Finite Group Databasebeta

← C2×C3×#5688 #11900821

Subgroups — C2×C3×#5688 #11900821

Order 1152.

2449 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
1C2×C32×#125C211yesmaximal, characteristic, complemented
2C3×#5688C2116yesmaximal, direct product factor
3C2×C3×#896C214yesmaximal
4C2×C3×#892C212yesmaximal
5C2×C3×#888C212yesmaximal, complemented
6C2×C3×#887C214yesmaximal, complemented
7C2×C3×C42×D3C211yesmaximal, characteristic
8C2×C3×#870C211yesmaximal, characteristic, complemented
9C2×C3×#533—31nomaximal
10C2×#5688C311yesmaximal, characteristic, direct product factor
11C22×C3×C4×D3C2212yes—
12C22×C3×C4×D3C2211yescharacteristic
13C22×C3×Dic6C2214yes—
14C32×#125C2218yescomplemented
15C2×C32×C4⋊C4C2214yes—
16C2×C32×C4⋊C4C2212yes—
17C2×C32×C42C2211yescharacteristic
18C2×C3×(C2×D3)⋊C4C2214yes—
19C2×C3×C4⋊Dic3C2212yes—
20C2×C3×Dic3⋊C4C2214yes—
21C2×C3×Dic3⋊C4C2214yes—
22C2×C3×C4×Dic3C2214yes—
23C2×C3×C4×Dic3C2212yes—
24C2×C3×C4×Dic3C2211yescharacteristic
25C3×#896C22132yes—
26C3×#892C22116yes—
27C3×#888C22116yescomplemented
28C3×#887C22132yescomplemented
29C3×C42×D3C2218yes—
30C3×#870C2218yescomplemented
31C3×#533—316no—
32C2×C3×C4⋊Q8—32no—
33C2×C3×#125—21no—
34C2×C3×#125—31no—
35C2×C3×#125D311yescharacteristic, complemented
36C2×C3×#125C2×C311yescharacteristic, complemented
37C2×C3×C22⋊Q8—34no—
38C2×C3×C4×Q8—34no—
39C2×C3×#117—32no—
40C22×C3×C42—31no—
41#5688C2×C3116yesdirect product factor
42C2×#896C2×C314yes—
43C2×#892C2×C312yes—
44C2×#888C2×C312yescomplemented
45C2×#887C2×C314yescomplemented
46C2×C42×D3C2×C311yescharacteristic
47C2×#870C2×C311yescharacteristic, complemented
48C23×C3×D3C2311yescharacteristic
49C22×C32×C4C2312yes—
50C22×C32×C4C2314yes—
51C22×C32×C4C2311yescharacteristic
52C22×C3×Dic3C2314yes—
53C22×C3×Dic3C2311yescharacteristic
54C22×C3×Dic3C2312yes—
55C2×C3×C4×D3—28no—
56C2×C3×C4×D3Q818yescomplemented
57C2×C3×C4×D3C2318yes—
58C2×C3×C4×D3C2314yes—
59C2×C3×Dic6—216no—
60C2×C3×Dic6C23116yes—
61C32×C4⋊C4C23116yes—
62C32×C4⋊C4C2318yes—
63C32×C42C2314yes—
64C3×(C2×D3)⋊C4C23116yes—
65C3×C4⋊Dic3C2318yes—
66C3×Dic3⋊C4C23116yes—
67C3×Dic3⋊C4C23116yes—
68C3×C4×Dic3C23116yes—
69C3×C4×Dic3C2318yes—
70C3×C4×Dic3C2314yes—
71C2×#533—31noSylow
72C22×C3×Q8—34no—
73C23×C3×C4—31no—
74C23×C3×C4—32no—
75C22×C4×D3C22×C312yes—
76C22×C4×D3C22×C311yescharacteristic
77C22×Dic6C22×C314yes—
78C3×C4⋊Q8—316no—
79C3×#125—28no—
80C3×#125—38no—
81C3×#125C2×D318yescomplemented
82C3×#125C22×C318yescomplemented
83C3×C22⋊Q8—332no—
84C3×C4×Q8—332no—
