Finite Group Databasebeta

← C3×#85846 #11905951

Subgroups — C3×#85846 #11905951

Order 1152.

3651 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1009C212yesmaximal, complemented
2C32×C4×#41C211yesmaximal, characteristic, complemented
3C3×#5577C214yesmaximal, complemented
4C3×D4×D12C2112yesmaximal, complemented
5C3×#5525C216yesmaximal, complemented
6C3×#5477C213yesmaximal, complemented
7C2×C3×#874C213yesmaximal, complemented
8C3×#3498—31nomaximal
9#85846C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C3×#1009—216no—
12C2×C3×#239C2214yescomplemented
13C2×C3×D3×D4C22112yescomplemented
14C2×C3×#235C2216yescomplemented
15C22×C3×D12C2216yescomplemented
16C22×C3×D12C22112yescomplemented
17C32×C4×Q8C2212yescomplemented
18C32×C4×D4C2216yescomplemented
19C32×#117C2213yescomplemented
20C2×C32×C42C2213yescomplemented
21C3×#930C22112yescomplemented
22C3×#895C22124yescomplemented
23C3×#882C22124yescomplemented
24C3×#875C2216yescomplemented
25C3×#874C2216yescomplemented
26C3×#874C22112yescomplemented
27C3×#874C2213yescomplemented
28C3×C4×D12C22112yescomplemented
29C3×#869C2211yescharacteristic, complemented
30C2×C3×H(5,𝔽2)—32no—
31C2×#1009C2×C312yescomplemented
32C3×#550—34no—
33C3×D42—312no—
34C3×#535—36no—
35C3×#532—33no—
36C2×C3×#127—33no—
37C3×C4×#41—21no—
38C3×C4×#41D311yescharacteristic, complemented
39C3×C4×#41C2×C311yescharacteristic, complemented
40#5577C2×C314yescomplemented
41D4×D12C2×C3112yescomplemented
42#5525C2×C316yescomplemented
43#5477C2×C313yescomplemented
44C2×#874C2×C313yescomplemented
45C23×C3×D3C23112yes—
46C32×#41D418yescomplemented
47C2×C32×Q8C2311yescharacteristic
48C2×C32×D4C2313yes—
49C22×C32×C4C2316yes—
50C22×C32×C4C2313yes—
51C2×C3×#67C23112yes—
52C3×#239—216no—
53C3×D3×D4—248no—
54C3×#235—224no—
55C2×C3×D12—224no—
56C2×C3×D12—424no—
57C2×C3×D12C23112yes—
58C2×C3×D12C23112yes—
59C2×C3×D12C2316yes—
60C2×C3×D12C23124yes—
61C2×C3×C4×D3C23112yes—
62C2×C3×Dic6C2312yes—
63C32×C4⋊C4C2316yes—
64C32×C22⋊C4C2316yes—
65C32×C42C2311yescharacteristic
66C32×C42C2313yes—
67C32×C42C2316yes—
68C3×(C2×D3)⋊C4C23124yes—
69C3×C4⋊Dic3C2314yes—
70#3498—31noSylow
71C3×H(5,𝔽2)—616no—
72C2×C3×#41—21no—
73C2×C3×#41—34no—
74C2×C3×#41—36no—
75C2×C3×#41C2×D311yescharacteristic
76C2×C3×#41C22×C311yescharacteristic
77C22×C3×D4—312no—
78C22×C3×D4—36no—
79C22×C3×D4—312no—
80#1009—216no—
81C2×#239C22×C314yescomplemented
82C2×D3×D4C22×C3112yescomplemented
83C2×#235C22×C316yescomplemented
84C22×D12C22×C316yescomplemented
85C22×D12C22×C3112yescomplemented
86C3×C4⋊Q8—31no—
87C3×#127—33no—
88C3×#127—36no—
89C3×#127—312no—
90C3×#124—36no—
91C3×#121—312no—
92C3×#121—324no—
93C3×#120—324no—
94C3×C4×Q8—22no—
95C3×C4×Q8C2×D312yescomplemented
96C3×C4×Q8C22×C312yescomplemented
97C3×C4×D4—26no—
98C3×C4×D4—312no—
99C3×C4×D4C2×D316yescomplemented
100C3×C4×D4C22×C316yescomplemented
101C3×#117—23no—
102C3×#117C2×D313yescomplemented
103C3×#117C22×C313yescomplemented
104C2×C3×C42—23no—
