Finite Group Databasebeta

← C3×#83298 #11870406

Subgroups — C3×#83298 #11870406

Order 1152.

1139 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
1C32×#532C211yesmaximal, characteristic, complemented
2C2×C3×C4×D12C211yesmaximal, characteristic, complemented
3C3×#5087C212yesmaximal, complemented
4C3×#5072C212yesmaximal, complemented
5C3×#5036C212yesmaximal, complemented
6C3×#5033C212yesmaximal, complemented
7C3×C4×#213C214yesmaximal, complemented
8C3×#4925C211yesmaximal, characteristic, complemented
9C3×#3100—31nomaximal
10#83298C311yesmaximal, characteristic, direct product factor
11C2×C32×#41—21no—
12C22×C3×D12C2211yescharacteristic
13C22×C3×C4×D3—21no—
14C32×C4⋊Q8C2211yescharacteristic, complemented
15C32×#127C2211yescharacteristic, complemented
16C32×#124—21no—
17C32×#121—21no—
18C32×#121C2212yes—
19C32×C4×D4—21no—
20C32×C4×D4C2212yescomplemented
21C2×C32×C42C2211yescharacteristic
22C2×C3×#213C2214yes—
23C2×C3×(C2×D3)⋊C4—21no—
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C3×C4×D12—21no—
26C3×C4×D12—21no—
27C3×C4×D12C2212yes—
28C3×C4×D12C2212yes—
29C3×#832C2214yescomplemented
30C3×#830C2212yes—
31C3×#809C2214yes—
32C3×#807C2214yescomplemented
33C3×C4⋊(C3⋊C8)C2212yescomplemented
34C3×C4×C3⋊C8C2212yescomplemented
35C3×#532—21no—
36C3×#532D311yescharacteristic, complemented
37C3×#532C2×C311yescharacteristic, complemented
38C2×C3×C4×D4—31no—
39C2×C4×D12C2×C311yescharacteristic, complemented
40C3×#480—32no—
41C3×#474—32no—
42C3×#459—32no—
43C3×#447—32no—
44C3×C4×D8—34no—
45C3×#431—31no—
46#5087C2×C312yescomplemented
47#5072C2×C312yescomplemented
48#5036C2×C312yescomplemented
49#5033C2×C312yescomplemented
50C4×#213C2×C314yescomplemented
51#4925C2×C311yescharacteristic, complemented
52C23×C3×D3—21no—
53C32×#41—44no—
54C2×C32×Q8—21no—
55C2×C32×D4—22no—
56C2×C32×D4—21no—
57C2×C32×D4—21no—
58C2×C32×D4C2312yes—
59C22×C32×C4—22no—
60C22×C32×C4D411yescharacteristic
61C22×C32×C4D411yescharacteristic
62C22×C32×C4C2311yescharacteristic
63C22×C3×Dic3—21no—
64C2×C3×D12—21no—
65C2×C3×D12—24no—
66C2×C3×D12C2312yes—
67C2×C3×C4×D3—22no—
68C2×C3×C4×D3—22no—
69C2×C3×C4×D3—44no—
70C32×C4⋊C4—21no—
71C32×C4⋊C4C2312yes—
72C32×C22⋊C4—22no—
73C32×C22⋊C4—22no—
74C32×C42D411yescharacteristic
75C32×C42D411yescharacteristic
76C32×C42C2311yescharacteristic
77C32×C42C2311yescharacteristic
78C3×#213—28no—
79C3×(C2×D3)⋊C4—22no—
80C3×(C2×D3)⋊C4—22no—
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3C2312yes—
83C2×C3×C3⋊C8C2314yes—
84#3100—31noSylow
85C2×C3×#41—22no—
86C2×C3×#41—21no—
87C2×C3×#41—21no—
88C22×C3×D4—31no—
89C23×C3×C4—61no—
90C22×D12C22×C311yescharacteristic
91C22×C4×D3—21no—
92C2×C3×D8—34no—
93C3×C4⋊Q8—21no—
94C3×C4⋊Q8C2×D311yescharacteristic, complemented
95C3×C4⋊Q8C22×C311yescharacteristic, complemented
96C3×#127—21no—
97C3×#127C2×D311yescharacteristic, complemented
98C3×#127C22×C311yescharacteristic, complemented
99C3×#124—22no—
100C3×#124—21no—
101C3×#124—21no—
102C3×#121—22no—
103C3×#121—21no—
104C3×#121—21no—
105C3×#121—22no—
106C3×#121C2×D312yes—
107C3×#121C22×C312yes—
108C3×C4×D4—22no—
109C3×C4×D4—21no—
110C3×C4×D4—21no—
111C3×C4×D4—22no—
112C3×C4×D4—32no—
113C3×C4×D4—32no—
114C3×C4×D4—61no—
115C3×C4×D4—61no—
116C3×C4×D4C2×D312yescomplemented
117C3×C4×D4C22×C312yescomplemented
118C2×C3×C4⋊C4—31no—
119C2×C3×C22⋊C4—61no—
120C2×C3×C42—21no—
121C2×C3×C42C2×D311yescharacteristic
122C2×C3×C42C22×C311yescharacteristic
123C2×#213C22×C314yes—
124C2×(C2×D3)⋊C4—21no—
125C2×C4⋊Dic3C22×C311yescharacteristic
126C4×D12—21no—
127C4×D12—21no—
128C4×D12C22×C312yes—
129C4×D12C22×C312yes—
130C3×#107—34no—
131C3×C4⋊C8—32no—
132C3×#102—34no—
133C3×#102—34no—
134C3×C22⋊C8—32no—
135C3×C4×C8—32no—
136#832C22×C314yescomplemented
137#830C22×C312yes—
138#809C22×C314yes—
139#807C22×C314yescomplemented
140C4⋊(C3⋊C8)C22×C312yescomplemented
141C4×C3⋊C8C22×C312yescomplemented
142C23×C32—22no—
143C23×C32C2×D411yescharacteristic
144C22×C3×D3—24no—
145C22×C3×D3—22no—
146C22×C3×D3—42no—
147C22×C3×D3—42no—
148C32×Q8—42no—
