Finite Group Databasebeta

← C3×#83670 #11898338

Subgroups — C3×#83670 #11898338

Order 1152.

826 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#556C211yesmaximal, characteristic, complemented
2C3×#5584C211yesmaximal, characteristic, complemented
3C3×#5092C211yesmaximal, characteristic, complemented
4C3×#5083C211yesmaximal, characteristic, complemented
5C3×#5059C212yesmaximal
6C3×#5054C212yesmaximal, complemented
7C3×C4×#216C212yesmaximal, complemented
8C3×#5028C212yesmaximal, complemented
9C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
10C3×#5019C211yesmaximal, characteristic, complemented
11C3×C4×#213C211yesmaximal, characteristic
12C3×#3393—31nomaximal
13#83670C311yesmaximal, characteristic, direct product factor
14C32×C4⋊Q8C2211yescharacteristic
15C32×#126—22no—
16C32×#124C2212yes—
17C32×C22⋊Q8—21no—
18C32×C4×Q8C2212yes—
19C32×C4×Q8C2211yescharacteristic
20C32×C4×D4C2211yescharacteristic
21C32×#117—21no—
22C3×#946—21no—
23C2×C3×#216C2212yes—
24C2×C3×#216C2212yes—
25C2×C3×#215C2212yes—
26C2×C3×#214C2212yes—
27C2×C3×#213C2211yescharacteristic
28C3×#898—22no—
29C3×#892—21no—
30C3×#875C2212yes—
31C3×C4×D12C2211yescharacteristic
32C3×#869C2211yescharacteristic
33C3×C4×Dic6C2212yes—
34C3×#835C2212yes—
35C3×#832C2211yescharacteristic
36C3×#810C2212yes—
37C3×#809C2211yescharacteristic
38C3×#807C2211yescharacteristic
39C3×C4⋊(C3⋊C8)C2211yescharacteristic
40C3×C4⋊(C3⋊C8)C2212yes—
41C3×C4×C3⋊C8C2211yescharacteristic
42C3×C4×C3⋊C8C2212yes—
43C3×#556—21no—
44C3×#556—31no—
45C3×#556D311yescharacteristic, complemented
46C3×#556C2×C311yescharacteristic, complemented
47#5584C2×C311yescharacteristic, complemented
48C3×#495—32no—
49C3×#494—31no—
50C3×#464—32no—
51C3×#464—32no—
52C3×#461—31no—
53C3×#461—31no—
54C3×C8×Q8—31no—
55C3×C4×Q16—32no—
56C3×C4×D8—31no—
57#5092C2×C311yescharacteristic, complemented
58#5083C2×C311yescharacteristic, complemented
59#5059C2×C312yes—
60#5054C2×C312yescomplemented
61C4×#216C2×C312yescomplemented
62#5028C2×C312yescomplemented
63Q8×C3⋊C8C2×C311yescharacteristic, complemented
64#5019C2×C311yescharacteristic, complemented
65C4×#213C2×C311yescharacteristic
66C2×C32×Q8D411yescharacteristic
67C2×C32×Q8C2312yes—
68C2×C32×D4C2311yescharacteristic
69C22×C32×C4—21no—
70C2×C3×D12C2311yescharacteristic
71C2×C3×C4×D3—21no—
72C2×C3×Dic6C2312yes—
73C32×C4⋊C4—21no—
74C32×C4⋊C4—22no—
75C32×C4⋊C4—21no—
76C32×C4⋊C4D411yescharacteristic
77C32×C4⋊C4D412yes—
78C32×C4⋊C4C2311yescharacteristic
79C32×C22⋊C4—22no—
80C32×C22⋊C4—21no—
81C32×C22⋊C4—22no—
82C32×C42—22no—
83C32×C42C2311yescharacteristic
84C32×C42C2312yes—
85C3×#216—24no—
86C3×#216—42no—
87C3×#215—42no—
88C3×#214—42no—
89C3×#213—22no—
90C3×(C2×D3)⋊C4—22no—
91C3×(C2×D3)⋊C4—21no—
92C3×(C2×D3)⋊C4—22no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3C2311yescharacteristic
96C3×Dic3⋊C4—22no—
97C3×C4×Dic3—22no—
98C2×C3×C3⋊C8C2312yes—
99C2×C3×C3⋊C8C2311yescharacteristic
100C2×C3×C3⋊C8C2311yescharacteristic
101#3393—31noSylow
102C2×C3×Q16—32no—
103C2×C3×Q16—32no—
104C2×C3×SD16—32no—
105C2×C3×SD16—32no—
106C2×C3×D8—31no—
107C3×C4⋊Q8—21no—
108C3×C4⋊Q8—31no—
109C3×C4⋊Q8C2×D311yescharacteristic
110C3×C4⋊Q8C22×C311yescharacteristic
111C3×#126—24no—
112C3×#126—22no—
113C3×#126—22no—
114C3×#126—62no—
115C3×#124—22no—
116C3×#124—32no—
117C3×#124C2×D312yes—
118C3×#124C22×C312yes—
119C3×C22⋊Q8—22no—
120C3×C22⋊Q8—21no—
121C3×C22⋊Q8—21no—
122C3×C22⋊Q8—61no—
123C3×C4×Q8—22no—
124C3×C4×Q8—21no—
125C3×C4×Q8—32no—
126C3×C4×Q8C2×D312yes—
127C3×C4×Q8C2×D311yescharacteristic
128C3×C4×Q8C22×C312yes—
129C3×C4×Q8C22×C311yescharacteristic
130C3×C4×D4—21no—
131C3×C4×D4—31no—
132C3×C4×D4C2×D311yescharacteristic
133C3×C4×D4C22×C311yescharacteristic
134C3×#117—22no—
135C3×#117—21no—
136C3×#117—21no—
137C3×#117—61no—
138#946—21no—
139C2×#216C22×C312yes—
140C2×#216C22×C312yes—
141C2×#215C22×C312yes—
142C2×#214C22×C312yes—
143C2×#213C22×C311yescharacteristic
144#898—22no—
145#892—21no—
146#875C22×C312yes—
147C4×D12C22×C311yescharacteristic
148#869C22×C311yescharacteristic
