Finite Group Databasebeta

← C3×#82563 #11857998

Subgroups — C3×#82563 #11857998

Order 1152.

1703 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1009C211yesmaximal, characteristic
2C32×#520C211yesmaximal, characteristic, complemented
3C2×C3×#948C211yesmaximal, characteristic, complemented
4C3×#5002C213yesmaximal, complemented
5C3×#5001C213yesmaximal, complemented
6C3×#4966C213yesmaximal, complemented
7C2×C3×#809C213yesmaximal, complemented
8C3×#2897—31nomaximal
9#82563C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C3×#1009—24no—
12C3×#1009C418yescomplemented
13C2×C3×#239—21no—
14C2×C3×D3×D4—23no—
15C2×C3×#235C2213yes—
16C22×C3×D12C2213yes—
17C32×C4×Q8—21no—
18C32×C4×D4—23no—
19C32×#117C2213yescomplemented
20C2×C32×C4⋊C4C2213yescomplemented
21C3×#948—24no—
22C2×C3×#208C2213yescomplemented
23C22×C3×C3⋊C8C2213yescomplemented
24C3×#810C2213yescomplemented
25C3×#810C2211yescharacteristic, complemented
26C3×#809C2213yescomplemented
27C3×#809C2216yescomplemented
28C3×#809C2213yescomplemented
29C2×C3×H(5,𝔽2)—31no—
30C2×#1009C2×C311yescharacteristic
31C2×C3×#131—31no—
32C3×#520—21no—
33C3×#520D311yescharacteristic, complemented
34C3×#520C2×C311yescharacteristic, complemented
35C2×#948C2×C311yescharacteristic, complemented
36C3×#418—33no—
37C3×#417—33no—
38C3×#416—33no—
39C2×C3×#102—33no—
40#5002C2×C313yescomplemented
41#5001C2×C313yescomplemented
42#4966C2×C313yescomplemented
43C2×#809C2×C313yescomplemented
44C23×C3×D3—23no—
45C32×#41C2×C414yes—
46C32×#41D414yes—
47C2×C32×Q8D411yescharacteristic
48C2×C32×D4D413yes—
49C22×C32×C4—23no—
50C22×C32×C4C2313yes—
51C2×C3×#67—23no—
52C3×#239—28no—
53C3×D3×D4—224no—
54C3×#235—26no—
55C3×#235C2×C4112yescomplemented
56C2×C3×D12—23no—
57C2×C3×D12—26no—
58C2×C3×D12C2×C4112yescomplemented
59C2×C3×D12C2313yes—
60C2×C3×C4×D3—23no—
61C2×C3×Dic6C2311yescharacteristic
62C32×C4⋊C4—23no—
63C32×C4⋊C4C2313yes—
64C32×C4⋊C4C2311yescharacteristic
65C32×C22⋊C4—23no—
66C32×C42—23no—
67C3×#208—26no—
68C2×C3×C3⋊C8—26no—
69C2×C3×C3⋊C8C2311yescharacteristic
70C2×C3×C3⋊C8C2313yes—
71#2897—31noSylow
72C3×H(5,𝔽2)—38no—
73C3×H(5,𝔽2)—64no—
74C2×C3×#41—21no—
75C2×C3×#41—33no—
76C2×C3×#41—61no—
77C2×C3×#41C2×D311yescharacteristic
78C2×C3×#41C22×C311yescharacteristic
79C22×C3×D4—33no—
80C22×C3×D4—63no—
81#1009—24no—
82#1009C3×C418yescomplemented
83C2×#239—21no—
84C2×D3×D4—23no—
85C2×#235C22×C313yes—
86C22×D12C22×C313yes—
87C3×#131—64no—
88C2×C3×M16—33no—
89C22×C3×C8—33no—
90C3×C4×Q8—22no—
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×C4×D4—26no—
94C3×C4×D4—23no—
95C3×C4×D4—23no—
96C3×#117—23no—
97C3×#117C2×D313yescomplemented
98C3×#117C22×C313yescomplemented
99C2×C3×C4⋊C4—23no—
100C2×C3×C4⋊C4C2×D313yescomplemented
101C2×C3×C4⋊C4C22×C313yescomplemented
102#948—24no—
103C2×#208C22×C313yescomplemented
104C22×C3⋊C8C22×C313yescomplemented
105C3×#103—31no—
106C3×#103—33no—
107C3×#102—33no—
108C3×#102—36no—
109C3×#102—33no—
110#810C22×C313yescomplemented
111#810C22×C311yescharacteristic, complemented
112#809C22×C313yescomplemented
113#809C22×C316yescomplemented
114#809C22×C313yescomplemented
115C23×C32C2×D413yes—
116C22×C3×D3—23no—
117C22×C3×D3—212no—
118C22×C3×D3—83no—
119C22×C3×D3—83no—
120C32×Q8C22⋊C414yes—
121C32×D4C22⋊C4112yes—
122C2×C32×C4—21no—
123C2×C32×C4—23no—
