Finite Group Databasebeta

← C2×C3×#5698 #11901555

Subgroups — C2×C3×#5698 #11901555

Order 1152.

2121 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#125C211yesmaximal, characteristic, complemented
2C3×#5698C2116yesmaximal, direct product factor
3C2×C3×#898C214yesmaximal
4C2×C3×#897C212yesmaximal
5C2×C3×#890C214yesmaximal, complemented
6C2×C3×#887C212yesmaximal, complemented
7C2×C3×#872C211yesmaximal, characteristic
8C2×C3×#869C211yesmaximal, characteristic, complemented
9C2×C3×#541—31nomaximal
10C2×#5698C311yesmaximal, characteristic, direct product factor
11C22×C3×C4×D3C2211yescharacteristic
12C22×C3×Dic6C2212yes—
13C32×#125C2218yescomplemented
14C2×C32×C4⋊C4C2212yes—
15C2×C32×C4⋊C4C2214yes—
16C2×C32×C42C2211yescharacteristic
17C2×C3×(C2×D3)⋊C4C2214yes—
18C2×C3×(C2×D3)⋊C4C2212yes—
19C2×C3×C4⋊Dic3C2214yes—
20C2×C3×C4⋊Dic3C2214yes—
21C2×C3×Dic3⋊C4C2214yes—
22C2×C3×Dic3⋊C4C2212yes—
23C2×C3×C4×Dic3C2211yescharacteristic
24C2×C3×C4×Dic3C2214yes—
25C3×#898C22132yes—
26C3×#897C22116yes—
27C3×#890C22132yescomplemented
28C3×#887C22116yescomplemented
29C3×#872C2218yes—
30C3×#869C2218yescomplemented
31C3×#541—316no—
32C2×C3×C4⋊Q8—31no—
33C2×C3×#126—34no—
34C2×C3×#125—21no—
35C2×C3×#125—34no—
36C2×C3×#125D311yescharacteristic, complemented
37C2×C3×#125C2×C311yescharacteristic, complemented
38C2×C3×C22⋊Q8—32no—
39C2×C3×C4×Q8—32no—
40C2×C3×#117—31no—
41#5698C2×C3116yesdirect product factor
42C2×#898C2×C314yes—
43C2×#897C2×C312yes—
44C2×#890C2×C314yescomplemented
45C2×#887C2×C312yescomplemented
46C2×#872C2×C311yescharacteristic
47C2×#869C2×C311yescharacteristic, complemented
48C23×C3×D3C2311yescharacteristic
49C22×C32×C4C2314yes—
50C22×C32×C4C2312yes—
51C22×C32×C4C2311yescharacteristic
52C22×C3×Dic3C2312yes—
53C22×C3×Dic3C2314yes—
54C22×C3×Dic3C2311yescharacteristic
55C2×C3×C4×D3—24no—
56C2×C3×C4×D3C2314yes—
57C2×C3×Dic6—28no—
58C2×C3×Dic6C2318yes—
59C32×C4⋊C4C2318yes—
60C32×C4⋊C4C23116yes—
61C32×C42C2314yes—
62C3×(C2×D3)⋊C4C23116yes—
63C3×(C2×D3)⋊C4C2318yes—
64C3×C4⋊Dic3C23116yes—
65C3×C4⋊Dic3C23116yes—
66C3×Dic3⋊C4C23116yes—
67C3×Dic3⋊C4C2318yes—
68C3×C4×Dic3C2314yes—
69C3×C4×Dic3C23116yes—
70C2×#541—31noSylow
71C22×C3×Q8—32no—
72C23×C3×C4—31no—
73C22×C4×D3C22×C311yescharacteristic
74C22×Dic6C22×C312yes—
75C3×C4⋊Q8—38no—
76C3×#126—332no—
77C3×#125—28no—
78C3×#125—332no—
79C3×#125C2×D318yescomplemented
80C3×#125C22×C318yescomplemented
81C3×C22⋊Q8—316no—
82C3×C4×Q8—316no—
83C3×#117—38no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4—32no—
87C2×C3×C4⋊C4—34no—
88C2×C3×C4⋊C4—34no—
89C2×C3×C4⋊C4—34no—
90C2×C3×C4⋊C4C2×D312yes—
91C2×C3×C4⋊C4C2×D314yes—
92C2×C3×C4⋊C4C22×C312yes—
93C2×C3×C4⋊C4C22×C314yes—
94C2×C3×C22⋊C4—32no—
95C2×C3×C22⋊C4—34no—
96C2×C3×C42—21no—
97C2×C3×C42—31no—
98C2×C3×C42—34no—
99C2×C3×C42C2×D311yescharacteristic
100C2×C3×C42C22×C311yescharacteristic
101C2×(C2×D3)⋊C4C22×C312yes—
102C2×(C2×D3)⋊C4C22×C314yes—
103C2×C4⋊Dic3C22×C314yes—
