Finite Group Databasebeta

← C2×C3×C4×#627 #13174630

Subgroups — C2×C3×C4×#627 #13174630

Order 1728.

1570 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×#627C211yesmaximal, characteristic
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3C2×C32×C4×Dic3C212yesmaximal
4C2×C3×#3724C211yesmaximal, characteristic
5C2×C3×Dic3⋊Dic3C212yesmaximal
6C3×C4×#627C218yesmaximal, direct product factor
7C2×C3×C4×Dic6—32nomaximal
8C2×C4×#627C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2212yes—
14C2×C3×#627C418yesdirect product factor
15C2×C3×#627C2214yes—
16C3×C4×C3⋊Dic3C2214yes—
17C32×C4×Dic3C2218yes—
18C3×#3724C2214yes—
19C3×Dic3⋊Dic3C2218yes—
20C22×C3×Dic6—32no—
21C22×#627C2×C311yescharacteristic
22C2×C32×C42—32no—
23C2×C4×C3⋊Dic3C2×C311yescharacteristic
24C2×C3×C4⋊Dic3—32no—
25C2×C3×Dic3⋊C4—32no—
26C2×C3×Dic3⋊C4—32no—
27C2×C3×C4×Dic3—22no—
28C2×C3×C4×Dic3—32no—
29C2×C3×C4×Dic3—61no—
30C2×C3×C4×Dic3D312yes—
31C2×C3×C4×Dic3C2×C312yes—
32C3×C4×Dic6—316no—
33C2×#3724C2×C311yescharacteristic
34C2×Dic3⋊Dic3C2×C312yes—
35C4×#627C2×C318yesdirect product factor
36C23×C33C2311yescharacteristic
37C2×C33×C4Q814yescomplemented
38C2×C33×C4C2312yes—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3C2×C414yes—
41C2×C3×C3⋊Dic3C2312yes—
42C2×C3×C3⋊Dic3C2312yes—
43C2×C32×Dic3—24no—
44C2×C32×Dic3C2×C418yes—
45C2×C32×Dic3C2314yes—
46C2×C32×Dic3C2314yes—
47C3×#627C2×C4116yesdirect product factor
48C2×C3×C4×Q8—91no—
49C2×C4×Dic6—32no—
50C22×C32×C4—21no—
51C22×C32×C4—22no—
52C22×C32×C4—32no—
53C22×C32×C4—32no—
54C22×C32×C4—41no—
55C22×C32×C4C2×D312yes—
56C22×C32×C4C22×C311yescharacteristic
57C22×C3⋊Dic3C22×C311yescharacteristic
58C22×C3⋊Dic3C22×C311yescharacteristic
59C22×C3×Dic3—22no—
60C22×C3×Dic3—22no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—61no—
64C22×C3×Dic3—61no—
65C22×C3×Dic3C2×D312yes—
66C22×C3×Dic3C2×D312yes—
67C22×C3×Dic3C22×C312yes—
68C22×C3×Dic3C22×C312yes—
69C2×C3×Dic6—316no—
70C2×C3×Dic6—38no—
71C2×#627C3×C418yesdirect product factor
72C2×#627C22×C314yes—
73C32×C42—38no—
74C4×C3⋊Dic3C22×C314yes—
75C3×C4⋊Dic3—38no—
76C3×Dic3⋊C4—38no—
77C3×Dic3⋊C4—38no—
78C3×C4×Dic3—28no—
79C3×C4×Dic3—38no—
80C3×C4×Dic3—64no—
81C3×C4×Dic3C2×D318yes—
82C3×C4×Dic3C22×C318yes—
83#3724C22×C314yes—
84Dic3⋊Dic3C22×C318yes—
85C22×C33C22×C412yes—
86C22×C33C2×Q812yes—
87C22×C33#4112yes—
88C22×C33C2411yescharacteristic
89C33×C4C2×Q814yescomplemented
90C3×C3⋊Dic3—22no—
91C3×C3⋊Dic3C22×C414yes—
92C32×Dic3—24no—
93C32×Dic3C22×C418yes—
94C22×C3×Q8—91no—
95C22×Dic6—32no—
96C3×C4×Q8—98no—
97C2×C3×C4⋊C4—91no—
