Finite Group Databasebeta

← C3×#82570 #11858054

Subgroups — C3×#82570 #11858054

Order 1152.

1599 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1010C211yesmaximal, characteristic
2C32×#520C211yesmaximal, characteristic, complemented
3C2×C3×#948C211yesmaximal, characteristic, complemented
4C3×#5004C213yesmaximal, complemented
5C3×#5002C213yesmaximal, complemented
6C2×C3×#810C213yesmaximal, complemented
7C3×#4966C213yesmaximal, complemented
8C3×#2898—31nomaximal
9#82570C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C3×#1010—24no—
12C3×#1010C418yescomplemented
13C2×C3×D3×Q8—21no—
14C2×C3×#237—23no—
15C2×C3×#235C2213yes—
16C22×C3×Dic6C2213yes—
17C32×C4×Q8—21no—
18C32×C4×D4—23no—
19C32×#117C2213yes—
20C2×C32×C4⋊C4C2213yes—
21C3×#948—24no—
22C2×C3×#208C2213yes—
23C22×C3×C3⋊C8C2213yes—
24C3×#810C2216yes—
25C3×#810C2213yes—
26C3×#810C2213yes—
27C3×#809C2213yes—
28C3×#809C2211yescharacteristic
29C2×C3×2−1+4—31no—
30C2×#1010C2×C311yescharacteristic
31C2×C3×#131—31no—
32C3×#520—21no—
33C3×#520D311yescharacteristic, complemented
34C3×#520C2×C311yescharacteristic, complemented
35C2×#948C2×C311yescharacteristic, complemented
36C3×#419—33no—
37C3×#417—33no—
38C3×#416—33no—
39C2×C3×#103—33no—
40#5004C2×C313yescomplemented
41#5002C2×C313yescomplemented
42C2×#810C2×C313yescomplemented
43#4966C2×C313yescomplemented
44C32×#41C2×C414yes—
45C32×#41D414yes—
46C2×C32×Q8D411yescharacteristic
47C2×C32×D4D413yes—
48C22×C32×C4—23no—
49C22×C32×C4C2313yes—
50C2×C3×#67—23no—
51C22×C3×Dic3—23no—
52C3×D3×Q8—28no—
53C3×#237—224no—
54C3×#235—26no—
55C3×#235C2×C4112yescomplemented
56C2×C3×D12C2311yescharacteristic
57C2×C3×C4×D3—23no—
58C2×C3×Dic6—23no—
59C2×C3×Dic6—26no—
60C2×C3×Dic6C2×C4112yescomplemented
61C2×C3×Dic6C2313yes—
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#2898—31noSylow
72C3×2−1+4—38no—
73C3×2−1+4—64no—
74C2×C3×#41—21no—
75C2×C3×#41—33no—
76C2×C3×#41—63no—
77C2×C3×#41C2×D311yescharacteristic
78C2×C3×#41C22×C311yescharacteristic
79C22×C3×Q8—33no—
80C22×C3×Q8—61no—
81#1010—24no—
82#1010C3×C418yescomplemented
83C2×D3×Q8—21no—
84C2×#237—23no—
85C2×#235C22×C313yes—
86C22×Dic6C22×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×D313yes—
98C3×#117C22×C313yes—
99C2×C3×C4⋊C4—23no—
100C2×C3×C4⋊C4C2×D313yes—
101C2×C3×C4⋊C4C22×C313yes—
102#948—24no—
103C2×#208C22×C313yes—
104C22×C3⋊C8C22×C313yes—
105C3×#103—33no—
106C3×#103—36no—
107C3×#103—33no—
108C3×#102—31no—
109C3×#102—33no—