85C3×#117—316no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—24no—
88C2×C3×C4⋊C4—34no—
89C2×C3×C4⋊C4—34no—
90C2×C3×C4⋊C4—32no—
91C2×C3×C4⋊C4C2×D312yes—
92C2×C3×C4⋊C4C2×D314yes—
93C2×C3×C4⋊C4C22×C314yes—
94C2×C3×C4⋊C4C22×C312yes—
95C2×C3×C22⋊C4—34no—
96C2×C3×C42—21no—
97C2×C3×C42—32no—
98C2×C3×C42—31no—
99C2×C3×C42—34no—
100C2×C3×C42—38no—
101C2×C3×C42C2×D311yescharacteristic
102C2×C3×C42C22×C311yescharacteristic
103C2×(C2×D3)⋊C4C22×C314yes—
104C2×C4⋊Dic3C22×C312yes—
105C2×Dic3⋊C4C22×C314yes—
106C2×Dic3⋊C4C22×C314yes—
107C2×C4×Dic3C22×C312yes—
108C2×C4×Dic3C22×C311yescharacteristic
109C2×C4×Dic3C22×C314yes—
110#896C22×C3132yes—
111#892C22×C3116yes—
112#888C22×C3116yescomplemented
113#887C22×C3132yescomplemented
114C42×D3C22×C318yes—
115#870C22×C318yescomplemented
116C23×C32C2411yescharacteristic
117C22×C3×D3—24no—
118C22×C3×D3C2×Q814yes—
119C22×C3×D3C2412yes—
120C2×C32×C4—28no—
121C2×C32×C4—24no—
122C2×C32×C4C2×Q814yes—
123C2×C32×C4C2414yes—
124C2×C32×C4C2418yes—
125C2×C32×C4C2412yes—
126C2×C3×Dic3—28no—
127C2×C3×Dic3C2×Q814yes—
128C2×C3×Dic3#4118yes—
129C2×C3×Dic3C2418yes—
130C2×C3×Dic3C2412yes—
131C2×C3×Dic3C2414yes—
132C3×C4×D3—216no—
133C3×C4×D3C2×Q8116yescomplemented
134C3×Dic6—232no—
135#533—316no—
136C2×C4⋊Q8—32no—
137C2×#125—31no—
138C2×#125C3×D311yescharacteristic, complemented
139C2×C22⋊Q8—34no—
140C2×C4×Q8—34no—
141C2×#117—32no—
142C22×C42—31no—
143C24×C3—31no—
144C23×D3C23×C311yescharacteristic
145C2×C3×Q8—316no—
146C2×C3×Q8—616no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—24no—
150C22×C3×C4—31no—
151C22×C3×C4—32no—
152C22×C3×C4—38no—
153C22×C3×C4—34no—
154C22×C3×C4—34no—
155C22×C3×C4—38no—
156C22×C3×C4—68no—
157C22×C3×C4C22×D312yes—
158C22×C3×C4C22×D314yes—
159C22×C3×C4C22×D311yescharacteristic
160C22×C3×C4C23×C312yes—
161C22×C3×C4C23×C314yes—
162C22×C3×C4C23×C311yescharacteristic
163C22×Dic3C23×C314yes—
164C22×Dic3C23×C311yescharacteristic
165C22×Dic3C23×C312yes—
166C2×C4×D3—28no—
167C2×C4×D3C3×Q818yescomplemented
168C2×C4×D3C23×C318yes—
169C2×C4×D3C23×C314yes—
170C2×Dic6—216no—
171C2×Dic6C23×C3116yes—
172C3×C4⋊C4—28no—
173C3×C4⋊C4—216no—
174C3×C4⋊C4—316no—
175C3×C4⋊C4—316no—
176C3×C4⋊C4—38no—
177C3×C4⋊C4C22×D318yes—
178C3×C4⋊C4C22×D3116yes—
179C3×C4⋊C4C23×C3116yes—
180C3×C4⋊C4C23×C318yes—
181C3×C22⋊C4—316no—
182C3×C42—24no—
183C3×C42—38no—
184C3×C42—34no—
185C3×C42—316no—
186C3×C42C22×D314yes—
187C3×C42C23×C314yes—
188(C2×D3)⋊C4C23×C3116yes—
189C4⋊Dic3C23×C318yes—
190Dic3⋊C4C23×C3116yes—
191Dic3⋊C4C23×C3116yes—
