Finite Group Databasebeta

← C2×C3×#63554 #13158091

Subgroups — C2×C3×#63554 #13158091

Order 1728.

898 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C2×C32×C4×Dic3C211yesmaximal, characteristic
3C22×C32×C3⋊C8C211yesmaximal, characteristic
4C3×#63554C214yesmaximal, direct product factor
5C2×C3×#815—31nomaximal
6C2×C3×#803—31nomaximal
7C2×#63554C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C32×Dic3C412yes—
10C2×C3×C3⋊(C3⋊C8)C414yescomplemented
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C32×C4×Dic3C412yes—
13C32×C4×Dic3C2212yes—
14C2×C32×C3⋊C8C414yescomplemented
15C2×C32×C3⋊C8C2212yes—
16C22×C32×C8—31no—
17C2×C32×C42—31no—
18C22×C3⋊(C3⋊C8)C2×C311yescharacteristic
19C2×C3×C4×Dic3—21no—
20C2×C3×C4×Dic3D311yescharacteristic
21C2×C3×C4×Dic3C2×C311yescharacteristic
22C22×C3×C3⋊C8—21no—
23C22×C3×C3⋊C8—31no—
24C22×C3×C3⋊C8—31no—
25C22×C3×C3⋊C8—61no—
26C22×C3×C3⋊C8D311yescharacteristic
27C22×C3×C3⋊C8C2×C311yescharacteristic
28C3×#815—34no—
29C3×#803—34no—
30#63554C2×C314yesdirect product factor
31C23×C33C2×C411yescharacteristic
32C2×C33×C4C2×C411yescharacteristic
33C2×C33×C4C2×C412yes—
34C2×C33×C4C2×C412yes—
35C2×C33×C4C2311yescharacteristic
36C2×C32×Dic3—24no—
37C2×C32×Dic3C2×C414yes—
38C3×C3⋊(C3⋊C8)C2×C414yescomplemented
39C32×C3⋊C8C2×C414yescomplemented
40C2×C3×#97—91no—
41C2×#815—31no—
42C2×#803—31no—
43C22×C32×C4—21no—
44C22×C32×C4—21no—
45C22×C32×C4—21no—
46C22×C32×C4—32no—
47C22×C32×C4—41no—
48C22×C32×C4C2×D311yescharacteristic
49C22×C32×C4C2×D311yescharacteristic
50C22×C32×C4C22×C311yescharacteristic
51C22×C3×Dic3—22no—
52C22×C3×Dic3Dic312yes—
53C22×C3×Dic3C3×C412yes—
54C2×C32×C8—34no—
55C2×C32×C8—32no—
56C32×C42—32no—
57C32×C42—32no—
58C2×C3⋊(C3⋊C8)C3×C414yescomplemented
59C2×C3⋊(C3⋊C8)C22×C312yes—
60C3×C4×Dic3—22no—
61C3×C4×Dic3—22no—
62C3×C4×Dic3Dic312yes—
63C3×C4×Dic3C3×C412yes—
64C3×C4×Dic3C2×D312yes—
65C3×C4×Dic3C22×C312yes—
66C2×C3×C3⋊C8—24no—
67C2×C3×C3⋊C8—22no—
68C2×C3×C3⋊C8—34no—
69C2×C3×C3⋊C8—34no—
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—32no—
72C2×C3×C3⋊C8—64no—
73C2×C3×C3⋊C8—62no—
74C2×C3×C3⋊C8Dic314yescomplemented
75C2×C3×C3⋊C8C3×C414yescomplemented
76C2×C3×C3⋊C8C2×D312yes—
77C2×C3×C3⋊C8C22×C312yes—
78C22×C33C4212yes—
79C22×C33M1614yes—
80C22×C33C22×C411yescharacteristic
81C33×C4C4212yes—
82C33×C4C22×C411yescharacteristic
83C33×C4C22×C411yescharacteristic
84C32×Dic3—24no—
85C22×C3×C8—31no—
86C22×C3×C8—31no—
87C22×C3×C8—61no—
88C22×C3×C8—91no—
89C2×C3×C42—31no—
90C2×C3×C42—31no—
91C2×C3×C42—61no—
92C2×C4×Dic3C3×D311yescharacteristic
93C22×C3⋊C8—31no—
94C22×C3⋊C8—31no—
95C22×C3⋊C8—61no—
96C22×C3⋊C8C3×D311yescharacteristic
97C3×#97—94no—
98#815—34no—
99#803—34no—
100C23×C32—21no—