124C2×C32×C4—83no—
125C2×C32×C4C22×C416yes—
126C2×C32×C4C2×D416yes—
127C2×C32×C4C2×D413yes—
128C2×C32×C4C2411yescharacteristic
129C3×#67—212no—
130C2×C3×Dic3—21no—
131C3×D12—212no—
132C3×D12—23no—
133C3×D12C22×C416yes—
134C3×C4×D3—212no—
135C3×Dic6—21no—
136C3×Dic6C22×C412yes—
137C3×C3⋊C8—21no—
138C3×C3⋊C8—23no—
139C2×H(5,𝔽2)—31no—
140C2×#131—31no—
141#520C3×D311yescharacteristic, complemented
142#418—33no—
143#417—33no—
144#416—33no—
145C2×#102—33no—
146C24×C3—63no—
147C23×D3—23no—
148C3×#41—24no—
149C3×#41—24no—
150C3×#41—312no—
151C3×#41—68no—
152C3×#41—66no—
153C3×#41C4×D314yes—
154C3×#41D1214yes—
155C3×#41C2×C3×C414yes—
156C3×#41C3×D414yes—
157C2×C3×Q8—21no—
158C2×C3×Q8—31no—
159C2×C3×Q8#6711yescharacteristic
160C2×C3×Q8C3×D411yescharacteristic
161C2×C3×D4—23no—
162C2×C3×D4—312no—
163C2×C3×D4—33no—
164C2×C3×D4—63no—
165C2×C3×D4—624no—
166C2×C3×D4—66no—
167C2×C3×D4—63no—
168C2×C3×D4#6713yes—
169C2×C3×D4C3×D413yes—
170C22×C3×C4—26no—
171C22×C3×C4—23no—
172C22×C3×C4—23no—
173C22×C3×C4—23no—
174C22×C3×C4—63no—
175C22×C3×C4C22×D313yes—
176C22×C3×C4C23×C313yes—
177C2×#67—23no—
178#239—28no—
179D3×D4—224no—
180#235—26no—
181#235C2×C3×C4112yescomplemented
182C2×D12—23no—
183C2×D12—26no—
184C2×D12C2×C3×C4112yescomplemented
185C2×D12C23×C313yes—
186C2×C4×D3—23no—
187C2×Dic6C23×C311yescharacteristic
188C3×M16—66no—
189C2×C3×C8—31no—
190C2×C3×C8—33no—
191C2×C3×C8—66no—
192C3×C4⋊C4—26no—
193C3×C4⋊C4—23no—
194C3×C4⋊C4—23no—
195C3×C4⋊C4—23no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4C22×D313yes—
198C3×C4⋊C4C22×D311yescharacteristic
199C3×C4⋊C4C23×C313yes—
200C3×C4⋊C4C23×C311yescharacteristic
201C3×C22⋊C4—26no—
202C3×C22⋊C4—23no—
203C3×C22⋊C4—23no—
204C3×C42—26no—
205C3×C42—23no—
206C3×C42—23no—
207#208—26no—
208C2×C3⋊C8—26no—
209C2×C3⋊C8C23×C311yescharacteristic
210C2×C3⋊C8C23×C313yes—
211C22×C32—23no—
212C22×C32—23no—
213C22×C32C2×C22⋊C416yes—
214C22×C32C22×D411yescharacteristic
215C2×C3×D3—26no—
216C2×C3×D3—43no—
217C2×C3×D3—43no—
218C2×C3×D3—812no—
219C32×C4—43no—
220C32×C4—41no—
221C32×C4C2×C22⋊C416yes—
222C32×C4C23×C411yescharacteristic
223C32×C4C22×D411yescharacteristic
224C3×Dic3—22no—
225H(5,𝔽2)—38no—
226H(5,𝔽2)—64no—
227C2×#41—33no—
228C2×#41—61no—
229C2×#41C2×C3×D311yescharacteristic
230C22×D4—33no—
231C22×D4—63no—
232#131—64no—
233C2×M16—33no—
234C22×C8—33no—
235C4×Q8—21no—
236C4×D4—23no—
237#117C2×C3×D313yescomplemented
238C2×C4⋊C4C2×C3×D313yescomplemented
239#103—31no—
240#103—33no—
241#102—33no—
242#102—36no—
243#102—33no—
244C23×C3—23no—
245C23×C3—63no—
246C23×C3—612no—
247C23×C3—243no—
248C23×C3—243no—
249C23×C3C2×#6713yes—
250C23×C3C2×C3×D413yes—
251C22×D3—23no—
252C22×D3—212no—
253C22×D3—83no—
254C22×D3—83no—
255C3×Q8—24no—
256C3×Q8—32no—
257C3×Q8—61no—
258C3×Q8(C2×D3)⋊C414yes—
259C3×Q8C3×C22⋊C414yes—
260C3×D4—212no—
261C3×D4—36no—
262C3×D4—612no—
263C3×D4—612no—
264C3×D4—63no—
265C3×D4(C2×D3)⋊C4112yes—
266C3×D4C3×C22⋊C4112yes—
267C2×C3×C4—22no—
268C2×C3×C4—26no—
269C2×C3×C4—21no—
270C2×C3×C4—23no—
271C2×C3×C4—21no—
272C2×C3×C4—23no—
273C2×C3×C4—21no—
274C2×C3×C4—26no—
275C2×C3×C4—26no—
276C2×C3×C4—23no—
277C2×C3×C4—61no—
278C2×C3×C4—612no—
279C2×C3×C4—83no—
280C2×C3×C4—83no—
281C2×C3×C4—86no—
282C2×C3×C4C2×C4×D316yes—
283C2×C3×C4C2×D1216yes—
284C2×C3×C4C2×#6713yes—
285C2×C3×C4C22×C3×C416yes—
