Finite Group Databasebeta

← C2×C3×#5510 #11901135

Subgroups — C2×C3×#5510 #11901135

Order 1152.

3489 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#235C212yesmaximal, complemented
2C22×C32×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#930C214yesmaximal, complemented
4C3×#5510C2116yesmaximal, direct product factor
5C2×C3×#897C214yesmaximal, complemented
6C2×C3×#895C214yesmaximal, complemented
7C2×C3×#537—31nomaximal
8C2×#5510C311yesmaximal, characteristic, direct product factor
9C23×C32×C4C2212yes—
10C23×C32×C4C2211yescharacteristic
11C22×C3×#67C2214yes—
12C2×C3×#235—216no—
13C2×C3×#235C22116yescomplemented
14C22×C3×D12C2212yes—
15C22×C3×D12C2214yes—
16C22×C3×C4×D3C2214yes—
17C22×C3×Dic6C2212yes—
18C2×C32×C4⋊C4C2214yes—
19C2×C32×C4⋊C4C2218yescomplemented
20C3×#930C22132yescomplemented
21C2×C3×(C2×D3)⋊C4C2218yes—
22C2×C3×C4⋊Dic3C2214yes—
23C3×#897C22132yescomplemented
24C3×#895C22132yescomplemented
25C22×C3×#41—32no—
26C22×#235C2×C312yescomplemented
27C3×#537—316no—
28C2×C3×C22⋊Q8—34no—
29C2×C3×#121—34no—
30C2×C3×#121—34no—
31C22×C3×C4⋊C4—21no—
32C22×C3×C4⋊C4D311yescharacteristic, complemented
33C22×C3×C4⋊C4C2×C311yescharacteristic, complemented
34C2×#930C2×C314yescomplemented
35#5510C2×C3116yesdirect product factor
36C2×#897C2×C314yescomplemented
37C2×#895C2×C314yescomplemented
38C24×C32C2311yescharacteristic
39C23×C3×D3C2314yes—
40C22×C32×C4—44no—
41C22×C32×C4D418yescomplemented
42C22×C32×C4C2314yes—
43C22×C32×C4C2318yes—
44C22×C32×C4C2312yes—
45C22×C32×C4C2314yes—
46C2×C3×#67—216no—
47C2×C3×#67C23116yes—
48C22×C3×Dic3C2314yes—
49C3×#235—264no—
50C2×C3×D12—28no—
51C2×C3×D12—48no—
52C2×C3×D12C2318yes—
53C2×C3×D12C23116yes—
54C2×C3×C4×D3—216no—
55C2×C3×C4×D3C23116yes—
56C2×C3×Dic6—28no—
57C2×C3×Dic6C2318yes—
58C32×C4⋊C4C23116yes—
59C3×(C2×D3)⋊C4C23132yes—
60C3×C4⋊Dic3C23116yes—
61C2×#537—31noSylow
62C2×C3×#41—316no—
63C2×C3×#41—616no—
64C22×C3×Q8—32no—
65C22×C3×D4—34no—
66C22×C3×D4—32no—
67C22×C3×D4—34no—
68C23×C3×C4—21no—
69C23×C3×C4—22no—
70C23×C3×C4—34no—
71C23×C3×C4C2×D312yes—
72C23×C3×C4C2×D311yescharacteristic
73C23×C3×C4C22×C312yes—
74C23×C3×C4C22×C311yescharacteristic
75C22×#67C22×C314yes—
76C2×#235—216no—
77C2×#235C22×C3116yescomplemented
78C22×D12C22×C312yes—
79C22×D12C22×C314yes—
80C22×C4×D3C22×C314yes—
81C22×Dic6C22×C312yes—
82C3×C22⋊Q8—332no—
83C3×#121—332no—
84C3×#121—332no—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4—28no—
87C2×C3×C4⋊C4—34no—
88C2×C3×C4⋊C4C2×D314yes—
89C2×C3×C4⋊C4C2×D318yescomplemented
90C2×C3×C4⋊C4C22×C314yes—
91C2×C3×C4⋊C4C22×C318yescomplemented
92C2×C3×C22⋊C4—38no—
93#930C22×C3132yescomplemented
