Finite Group Databasebeta

← C3×#80416 #11895836

Subgroups — C3×#80416 #11895836

Order 1152.

804 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5732C211yesmaximal, characteristic, complemented
2C3×D4×Dic6C211yesmaximal, characteristic, complemented
3C32×#461C211yesmaximal, characteristic, complemented
4C3×#5165C212yesmaximal
5C3×#5164C212yesmaximal, complemented
6C3×#5079C211yesmaximal, characteristic, complemented
7C3×C4×#213C211yesmaximal, characteristic
8C3×#4846C211yesmaximal, characteristic, complemented
9C3×#4831C211yesmaximal, characteristic, complemented
10C3×#4794C212yesmaximal, complemented
11C3×#4793C212yesmaximal, complemented
12C3×#3367—31nomaximal
13#80416C311yesmaximal, characteristic, direct product factor
14C22×C3×Dic6—21no—
15C2×C32×SD16C2212yes—
16C32×C4⋊Q8C2211yescharacteristic
17C32×C4×D4C2211yescharacteristic
18C3×#947C2212yes—
19C3×Q8×Dic3C2212yes—
20C3×D4×Dic3C2212yes—
21C2×C3×#213C2211yescharacteristic
22C3×#924—21no—
23C3×#898—22no—
24C3×#897—21no—
25C3×#892—21no—
26C3×#888C2212yes—
27C3×#878—22no—
28C3×C4×D12C2211yescharacteristic
29C3×#869C2211yescharacteristic
30C3×C4×Dic6C2211yescharacteristic
31C32×C4⋊C8C2211yescharacteristic
32C32×#103C2212yes—
33C3×#835C2212yes—
34C3×#835C2212yes—
35C3×#832C2211yescharacteristic
36C3×#821C2212yes—
37C3×#818C2212yes—
38C3×#815C2212yes—
39C3×Dic3⋊C8C2212yes—
40C3×#809C2211yescharacteristic
41C3×#807C2211yescharacteristic
42C3×C4×C3⋊C8C2211yescharacteristic
43C3×#556—31no—
44C3×D4×Q8—31no—
45#5732C2×C311yescharacteristic, complemented
46D4×Dic6C2×C311yescharacteristic, complemented
47C3×#489—32no—
48C3×#487—31no—
49C3×#477—32no—
50C3×#474—31no—
51C3×#463—32no—
52C3×#461—21no—
53C3×#461D311yescharacteristic, complemented
54C3×#461C2×C311yescharacteristic, complemented
55C3×#446—31no—
56C3×#440—32no—
57C3×C4×D8—31no—
58#5165C2×C312yes—
59#5164C2×C312yescomplemented
60#5079C2×C311yescharacteristic, complemented
61C4×#213C2×C311yescharacteristic
62#4846C2×C311yescharacteristic, complemented
63#4831C2×C311yescharacteristic, complemented
64#4794C2×C312yescomplemented
65#4793C2×C312yescomplemented
66C2×C32×Q8C2312yes—
67C2×C32×D4C2311yescharacteristic
68C22×C32×C4—21no—
69C22×C3×Dic3—22no—
70C2×C3×D12C2311yescharacteristic
71C2×C3×C4×D3—21no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6—22no—
74C2×C3×Dic6—44no—
75C2×C3×Dic6D411yescharacteristic
76C32×SD16—24no—
77C2×C32×C8C2312yes—
78C32×C4⋊C4—21no—
79C32×C4⋊C4—21no—
80C32×C4⋊C4C2311yescharacteristic
81C32×C22⋊C4—21no—
82C32×C42C2311yescharacteristic
83C3×#217—22no—
84C3×#213—22no—
85C3×(C2×D3)⋊C4—21no—
86C3×(C2×D3)⋊C4—22no—
87C3×(C2×D3)⋊C4—22no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—22no—
90C3×C4⋊Dic3D411yescharacteristic
91C3×C4⋊Dic3C2312yes—
92C3×C4⋊Dic3C2311yescharacteristic
93C3×Dic3⋊C4—22no—
94C3×Dic3⋊C4D412yes—
95C3×C4×Dic3—22no—
96C3×C4×Dic3C2312yes—
97C2×C3×C3⋊C8C2311yescharacteristic
98C2×C3×C3⋊C8C2311yescharacteristic
99#3367—31noSylow
100C22×C3×Q8—61no—
101C22×Dic6—21no—
102C2×C3×SD16—22no—
103C2×C3×SD16C2×D312yes—
104C2×C3×SD16C22×C312yes—
105C2×C3×D8—31no—
106C3×C4⋊Q8—21no—
107C3×C4⋊Q8—32no—
108C3×C4⋊Q8—31no—
109C3×C4⋊Q8C2×D311yescharacteristic