192C4×Dic3C23×C318yes—
193C4×Dic3C23×C314yes—
194C4×Dic3C23×C3116yes—
195C22×C32C22×Q812yes—
196C22×C32C2×#4114yes—
197C22×C32C2511yescharacteristic
198C2×C3×D3—28no—
199C2×C3×D3—24no—
200C2×C3×D3C22×Q814yes—
201C32×C4—28no—
202C32×C4—24no—
203C32×C4C22×Q814yes—
204C3×Dic3—28no—
205C3×Dic3C22×Q814yes—
206C3×Dic3C2×#4118yes—
207C22×Q8—34no—
208C23×C4—31no—
209C23×C4—32no—
210C4⋊Q8—316no—
211#125—38no—
212#125C2×C3×D318yescomplemented
213C22⋊Q8—332no—
214C4×Q8—332no—
215#117—316no—
216C2×C4⋊C4—32no—
217C2×C4⋊C4—34no—
218C2×C4⋊C4—34no—
219C2×C4⋊C4C2×C3×D312yes—
220C2×C4⋊C4C2×C3×D314yes—
221C2×C22⋊C4—34no—
222C2×C42—34no—
223C2×C42—32no—
224C2×C42—31no—
225C2×C42—38no—
226C2×C42C2×C3×D311yescharacteristic
227C23×C3—21no—
228C23×C3—34no—
229C23×C3—32no—
230C23×C3—64no—
231C23×C3C23×D311yescharacteristic
232C23×C3C24×C311yescharacteristic
233C22×D3—24no—
234C22×D3C2×C3×Q814yes—
235C22×D3C24×C312yes—
236C3×Q8—632no—
237C2×C3×C4—24no—
238C2×C3×C4—28no—
239C2×C3×C4—28no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—28no—
244C2×C3×C4—24no—
245C2×C3×C4—34no—
246C2×C3×C4—38no—
247C2×C3×C4—32no—
248C2×C3×C4—34no—
249C2×C3×C4—316no—
250C2×C3×C4—38no—
251C2×C3×C4—44no—
252C2×C3×C4—48no—
253C2×C3×C4—616no—
254C2×C3×C4—68no—
255C2×C3×C4D3×Q814yes—
256C2×C3×C4C2×C3×Q814yes—
257C2×C3×C4C23×D314yes—
258C2×C3×C4C23×D318yes—
259C2×C3×C4C23×D312yes—
260C2×C3×C4C24×C314yes—
261C2×C3×C4C24×C318yes—
262C2×C3×C4C24×C312yes—
263C2×Dic3—28no—
264C2×Dic3C2×C3×Q814yes—
265C2×Dic3C3×#4118yes—
266C2×Dic3C24×C318yes—
267C2×Dic3C24×C312yes—
268C2×Dic3C24×C314yes—
269C4×D3—216no—
270C4×D3C2×C3×Q8116yescomplemented
271Dic6—232no—
272C2×C32#53314yescomplemented
273C2×C32C23×Q811yescharacteristic
274C2×C32C22×#4112yes—
275C3×D3—24no—
276C24—31no—
277C2×Q8—316no—
278C2×Q8—616no—
279C22×C4—31no—
280C22×C4—32no—
281C22×C4—34no—
282C22×C4—38no—
283C22×C4—34no—
284C22×C4—38no—
285C22×C4—68no—
286C22×C4C22×C3×D312yes—
287C22×C4C22×C3×D314yes—
288C22×C4C22×C3×D311yescharacteristic
289C4⋊C4—38no—
290C4⋊C4—316no—
291C4⋊C4—316no—
292C4⋊C4C22×C3×D318yes—
293C4⋊C4C22×C3×D3116yes—
294C22⋊C4—316no—
295C42—38no—
296C42—34no—
297C42—316no—
298C42C22×C3×D314yes—
299C22×C3—22no—
300C22×C3—24no—
301C22×C3—21no—
302C22×C3—34no—
303C22×C3—68no—
304C22×C3—64no—
305C22×C3C2×D3×Q812yes—
306C22×C3D3×#4114yes—
307C22×C3C22×C3×Q812yes—
308C22×C3C2×C3×#4114yes—
309C22×C3C24×D311yescharacteristic
310C22×C3C25×C311yescharacteristic
311C2×D3—24no—
312C2×D3—28no—
313C2×D3C22×C3×Q814yes—
314C3×C4—24no—
315C3×C4—28no—
