Finite Group Databasebeta

← C3×#69968 #11773916

Subgroups — C3×#69968 #11773916

Order 1152.

727 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#67C211yesmaximal, characteristic
2C32×#407C211yesmaximal, characteristic, complemented
3C3×#5207C211yesmaximal, characteristic, complemented
4C3×#4469C212yesmaximal, complemented
5C3×#4468C212yesmaximal, complemented
6C3×#1465—31nomaximal
7#69968C311yesmaximal, characteristic, direct product factor
8C23×C32×C4C2211yescharacteristic
9C22×C3×#67C411yescharacteristic
10C22×C3×C4×D3C411yescharacteristic
11C2×C32×M16—21no—
12C2×C3×#217C411yescharacteristic
13C3×C4×#67—21no—
14C3×C4×#67—21no—
15C3×C4×#67—21no—
16C3×C4×#67—21no—
17C2×C3×(C2×D3)⋊C4C2211yescharacteristic
18C2×C3×Dic3⋊C4C2211yescharacteristic
19C2×C3×C4×Dic3C411yescharacteristic
20C2×C3×#208—21no—
21C32×C22⋊C8—21no—
22C32×C22⋊C8C2212yes—
23C3×#830—21no—
24C3×#830C2212yes—
25C2×C3×C4×D4—31no—
26C2×C4×#67C2×C311yescharacteristic
27C3×#407—21no—
28C3×#407—31no—
29C3×#407D311yescharacteristic, complemented
30C3×#407C2×C311yescharacteristic, complemented
31#5207C2×C311yescharacteristic, complemented
32C3×#324—32no—
33C3×#323—32no—
34#4469C2×C312yescomplemented
35#4468C2×C312yescomplemented
36C24×C32C2×C411yescharacteristic
37C23×C3×D3C2×C411yescharacteristic
38C22×C32×C4—42no—
39C22×C32×C4C2×C411yescharacteristic
40C22×C32×C4D411yescharacteristic
41C22×C32×C4D411yescharacteristic
42C22×C32×C4D411yescharacteristic
43C22×C32×C4D411yescharacteristic
44C22×C32×C4C2311yescharacteristic
45C2×C3×#67—21no—
46C2×C3×#67—21no—
47C2×C3×#67—22no—
48C2×C3×#67—22no—
49C22×C3×Dic3C2×C411yescharacteristic
50C22×C3×Dic3C2×C411yescharacteristic
51C22×C3×Dic3C2×C411yescharacteristic
52C2×C3×C4×D3—21no—
53C2×C3×C4×D3—21no—
54C2×C3×C4×D3—42no—
55C32×M16—42no—
56C2×C32×C8—22no—
57C3×#217—21no—
58C3×#217—21no—
59C3×(C2×D3)⋊C4—21no—
60C3×(C2×D3)⋊C4—21no—
61C3×Dic3⋊C4—21no—
62C3×Dic3⋊C4—21no—
63C3×C4×Dic3—21no—
64C3×C4×Dic3—21no—
65C3×#208—42no—
66C2×C3×C3⋊C8—22no—
67#1465—31noSylow
68C22×C3×D4—31no—
69C23×C3×C4—21no—
70C23×C3×C4—31no—
71C23×C3×C4C2×D311yescharacteristic
72C23×C3×C4C22×C311yescharacteristic
73C22×#67C3×C411yescharacteristic
74C22×C4×D3C3×C411yescharacteristic
75C2×C3×M16—22no—
76C2×C3×M16—21no—
77C2×C3×M16—21no—
78C2×C3×M16—61no—
79C3×C4×D4—61no—
80C3×C4×D4—61no—
81C3×C4×D4—61no—
82C3×C4×D4—61no—
83C2×C3×C4⋊C4—31no—
84C2×C3×C22⋊C4—31no—
85C2×C3×C22⋊C4—31no—
86C2×C3×C42—31no—
87C2×#217C3×C411yescharacteristic
88C4×#67—21no—
89C4×#67—21no—
90C4×#67—21no—
91C4×#67—21no—
92C2×(C2×D3)⋊C4C22×C311yescharacteristic
93C2×Dic3⋊C4C22×C311yescharacteristic
94C2×C4×Dic3C3×C411yescharacteristic
95C2×#208—21no—
96C3×C22⋊C8—22no—
97C3×C22⋊C8—21no—
98C3×C22⋊C8—21no—
99C3×C22⋊C8—22no—
100C3×C22⋊C8—32no—
101C3×C22⋊C8—61no—
102C3×C22⋊C8C2×D312yes—
