Finite Group Databasebeta

← C3×#85881 #11904479

Subgroups — C3×#85881 #11904479

Order 1152.

3117 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×C3×#1007C211yesmaximal, characteristic, complemented
2C22×C3×D3×Q8C211yesmaximal, characteristic, complemented
3C32×#519C211yesmaximal, characteristic, complemented
4C3×#5576C218yesmaximal, complemented
5C3×Q8×D12C218yesmaximal, complemented
6C3×#5526C216yesmaximal, complemented
7C3×#5524C216yesmaximal, complemented
8C3×#3486—31nomaximal
9#85881C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2211yescharacteristic
11C3×#1007—28no—
12C2×C3×#239C2214yescomplemented
13C2×C3×D3×Q8—28no—
14C2×C3×D3×Q8C2218yes—
15C2×C3×D3×Q8C2214yescomplemented
16C2×C3×#235C2216yes—
17C22×C3×C4×D3C2213yes—
18C22×C3×Dic6C2213yes—
19C32×C4×Q8C2218yescomplemented
20C32×#117C2216yes—
21C3×#897C22112yes—
22C3×#897C22124yescomplemented
23C3×#895C22112yes—
24C3×#886C22112yes—
25C3×#882C2214yes—
26C3×#875C22112yescomplemented
27C3×C4×D12C22124yes—
28C3×#869C22112yescomplemented
29C2×C3×2−1+4—31no—
30C23×C3×Q8—31no—
31C2×#1007C2×C311yescharacteristic, complemented
32C22×D3×Q8C2×C311yescharacteristic, complemented
33C3×D4×Q8—38no—
34C3×#548—38no—
35C3×#536—36no—
36C3×#525—36no—
37C3×#519—21no—
38C3×#519D311yescharacteristic, complemented
39C3×#519C2×C311yescharacteristic, complemented
40#5576C2×C318yescomplemented
41Q8×D12C2×C318yescomplemented
42#5526C2×C316yescomplemented
43#5524C2×C316yescomplemented
44C23×C3×D3C2311yescharacteristic
45C2×C32×Q8D418yescomplemented
46C2×C32×Q8C2314yes—
47C22×C32×C4C2313yes—
48C2×C3×#67C2314yes—
49C22×C3×Dic3C2313yes—
50C3×#239—216no—
51C3×D3×Q8—232no—
52C3×D3×Q8—216no—
53C3×#235—224no—
54C2×C3×D12C2318yes—
55C2×C3×D12C2316yes—
56C2×C3×C4×D3—212no—
57C2×C3×C4×D3C23112yes—
58C2×C3×C4×D3C23112yes—
59C2×C3×Dic6—212no—
60C2×C3×Dic6C2316yes—
61C2×C3×Dic6C23112yes—
62C32×C4⋊C4C23112yes—
63C32×C22⋊C4C2314yes—
64C32×C42C23112yes—
65C3×(C2×D3)⋊C4C2318yes—
66C3×(C2×D3)⋊C4C23124yes—
67C3×C4⋊Dic3C23124yes—
68#3486—31noSylow
69C3×2−1+4—68no—
70C2×C3×#41—34no—
71C2×C3×#41—36no—
72C22×C3×Q8—21no—
73C22×C3×Q8—38no—
74C22×C3×Q8—34no—
75C22×C3×Q8—33no—
76C22×C3×Q8—68no—
77C22×C3×Q8C2×D311yescharacteristic
78C22×C3×Q8C22×C311yescharacteristic
79C23×C3×C4—33no—
80#1007—28no—
81C2×#239C22×C314yescomplemented
82C2×D3×Q8—28no—
83C2×D3×Q8C22×C318yes—
84C2×D3×Q8C22×C314yescomplemented
85C2×#235C22×C316yes—
86C22×C4×D3C22×C313yes—
