Finite Group Databasebeta

← C3×#69976 #11775114

Subgroups — C3×#69976 #11775114

Order 1152.

763 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5791C211yesmaximal, characteristic
2C32×#407C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊#95C211yesmaximal, characteristic, complemented
4C3×#4470C212yesmaximal, complemented
5C3×#4467C212yesmaximal, complemented
6C3×#1504—31nomaximal
7#69976C311yesmaximal, characteristic, direct product factor
8C23×C32×C4C2211yescharacteristic
9C23×C3×Dic3—21no—
10C2×C3×#235C411yescharacteristic, complemented
11C2×C32×M16—21no—
12C3×#953—21no—
13C3×#930C411yescharacteristic, complemented
14C3×#930C2211yescharacteristic
15C3×#929—21no—
16C3×C4×#67—21no—
17C3×C4×#67—21no—
18C3×#926C411yescharacteristic, complemented
19C3×#924C411yescharacteristic, complemented
20C3×#924C2211yescharacteristic
21C32×C22⋊C8—21no—
22C32×C22⋊C8C2212yes—
23C3×C3⋊#95—21no—
24C3×C3⋊#95C2212yes—
25C3×#525—31no—
26#5791C2×C311yescharacteristic
27C3×#407—21no—
28C3×#407D311yescharacteristic, complemented
29C3×#407C2×C311yescharacteristic, complemented
30C2×C3×#95—31no—
31C2×C3⋊#95C2×C311yescharacteristic, complemented
32C3×#328—32no—
33C3×#327—32no—
34#4470C2×C312yescomplemented
35#4467C2×C312yescomplemented
36C24×C32D411yescharacteristic
37C22×C32×C4—42no—
38C22×C32×C4C2×C411yescharacteristic
39C22×C32×C4C2×C411yescharacteristic
40C22×C32×C4D411yescharacteristic
41C22×C32×C4D411yescharacteristic
42C22×C32×C4D411yescharacteristic
43C22×C32×C4C2311yescharacteristic
44C2×C3×#67—21no—
45C2×C3×#67—21no—
46C2×C3×#67—21no—
47C22×C3×Dic3—24no—
48C22×C3×Dic3—22no—
49C22×C3×Dic3—42no—
50C22×C3×Dic3—42no—
51C3×#235—42no—
52C2×C3×D12C2×C411yescharacteristic, complemented
53C2×C3×C4×D3—21no—
54C2×C3×Dic6C2×C411yescharacteristic, complemented
55C32×M16—42no—
56C2×C32×C8—22no—
57C3×#217—21no—
58C3×#217—21no—
59C3×#217—21no—
60C3×(C2×D3)⋊C4—21no—
61C3×(C2×D3)⋊C4—21no—
62C3×C4⋊Dic3C2×C411yescharacteristic, complemented
63C3×C4⋊Dic3C2×C411yescharacteristic, complemented
64C3×Dic3⋊C4—21no—
65C3×Dic3⋊C4—21no—
66C3×C4×Dic3—21no—
67#1504—31noSylow
68C2×C3×#41—31no—
69C23×C3×C4—21no—
70C23×C3×C4—61no—
71C23×C3×C4C2×D311yescharacteristic
72C23×C3×C4C22×C311yescharacteristic
73C23×Dic3—21no—
74C2×#235C3×C411yescharacteristic, complemented
75C2×C3×M16—22no—
76C2×C3×M16—21no—
77C2×C3×M16—21no—
78C3×#123—61no—
79C3×C22⋊Q8—31no—
80C3×C22⋊Q8—31no—
81C3×#121—31no—
82C3×#121—31no—
83C3×#120—61no—
84C3×C4×D4—61no—
85C3×C4×D4—61no—
86C3×#117—31no—
87#953—21no—
88#930C3×C411yescharacteristic, complemented
89#930C22×C311yescharacteristic
90#929—21no—
91C4×#67—21no—
92C4×#67—21no—
93#926C3×C411yescharacteristic, complemented
94#924C3×C411yescharacteristic, complemented
95#924C22×C311yescharacteristic
96C3×C22⋊C8—22no—
97C3×C22⋊C8—21no—
98C3×C22⋊C8—21no—
99C3×C22⋊C8—22no—
100C3×C22⋊C8C2×D312yes—
101C3×C22⋊C8C22×C312yes—
102C3×#95—32no—
103C3×#95—61no—
104C3⋊#95—21no—
105C3⋊#95C22×C312yes—
106C23×C32—22no—
107C23×C32—22no—
108C23×C32—21no—
109C23×C32—21no—
110C23×C32C22⋊C411yescharacteristic
111C23×C32C22⋊C411yescharacteristic
112C23×C32C2×D411yescharacteristic
113C22×C3×D3—21no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C2×C32×C4—41no—
120C2×C32×C4C22⋊C411yescharacteristic
121C2×C32×C4C22⋊C411yescharacteristic
122C2×C32×C4C22×C411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C3×#67—42no—
125C3×#67—42no—
126C3×#67—41no—
127C3×#67—41no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—48no—
133C2×C3×Dic3—42no—
134C2×C3×Dic3—42no—
135C3×D12—41no—
136C3×C4×D3—42no—
137C3×Dic6—41no—
138C32×C8—42no—
