Finite Group Databasebeta

← C3×#10590349 #13157363

Subgroups — C3×#10590349 #13157363

Order 1728.

830 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
1C3×#64083C211yesmaximal, characteristic, complemented
2C32×C4×D12C211yesmaximal, characteristic, complemented
3C2×C3×#3714C212yesmaximal, complemented
4C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×#63564C212yesmaximal, complemented
6C3×#5015—31nomaximal
7C3×#4843—31nomaximal
8#10590349C311yesmaximal, characteristic, direct product factor
9C2×C3×C4⋊(C3⋊D3)—21no—
10C2×C3×C4⋊(C3⋊D3)—21no—
11C2×C3×C4⋊(C3⋊D3)C2212yes—
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3—21no—
14C33×C42C2211yescharacteristic
15C32×(C2×D3)⋊C4—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2212yes—
18C3×#3714—24no—
19C32×C4×D4—31no—
20#64083C2×C311yescharacteristic, complemented
21C2×C3×#213—32no—
22C2×C3×D24—32no—
23C3×#874—31no—
24C3×#874—31no—
25C3×#874—61no—
26C3×C4×D12—21no—
27C3×C4×D12D311yescharacteristic, complemented
28C3×C4×D12C2×C311yescharacteristic, complemented
29C2×#3714C2×C312yescomplemented
30C32×C4⋊C8—31no—
31C3×#821—32no—
32C3×#809—32no—
33C3×C4⋊(C3⋊C8)—21no—
34C3×C4⋊(C3⋊C8)D311yescharacteristic, complemented
35C3×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
36#63564C2×C312yescomplemented
37C22×C3×C3⋊D3—22no—
38C22×C32×D3—21no—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C3×C4⋊(C3⋊D3)—22no—
43C3×C4⋊(C3⋊D3)—22no—
44C3×C4⋊(C3⋊D3)—42no—
45C3×C4⋊(C3⋊D3)—42no—
46C2×C32×Dic3—21no—
47C32×D12—21no—
48C32×D12D412yescomplemented
49C32×C4×D3—42no—
50C32×C3⋊C8—22no—
51C3×#459—91no—
52#5015—31no—
53#4843—31no—
54C2×C32×D4—31no—
55C22×C32×C4—61no—
56C2×C4⋊(C3⋊D3)—21no—
57C2×C4⋊(C3⋊D3)—21no—
58C2×C4⋊(C3⋊D3)C22×C312yes—
59C2×C3×D12—21no—
60C2×C3×D12—32no—
61C2×C3×D12—32no—
62C2×C3×D12—62no—
63C2×C3×D12—62no—
64C2×C3×D12—62no—
65C2×C3×D12—61no—
66C2×C3×D12—61no—
67C2×C3×D12—61no—
68C2×C3×D12—61no—
69C2×C3×D12C2×D311yescharacteristic
70C2×C3×D12C22×C311yescharacteristic
71C2×C3×C4×D3—22no—
72C2×C3×C4×D3—21no—
73C2×C3×C4×D3—21no—
74C2×C32×C8—32no—
75C32×C4⋊C4—31no—
76C32×C22⋊C4—61no—
77C32×C42—21no—
78C32×C42—21no—
79C32×C42—21no—
80C32×C42—41no—
81C32×C42C2×D311yescharacteristic
82C32×C42C2×D311yescharacteristic
83C32×C42C22×C311yescharacteristic
84C3×#213—64no—
85C3×(C2×D3)⋊C4—22no—
86C3×(C2×D3)⋊C4—21no—
87C3×(C2×D3)⋊C4—21no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C22×C311yescharacteristic
91C2×C3×C3⋊C8—22no—
92C2×C3×C3⋊C8C2×D312yes—
93C2×C3×C3⋊C8C22×C312yes—
94C3×D24—64no—
95#3714—24no—
96C22×C33C2×D411yescharacteristic
97C2×C3×C3⋊D3—42no—
98C2×C3×C3⋊D3—42no—
99C2×C3×C3⋊D3—42no—
100C2×C32×D3—22no—
101C2×C32×D3—41no—
102C2×C32×D3—41no—
103C33×C4—21no—
104C33×C4D812yescomplemented
105C33×C4C2×D411yescharacteristic
106C33×C4#4111yescharacteristic
107C32×Dic3—41no—
108C2×C3×D8—92no—
109C3×#127—91no—
110C3×C4×D4—31no—
111C3×C4×D4—31no—
112C3×C4×D4—61no—
113C2×#213—32no—
114C2×D24—32no—
115#874—31no—
116#874—31no—
117#874—61no—
118C4×D12C3×D311yescharacteristic, complemented
119C3×C4⋊C8—31no—
120C3×C4⋊C8—31no—
121C3×C4⋊C8—61no—
122C3×#102—92no—
123#821—32no—
