Finite Group Databasebeta

← C3×D3×#803 #13157332

Subgroups — C3×D3×#803 #13157332

Order 1728.

802 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×#63630C211yesmaximal, characteristic, complemented
4C32×#803C211yesmaximal, characteristic, complemented
5C3×#63554C212yesmaximal, complemented
6C2×C3×#803—31nomaximal
7C3×D3×#97—31nomaximal
8D3×#803C311yesmaximal, 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⋊C8C418yescomplemented
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×#97—31no—
25#63630C2×C311yescharacteristic, complemented
26C3×#815—32no—
27C3×#803—21no—
28C3×#803—31no—
29C3×#803—31no—
30C3×#803—32no—
31C3×#803—61no—
32C3×#803D311yescharacteristic, direct product factor
33C3×#803C2×C311yescharacteristic, complemented
34#63554C2×C312yescomplemented
35C22×C32×D3C2×C411yescharacteristic
36C2×C33×C4C2×C412yes—
37C2×C33×C4C2311yescharacteristic
38C2×C32×Dic3C2×C412yes—
39C2×C32×Dic3C2×C411yescharacteristic
40C32×C4×D3—24no—
41C32×C4×D3C2×C414yes—
42C3×C3⋊(C3⋊C8)C2×C414yescomplemented
43C32×C3⋊C8C2×C414yescomplemented
44C2×C3×#97—91no—
45C2×#803—31no—
46D3×#97—31no—
47C22×C32×C4—31no—
48C22×C32×C4—32no—
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×C418yescomplemented
81C22×C33C22×C411yescharacteristic
82C2×C32×D3C4212yes—
83C2×C32×D3M1614yes—
84C33×C4—22no—
85C33×C4C22×C412yes—
86C32×Dic3—22no—
87C32×Dic3C4212yes—
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×#97—31no—
96C3×#97—31no—
97C3×#97—61no—
98C3×#97—92no—
99C3×#97—91no—
100#815—32no—
101#803—31no—
102#803—31no—
103#803—32no—
104#803—61no—
105#803C3×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—34no—
118C2×C32×C4—32no—
119C2×C32×C4—42no—
120C2×C32×C4—41no—
121C2×C32×C4—64no—
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×C412yes—
133C2×C3×Dic3C2×C3×C411yescharacteristic
134C3×C4×D3—28no—
135C3×C4×D3—24no—
136C3×C4×D3—24no—
137C3×C4×D3—24no—
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×D314yescomplemented
147C3×C3⋊C8C2×C3×C414yescomplemented
148C2×#97—91noSylow
149C2×C33C2×C4211yescharacteristic
150C2×C33C2×M1612yes—
151C32×D3#9714yescomplemented
152C22×C3×C4—31no—
153C22×C3×C4—31no—
154C22×C3×C4—32no—
155C22×C3×C4—32no—
156C22×C3×C4—61no—
157C22×C3×C4—62no—
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×D3#20814yes—
198C2×C3×D3C4×Dic312yes—
199C2×C3×D3C3×C4212yes—
200C2×C3×D3C3×M1614yes—
201C32×C4—22no—
202C32×C4—24no—
203C32×C4—22no—
204C32×C4—22no—
205C32×C4—22no—
206C32×C4—22no—
207C32×C4—22no—
208C32×C4—32no—
209C32×C4—42no—
210C32×C4—42no—
211C32×C4—42no—
212C32×C4—44no—
213C32×C4—62no—
214C32×C4C2×C4×D312yes—
215C32×C4C2×C4×D312yes—
216C32×C4C22×C3×C412yes—
217C3×Dic3—24no—
218C3×Dic3—22no—
219C3×Dic3—22no—
220C3×Dic3—22no—
221C3×Dic3C4×Dic312yes—
222C3×Dic3C3×C4212yes—
223C22×C8—92no—
224C2×C42—31no—
225#97—31no—
226#97—92no—
227#97—91no—
228C33C2×#9711yescharacteristic, 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—31no—
240C2×C3×C4—31no—
241C2×C3×C4—34no—
242C2×C3×C4—34no—
243C2×C3×C4—32no—
244C2×C3×C4—32no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—64no—