94C2×(C2×D3)⋊C4C22×C318yes—
95C2×C4⋊Dic3C22×C314yes—
96#897C22×C3132yescomplemented
97#895C22×C3132yescomplemented
98C23×C32—22no—
99C23×C32—22no—
100C23×C32C2×D414yes—
101C23×C32C2412yes—
102C23×C32C2411yescharacteristic
103C22×C3×D3—28no—
104C22×C3×D3—28no—
105C22×C3×D3—28no—
106C22×C3×D3C2418yes—
107C2×C32×C4—28no—
108C2×C32×C4—48no—
109C2×C32×C4C2×D418yes—
110C2×C32×C4C2×D4116yescomplemented
111C2×C32×C4C2418yes—
112C2×C32×C4C2414yes—
113C3×#67—232no—
114C2×C3×Dic3—28no—
115C2×C3×Dic3C2418yes—
116C3×D12—216no—
117C3×D12—416no—
118C3×C4×D3—232no—
119C3×Dic6—216no—
120C22×#41—32no—
121#537—316no—
122C2×C22⋊Q8—34no—
123C2×#121—34no—
124C2×#121—34no—
125C22×C4⋊C4C3×D311yescharacteristic, complemented
126C24×C3—21no—
127C24×C3—34no—
128C24×C3C22×D311yescharacteristic
129C24×C3C23×C311yescharacteristic
130C23×D3C23×C314yes—
131C3×#41—664no—
132C2×C3×Q8—38no—
133C2×C3×Q8—68no—
134C2×C3×D4—316no—
135C2×C3×D4—38no—
136C2×C3×D4—316no—
137C2×C3×D4—616no—
138C2×C3×D4—68no—
139C2×C3×D4—128no—
140C22×C3×C4—28no—
141C22×C3×C4—22no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—28no—
145C22×C3×C4—34no—
146C22×C3×C4—316no—
147C22×C3×C4—44no—
148C22×C3×C4—44no—
149C22×C3×C4—48no—
150C22×C3×C4—616no—
151C22×C3×C4D1218yescomplemented
152C22×C3×C4C3×D418yescomplemented
153C22×C3×C4C22×D314yes—
154C22×C3×C4C22×D318yes—
155C22×C3×C4C22×D312yes—
156C22×C3×C4C22×D314yes—
157C22×C3×C4C23×C314yes—
158C22×C3×C4C23×C318yes—
159C22×C3×C4C23×C312yes—
160C22×C3×C4C23×C314yes—
161C2×#67—216no—
162C2×#67C23×C3116yes—
163C22×Dic3C23×C314yes—
164#235—264no—
165C2×D12—28no—
166C2×D12—48no—
167C2×D12C23×C318yes—
168C2×D12C23×C3116yes—
169C2×C4×D3—216no—
170C2×C4×D3C23×C3116yes—
171C2×Dic6—28no—
172C2×Dic6C23×C318yes—
173C3×C4⋊C4—216no—
174C3×C4⋊C4—316no—
175C3×C4⋊C4C22×D3116yes—
176C3×C4⋊C4C23×C3116yes—
177C3×C22⋊C4—332no—
178(C2×D3)⋊C4C23×C3132yes—
179C4⋊Dic3C23×C3116yes—
180C22×C32—28no—
181C22×C32—22no—
182C22×C32—22no—
183C22×C32C22×D414yes—
184C22×C32C22×D412yes—
185C22×C32H(5,𝔽2)12yes—
186C22×C322−1+412yes—
187C22×C32C2511yescharacteristic
188C2×C3×D3—28no—
189C2×C3×D3—28no—
190C2×C3×D3—28no—
191C2×C3×D3—416no—
192C32×C4—28no—
193C32×C4C22×D418yes—
194C3×Dic3—28no—
195C2×#41—316no—
196C2×#41—616no—
197C22×Q8—32no—
198C22×D4—34no—
199C22×D4—32no—
200C22×D4—34no—
201C23×C4—34no—
202C23×C4C2×C3×D312yes—
203C23×C4C2×C3×D311yescharacteristic
204C22⋊Q8—332no—
205#121—332no—
206#121—332no—
207C2×C4⋊C4—34no—
208C2×C4⋊C4C2×C3×D314yes—
209C2×C4⋊C4C2×C3×D318yescomplemented
210C2×C22⋊C4—38no—
211C23×C3—24no—
212C23×C3—24no—
213C23×C3—22no—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—21no—