103C3×C22⋊C8C22×C312yes—
104#830—21no—
105#830C22×C312yes—
106C23×C32—21no—
107C23×C32—21no—
108C23×C32—41no—
109C23×C32—41no—
110C23×C32C22⋊C411yescharacteristic
111C23×C32C22⋊C411yescharacteristic
112C23×C32C22×C411yescharacteristic
113C22×C3×D3—21no—
114C22×C3×D3—21no—
115C22×C3×D3—41no—
116C22×C3×D3—41no—
117C22×C3×D3M1612yescomplemented
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—41no—
122C2×C32×C4—41no—
123C2×C32×C4—41no—
124C2×C32×C4C22⋊C411yescharacteristic
125C2×C32×C4C22⋊C411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2×D411yescharacteristic
128C3×#67—24no—
129C3×#67—24no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—41no—
135C2×C3×Dic3—41no—
136C2×C3×Dic3M1612yescomplemented
137C3×C4×D3—42no—
138C3×C4×D3—42no—
139C32×C8—42no—
140C3×C3⋊C8—42no—
141C2×C4×D4—31no—
142#407—31no—
143#407C3×D311yescharacteristic, complemented
144#324—32no—
145#323—32no—
146C24×C3—21no—
147C24×C3—31no—
148C24×C3C4×D311yescharacteristic
149C24×C3C2×C3×C411yescharacteristic
150C23×D3C2×C3×C411yescharacteristic
151C2×C3×D4—62no—
152C2×C3×D4—62no—
153C2×C3×D4—61no—
154C2×C3×D4—61no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—31no—
162C22×C3×C4—31no—
163C22×C3×C4—31no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—44no—
167C22×C3×C4—61no—
168C22×C3×C4—61no—
169C22×C3×C4—122no—
170C22×C3×C4C4×D311yescharacteristic
171C22×C3×C4D1211yescharacteristic
172C22×C3×C4D1211yescharacteristic
173C22×C3×C4#6711yescharacteristic
174C22×C3×C4#6711yescharacteristic
175C22×C3×C4C2×C3×C411yescharacteristic
176C22×C3×C4C3×D411yescharacteristic
177C22×C3×C4C3×D411yescharacteristic
178C22×C3×C4C3×D411yescharacteristic
179C22×C3×C4C3×D411yescharacteristic
180C22×C3×C4C22×D311yescharacteristic
181C22×C3×C4C23×C311yescharacteristic
182C2×#67—21no—
183C2×#67—21no—
184C2×#67—22no—
185C2×#67—22no—
186C22×Dic3C2×C3×C411yescharacteristic
187C22×Dic3C2×C3×C411yescharacteristic
188C22×Dic3C2×C3×C411yescharacteristic
189C2×C4×D3—21no—
190C2×C4×D3—21no—
191C2×C4×D3—42no—
192C3×M16—44no—
193C3×M16—42no—
194C3×M16—42no—
195C3×M16—122no—
196C2×C3×C8—24no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—62no—
200C3×C4⋊C4—61no—
201C3×C4⋊C4—61no—
202C3×C22⋊C4—61no—
203C3×C22⋊C4—61no—
204C3×C22⋊C4—61no—
205C3×C22⋊C4—61no—
206C3×C42—61no—
207C3×C42—61no—
208#217—21no—
209#217—21no—
210(C2×D3)⋊C4—21no—
211(C2×D3)⋊C4—21no—
212Dic3⋊C4—21no—
213Dic3⋊C4—21no—
214C4×Dic3—21no—
215C4×Dic3—21no—
216#208—42no—
217C2×C3⋊C8—22no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32C2×C22⋊C411yescharacteristic
229C22×C32C2×M1611yescharacteristic
230C22×C32C2×M1611yescharacteristic
231C2×C3×D3—22no—
232C2×C3×D3—42no—
233C2×C3×D3—41no—
234C2×C3×D3—41no—
235C2×C3×D3—42no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4#9912yes—
240C3×Dic3—22no—
