#63840C2direct (N × H)
8. C2×#63840C3direct (N × H)
36. #63840C2×C3direct (N × H)
46. C2×C32×Dic3Q8semidirect
97. C32×Dic3C2×Q8semidirect
152. C2×C3×Dic3Dic6semidirect
153. C2×C3×Dic3C3×Q8semidirect
163. C2×C33C4⋊Q8semidirect
238. C3×Dic3C2×Dic6semidirect
239. C3×Dic3C2×C3×Q8semidirect
246. C33C2×C4⋊Q8semidirect
289. C2×Dic3C3×Dic6semidirect
305. C2×C32#888semidirect
306. C2×C32C3×C4⋊Q8semidirect
359. Dic3C2×C3×Dic6semidirect
363. C32C2×#888semidirect
364. C32C2×C3×C4⋊Q8semidirect
388. C2×C3#63840direct (N × H)
389. C2×C3C3×#888semidirect
406. C3C2×#63840direct (N × H)
407. C3C2×C3×#888semidirect
408. C2C3×#63840direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C22×C3×#6272
C2×C32×Dic3⋊C42
C2×C3×Dic3⋊Dic32
C2×C3×Dic322
C3×#638402
C2×C3×#8883
C2×#638403