Finite Group Databasebeta

← C2×C3×#5731 #11901549

Subgroups — C2×C3×#5731 #11901549

Order 1152.

2153 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#946C212yesmaximal
3C2×C3×Q8×Dic3C212yesmaximal, complemented
4C3×#5731C2116yesmaximal, direct product factor
5C2×C3×#898C214yesmaximal
6C2×C3×#889C214yesmaximal, complemented
7C2×C3×#872C211yesmaximal, characteristic
8C2×C3×#870C211yesmaximal, characteristic, complemented
9C2×C3×#541—31nomaximal
10C2×#5731C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2212yes—
12C22×C3×C4×D3C2211yescharacteristic
13C32×C4⋊Q8C2218yescomplemented
14C2×C32×C4⋊C4C2214yes—
15C2×C32×C42C2211yescharacteristic
16C3×#946C22116yes—
17C3×Q8×Dic3C22116yescomplemented
18C2×C3×(C2×D3)⋊C4C2214yes—
19C2×C3×(C2×D3)⋊C4C2212yes—
20C2×C3×C4⋊Dic3C2214yes—
21C2×C3×C4⋊Dic3C2212yes—
22C2×C3×Dic3⋊C4C2214yes—
23C2×C3×Dic3⋊C4C2212yes—
24C2×C3×Dic3⋊C4C2214yes—
25C2×C3×C4×Dic3C2214yes—
26C2×C3×C4×Dic3C2211yescharacteristic
27C3×#898C22132yes—
28C3×#889C22132yescomplemented
29C3×#872C2218yes—
30C3×#870C2218yescomplemented
31C3×#541—316no—
32C2×C3×C4⋊Q8—21no—
33C2×C3×C4⋊Q8D311yescharacteristic, complemented
34C2×C3×C4⋊Q8C2×C311yescharacteristic, complemented
35C2×C3×#126—34no—
36C2×C3×#125—34no—
37C2×C3×#125—31no—
38C2×C3×C22⋊Q8—32no—
39C2×C3×C4×Q8—32no—
40C2×C3×#117—31no—
41C2×#946C2×C312yes—
42C2×Q8×Dic3C2×C312yescomplemented
43#5731C2×C3116yesdirect product factor
44C2×#898C2×C314yes—
45C2×#889C2×C314yescomplemented
46C2×#872C2×C311yescharacteristic
47C2×#870C2×C311yescharacteristic, complemented
48C23×C3×D3C2311yescharacteristic
49C2×C32×Q8—28no—
50C2×C32×Q8C2318yes—
51C22×C32×C4C2314yes—
52C22×C32×C4C2312yes—
53C22×C32×C4C2311yescharacteristic
54C22×C3×Dic3C2312yes—
55C22×C3×Dic3C2314yes—
56C22×C3×Dic3C2311yescharacteristic
57C2×C3×C4×D3—24no—
58C2×C3×C4×D3C2314yes—
59C32×C4⋊C4C23116yes—
60C32×C42C2314yes—
61C3×(C2×D3)⋊C4C23116yes—
62C3×(C2×D3)⋊C4C2318yes—
63C3×C4⋊Dic3C23116yes—
64C3×C4⋊Dic3C2318yes—
65C3×Dic3⋊C4C23116yes—
66C3×Dic3⋊C4C2318yes—
67C3×Dic3⋊C4C23116yes—
68C3×C4×Dic3C23116yes—
69C3×C4×Dic3C2314yes—
70C2×#541—31noSylow
71C22×C3×Q8—22no—
72C22×C3×Q8C2×D312yes—
73C22×C3×Q8C22×C312yes—
74C23×C3×C4—31no—
75C22×C4×D3C22×C311yescharacteristic
76C3×C4⋊Q8—28no—
77C3×C4⋊Q8C2×D318yescomplemented
78C3×C4⋊Q8C22×C318yescomplemented
79C3×#126—332no—
80C3×#125—332no—
81C3×#125—38no—
82C3×C22⋊Q8—316no—