149C4×Dic6C22×C312yes—
150C3×#107—31no—
151C3×C4⋊C8—31no—
152C3×C4⋊C8—32no—
153C3×#103—32no—
154C3×#103—32no—
155C3×#102—31no—
156C3×#102—31no—
157C3×C4×C8—31no—
158C3×C4×C8—32no—
159#835C22×C312yes—
160#832C22×C311yescharacteristic
161#810C22×C312yes—
162#809C22×C311yescharacteristic
163#807C22×C311yescharacteristic
164C4⋊(C3⋊C8)C22×C311yescharacteristic
165C4⋊(C3⋊C8)C22×C312yes—
166C4×C3⋊C8C22×C311yescharacteristic
167C4×C3⋊C8C22×C312yes—
168C23×C32—21no—
169C22×C3×D3—21no—
170C32×Q8—22no—
171C32×Q8—22no—
172C32×Q8—22no—
173C32×D4—21no—
174C32×D4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—22no—
177C2×C32×C4—42no—
178C2×C32×C4C2×D412yes—
179C2×C32×C4C2×D412yes—
180C2×C32×C4C2×D411yescharacteristic
181C2×C32×C4C2×D411yescharacteristic
182C2×C32×C4C2411yescharacteristic
183C2×C3×Dic3—21no—
184C2×C3×Dic3—22no—
185C3×D12—21no—
186C3×D12—21no—
187C3×C4×D3—42no—
188C3×Dic6—22no—
189C3×Dic6—22no—
190C3×C3⋊C8—22no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8#4112yes—
193#556—31no—
194#556C3×D311yescharacteristic, complemented
195#495—32no—
196#494—31no—
197#464—32no—
198#464—32no—
199#461—31no—
200#461—31no—
201C8×Q8—31no—
202C4×Q16—32no—
203C4×D8—31no—
204C2×C3×Q8—21no—
205C2×C3×Q8—22no—
206C2×C3×Q8—32no—
207C2×C3×Q8#6711yescharacteristic
208C2×C3×Q8C3×D411yescharacteristic
209C2×C3×Q8C22×D312yes—
210C2×C3×Q8C23×C312yes—
211C2×C3×D4—21no—
212C2×C3×D4—31no—
213C2×C3×D4C22×D311yescharacteristic
214C2×C3×D4C23×C311yescharacteristic
215C22×C3×C4—22no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—61no—
219C2×D12C23×C311yescharacteristic
220C2×C4×D3—21no—
221C2×Dic6C23×C312yes—
222C3×Q16—64no—
223C3×Q16—122no—
224C3×SD16—122no—
225C3×SD16—122no—
226C3×D8—62no—
227C2×C3×C8—31no—
228C2×C3×C8—31no—
229C2×C3×C8—32no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—62no—
244C3×C4⋊C4—61no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4#6711yescharacteristic
247C3×C4⋊C4#6712yes—
248C3×C4⋊C4C3×D411yescharacteristic
249C3×C4⋊C4C3×D412yes—
250C3×C4⋊C4C22×D311yescharacteristic
251C3×C4⋊C4C23×C311yescharacteristic
252C3×C22⋊C4—24no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—21no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—21no—
260C3×C22⋊C4—22no—
261C3×C22⋊C4—62no—
262C3×C22⋊C4—61no—
263C3×C22⋊C4—62no—
264C3×C42—24no—
265C3×C42—22no—
266C3×C42—22no—
267C3×C42—21no—
268C3×C42—22no—
269C3×C42—62no—
270C3×C42C22×D311yescharacteristic
271C3×C42C22×D312yes—
272C3×C42C23×C312yes—
273C3×C42C23×C311yescharacteristic
274#216—24no—
275#216—42no—
276#215—42no—
277#214—42no—
278#213—22no—
279(C2×D3)⋊C4—21no—
280(C2×D3)⋊C4—22no—
281(C2×D3)⋊C4—22no—
282C4⋊Dic3—21no—
283C4⋊Dic3—21no—
284C4⋊Dic3C23×C311yescharacteristic
285Dic3⋊C4—22no—
286C4×Dic3—22no—
287C2×C3⋊C8C23×C311yescharacteristic
288C2×C3⋊C8C23×C312yes—
289C2×C3⋊C8C23×C311yescharacteristic
290C22×C32—41no—
291C22×C32—41no—
292C22×C32—41no—
293C22×C32C22×D411yescharacteristic
294C2×C3×D3—41no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C32×C4—22no—
298C32×C4—22no—
299C32×C4—21no—
300C32×C4—42no—
301C32×C4—41no—
302C32×C4#13512yes—
303C32×C4C2×#4111yescharacteristic
304C32×C4H(5,𝔽2)11yescharacteristic
305C3×Dic3—42no—
306C3×Dic3—41no—
307C2×Q16—32no—
308C2×Q16—32no—
309C2×SD16—32no—
310C2×SD16—32no—
311C2×D8—31no—
312C4⋊Q8—31no—
313C4⋊Q8C2×C3×D311yescharacteristic
314#126—22no—
315#126—62no—
316#124—32no—
317#124C2×C3×D312yes—
318C22⋊Q8—21no—
319C22⋊Q8—61no—
320C4×Q8—32no—
321C4×Q8C2×C3×D312yes—
322C4×Q8C2×C3×D311yescharacteristic
323C4×D4—31no—
324C4×D4C2×C3×D311yescharacteristic
325#117—21no—
326#117—61no—
327#107—31no—
328C4⋊C8—31no—
329C4⋊C8—32no—
330#103—32no—
331#103—32no—
332#102—31no—
333#102—31no—
334C4×C8—31no—
335C4×C8—32no—
336C23×C3—22no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—61no—
340C22×D3—21no—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—22no—
346C3×Q8—22no—
347C3×Q8—42no—
348C3×Q8—42no—
349C3×Q8—42no—
350C3×Q8—62no—
351C3×Q8—62no—
352C3×D4—21no—
353C3×D4—21no—
