Finite Group Databasebeta

← C3×#82308 #11870418

Subgroups — C3×#82308 #11870418

Order 1152.

1197 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic, complemented
2C3×#5477C211yesmaximal, characteristic, complemented
3C3×#5217C212yesmaximal, complemented
4C3×#5015C212yesmaximal, complemented
5C3×C4×#213C214yesmaximal, complemented
6C3×#5007C212yesmaximal, complemented
7C3×#4967C212yesmaximal, complemented
8C3×#4925C211yesmaximal, characteristic, complemented
9C3×#3100—31nomaximal
10#82308C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C23×C32×C4—21no—
13C2×C3×#235—21no—
14C32×C4×D4—21no—
15C32×C4×D4—21no—
16C32×C4×D4C2212yes—
17C32×C4×D4C2212yes—
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C32×C22⋊C4—21no—
20C2×C32×C42C2211yescharacteristic
21C2×C3×#213C2214yes—
22C3×#930—21no—
23C3×#930C2212yes—
24C3×#875—21no—
25C3×#874C2211yescharacteristic, complemented
26C3×C4×D12—21no—
27C3×C4×D12C2212yescomplemented
28C3×#869C2211yescharacteristic, complemented
29C3×#832C2214yes—
30C3×#830C2212yes—
31C3×#809C2214yescomplemented
32C3×#807C2214yescomplemented
33C3×C4⋊(C3⋊C8)C2212yescomplemented
34C3×C4×C3⋊C8C2212yescomplemented
35C3×#532—31no—
36C2×C3×C4×D4—21no—
37C2×C3×C4×D4D311yescharacteristic, complemented
38C2×C3×C4×D4C2×C311yescharacteristic, complemented
39#5477C2×C311yescharacteristic, complemented
40C3×#480—32no—
41C3×#474—32no—
42C3×#459—32no—
43C3×#447—32no—
44C3×C4×D8—34no—
45C3×#431—31no—
46#5217C2×C312yescomplemented
47#5015C2×C312yescomplemented
48C4×#213C2×C314yescomplemented
49#5007C2×C312yescomplemented
50#4967C2×C312yescomplemented
51#4925C2×C311yescharacteristic, complemented
52C24×C32—21no—
53C2×C32×D4—24no—
54C2×C32×D4—21no—
55C2×C32×D4C2312yes—
56C22×C32×C4—21no—
57C22×C32×C4—22no—
58C22×C32×C4—22no—
59C22×C32×C4—44no—
60C22×C32×C4D411yescharacteristic
61C22×C32×C4D411yescharacteristic
62C22×C32×C4C2311yescharacteristic
63C2×C3×#67—22no—
64C3×#235—44no—
65C2×C3×D12—21no—
66C2×C3×D12—21no—
67C2×C3×D12C2312yes—
68C2×C3×C4×D3—22no—
69C2×C3×Dic6—21no—
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—61no—
86C22×C3×D4—21no—
87C22×C3×D4C2×D311yescharacteristic
88C22×C3×D4C22×C311yescharacteristic
89C23×C3×C4—22no—
90C23×C3×C4—21no—
91C23×C3×C4—21no—
92C2×#235—21no—
93C2×C3×D8—34no—
94C3×C4⋊Q8—31no—
95C3×#127—31no—
96C3×#124—61no—
97C3×#121—32no—
98C3×#121—61no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×C4×D4—21no—
104C3×C4×D4—21no—
105C3×C4×D4—22no—
106C3×C4×D4—22no—
107C3×C4×D4—32no—
108C3×C4×D4—61no—
109C3×C4×D4C2×D312yes—
110C3×C4×D4C2×D312yes—
111C3×C4×D4C22×C312yes—
112C3×C4×D4C22×C312yes—
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4C2×D311yescharacteristic
115C2×C3×C4⋊C4C22×C311yescharacteristic
116C2×C3×C22⋊C4—22no—
117C2×C3×C22⋊C4—21no—
118C2×C3×C22⋊C4—21no—
119C2×C3×C42—21no—
120C2×C3×C42C2×D311yescharacteristic
121C2×C3×C42C22×C311yescharacteristic
122C2×#213C22×C314yes—
123#930—21no—
124#930C22×C312yes—
125#875—21no—
126#874C22×C311yescharacteristic, complemented
127C4×D12—21no—
128C4×D12C22×C312yescomplemented
129#869C22×C311yescharacteristic, complemented
130C3×#107—34no—
131C3×C4⋊C8—32no—
132C3×#102—34no—