219C23×C3—24no—
220C23×C3—38no—
221C23×C3—68no—
222C23×C3—68no—
223C23×C3—68no—
224C23×C3C2×D1214yes—
225C23×C3C2×C3×D414yes—
226C23×C3C23×D312yes—
227C23×C3C23×D311yescharacteristic
228C23×C3C24×C312yes—
229C23×C3C24×C311yescharacteristic
230C22×D3—28no—
231C22×D3—28no—
232C22×D3—28no—
233C22×D3C24×C318yes—
234C3×Q8—616no—
235C3×D4—632no—
236C3×D4—616no—
237C3×D4—1216no—
238C2×C3×C4—28no—
239C2×C3×C4—28no—
240C2×C3×C4—24no—
241C2×C3×C4—28no—
242C2×C3×C4—28no—
243C2×C3×C4—216no—
244C2×C3×C4—38no—
245C2×C3×C4—48no—
246C2×C3×C4—48no—
247C2×C3×C4—48no—
248C2×C3×C4—416no—
249C2×C3×C4—68no—
250C2×C3×C4—632no—
251C2×C3×C4C2×D1218yes—
252C2×C3×C4C2×D12116yescomplemented
253C2×C3×C4C2×C3×D418yes—
254C2×C3×C4C2×C3×D4116yescomplemented
255C2×C3×C4C23×D318yes—
256C2×C3×C4C23×D314yes—
257C2×C3×C4C24×C318yes—
258C2×C3×C4C24×C314yes—
259#67—232no—
260C2×Dic3—28no—
261C2×Dic3C24×C318yes—
262D12—216no—
263D12—416no—
264C4×D3—232no—
265Dic6—216no—
266C2×C32—24no—
267C2×C32#53714yescomplemented
268C2×C32C23×D411yescharacteristic
269C2×C32C2×H(5,𝔽2)11yescharacteristic
270C2×C32C2×2−1+411yescharacteristic
271C3×D3—48no—
272C24—34no—
273C24C22×C3×D311yescharacteristic
274#41—664no—
275C2×Q8—38no—
276C2×Q8—68no—
277C2×D4—316no—
278C2×D4—38no—
279C2×D4—316no—
280C2×D4—616no—
281C2×D4—68no—
282C2×D4—128no—
283C22×C4—34no—
284C22×C4—316no—
285C22×C4—44no—
286C22×C4—616no—
287C22×C4C3×D1218yescomplemented
288C22×C4C22×C3×D314yes—
289C22×C4C22×C3×D318yes—
290C22×C4C22×C3×D312yes—
291C22×C4C22×C3×D314yes—
292C4⋊C4—316no—
293C4⋊C4C22×C3×D3116yes—
294C22⋊C4—332no—
295C22×C3—24no—
296C22×C3—216no—
297C22×C3—24no—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—28no—
303C22×C3—28no—
304C22×C3—24no—
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—22no—
308C22×C3—21no—
309C22×C3—68no—
310C22×C3—68no—
311C22×C3—68no—
312C22×C3—1216no—
313C22×C3C22×D1214yes—
314C22×C3C22×D1212yes—
315C22×C3#100412yes—
316C22×C3#100712yes—
317C22×C3C22×C3×D414yes—
318C22×C3C22×C3×D412yes—
319C22×C3C3×H(5,𝔽2)12yes—
320C22×C3C3×2−1+412yes—
321C22×C3C24×D311yescharacteristic
322C22×C3C25×C311yescharacteristic
323C2×D3—28no—
324C2×D3—28no—
325C2×D3—28no—
326C2×D3—416no—
327C3×C4—28no—
328C3×C4—28no—
329C3×C4—28no—
330C3×C4—48no—
331C3×C4—68no—
332C3×C4C22×D1218yes—
333C3×C4C22×C3×D418yes—
334Dic3—28no—
335C32C2×#53711yescharacteristic, complemented, Sylow
336C23—22no—
337C23—22no—
338C23—38no—
339C23—68no—
340C23—68no—
341C23—68no—
342C23C2×C3×D1214yes—
343C23C23×C3×D312yes—
344C23C23×C3×D311yescharacteristic
345Q8—616no—
346D4—632no—
347D4—616no—
348D4—1216no—
349C2×C4—28no—
350C2×C4—38no—
351C2×C4—48no—
