Finite Group Databasebeta

← C2×C3×#5700 #11910238

Subgroups — C2×C3×#5700 #11910238

Order 1152.

2417 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#125C211yesmaximal, characteristic, complemented
2C3×#5700C2116yesmaximal, direct product factor
3C2×C3×#896C214yesmaximal, complemented
4C2×C3×#895C214yesmaximal, complemented
5C2×C3×#894C214yesmaximal, complemented
6C2×C3×#875C212yesmaximal, complemented
7C2×C3×#562—31nomaximal
8C2×#5700C311yesmaximal, characteristic, direct product factor
9C22×C3×D12C2214yes—
10C22×C3×D12C2212yes—
11C22×C3×C4×D3C2214yes—
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)⋊C4C2214yes—
19C2×C3×(C2×D3)⋊C4C2214yes—
20C2×C3×Dic3⋊C4C2214yes—
21C3×#896C22132yescomplemented
22C3×#895C22132yescomplemented
23C3×#894C22132yescomplemented
24C3×#875C22116yescomplemented
25C3×#562—316no—
26C2×C3×#125—21no—
27C2×C3×#125D311yescharacteristic, complemented
28C2×C3×#125C2×C311yescharacteristic, complemented
29C2×C3×#124—32no—
30C2×C3×#123—34no—
31C2×C3×C22⋊Q8—34no—
32C2×C3×#121—34no—
33#5700C2×C3116yesdirect product factor
34C2×#896C2×C314yescomplemented
35C2×#895C2×C314yescomplemented
36C2×#894C2×C314yescomplemented
37C2×#875C2×C312yescomplemented
38C23×C3×D3C2314yes—
39C22×C32×C4C2314yes—
40C22×C32×C4C2311yescharacteristic
41C22×C32×C4C2312yes—
42C22×C3×Dic3C2314yes—
43C2×C3×D12—44no—
44C2×C3×D12—48no—
45C2×C3×D12C23116yes—
46C2×C3×D12C2318yes—
47C2×C3×C4×D3—48no—
48C2×C3×C4×D3C23116yes—
49C2×C3×Dic6—44no—
50C2×C3×Dic6C2318yes—
51C32×C4⋊C4C2318yes—
52C32×C4⋊C4C23116yes—
53C32×C42C2314yes—
54C3×(C2×D3)⋊C4C23116yes—
55C3×(C2×D3)⋊C4C23116yes—
56C3×(C2×D3)⋊C4C23116yes—
57C3×Dic3⋊C4C23116yes—
58C2×#562—31noSylow
59C22×C3×Q8—32no—
60C22×C3×D4—34no—
61C22×C3×D4—32no—
62C23×C3×C4—34no—
63C22×D12C22×C312yes—
64C22×D12C22×C314yes—
65C22×C4×D3C22×C314yes—
66C22×Dic6C22×C312yes—
67C3×#125—28no—
68C3×#125C2×D318yescomplemented
69C3×#125C22×C318yescomplemented
70C3×#124—316no—
71C3×#123—332no—
72C3×C22⋊Q8—332no—
73C3×#121—332no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—34no—
77C2×C3×C4⋊C4C2×D314yes—
78C2×C3×C4⋊C4C2×D312yes—
79C2×C3×C4⋊C4C22×C312yes—
80C2×C3×C4⋊C4C22×C314yes—
81C2×C3×C22⋊C4—34no—
82C2×C3×C22⋊C4—34no—
83C2×C3×C22⋊C4—34no—
84C2×C3×C42—21no—
85C2×C3×C42C2×D311yescharacteristic
86C2×C3×C42C22×C311yescharacteristic
87C2×(C2×D3)⋊C4C22×C314yes—
88C2×(C2×D3)⋊C4C22×C314yes—
89C2×(C2×D3)⋊C4C22×C314yes—
90C2×Dic3⋊C4C22×C314yes—
91#896C22×C3132yescomplemented
92#895C22×C3132yescomplemented
93#894C22×C3132yescomplemented
94#875C22×C3116yescomplemented
95C23×C32C2411yescharacteristic
96C22×C3×D3—28no—
97C22×C3×D3—28no—
98C22×C3×D3—28no—
99C22×C3×D3C2418yes—
100C2×C32×C4—22no—
101C2×C32×C4—24no—
102C2×C32×C4—28no—
103C2×C32×C4C2418yes—
104C2×C32×C4C2412yes—
105C2×C32×C4C2414yes—
106C2×C3×Dic3—28no—
107C2×C3×Dic3C2418yes—
108C3×D12—48no—
109C3×D12—416no—
110C3×C4×D3—416no—