133C3×#102—34no—
134C3×C22⋊C8—32no—
135C3×C4×C8—32no—
136#832C22×C314yes—
137#830C22×C312yes—
138#809C22×C314yescomplemented
139#807C22×C314yescomplemented
140C4⋊(C3⋊C8)C22×C312yescomplemented
141C4×C3⋊C8C22×C312yescomplemented
142C23×C32—24no—
143C23×C32—22no—
144C23×C32—42no—
145C23×C32—42no—
146C23×C32C2×D411yescharacteristic
147C22×C3×D3—22no—
148C32×D4—24no—
149C32×D4—22no—
150C32×D4#4114yescomplemented
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—41no—
154C2×C32×C4—42no—
155C2×C32×C4—44no—
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
164C3×#67—44no—
165C2×C3×Dic3—22no—
166C3×D12—24no—
167C3×D12—42no—
168C3×D12—42no—
169C3×C4×D3—44no—
170C3×Dic6—42no—
171C3×C3⋊C8—24no—
172#532—31no—
173C2×C4×D4C3×D311yescharacteristic, complemented
174#480—32no—
175#474—32no—
176#459—32no—
177#447—32no—
178C4×D8—34no—
179#431—31no—
180C24×C3—22no—
181C24×C3—21no—
182C24×C3—21no—
183C3×#41—124no—
184C2×C3×Q8—61no—
185C2×C3×D4—22no—
186C2×C3×D4—28no—
187C2×C3×D4—24no—
188C2×C3×D4—24no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—22no—
192C2×C3×D4—32no—
193C2×C3×D4—62no—
194C2×C3×D4—61no—
195C2×C3×D4—61no—
196C2×C3×D4C22×D312yes—
197C2×C3×D4C23×C312yes—
198C22×C3×C4—22no—
199C22×C3×C4—24no—
200C22×C3×C4—24no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—48no—
211C22×C3×C4—44no—
212C22×C3×C4—44no—
213C22×C3×C4—62no—
214C22×C3×C4#6711yescharacteristic
215C22×C3×C4#6711yescharacteristic
216C22×C3×C4C3×D411yescharacteristic
217C22×C3×C4C3×D411yescharacteristic
218C22×C3×C4C22×D311yescharacteristic
219C22×C3×C4C23×C311yescharacteristic
220C2×#67—22no—
221#235—44no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12C23×C312yes—
225C2×C4×D3—22no—
226C2×Dic6—21no—
227C3×D8—68no—
228C2×C3×C8—34no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—32no—
234C3×C4⋊C4—61no—
235C3×C4⋊C4C22×D312yes—
236C3×C4⋊C4C23×C312yes—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—62no—
244C3×C22⋊C4—62no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42#6711yescharacteristic
250C3×C42#6711yescharacteristic
251C3×C42C3×D411yescharacteristic
252C3×C42C3×D411yescharacteristic
253C3×C42C22×D311yescharacteristic
254C3×C42C22×D311yescharacteristic
255C3×C42C23×C311yescharacteristic
256C3×C42C23×C311yescharacteristic
257#213—28no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—22no—
260C4⋊Dic3—21no—
261C4⋊Dic3C23×C312yes—
262C2×C3⋊C8C23×C314yes—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—24no—
266C22×C32—48no—
267C22×C32—42no—
268C22×C32—42no—
269C22×C32C2×D812yes—
270C22×C32C22×D411yescharacteristic
271C2×C3×D3—42no—
272C2×C3×D3—42no—
273C2×C3×D3—42no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—42no—
278C32×C4C2×D812yes—
279C32×C4C2×D812yes—
280C32×C4C2×#4112yes—
281C3×Dic3—42no—
282C2×#41—61no—
283C22×D4C2×C3×D311yescharacteristic
284C23×C4—21no—
285C2×D8—34no—
286C4⋊Q8—31no—
287#127—31no—
288#124—61no—
289#121—32no—
290#121—61no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—32no—
294C4×D4—61no—
295C4×D4C2×C3×D312yes—
296C4×D4C2×C3×D312yes—
297C2×C4⋊C4C2×C3×D311yescharacteristic
298C2×C22⋊C4—21no—
299C2×C42C2×C3×D311yescharacteristic
300#107—34no—