139#525—31no—
140#407C3×D311yescharacteristic, complemented
141C2×#95—31no—
142#328—32no—
143#327—32no—
144C24×C3—21no—
145C24×C3D1211yescharacteristic
146C24×C3C3×D411yescharacteristic
147C3×#41—122no—
148C2×C3×Q8—31no—
149C2×C3×D4—31no—
150C2×C3×D4—61no—
151C2×C3×D4—61no—
152C2×C3×D4—61no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—44no—
162C22×C3×C4—64no—
163C22×C3×C4—62no—
164C22×C3×C4—61no—
165C22×C3×C4—122no—
166C22×C3×C4—122no—
167C22×C3×C4C4×D311yescharacteristic
168C22×C3×C4C4×D311yescharacteristic
169C22×C3×C4D1211yescharacteristic
170C22×C3×C4#6711yescharacteristic
171C22×C3×C4#6711yescharacteristic
172C22×C3×C4C2×C3×C411yescharacteristic
173C22×C3×C4C2×C3×C411yescharacteristic
174C22×C3×C4C3×D411yescharacteristic
175C22×C3×C4C3×D411yescharacteristic
176C22×C3×C4C3×D411yescharacteristic
177C22×C3×C4C22×D311yescharacteristic
178C22×C3×C4C23×C311yescharacteristic
179C2×#67—21no—
180C2×#67—21no—
181C2×#67—21no—
182C22×Dic3—24no—
183C22×Dic3—22no—
184C22×Dic3—42no—
185C22×Dic3—42no—
186#235—42no—
187C2×D12C2×C3×C411yescharacteristic, complemented
188C2×C4×D3—21no—
189C2×Dic6C2×C3×C411yescharacteristic, complemented
190C3×M16—44no—
191C3×M16—42no—
192C3×M16—42no—
193C2×C3×C8—24no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C3×C4⋊C4—31no—
197C3×C4⋊C4—31no—
198C3×C4⋊C4—61no—
199C3×C4⋊C4—61no—
200C3×C22⋊C4—61no—
201C3×C22⋊C4—61no—
202C3×C22⋊C4—61no—
203C3×C22⋊C4—61no—
204C3×C22⋊C4—61no—
205C3×C42—61no—
206#217—21no—
207#217—21no—
208#217—21no—
209(C2×D3)⋊C4—21no—
210(C2×D3)⋊C4—21no—
211C4⋊Dic3C2×C3×C411yescharacteristic, complemented
212C4⋊Dic3C2×C3×C411yescharacteristic, complemented
213Dic3⋊C4—21no—
214Dic3⋊C4—21no—
215C4×Dic3—21no—
216C22×C32—24no—
217C22×C32—24no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32#9912yes—
225C22×C32#10412yes—
226C22×C32C2×C22⋊C411yescharacteristic
227C22×C32AGL(1,ℤ/8)11yescharacteristic
228C22×C32#13711yescharacteristic
229C2×C3×D3—41no—
230C2×C3×D3—41no—
231C2×C3×D3—41no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—21no—
236C3×Dic3—44no—
237C3×Dic3—41no—
238C3×Dic3—41no—
239C3×Dic3—41no—
240C2×#41—31no—
241C23×C4—61no—
242C23×C4C2×C3×D311yescharacteristic
243C2×M16—21no—
244#123—61no—
245C22⋊Q8—31no—
246C22⋊Q8—31no—
247#121—31no—
248#121—31no—
249#120—61no—
250C4×D4—61no—
251C4×D4—61no—
252#117—31no—
253C22⋊C8—21no—
254C22⋊C8C2×C3×D312yes—
255#95—32no—
256#95—61no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—42no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—42no—
272C23×C3—61no—
273C23×C3(C2×D3)⋊C411yescharacteristic
274C23×C3(C2×D3)⋊C411yescharacteristic
275C23×C3C3×C22⋊C411yescharacteristic
276C23×C3C3×C22⋊C411yescharacteristic
277C23×C3C2×D1211yescharacteristic
278C23×C3C2×C3×D411yescharacteristic
279C22×D3—21no—
280C3×Q8—121no—
281C3×D4—121no—
282C3×D4—122no—
283C3×D4—122no—
284C3×D4—121no—
285C3×D4—121no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—64no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—122no—
313C2×C3×C4—128no—
314C2×C3×C4—122no—
315C2×C3×C4—122no—
316C2×C3×C4(C2×D3)⋊C411yescharacteristic
317C2×C3×C4(C2×D3)⋊C411yescharacteristic
318C2×C3×C4C3×C22⋊C411yescharacteristic
319C2×C3×C4C3×C22⋊C411yescharacteristic
320C2×C3×C4C2×C4×D311yescharacteristic
321C2×C3×C4C2×#6711yescharacteristic
322C2×C3×C4C22×C3×C411yescharacteristic
323C2×C3×C4C2×C3×D411yescharacteristic
324#67—42no—
325#67—42no—
326#67—41no—
327#67—41no—
328C2×Dic3—24no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—48no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335D12—41no—
336C4×D3—42no—