104C2×C4⋊Dic3C22×C314yes—
105C2×Dic3⋊C4C22×C312yes—
106C2×Dic3⋊C4C22×C314yes—
107C2×C4×Dic3C22×C311yescharacteristic
108C2×C4×Dic3C22×C314yes—
109#898C22×C3132yes—
110#897C22×C3116yes—
111#890C22×C3132yescomplemented
112#887C22×C3116yescomplemented
113#872C22×C318yes—
114#869C22×C318yescomplemented
115C23×C32C2411yescharacteristic
116C22×C3×D3—22no—
117C22×C3×D3—24no—
118C22×C3×D3C2412yes—
119C2×C32×C4—28no—
120C2×C32×C4—24no—
121C2×C32×C4#4114yes—
122C2×C32×C4C2418yes—
123C2×C32×C4C2414yes—
124C2×C32×C4C2412yes—
125C2×C3×Dic3—28no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3C2414yes—
129C2×C3×Dic3C2418yes—
130C2×C3×Dic3C2412yes—
131C3×C4×D3—28no—
132C3×Dic6—216no—
133#541—316no—
134C2×C4⋊Q8—31no—
135C2×#126—34no—
136C2×#125—34no—
137C2×#125C3×D311yescharacteristic, complemented
138C2×C22⋊Q8—32no—
139C2×C4×Q8—32no—
140C2×#117—31no—
141C24×C3—31no—
142C23×D3C23×C311yescharacteristic
143C2×C3×Q8—38no—
144C2×C3×Q8—68no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—24no—
148C22×C3×C4—32no—
149C22×C3×C4—34no—
150C22×C3×C4—31no—
151C22×C3×C4—34no—
152C22×C3×C4—64no—
153C22×C3×C4C22×D314yes—
154C22×C3×C4C22×D312yes—
155C22×C3×C4C22×D311yescharacteristic
156C22×C3×C4C23×C314yes—
157C22×C3×C4C23×C312yes—
158C22×C3×C4C23×C311yescharacteristic
159C22×Dic3C23×C312yes—
160C22×Dic3C23×C314yes—
161C22×Dic3C23×C311yescharacteristic
162C2×C4×D3—24no—
163C2×C4×D3C23×C314yes—
164C2×Dic6—28no—
165C2×Dic6C23×C318yes—
166C3×C4⋊C4—28no—
167C3×C4⋊C4—216no—
168C3×C4⋊C4—38no—
169C3×C4⋊C4—316no—
170C3×C4⋊C4—316no—
171C3×C4⋊C4—316no—
172C3×C4⋊C4C22×D318yes—
173C3×C4⋊C4C22×D3116yes—
174C3×C4⋊C4C23×C318yes—
175C3×C4⋊C4C23×C3116yes—
176C3×C22⋊C4—38no—
177C3×C22⋊C4—316no—
178C3×C42—24no—
179C3×C42—34no—
180C3×C42—316no—
181C3×C42C22×D314yes—
182C3×C42C23×C314yes—
183(C2×D3)⋊C4C23×C318yes—
184(C2×D3)⋊C4C23×C3116yes—
185C4⋊Dic3C23×C3116yes—
186C4⋊Dic3C23×C3116yes—
187Dic3⋊C4C23×C318yes—
188Dic3⋊C4C23×C3116yes—
189C4×Dic3C23×C314yes—
190C4×Dic3C23×C3116yes—
191C22×C32C2×#4112yes—
192C22×C322−1+414yes—
193C22×C32C2511yescharacteristic
194C2×C3×D3—22no—
195C2×C3×D3—24no—
196C2×C3×D3—44no—
197C32×C4—28no—
198C32×C4—24no—
199C32×C4C2×#4114yes—
200C3×Dic3—28no—
201C3×Dic3—22no—
202C3×Dic3—24no—
203C22×Q8—32no—
204C23×C4—31no—
205C4⋊Q8—38no—
206#126—332no—
207#125—332no—
208#125C2×C3×D318yescomplemented
209C22⋊Q8—316no—
210C4×Q8—316no—
211#117—38no—
212C2×C4⋊C4—32no—
213C2×C4⋊C4—34no—
214C2×C4⋊C4—34no—
215C2×C4⋊C4—34no—
216C2×C4⋊C4C2×C3×D312yes—
217C2×C4⋊C4C2×C3×D314yes—
218C2×C22⋊C4—32no—
219C2×C22⋊C4—34no—
220C2×C42—31no—
221C2×C42—34no—
222C2×C42C2×C3×D311yescharacteristic
223C23×C3—21no—
224C23×C3—32no—
225C23×C3—62no—
226C23×C3—64no—
227C23×C3C23×D311yescharacteristic
228C23×C3C24×C311yescharacteristic
229C22×D3—22no—
230C22×D3—24no—
231C22×D3C24×C312yes—
232C3×Q8—616no—
233C2×C3×C4—28no—
234C2×C3×C4—24no—
235C2×C3×C4—28no—
236C2×C3×C4—24no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—28no—
