Finite Group Databasebeta

← C2×C4×#6294 #13178517

Subgroups — C2×C4×#6294 #13178517

Order 1728.

2498 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C22×#6294C212yesmaximal
3C2×C3×C4×C3⋊Dic3C212yesmaximal, complemented
4C2×#88813C212yesmaximal, complemented
5C4×#6294C218yesmaximal, direct product factor
6C2×C4×#624—31nomaximal
7C2×C4×D3×Dic3—32nomaximal
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2214yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C2×C3×C4×C3⋊D3C418yescomplemented
13C2×C3×C4×C3⋊D3C2214yes—
14C2×#6294C4116yesdirect product factor
15C2×#6294C2218yes—
16C3×C4×C3⋊Dic3C2218yescomplemented
17#88813C2218yescomplemented
18C22×C4×C3⋊D3D311yescharacteristic
19C22×C3×C4×D3—32no—
20C22×C3×C4×D3—61no—
21C22×#624—32no—
22C22×D3×Dic3—34no—
23C2×C4×C3⋊Dic3D312yescomplemented
24C2×C3×C4×Dic3—32no—
25C2×C3×C4×Dic3—32no—
26C2×C3×C4×Dic3—62no—
27C2×Dic32—34no—
28C2×Dic32—32no—
29C4×#624—38no—
30C4×D3×Dic3—316no—
31C23×C33C2311yescharacteristic
32C22×C3×C3⋊D3C2×C418yes—
33C22×C3×C3⋊D3C2×C414yes—
34C22×C3×C3⋊D3C2312yes—
35C2×C33×C4C2×C414yescomplemented
36C2×C33×C4C2312yes—
37C2×C3×C3⋊Dic3C2×C4116yescomplemented
38C2×C3×C3⋊Dic3C2×C414yescomplemented
39C2×C3×C3⋊Dic3C2318yes—
40C2×C3×C3⋊Dic3C2312yes—
41C3×C4×C3⋊D3C2×C4116yescomplemented
42#6294C2×C4132yesdirect product factor
43C22×C4×Dic3—91no—
44C2×C42×D3—92no—
45C23×C3⋊D3C2×D311yescharacteristic
46C23×C3×D3—32no—
47C23×C3×D3—61no—
48C22×C32×C4—22no—
49C22×C32×C4—21no—
50C22×C32×C4—41no—
51C22×C32×C4C2×D312yes—
52C22×C32×C4C2×D311yescharacteristic
53C22×C3⋊Dic3C2×D314yes—
54C22×C3⋊Dic3C2×D311yescharacteristic
55C2×C4×C3⋊D3Dic318yescomplemented
56C2×C4×C3⋊D3C2×D314yes—
57C22×C3×Dic3—34no—
58C22×C3×Dic3—34no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—64no—
61C22×C3×Dic3—61no—
62C2×C3×C4×D3—316no—
63C2×C3×C4×D3—38no—
64C2×C3×C4×D3—68no—
65C2×C3×C4×D3—64no—
66C2×#624—316no—
67C2×#624—38no—
68C2×D3×Dic3—332no—
69C2×D3×Dic3—316no—
70C4×C3⋊Dic3C2×D318yescomplemented
71C3×C4×Dic3—38no—
72C3×C4×Dic3—38no—
73C3×C4×Dic3—68no—
74Dic32—316no—
75Dic32—38no—
76C22×C33C22×C414yes—
77C22×C33C22×C412yes—
78C22×C33C2411yescharacteristic
79C2×C3×C3⋊D3C42116yescomplemented
80C2×C3×C3⋊D3C22×C418yes—
81C2×C3×C3⋊D3C22×C414yes—
82C33×C4C22×C414yescomplemented
83C3×C3⋊Dic3C22×C4116yescomplemented
84C3×C3⋊Dic3C22×C414yescomplemented
85C23×C3×C4—91no—
86C23×Dic3—92no—
87C22×C4×D3—32no—
88C22×C4×D3—61no—
89C22×C4×D3—94no—
90C2×C3×C42—92no—
91C2×C4×Dic3—32no—
92C2×C4×Dic3—32no—
93C2×C4×Dic3—62no—
94C2×C4×Dic3—98no—
95C2×C4×Dic3—94no—