83C3×C4×Q8—316no—
84C3×#117—38no—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4—32no—
87C2×C3×C4⋊C4—34no—
88C2×C3×C4⋊C4—34no—
89C2×C3×C4⋊C4—32no—
90C2×C3×C4⋊C4—34no—
91C2×C3×C4⋊C4C2×D314yes—
92C2×C3×C4⋊C4C22×C314yes—
93C2×C3×C22⋊C4—32no—
94C2×C3×C22⋊C4—34no—
95C2×C3×C42—21no—
96C2×C3×C42—31no—
97C2×C3×C42—34no—
98C2×C3×C42C2×D311yescharacteristic
99C2×C3×C42C22×C311yescharacteristic
100#946C22×C3116yes—
101Q8×Dic3C22×C3116yescomplemented
102C2×(C2×D3)⋊C4C22×C312yes—
103C2×(C2×D3)⋊C4C22×C314yes—
104C2×C4⋊Dic3C22×C312yes—
105C2×C4⋊Dic3C22×C314yes—
106C2×Dic3⋊C4C22×C314yes—
107C2×Dic3⋊C4C22×C312yes—
108C2×Dic3⋊C4C22×C314yes—
109C2×C4×Dic3C22×C311yescharacteristic
110C2×C4×Dic3C22×C314yes—
111#898C22×C3132yes—
112#889C22×C3132yescomplemented
113#872C22×C318yes—
114#870C22×C318yescomplemented
115C23×C32C2411yescharacteristic
116C22×C3×D3—22no—
117C22×C3×D3—24no—
118C22×C3×D3C2412yes—
119C32×Q8—216no—
120C2×C32×C4—24no—
121C2×C32×C4—28no—
122C2×C32×C4#4114yes—
123C2×C32×C4C2418yes—
124C2×C32×C4C2414yes—
125C2×C32×C4C2412yes—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—28no—
129C2×C3×Dic3C2414yes—
130C2×C3×Dic3C2418yes—
131C2×C3×Dic3C2412yes—
132C3×C4×D3—28no—
133#541—316no—
134C2×C4⋊Q8C3×D311yescharacteristic, complemented
135C2×#126—34no—
136C2×#125—34no—
137C2×#125—31no—
138C2×C22⋊Q8—32no—
139C2×C4×Q8—32no—
140C2×#117—31no—
141C24×C3—31no—
142C23×D3C23×C311yescharacteristic
143C2×C3×Q8—28no—
144C2×C3×Q8—28no—
145C2×C3×Q8—28no—
146C2×C3×Q8—48no—
147C2×C3×Q8C22×D318yes—
148C2×C3×Q8C23×C318yes—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—24no—
152C22×C3×C4—32no—
153C22×C3×C4—34no—
154C22×C3×C4—31no—
155C22×C3×C4—34no—
156C22×C3×C4—64no—
157C22×C3×C4C22×D314yes—
158C22×C3×C4C22×D312yes—
159C22×C3×C4C22×D311yescharacteristic
160C22×C3×C4C23×C314yes—
161C22×C3×C4C23×C312yes—
162C22×C3×C4C23×C311yescharacteristic
163C22×Dic3C23×C312yes—
164C22×Dic3C23×C314yes—
165C22×Dic3C23×C311yescharacteristic
166C2×C4×D3—24no—
167C2×C4×D3C23×C314yes—
168C3×C4⋊C4—216no—
169C3×C4⋊C4—38no—
170C3×C4⋊C4—316no—
171C3×C4⋊C4—316no—
172C3×C4⋊C4—38no—
173C3×C4⋊C4—316no—
174C3×C4⋊C4C22×D3116yes—
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×C318yes—
186C4⋊Dic3C23×C3116yes—
187Dic3⋊C4C23×C3116yes—
188Dic3⋊C4C23×C318yes—
189Dic3⋊C4C23×C3116yes—
190C4×Dic3C23×C314yes—
191C4×Dic3C23×C3116yes—
192C22×C32C2×#4112yes—
193C22×C322−1+414yes—