105C2×C3×C42C2×D313yescomplemented
106C2×C3×C42C22×C313yescomplemented
107#930C22×C3112yescomplemented
108#895C22×C3124yescomplemented
109#882C22×C3124yescomplemented
110#875C22×C316yescomplemented
111#874C22×C313yescomplemented
112#874C22×C316yescomplemented
113#874C22×C3112yescomplemented
114C4×D12C22×C3112yescomplemented
115#869C22×C311yescharacteristic, complemented
116C23×C32C2413yes—
117C22×C3×D3—224no—
118C22×C3×D3—424no—
119C22×C3×D3C24112yes—
120C32×Q8C2×D414yescomplemented
121C32×D4C2×D4112yescomplemented
122C2×C32×C4—212no—
123C2×C32×C4C2×D4112yescomplemented
124C2×C32×C4C2416yes—
125C2×C32×C4C2411yescharacteristic
126C2×C32×C4C2413yes—
127C2×C32×C4C2412yes—
128C3×#67—224no—
129C2×C3×Dic3C2414yes—
130C3×D12—212no—
131C3×D12—224no—
132C3×D12—224no—
133C3×D12—424no—
134C3×C4×D3—224no—
135C3×Dic6—24no—
136C2×H(5,𝔽2)—32no—
137#550—34no—
138D42—312no—
139#535—36no—
140#532—33no—
141C2×#127—33no—
142C4×#41C3×D311yescharacteristic, complemented
143C24×C3—312no—
144C23×D3C23×C3112yes—
145C3×#41—28no—
146C3×#41—616no—
147C3×#41—624no—
148C3×#41D1218yescomplemented
149C3×#41C3×D418yescomplemented
150C2×C3×Q8—21no—
151C2×C3×Q8—32no—
152C2×C3×Q8C22×D311yescharacteristic
153C2×C3×Q8C23×C311yescharacteristic
154C2×C3×D4—23no—
155C2×C3×D4—312no—
156C2×C3×D4—312no—
157C2×C3×D4—312no—
158C2×C3×D4—36no—
159C2×C3×D4—324no—
160C2×C3×D4—648no—
161C2×C3×D4—624no—
162C2×C3×D4—1224no—
163C2×C3×D4C22×D313yes—
164C2×C3×D4C23×C313yes—
165C22×C3×C4—23no—
166C22×C3×C4—26no—
167C22×C3×C4—312no—
168C22×C3×C4C22×D316yes—
169C22×C3×C4C22×D313yes—
170C22×C3×C4C23×C316yes—
171C22×C3×C4C23×C313yes—
172C2×#67C23×C3112yes—
173#239—216no—
174D3×D4—248no—
175#235—224no—
176C2×D12—224no—
177C2×D12—424no—
178C2×D12C23×C3112yes—
179C2×D12C23×C316yes—
180C2×D12C23×C3112yes—
181C2×D12C23×C3124yes—
182C2×C4×D3C23×C3112yes—
183C2×Dic6C23×C312yes—
184C3×C4⋊C4—26no—
185C3×C4⋊C4—34no—
186C3×C4⋊C4C22×D316yes—
187C3×C4⋊C4C23×C316yes—
188C3×C22⋊C4—26no—
189C3×C22⋊C4—324no—
190C3×C22⋊C4C22×D316yes—
191C3×C22⋊C4C23×C316yes—
192C3×C42—21no—
193C3×C42—23no—
194C3×C42—26no—
195C3×C42C22×D311yescharacteristic
196C3×C42C22×D313yes—
197C3×C42C22×D316yes—
198C3×C42C23×C311yescharacteristic
199C3×C42C23×C316yes—
200C3×C42C23×C313yes—
201(C2×D3)⋊C4C23×C3124yes—
202C4⋊Dic3C23×C314yes—
203C22×C32—26no—
204C22×C32C22×D416yes—
205C22×C32C2511yescharacteristic
206C2×C3×D3—212no—
207C2×C3×D3—224no—
208C2×C3×D3—824no—
209C32×C4—26no—
210C32×C4C22×D412yes—
211C32×C4C22×D416yes—
212C32×C4H(5,𝔽2)14yescomplemented
213C3×Dic3—24no—
214H(5,𝔽2)—616no—
215C2×#41—34no—
216C2×#41—36no—
217C2×#41C2×C3×D311yescharacteristic
218C22×D4—312no—
219C22×D4—36no—
220C22×D4—312no—
221C4⋊Q8—31no—
222#127—33no—
223#127—36no—
224#127—312no—
225#124—36no—
226#121—312no—
227#121—324no—
228#120—324no—
229C4×Q8C2×C3×D312yescomplemented
230C4×D4—312no—
231C4×D4C2×C3×D316yescomplemented
232#117C2×C3×D313yescomplemented
233C2×C42C2×C3×D313yescomplemented
234C23×C3—23no—
