Finite Group Databasebeta

← C3×#87291 #11905877

Subgroups — C3×#87291 #11905877

Order 1152.

3381 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×2−1+4C211yesmaximal, characteristic, complemented
2C2×C3×#1007C211yesmaximal, characteristic, complemented
3C3×#5834C214yesmaximal, complemented
4C3×#5833C216yesmaximal, complemented
5C3×#5832C216yesmaximal, complemented
6C3×#5831C214yesmaximal, complemented
7C3×#5817C214yesmaximal, complemented
8C3×Q8×#67C214yesmaximal, complemented
9C2×C3×Q8×Dic3C211yesmaximal, characteristic, complemented
10C3×#3497—31nomaximal
11#87291C311yesmaximal, characteristic, direct product factor
12C32×2−1+4—28no—
13C2×C32×#41C2214yes—
14C2×C32×#41C2216yes—
15C22×C32×Q8C2214yes—
16C22×C32×Q8C2211yescharacteristic
17C3×#1007—28no—
18C2×C3×#239C2214yes—
19C2×C3×D3×Q8C2214yes—
20C2×C3×#235C2216yes—
21C3×#947C2216yes—
22C3×#946C22112yes—
23C3×Q8×Dic3C2214yes—
24C3×Q8×Dic3C2214yes—
25C3×#944C2216yes—
26C3×#940C2216yes—
27C3×#939C2214yes—
28C3×#939C22112yes—
29C3×#938C22112yes—
30C3×#936C2216yes—
31C3×#930C22112yes—
32C3×C4×#67C22112yes—
33C3×C4×#67C22112yes—
34C2×C3×C4⋊Dic3C2213yes—
35C3×#924C22112yes—
36C2×C3×C4×Dic3C2213yes—
37C2×C3×2−1+4—21no—
38C2×C3×2−1+4—31no—
39C2×C3×2−1+4D311yescharacteristic, complemented
40C2×C3×2−1+4C2×C311yescharacteristic, complemented
41C2×#1007C2×C311yescharacteristic, complemented
42C3×#550—34no—
43C3×D4×Q8—34no—
44C3×#548—34no—
45C3×#548—34no—
46C3×#537—36no—
47C3×#532—36no—
48C2×C3×C4×Q8—31no—
49#5834C2×C314yescomplemented
50#5833C2×C316yescomplemented
51#5832C2×C316yescomplemented
52#5831C2×C314yescomplemented
53#5817C2×C314yescomplemented
54Q8×#67C2×C314yescomplemented
55C2×Q8×Dic3C2×C311yescharacteristic, complemented
56C32×#41—216no—
57C32×#41—224no—
58C2×C32×Q8—216no—
59C2×C32×Q8D418yescomplemented
60C2×C32×Q8C2316yes—
61C2×C32×Q8C2314yes—
62C2×C32×D4C2316yes—
63C2×C32×D4C2314yes—
64C22×C32×C4C23112yes—
65C22×C32×C4C2313yes—
66C2×C3×#67C2314yes—
67C2×C3×#67C2314yes—
68C2×C3×#67C23112yes—
69C22×C3×Dic3C2311yescharacteristic
70C22×C3×Dic3C2313yes—
71C3×#239—216no—
72C3×D3×Q8—216no—
73C3×#235—224no—
74C2×C3×D12C2316yes—
75C2×C3×C4×D3C23112yes—
76C2×C3×Dic6C2316yes—
77C3×#217C23112yes—
78C3×#217C2314yes—
79C3×(C2×D3)⋊C4C2314yes—
80C3×(C2×D3)⋊C4C23112yes—
81C3×C4⋊Dic3C23112yes—
82C3×Dic3⋊C4C23112yes—
83C3×Dic3⋊C4C2314yes—
84C3×C4×Dic3C2316yes—
85C3×C4×Dic3C2316yes—
86#3497—31noSylow
87C3×2−1+4—216no—
88C3×2−1+4—28no—
89C3×2−1+4—28no—
90C3×2−1+4—68no—
91C2×C3×#41—26no—
92C2×C3×#41—24no—
93C2×C3×#41—34no—
94C2×C3×#41—36no—
95C2×C3×#41C2×D316yes—
96C2×C3×#41C2×D314yes—
97C2×C3×#41C22×C314yes—
98C2×C3×#41C22×C316yes—
99C22×C3×Q8—24no—
100C22×C3×Q8—21no—
101C22×C3×Q8—34no—
102C22×C3×Q8C2×D314yes—
103C22×C3×Q8C2×D311yescharacteristic
104C22×C3×Q8C22×C314yes—
105C22×C3×Q8C22×C311yescharacteristic
106#1007—28no—
107C2×#239C22×C314yes—