352C2×C4—68no—
353C2×C4—632no—
354C2×C4C2×C3×D1218yes—
355C2×C4C2×C3×D12116yescomplemented
356C2×C4C23×C3×D318yes—
357C2×C4C23×C3×D314yes—
358C2×C3—28no—
359C2×C3—24no—
360C2×C3—24no—
361C2×C3—24no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—128no—
366C2×C3#551014yesdirect product factor
367C2×C3C3×#53714yescomplemented
368C2×C3C23×D1211yescharacteristic
369C2×C3C2×#100411yescharacteristic
370C2×C3C2×#100711yescharacteristic
371C2×C3C23×C3×D411yescharacteristic
372C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
373C2×C3C2×C3×2−1+411yescharacteristic
374D3—48no—
375C22—28no—
376C22—22no—
377C22—22no—
378C22—68no—
379C22—68no—
380C22—68no—
381C22—1216no—
382C22C22×C3×D1214yes—
383C22C22×C3×D1212yes—
384C22C3×#100412yes—
385C22C3×#100712yes—
386C22C24×C3×D311yescharacteristic
387C4—28no—
388C4—68no—
389C4C22×C3×D1218yes—
390C3—21no—
391C3C2×#551011yescharacteristic, direct product factor
392C3C2×C3×#53711yescharacteristic, complemented
393C2—24no—
394C2—128no—
395C2C3×#551014yesdirect product factor
396C2C23×C3×D1211yescharacteristic
397C2C2×C3×#100411yescharacteristic
398C2C2×C3×#100711yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC22×C32×C4⋊C4C2
socleC23×C32C24
hypercenterC22×C3×C4⋊C4D3
hypocenterC3C2×C3×#537
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#235C2semidirect
2. C22×C32×C4⋊C4C2semidirect
3. C2×C3×#930C2semidirect
4. C3×#5510C2direct (N × H)
5. C2×C3×#897C2semidirect
6. C2×C3×#895C2semidirect
8. C2×#5510C3direct (N × H)
13. C2×C3×#235C22semidirect
19. C2×C32×C4⋊C4C22semidirect
20. C3×#930C22semidirect
23. C3×#897C22semidirect
24. C3×#895C22semidirect
26. C22×#235C2×C3semidirect
32. C22×C3×C4⋊C4D3semidirect
33. C22×C3×C4⋊C4C2×C3semidirect
34. C2×#930C2×C3semidirect
35. #5510C2×C3direct (N × H)
36. C2×#897C2×C3semidirect
37. C2×#895C2×C3semidirect
41. C22×C32×C4D4semidirect
77. C2×#235C22×C3semidirect
89. C2×C3×C4⋊C4C2×D3semidirect
91. C2×C3×C4⋊C4C22×C3semidirect
93. #930C22×C3semidirect
96. #897C22×C3semidirect
97. #895C22×C3semidirect
110. C2×C32×C4C2×D4semidirect
125. C22×C4⋊C4C3×D3semidirect
151. C22×C3×C4D12semidirect
152. C22×C3×C4C3×D4semidirect
209. C2×C4⋊C4C2×C3×D3semidirect
252. C2×C3×C4C2×D12semidirect
254. C2×C3×C4C2×C3×D4semidirect
267. C2×C32#537semidirect
287. C22×C4C3×D12semidirect
335. C32C2×#537semidirect
355. C2×C4C2×C3×D12semidirect
366. C2×C3#5510direct (N × H)
367. C2×C3C3×#537semidirect
391. C3C2×#5510direct (N × H)
392. C3C2×C3×#537semidirect
395. C2C3×#5510direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2352
C22×C32×C4⋊C42
C2×C3×#9302
C3×#55102
C2×C3×#8972
C2×C3×#8952
C2×C3×#5373
C2×#55103