96C2×C4×Dic3—92no—
97C42×D3—916no—
98C23×C32—21no—
99C23×C32—22no—
100C23×C32—41no—
101C23×C32C22×D312yes—
102C23×C32C22×D311yescharacteristic
103C22×C3⋊D3C4×D318yes—
104C22×C3⋊D3C2×Dic314yes—
105C22×C3⋊D3C22×D312yes—
106C22×C3×D3—316no—
107C22×C3×D3—38no—
108C22×C3×D3—34no—
109C22×C3×D3—68no—
110C22×C3×D3—64no—
111C22×C3×D3—62no—
112C2×C32×C4—28no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4—22no—
116C2×C32×C4—44no—
117C2×C32×C4—42no—
118C2×C32×C4C4×D318yescomplemented
119C2×C32×C4C2×Dic314yescomplemented
120C2×C32×C4C22×D314yes—
121C2×C32×C4C22×D312yes—
122C2×C3⋊Dic3C4×D3116yescomplemented
123C2×C3⋊Dic3C2×Dic314yescomplemented
124C2×C3⋊Dic3C22×D318yes—
125C2×C3⋊Dic3C22×D312yes—
126C4×C3⋊D3C2×Dic3116yescomplemented
127C2×C3×Dic3—316no—
128C2×C3×Dic3—316no—
129C2×C3×Dic3—38no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—38no—
132C2×C3×Dic3—38no—
133C2×C3×Dic3—68no—
134C2×C3×Dic3—64no—
135C2×C3×Dic3—616no—
136C2×C3×Dic3—62no—
137C3×C4×D3—332no—
138C3×C4×D3—616no—
139#624—332no—
140D3×Dic3—364no—
141C22×C42—271noSylow
142C2×C33C2×C4214yescomplemented
143C2×C33C23×C412yes—
144C2×C33C23×C411yescharacteristic
145C3×C3⋊D3C2×C4218yescomplemented
146C24×C3—91no—
147C23×D3—32no—
148C23×D3—61no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—41no—
152C22×C3×C4—91no—
153C22×C3×C4—94no—
154C22×C3×C4—98no—
155C22×C3×C4—94no—
156C22×C3×C4D3212yes—
157C22×C3×C4D3211yescharacteristic
158C22×Dic3—34no—
159C22×Dic3—34no—
160C22×Dic3—32no—
161C22×Dic3—64no—
162C22×Dic3—61no—
163C22×Dic3—98no—
164C22×Dic3—916no—
165C2×C4×D3—316no—
166C2×C4×D3—38no—
167C2×C4×D3—68no—
168C2×C4×D3—64no—
169C2×C4×D3—916no—
170C2×C4×D3—932no—
171C3×C42—98no—
172C4×Dic3—38no—
173C4×Dic3—38no—
174C4×Dic3—68no—
175C4×Dic3—916no—
176C4×Dic3—98no—
177C22×C32—22no—
178C22×C32—21no—
179C22×C32—28no—
180C22×C32—24no—
181C22×C32—24no—
182C22×C32—22no—
183C22×C32—42no—
184C22×C32—44no—
185C22×C32—41no—
186C22×C32C2×C4×D318yes—
187C22×C32C2×C4×D314yes—
188C22×C32C2×C4×D314yes—
189C22×C32C22×Dic312yes—
190C22×C32C23×D312yes—
191C22×C32C23×D311yescharacteristic
192C2×C3⋊D3C4×Dic3116yescomplemented
193C2×C3⋊D3C2×C4×D318yes—
194C2×C3⋊D3C22×Dic314yes—
195C2×C3×D3—332no—
196C2×C3×D3—316no—
197C2×C3×D3—38no—
198C2×C3×D3—616no—
199C2×C3×D3—64no—
200C2×C3×D3—68no—
201C32×C4—28no—
202C32×C4—24no—
203C32×C4—44no—
204C32×C4C2×C4×D318yescomplemented
205C32×C4C22×Dic314yescomplemented
206C3⋊Dic3C2×C4×D3116yescomplemented
207C3⋊Dic3C22×Dic314yescomplemented
208C3×Dic3—316no—
209C3×Dic3—316no—
210C3×Dic3—38no—
211C3×Dic3—64no—