108C2×D3×Q8C22×C314yes—
109C2×#235C22×C316yes—
110C3×C4⋊Q8—36no—
111C3×#127—36no—
112C3×#124—36no—
113C3×#124—36no—
114C3×C22⋊Q8—312no—
115C3×C22⋊Q8—312no—
116C3×#121—312no—
117C3×#121—34no—
118C3×#121—312no—
119C3×#121—312no—
120C3×C4×Q8—34no—
121C3×C4×Q8—34no—
122C3×C4×D4—312no—
123C3×C4×D4—312no—
124C2×C3×C4⋊C4—33no—
125C2×C3×C42—33no—
126#947C22×C316yes—
127#946C22×C3112yes—
128Q8×Dic3C22×C314yes—
129Q8×Dic3C22×C314yes—
130#944C22×C316yes—
131#940C22×C316yes—
132#939C22×C314yes—
133#939C22×C3112yes—
134#938C22×C3112yes—
135#936C22×C316yes—
136#930C22×C3112yes—
137C4×#67C22×C3112yes—
138C4×#67C22×C3112yes—
139C2×C4⋊Dic3C22×C313yes—
140#924C22×C3112yes—
141C2×C4×Dic3C22×C313yes—
142C23×C32C2414yes—
143C23×C32C2411yescharacteristic
144C22×C3×D3C2414yes—
145C32×Q8—212no—
146C32×Q8C2×D4116yes—
147C32×D4—28no—
148C32×D4—212no—
149C2×C32×C4—224no—
150C2×C32×C4C2×D4112yes—
151C2×C32×C4C2414yes—
152C2×C32×C4C2416yes—
153C3×#67—28no—
154C3×#67—28no—
155C3×#67—412no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—43no—
158C2×C3×Dic3C2411yescharacteristic
159C2×C3×Dic3C2416yes—
160C2×C3×Dic3C2414yes—
161C2×C3×Dic3C2411yescharacteristic
162C3×D12—212no—
163C3×C4×D3—224no—
164C3×Dic6—212no—
165C2×2−1+4—31no—
166C2×2−1+4C3×D311yescharacteristic, complemented
167#550—34no—
168D4×Q8—34no—
169#548—34no—
170#548—34no—
171#537—36no—
172#532—36no—
173C2×C4×Q8—31no—
174C3×#41—232no—
175C3×#41—248no—
176C3×#41—216no—
177C3×#41—216no—
178C3×#41—224no—
179C3×#41—224no—
180C3×#41—616no—
181C3×#41—624no—
182C2×C3×Q8—232no—
183C2×C3×Q8—216no—
184C2×C3×Q8—216no—
185C2×C3×Q8—28no—
186C2×C3×Q8—26no—
187C2×C3×Q8—24no—
188C2×C3×Q8—36no—
189C2×C3×Q8—616no—
190C2×C3×Q8#6718yescomplemented
191C2×C3×Q8C3×D418yescomplemented
192C2×C3×Q8C22×D316yes—
193C2×C3×Q8C22×D314yes—
194C2×C3×Q8C23×C316yes—
195C2×C3×Q8C23×C314yes—
196C2×C3×D4—24no—
197C2×C3×D4—26no—
198C2×C3×D4—34no—
199C2×C3×D4—34no—
200C2×C3×D4—312no—
201C2×C3×D4—36no—
202C2×C3×D4C22×D314yes—
203C2×C3×D4C22×D316yes—
204C2×C3×D4C23×C316yes—
205C2×C3×D4C23×C314yes—
206C22×C3×C4—212no—
207C22×C3×C4—23no—
208C22×C3×C4—31no—
209C22×C3×C4—33no—
210C22×C3×C4—312no—
211C22×C3×C4C22×D3112yes—
212C22×C3×C4C22×D313yes—
213C22×C3×C4C23×C3112yes—
214C22×C3×C4C23×C313yes—
215C2×#67C23×C314yes—
216C2×#67C23×C314yes—
217C2×#67C23×C3112yes—
218C22×Dic3C23×C311yescharacteristic
219C22×Dic3C23×C313yes—
220#239—216no—
221D3×Q8—216no—
222#235—224no—
223C2×D12C23×C316yes—
224C2×C4×D3C23×C3112yes—
225C2×Dic6C23×C316yes—
226C3×C4⋊C4—34no—
227C3×C4⋊C4—312no—
228C3×C4⋊C4—312no—
229C3×C22⋊C4—34no—
230C3×C22⋊C4—312no—
231C3×C22⋊C4—34no—
232C3×C22⋊C4—312no—
233C3×C42—36no—
234C3×C42—36no—
235#217C23×C3112yes—
236#217C23×C314yes—
237(C2×D3)⋊C4C23×C314yes—
238(C2×D3)⋊C4C23×C3112yes—
239C4⋊Dic3C23×C3112yes—
240Dic3⋊C4C23×C3112yes—
241Dic3⋊C4C23×C314yes—
242C4×Dic3C23×C316yes—
243C4×Dic3C23×C316yes—