110#810C22×C316yes—
111#810C22×C313yes—
112#810C22×C313yes—
113#809C22×C313yes—
114#809C22×C311yescharacteristic
115C23×C32C2×D413yes—
116C22×C3×D3—21no—
117C32×Q8C22⋊C414yes—
118C32×D4C22⋊C4112yes—
119C2×C32×C4—21no—
120C2×C32×C4—23no—
121C2×C32×C4—83no—
122C2×C32×C4C22×C416yes—
123C2×C32×C4C2×D416yes—
124C2×C32×C4C2×D413yes—
125C2×C32×C4C2411yescharacteristic
126C3×#67—212no—
127C2×C3×Dic3—212no—
128C2×C3×Dic3—23no—
129C3×D12—21no—
130C3×D12C22×C412yes—
131C3×C4×D3—212no—
132C3×Dic6—212no—
133C3×Dic6—23no—
134C3×Dic6C22×C416yes—
135C3×C3⋊C8—21no—
136C3×C3⋊C8—23no—
137C2×2−1+4—31no—
138C2×#131—31no—
139#520C3×D311yescharacteristic, complemented
140#419—33no—
141#417—33no—
142#416—33no—
143C2×#103—33no—
144C3×#41—24no—
145C3×#41—24no—
146C3×#41—312no—
147C3×#41—624no—
148C3×#41—66no—
149C3×#41C4×D314yes—
150C3×#41D1214yes—
151C3×#41C2×C3×C414yes—
152C3×#41C3×D414yes—
153C2×C3×Q8—21no—
154C2×C3×Q8—312no—
155C2×C3×Q8—33no—
156C2×C3×Q8—68no—
157C2×C3×Q8—66no—
158C2×C3×Q8—63no—
159C2×C3×Q8#6711yescharacteristic
160C2×C3×Q8C3×D411yescharacteristic
161C2×C3×D4—23no—
162C2×C3×D4—31no—
163C2×C3×D4—63no—
164C2×C3×D4#6713yes—
165C2×C3×D4C3×D413yes—
166C22×C3×C4—26no—
167C22×C3×C4—23no—
168C22×C3×C4—23no—
169C22×C3×C4—23no—
170C22×C3×C4—63no—
171C22×C3×C4—63no—
172C22×C3×C4C22×D313yes—
173C22×C3×C4C23×C313yes—
174C2×#67—23no—
175C22×Dic3—23no—
176D3×Q8—28no—
177#237—224no—
178#235—26no—
179#235C2×C3×C4112yescomplemented
180C2×D12C23×C311yescharacteristic
181C2×C4×D3—23no—
182C2×Dic6—23no—
183C2×Dic6—26no—
184C2×Dic6C2×C3×C4112yescomplemented
185C2×Dic6C23×C313yes—
186C3×M16—66no—
187C2×C3×C8—31no—
188C2×C3×C8—33no—
189C2×C3×C8—66no—
190C3×C4⋊C4—26no—
191C3×C4⋊C4—23no—
192C3×C4⋊C4—23no—
193C3×C4⋊C4—23no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4C22×D313yes—
196C3×C4⋊C4C22×D311yescharacteristic
197C3×C4⋊C4C23×C313yes—
198C3×C4⋊C4C23×C311yescharacteristic
199C3×C22⋊C4—26no—
200C3×C22⋊C4—23no—
201C3×C22⋊C4—23no—
202C3×C42—26no—
203C3×C42—23no—
204C3×C42—23no—
205#208—26no—
206C2×C3⋊C8—26no—
207C2×C3⋊C8C23×C311yescharacteristic
208C2×C3⋊C8C23×C313yes—
209C22×C32—23no—
210C22×C32—23no—
211C22×C32C2×C22⋊C416yes—
212C22×C32C22×D411yescharacteristic
213C2×C3×D3—22no—
214C2×C3×D3—41no—
215C2×C3×D3—41no—
216C32×C4—43no—
217C32×C4—41no—
218C32×C4C2×C22⋊C416yes—
219C32×C4C23×C411yescharacteristic
220C32×C4C22×D411yescharacteristic
221C3×Dic3—26no—
2222−1+4—38no—
2232−1+4—64no—
224C2×#41—33no—
225C2×#41—63no—