149C32×D4—24no—
150C32×D4—42no—
151C32×D4—44no—
152C32×D4—42no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—41no—
156C2×C32×C4—44no—
157C2×C32×C4D814yescomplemented
158C2×C32×C4C2×D411yescharacteristic
159C2×C32×C4C2×D411yescharacteristic
160C2×C32×C4C2×D411yescharacteristic
161C2×C32×C4C2×D411yescharacteristic
162C2×C32×C4C2×D411yescharacteristic
163C2×C32×C4C2411yescharacteristic
164C2×C3×Dic3—22no—
165C2×C3×Dic3—42no—
166C3×D12—24no—
167C3×D12—22no—
168C3×D12#4114yescomplemented
169C3×C4×D3—44no—
170C3×C4×D3—44no—
171C3×C3⋊C8—24no—
172#532C3×D311yescharacteristic, complemented
173C2×C4×D4—31no—
174#480—32no—
175#474—32no—
176#459—32no—
177#447—32no—
178C4×D8—34no—
179#431—31no—
180C24×C3—61no—
181C23×D3—21no—
182C3×#41—48no—
183C3×#41—44no—
184C3×#41—44no—
185C2×C3×Q8—22no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×D4—22no—
189C2×C3×D4—24no—
190C2×C3×D4—22no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—22no—
198C2×C3×D4—32no—
199C2×C3×D4—64no—
200C2×C3×D4—61no—
201C2×C3×D4C22×D312yes—
202C2×C3×D4C23×C312yes—
203C22×C3×C4—24no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—61no—
210C22×C3×C4—62no—
211C22×C3×C4—62no—
212C22×C3×C4—124no—
213C22×C3×C4#6711yescharacteristic
214C22×C3×C4#6711yescharacteristic
215C22×C3×C4C3×D411yescharacteristic
216C22×C3×C4C3×D411yescharacteristic
217C22×C3×C4C22×D311yescharacteristic
218C22×C3×C4C23×C311yescharacteristic
219C22×Dic3—21no—
220C2×D12—21no—
221C2×D12—24no—
222C2×D12C23×C312yes—
223C2×C4×D3—22no—
224C2×C4×D3—22no—
225C2×C4×D3—44no—
226C3×D8—68no—
227C2×C3×C8—34no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—32no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4C22×D312yes—
235C3×C4⋊C4C23×C312yes—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—62no—
243C3×C22⋊C4—62no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42#6711yescharacteristic
249C3×C42#6711yescharacteristic
250C3×C42C3×D411yescharacteristic
251C3×C42C3×D411yescharacteristic
252C3×C42C22×D311yescharacteristic
253C3×C42C22×D311yescharacteristic
254C3×C42C23×C311yescharacteristic
255C3×C42C23×C311yescharacteristic
256#213—28no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259C4⋊Dic3—21no—
260C4⋊Dic3C23×C312yes—
261C2×C3⋊C8C23×C314yes—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—42no—
265C22×C32—42no—
266C22×C32—42no—
267C22×C32C2×D812yes—
268C22×C32C22×D411yescharacteristic
269C2×C3×D3—24no—
270C2×C3×D3—48no—
271C2×C3×D3—42no—
272C2×C3×D3—42no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—42no—
277C32×C4C2×D812yes—
278C32×C4C2×D812yes—
279C32×C4C2×#4112yes—
280C3×Dic3—42no—
281C2×#41—21no—
282C22×D4—31no—
283C23×C4—61no—
284C2×D8—34no—
285C4⋊Q8C2×C3×D311yescharacteristic, complemented
286#127C2×C3×D311yescharacteristic, complemented
287#124—21no—
288#121—21no—
289#121C2×C3×D312yes—
290C4×D4—21no—
291C4×D4—32no—
292C4×D4—32no—
293C4×D4—61no—
294C4×D4—61no—
295C4×D4C2×C3×D312yescomplemented
296C2×C4⋊C4—31no—
297C2×C22⋊C4—61no—
298C2×C42C2×C3×D311yescharacteristic
299#107—34no—
300C4⋊C8—32no—
301#102—34no—
302#102—34no—
303C22⋊C8—32no—
304C4×C8—32no—
305C23×C3—24no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—21no—
309C23×C3—64no—
310C23×C3—62no—
311C23×C3—122no—
312C23×C3—122no—
313C23×C3C2×#6711yescharacteristic
314C23×C3C2×C3×D411yescharacteristic
315C22×D3—24no—
316C22×D3—22no—
317C22×D3—42no—
318C22×D3—42no—
319C3×Q8—44no—
320C3×Q8—42no—
321C3×Q8—42no—
322C3×D4—24no—
323C3×D4—24no—
324C3×D4—34no—
325C3×D4—44no—
326C3×D4—44no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4—44no—
330C3×D4—48no—
331C3×D4—44no—
332C3×D4—44no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—64no—
336C3×D4—62no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—24no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4—48no—
354C2×C3×C4—44no—
355C2×C3×C4—44no—
356C2×C3×C4—62no—
357C2×C3×C4—122no—
358C2×C3×C4—124no—
359C2×C3×C4—124no—
360C2×C3×C4#21314yescomplemented
361C2×C3×C4C3×D814yescomplemented