212C3×Dic3—616no—
213C23×C4—91no—
214C23×C4—272no—
215C2×C42—92no—
216C2×C42—278no—
217C2×C42—272no—
218C33C22×C4211yescharacteristic, complemented, Sylow
219C23×C3—22no—
220C23×C3—21no—
221C23×C3—41no—
222C23×C3—92no—
223C23×C3—98no—
224C23×C3—94no—
225C23×C3C2×D3212yes—
226C23×C3C2×D3211yescharacteristic
227C22×D3—34no—
228C22×D3—316no—
229C22×D3—38no—
230C22×D3—68no—
231C22×D3—64no—
232C22×D3—62no—
233C2×C3×C4—28no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—22no—
237C2×C3×C4—44no—
238C2×C3×C4—42no—
239C2×C3×C4—92no—
240C2×C3×C4—98no—
241C2×C3×C4—916no—
242C2×C3×C4—94no—
243C2×C3×C4—916no—
244C2×C3×C4D3×Dic318yescomplemented
245C2×C3×C4#62414yescomplemented
246C2×C3×C4C2×D3214yes—
247C2×C3×C4C2×D3212yes—
248C2×Dic3—38no—
249C2×Dic3—38no—
250C2×Dic3—34no—
251C2×Dic3—316no—
252C2×Dic3—316no—
253C2×Dic3—38no—
254C2×Dic3—616no—
255C2×Dic3—64no—
256C2×Dic3—68no—
257C2×Dic3—62no—
258C2×Dic3—932no—
259C4×D3—332no—
260C4×D3—616no—
261C4×D3—964no—
262C2×C32—24no—
263C2×C32—22no—
264C2×C32—22no—
265C2×C32—21no—
266C2×C32—28no—
267C2×C32—24no—
268C2×C32—44no—
269C2×C32—41no—
270C2×C32—42no—
271C2×C32C42×D318yescomplemented
272C2×C32C2×C4×Dic314yescomplemented
273C2×C32C22×C4×D314yes—
274C2×C32C22×C4×D312yes—
275C2×C32C22×C4×D312yes—
276C2×C32C23×Dic311yescharacteristic
277C3⋊D3C2×C4×Dic318yescomplemented
278C3×D3—316no—
279C3×D3—68no—
280C24—91no—
281C22×C4—94no—
282C22×C4—91no—
283C22×C4—98no—
284C22×C4—94no—
285C22×C4—278no—
286C22×C4—2716no—
287C22×C4#111311yescharacteristic
288C42—98no—
289C42—278no—
290C22×C3—22no—
291C22×C3—21no—
292C22×C3—28no—
293C22×C3—24no—
294C22×C3—22no—
295C22×C3—24no—
296C22×C3—41no—
297C22×C3—44no—
298C22×C3—42no—
299C22×C3—98no—
300C22×C3—94no—
301C22×C3—916no—
302C22×C3C4×D3218yes—
303C22×C3C4×D3214yes—
304C22×C3C2×D3×Dic314yes—
305C22×C3C2×#62412yes—
306C22×C3C22×D3212yes—
307C22×C3C22×D3211yescharacteristic
308C2×D3—316no—
309C2×D3—38no—
310C2×D3—332no—
311C2×D3—64no—
312C2×D3—616no—
313C2×D3—68no—
314C3×C4—24no—
315C3×C4—28no—
316C3×C4—44no—
317C3×C4—916no—
318C3×C4—94no—
319C3×C4C2×D3×Dic318yescomplemented
320C3×C4C2×#62414yescomplemented
321Dic3—316no—
322Dic3—316no—
323Dic3—38no—
324Dic3—64no—
325Dic3—616no—
326C32—22no—
327C32—21no—
328C32—41no—
329C32C2×C42×D312yescomplemented
330C32C22×C4×Dic311yescharacteristic, complemented
331C23—92no—
332C23—98no—
333C23—94no—
334C23C2×#111311yescharacteristic
335C2×C4—98no—
336C2×C4—92no—
337C2×C4—916no—
338C2×C4—94no—
339C2×C4—916no—
340C2×C4—2732no—
341C2×C4#629414yesdirect product factor
342C2×C4C2×#111312yes—