194C22×C32C2511yescharacteristic
195C2×C3×D3—22no—
196C2×C3×D3—24no—
197C2×C3×D3—44no—
198C32×C4—24no—
199C32×C4—28no—
200C32×C4C2×#4114yes—
201C3×Dic3—22no—
202C3×Dic3—24no—
203C3×Dic3—28no—
204C22×Q8C2×C3×D312yes—
205C23×C4—31no—
206C4⋊Q8C2×C3×D318yescomplemented
207#126—332no—
208#125—332no—
209#125—38no—
210C22⋊Q8—316no—
211C4×Q8—316no—
212#117—38no—
213C2×C4⋊C4—32no—
214C2×C4⋊C4—32no—
215C2×C4⋊C4—34no—
216C2×C4⋊C4—34no—
217C2×C4⋊C4—34no—
218C2×C4⋊C4C2×C3×D314yes—
219C2×C22⋊C4—32no—
220C2×C22⋊C4—34no—
221C2×C42—31no—
222C2×C42—34no—
223C2×C42C2×C3×D311yescharacteristic
224C23×C3—21no—
225C23×C3—32no—
226C23×C3—62no—
227C23×C3—64no—
228C23×C3C23×D311yescharacteristic
229C23×C3C24×C311yescharacteristic
230C22×D3—22no—
231C22×D3—24no—
232C22×D3C24×C312yes—
233C3×Q8—216no—
234C3×Q8—216no—
235C3×Q8—416no—
236C2×C3×C4—28no—
237C2×C3×C4—28no—
238C2×C3×C4—24no—
239C2×C3×C4—28no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—28no—
244C2×C3×C4—24no—
245C2×C3×C4—34no—
246C2×C3×C4—38no—
247C2×C3×C4—32no—
248C2×C3×C4—48no—
249C2×C3×C4—62no—
250C2×C3×C4—64no—
251C2×C3×C4—68no—
252C2×C3×C4—68no—
253C2×C3×C4#23714yes—
254C2×C3×C4C3×#4114yes—
255C2×C3×C4C23×D318yes—
256C2×C3×C4C23×D314yes—
257C2×C3×C4C23×D312yes—
258C2×C3×C4C24×C318yes—
259C2×C3×C4C24×C314yes—
260C2×C3×C4C24×C312yes—
261C2×Dic3—22no—
262C2×Dic3—24no—
263C2×Dic3—28no—
264C2×Dic3C24×C314yes—
265C2×Dic3C24×C318yes—
266C2×Dic3C24×C312yes—
267C4×D3—28no—
268C2×C32#54114yescomplemented
269C2×C32C22×#4111yescharacteristic
270C2×C32C2×2−1+412yes—
271C3×D3—42no—
272C24—31no—
273C2×Q8—28no—
274C2×Q8C22×C3×D318yes—
275C22×C4—32no—
276C22×C4—34no—
277C22×C4—31no—
278C22×C4—34no—
279C22×C4—64no—
280C22×C4C22×C3×D314yes—
281C22×C4C22×C3×D312yes—
282C22×C4C22×C3×D311yescharacteristic
283C4⋊C4—38no—
284C4⋊C4—38no—
285C4⋊C4—316no—
286C4⋊C4—316no—
287C4⋊C4—316no—
288C4⋊C4C22×C3×D3116yes—
289C22⋊C4—38no—
290C22⋊C4—316no—
291C42—34no—
292C42—316no—
293C42C22×C3×D314yes—
294C22×C3—22no—
295C22×C3—24no—
296C22×C3—21no—
297C22×C3—62no—
298C22×C3—64no—
299C22×C3—124no—
300C22×C3C2×#23712yes—
301C22×C3#100714yes—
302C22×C3C2×C3×#4112yes—
303C22×C3C3×2−1+414yes—
304C22×C3C24×D311yescharacteristic
305C22×C3C25×C311yescharacteristic
306C2×D3—22no—
307C2×D3—24no—
308C2×D3—44no—
309C3×C4—28no—
310C3×C4—28no—
311C3×C4—24no—
312C3×C4—28no—
313C3×C4—24no—
314C3×C4—24no—
315C3×C4—48no—
316C3×C4—62no—