87C22×Dic6C22×C313yes—
88C3×C4⋊Q8—312no—
89C3×#124—312no—
90C3×#123—312no—
91C3×C22⋊Q8—312no—
92C3×C22⋊Q8—324no—
93C3×#121—312no—
94C3×#120—34no—
95C3×C4×Q8—28no—
96C3×C4×Q8C2×D318yescomplemented
97C3×C4×Q8C22×C318yescomplemented
98C3×C4×D4—324no—
99C3×#117—26no—
100C3×#117C2×D316yes—
101C3×#117C22×C316yes—
102#897C22×C3112yes—
103#897C22×C3124yescomplemented
104#895C22×C3112yes—
105#886C22×C3112yes—
106#882C22×C314yes—
107#875C22×C3112yescomplemented
108C4×D12C22×C3124yes—
109#869C22×C3112yescomplemented
110C23×C32C2411yescharacteristic
111C22×C3×D3—22no—
112C22×C3×D3—24no—
113C22×C3×D3C2414yes—
114C22×C3×D3C2412yes—
115C32×Q8C2×D4116yes—
116C2×C32×C4C2×D4112yes—
117C2×C32×C4C2418yes—
118C2×C32×C4C2416yes—
119C3×#67—28no—
120C2×C3×Dic3—26no—
121C2×C3×Dic3C2414yes—
122C2×C3×Dic3C2416yes—
123C3×D12—216no—
124C3×D12—212no—
125C3×C4×D3—224no—
126C3×C4×D3—224no—
127C3×Dic6—224no—
128C3×Dic6—212no—
129C2×2−1+4—31no—
130C23×Q8—31no—
131D4×Q8—38no—
132#548—38no—
133#536—36no—
134#525—36no—
135#519C3×D311yescharacteristic, complemented
136C24×C3—31no—
137C23×D3C23×C311yescharacteristic
138C3×#41—616no—
139C3×#41—624no—
140C2×C3×Q8—28no—
141C2×C3×Q8—24no—
142C2×C3×Q8—312no—
143C2×C3×Q8—36no—
144C2×C3×Q8—632no—
145C2×C3×Q8—616no—
146C2×C3×Q8—612no—
147C2×C3×Q8D1218yescomplemented
148C2×C3×Q8C3×D418yescomplemented
149C2×C3×Q8C22×D314yes—
150C2×C3×Q8C23×C314yes—
151C2×C3×D4—34no—
152C2×C3×D4—38no—
153C2×C3×D4—36no—
154C22×C3×C4—23no—
155C22×C3×C4—33no—
156C22×C3×C4—312no—
157C22×C3×C4—312no—
158C22×C3×C4—612no—
159C22×C3×C4C22×D313yes—
160C22×C3×C4C23×C313yes—
161C2×#67C23×C314yes—
162C22×Dic3C23×C313yes—
163#239—216no—
164D3×Q8—232no—
165D3×Q8—216no—
166#235—224no—
167C2×D12C23×C318yes—
168C2×D12C23×C316yes—
169C2×C4×D3—212no—
170C2×C4×D3C23×C3112yes—
171C2×C4×D3C23×C3112yes—
172C2×Dic6—212no—
173C2×Dic6C23×C316yes—
174C2×Dic6C23×C3112yes—
175C3×C4⋊C4—212no—
176C3×C4⋊C4—324no—
177C3×C4⋊C4C22×D3112yes—
178C3×C4⋊C4C23×C3112yes—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—38no—
181C3×C22⋊C4—324no—
182C3×C22⋊C4C22×D314yes—
183C3×C22⋊C4C23×C314yes—
184C3×C42—212no—
185C3×C42C22×D3112yes—
186C3×C42C23×C3112yes—
187(C2×D3)⋊C4C23×C318yes—
188(C2×D3)⋊C4C23×C3124yes—
189C4⋊Dic3C23×C3124yes—
190C22×C32—22no—
191C22×C32C22×D412yes—
192C22×C32C2511yescharacteristic
193C2×C3×D3—28no—
194C2×C3×D3—22no—
195C2×C3×D3—22no—
196C2×C3×D3—24no—
197C2×C3×D3—28no—
198C2×C3×D32−1+414yescomplemented