110C3×C4⋊Q8C22×C311yescharacteristic
111C3×#126—62no—
112C3×#124—32no—
113C3×C22⋊Q8—62no—
114C3×C22⋊Q8—61no—
115C3×C22⋊Q8—61no—
116C3×C4×Q8—31no—
117C3×C4×Q8—32no—
118C3×C4×D4—21no—
119C3×C4×D4—32no—
120C3×C4×D4—31no—
121C3×C4×D4C2×D311yescharacteristic
122C3×C4×D4C22×C311yescharacteristic
123C3×#117—61no—
124#947C22×C312yes—
125Q8×Dic3C22×C312yes—
126D4×Dic3C22×C312yes—
127C2×#213C22×C311yescharacteristic
128#924—21no—
129#898—22no—
130#897—21no—
131#892—21no—
132#888C22×C312yes—
133#878—22no—
134C4×D12C22×C311yescharacteristic
135#869C22×C311yescharacteristic
136C4×Dic6C22×C311yescharacteristic
137C3×#107—32no—
138C3×#107—31no—
139C3×C4⋊C8—21no—
140C3×C4⋊C8—32no—
141C3×C4⋊C8C2×D311yescharacteristic
142C3×C4⋊C8C22×C311yescharacteristic
143C3×#103—22no—
144C3×#103—32no—
145C3×#103—32no—
146C3×#103C2×D312yes—
147C3×#103C22×C312yes—
148C3×#102—31no—
149C3×#102—32no—
150C3×#102—31no—
151C3×#97—32no—
152C3×C4×C8—31no—
153#835C22×C312yes—
154#835C22×C312yes—
155#832C22×C311yescharacteristic
156#821C22×C312yes—
157#818C22×C312yes—
158#815C22×C312yes—
159Dic3⋊C8C22×C312yes—
160#809C22×C311yescharacteristic
161#807C22×C311yescharacteristic
162C4×C3⋊C8C22×C311yescharacteristic
163C23×C32—21no—
164C22×C3×D3—21no—
165C32×Q8—22no—
166C32×Q8—22no—
167C32×D4—21no—
168C32×D4#4112yes—
169C2×C32×C4—22no—
170C2×C32×C4—21no—
171C2×C32×C4—42no—
172C2×C32×C4C2×D411yescharacteristic
173C2×C32×C4C2×D411yescharacteristic
174C2×C32×C4C2411yescharacteristic
175C2×C3×Dic3—21no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—44no—
178C2×C3×Dic3C2×D412yes—
179C2×C3×Dic3C2×D412yes—
180C3×D12—21no—
181C3×D12—21no—
182C3×C4×D3—42no—
183C3×Dic6—22no—
184C3×Dic6—44no—
185C3×Dic6—42no—
186C32×C8—22no—
187C3×C3⋊C8—21no—
188C3×C3⋊C8—21no—
189#556—31no—
190D4×Q8—31no—
191#489—32no—
192#487—31no—
193#477—32no—
194#474—31no—
195#463—32no—
196#461C3×D311yescharacteristic, complemented
197#446—31no—
198#440—32no—
199C4×D8—31no—
200C2×C3×Q8—22no—
201C2×C3×Q8—31no—
202C2×C3×Q8—62no—
203C2×C3×Q8—61no—
204C2×C3×Q8—124no—
205C2×C3×Q8C22×D312yes—
206C2×C3×Q8C23×C312yes—
207C2×C3×D4—21no—
208C2×C3×D4—31no—
209C2×C3×D4C22×D311yescharacteristic
210C2×C3×D4C23×C311yescharacteristic
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—41no—
214C22×C3×C4—62no—
215C22×C3×C4—61no—
216C22×Dic3—22no—
217C2×D12C23×C311yescharacteristic
218C2×C4×D3—21no—
219C2×Dic6—21no—
220C2×Dic6—22no—
221C2×Dic6—44no—
222C2×Dic6C3×D411yescharacteristic
223C3×SD16—24no—
224C3×SD16—24no—
225C3×SD16—44no—
226C3×D8—62no—
227C2×C3×C8—22no—
228C2×C3×C8—31no—
229C2×C3×C8—31no—
230C2×C3×C8C22×D312yes—
231C2×C3×C8C23×C312yes—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—32no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—32no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—41no—
242C3×C4⋊C4—41no—
243C3×C4⋊C4—62no—
244C3×C4⋊C4—62no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4C22×D311yescharacteristic
247C3×C4⋊C4C23×C311yescharacteristic
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—41no—
251C3×C22⋊C4—62no—
252C3×C22⋊C4—62no—
253C3×C22⋊C4—62no—
254C3×C22⋊C4—61no—
255C3×C42—21no—
256C3×C42—32no—
257C3×C42—62no—