362C2×C3×C4C2×#6711yescharacteristic
363C2×C3×C4C2×#6711yescharacteristic
364C2×C3×C4C2×#6711yescharacteristic
365C2×C3×C4C2×#6711yescharacteristic
366C2×C3×C4C2×#6711yescharacteristic
367C2×C3×C4C2×C3×D411yescharacteristic
368C2×C3×C4C2×C3×D411yescharacteristic
369C2×C3×C4C2×C3×D411yescharacteristic
370C2×C3×C4C2×C3×D411yescharacteristic
371C2×C3×C4C2×C3×D411yescharacteristic
372C2×C3×C4C23×D311yescharacteristic
373C2×C3×C4C24×C311yescharacteristic
374C2×Dic3—22no—
375C2×Dic3—42no—
376D12—24no—
377D12—22no—
378D12C3×#4114yescomplemented
379C4×D3—44no—
380C4×D3—44no—
381C3×C8—64no—
382C3⋊C8—24no—
383C2×C32—22no—
384C2×C32—82no—
385C2×C32#52511yescharacteristic
386C2×C32C22×D811yescharacteristic
387C2×C32#57511yescharacteristic
388C3×D3—44no—
389C24—61no—
390#41—44no—
391C2×Q8—21no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—32no—
396C2×D4—64no—
397C2×D4—61no—
398C2×D4C22×C3×D312yes—
399C22×C4—22no—
400C22×C4—61no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—124no—
404C22×C4C3×#6711yescharacteristic
405C22×C4C3×#6711yescharacteristic
406C22×C4C22×C3×D311yescharacteristic
407D8—68no—
408C2×C8—34no—
409C4⋊C4—21no—
410C4⋊C4—32no—
411C4⋊C4—61no—
412C4⋊C4C22×C3×D312yes—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—62no—
416C22⋊C4—62no—
417C42C3×#6711yescharacteristic
418C42C3×#6711yescharacteristic
419C42C22×C3×D311yescharacteristic
420C42C22×C3×D311yescharacteristic
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—22no—
426C22×C3—21no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—44no—
430C22×C3—44no—
431C22×C3—44no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—64no—
439C22×C3—128no—
440C22×C3—122no—
441C22×C3—122no—
442C22×C3C2×#21312yes—
443C22×C3C2×C3×D812yes—
444C22×C3C22×#6711yescharacteristic
445C22×C3C22×C3×D411yescharacteristic
446C2×D3—24no—
447C2×D3—48no—
448C2×D3—42no—
449C2×D3—42no—
450C3×C4—22no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—41no—
461C3×C4—41no—
462C3×C4—44no—
463C3×C4—42no—
464C3×C4—42no—
465C3×C4—122no—
466C3×C4C2×#21312yes—
467C3×C4C2×#21312yes—
468C3×C4C2×C3×D812yes—
469C3×C4C2×C3×D812yes—
470C3×C4D3×#4112yes—
471C3×C4C2×C3×#4112yes—
472Dic3—42no—
473C32#310011yescharacteristic, complemented, Sylow
474C23—22no—
475C23—64no—
476C23—62no—
477C23—122no—
478C23—122no—
479C23C2×C3×#6711yescharacteristic
480Q8—42no—
481D4—24no—
482D4—34no—
483D4—42no—
484D4—44no—
485D4—42no—
486D4—64no—
487D4—62no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—41no—
491C2×C4—44no—
492C2×C4—62no—
493C2×C4—122no—
494C2×C4—124no—
495C2×C4—124no—
496C2×C4C3×#21314yescomplemented
497C2×C4C2×C3×#6711yescharacteristic
498C2×C4C2×C3×#6711yescharacteristic
499C2×C4C2×C3×#6711yescharacteristic
500C2×C4C2×C3×#6711yescharacteristic
501C2×C4C2×C3×#6711yescharacteristic
502C2×C4C23×C3×D311yescharacteristic
503C8—64no—
504C2×C3—22no—
505C2×C3—22no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—42no—
510C2×C3—84no—
511C2×C3—82no—
512C2×C3—82no—
513C2×C3—124no—
514C2×C3C22×#21311yescharacteristic
515C2×C3#582211yescharacteristic
516C2×C3#582811yescharacteristic
517C2×C3C3×#52511yescharacteristic
518C2×C3C22×C3×D811yescharacteristic
519C2×C3C3×#57511yescharacteristic
520D3—44no—
521C22—21no—
522C22—21no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—64no—
527C22—128no—
528C22—122no—
529C22—122no—
530C22C2×C3×#21312yes—
531C22C22×C3×#6711yescharacteristic
532C4—21no—
533C4—21no—
534C4—21no—
535C4—42no—
536C4—122no—
537C4C2×C3×#21312yes—
538C4C2×C3×#21312yes—
539C4C3×D3×#4112yes—
540C3—21no—
541C3#8329811yescharacteristic, direct product factor
542C3C3×#310011yescharacteristic, complemented
543C2—22no—
544C2—82no—
545C2—124no—
546C2C22×C3×#21311yescharacteristic
547C2C3×#582211yescharacteristic
548C2C3×#582811yescharacteristic
subgroupG/H
centerC2×C3×C4C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#532C2
socleC22×C32C22×D4
hypercenterC3×#532D3
hypocenterC3C3×#3100
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ C3×#532