354C3×D4—21no—
355C3×D4—21no—
356C3×D4—41no—
357C3×D4—41no—
358C3×D4—61no—
359C3×D4—61no—
360C2×C3×C4—22no—
361C2×C3×C4—24no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—44no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—61no—
375C2×C3×C4—62no—
376C2×C3×C4—122no—
377C2×C3×C4C2×#6712yes—
378C2×C3×C4C2×#6711yescharacteristic
379C2×C3×C4C2×#6712yes—
380C2×C3×C4C2×#6711yescharacteristic
381C2×C3×C4C2×C3×D412yes—
382C2×C3×C4C2×C3×D412yes—
383C2×C3×C4C2×C3×D411yescharacteristic
384C2×C3×C4C2×C3×D411yescharacteristic
385C2×C3×C4C23×D311yescharacteristic
386C2×C3×C4C24×C311yescharacteristic
387C2×Dic3—21no—
388C2×Dic3—22no—
389D12—21no—
390D12—21no—
391C4×D3—42no—
392Dic6—22no—
393Dic6—22no—
394C3×C8—32no—
395C3×C8—62no—
396C3×C8—61no—
397C3⋊C8—22no—
398C3⋊C8—21no—
399C3⋊C8C3×#4112yes—
400C2×C32—81no—
401C2×C32#55011yescharacteristic
402C2×C32C2×#13511yescharacteristic
403C2×C32#57811yescharacteristic
404C3×D3—81no—
405C2×Q8—32no—
406C2×Q8C3×#6711yescharacteristic
407C2×Q8C22×C3×D312yes—
408C2×D4—31no—
409C2×D4C22×C3×D311yescharacteristic
410C22×C4—21no—
411C22×C4—61no—
412Q16—64no—
413Q16—122no—
414SD16—122no—
415SD16—122no—
416D8—62no—
417C2×C8—31no—
418C2×C8—31no—
419C2×C8—32no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—31no—
424C4⋊C4—61no—
425C4⋊C4—62no—
426C4⋊C4—61no—
427C4⋊C4C3×#6711yescharacteristic
428C4⋊C4C3×#6712yes—
429C4⋊C4C22×C3×D311yescharacteristic
430C22⋊C4—22no—
431C22⋊C4—21no—
432C22⋊C4—22no—
433C22⋊C4—62no—
434C22⋊C4—61no—
435C22⋊C4—62no—
436C42—22no—
437C42—62no—
438C42C22×C3×D311yescharacteristic
439C42C22×C3×D312yes—
440C22×C3—21no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—41no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—41no—
450C22×C3—121no—
451C22×C3—121no—
452C22×C3—121no—
453C22×C3C22×#6711yescharacteristic
454C22×C3C22×C3×D411yescharacteristic
455C2×D3—41no—
456C2×D3—41no—
457C2×D3—41no—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—21no—
461C3×C4—22no—
462C3×C4—22no—
463C3×C4—21no—
464C3×C4—22no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—42no—
468C3×C4—42no—
469C3×C4—41no—
470C3×C4—42no—
471C3×C4—44no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—42no—
475C3×C4—42no—
476C3×C4—121no—
477C3×C4—122no—
478C3×C4#95012yes—
479C3×C4C3×#13512yes—
480C3×C4D3×#4111yescharacteristic
481C3×C4#100911yescharacteristic
482C3×C4C2×C3×#4111yescharacteristic
483C3×C4C3×H(5,𝔽2)11yescharacteristic
484Dic3—41no—
485Dic3—42no—
486C32#339311yescharacteristic, complemented, Sylow
487C23—21no—
488C23—61no—
489Q8—22no—
490Q8—22no—
491Q8—22no—
492Q8—62no—
493Q8—62no—
494D4—21no—
495D4—21no—
496D4—61no—
497D4—61no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—42no—
501C2×C4—61no—
502C2×C4—62no—
503C2×C4—122no—
504C2×C4C2×C3×#6712yes—
505C2×C4C2×C3×#6711yescharacteristic
506C2×C4C2×C3×#6712yes—
507C2×C4C2×C3×#6711yescharacteristic
508C2×C4C23×C3×D311yescharacteristic
509C8—32no—
510C8—62no—
511C8—61no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—82no—
516C2×C3—81no—
517C2×C3—81no—
518C2×C3—241no—
519C2×C3C2×#95011yescharacteristic
520C2×C3#583411yescharacteristic
521C2×C3#583811yescharacteristic
522C2×C3C3×#55011yescharacteristic
523C2×C3C2×C3×#13511yescharacteristic
524C2×C3C3×#57811yescharacteristic
525D3—81no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—121no—
530C22—121no—
531C22—121no—
532C22C22×C3×#6711yescharacteristic
533C4—22no—
534C4—22no—
535C4—21no—
536C4—41no—
537C4—42no—
538C4—121no—
539C4—122no—
540C4C3×#95012yes—
541C4C3×D3×#4111yescharacteristic
542C4C3×#100911yescharacteristic
543C3—21no—
544C3#8367011yescharacteristic, direct product factor
545C3C3×#339311yescharacteristic, complemented
546C2—81no—
547C2—241no—
548C2C2×C3×#95011yescharacteristic
549C2C3×#583411yescharacteristic
550C2C3×#583811yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#556C2
socleC22×C32C22×D4
prime radicalC2×C3×#213C22
hypercenterC3×#556D3
hypocenterC3C3×#3393
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ C3×#556