244C22×C32—24no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—24no—
248C22×C32—24no—
249C22×C32C22×D412yes—
250C22×C32C2511yescharacteristic
251C2×C3×D3—24no—
252C2×C3×D3—24no—
253C2×C3×D3—24no—
254C32×C4—24no—
255C32×C4C22×D4112yes—
256C3×Dic3—24no—
257C3×Dic3—21no—
258C3×Dic3—26no—
259C3×Dic32−1+412yescomplemented
2602−1+4—28no—
2612−1+4—68no—
262C2×#41—34no—
263C2×#41—36no—
264C2×#41C2×C3×D316yes—
265C2×#41C2×C3×D314yes—
266C22×Q8—34no—
267C22×Q8C2×C3×D314yes—
268C22×Q8C2×C3×D311yescharacteristic
269C4⋊Q8—36no—
270#127—36no—
271#124—36no—
272#124—36no—
273C22⋊Q8—312no—
274C22⋊Q8—312no—
275#121—34no—
276#121—312no—
277#121—312no—
278#121—312no—
279C4×Q8—34no—
280C4×Q8—34no—
281C4×D4—312no—
282C4×D4—312no—
283C2×C4⋊C4—33no—
284C2×C42—33no—
285C23×C3—24no—
286C23×C3—21no—
287C23×C3—34no—
288C23×C3C23×D314yes—
289C23×C3C23×D311yescharacteristic
290C23×C3C24×C314yes—
291C23×C3C24×C311yescharacteristic
292C22×D3C24×C314yes—
293C3×Q8—224no—
294C3×Q8—212no—
295C3×Q8—212no—
296C3×Q8—216no—
297C3×Q8—612no—
298C3×Q8C2×#67116yes—
299C3×Q8C2×C3×D4116yes—
300C3×D4—216no—
301C3×D4—224no—
302C3×D4—28no—
303C3×D4—28no—
304C3×D4—212no—
305C3×D4—212no—
306C3×D4—68no—
307C3×D4—68no—
308C3×D4—612no—
309C3×D4—1212no—
310C2×C3×C4—248no—
311C2×C3×C4—224no—
312C2×C3×C4—224no—
313C2×C3×C4—24no—
314C2×C3×C4—26no—
315C2×C3×C4—212no—
316C2×C3×C4—31no—
317C2×C3×C4—31no—
318C2×C3×C4—36no—
319C2×C3×C4—34no—
320C2×C3×C4—62no—
321C2×C3×C4—624no—
322C2×C3×C4—123no—
323C2×C3×C4C2×#67112yes—
324C2×C3×C4C2×C3×D4112yes—
325C2×C3×C4C23×D314yes—
326C2×C3×C4C23×D316yes—
327C2×C3×C4C24×C314yes—
328C2×C3×C4C24×C316yes—
329#67—28no—
330#67—28no—
331#67—412no—
332C2×Dic3—22no—
333C2×Dic3—43no—
334C2×Dic3C24×C311yescharacteristic
335C2×Dic3C24×C316yes—
336C2×Dic3C24×C314yes—
337C2×Dic3C24×C311yescharacteristic
338D12—212no—
339C4×D3—224no—
340Dic6—212no—
341C2×C32—22no—
342C2×C32—44no—
343C2×C32C23×D411yescharacteristic
344C2×C32C2×2−1+411yescharacteristic
345C2×C32#58511yescharacteristic
346C3×D3—44no—
347#41—216no—
348#41—224no—
349#41—616no—
350#41—624no—
351C2×Q8—216no—
352C2×Q8—36no—
353C2×Q8—616no—
354C2×Q8C3×#6718yescomplemented
355C2×Q8C22×C3×D316yes—
356C2×Q8C22×C3×D314yes—
357C2×D4—34no—
358C2×D4—34no—
359C2×D4—312no—
360C2×D4—36no—
361C2×D4C22×C3×D314yes—
362C2×D4C22×C3×D316yes—
363C22×C4—31no—
364C22×C4—33no—
365C22×C4—312no—
366C22×C4C22×C3×D3112yes—
367C22×C4C22×C3×D313yes—
368C4⋊C4—312no—
369C4⋊C4—34no—
370C4⋊C4—312no—
371C22⋊C4—34no—
372C22⋊C4—312no—
373C22⋊C4—34no—
374C22⋊C4—312no—
375C42—36no—
376C42—36no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—28no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—24no—
385C22×C3—24no—
386C22×C3—24no—
387C22×C3—24no—
388C22×C3—22no—
389C22×C3—21no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—64no—
395C22×C3—64no—
396C22×C3—64no—
397C22×C3C22×#6712yes—
398C22×C3C22×C3×D412yes—
