Finite Group Databasebeta

← C3×C4×D3×C3⋊C8 #13157331

Subgroups — C3×C4×D3×C3⋊C8 #13157331

Order 1728.

874 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C42×D3C211yesmaximal, characteristic
2C2×C3×D3×C3⋊C8C212yesmaximal
3C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C32×C4×C3⋊C8C211yesmaximal, characteristic, complemented
5C3×Dic3×C3⋊C8C212yesmaximal, complemented
6C2×C3×C4×C3⋊C8—31nomaximal
7C3×C4×D3×C8—31nomaximal
8C4×D3×C3⋊C8C311yesmaximal, characteristic, direct product factor
9C2×C32×C4×D3C412yes—
10C2×C32×C4×D3C2211yescharacteristic
11C33×C42C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4×Dic3C412yes—
14C32×C4×Dic3C2211yescharacteristic
15C2×C32×C3⋊C8C2212yes—
16C3×D3×C3⋊C8C418yesdirect product factor
17C2×C32×C42—31no—
18C22×C3×C3⋊C8—32no—
19C2×C3×D3×C8—32no—
20C3×C42×D3—21no—
21C3×C42×D3D311yescharacteristic
22C3×C42×D3C2×C311yescharacteristic
23C2×D3×C3⋊C8C2×C312yes—
24C32×C4×C8—31no—
25C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
26C3×C8×Dic3—32no—
27C3×C4×C3⋊C8—21no—
28C3×C4×C3⋊C8—31no—
29C3×C4×C3⋊C8—31no—
30C3×C4×C3⋊C8—32no—
31C3×C4×C3⋊C8—61no—
32C3×C4×C3⋊C8D311yescharacteristic, direct product factor
33C3×C4×C3⋊C8C2×C311yescharacteristic, complemented
34Dic3×C3⋊C8C2×C312yescomplemented
35C22×C32×D3C2×C411yescharacteristic
36C2×C33×C4C2×C412yes—
37C2×C33×C4C2311yescharacteristic
38C2×C32×Dic3C2×C411yescharacteristic
39C2×C32×Dic3C2×C412yes—
40C32×C4×D3C818yescomplemented
41C32×C4×D3C2×C414yes—
42C3×C3⋊(C3⋊C8)C2×C414yescomplemented
43C32×C3⋊C8C2×C414yescomplemented
44C2×C3×C4×C8—91no—
45C2×C4×C3⋊C8—31no—
46C4×D3×C8—31no—
47C22×C32×C4—32no—
48C22×C32×C4—31no—
49C2×C3×C4×D3—22no—
50C2×C3×C4×D3—21no—
51C2×C3×C4×D3Dic312yes—
52C2×C3×C4×D3C3×C412yes—
53C2×C3×C4×D3C2×D311yescharacteristic
54C2×C3×C4×D3C22×C311yescharacteristic
55C2×C32×C8—32no—
56C32×C42—21no—
57C32×C42—21no—
58C32×C42—21no—
59C32×C42—32no—
60C32×C42—31no—
61C32×C42—41no—
62C32×C42C2×D311yescharacteristic
63C32×C42C2×D311yescharacteristic
64C32×C42C22×C311yescharacteristic
65C2×C3⋊(C3⋊C8)C22×C312yes—
66C3×C4×Dic3—22no—
67C3×C4×Dic3—21no—
68C3×C4×Dic3Dic312yes—
69C3×C4×Dic3C3×C412yes—
70C3×C4×Dic3C2×D311yescharacteristic
71C3×C4×Dic3C22×C311yescharacteristic
72C2×C3×C3⋊C8—22no—
73C2×C3×C3⋊C8—32no—
74C2×C3×C3⋊C8—32no—
75C2×C3×C3⋊C8—38no—
76C2×C3×C3⋊C8—62no—
77C2×C3×C3⋊C8C2×D312yes—
78C2×C3×C3⋊C8C22×C312yes—
79C3×D3×C8—38no—
80D3×C3⋊C8C3×C418yesdirect product factor
81C22×C33C22×C411yescharacteristic
82C2×C32×D3C4212yes—
83C2×C32×D3C2×C814yes—
84C33×C4C2×C814yescomplemented
85C33×C4C22×C412yes—
86C32×Dic3C4212yes—
87C32×Dic3C2×C814yescomplemented
88C22×C3×C8—92no—
89C2×C3×C42—31no—
90C2×C3×C42—31no—
91C2×C3×C42—61no—
92C22×C3⋊C8—32no—
93C2×D3×C8—32no—
94C42×D3C3×D311yescharacteristic