337Dic6—41no—
338C3×C8—44no—
339C3×C8—42no—
340C3×C8—42no—
341C2×C32—21no—
342C2×C32—22no—
343C2×C32—22no—
344C2×C32—21no—
345C2×C32C2×#9911yescharacteristic
346C2×C32#41711yescharacteristic
347C2×C32C2×#10411yescharacteristic
348C3×D3—81no—
349C24C3×D1211yescharacteristic
350#41—122no—
351C2×Q8—31no—
352C2×D4—31no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—61no—
356C22×C4—42no—
357C22×C4—64no—
358C22×C4—62no—
359C22×C4—61no—
360C22×C4—122no—
361C22×C4—122no—
362C22×C4C3×C4×D311yescharacteristic
363C22×C4C3×C4×D311yescharacteristic
364C22×C4C3×D1211yescharacteristic
365C22×C4C3×#6711yescharacteristic
366C22×C4C3×#6711yescharacteristic
367C22×C4C22×C3×D311yescharacteristic
368M16—42no—
369C2×C8—22no—
370C4⋊C4—31no—
371C4⋊C4—31no—
372C4⋊C4—61no—
373C4⋊C4—61no—
374C22⋊C4—61no—
375C22⋊C4—61no—
376C22⋊C4—61no—
377C22⋊C4—61no—
378C22⋊C4—61no—
379C42—61no—
380C22×C3—24no—
381C22×C3—24no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—121no—
410C22×C3—121no—
411C22×C3—121no—
412C22×C3#80612yes—
413C22×C3#82512yes—
414C22×C3C3×#9912yes—
415C22×C3C3×#10412yes—
416C22×C3#90811yescharacteristic
417C22×C3#90911yescharacteristic
418C22×C3C2×(C2×D3)⋊C411yescharacteristic
419C22×C3C2×C3×C22⋊C411yescharacteristic
420C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
421C22×C3C3×#13711yescharacteristic
422C2×D3—41no—
423C2×D3—41no—
424C2×D3—41no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—124no—
438C3×C4—121no—
439C3×C4—121no—
440C3×C4—121no—
441Dic3—44no—
442Dic3—41no—
443Dic3—41no—
444Dic3—41no—
445C32#150411yescharacteristic, complemented, Sylow
446C23—21no—
447C23—21no—
448C23—22no—
449C23—22no—
450C23—61no—
451C23C3×(C2×D3)⋊C411yescharacteristic
452C23C3×(C2×D3)⋊C411yescharacteristic
453C23C2×C3×D1211yescharacteristic
454Q8—121no—
455D4—121no—
456D4—121no—
457D4—121no—
458D4—122no—
459D4—122no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—64no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—122no—
471C2×C4—128no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4C3×(C2×D3)⋊C411yescharacteristic
475C2×C4C3×(C2×D3)⋊C411yescharacteristic
476C2×C4C2×C3×C4×D311yescharacteristic
477C2×C4C2×C3×#6711yescharacteristic
478C8—42no—
479C2×C3—21no—
480C2×C3—22no—
481C2×C3—22no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—22no—
485C2×C3—22no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—41no—
491C2×C3—41no—
492C2×C3—42no—
493C2×C3—42no—
494C2×C3—241no—
495C2×C3C2×#80611yescharacteristic
496C2×C3#513311yescharacteristic
497C2×C3C2×#82511yescharacteristic
498C2×C3C2×C3×#9911yescharacteristic
499C2×C3C3×#41711yescharacteristic
500C2×C3C2×C3×#10411yescharacteristic
501D3—81no—
502C22—24no—
503C22—24no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—121no—
511C22—121no—
512C22—121no—
513C22C3×#80612yes—
514C22C3×#82512yes—
515C22C3×#90811yescharacteristic
516C22C3×#90911yescharacteristic
517C22C2×C3×(C2×D3)⋊C411yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—124no—
523C4—121no—
524C4—121no—
525C4—121no—
526C3—21no—
527C3#6997611yescharacteristic, direct product factor
528C3C3×#150411yescharacteristic, complemented
529C2—21no—
530C2—21no—
531C2—22no—
532C2—22no—
533C2—241no—
534C2C2×C3×#80611yescharacteristic
535C2C3×#513311yescharacteristic
536C2C2×C3×#82511yescharacteristic
subgroupG/H
centerC22×C3C2×(C2×D3)⋊C4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#407C2
socleC22×C32C2×C22⋊C4
prime radicalC3×#5791C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#407D3
hypocenterC3C3×#1504
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C3×#407