286C2×C3×C4C2×C3×D416yes—
287C2×C3×C4C2×C3×D413yes—
288C2×C3×C4C23×D311yescharacteristic
289C2×C3×C4C24×C311yescharacteristic
290#67—212no—
291C2×Dic3—21no—
292D12—212no—
293D12—23no—
294D12C22×C3×C416yes—
295C4×D3—212no—
296Dic6—21no—
297Dic6C22×C3×C412yes—
298C3×C8—61no—
299C3×C8—63no—
300C3⋊C8—21no—
301C3⋊C8—23no—
302C2×C32—26no—
303C2×C32C22×C22⋊C411yescharacteristic
304C2×C32#57611yescharacteristic
305C2×C32#57711yescharacteristic
306C3×D3—46no—
307C24—63no—
308#41—312no—
309#41—68no—
310#41—66no—
311#41C3×C4×D314yes—
312#41C3×D1214yes—
313C2×Q8—31no—
314C2×Q8C3×#6711yescharacteristic
315C2×D4—33no—
316C2×D4—312no—
317C2×D4—63no—
318C2×D4—624no—
319C2×D4—66no—
320C2×D4—63no—
321C2×D4C3×#6713yes—
322C22×C4—23no—
323C22×C4—63no—
324C22×C4C22×C3×D313yes—
325M16—66no—
326C2×C8—31no—
327C2×C8—33no—
328C2×C8—66no—
329C4⋊C4—23no—
330C4⋊C4C22×C3×D313yes—
331C4⋊C4C22×C3×D311yescharacteristic
332C22⋊C4—23no—
333C42—23no—
334C22×C3—23no—
335C22×C3—23no—
336C22×C3—23no—
337C22×C3—23no—
338C22×C3—26no—
339C22×C3—21no—
340C22×C3—43no—
341C22×C3—43no—
342C22×C3—66no—
343C22×C3—123no—
344C22×C3—123no—
345C22×C3—2412no—
346C22×C3C2×(C2×D3)⋊C416yes—
347C22×C3C2×C3×C22⋊C416yes—
348C22×C3C22×#6711yescharacteristic
349C22×C3C22×C3×D411yescharacteristic
350C2×D3—26no—
351C2×D3—43no—
352C2×D3—43no—
353C2×D3—812no—
354C3×C4—26no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—46no—
358C3×C4—42no—
359C3×C4—43no—
360C3×C4—41no—
361C3×C4—43no—
362C3×C4—41no—
363C3×C4—62no—
364C3×C4C2×(C2×D3)⋊C416yes—
365C3×C4C2×C3×C22⋊C416yes—
366C3×C4C22×C4×D311yescharacteristic
367C3×C4C22×D1211yescharacteristic
368C3×C4C23×C3×C411yescharacteristic
369C3×C4C22×C3×D411yescharacteristic
370Dic3—22no—
371C32#289711yescharacteristic, complemented, Sylow
372C23—63no—
373C23—612no—
374C23—243no—
375C23—243no—
376C23C2×C3×#6713yes—
377Q8—32no—
378Q8—61no—
379Q8C3×(C2×D3)⋊C414yes—
380D4—36no—
381D4—612no—
382D4—612no—
383D4—63no—
384D4C3×(C2×D3)⋊C4112yes—
385C2×C4—21no—
386C2×C4—23no—
387C2×C4—61no—
388C2×C4—612no—
389C2×C4—83no—
390C2×C4C2×C3×C4×D316yes—
391C2×C4C2×C3×D1216yes—
392C2×C4C2×C3×#6713yes—
393C2×C4C23×C3×D311yescharacteristic
394C8—61no—
395C8—63no—
396C2×C3—26no—
397C2×C3—26no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—46no—
402C2×C3—126no—
403C2×C3C22×(C2×D3)⋊C411yescharacteristic
404C2×C3#583511yescharacteristic
405C2×C3#583711yescharacteristic
406C2×C3C22×C3×C22⋊C411yescharacteristic
407C2×C3C3×#57611yescharacteristic
408C2×C3C3×#57711yescharacteristic
409D3—46no—
410C22—23no—
411C22—23no—
412C22—66no—
413C22—123no—
414C22—123no—
415C22—2412no—
416C22C2×C3×(C2×D3)⋊C416yes—
417C22C22×C3×#6711yescharacteristic
418C4—43no—
419C4—41no—
420C4—62no—
421C4C2×C3×(C2×D3)⋊C416yes—
422C4C22×C3×C4×D311yescharacteristic
423C4C22×C3×D1211yescharacteristic
424C3—21no—
425C3#8256311yescharacteristic, direct product factor
426C3C3×#289711yescharacteristic, complemented
427C2—26no—
428C2—126no—
429C2C22×C3×(C2×D3)⋊C411yescharacteristic
430C2C3×#583511yescharacteristic
431C2C3×#583711yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC32×#520C2
socleC22×C32C22×D4
prime radicalC2×C3×#1009C2
cosocleC2×C3×C4C24×C3
hypercenterC3×#520D3
hypocenterC3C3×#2897
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C3×#520