101C23×C32—21no—
102C23×C32—21no—
103C23×C32—41no—
104C23×C32C4×D311yescharacteristic
105C23×C32C2×Dic311yescharacteristic
106C23×C32C2×C3×C411yescharacteristic
107C2×C32×C4—21no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—21no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—34no—
120C2×C32×C4—41no—
121C2×C32×C4—42no—
122C2×C32×C4—42no—
123C2×C32×C4—41no—
124C2×C32×C4—64no—
125C2×C32×C4C4×D311yescharacteristic
126C2×C32×C4C4×D312yes—
127C2×C32×C4C4×D312yes—
128C2×C32×C4C4×D312yes—
129C2×C32×C4C2×Dic311yescharacteristic
130C2×C32×C4C2×Dic312yes—
131C2×C32×C4C2×C3×C412yes—
132C2×C32×C4C2×C3×C411yescharacteristic
133C2×C32×C4C2×C3×C412yes—
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C23×C311yescharacteristic
137C2×C3×Dic3—28no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3C2×Dic314yes—
142C2×C3×Dic3C2×C3×C414yes—
143C32×C8—34no—
144C3⋊(C3⋊C8)C2×C3×C414yescomplemented
145C3×C3⋊C8—24no—
146C3×C3⋊C8—34no—
147C3×C3⋊C8—34no—
148C3×C3⋊C8—64no—
149C3×C3⋊C8C2×Dic314yescomplemented
150C3×C3⋊C8C2×C3×C414yescomplemented
151C2×#97—91noSylow
152C2×C33#9714yescomplemented
153C2×C33C2×C4211yescharacteristic
154C2×C33C2×M1612yes—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—32no—
159C22×C3×C4—32no—
160C22×C3×C4—41no—
161C22×C3×C4—62no—
162C22×C3×C4D3211yescharacteristic
163C22×C3×C4C2×C3×D311yescharacteristic
164C22×C3×C4C2×C3×D311yescharacteristic
165C22×Dic3C3×Dic312yes—
166C2×C3×C8—34no—
167C2×C3×C8—34no—
168C2×C3×C8—32no—
169C2×C3×C8—32no—
170C2×C3×C8—64no—
171C2×C3×C8—62no—
172C2×C3×C8—94no—
173C2×C3×C8—92no—
174C3×C42—32no—
175C3×C42—32no—
176C3×C42—32no—
177C3×C42—32no—
178C3×C42—62no—
179C3×C42—62no—
180C4×Dic3C3×Dic312yes—
181C4×Dic3C2×C3×D312yes—
182C2×C3⋊C8—34no—
183C2×C3⋊C8—34no—
184C2×C3⋊C8—32no—
185C2×C3⋊C8—32no—
186C2×C3⋊C8—64no—
187C2×C3⋊C8—62no—
188C2×C3⋊C8C3×Dic314yescomplemented
189C2×C3⋊C8C2×C3×D312yes—
190C22×C32—24no—
191C22×C32—22no—
192C22×C32—24no—
193C22×C32—24no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—42no—
200C22×C32—44no—
201C22×C32—41no—
202C22×C32ΓL(1,𝔽25)14yes—
203C22×C32#20814yes—
204C22×C32C4×Dic312yes—
205C22×C32C4×Dic312yes—
206C22×C32C3×C4212yes—
207C22×C32C3×M1614yes—
208C22×C32C2×C4×D311yescharacteristic
209C22×C32C22×Dic311yescharacteristic
210C22×C32C22×C3×C411yescharacteristic
211C32×C4—22no—
212C32×C4—22no—
213C32×C4—22no—
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—42no—
221C32×C4—41no—
222C32×C4—41no—
223C32×C4—64no—
224C32×C4C4×Dic312yes—
225C32×C4C4×Dic312yes—
226C32×C4C3×C4212yes—
227C32×C4C2×C4×D311yescharacteristic
228C32×C4C2×C4×D311yescharacteristic
229C32×C4C2×C4×D311yescharacteristic
230C32×C4C22×Dic311yescharacteristic
231C32×C4C22×C3×C411yescharacteristic