199C32×C4—28no—
200C32×C4C22×D4112yes—
201C3×Dic3—26no—
202C3×Dic3—24no—
2032−1+4—68no—
204C2×#41—34no—
205C2×#41—36no—
206C22×Q8—38no—
207C22×Q8—34no—
208C22×Q8—33no—
209C22×Q8—68no—
210C22×Q8C2×C3×D311yescharacteristic
211C23×C4—33no—
212C4⋊Q8—312no—
213#124—312no—
214#123—312no—
215C22⋊Q8—312no—
216C22⋊Q8—324no—
217#121—312no—
218#120—34no—
219C4×Q8C2×C3×D318yescomplemented
220C4×D4—324no—
221#117C2×C3×D316yes—
222C23×C3—21no—
223C23×C3—32no—
224C23×C3—34no—
225C23×C3—64no—
226C23×C3—62no—
227C23×C3C23×D311yescharacteristic
228C23×C3C24×C311yescharacteristic
229C22×D3—22no—
230C22×D3—24no—
231C22×D3C24×C314yes—
232C22×D3C24×C312yes—
233C3×Q8—216no—
234C3×Q8—624no—
235C3×Q8—612no—
236C3×Q8C2×D12116yes—
237C3×Q8C2×C3×D4116yes—
238C3×D4—616no—
239C3×D4—68no—
240C3×D4—612no—
241C2×C3×C4—26no—
242C2×C3×C4—28no—
243C2×C3×C4—212no—
244C2×C3×C4—36no—
245C2×C3×C4—34no—
246C2×C3×C4—66no—
247C2×C3×C4—624no—
248C2×C3×C4—624no—
249C2×C3×C4C2×D12112yes—
250C2×C3×C4C2×C3×D4112yes—
251C2×C3×C4C23×D318yes—
252C2×C3×C4C23×D316yes—
253C2×C3×C4C24×C318yes—
254C2×C3×C4C24×C316yes—
255#67—28no—
256C2×Dic3—26no—
257C2×Dic3C24×C314yes—
258C2×Dic3C24×C316yes—
259D12—216no—
260D12—212no—
261C4×D3—224no—
262C4×D3—224no—
263Dic6—224no—
264Dic6—212no—
265C2×C32—22no—
266C2×C32C23×D411yescharacteristic
267C2×C32C2×2−1+412yes—
268C3×D3—24no—
269C3×D3—44no—
270C24—31no—
271#41—616no—
272#41—624no—
273C2×Q8—312no—
274C2×Q8—36no—
275C2×Q8—632no—
276C2×Q8—616no—
277C2×Q8—612no—
278C2×Q8C3×D1218yescomplemented
279C2×Q8C22×C3×D314yes—
280C2×D4—34no—
281C2×D4—38no—
282C2×D4—36no—
283C22×C4—33no—
284C22×C4—312no—
285C22×C4—312no—
286C22×C4—612no—
287C22×C4C22×C3×D313yes—
288C4⋊C4—324no—
289C4⋊C4C22×C3×D3112yes—
290C22⋊C4—38no—
291C22⋊C4—324no—
292C22⋊C4C22×C3×D314yes—
293C42C22×C3×D3112yes—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3—21no—
298C22×C3—34no—
299C22×C3—42no—
300C22×C3—68no—
301C22×C3—62no—
302C22×C3—62no—
303C22×C3—64no—
304C22×C3—68no—
305C22×C3C22×D1212yes—
306C22×C3C22×C3×D412yes—
307C22×C3C24×D311yescharacteristic
308C22×C3C25×C311yescharacteristic
309C2×D3—28no—
310C2×D3—22no—
311C2×D3—22no—
312C2×D3—24no—
313C2×D3—28no—
314C2×D3C3×2−1+414yescomplemented
315C3×C4—28no—
316C3×C4—28no—
317C3×C4—212no—
318C3×C4—48no—
319C3×C4—64no—
320C3×C4—66no—
321C3×C4C22×D12112yes—
322C3×C4C22×C3×D4112yes—
323Dic3—26no—
324Dic3—24no—
325C32#348611yescharacteristic, complemented, Sylow