316C3×C4—28no—
317C3×C4—24no—
318C3×C4—24no—
319C3×C4—34no—
320C3×C4—38no—
321C3×C4—44no—
322C3×C4—48no—
323C3×C4—68no—
324C3×C4C2×D3×Q814yes—
325C3×C4C22×C3×Q814yes—
326Dic3—28no—
327Dic3C22×C3×Q814yes—
328Dic3C2×C3×#4118yes—
329C32C2×#53311yescharacteristic, complemented, Sylow
330C23—34no—
331C23—32no—
332C23—64no—
333C23C23×C3×D311yescharacteristic
334Q8—632no—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—34no—
338C2×C4—38no—
339C2×C4—32no—
340C2×C4—34no—
341C2×C4—316no—
342C2×C4—38no—
343C2×C4—616no—
344C2×C4—68no—
345C2×C4C3×D3×Q814yes—
346C2×C4C23×C3×D314yes—
347C2×C4C23×C3×D318yes—
348C2×C4C23×C3×D312yes—
349C2×C3—21no—
350C2×C3—22no—
351C2×C3—24no—
352C2×C3—64no—
353C2×C3#568814yesdirect product factor
354C2×C3C3×#53314yescomplemented
355C2×C3C22×D3×Q811yescharacteristic
356C2×C3C2×D3×#4112yes—
357C2×C3C23×C3×Q811yescharacteristic
358C2×C3C22×C3×#4112yes—
359D3—24no—
360C22—34no—
361C22—64no—
362C22—68no—
363C22C2×C3×D3×Q812yes—
364C22C3×D3×#4114yes—
365C22C24×C3×D311yescharacteristic
366C4—24no—
367C4—28no—
368C4—34no—
369C4—38no—
370C4—68no—
371C4C2×C3×D3×Q814yes—
372C3—21no—
373C3C2×#568811yescharacteristic, direct product factor
374C3C2×C3×#53311yescharacteristic, complemented
375C2—64no—
376C2C3×#568814yesdirect product factor
377C2C22×C3×D3×Q811yescharacteristic
378C2C2×C3×D3×#4112yes—
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×#533
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×#5688C2direct (N × H)
5. C2×C3×#888C2semidirect
6. C2×C3×#887C2semidirect
8. C2×C3×#870C2semidirect
10. C2×#5688C3direct (N × H)
14. C32×#125C22semidirect
27. C3×#888C22semidirect
28. C3×#887C22semidirect
30. C3×#870C22semidirect
35. C2×C3×#125D3semidirect
36. C2×C3×#125C2×C3semidirect
41. #5688C2×C3direct (N × H)
44. C2×#888C2×C3semidirect
45. C2×#887C2×C3semidirect
47. C2×#870C2×C3semidirect
56. C2×C3×C4×D3Q8semidirect
81. C3×#125C2×D3semidirect
82. C3×#125C22×C3semidirect
112. #888C22×C3semidirect
113. #887C22×C3semidirect
115. #870C22×C3semidirect
133. C3×C4×D3C2×Q8semidirect
138. C2×#125C3×D3semidirect
167. C2×C4×D3C3×Q8semidirect
212. #125C2×C3×D3semidirect
270. C4×D3C2×C3×Q8semidirect
272. C2×C32#533semidirect
329. C32C2×#533semidirect
353. C2×C3#5688direct (N × H)
354. C2×C3C3×#533semidirect
373. C3C2×#5688direct (N × H)
374. C3C2×C3×#533semidirect
376. C2C3×#5688direct (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×#56882
C2×C3×#8962
C2×C3×#8922
C2×C3×#8882
C2×C3×#8872
C2×C3×C42×D32
C2×C3×#8702
C2×C3×#5333
C2×#56883