124#809—32no—
125C4⋊(C3⋊C8)C3×D311yescharacteristic, complemented
126C23×C32—61no—
127C22×C3⋊D3—22no—
128C22×C3×D3—22no—
129C22×C3×D3—21no—
130C22×C3×D3—21no—
131C22×C3×D3—64no—
132C22×C3×D3—62no—
133C22×C3×D3—62no—
134C32×D4—32no—
135C32×D4—61no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—41no—
146C2×C32×C4—41no—
147C2×C32×C4—41no—
148C2×C32×C4—61no—
149C2×C32×C4—122no—
150C2×C32×C4D1211yescharacteristic
151C2×C32×C4D1211yescharacteristic
152C2×C32×C4#6711yescharacteristic
153C2×C32×C4#6711yescharacteristic
154C2×C32×C4C3×D411yescharacteristic
155C2×C32×C4C3×D411yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C23×C311yescharacteristic
159C4⋊(C3⋊D3)—22no—
160C4⋊(C3⋊D3)—22no—
161C4⋊(C3⋊D3)—42no—
162C4⋊(C3⋊D3)—42no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C3×D12—22no—
167C3×D12—21no—
168C3×D12—21no—
169C3×D12—22no—
170C3×D12—62no—
171C3×D12—62no—
172C3×D12—62no—
173C3×D12—62no—
174C3×D12—122no—
175C3×D12—122no—
176C3×D12—124no—
177C3×D12—122no—
178C3×D12—122no—
179C3×D12—124no—
180C3×D12—122no—
181C3×D12—122no—
182C3×D12D1212yescomplemented
183C3×D12C3×D412yescomplemented
184C3×C4×D3—44no—
185C3×C4×D3—42no—
186C3×C4×D3—42no—
187C32×C8—62no—
188C3×C3⋊C8—22no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8—42no—
191#459—91noSylow
192C2×C33#12111yescharacteristic
193C2×C33C2×D811yescharacteristic
194C2×C33AGL(1,ℤ/8)11yescharacteristic
195C3×C3⋊D3—82no—
196C32×D3—42no—
197C2×C3×D4—31no—
198C2×C3×D4—31no—
199C2×C3×D4—61no—
200C2×C3×D4—92no—
201C2×C3×D4—181no—
202C2×C3×D4—181no—
203C22×C3×C4—62no—
204C22×C3×C4—61no—
205C22×C3×C4—61no—
206C2×D12—32no—
207C2×D12—32no—
208C2×D12—62no—
209C2×D12—62no—
210C2×D12—62no—
211C2×D12—61no—
212C2×D12—61no—
213C2×D12—61no—
214C2×D12—61no—
215C2×D12C2×C3×D311yescharacteristic
216C2×C4×D3—21no—
217C3×D8—184no—
218C2×C3×C8—32no—
219C2×C3×C8—32no—
220C2×C3×C8—62no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—61no—
224C3×C22⋊C4—62no—
225C3×C22⋊C4—61no—
226C3×C22⋊C4—61no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—41no—
231C3×C42D3211yescharacteristic
232C3×C42C2×C3×D311yescharacteristic
233C3×C42C2×C3×D311yescharacteristic
234#213—64no—
235(C2×D3)⋊C4—21no—
236C4⋊Dic3C2×C3×D311yescharacteristic
237C2×C3⋊C8C2×C3×D312yes—
238D24—64no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—41no—
243C22×C32—62no—
244C22×C32—121no—
245C22×C32—121no—
246C22×C32C2×D1211yescharacteristic
247C22×C32C2×#6711yescharacteristic
248C22×C32C2×C3×D411yescharacteristic
249C2×C3⋊D3—42no—
250C2×C3⋊D3—42no—
251C2×C3⋊D3—42no—
252C2×C3×D3—24no—
253C2×C3×D3—22no—
254C2×C3×D3—22no—
255C2×C3×D3—42no—
256C2×C3×D3—42no—
257C2×C3×D3—41no—
258C2×C3×D3—41no—
259C2×C3×D3—41no—
260C2×C3×D3—41no—
261C2×C3×D3—124no—
262C2×C3×D3—124no—
263C2×C3×D3—124no—
264C2×C3×D3—122no—
265C2×C3×D3—122no—
266C2×C3×D3—122no—
267C2×C3×D3—122no—
268C2×C3×D3—122no—
269C2×C3×D3—122no—
270C32×C4—21no—
271C32×C4—22no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—22no—
275C32×C4—22no—
276C32×C4—22no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—42no—
284C32×C4—41no—
285C32×C4—41no—
286C32×C4—41no—