235C23×C3—312no—
236C23×C3—624no—
237C23×C3—1224no—
238C23×C3C23×D313yes—
239C23×C3C24×C313yes—
240C22×D3—224no—
241C22×D3—424no—
242C22×D3C24×C3112yes—
243C3×Q8—24no—
244C3×Q8—64no—
245C3×Q8C2×D1214yescomplemented
246C3×Q8C2×C3×D414yescomplemented
247C3×D4—212no—
248C3×D4—624no—
249C3×D4—624no—
250C3×D4—612no—
251C3×D4—624no—
252C3×D4—1224no—
253C3×D4C2×D12112yescomplemented
254C3×D4C2×C3×D4112yescomplemented
255C2×C3×C4—212no—
256C2×C3×C4—212no—
257C2×C3×C4—21no—
258C2×C3×C4—22no—
259C2×C3×C4—23no—
260C2×C3×C4—26no—
261C2×C3×C4—212no—
262C2×C3×C4—34no—
263C2×C3×C4—412no—
264C2×C3×C4—624no—
265C2×C3×C4C2×D12112yescomplemented
266C2×C3×C4C2×C3×D4112yescomplemented
267C2×C3×C4C23×D316yes—
268C2×C3×C4C23×D311yescharacteristic
269C2×C3×C4C23×D313yes—
270C2×C3×C4C23×D312yes—
271C2×C3×C4C24×C316yes—
272C2×C3×C4C24×C311yescharacteristic
273C2×C3×C4C24×C313yes—
274C2×C3×C4C24×C312yes—
275#67—224no—
276C2×Dic3C24×C314yes—
277D12—212no—
278D12—224no—
279D12—224no—
280D12—424no—
281C4×D3—224no—
282Dic6—24no—
283C2×C32—26no—
284C2×C32C23×D411yescharacteristic
285C2×C32C2×H(5,𝔽2)12yes—
286C3×D3—412no—
287C24—312no—
288#41—616no—
289#41—624no—
290#41C3×D1218yescomplemented
291C2×Q8—32no—
292C2×Q8C22×C3×D311yescharacteristic
293C2×D4—312no—
294C2×D4—312no—
295C2×D4—312no—
296C2×D4—36no—
297C2×D4—324no—
298C2×D4—648no—
299C2×D4—624no—
300C2×D4—1224no—
301C2×D4C22×C3×D313yes—
302C22×C4—312no—
303C22×C4C22×C3×D316yes—
304C22×C4C22×C3×D313yes—
305C4⋊C4—34no—
306C4⋊C4C22×C3×D316yes—
307C22⋊C4—324no—
308C22⋊C4C22×C3×D316yes—
309C42C22×C3×D311yescharacteristic
310C42C22×C3×D313yes—
311C42C22×C3×D316yes—
312C22×C3—26no—
313C22×C3—26no—
314C22×C3—26no—
315C22×C3—21no—
316C22×C3—46no—
317C22×C3—612no—
318C22×C3—624no—
319C22×C3—2424no—
320C22×C3C22×D1216yes—
321C22×C3C22×C3×D416yes—
322C22×C3C24×D311yescharacteristic
323C22×C3C25×C311yescharacteristic
324C2×D3—212no—
325C2×D3—224no—
326C2×D3—824no—
327C3×C4—26no—
328C3×C4—26no—
329C3×C4—22no—
330C3×C4—26no—
331C3×C4—24no—
332C3×C4—46no—
333C3×C4—64no—
334C3×C4C22×D1216yes—
335C3×C4C22×D1212yes—
336C3×C4#100914yescomplemented
337C3×C4C22×C3×D412yes—
338C3×C4C22×C3×D416yes—
339C3×C4C3×H(5,𝔽2)14yescomplemented
340Dic3—24no—
341C32#349811yescharacteristic, complemented, Sylow
342C23—312no—
343C23—624no—
344C23—1224no—
345C23C23×C3×D313yes—
346Q8—64no—
347Q8C2×C3×D1214yescomplemented
348D4—624no—
349D4—624no—
350D4—612no—
351D4—624no—
352D4—1224no—
353D4C2×C3×D12112yescomplemented
354C2×C4—212no—
355C2×C4—34no—
356C2×C4—624no—
357C2×C4C2×C3×D12112yescomplemented
358C2×C4C23×C3×D316yes—
359C2×C4C23×C3×D311yescharacteristic
360C2×C4C23×C3×D313yes—
361C2×C4C23×C3×D312yes—
362C2×C3—26no—
363C2×C3—26no—
364C2×C3—21no—
365C2×C3—22no—
366C2×C3—46no—
367C2×C3—1212no—
368C2×C3C23×D1211yescharacteristic
369C2×C3C2×#100912yes—
370C2×C3C23×C3×D411yescharacteristic
371C2×C3C2×C3×H(5,𝔽2)12yes—
372D3—412no—
373C22—26no—
374C22—612no—
375C22—624no—
376C22—2424no—