301C4⋊C8—32no—
302#102—34no—
303#102—34no—
304C22⋊C8—32no—
305C4×C8—32no—
306C23×C3—24no—
307C23×C3—28no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—21no—
313C23×C3—44no—
314C23×C3—44no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—62no—
320C23×C3C2×#6711yescharacteristic
321C23×C3C2×C3×D411yescharacteristic
322C22×D3—22no—
323C3×Q8—122no—
324C3×D4—28no—
325C3×D4—24no—
326C3×D4—24no—
327C3×D4—24no—
328C3×D4—22no—
329C3×D4—22no—
330C3×D4—24no—
331C3×D4—64no—
332C3×D4—122no—
333C3×D4—124no—
334C3×D4—122no—
335C3×D4#23514yescomplemented
336C3×D4C3×#4114yescomplemented
337C2×C3×C4—24no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—24no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—24no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—48no—
353C2×C3×C4—48no—
354C2×C3×C4—42no—
355C2×C3×C4—44no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—44no—
359C2×C3×C4—44no—
360C2×C3×C4—44no—
361C2×C3×C4—44no—
362C2×C3×C4—62no—
363C2×C3×C4—124no—
364C2×C3×C4#21314yescomplemented
365C2×C3×C4C3×D814yescomplemented
366C2×C3×C4C2×#6711yescharacteristic
367C2×C3×C4C2×#6711yescharacteristic
368C2×C3×C4C2×#6711yescharacteristic
369C2×C3×C4C2×#6711yescharacteristic
370C2×C3×C4C2×#6711yescharacteristic
371C2×C3×C4C2×C3×D411yescharacteristic
372C2×C3×C4C2×C3×D411yescharacteristic
373C2×C3×C4C2×C3×D411yescharacteristic
374C2×C3×C4C2×C3×D411yescharacteristic
375C2×C3×C4C2×C3×D411yescharacteristic
376C2×C3×C4C23×D311yescharacteristic
377C2×C3×C4C24×C311yescharacteristic
378#67—44no—
379C2×Dic3—22no—
380D12—24no—
381D12—42no—
382D12—42no—
383C4×D3—44no—
384Dic6—42no—
385C3×C8—64no—
386C3⋊C8—24no—
387C2×C32—22no—
388C2×C32—44no—
389C2×C32#52511yescharacteristic
390C2×C32C22×D811yescharacteristic
391C2×C32#57511yescharacteristic
392C3×D3—82no—
393C24—21no—
394#41—124no—
395C2×Q8—61no—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—32no—
399C2×D4—62no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4C22×C3×D312yes—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—44no—
407C22×C4—62no—
408C22×C4C3×#6711yescharacteristic
409C22×C4C3×#6711yescharacteristic
410C22×C4C22×C3×D311yescharacteristic
411D8—68no—
412C2×C8—34no—
413C4⋊C4—21no—
414C4⋊C4—32no—
415C4⋊C4—61no—
416C4⋊C4C22×C3×D312yes—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—62no—
420C22⋊C4—62no—
421C42C3×#6711yescharacteristic
422C42C3×#6711yescharacteristic
423C42C22×C3×D311yescharacteristic
424C42C22×C3×D311yescharacteristic
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—28no—
432C22×C3—24no—
433C22×C3—24no—
434C22×C3—22no—
435C22×C3—21no—
436C22×C3—44no—
437C22×C3—416no—
438C22×C3—48no—
439C22×C3—48no—
440C22×C3—44no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—122no—
446C22×C3—122no—
447C22×C3—122no—
448C22×C3C2×#21312yes—
449C22×C3C2×C3×D812yes—
450C22×C3C22×#6711yescharacteristic
451C22×C3C22×C3×D411yescharacteristic
452C2×D3—42no—
453C2×D3—42no—
454C2×D3—42no—
455C3×C4—22no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—22no—
465C3×C4—41no—
466C3×C4—41no—
467C3×C4—44no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—122no—
471C3×C4C2×#21312yes—
472C3×C4C2×#21312yes—