240C2×C3×C4—24no—
241C2×C3×C4—34no—
242C2×C3×C4—38no—
243C2×C3×C4—32no—
244C2×C3×C4—44no—
245C2×C3×C4—48no—
246C2×C3×C4—62no—
247C2×C3×C4—68no—
248C2×C3×C4—64no—
249C2×C3×C4—68no—
250C2×C3×C4#23914yes—
251C2×C3×C4C3×#4114yes—
252C2×C3×C4C23×D318yes—
253C2×C3×C4C23×D314yes—
254C2×C3×C4C23×D312yes—
255C2×C3×C4C24×C318yes—
256C2×C3×C4C24×C314yes—
257C2×C3×C4C24×C312yes—
258C2×Dic3—28no—
259C2×Dic3—22no—
260C2×Dic3—24no—
261C2×Dic3C24×C314yes—
262C2×Dic3C24×C318yes—
263C2×Dic3C24×C312yes—
264C4×D3—28no—
265Dic6—216no—
266C2×C32#54114yescomplemented
267C2×C32C22×#4111yescharacteristic
268C2×C32C2×2−1+412yes—
269C3×D3—42no—
270C24—31no—
271C2×Q8—38no—
272C2×Q8—68no—
273C22×C4—32no—
274C22×C4—34no—
275C22×C4—31no—
276C22×C4—34no—
277C22×C4—64no—
278C22×C4C22×C3×D314yes—
279C22×C4C22×C3×D312yes—
280C22×C4C22×C3×D311yescharacteristic
281C4⋊C4—38no—
282C4⋊C4—316no—
283C4⋊C4—316no—
284C4⋊C4—316no—
285C4⋊C4C22×C3×D318yes—
286C4⋊C4C22×C3×D3116yes—
287C22⋊C4—38no—
288C22⋊C4—316no—
289C42—34no—
290C42—316no—
291C42C22×C3×D314yes—
292C22×C3—22no—
293C22×C3—24no—
294C22×C3—21no—
295C22×C3—62no—
296C22×C3—64no—
297C22×C3—124no—
298C22×C3C2×#23912yes—
299C22×C3#101014yes—
300C22×C3C2×C3×#4112yes—
301C22×C3C3×2−1+414yes—
302C22×C3C24×D311yescharacteristic
303C22×C3C25×C311yescharacteristic
304C2×D3—22no—
305C2×D3—24no—
306C2×D3—44no—
307C3×C4—28no—
308C3×C4—24no—
309C3×C4—28no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—44no—
313C3×C4—48no—
314C3×C4—62no—
315C3×C4—68no—
316C3×C4—64no—
317C3×C4C2×#23914yes—
318C3×C4C2×C3×#4114yes—
319Dic3—28no—
320Dic3—22no—
321Dic3—24no—
322C32C2×#54111yescharacteristic, complemented, Sylow
323C23—32no—
324C23—62no—
325C23—64no—
326C23C23×C3×D311yescharacteristic
327Q8—616no—
328C2×C4—28no—
329C2×C4—24no—
330C2×C4—34no—
331C2×C4—38no—
332C2×C4—32no—
333C2×C4—62no—
334C2×C4—68no—
335C2×C4—64no—
336C2×C4—68no—
337C2×C4C3×#23914yes—
338C2×C4C23×C3×D318yes—
339C2×C4C23×C3×D314yes—
340C2×C4C23×C3×D312yes—
341C2×C3—21no—
342C2×C3—22no—
343C2×C3—24no—
344C2×C3—122no—
345C2×C3#569814yesdirect product factor
346C2×C3C3×#54114yescomplemented
347C2×C3C22×#23911yescharacteristic
348C2×C3C2×#101012yes—
349C2×C3C22×C3×#4111yescharacteristic
350C2×C3C2×C3×2−1+412yes—
351D3—42no—
352C22—62no—
353C22—64no—
354C22—124no—
355C22C2×C3×#23912yes—
356C22C3×#101014yes—
357C22C24×C3×D311yescharacteristic
358C4—28no—
359C4—24no—
360C4—62no—
361C4—68no—
362C4—64no—
363C4C2×C3×#23914yes—
364C3—21no—
365C3C2×#569811yescharacteristic, direct product factor
366C3C2×C3×#54111yescharacteristic, complemented
367C2—122no—
368C2C3×#569814yesdirect product factor
369C2C22×C3×#23911yescharacteristic
370C2C2×C3×#101012yes—
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×#125C2
socleC23×C32C24
prime radicalC23×C3×D3C23
hypercenterC2×C3×#125D3
hypocenterC3C2×C3×#541
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×#125