343C2×C3—24no—
344C2×C3—22no—
345C2×C3—21no—
346C2×C3—22no—
347C2×C3—24no—
348C2×C3—28no—
349C2×C3—42no—
350C2×C3—41no—
351C2×C3—44no—
352C2×C3—98no—
353C2×C3C4×D3×Dic318yescomplemented
354C2×C3C4×#62414yescomplemented
355C2×C3C2×C4×D3214yes—
356C2×C3C2×C4×D3212yes—
357C2×C3C22×D3×Dic312yes—
358C2×C3C22×#62411yescharacteristic
359D3—316no—
360D3—68no—
361C22—98no—
362C22—94no—
363C22—916no—
364C22C4×#111314yes—
365C22C2×#629412yes—
366C22C22×#111311yescharacteristic
367C4—916no—
368C4—94no—
369C4C2×#629414yesdirect product factor
370C3—21no—
371C3—22no—
372C3—41no—
373C3C2×C4×D3×Dic312yescomplemented
374C3C2×C4×#62411yescharacteristic, complemented
375C2—98no—
376C2C4×#629414yesdirect product factor
377C2C2×C4×#111312yes—
378C2C22×#629411yescharacteristic
subgroupG/H
center, hypercenterC22×C4#1113
commutator, hypocenterC33C22×C42
FrattiniC22C22×#1113
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C33C24
LCSG ▷ C33
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4×C3⋊Dic3C2semidirect
4. C2×#88813C2semidirect
5. C4×#6294C2direct (N × H)
12. C2×C3×C4×C3⋊D3C4semidirect
14. C2×#6294C4direct (N × H)
16. C3×C4×C3⋊Dic3C22semidirect
17. #88813C22semidirect
23. C2×C4×C3⋊Dic3D3semidirect
35. C2×C33×C4C2×C4semidirect
37. C2×C3×C3⋊Dic3C2×C4semidirect
38. C2×C3×C3⋊Dic3C2×C4semidirect
41. C3×C4×C3⋊D3C2×C4semidirect
42. #6294C2×C4direct (N × H)
55. C2×C4×C3⋊D3Dic3semidirect
70. C4×C3⋊Dic3C2×D3semidirect
79. C2×C3×C3⋊D3C42semidirect
82. C33×C4C22×C4semidirect
83. C3×C3⋊Dic3C22×C4semidirect
84. C3×C3⋊Dic3C22×C4semidirect
118. C2×C32×C4C4×D3semidirect
119. C2×C32×C4C2×Dic3semidirect
122. C2×C3⋊Dic3C4×D3semidirect
123. C2×C3⋊Dic3C2×Dic3semidirect
126. C4×C3⋊D3C2×Dic3semidirect
142. C2×C33C2×C42semidirect
145. C3×C3⋊D3C2×C42semidirect
192. C2×C3⋊D3C4×Dic3semidirect
204. C32×C4C2×C4×D3semidirect
205. C32×C4C22×Dic3semidirect
206. C3⋊Dic3C2×C4×D3semidirect
207. C3⋊Dic3C22×Dic3semidirect
218. C33C22×C42semidirect
244. C2×C3×C4D3×Dic3semidirect
245. C2×C3×C4#624semidirect
271. C2×C32C42×D3semidirect
272. C2×C32C2×C4×Dic3semidirect
277. C3⋊D3C2×C4×Dic3semidirect
319. C3×C4C2×D3×Dic3semidirect
320. C3×C4C2×#624semidirect
329. C32C2×C42×D3semidirect
330. C32C22×C4×Dic3semidirect
341. C2×C4#6294direct (N × H)
353. C2×C3C4×D3×Dic3semidirect
354. C2×C3C4×#624semidirect
369. C4C2×#6294direct (N × H)
373. C3C2×C4×D3×Dic3semidirect
374. C3C2×C4×#624semidirect
376. C2C4×#6294direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C22×#62942
C2×C3×C4×C3⋊Dic32
C2×#888132
C4×#62942
C2×C4×#6243
C2×C4×D3×Dic33