226C2×#41C2×C3×D311yescharacteristic
227C22×Q8—33no—
228C22×Q8—61no—
229#131—64no—
230C2×M16—33no—
231C22×C8—33no—
232C4×Q8—21no—
233C4×D4—23no—
234#117C2×C3×D313yes—
235C2×C4⋊C4C2×C3×D313yes—
236#103—33no—
237#103—36no—
238#103—33no—
239#102—31no—
240#102—33no—
241C23×C3—23no—
242C23×C3—61no—
243C23×C3C2×#6713yes—
244C23×C3C2×C3×D413yes—
245C22×D3—21no—
246C3×Q8—24no—
247C3×Q8—36no—
248C3×Q8—612no—
249C3×Q8—63no—
250C3×Q8(C2×D3)⋊C414yes—
251C3×Q8C3×C22⋊C414yes—
252C3×D4—212no—
253C3×D4—32no—
254C3×D4—612no—
255C3×D4—61no—
256C3×D4(C2×D3)⋊C4112yes—
257C3×D4C3×C22⋊C4112yes—
258C2×C3×C4—22no—
259C2×C3×C4—26no—
260C2×C3×C4—21no—
261C2×C3×C4—23no—
262C2×C3×C4—21no—
263C2×C3×C4—23no—
264C2×C3×C4—21no—
265C2×C3×C4—26no—
266C2×C3×C4—26no—
267C2×C3×C4—23no—
268C2×C3×C4—612no—
269C2×C3×C4—63no—
270C2×C3×C4—612no—
271C2×C3×C4—83no—
272C2×C3×C4—83no—
273C2×C3×C4—86no—
274C2×C3×C4C2×C4×D316yes—
275C2×C3×C4C2×D1216yes—
276C2×C3×C4C2×#6713yes—
277C2×C3×C4C22×C3×C416yes—
278C2×C3×C4C2×C3×D416yes—
279C2×C3×C4C2×C3×D413yes—
280C2×C3×C4C23×D311yescharacteristic
281C2×C3×C4C24×C311yescharacteristic
282#67—212no—
283C2×Dic3—212no—
284C2×Dic3—23no—
285D12—21no—
286D12C22×C3×C412yes—
287C4×D3—212no—
288Dic6—212no—
289Dic6—23no—
290Dic6C22×C3×C416yes—
291C3×C8—61no—
292C3×C8—63no—
293C3⋊C8—21no—
294C3⋊C8—23no—
295C2×C32—26no—
296C2×C32C22×C22⋊C411yescharacteristic
297C2×C32#57711yescharacteristic
298C2×C32#57811yescharacteristic
299C3×D3—42no—
300#41—312no—
301#41—624no—
302#41—66no—
303#41C3×C4×D314yes—
304#41C3×D1214yes—
305C2×Q8—33no—
306C2×Q8—312no—
307C2×Q8—68no—
308C2×Q8—66no—
309C2×Q8—63no—
310C2×Q8C3×#6711yescharacteristic
311C2×D4—31no—
312C2×D4—63no—
313C2×D4C3×#6713yes—
314C22×C4—23no—
315C22×C4—63no—
316C22×C4—63no—
317C22×C4C22×C3×D313yes—
318M16—66no—
319C2×C8—31no—
320C2×C8—33no—
321C2×C8—66no—
322C4⋊C4—23no—
323C4⋊C4C22×C3×D313yes—
324C4⋊C4C22×C3×D311yescharacteristic
325C22⋊C4—23no—
326C42—23no—
327C22×C3—23no—
328C22×C3—23no—
329C22×C3—23no—
330C22×C3—23no—
331C22×C3—26no—
332C22×C3—21no—
333C22×C3—43no—
334C22×C3—43no—
335C22×C3—62no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3C2×(C2×D3)⋊C416yes—
339C22×C3C2×C3×C22⋊C416yes—
340C22×C3C22×#6711yescharacteristic
341C22×C3C22×C3×D411yescharacteristic
342C2×D3—22no—
343C2×D3—41no—
344C2×D3—41no—
345C3×C4—26no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—46no—
349C3×C4—42no—
350C3×C4—43no—
351C3×C4—41no—
352C3×C4—43no—
353C3×C4—41no—