Complements

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

NH ≅ G/Ntype
1. C32×#532C2semidirect
2. C2×C3×C4×D12C2semidirect
3. C3×#5087C2semidirect
4. C3×#5072C2semidirect
5. C3×#5036C2semidirect
6. C3×#5033C2semidirect
7. C3×C4×#213C2semidirect
8. C3×#4925C2semidirect
10. #83298C3direct (N × H)
14. C32×C4⋊Q8C22semidirect
15. C32×#127C22semidirect
20. C32×C4×D4C22semidirect
29. C3×#832C22semidirect
32. C3×#807C22semidirect
33. C3×C4⋊(C3⋊C8)C22semidirect
34. C3×C4×C3⋊C8C22semidirect
36. C3×#532D3semidirect
37. C3×#532C2×C3semidirect
39. C2×C4×D12C2×C3semidirect
46. #5087C2×C3semidirect
47. #5072C2×C3semidirect
48. #5036C2×C3semidirect
49. #5033C2×C3semidirect
50. C4×#213C2×C3semidirect
51. #4925C2×C3semidirect
94. C3×C4⋊Q8C2×D3semidirect
95. C3×C4⋊Q8C22×C3semidirect
97. C3×#127C2×D3semidirect
98. C3×#127C22×C3semidirect
116. C3×C4×D4C2×D3semidirect
117. C3×C4×D4C22×C3semidirect
136. #832C22×C3semidirect
139. #807C22×C3semidirect
140. C4⋊(C3⋊C8)C22×C3semidirect
141. C4×C3⋊C8C22×C3semidirect
157. C2×C32×C4D8semidirect
168. C3×D12#41semidirect
172. #532C3×D3semidirect
285. C4⋊Q8C2×C3×D3semidirect
286. #127C2×C3×D3semidirect
295. C4×D4C2×C3×D3semidirect
360. C2×C3×C4#213semidirect
361. C2×C3×C4C3×D8semidirect
378. D12C3×#41semidirect
473. C32#3100semidirect
496. C2×C4C3×#213semidirect
541. C3#83298direct (N × H)
542. C3C3×#3100semidirect

Maximal subgroups

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

subgroupindex
C32×#5322
C2×C3×C4×D122
C3×#50872
C3×#50722
C3×#50362
C3×#50332
C3×C4×#2132
C3×#49252
C3×#31003
#832983