473C3×C4C2×C3×D812yes—
474C3×C4C2×C3×D812yes—
475C3×C4C2×#23512yes—
476C3×C4C2×C3×#4112yes—
477Dic3—42no—
478C32#310011yescharacteristic, complemented, Sylow
479C23—24no—
480C23—22no—
481C23—42no—
482C23—42no—
483C23—62no—
484C23C2×C3×#6711yescharacteristic
485Q8—122no—
486D4—24no—
487D4—22no—
488D4—64no—
489D4—122no—
490D4—124no—
491D4—122no—
492D4C3×#23514yescomplemented
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—62no—
500C2×C4—124no—
501C2×C4C3×#21314yescomplemented
502C2×C4C2×C3×#6711yescharacteristic
503C2×C4C2×C3×#6711yescharacteristic
504C2×C4C2×C3×#6711yescharacteristic
505C2×C4C2×C3×#6711yescharacteristic
506C2×C4C2×C3×#6711yescharacteristic
507C2×C4C23×C3×D311yescharacteristic
508C8—64no—
509C2×C3—24no—
510C2×C3—22no—
511C2×C3—22no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—48no—
516C2×C3—44no—
517C2×C3—44no—
518C2×C3—242no—
519C2×C3#579111yescharacteristic
520C2×C3C22×#21311yescharacteristic
521C2×C3#582211yescharacteristic
522C2×C3C3×#52511yescharacteristic
523C2×C3C22×C3×D811yescharacteristic
524C2×C3C3×#57511yescharacteristic
525D3—82no—
526C22—21no—
527C22—21no—
528C22—24no—
529C22—48no—
530C22—42no—
531C22—42no—
532C22—122no—
533C22—122no—
534C22—122no—
535C22C2×C3×#21312yes—
536C22C22×C3×#6711yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—42no—
541C4—122no—
542C4C2×C3×#21312yes—
543C4C2×C3×#21312yes—
544C4C2×C3×#23512yes—
545C3—21no—
546C3#8230811yescharacteristic, direct product factor
547C3C3×#310011yescharacteristic, complemented
548C2—22no—
549C2—44no—
550C2—242no—
551C2C3×#579111yescharacteristic
552C2C22×C3×#21311yescharacteristic
553C2C3×#582211yescharacteristic
subgroupG/H
centerC2×C3×C4C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C4×D4C2
socleC22×C32C22×D4
hypercenterC2×C3×C4×D4D3
hypocenterC3C3×#3100
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×D4C2semidirect
2. C3×#5477C2semidirect
3. C3×#5217C2semidirect
4. C3×#5015C2semidirect
5. C3×C4×#213C2semidirect
6. C3×#5007C2semidirect
7. C3×#4967C2semidirect
8. C3×#4925C2semidirect
10. #82308C3direct (N × H)
25. C3×#874C22semidirect
27. C3×C4×D12C22semidirect
28. C3×#869C22semidirect
31. C3×#809C22semidirect
32. C3×#807C22semidirect
33. C3×C4⋊(C3⋊C8)C22semidirect
34. C3×C4×C3⋊C8C22semidirect
37. C2×C3×C4×D4D3semidirect
38. C2×C3×C4×D4C2×C3semidirect
39. #5477C2×C3semidirect
46. #5217C2×C3semidirect
47. #5015C2×C3semidirect
48. C4×#213C2×C3semidirect
49. #5007C2×C3semidirect
50. #4967C2×C3semidirect
51. #4925C2×C3semidirect
126. #874C22×C3semidirect
128. C4×D12C22×C3semidirect
129. #869C22×C3semidirect
138. #809C22×C3semidirect
139. #807C22×C3semidirect
140. C4⋊(C3⋊C8)C22×C3semidirect
141. C4×C3⋊C8C22×C3semidirect
150. C32×D4#41semidirect
157. C2×C32×C4D8semidirect
173. C2×C4×D4C3×D3semidirect
335. C3×D4#235semidirect
336. C3×D4C3×#41semidirect
364. C2×C3×C4#213semidirect
365. C2×C3×C4C3×D8semidirect
478. C32#3100semidirect
492. D4C3×#235semidirect
501. C2×C4C3×#213semidirect
546. C3#82308direct (N × H)
547. C3C3×#3100semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×D42
C3×#54772
C3×#52172
C3×#50152
C3×C4×#2132
C3×#50072
C3×#49672
C3×#49252
C3×#31003
#823083