232C32×C4C22×C3×C411yescharacteristic
233C3×Dic3—28no—
234C3×Dic3—24no—
235C3×Dic3—24no—
236C22×C8—31no—
237C22×C8—91no—
238C2×C42—31no—
239#97—94no—
240C33C2×#9711yescharacteristic, complemented, Sylow
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—41no—
245C23×C3D3×Dic311yescharacteristic
246C23×C3C3×C4×D311yescharacteristic
247C23×C3C2×C3×Dic311yescharacteristic
248C2×C3×C4—21no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—22no—
253C2×C3×C4—21no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—34no—
261C2×C3×C4—34no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—64no—
267C2×C3×C4—68no—
268C2×C3×C4—64no—
269C2×C3×C4—64no—
270C2×C3×C4D3×Dic311yescharacteristic
271C2×C3×C4D3×Dic312yes—
272C2×C3×C4#62412yes—
273C2×C3×C4C3×C4×D311yescharacteristic
274C2×C3×C4C3×C4×D312yes—
275C2×C3×C4C3×C4×D312yes—
276C2×C3×C4C3×C4×D312yes—
277C2×C3×C4C2×C3×Dic311yescharacteristic
278C2×C3×C4C2×C3×Dic312yes—
279C2×C3×C4C2×D3211yescharacteristic
280C2×C3×C4C22×C3×D311yescharacteristic
281C2×C3×C4C22×C3×D311yescharacteristic
282C2×Dic3—24no—
283C2×Dic3C2×C3×Dic314yes—
284C3×C8—34no—
285C3×C8—34no—
286C3×C8—64no—
287C3×C8—94no—
288C3⋊C8—34no—
289C3⋊C8—34no—
290C3⋊C8—64no—
291C3⋊C8C2×C3×Dic314yescomplemented
292C2×C32—22no—
293C2×C32—22no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—24no—
297C2×C32—22no—
298C2×C32—24no—
299C2×C32—24no—
300C2×C32—21no—
301C2×C32—44no—
302C2×C32—41no—
303C2×C32—42no—
304C2×C32#80314yescomplemented
305C2×C32#81514yescomplemented
306C2×C32C3×#9714yescomplemented
307C2×C32C2×ΓL(1,𝔽25)12yes—
308C2×C32C2×#20812yes—
309C2×C32C2×C4×Dic311yescharacteristic
310C2×C32C2×C4×Dic311yescharacteristic
311C2×C32C2×C3×C4211yescharacteristic
312C2×C32C2×C3×M1612yes—
313C22×C4—32no—
314C22×C4C3×D3211yescharacteristic
315C2×C8—34no—
316C2×C8—32no—
317C2×C8—94no—
318C2×C8—92no—
319C42—32no—
320C42—32no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—24no—
325C22×C3—24no—
326C22×C3—24no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—41no—
331C22×C3—44no—
332C22×C3—42no—
333C22×C3#371114yes—
334C22×C3Dic3212yes—
335C22×C3C3×ΓL(1,𝔽25)14yes—
336C22×C3C3×#20814yes—
337C22×C3C3×C4×Dic312yes—
338C22×C3C3×C4×Dic312yes—
339C22×C3C2×D3×Dic311yescharacteristic
340C22×C3C2×C3×C4×D311yescharacteristic
341C22×C3C22×C3×Dic311yescharacteristic
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—41no—
352C3×C4—41no—
353C3×C4—42no—
354C3×C4—68no—
355C3×C4—64no—
356C3×C4—64no—
357C3×C4Dic3212yes—
358C3×C4C3×C4×Dic312yes—
359C3×C4C3×C4×Dic312yes—
360C3×C4C2×D3×Dic311yescharacteristic
361C3×C4C2×#62411yescharacteristic
362C3×C4C2×C3×C4×D311yescharacteristic
363C3×C4C2×C3×C4×D311yescharacteristic