258C3×C42C22×D311yescharacteristic
259C3×C42C23×C311yescharacteristic
260#217—22no—
261#213—22no—
262(C2×D3)⋊C4—21no—
263(C2×D3)⋊C4—22no—
264(C2×D3)⋊C4—22no—
265C4⋊Dic3—21no—
266C4⋊Dic3—22no—
267C4⋊Dic3C3×D411yescharacteristic
268C4⋊Dic3C23×C311yescharacteristic
269C4⋊Dic3C23×C312yes—
270Dic3⋊C4—22no—
271Dic3⋊C4C3×D412yes—
272C4×Dic3—22no—
273C4×Dic3C23×C312yes—
274C2×C3⋊C8C23×C311yescharacteristic
275C2×C3⋊C8C23×C311yescharacteristic
276C22×C32—22no—
277C22×C32—41no—
278C22×C32—41no—
279C22×C32C22×D411yescharacteristic
280C2×C3×D3—41no—
281C2×C3×D3—41no—
282C2×C3×D3—41no—
283C32×C4—21no—
284C32×C4—42no—
285C32×C4—41no—
286C32×C4AGL(1,ℤ/8)12yes—
287C32×C4C2×#4111yescharacteristic
288C32×C42−1+411yescharacteristic
289C3×Dic3—22no—
290C3×Dic3—22no—
291C3×Dic3—42no—
292C3×Dic3—41no—
293C22×Q8—61no—
294C2×SD16C2×C3×D312yes—
295C2×D8—31no—
296C4⋊Q8—32no—
297C4⋊Q8—31no—
298C4⋊Q8C2×C3×D311yescharacteristic
299#126—62no—
300#124—32no—
301C22⋊Q8—62no—
302C22⋊Q8—61no—
303C22⋊Q8—61no—
304C4×Q8—31no—
305C4×Q8—32no—
306C4×D4—32no—
307C4×D4—31no—
308C4×D4C2×C3×D311yescharacteristic
309#117—61no—
310#107—32no—
311#107—31no—
312C4⋊C8—32no—
313C4⋊C8C2×C3×D311yescharacteristic
314#103—32no—
315#103—32no—
316#103C2×C3×D312yes—
317#102—31no—
318#102—32no—
319#102—31no—
320#97—32no—
321C4×C8—31no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—41no—
325C23×C3—61no—
326C22×D3—21no—
327C3×Q8—22no—
328C3×Q8—22no—
329C3×Q8—22no—
330C3×Q8—22no—
331C3×Q8—42no—
332C3×Q8—42no—
333C3×Q8—62no—
334C3×Q8—124no—
335C3×Q8—122no—
336C3×D4—21no—
337C3×D4—21no—
338C3×D4—22no—
339C3×D4—41no—
340C3×D4—61no—
341C3×D4—61no—
342C3×D4#23712yes—
343C3×D4C3×#4112yes—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—32no—
352C2×C3×C4—32no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—61no—
358C2×C3×C4—62no—
359C2×C3×C4—82no—
360C2×C3×C4—124no—
361C2×C3×C4—122no—
362C2×C3×C4D3×D411yescharacteristic
363C2×C3×C4D3×D411yescharacteristic
364C2×C3×C4C2×C3×D411yescharacteristic
365C2×C3×C4C2×C3×D411yescharacteristic
366C2×C3×C4C23×D311yescharacteristic
367C2×C3×C4C24×C311yescharacteristic
368C2×Dic3—21no—
369C2×Dic3—22no—
370C2×Dic3—44no—
371C2×Dic3C2×C3×D412yes—
372C2×Dic3C2×C3×D412yes—
373D12—21no—
374D12—21no—
375C4×D3—42no—
376Dic6—22no—
377Dic6—44no—
378Dic6—42no—
379C3×C8—22no—
380C3×C8—22no—
381C3×C8—42no—
382C3×C8—61no—
383C3×C8—61no—
384C3⋊C8—21no—
385C3⋊C8—21no—
386C2×C32—42no—
387C2×C32#54811yescharacteristic
388C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
389C2×C32#57811yescharacteristic
390C3×D3—81no—
391C2×Q8—31no—
392C2×Q8—62no—
393C2×Q8—61no—
394C2×Q8—124no—
395C2×Q8C22×C3×D312yes—
396C2×D4—31no—
397C2×D4C22×C3×D311yescharacteristic
398C22×C4—21no—
399C22×C4—62no—
400C22×C4—61no—
401SD16—24no—
402D8—62no—
403C2×C8—31no—
404C2×C8—31no—
405C2×C8C22×C3×D312yes—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—31no—
409C4⋊C4—32no—
410C4⋊C4—32no—
411C4⋊C4—31no—
412C4⋊C4—62no—
413C4⋊C4—62no—
414C4⋊C4—61no—
415C4⋊C4C22×C3×D311yescharacteristic
416C22⋊C4—21no—
417C22⋊C4—62no—
418C22⋊C4—62no—
419C22⋊C4—62no—
420C22⋊C4—61no—
421C42—32no—
422C42—62no—
423C42C22×C3×D311yescharacteristic
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—21no—