95C3×C4×C8—31no—
96C3×C4×C8—31no—
97C3×C4×C8—61no—
98C3×C4×C8—92no—
99C3×C4×C8—91no—
100C8×Dic3—32no—
101C4×C3⋊C8—31no—
102C4×C3⋊C8—31no—
103C4×C3⋊C8—32no—
104C4×C3⋊C8—61no—
105C4×C3⋊C8C3×D311yescharacteristic, direct product factor
106C23×C32—31no—
107C22×C3×D3—21no—
108C22×C3×D3C2×Dic311yescharacteristic
109C22×C3×D3C2×C3×C411yescharacteristic
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—31no—
117C2×C32×C4—32no—
118C2×C32×C4—38no—
119C2×C32×C4—34no—
120C2×C32×C4—42no—
121C2×C32×C4—41no—
122C2×C32×C4C4×D312yes—
123C2×C32×C4C2×Dic312yes—
124C2×C32×C4C2×C3×C412yes—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C23×C311yescharacteristic
128C2×C3×Dic3—22no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3C2×Dic312yes—
131C2×C3×Dic3C2×Dic311yescharacteristic
132C2×C3×Dic3C2×C3×C411yescharacteristic
133C2×C3×Dic3C2×C3×C412yes—
134C3×C4×D3—28no—
135C3×C4×D3—24no—
136C3×C4×D3C3⋊C818yesdirect product factor
137C3×C4×D3C3×C818yescomplemented
138C3×C4×D3C4×D314yes—
139C3×C4×D3C2×C3×C414yes—
140C32×C8—34no—
141C3⋊(C3⋊C8)C2×C3×C414yescomplemented
142C3×C3⋊C8—24no—
143C3×C3⋊C8—34no—
144C3×C3⋊C8—34no—
145C3×C3⋊C8—64no—
146C3×C3⋊C8C4×D314yesdirect product factor
147C3×C3⋊C8C2×C3×C414yescomplemented
148C2×C4×C8—91noSylow
149C2×C33C2×C4211yescharacteristic
150C2×C33C22×C812yes—
151C32×D3C4×C814yescomplemented
152C22×C3×C4—32no—
153C22×C3×C4—32no—
154C22×C3×C4—31no—
155C22×C3×C4—31no—
156C22×C3×C4—62no—
157C22×C3×C4—61no—
158C2×C4×D3C3×Dic312yes—
159C2×C4×D3C2×C3×D311yescharacteristic
160C2×C3×C8—32no—
161C2×C3×C8—32no—
162C2×C3×C8—62no—
163C2×C3×C8—98no—
164C2×C3×C8—92no—
165C3×C42—21no—
166C3×C42—21no—
167C3×C42—21no—
168C3×C42—32no—
169C3×C42—32no—
170C3×C42—31no—
171C3×C42—31no—
172C3×C42—41no—
173C3×C42—62no—
174C3×C42—61no—
175C3×C42D3211yescharacteristic
176C3×C42C2×C3×D311yescharacteristic
177C3×C42C2×C3×D311yescharacteristic
178C4×Dic3C3×Dic312yes—
179C4×Dic3C2×C3×D311yescharacteristic
180C2×C3⋊C8—32no—
181C2×C3⋊C8—32no—
182C2×C3⋊C8—38no—
183C2×C3⋊C8—62no—
184C2×C3⋊C8C2×C3×D312yes—
185D3×C8—38no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—32no—
190C22×C32—34no—
191C22×C32—41no—
192C22×C32C2×C4×D311yescharacteristic
193C22×C32C22×Dic311yescharacteristic
194C22×C32C22×C3×C411yescharacteristic
195C2×C3×D3—24no—
196C2×C3×D3—22no—
197C2×C3×D3C2×C3⋊C814yes—
198C2×C3×D3C4×Dic312yes—
199C2×C3×D3C3×C4212yes—
200C2×C3×D3C2×C3×C814yes—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—24no—
204C32×C4—22no—
205C32×C4—22no—
206C32×C4—22no—
207C32×C4—32no—
208C32×C4—34no—
209C32×C4—44no—
210C32×C4—42no—
211C32×C4D3×C814yescomplemented
212C32×C4C2×C3⋊C814yescomplemented
213C32×C4C2×C3×C814yescomplemented
214C32×C4C2×C4×D312yes—
215C32×C4C2×C4×D312yes—
216C32×C4C22×C3×C412yes—
217C3×Dic3—24no—