98C2×C3×C4⋊C4—92no—
99C2×C3×C42—32no—
100C2×C3×C42—32no—
101C2×C3×C42—62no—
102C2×C3×C42—91no—
103C2×C4⋊Dic3—32no—
104C2×Dic3⋊C4—32no—
105C2×Dic3⋊C4—32no—
106C2×C4×Dic3—32no—
107C2×C4×Dic3—61no—
108C2×C4×Dic3C3×D312yes—
109C4×Dic6—316no—
110C23×C32—21no—
111C23×C32—22no—
112C23×C32—41no—
113C23×C32C22×D312yes—
114C23×C32C23×C311yescharacteristic
115C2×C32×C4—24no—
116C2×C32×C4—28no—
117C2×C32×C4—22no—
118C2×C32×C4—24no—
119C2×C32×C4—38no—
120C2×C32×C4—34no—
121C2×C32×C4—34no—
122C2×C32×C4—44no—
123C2×C32×C4—42no—
124C2×C32×C4—64no—
125C2×C32×C4Dic618yescomplemented
126C2×C32×C4C3×Q814yescomplemented
127C2×C32×C4C22×D314yes—
128C2×C32×C4C23×C312yes—
129C2×C3⋊Dic3—22no—
130C2×C3⋊Dic3C2×C3×C414yes—
131C2×C3⋊Dic3C23×C312yes—
132C2×C3⋊Dic3C23×C312yes—
133C2×C3×Dic3—28no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—28no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—38no—
140C2×C3×Dic3—34no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3C4×D318yes—
148C2×C3×Dic3C2×C3×C418yes—
149C2×C3×Dic3C22×D314yes—
150C2×C3×Dic3C22×D314yes—
151C2×C3×Dic3C23×C314yes—
152C2×C3×Dic3C23×C314yes—
153C3×Dic6—332no—
154#627C2×C3×C4116yesdirect product factor
155C2×C4×Q8—91noSylow
156C2×C33C4×Q814yescomplemented
157C2×C33C23×C411yescharacteristic
158C2×C33C22×Q811yescharacteristic
159C2×C33C2×#4111yescharacteristic
160C2×C3×Q8—94no—
161C2×C3×Q8—98no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—32no—
165C22×C3×C4—32no—
166C22×C3×C4—32no—
167C22×C3×C4—32no—
168C22×C3×C4—41no—
169C22×C3×C4—62no—
170C22×C3×C4—62no—
171C22×C3×C4—91no—
172C22×C3×C4—91no—
173C22×C3×C4D3211yescharacteristic
174C22×C3×C4C2×C3×D312yes—
175C22×Dic3—32no—
176C22×Dic3—32no—
177C22×Dic3—61no—
178C22×Dic3—61no—
179C22×Dic3C2×C3×D312yes—
180C22×Dic3C2×C3×D312yes—
181C2×Dic6—316no—
182C2×Dic6—38no—
183C3×C4⋊C4—94no—
184C3×C4⋊C4—98no—
185C3×C42—38no—
186C3×C42—38no—
187C3×C42—68no—
188C3×C42—94no—
189C4⋊Dic3—38no—
190Dic3⋊C4—38no—
191Dic3⋊C4—38no—
192C4×Dic3—38no—
193C4×Dic3—64no—
194C4×Dic3C2×C3×D318yes—
195C22×C32—21no—
196C22×C32—22no—
197C22×C32—22no—
198C22×C32—22no—
199C22×C32—22no—
200C22×C32—24no—
201C22×C32—24no—
202C22×C32—24no—
203C22×C32—42no—
204C22×C32—42no—
205C22×C32—42no—
206C22×C32—41no—
207C22×C32C2×Dic614yes—
208C22×C32C2×C4×D314yes—
209C22×C32#23514yes—
210C22×C32C22×C3×C412yes—
211C22×C32C2×C3×Q812yes—
212C22×C32C3×#4112yes—
213C22×C32C23×D312yes—
214C22×C32C24×C311yescharacteristic
215C32×C4—24no—
216C32×C4—28no—
217C32×C4—38no—
218C32×C4—44no—
219C32×C4—64no—
220C32×C4C2×Dic618yescomplemented