241C3×Dic3—41no—
242C3×Dic3—41no—
243C22×D4—31no—
244C23×C4—31no—
245C23×C4C2×C3×D311yescharacteristic
246C2×M16—21no—
247C2×M16—61no—
248C4×D4—61no—
249C4×D4—61no—
250C4×D4—61no—
251C4×D4—61no—
252C2×C4⋊C4—31no—
253C2×C22⋊C4—31no—
254C2×C22⋊C4—31no—
255C2×C42—31no—
256C22⋊C8—21no—
257C22⋊C8—32no—
258C22⋊C8—61no—
259C22⋊C8C2×C3×D312yes—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—32no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—61no—
277C23×C3—61no—
278C23×C3—121no—
279C23×C3—121no—
280C23×C3(C2×D3)⋊C411yescharacteristic
281C23×C3(C2×D3)⋊C411yescharacteristic
282C23×C3C3×C22⋊C411yescharacteristic
283C23×C3C3×C22⋊C411yescharacteristic
284C23×C3C2×C4×D311yescharacteristic
285C23×C3C22×C3×C411yescharacteristic
286C22×D3—21no—
287C22×D3—21no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3C3×M1612yescomplemented
291C3×D4—64no—
292C3×D4—64no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—32no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—122no—
321C2×C3×C4—122no—
322C2×C3×C4—121no—
323C2×C3×C4—121no—
324C2×C3×C4(C2×D3)⋊C411yescharacteristic
325C2×C3×C4(C2×D3)⋊C411yescharacteristic
326C2×C3×C4C3×C22⋊C411yescharacteristic
327C2×C3×C4C3×C22⋊C411yescharacteristic
328C2×C3×C4C2×D1211yescharacteristic
329C2×C3×C4C2×#6711yescharacteristic
330C2×C3×C4C2×C3×D411yescharacteristic
331C2×C3×C4C2×C3×D411yescharacteristic
332#67—24no—
333#67—24no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3C3×M1612yescomplemented
341C4×D3—42no—
342C4×D3—42no—
343C3×C8—44no—
344C3×C8—42no—
345C3×C8—42no—
346C3×C8—122no—
347C3⋊C8—42no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32#40711yescharacteristic
353C2×C32C2×#9911yescharacteristic
354C2×C32#41311yescharacteristic
355C3×D3—42no—
356C24—31no—
357C24C3×C4×D311yescharacteristic
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—61no—
361C2×D4—61no—
362C22×C4—31no—
363C22×C4—31no—
364C22×C4—31no—
365C22×C4—42no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—122no—
369C22×C4C3×C4×D311yescharacteristic
370C22×C4C3×D1211yescharacteristic
371C22×C4C3×D1211yescharacteristic
372C22×C4C3×#6711yescharacteristic
373C22×C4C3×#6711yescharacteristic
374C22×C4C22×C3×D311yescharacteristic
375M16—42no—
376M16—122no—
377C2×C8—22no—
378C2×C8—62no—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C22⋊C4—61no—
382C22⋊C4—61no—
383C22⋊C4—61no—
384C22⋊C4—61no—
385C42—61no—
386C42—61no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3—62no—
421C22×C3—122no—
422C22×C3—122no—
423C22×C3—121no—
424C22×C3—121no—
425C22×C3D3×M1611yescharacteristic
426C22×C3D3×M1611yescharacteristic
427C22×C3C2×(C2×D3)⋊C411yescharacteristic
428C22×C3C2×C3×C22⋊C411yescharacteristic
429C22×C3C2×C3×M1611yescharacteristic
430C22×C3C2×C3×M1611yescharacteristic
431C2×D3—22no—
432C2×D3—42no—