364C3×C4C2×C3×C4×D311yescharacteristic
365C3×C4C22×C3×Dic311yescharacteristic
366Dic3—24no—
367C32—21no—
368C32—21no—
369C32—21no—
370C32—41no—
371C32C2×#80311yescharacteristic, complemented
372C32C2×#81511yescharacteristic, complemented
373C32C2×C3×#9711yescharacteristic, complemented
374C23C3×D3×Dic311yescharacteristic
375C2×C4—34no—
376C2×C4—64no—
377C2×C4C3×D3×Dic312yes—
378C2×C4C3×D3×Dic311yescharacteristic
379C2×C4C3×#62412yes—
380C2×C4C2×C3×D3211yescharacteristic
381C8—34no—
382C8—94no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—22no—
387C2×C3—22no—
388C2×C3—22no—
389C2×C3—24no—
390C2×C3—24no—
391C2×C3—24no—
392C2×C3—41no—
393C2×C3—42no—
394C2×C3—44no—
395C2×C3#6355414yesdirect product factor
396C2×C3C3×#80314yescomplemented
397C2×C3C3×#81514yescomplemented
398C2×C3C2×#371112yes—
399C2×C3C2×Dic3211yescharacteristic
400C2×C3C2×C3×ΓL(1,𝔽25)12yes—
401C2×C3C2×C3×#20812yes—
402C2×C3C2×C3×C4×Dic311yescharacteristic
403C2×C3C2×C3×C4×Dic311yescharacteristic
404C22C3×#371114yes—
405C22C3×Dic3212yes—
406C22C2×C3×D3×Dic311yescharacteristic
407C4—64no—
408C4C3×Dic3212yes—
409C4C2×C3×D3×Dic311yescharacteristic
410C4C2×C3×#62411yescharacteristic
411C3—21no—
412C3—21no—
413C3—21no—
414C3—41no—
415C3C2×#6355411yescharacteristic, direct product factor
416C3C2×C3×#80311yescharacteristic, complemented
417C3C2×C3×#81511yescharacteristic, complemented
418C2C3×#6355414yesdirect product factor
419C2C2×C3×#371112yes—
420C2C2×C3×Dic3211yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutatorC2×C32C2×C3×C42
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C2×C4
cosocleC2×C32×C4C23×C3
hypocenterC32C2×C3×#97
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63554C2direct (N × H)
7. C2×#63554C3direct (N × H)
10. C2×C3×C3⋊(C3⋊C8)C4semidirect
14. C2×C32×C3⋊C8C4semidirect
30. #63554C2×C3direct (N × H)
38. C3×C3⋊(C3⋊C8)C2×C4semidirect
39. C32×C3⋊C8C2×C4semidirect
58. C2×C3⋊(C3⋊C8)C3×C4semidirect
74. C2×C3×C3⋊C8Dic3semidirect
75. C2×C3×C3⋊C8C3×C4semidirect
144. C3⋊(C3⋊C8)C2×C3×C4semidirect
149. C3×C3⋊C8C2×Dic3semidirect
150. C3×C3⋊C8C2×C3×C4semidirect
152. C2×C33#97semidirect
188. C2×C3⋊C8C3×Dic3semidirect
240. C33C2×#97semidirect
291. C3⋊C8C2×C3×Dic3semidirect
304. C2×C32#803semidirect
305. C2×C32#815semidirect
306. C2×C32C3×#97semidirect
371. C32C2×#803semidirect
372. C32C2×#815semidirect
373. C32C2×C3×#97semidirect
395. C2×C3#63554direct (N × H)
396. C2×C3C3×#803semidirect
397. C2×C3C3×#815semidirect
415. C3C2×#63554direct (N × H)
416. C3C2×C3×#803semidirect
417. C3C2×C3×#815semidirect
418. C2C3×#63554direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊(C3⋊C8)2
C2×C32×C4×Dic32
C22×C32×C3⋊C82
C3×#635542
C2×C3×#8153
C2×C3×#8033
C2×#635543