221C32×C4C2×C3×Q814yescomplemented
222C3⋊Dic3—22no—
223C3⋊Dic3C22×C3×C414yes—
224C3×Dic3—28no—
225C3×Dic3—24no—
226C3×Dic3—24no—
227C3×Dic3—28no—
228C3×Dic3—38no—
229C3×Dic3—64no—
230C3×Dic3—64no—
231C3×Dic3—64no—
232C3×Dic3C2×C4×D318yes—
233C3×Dic3C22×C3×C418yes—
234C22×Q8—91no—
235C4×Q8—98no—
236C2×C4⋊C4—91no—
237C2×C4⋊C4—92no—
238C2×C42—32no—
239C2×C42—91no—
240C33C2×C4×Q811yescharacteristic, complemented, Sylow
241C23×C3—22no—
242C23×C3—21no—
243C23×C3—41no—
244C23×C3C2×D3211yescharacteristic
245C23×C3C22×C3×D312yes—
246C3×Q8—916no—
247C2×C3×C4—28no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—34no—
252C2×C3×C4—34no—
253C2×C3×C4—38no—
254C2×C3×C4—38no—
255C2×C3×C4—34no—
256C2×C3×C4—34no—
257C2×C3×C4—44no—
258C2×C3×C4—42no—
259C2×C3×C4—64no—
260C2×C3×C4—64no—
261C2×C3×C4—68no—
262C2×C3×C4—68no—
263C2×C3×C4—64no—
264C2×C3×C4—64no—
265C2×C3×C4—92no—
266C2×C3×C4—94no—
267C2×C3×C4—92no—
268C2×C3×C4—182no—
269C2×C3×C4#62714yesdirect product factor
270C2×C3×C4C3×Dic618yescomplemented
271C2×C3×C4C2×D3212yes—
272C2×C3×C4C22×C3×D314yes—
273C2×Dic3—24no—
274C2×Dic3—34no—
275C2×Dic3—38no—
276C2×Dic3—34no—
277C2×Dic3—64no—
278C2×Dic3—62no—
279C2×Dic3—62no—
280C2×Dic3—64no—
281C2×Dic3—64no—
282C2×Dic3C3×C4×D318yes—
283C2×Dic3C22×C3×D314yes—
284C2×Dic3C22×C3×D314yes—
285Dic6—332no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—22no—
290C2×C32—22no—
291C2×C32—22no—
292C2×C32—24no—
293C2×C32—28no—
294C2×C32—44no—
295C2×C32—41no—
296C2×C32—41no—
297C2×C32—41no—
298C2×C32C4×Dic618yescomplemented
299C2×C32C3×C4×Q814yescomplemented
300C2×C32C22×Dic612yes—
301C2×C32C22×C4×D312yes—
302C2×C32C2×#23512yes—
303C2×C32C23×C3×C411yescharacteristic
304C2×C32C22×C3×Q811yescharacteristic
305C2×C32C2×C3×#4111yescharacteristic
306C2×Q8—94no—
307C2×Q8—98no—
308C22×C4—32no—
309C22×C4—32no—
310C22×C4—91no—
311C22×C4—91no—
312C22×C4C3×D3211yescharacteristic
313C4⋊C4—94no—
314C4⋊C4—98no—
315C42—38no—
316C42—94no—
317C22×C3—21no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—24no—
321C22×C3—22no—
322C22×C3—24no—
323C22×C3—24no—
324C22×C3—22no—
325C22×C3—41no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3#379812yes—
330C22×C3C4×D3212yes—
331C22×C3C2×#62712yes—
332C22×C3C2×C3×Dic614yes—
333C22×C3C2×C3×C4×D314yes—
334C22×C3C3×#23514yes—
335C22×C3C22×D3211yescharacteristic
336C22×C3C23×C3×D312yes—
337C3×C4—28no—
338C3×C4—24no—
339C3×C4—38no—
340C3×C4—38no—
341C3×C4—44no—
342C3×C4—68no—
343C3×C4—68no—
344C3×C4—64no—
345C3×C4—64no—
346C3×C4—94no—
347C3×C4—182no—
348C3×C4C2×#62714yesdirect product factor