248C2×C3×C4—61no—
249C2×C3×C4—62no—
250C2×C3×C4—64no—
251C2×C3×C4—64no—
252C2×C3×C4—68no—
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×D3—24no—
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×D314yescomplemented
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×C32D3×M1612yes—
282C2×C32C2×#20812yes—
283C2×C32C2×C4×Dic311yescharacteristic
284C2×C32C2×C3×C4211yescharacteristic
285C2×C32C2×C3×M1612yes—
286C3×D3—24no—
287C3×D3#80314yesdirect product factor
288C3×D3C3×#9714yescomplemented
289C22×C4—31no—
290C22×C4—32no—
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×D3C3×#20814yes—
311C2×D3C3×C4×Dic312yes—
312C3×C4—24no—
313C3×C4—22no—
314C3×C4—22no—
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—22no—
319C3×C4—32no—
320C3×C4—32no—
321C3×C4—42no—
322C3×C4—42no—
323C3×C4—44no—
324C3×C4—42no—
325C3×C4—62no—
326C3×C4—64no—
327C3×C4—62no—
328C3×C4—62no—
329C3×C4C4×D3212yes—
330C3×C4C2×C3×C4×D312yes—
331C3×C4C2×C3×C4×D312yes—
332Dic3—22no—
333Dic3C3×C4×Dic312yes—
334C32—21no—
335C32—21no—
336C32—21no—
337C32—41no—
338C32D3×#9711yescharacteristic, complemented
339C32C2×#80311yescharacteristic, complemented
340C32C2×C3×#9711yescharacteristic, complemented
341C23—31no—
342C2×C4—31no—
343C2×C4—34no—
344C2×C4—32no—
345C2×C4—64no—
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×C3D3×#20812yes—
362C2×C3C4×D3×Dic311yescharacteristic
363C2×C3C3×C42×D311yescharacteristic
364C2×C3C3×D3×M1612yes—
365C2×C3C2×C3×#20812yes—
366C2×C3C2×C3×C4×Dic311yescharacteristic
367D3C3×#80314yesdirect product factor
368C22—34no—
369C22—32no—
370C22C2×C3×D3×Dic311yescharacteristic
371C4—22no—
372C4—32no—
373C4—62no—
374C4C3×C4×D3212yes—
375C3—21no—
376C3—21no—
377C3—21no—
378C3—41no—
379C3D3×#80311yescharacteristic, direct product factor
380C3C3×D3×#9711yescharacteristic, complemented
381C3C2×C3×#80311yescharacteristic, complemented
382C2—34no—
383C2C3×D3×#20812yes—
384C2C3×C4×D3×Dic311yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC2×C32C2×C3×C42
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C32×D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C2×C3×#97
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
3. C3×#63630C2semidirect
4. C32×#803C2semidirect
5. C3×#63554C2semidirect
8. D3×#803C3direct (N × H)
16. C3×D3×C3⋊C8C4semidirect
25. #63630C2×C3semidirect
32. C3×#803D3direct (N × H)
33. C3×#803C2×C3semidirect
34. #63554C2×C3semidirect
42. C3×C3⋊(C3⋊C8)C2×C4semidirect
43. C32×C3⋊C8C2×C4semidirect
80. D3×C3⋊C8C3×C4semidirect
105. #803C3×D3direct (N × H)
141. C3⋊(C3⋊C8)C2×C3×C4semidirect
146. C3×C3⋊C8C4×D3semidirect
147. C3×C3⋊C8C2×C3×C4semidirect
151. C32×D3#97semidirect
228. C33C2×#97semidirect
270. C3⋊C8C3×C4×D3semidirect
287. C3×D3#803direct (N × H)
288. C3×D3C3×#97semidirect
338. C32D3×#97semidirect
339. C32C2×#803semidirect
340. C32C2×C3×#97semidirect
367. D3C3×#803direct (N × H)
379. C3D3×#803direct (N × H)
380. C3C3×D3×#97semidirect
381. C3C2×C3×#803semidirect

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×#636302
C32×#8032
C3×#635542
C2×C3×#8033
C3×D3×#973
D3×#8033