427C22×C3—42no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3—121no—
435C22×C3—121no—
436C22×C3—121no—
437C22×C3C2×D3×D411yescharacteristic
438C22×C3C22×C3×D411yescharacteristic
439C2×D3—41no—
440C2×D3—41no—
441C2×D3—41no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—22no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—62no—
453C3×C4—62no—
454C3×C4—81no—
455C3×C4—82no—
456C3×C4—122no—
457C3×C4—121no—
458C3×C4#91412yes—
459C3×C4C3×AGL(1,ℤ/8)12yes—
460C3×C4C2×#23711yescharacteristic
461C3×C4#101011yescharacteristic
462C3×C4C2×C3×#4111yescharacteristic
463C3×C4C3×2−1+411yescharacteristic
464Dic3—22no—
465Dic3—22no—
466Dic3—42no—
467Dic3—41no—
468C32#336711yescharacteristic, complemented, Sylow
469C23—21no—
470C23—61no—
471Q8—22no—
472Q8—22no—
473Q8—62no—
474Q8—124no—
475Q8—122no—
476D4—21no—
477D4—61no—
478D4—61no—
479D4C3×#23712yes—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—32no—
483C2×C4—32no—
484C2×C4—42no—
485C2×C4—61no—
486C2×C4—62no—
487C2×C4—124no—
488C2×C4—122no—
489C2×C4C3×D3×D411yescharacteristic
490C2×C4C3×D3×D411yescharacteristic
491C2×C4C23×C3×D311yescharacteristic
492C8—22no—
493C8—61no—
494C8—61no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—42no—
499C2×C3—42no—
500C2×C3—82no—
501C2×C3—241no—
502C2×C3#559811yescharacteristic
503C2×C3C2×#91411yescharacteristic
504C2×C3#577911yescharacteristic
505C2×C3C3×#54811yescharacteristic
506C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
507C2×C3C3×#57811yescharacteristic
508D3—81no—
509C22—22no—
510C22—41no—
511C22—41no—
512C22—121no—
513C22—121no—
514C22—121no—
515C22C2×C3×D3×D411yescharacteristic
516C4—21no—
517C4—41no—
518C4—42no—
519C4—62no—
520C4—62no—
521C4—122no—
522C4—121no—
523C4C3×#91412yes—
524C4C2×C3×#23711yescharacteristic
525C4C3×#101011yescharacteristic
526C3—21no—
527C3#8041611yescharacteristic, direct product factor
528C3C3×#336711yescharacteristic, complemented
529C2—42no—
530C2—241no—
531C2C3×#559811yescharacteristic
532C2C2×C3×#91411yescharacteristic
533C2C3×#577911yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#461C2
socleC22×C32C22×D4
prime radicalC2×C3×#213C22
hypercenterC3×#461D3
hypocenterC3C3×#3367
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#461

Complements

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

NH ≅ G/Ntype
1. C3×#5732C2semidirect
2. C3×D4×Dic6C2semidirect
3. C32×#461C2semidirect
5. C3×#5164C2semidirect
6. C3×#5079C2semidirect
8. C3×#4846C2semidirect
9. C3×#4831C2semidirect
10. C3×#4794C2semidirect
11. C3×#4793C2semidirect
13. #80416C3direct (N × H)
45. #5732C2×C3semidirect
46. D4×Dic6C2×C3semidirect
53. C3×#461D3semidirect
54. C3×#461C2×C3semidirect
59. #5164C2×C3semidirect
60. #5079C2×C3semidirect
62. #4846C2×C3semidirect
63. #4831C2×C3semidirect
64. #4794C2×C3semidirect
65. #4793C2×C3semidirect
196. #461C3×D3semidirect
468. C32#3367semidirect
527. C3#80416direct (N × H)
528. C3C3×#3367semidirect

Maximal subgroups

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

subgroupindex
C3×#57322
C3×D4×Dic62
C32×#4612
C3×#51652
C3×#51642
C3×#50792
C3×C4×#2132
C3×#48462
C3×#48312
C3×#47942
C3×#47932
C3×#33673
#804163