349C3×C4C2×C3×Dic618yescomplemented
350Dic3—24no—
351Dic3—38no—
352Dic3—64no—
353Dic3—64no—
354Dic3—64no—
355Dic3C2×C3×C4×D318yes—
356C32—21no—
357C32—22no—
358C32—41no—
359C32C2×C4×Dic612yescomplemented
360C32C2×C3×C4×Q811yescharacteristic, complemented
361C23C2×C3×D3211yescharacteristic
362Q8—916no—
363C2×C4—34no—
364C2×C4—38no—
365C2×C4—34no—
366C2×C4—64no—
367C2×C4—92no—
368C2×C4—94no—
369C2×C4—92no—
370C2×C4—182no—
371C2×C4C3×#62714yesdirect product factor
372C2×C4C2×C3×D3212yes—
373C2×C3—22no—
374C2×C3—21no—
375C2×C3—28no—
376C2×C3—24no—
377C2×C3—21no—
378C2×C3—22no—
379C2×C3—21no—
380C2×C3—22no—
381C2×C3—41no—
382C2×C3—44no—
383C2×C3—41no—
384C2×C3—41no—
385C2×C3C4×#62714yesdirect product factor
386C2×C3C3×C4×Dic618yescomplemented
387C2×C3C2×#379811yescharacteristic
388C2×C3C2×C4×D3211yescharacteristic
389C2×C3C22×#62711yescharacteristic
390C2×C3C22×C3×Dic612yes—
391C2×C3C22×C3×C4×D312yes—
392C2×C3C2×C3×#23512yes—
393C22C3×#379812yes—
394C22C3×C4×D3212yes—
395C22C2×C3×#62712yes—
396C22C22×C3×D3211yescharacteristic
397C4—38no—
398C4—64no—
399C4—94no—
400C4—182no—
401C4C2×C3×#62714yesdirect product factor
402C3—22no—
403C3—21no—
404C3—41no—
405C3C2×C4×#62711yescharacteristic, direct product factor
406C3C2×C3×C4×Dic612yescomplemented
407C2C3×C4×#62714yesdirect product factor
408C2C2×C3×#379811yescharacteristic
409C2C2×C3×C4×D3211yescharacteristic
410C2C22×C3×#62711yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC22×C32C24×C3
hypocenterC32C2×C3×C4×Q8
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
6. C3×C4×#627C2direct (N × H)
8. C2×C4×#627C3direct (N × H)
14. C2×C3×#627C4direct (N × H)
35. C4×#627C2×C3direct (N × H)
37. C2×C33×C4Q8semidirect
47. C3×#627C2×C4direct (N × H)
71. C2×#627C3×C4direct (N × H)
89. C33×C4C2×Q8semidirect
125. C2×C32×C4Dic6semidirect
126. C2×C32×C4C3×Q8semidirect
154. #627C2×C3×C4direct (N × H)
156. C2×C33C4×Q8semidirect
220. C32×C4C2×Dic6semidirect
221. C32×C4C2×C3×Q8semidirect
240. C33C2×C4×Q8semidirect
269. C2×C3×C4#627direct (N × H)
270. C2×C3×C4C3×Dic6semidirect
298. C2×C32C4×Dic6semidirect
299. C2×C32C3×C4×Q8semidirect
348. C3×C4C2×#627direct (N × H)
349. C3×C4C2×C3×Dic6semidirect
359. C32C2×C4×Dic6semidirect
360. C32C2×C3×C4×Q8semidirect
371. C2×C4C3×#627direct (N × H)
385. C2×C3C4×#627direct (N × H)
386. C2×C3C3×C4×Dic6semidirect
401. C4C2×C3×#627direct (N × H)
405. C3C2×C4×#627direct (N × H)
406. C3C2×C3×C4×Dic6semidirect
407. C2C3×C4×#627direct (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
C2×C3×Dic3⋊Dic32
C3×C4×#6272
C2×C3×C4×Dic63
C2×C4×#6273