111C3×Dic6—48no—
112#562—316no—
113C2×#125C3×D311yescharacteristic, complemented
114C2×#124—32no—
115C2×#123—34no—
116C2×C22⋊Q8—34no—
117C2×#121—34no—
118C24×C3—34no—
119C23×D3C23×C314yes—
120C2×C3×Q8—38no—
121C2×C3×Q8—124no—
122C2×C3×D4—316no—
123C2×C3×D4—38no—
124C2×C3×D4—128no—
125C2×C3×D4—124no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—24no—
129C22×C3×C4—34no—
130C22×C3×C4—316no—
131C22×C3×C4—128no—
132C22×C3×C4C22×D311yescharacteristic
133C22×C3×C4C22×D312yes—
134C22×C3×C4C22×D314yes—
135C22×C3×C4C23×C314yes—
136C22×C3×C4C23×C311yescharacteristic
137C22×C3×C4C23×C312yes—
138C22×Dic3C23×C314yes—
139C2×D12—44no—
140C2×D12—48no—
141C2×D12C23×C318yes—
142C2×D12C23×C3116yes—
143C2×C4×D3—48no—
144C2×C4×D3C23×C3116yes—
145C2×Dic6—44no—
146C2×Dic6C23×C318yes—
147C3×C4⋊C4—216no—
148C3×C4⋊C4—28no—
149C3×C4⋊C4—316no—
150C3×C4⋊C4C22×D3116yes—
151C3×C4⋊C4C22×D318yes—
152C3×C4⋊C4C23×C318yes—
153C3×C4⋊C4C23×C3116yes—
154C3×C22⋊C4—316no—
155C3×C22⋊C4—316no—
156C3×C22⋊C4—316no—
157C3×C42—24no—
158C3×C42C22×D314yes—
159C3×C42C23×C314yes—
160(C2×D3)⋊C4C23×C3116yes—
161(C2×D3)⋊C4C23×C3116yes—
162(C2×D3)⋊C4C23×C3116yes—
163Dic3⋊C4C23×C3116yes—
164C22×C32H(5,𝔽2)14yes—
165C22×C322−1+412yes—
166C22×C32C2511yescharacteristic
167C2×C3×D3—28no—
168C2×C3×D3—28no—
169C2×C3×D3—28no—
170C2×C3×D3—416no—
171C32×C4—22no—
172C32×C4—24no—
173C32×C4—28no—
174C3×Dic3—28no—
175C22×Q8—32no—
176C22×D4—34no—
177C22×D4—32no—
178C23×C4—34no—
179#125C2×C3×D318yescomplemented
180#124—316no—
181#123—332no—
182C22⋊Q8—332no—
183#121—332no—
184C2×C4⋊C4—34no—
185C2×C4⋊C4C2×C3×D314yes—
186C2×C4⋊C4C2×C3×D312yes—
187C2×C22⋊C4—34no—
188C2×C22⋊C4—34no—
189C2×C22⋊C4—34no—
190C2×C42C2×C3×D311yescharacteristic
191C23×C3—21no—
192C23×C3—38no—
193C23×C3—68no—
194C23×C3—68no—
195C23×C3—68no—
196C23×C3C23×D311yescharacteristic
197C23×C3C24×C311yescharacteristic
198C22×D3—28no—
199C22×D3—28no—
200C22×D3—28no—
201C22×D3C24×C318yes—
202C3×Q8—128no—
203C3×D4—1216no—
204C3×D4—128no—
205C2×C3×C4—24no—
206C2×C3×C4—22no—
207C2×C3×C4—28no—
208C2×C3×C4—24no—
209C2×C3×C4—24no—
210C2×C3×C4—28no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—24no—
214C2×C3×C4—28no—
215C2×C3×C4—38no—
216C2×C3×C4—48no—
217C2×C3×C4—44no—
218C2×C3×C4—68no—
219C2×C3×C4—1216no—
220C2×C3×C4C23×D312yes—
221C2×C3×C4C23×D314yes—
222C2×C3×C4C23×D318yes—
223C2×C3×C4C24×C318yes—
224C2×C3×C4C24×C312yes—
225C2×C3×C4C24×C314yes—
226C2×Dic3—28no—
227C2×Dic3C24×C318yes—
228D12—48no—
229D12—416no—
230C4×D3—416no—
231Dic6—48no—
232C2×C32#56214yescomplemented
233C2×C32C2×H(5,𝔽2)12yes—
234C2×C32C2×2−1+411yescharacteristic
235C3×D3—48no—
236C24—34no—
237C2×Q8—38no—
238C2×Q8—124no—
239C2×D4—316no—
240C2×D4—38no—
241C2×D4—128no—
242C2×D4—124no—
243C22×C4—34no—
244C22×C4—316no—
245C22×C4—128no—
246C22×C4C22×C3×D311yescharacteristic
247C22×C4C22×C3×D312yes—
248C22×C4C22×C3×D314yes—