287C32×C4—41no—
288C32×C4—42no—
289C32×C4—121no—
290C32×C4D2412yescomplemented
291C32×C4#21312yescomplemented
292C32×C4C3×D812yescomplemented
293C32×C4C2×D1211yescharacteristic
294C32×C4#23511yescharacteristic
295C32×C4D3×D411yescharacteristic
296C32×C4#23911yescharacteristic
297C32×C4C2×C3×D411yescharacteristic
298C32×C4C3×#4111yescharacteristic
299C3×Dic3—42no—
300C3×Dic3—41no—
301C3×Dic3—41no—
302C2×D8—92no—
303#127—91no—
304C4×D4—31no—
305C4⋊C8—31no—
306#102—92no—
307C33#45911yescharacteristic, complemented, Sylow
308C23×C3—62no—
309C23×C3—61no—
310C23×C3—61no—
311C23×C3—182no—
312C22×D3—21no—
313C22×D3—64no—
314C22×D3—62no—
315C22×D3—62no—
316C3×D4—32no—
317C3×D4—32no—
318C3×D4—62no—
319C3×D4—62no—
320C3×D4—61no—
321C3×D4—61no—
322C3×D4—182no—
323C3×D4—182no—
324C3×D4—362no—
325C3×D4—362no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—62no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—124no—
342C2×C3×C4—122no—
343C2×C3×C4—122no—
344C2×C3×C4#62611yescharacteristic
345C2×C3×C4#62611yescharacteristic
346C2×C3×C4C3×D1211yescharacteristic
347C2×C3×C4C3×D1211yescharacteristic
348C2×C3×C4C3×#6711yescharacteristic
349C2×C3×C4C3×#6711yescharacteristic
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C22×C3×D311yescharacteristic
352C2×C3×C4C22×C3×D311yescharacteristic
353C2×Dic3—21no—
354D12—21no—
355D12—62no—
356D12—62no—
357D12—62no—
358D12—62no—
359D12—122no—
360D12—122no—
361D12—124no—
362D12—122no—
363D12—122no—
364D12—124no—
365D12—122no—
366D12—122no—
367D12C3×D1212yescomplemented
368C4×D3—42no—
369C3×C8—62no—
370C3×C8—62no—
371C3×C8—122no—
372C3⋊C8—22no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32—122no—
386C2×C32#89511yescharacteristic
387C2×C32C2×D2411yescharacteristic
388C2×C32#90811yescharacteristic
389C2×C32#93011yescharacteristic
390C2×C32C2×#21311yescharacteristic
391C2×C32#94911yescharacteristic
392C2×C32C3×#12111yescharacteristic
393C2×C32C2×C3×D811yescharacteristic
394C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
395C3⋊D3—82no—
396C3×D3—44no—
397C3×D3—42no—
398C3×D3—42no—
399C3×D3—244no—
400C3×D3—242no—
401C3×D3—242no—
402C2×D4—31no—
403C2×D4—92no—
404C2×D4—181no—
405C2×D4—181no—
406C22×C4—61no—
407D8—184no—
408C2×C8—32no—
409C4⋊C4—31no—
410C22⋊C4—61no—
411C42C3×D3211yescharacteristic
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—41no—
416C22×C3—64no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3—122no—
420C22×C3—122no—
421C22×C3—121no—
422C22×C3—121no—
423C22×C3—121no—
424C22×C3—121no—
425C22×C3—362no—
426C22×C3—362no—
427C22×C3—362no—
428C22×C3C2×#62611yescharacteristic
429C22×C3C2×C3×D1211yescharacteristic
430C22×C3C2×C3×#6711yescharacteristic
431C2×D3—22no—
432C2×D3—41no—
433C2×D3—41no—
434C2×D3—124no—
435C2×D3—124no—
436C2×D3—124no—
437C2×D3—122no—
438C2×D3—122no—
439C2×D3—122no—
440C2×D3—122no—
441C2×D3—122no—
442C2×D3—122no—
443C3×C4—22no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—22no—
454C3×C4—22no—
455C3×C4—22no—
456C3×C4—41no—
457C3×C4—41no—
458C3×C4—42no—
459C3×C4—41no—
460C3×C4—41no—
461C3×C4—42no—
462C3×C4—122no—
463C3×C4—121no—
464C3×C4—121no—