326C23—32no—
327C23—34no—
328C23—64no—
329C23—62no—
330C23C23×C3×D311yescharacteristic
331Q8—624no—
332Q8—612no—
333Q8C2×C3×D12116yes—
334D4—616no—
335D4—68no—
336D4—612no—
337C2×C4—36no—
338C2×C4—34no—
339C2×C4—66no—
340C2×C4—624no—
341C2×C4—624no—
342C2×C4C2×C3×D12112yes—
343C2×C4C23×C3×D318yes—
344C2×C4C23×C3×D316yes—
345C2×C3—22no—
346C2×C3—22no—
347C2×C3—21no—
348C2×C3—22no—
349C2×C3—42no—
350C2×C3—64no—
351C2×C3—124no—
352C2×C3C23×D1211yescharacteristic
353C2×C3D3×2−1+412yes—
354C2×C3C23×C3×D411yescharacteristic
355C2×C3C2×C3×2−1+412yes—
356D3—24no—
357D3—44no—
358C22—22no—
359C22—34no—
360C22—68no—
361C22—62no—
362C22—62no—
363C22—64no—
364C22—68no—
365C22C22×C3×D1212yes—
366C22C24×C3×D311yescharacteristic
367C4—28no—
368C4—64no—
369C4—66no—
370C4C22×C3×D12112yes—
371C3—21no—
372C3#8588111yescharacteristic, direct product factor
373C3C3×#348611yescharacteristic, complemented
374C2—22no—
375C2—64no—
376C2—124no—
377C2C23×C3×D1211yescharacteristic
378C2C3×D3×2−1+412yes—
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#519C2
socleC22×C32C25
hypercenterC3×#519D3
hypocenterC3C3×#3486
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#519

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1007C2semidirect
2. C22×C3×D3×Q8C2semidirect
3. C32×#519C2semidirect
4. C3×#5576C2semidirect
5. C3×Q8×D12C2semidirect
6. C3×#5526C2semidirect
7. C3×#5524C2semidirect
9. #85881C3direct (N × H)
12. C2×C3×#239C22semidirect
15. C2×C3×D3×Q8C22semidirect
19. C32×C4×Q8C22semidirect
22. C3×#897C22semidirect
26. C3×#875C22semidirect
28. C3×#869C22semidirect
31. C2×#1007C2×C3semidirect
32. C22×D3×Q8C2×C3semidirect
38. C3×#519D3semidirect
39. C3×#519C2×C3semidirect
40. #5576C2×C3semidirect
41. Q8×D12C2×C3semidirect
42. #5526C2×C3semidirect
43. #5524C2×C3semidirect
45. C2×C32×Q8D4semidirect
81. C2×#239C22×C3semidirect
84. C2×D3×Q8C22×C3semidirect
96. C3×C4×Q8C2×D3semidirect
97. C3×C4×Q8C22×C3semidirect
103. #897C22×C3semidirect
107. #875C22×C3semidirect
109. #869C22×C3semidirect
135. #519C3×D3semidirect
147. C2×C3×Q8D12semidirect
148. C2×C3×Q8C3×D4semidirect
198. C2×C3×D32−1+4semidirect
219. C4×Q8C2×C3×D3semidirect
278. C2×Q8C3×D12semidirect
314. C2×D3C3×2−1+4semidirect
325. C32#3486semidirect
372. C3#85881direct (N × H)
373. C3C3×#3486semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10072
C22×C3×D3×Q82
C32×#5192
C3×#55762
C3×Q8×D122
C3×#55262
C3×#55242
C3×#34863
#858813