Complements

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

NH ≅ G/Ntype
1. C32×#556C2semidirect
2. C3×#5584C2semidirect
3. C3×#5092C2semidirect
4. C3×#5083C2semidirect
6. C3×#5054C2semidirect
7. C3×C4×#216C2semidirect
8. C3×#5028C2semidirect
9. C3×Q8×C3⋊C8C2semidirect
10. C3×#5019C2semidirect
13. #83670C3direct (N × H)
45. C3×#556D3semidirect
46. C3×#556C2×C3semidirect
47. #5584C2×C3semidirect
57. #5092C2×C3semidirect
58. #5083C2×C3semidirect
60. #5054C2×C3semidirect
61. C4×#216C2×C3semidirect
62. #5028C2×C3semidirect
63. Q8×C3⋊C8C2×C3semidirect
64. #5019C2×C3semidirect
194. #556C3×D3semidirect
486. C32#3393semidirect
544. C3#83670direct (N × H)
545. C3C3×#3393semidirect

Maximal subgroups

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

subgroupindex
C32×#5562
C3×#55842
C3×#50922
C3×#50832
C3×#50592
C3×#50542
C3×C4×#2162
C3×#50282
C3×Q8×C3⋊C82
C3×#50192
C3×C4×#2132
C3×#33933
#836703