399C22×C3C24×D311yescharacteristic
400C22×C3C25×C311yescharacteristic
401C2×D3—24no—
402C2×D3—24no—
403C2×D3—24no—
404C3×C4—28no—
405C3×C4—24no—
406C3×C4—24no—
407C3×C4—212no—
408C3×C4—32no—
409C3×C4—61no—
410C3×C4—66no—
411C3×C4—64no—
412C3×C4C22×#67112yes—
413C3×C4C22×C3×D4112yes—
414Dic3—24no—
415Dic3—21no—
416Dic3—26no—
417Dic3C3×2−1+412yescomplemented
418C32#349711yescharacteristic, complemented, Sylow
419C23—34no—
420C23C23×C3×D314yes—
421C23C23×C3×D311yescharacteristic
422Q8—212no—
423Q8—612no—
424Q8C2×C3×#67116yes—
425D4—28no—
426D4—212no—
427D4—68no—
428D4—68no—
429D4—612no—
430D4—1212no—
431C2×C4—224no—
432C2×C4—31no—
433C2×C4—31no—
434C2×C4—36no—
435C2×C4—34no—
436C2×C4—62no—
437C2×C4—624no—
438C2×C4—123no—
439C2×C4C2×C3×#67112yes—
440C2×C4C23×C3×D314yes—
441C2×C4C23×C3×D316yes—
442C2×C3—22no—
443C2×C3—22no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—42no—
448C2×C3—48no—
449C2×C3—44no—
450C2×C3—44no—
451C2×C3—124no—
452C2×C3D3×2−1+411yescharacteristic
453C2×C3#596811yescharacteristic
454C2×C3C23×#6711yescharacteristic
455C2×C3C23×C3×D411yescharacteristic
456C2×C3C2×C3×2−1+411yescharacteristic
457C2×C3C3×#58511yescharacteristic
458D3—44no—
459C22—21no—
460C22—21no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—64no—
465C22—64no—
466C22—64no—
467C22C22×C3×#6712yes—
468C22C24×C3×D311yescharacteristic
469C4—24no—
470C4—32no—
471C4—61no—
472C4—66no—
473C4—64no—
474C4C22×C3×#67112yes—
475C3—21no—
476C3#8729111yescharacteristic, direct product factor
477C3C3×#349711yescharacteristic, complemented
478C2—22no—
479C2—44no—
480C2—124no—
481C2C3×D3×2−1+411yescharacteristic
482C2C3×#596811yescharacteristic
483C2C23×C3×#6711yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×2−1+4C2
socleC22×C32C25
hypercenterC2×C3×2−1+4D3
hypocenterC3C3×#3497
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×C32×2−1+4C2semidirect
2. C2×C3×#1007C2semidirect
3. C3×#5834C2semidirect
4. C3×#5833C2semidirect
5. C3×#5832C2semidirect
6. C3×#5831C2semidirect
7. C3×#5817C2semidirect
8. C3×Q8×#67C2semidirect
9. C2×C3×Q8×Dic3C2semidirect
11. #87291C3direct (N × H)
39. C2×C3×2−1+4D3semidirect
40. C2×C3×2−1+4C2×C3semidirect
41. C2×#1007C2×C3semidirect
49. #5834C2×C3semidirect
50. #5833C2×C3semidirect
51. #5832C2×C3semidirect
52. #5831C2×C3semidirect
53. #5817C2×C3semidirect
54. Q8×#67C2×C3semidirect
55. C2×Q8×Dic3C2×C3semidirect
59. C2×C32×Q8D4semidirect
166. C2×2−1+4C3×D3semidirect
190. C2×C3×Q8#67semidirect
191. C2×C3×Q8C3×D4semidirect
259. C3×Dic32−1+4semidirect
354. C2×Q8C3×#67semidirect
417. Dic3C3×2−1+4semidirect
418. C32#3497semidirect
476. C3#87291direct (N × H)
477. C3C3×#3497semidirect

Maximal subgroups

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

subgroupindex
C2×C32×2−1+42
C2×C3×#10072
C3×#58342
C3×#58332
C3×#58322
C3×#58312
C3×#58172
C3×Q8×#672
C2×C3×Q8×Dic32
C3×#34973
#872913