Complements

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

NH ≅ G/Ntype
2. C32×#520C2semidirect
3. C2×C3×#948C2semidirect
4. C3×#5002C2semidirect
5. C3×#5001C2semidirect
6. C3×#4966C2semidirect
7. C2×C3×#809C2semidirect
9. #82563C3direct (N × H)
12. C3×#1009C4semidirect
19. C32×#117C22semidirect
20. C2×C32×C4⋊C4C22semidirect
22. C2×C3×#208C22semidirect
23. C22×C3×C3⋊C8C22semidirect
24. C3×#810C22semidirect
25. C3×#810C22semidirect
26. C3×#809C22semidirect
27. C3×#809C22semidirect
28. C3×#809C22semidirect
33. C3×#520D3semidirect
34. C3×#520C2×C3semidirect
35. C2×#948C2×C3semidirect
40. #5002C2×C3semidirect
41. #5001C2×C3semidirect
42. #4966C2×C3semidirect
43. C2×#809C2×C3semidirect
55. C3×#235C2×C4semidirect
58. C2×C3×D12C2×C4semidirect
82. #1009C3×C4semidirect
97. C3×#117C2×D3semidirect
98. C3×#117C22×C3semidirect
100. C2×C3×C4⋊C4C2×D3semidirect
101. C2×C3×C4⋊C4C22×C3semidirect
103. C2×#208C22×C3semidirect
104. C22×C3⋊C8C22×C3semidirect
110. #810C22×C3semidirect
111. #810C22×C3semidirect
112. #809C22×C3semidirect
113. #809C22×C3semidirect
114. #809C22×C3semidirect
141. #520C3×D3semidirect
181. #235C2×C3×C4semidirect
184. C2×D12C2×C3×C4semidirect
237. #117C2×C3×D3semidirect
238. C2×C4⋊C4C2×C3×D3semidirect
371. C32#2897semidirect
425. C3#82563direct (N × H)
426. C3C3×#2897semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10092
C32×#5202
C2×C3×#9482
C3×#50022
C3×#50012
C3×#49662
C2×C3×#8092
C3×#28973
#825633