218C3×Dic3—22no—
219C3×Dic3C2×C3⋊C814yescomplemented
220C3×Dic3C4×Dic312yes—
221C3×Dic3C3×C4212yes—
222C3×Dic3C2×C3×C814yescomplemented
223C22×C8—92no—
224C2×C42—31no—
225C4×C8—31no—
226C4×C8—92no—
227C4×C8—91no—
228C33C2×C4×C811yescharacteristic, complemented, Sylow
229C23×C3—31no—
230C23×C3—31no—
231C23×C3—61no—
232C22×D3C2×C3×Dic311yescharacteristic
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—32no—
240C2×C3×C4—31no—
241C2×C3×C4—32no—
242C2×C3×C4—31no—
243C2×C3×C4—38no—
244C2×C3×C4—38no—
245C2×C3×C4—34no—
246C2×C3×C4—34no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—64no—
250C2×C3×C4—68no—
251C2×C3×C4—62no—
252C2×C3×C4—61no—
253C2×C3×C4D3×Dic312yes—
254C2×C3×C4C3×C4×D312yes—
255C2×C3×C4C2×C3×Dic312yes—
256C2×C3×C4C2×D3211yescharacteristic
257C2×C3×C4C22×C3×D311yescharacteristic
258C2×C3×C4C22×C3×D311yescharacteristic
259C2×Dic3C2×C3×Dic312yes—
260C2×Dic3C2×C3×Dic311yescharacteristic
261C4×D3C3×C3⋊C818yesdirect product factor
262C4×D3C3×C4×D314yes—
263C3×C8—34no—
264C3×C8—34no—
265C3×C8—64no—
266C3×C8—94no—
267C3⋊C8—34no—
268C3⋊C8—34no—
269C3⋊C8—64no—
270C3⋊C8C3×C4×D314yesdirect product factor
271C2×C32—22no—
272C2×C32—22no—
273C2×C32—21no—
274C2×C32—22no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—34no—
278C2×C32—41no—
279C2×C32—42no—
280C2×C32C42×D311yescharacteristic
281C2×C32C2×D3×C812yes—
282C2×C32C22×C3⋊C812yes—
283C2×C32C2×C4×Dic311yescharacteristic
284C2×C32C2×C3×C4211yescharacteristic
285C2×C32C22×C3×C812yes—
286C3×D3—24no—
287C3×D3C4×C3⋊C814yesdirect product factor
288C3×D3C3×C4×C814yescomplemented
289C22×C4—32no—
290C22×C4—31no—
291C2×C8—32no—
292C2×C8—98no—
293C2×C8—92no—
294C42—32no—
295C42—31no—
296C42C3×D3211yescharacteristic
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—34no—
301C22×C3—34no—
302C22×C3—32no—
303C22×C3—32no—
304C22×C3—41no—
305C22×C3—64no—
306C22×C3—62no—
307C22×C3C2×D3×Dic311yescharacteristic
308C22×C3C2×C3×C4×D311yescharacteristic
309C22×C3C22×C3×Dic311yescharacteristic
310C2×D3C2×C3×C3⋊C814yes—
311C2×D3C3×C4×Dic312yes—
312C3×C4—22no—
313C3×C4—22no—
314C3×C4—22no—
315C3×C4—24no—
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—34no—
319C3×C4—34no—
320C3×C4—32no—
321C3×C4—32no—
322C3×C4—42no—
323C3×C4—44no—
324C3×C4—64no—
325C3×C4—62no—
326C3×C4D3×C3⋊C814yesdirect product factor
327C3×C4C3×D3×C814yescomplemented
328C3×C4C2×C3×C3⋊C814yescomplemented
329C3×C4C4×D3212yes—
330C3×C4C2×C3×C4×D312yes—
331C3×C4C2×C3×C4×D312yes—
332Dic3C2×C3×C3⋊C814yescomplemented
333Dic3C3×C4×Dic312yes—
334C32—21no—
335C32—21no—
336C32—21no—
337C32—41no—
338C32C4×D3×C811yescharacteristic, complemented
339C32C2×C4×C3⋊C811yescharacteristic, complemented
340C32C2×C3×C4×C811yescharacteristic, complemented
341C23—31no—
342C2×C4—32no—
343C2×C4—31no—