Complements

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

NH ≅ G/Ntype
1. C2×C32×#125C2semidirect
2. C3×#5698C2direct (N × H)
5. C2×C3×#890C2semidirect
6. C2×C3×#887C2semidirect
8. C2×C3×#869C2semidirect
10. C2×#5698C3direct (N × H)
13. C32×#125C22semidirect
27. C3×#890C22semidirect
28. C3×#887C22semidirect
30. C3×#869C22semidirect
36. C2×C3×#125D3semidirect
37. C2×C3×#125C2×C3semidirect
41. #5698C2×C3direct (N × H)
44. C2×#890C2×C3semidirect
45. C2×#887C2×C3semidirect
47. C2×#869C2×C3semidirect
79. C3×#125C2×D3semidirect
80. C3×#125C22×C3semidirect
111. #890C22×C3semidirect
112. #887C22×C3semidirect
114. #869C22×C3semidirect
137. C2×#125C3×D3semidirect
208. #125C2×C3×D3semidirect
266. C2×C32#541semidirect
322. C32C2×#541semidirect
345. C2×C3#5698direct (N × H)
346. C2×C3C3×#541semidirect
365. C3C2×#5698direct (N × H)
366. C3C2×C3×#541semidirect
368. C2C3×#5698direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1252
C3×#56982
C2×C3×#8982
C2×C3×#8972
C2×C3×#8902
C2×C3×#8872
C2×C3×#8722
C2×C3×#8692
C2×C3×#5413
C2×#56983