249C4⋊C4—316no—
250C4⋊C4C22×C3×D3116yes—
251C4⋊C4C22×C3×D318yes—
252C22⋊C4—316no—
253C22⋊C4—316no—
254C22⋊C4—316no—
255C42C22×C3×D314yes—
256C22×C3—22no—
257C22×C3—24no—
258C22×C3—21no—
259C22×C3—68no—
260C22×C3—68no—
261C22×C3—68no—
262C22×C3—1216no—
263C22×C3#100712yes—
264C22×C3#100914yes—
265C22×C3C3×H(5,𝔽2)14yes—
266C22×C3C3×2−1+412yes—
267C22×C3C24×D311yescharacteristic
268C22×C3C25×C311yescharacteristic
269C2×D3—28no—
270C2×D3—28no—
271C2×D3—28no—
272C2×D3—416no—
273C3×C4—24no—
274C3×C4—22no—
275C3×C4—28no—
276C3×C4—24no—
277C3×C4—24no—
278C3×C4—28no—
279C3×C4—22no—
280C3×C4—48no—
281C3×C4—44no—
282C3×C4—68no—
283Dic3—28no—
284C32C2×#56211yescharacteristic, complemented, Sylow
285C23—38no—
286C23—68no—
287C23—68no—
288C23—68no—
289C23C23×C3×D311yescharacteristic
290Q8—128no—
291D4—1216no—
292D4—128no—
293C2×C4—22no—
294C2×C4—28no—
295C2×C4—24no—
296C2×C4—38no—
297C2×C4—68no—
298C2×C4—1216no—
299C2×C4C23×C3×D312yes—
300C2×C4C23×C3×D314yes—
301C2×C4C23×C3×D318yes—
302C2×C3—21no—
303C2×C3—22no—
304C2×C3—24no—
305C2×C3—128no—
306C2×C3#570014yesdirect product factor
307C2×C3C3×#56214yescomplemented
308C2×C3C2×#100711yescharacteristic
309C2×C3C2×#100912yes—
310C2×C3C2×C3×H(5,𝔽2)12yes—
311C2×C3C2×C3×2−1+411yescharacteristic
312D3—48no—
313C22—68no—
314C22—68no—
315C22—68no—
316C22—1216no—
317C22C3×#100712yes—
318C22C3×#100914yes—
319C22C24×C3×D311yescharacteristic
320C4—22no—
321C4—28no—
322C4—24no—
323C4—68no—
324C3—21no—
325C3C2×#570011yescharacteristic, direct product factor
326C3C2×C3×#56211yescharacteristic, complemented
327C2—128no—
328C2C3×#570014yesdirect product factor
329C2C2×C3×#100711yescharacteristic
330C2C2×C3×#100912yes—
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×#125C2
socleC23×C32C24
hypercenterC2×C3×#125D3
hypocenterC3C2×C3×#562
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×#5700C2direct (N × H)
3. C2×C3×#896C2semidirect
4. C2×C3×#895C2semidirect
5. C2×C3×#894C2semidirect
6. C2×C3×#875C2semidirect
8. C2×#5700C3direct (N × H)
13. C32×#125C22semidirect
21. C3×#896C22semidirect
22. C3×#895C22semidirect
23. C3×#894C22semidirect
24. C3×#875C22semidirect
27. C2×C3×#125D3semidirect
28. C2×C3×#125C2×C3semidirect
33. #5700C2×C3direct (N × H)
34. C2×#896C2×C3semidirect
35. C2×#895C2×C3semidirect
36. C2×#894C2×C3semidirect
37. C2×#875C2×C3semidirect
68. C3×#125C2×D3semidirect
69. C3×#125C22×C3semidirect
91. #896C22×C3semidirect
92. #895C22×C3semidirect
93. #894C22×C3semidirect
94. #875C22×C3semidirect
113. C2×#125C3×D3semidirect
179. #125C2×C3×D3semidirect
232. C2×C32#562semidirect
284. C32C2×#562semidirect
306. C2×C3#5700direct (N × H)
307. C2×C3C3×#562semidirect
325. C3C2×#5700direct (N × H)
326. C3C2×C3×#562semidirect
328. C2C3×#5700direct (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×#57002
C2×C3×#8962
C2×C3×#8952
C2×C3×#8942
C2×C3×#8752
C2×C3×#5623
C2×#57003