317C3×C4—64no—
318C3×C4—68no—
319C3×C4C2×#23714yes—
320C3×C4C2×C3×#4114yes—
321Dic3—22no—
322Dic3—24no—
323Dic3—28no—
324C32C2×#54111yescharacteristic, complemented, Sylow
325C23—32no—
326C23—62no—
327C23—64no—
328C23C23×C3×D311yescharacteristic
329Q8—216no—
330C2×C4—28no—
331C2×C4—24no—
332C2×C4—34no—
333C2×C4—38no—
334C2×C4—32no—
335C2×C4—62no—
336C2×C4—64no—
337C2×C4—68no—
338C2×C4—68no—
339C2×C4C3×#23714yes—
340C2×C4C23×C3×D318yes—
341C2×C4C23×C3×D314yes—
342C2×C4C23×C3×D312yes—
343C2×C3—21no—
344C2×C3—22no—
345C2×C3—24no—
346C2×C3—122no—
347C2×C3#573114yesdirect product factor
348C2×C3C3×#54114yescomplemented
349C2×C3C22×#23711yescharacteristic
350C2×C3C2×#100712yes—
351C2×C3C22×C3×#4111yescharacteristic
352C2×C3C2×C3×2−1+412yes—
353D3—42no—
354C22—62no—
355C22—64no—
356C22—124no—
357C22C2×C3×#23712yes—
358C22C3×#100714yes—
359C22C24×C3×D311yescharacteristic
360C4—28no—
361C4—24no—
362C4—62no—
363C4—64no—
364C4—68no—
365C4C2×C3×#23714yes—
366C3—21no—
367C3C2×#573111yescharacteristic, direct product factor
368C3C2×C3×#54111yescharacteristic, complemented
369C2—122no—
370C2C3×#573114yesdirect product factor
371C2C22×C3×#23711yescharacteristic
372C2C2×C3×#100712yes—
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C4⋊Q8C2
socleC23×C32C24
prime radicalC23×C3×D3C23
hypercenterC2×C3×C4⋊Q8D3
hypocenterC3C2×C3×#541
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4⋊Q8C2semidirect
3. C2×C3×Q8×Dic3C2semidirect
4. C3×#5731C2direct (N × H)
6. C2×C3×#889C2semidirect
8. C2×C3×#870C2semidirect
10. C2×#5731C3direct (N × H)
13. C32×C4⋊Q8C22semidirect
17. C3×Q8×Dic3C22semidirect
28. C3×#889C22semidirect
30. C3×#870C22semidirect
33. C2×C3×C4⋊Q8D3semidirect
34. C2×C3×C4⋊Q8C2×C3semidirect
42. C2×Q8×Dic3C2×C3semidirect
43. #5731C2×C3direct (N × H)
45. C2×#889C2×C3semidirect
47. C2×#870C2×C3semidirect
77. C3×C4⋊Q8C2×D3semidirect
78. C3×C4⋊Q8C22×C3semidirect
101. Q8×Dic3C22×C3semidirect
112. #889C22×C3semidirect
114. #870C22×C3semidirect
134. C2×C4⋊Q8C3×D3semidirect
206. C4⋊Q8C2×C3×D3semidirect
268. C2×C32#541semidirect
324. C32C2×#541semidirect
347. C2×C3#5731direct (N × H)
348. C2×C3C3×#541semidirect
367. C3C2×#5731direct (N × H)
368. C3C2×C3×#541semidirect
370. C2C3×#5731direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C4⋊Q82
C2×C3×#9462
C2×C3×Q8×Dic32
C3×#57312
C2×C3×#8982
C2×C3×#8892
C2×C3×#8722
C2×C3×#8702
C2×C3×#5413
C2×#57313