354C3×C4—66no—
355C3×C4C2×(C2×D3)⋊C416yes—
356C3×C4C2×C3×C22⋊C416yes—
357C3×C4C22×C4×D311yescharacteristic
358C3×C4C22×D1211yescharacteristic
359C3×C4C23×C3×C411yescharacteristic
360C3×C4C22×C3×D411yescharacteristic
361Dic3—26no—
362C32#289811yescharacteristic, complemented, Sylow
363C23—61no—
364C23C2×C3×#6713yes—
365Q8—36no—
366Q8—612no—
367Q8—63no—
368Q8C3×(C2×D3)⋊C414yes—
369D4—32no—
370D4—612no—
371D4—61no—
372D4C3×(C2×D3)⋊C4112yes—
373C2×C4—21no—
374C2×C4—23no—
375C2×C4—612no—
376C2×C4—63no—
377C2×C4—612no—
378C2×C4—83no—
379C2×C4C2×C3×C4×D316yes—
380C2×C4C2×C3×D1216yes—
381C2×C4C2×C3×#6713yes—
382C2×C4C23×C3×D311yescharacteristic
383C8—61no—
384C8—63no—
385C2×C3—26no—
386C2×C3—26no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—46no—
391C2×C3—122no—
392C2×C3C22×(C2×D3)⋊C411yescharacteristic
393C2×C3#583611yescharacteristic
394C2×C3#583811yescharacteristic
395C2×C3C22×C3×C22⋊C411yescharacteristic
396C2×C3C3×#57711yescharacteristic
397C2×C3C3×#57811yescharacteristic
398D3—42no—
399C22—23no—
400C22—23no—
401C22—62no—
402C22—121no—
403C22—121no—
404C22C2×C3×(C2×D3)⋊C416yes—
405C22C22×C3×#6711yescharacteristic
406C4—43no—
407C4—41no—
408C4—66no—
409C4C2×C3×(C2×D3)⋊C416yes—
410C4C22×C3×C4×D311yescharacteristic
411C4C22×C3×D1211yescharacteristic
412C3—21no—
413C3#8257011yescharacteristic, direct product factor
414C3C3×#289811yescharacteristic, complemented
415C2—26no—
416C2—122no—
417C2C22×C3×(C2×D3)⋊C411yescharacteristic
418C2C3×#583611yescharacteristic
419C2C3×#583811yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC32×#520C2
socleC22×C32C22×D4
prime radicalC2×C3×#1010C2
cosocleC2×C3×C4C24×C3
hypercenterC3×#520D3
hypocenterC3C3×#2898
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×#5004C2semidirect
5. C3×#5002C2semidirect
6. C2×C3×#810C2semidirect
7. C3×#4966C2semidirect
9. #82570C3direct (N × H)
12. C3×#1010C4semidirect
33. C3×#520D3semidirect
34. C3×#520C2×C3semidirect
35. C2×#948C2×C3semidirect
40. #5004C2×C3semidirect
41. #5002C2×C3semidirect
42. C2×#810C2×C3semidirect
43. #4966C2×C3semidirect
55. C3×#235C2×C4semidirect
60. C2×C3×Dic6C2×C4semidirect
82. #1010C3×C4semidirect
139. #520C3×D3semidirect
179. #235C2×C3×C4semidirect
184. C2×Dic6C2×C3×C4semidirect
362. C32#2898semidirect
413. C3#82570direct (N × H)
414. C3C3×#2898semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10102
C32×#5202
C2×C3×#9482
C3×#50042
C3×#50022
C2×C3×#8102
C3×#49662
C3×#28983
#825703