465C3×C4#371412yescomplemented
466C3×C4C3×D2412yescomplemented
467C3×C4C3×#21312yescomplemented
468C3×C4#379911yescharacteristic
469C3×C4D3×D1211yescharacteristic
470C3×C4C2×C3×D1211yescharacteristic
471C3×C4C3×#23511yescharacteristic
472C3×C4C3×D3×D411yescharacteristic
473C3×C4C3×#23911yescharacteristic
474Dic3—41no—
475C32—21no—
476C32—21no—
477C32—21no—
478C32—41no—
479C32#484311yescharacteristic, complemented
480C32#501511yescharacteristic, complemented
481C32C3×#45911yescharacteristic, complemented
482C23—61no—
483C23—182no—
484D4—32no—
485D4—61no—
486D4—182no—
487D4—182no—
488D4—362no—
489D4—362no—
490C2×C4—61no—
491C2×C4—122no—
492C2×C4C3×#62611yescharacteristic
493C2×C4C3×#62611yescharacteristic
494C2×C4C2×C3×D3211yescharacteristic
495C8—62no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—41no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—124no—
509C2×C3—122no—
510C2×C3—122no—
511C2×C3—722no—
512C2×C3C2×#371411yescharacteristic
513C2×C3#6382511yescharacteristic
514C2×C3#6390911yescharacteristic
515C2×C3C3×#89511yescharacteristic
516C2×C3C2×C3×D2411yescharacteristic
517C2×C3C3×#90811yescharacteristic
518C2×C3C3×#93011yescharacteristic
519C2×C3C2×C3×#21311yescharacteristic
520C2×C3C3×#94911yescharacteristic
521D3—42no—
522D3—244no—
523D3—242no—
524D3—242no—
525C22—62no—
526C22—121no—
527C22—121no—
528C22—362no—
529C22—362no—
530C22—362no—
531C22C2×C3×#62611yescharacteristic
532C4—21no—
533C4—121no—
534C4C3×#371412yescomplemented
535C4C3×#379911yescharacteristic
536C4C3×D3×D1211yescharacteristic
537C3—21no—
538C3—21no—
539C3—21no—
540C3—41no—
541C3#1059034911yescharacteristic, direct product factor
542C3C3×#484311yescharacteristic, complemented
543C3C3×#501511yescharacteristic, complemented
544C2—122no—
545C2—722no—
546C2C2×C3×#371411yescharacteristic
547C2C3×#6382511yescharacteristic
548C2C3×#6390911yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
hypercenterC3×C42D32
hypocenterC32C3×#459
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
1. C3×#64083C2semidirect
2. C32×C4×D12C2semidirect
3. C2×C3×#3714C2semidirect
4. C32×C4⋊(C3⋊C8)C2semidirect
5. C3×#63564C2semidirect
8. #10590349C3direct (N × H)
20. #64083C2×C3semidirect
27. C3×C4×D12D3semidirect
28. C3×C4×D12C2×C3semidirect
29. C2×#3714C2×C3semidirect
34. C3×C4⋊(C3⋊C8)D3semidirect
35. C3×C4⋊(C3⋊C8)C2×C3semidirect
36. #63564C2×C3semidirect
48. C32×D12D4semidirect
104. C33×C4D8semidirect
118. C4×D12C3×D3semidirect
125. C4⋊(C3⋊C8)C3×D3semidirect
182. C3×D12D12semidirect
183. C3×D12C3×D4semidirect
290. C32×C4D24semidirect
291. C32×C4#213semidirect
292. C32×C4C3×D8semidirect
307. C33#459semidirect
367. D12C3×D12semidirect
465. C3×C4#3714semidirect
466. C3×C4C3×D24semidirect
467. C3×C4C3×#213semidirect
479. C32#4843semidirect
480. C32#5015semidirect
481. C32C3×#459semidirect
534. C4C3×#3714semidirect
541. C3#10590349direct (N × H)
542. C3C3×#4843semidirect
543. C3C3×#5015semidirect

Maximal subgroups

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

subgroupindex
C3×#640832
C32×C4×D122
C2×C3×#37142
C32×C4⋊(C3⋊C8)2
C3×#635642
C3×#50153
C3×#48433
#105903493