Complements

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

NH ≅ G/Ntype
2. C32×#407C2semidirect
3. C2×C3×C3⋊#95C2semidirect
4. C3×#4470C2semidirect
5. C3×#4467C2semidirect
7. #69976C3direct (N × H)
10. C2×C3×#235C4semidirect
13. C3×#930C4semidirect
18. C3×#926C4semidirect
19. C3×#924C4semidirect
28. C3×#407D3semidirect
29. C3×#407C2×C3semidirect
31. C2×C3⋊#95C2×C3semidirect
34. #4470C2×C3semidirect
35. #4467C2×C3semidirect
52. C2×C3×D12C2×C4semidirect
54. C2×C3×Dic6C2×C4semidirect
62. C3×C4⋊Dic3C2×C4semidirect
63. C3×C4⋊Dic3C2×C4semidirect
74. C2×#235C3×C4semidirect
88. #930C3×C4semidirect
93. #926C3×C4semidirect
94. #924C3×C4semidirect
140. #407C3×D3semidirect
187. C2×D12C2×C3×C4semidirect
189. C2×Dic6C2×C3×C4semidirect
211. C4⋊Dic3C2×C3×C4semidirect
212. C4⋊Dic3C2×C3×C4semidirect
445. C32#1504semidirect
527. C3#69976direct (N × H)
528. C3C3×#1504semidirect

Maximal subgroups

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

subgroupindex
C3×#57912
C32×#4072
C2×C3×C3⋊#952
C3×#44702
C3×#44672
C3×#15043
#699763