344C2×C4—34no—
345C2×C4—38no—
346C2×C4C3×D3×Dic312yes—
347C2×C4C2×C3×D3211yescharacteristic
348C8—34no—
349C8—94no—
350C2×C3—22no—
351C2×C3—22no—
352C2×C3—22no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—34no—
357C2×C3—34no—
358C2×C3—42no—
359C2×C3—41no—
360C2×C3—64no—
361C2×C3C2×D3×C3⋊C812yes—
362C2×C3C4×D3×Dic311yescharacteristic
363C2×C3C3×C42×D311yescharacteristic
364C2×C3C2×C3×D3×C812yes—
365C2×C3C22×C3×C3⋊C812yes—
366C2×C3C2×C3×C4×Dic311yescharacteristic
367D3C3×C4×C3⋊C814yesdirect product factor
368C22—34no—
369C22—32no—
370C22C2×C3×D3×Dic311yescharacteristic
371C4—34no—
372C4—32no—
373C4C3×D3×C3⋊C814yesdirect product factor
374C4C3×C4×D3212yes—
375C3—21no—
376C3—21no—
377C3—21no—
378C3—41no—
379C3C4×D3×C3⋊C811yescharacteristic, direct product factor
380C3C3×C4×D3×C811yescharacteristic, complemented
381C3C2×C3×C4×C3⋊C811yescharacteristic, complemented
382C2—34no—
383C2C2×C3×D3×C3⋊C812yes—
384C2C3×C4×D3×Dic311yescharacteristic
subgroupG/H
center, hypercenterC3×C42D32
commutator, hypocenterC32C2×C3×C4×C8
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C32×D3C2×C4
cosocleC2×C32×C4C23×C3
LCSG ▷ C32
UCSC1 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
3. C3×C4×C3⋊(C3⋊C8)C2semidirect
4. C32×C4×C3⋊C8C2semidirect
5. C3×Dic3×C3⋊C8C2semidirect
8. C4×D3×C3⋊C8C3direct (N × H)
16. C3×D3×C3⋊C8C4direct (N × H)
25. C4×C3⋊(C3⋊C8)C2×C3semidirect
32. C3×C4×C3⋊C8D3direct (N × H)
33. C3×C4×C3⋊C8C2×C3semidirect
34. Dic3×C3⋊C8C2×C3semidirect
40. C32×C4×D3C8semidirect
42. C3×C3⋊(C3⋊C8)C2×C4semidirect
43. C32×C3⋊C8C2×C4semidirect
80. D3×C3⋊C8C3×C4direct (N × H)
84. C33×C4C2×C8semidirect
87. C32×Dic3C2×C8semidirect
105. C4×C3⋊C8C3×D3direct (N × H)
136. C3×C4×D3C3⋊C8direct (N × H)
137. C3×C4×D3C3×C8semidirect
141. C3⋊(C3⋊C8)C2×C3×C4semidirect
146. C3×C3⋊C8C4×D3direct (N × H)
147. C3×C3⋊C8C2×C3×C4semidirect
151. C32×D3C4×C8semidirect
211. C32×C4D3×C8semidirect
212. C32×C4C2×C3⋊C8semidirect
213. C32×C4C2×C3×C8semidirect
219. C3×Dic3C2×C3⋊C8semidirect
222. C3×Dic3C2×C3×C8semidirect
228. C33C2×C4×C8semidirect
261. C4×D3C3×C3⋊C8direct (N × H)
270. C3⋊C8C3×C4×D3direct (N × H)
287. C3×D3C4×C3⋊C8direct (N × H)
288. C3×D3C3×C4×C8semidirect
326. C3×C4D3×C3⋊C8direct (N × H)
327. C3×C4C3×D3×C8semidirect
328. C3×C4C2×C3×C3⋊C8semidirect
332. Dic3C2×C3×C3⋊C8semidirect
338. C32C4×D3×C8semidirect
339. C32C2×C4×C3⋊C8semidirect
340. C32C2×C3×C4×C8semidirect
367. D3C3×C4×C3⋊C8direct (N × H)
373. C4C3×D3×C3⋊C8direct (N × H)
379. C3C4×D3×C3⋊C8direct (N × H)
380. C3C3×C4×D3×C8semidirect
381. C3C2×C3×C4×C3⋊C8semidirect

Maximal subgroups

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

subgroupindex
C32×C42×D32
C2×C3×D3×C3⋊C82
C3×C4×C3⋊(C3⋊C8)2
C32×C4×C3⋊C82
C3×Dic3×C3⋊C82
C2×C3×C4×C3⋊C83
C3×C4×D3×C83
C4×D3×C3⋊C83