433C2×D3—41no—
434C2×D3—41no—
435C2×D3—42no—
436C3×C4—22no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—22no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—62no—
447C3×C4—121no—
448C3×C4—121no—
449C3×C4#80612yes—
450C3×C4C3×#9912yes—
451Dic3—22no—
452Dic3—41no—
453Dic3—41no—
454C32#146511yescharacteristic, complemented, Sylow
455C23—21no—
456C23—21no—
457C23—32no—
458C23—41no—
459C23—41no—
460C23—61no—
461C23—61no—
462C23—121no—
463C23—121no—
464C23C3×(C2×D3)⋊C411yescharacteristic
465C23C3×(C2×D3)⋊C411yescharacteristic
466C23C2×C3×C4×D311yescharacteristic
467D4—64no—
468D4—64no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—32no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4C3×(C2×D3)⋊C411yescharacteristic
485C2×C4C3×(C2×D3)⋊C411yescharacteristic
486C2×C4C2×C3×D1211yescharacteristic
487C2×C4C2×C3×#6711yescharacteristic
488C8—42no—
489C8—122no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—41no—
498C2×C3—41no—
499C2×C3—42no—
500C2×C3—42no—
501C2×C3—41no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—41no—
505C2×C3—122no—
506C2×C3C2×#80611yescharacteristic
507C2×C3#512611yescharacteristic
508C2×C3#513211yescharacteristic
509C2×C3C3×#40711yescharacteristic
510C2×C3C2×C3×#9911yescharacteristic
511C2×C3C3×#41311yescharacteristic
512D3—42no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—42no—
522C22—42no—
523C22—62no—
524C22—122no—
525C22—122no—
526C22—121no—
527C22—121no—
528C22C3×D3×M1611yescharacteristic
529C22C3×D3×M1611yescharacteristic
530C22C2×C3×(C2×D3)⋊C411yescharacteristic
531C4—21no—
532C4—21no—
533C4—21no—
534C4—62no—
535C4—121no—
536C4—121no—
537C4C3×#80612yes—
538C3—21no—
539C3#6996811yescharacteristic, direct product factor
540C3C3×#146511yescharacteristic, complemented
541C2—21no—
542C2—21no—
543C2—41no—
544C2—41no—
545C2—122no—
546C2C2×C3×#80611yescharacteristic
547C2C3×#512611yescharacteristic
548C2C3×#513211yescharacteristic
subgroupG/H
centerC22×C3C2×(C2×D3)⋊C4
commutatorC23×C3C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#407C2
socleC22×C32C2×C22⋊C4
prime radicalC22×C3×#67C4
cosocleC22×C3×C4C23×C3
hypercenterC3×#407D3
hypocenterC3C3×#1465
LCSG ▷ C23×C3 ▷ C2×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. C3×#5207C2semidirect
4. C3×#4469C2semidirect
5. C3×#4468C2semidirect
7. #69968C3direct (N × H)
29. C3×#407D3semidirect
30. C3×#407C2×C3semidirect
31. #5207C2×C3semidirect
34. #4469C2×C3semidirect
35. #4468C2×C3semidirect
117. C22×C3×D3M16semidirect
136. C2×C3×Dic3M16semidirect
143. #407C3×D3semidirect
290. C22×D3C3×M16semidirect
340. C2×Dic3C3×M16semidirect
454. C32#1465semidirect
539. C3#69968direct (N × H)
540. C3C3×#1465semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×#672
C32×#4072
C3×#52072
C3×#44692
C3×#44682
C3×#14653
#699683