377C22C22×C3×D1216yes—
378C22C24×C3×D311yescharacteristic
379C4—26no—
380C4—64no—
381C4C22×C3×D1216yes—
382C4C22×C3×D1212yes—
383C4C3×#100914yescomplemented
384C3—21no—
385C3#8584611yescharacteristic, direct product factor
386C3C3×#349811yescharacteristic, complemented
387C2—26no—
388C2—1212no—
389C2C23×C3×D1211yescharacteristic
390C2C2×C3×#100912yes—
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×C4×#41C2
socleC22×C32C25
hypercenterC3×C4×#41D3
hypocenterC3C3×#3498
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1009C2semidirect
2. C32×C4×#41C2semidirect
3. C3×#5577C2semidirect
4. C3×D4×D12C2semidirect
5. C3×#5525C2semidirect
6. C3×#5477C2semidirect
7. C2×C3×#874C2semidirect
9. #85846C3direct (N × H)
12. C2×C3×#239C22semidirect
13. C2×C3×D3×D4C22semidirect
14. C2×C3×#235C22semidirect
15. C22×C3×D12C22semidirect
16. C22×C3×D12C22semidirect
17. C32×C4×Q8C22semidirect
18. C32×C4×D4C22semidirect
19. C32×#117C22semidirect
20. C2×C32×C42C22semidirect
21. C3×#930C22semidirect
22. C3×#895C22semidirect
23. C3×#882C22semidirect
24. C3×#875C22semidirect
25. C3×#874C22semidirect
26. C3×#874C22semidirect
27. C3×#874C22semidirect
28. C3×C4×D12C22semidirect
29. C3×#869C22semidirect
31. C2×#1009C2×C3semidirect
38. C3×C4×#41D3semidirect
39. C3×C4×#41C2×C3semidirect
40. #5577C2×C3semidirect
41. D4×D12C2×C3semidirect
42. #5525C2×C3semidirect
43. #5477C2×C3semidirect
44. C2×#874C2×C3semidirect
46. C32×#41D4semidirect
81. C2×#239C22×C3semidirect
82. C2×D3×D4C22×C3semidirect
83. C2×#235C22×C3semidirect
84. C22×D12C22×C3semidirect
85. C22×D12C22×C3semidirect
95. C3×C4×Q8C2×D3semidirect
96. C3×C4×Q8C22×C3semidirect
99. C3×C4×D4C2×D3semidirect
100. C3×C4×D4C22×C3semidirect
102. C3×#117C2×D3semidirect
103. C3×#117C22×C3semidirect
105. C2×C3×C42C2×D3semidirect
106. C2×C3×C42C22×C3semidirect
107. #930C22×C3semidirect
108. #895C22×C3semidirect
109. #882C22×C3semidirect
110. #875C22×C3semidirect
111. #874C22×C3semidirect
112. #874C22×C3semidirect
113. #874C22×C3semidirect
114. C4×D12C22×C3semidirect
115. #869C22×C3semidirect
120. C32×Q8C2×D4semidirect
121. C32×D4C2×D4semidirect
123. C2×C32×C4C2×D4semidirect
142. C4×#41C3×D3semidirect
148. C3×#41D12semidirect
149. C3×#41C3×D4semidirect
212. C32×C4H(5,𝔽2)semidirect
229. C4×Q8C2×C3×D3semidirect
231. C4×D4C2×C3×D3semidirect
232. #117C2×C3×D3semidirect
233. C2×C42C2×C3×D3semidirect
245. C3×Q8C2×D12semidirect
246. C3×Q8C2×C3×D4semidirect
253. C3×D4C2×D12semidirect
254. C3×D4C2×C3×D4semidirect
265. C2×C3×C4C2×D12semidirect
266. C2×C3×C4C2×C3×D4semidirect
290. #41C3×D12semidirect
336. C3×C4#1009semidirect
339. C3×C4C3×H(5,𝔽2)semidirect
341. C32#3498semidirect
347. Q8C2×C3×D12semidirect
353. D4C2×C3×D12semidirect
357. C2×C4C2×C3×D12semidirect
383. C4C3×#1009semidirect
385. C3#85846direct (N × H)
386. C3C3×#3498semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10092
C32×C4×#412
C3×#55772
C3×D4×D122
C3×#55252
C3×#54772
C2×C3×#8742
C3×#34983
#858463