Finite Group Databasebeta

← C3×#10590704 #13157804

Subgroups — C3×#10590704 #13157804

Order 1728.

582 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63909C211yesmaximal, characteristic
2C3×#63841C211yesmaximal, characteristic, complemented
3C3×#63648C211yesmaximal, characteristic, complemented
4C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
5C32×#817C211yesmaximal, characteristic, complemented
6C3×#63575C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8C3×#4866—31nomaximal
9C3×#4737—31nomaximal
10#10590704C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C3×#626—21no—
14C2×C33×C8C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C3×#3722—21no—
20C3×(C2×D3)⋊Dic3—21no—
21C3×Dic32—21no—
22C3×#930—31no—
23C3×#926—31no—
24C3×#895—31no—
25C3×#892—31no—
26#63909C2×C311yescharacteristic
27#63841C2×C311yescharacteristic, complemented
28C32×#106—31no—
29C32×C22⋊C8—31no—
30#63648C2×C311yescharacteristic, complemented
31C3×#821—31no—
32C3×#821—31no—
33C3×#821—31no—
34C3×#821—61no—
35C3×(C2×D3)⋊C8—21no—
36C3×(C2×D3)⋊C8D311yescharacteristic, complemented
37C3×(C2×D3)⋊C8C2×C311yescharacteristic, complemented
38C3×#818—31no—
39C3×#817—21no—
40C3×#817D311yescharacteristic, complemented
41C3×#817C2×C311yescharacteristic, complemented
42C3×#809—31no—
43C3×#807—31no—
44#63575C2×C311yescharacteristic, complemented
45#63564C2×C311yescharacteristic, complemented
46C22×C3×C3⋊D3—21no—
47C22×C32×D3D411yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C32×C4×D3—21no—
56C32×C4×D3—21no—
57C3×#626—42no—
58C33×C8—21no—
59C32×C3⋊C8—21no—
60C3×#482—91no—
61#4866—31no—
62#4737—31no—
63C22×C32×C4—31no—
64C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
65C2×C3×#67—61no—
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×D12—61no—
70C2×C3×C4×D3—21no—
71C2×C3×C4×D3C2×D311yescharacteristic
72C2×C3×C4×D3C22×C311yescharacteristic
73C2×#626—21no—
74C2×C32×C8—21no—
75C2×C32×C8—21no—
76C2×C32×C8—21no—
77C2×C32×C8—31no—
78C2×C32×C8—41no—
79C2×C32×C8C2×D311yescharacteristic
80C2×C32×C8C2×D311yescharacteristic
81C2×C32×C8C22×C311yescharacteristic
82C32×C4⋊C4—31no—
83C32×C4⋊C4—31no—
84C4⋊(C3⋊Dic3)C22×C311yescharacteristic
85C3×#217—61no—
86C3×(C2×D3)⋊C4—61no—
87C3×(C2×D3)⋊C4—61no—
88C3×(C2×D3)⋊C4—61no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C22×C311yescharacteristic
97C3×C4⋊Dic3C22×C311yescharacteristic
98C3×C4×Dic3—61no—
99C3×C4×Dic3—61no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C22×C311yescharacteristic
103#3722—21no—
104(C2×D3)⋊Dic3—21no—
105Dic32—21no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—41no—
108C2×C3×C3⋊D3—41no—
109C2×C3×C3⋊D3—41no—
110C2×C32×D3—21no—
111C2×C32×D3—21no—
112C2×C32×D3—21no—
113C33×C4#4111yescharacteristic
114C33×C4#4111yescharacteristic
115C3×C3⋊Dic3—41no—
116C32×Dic3—21no—
117C32×Dic3—41no—
118C32×Dic3—41no—
119C3×#121—91no—
120C3×#117—91no—
121#930—31no—
122#926—31no—
123#895—31no—
124#892—31no—
125C3×#107—91no—
126C3×#106—31no—
127C3×#106—31no—
128C3×#106—61no—
129C3×#102—91no—
130C3×#102—91no—
131C3×C22⋊C8—31no—
132C3×C22⋊C8—31no—
133C3×C22⋊C8—61no—
134#821—31no—
135#821—31no—
136#821—31no—
137#821—61no—
138(C2×D3)⋊C8C3×D311yescharacteristic, complemented
139#818—31no—
140#817C3×D311yescharacteristic, complemented
141#809—31no—
142#807—31no—
143C23×C32—31no—
144C22×C3⋊D3—21no—
145C22×C3×D3—21no—
146C22×C3×D3—61no—
147C22×C3×D3—61no—
148C22×C3×D3—121no—
149C22×C3×D3D1211yescharacteristic
150C22×C3×D3C3×D411yescharacteristic
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—31no—
155C2×C32×C4—41no—
156C2×C32×C4—61no—
157C2×C32×C4—61no—
158C2×C32×C4—61no—
159C2×C32×C4—61no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C23×C311yescharacteristic
163C2×C3⋊Dic3—21no—
164C4⋊(C3⋊D3)—21no—
165C4⋊(C3⋊D3)—21no—
166C3×#67—122no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—41no—
173C2×C3×Dic3—41no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—121no—
177C2×C3×Dic3D1211yescharacteristic
178C2×C3×Dic3C3×D411yescharacteristic
179C3×D12—62no—
180C3×D12—61no—
181C3×D12—61no—
182C3×D12—61no—
183C3×D12—61no—
184C3×D12—121no—
185C3×D12—122no—
186C3×C4×D3—21no—
187C3×C4×D3—21no—
188C3×C4×D3—21no—
189C3×C4×D3—21no—
190C3×C4×D3—41no—
191C3×C4×D3—41no—
192#626—42no—
193C32×C8—21no—
194C32×C8—22no—
195C32×C8—21no—
196C32×C8—21no—
197C32×C8—41no—
198C32×C8—41no—
199C32×C8—42no—
200C32×C8—61no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—41no—
204#482—91noSylow
205C2×C33#12311yescharacteristic
206C2×C33#13511yescharacteristic
207C2×C33AGL(1,ℤ/8)11yescharacteristic
208C3×C3⋊D3—81no—
209C32×D3—41no—
210C2×C3×D4—91no—
211C2×C3×D4—181no—
212C22×C3×C4—31no—
213C22×C3×C4—31no—
214C22×C3×C4—61no—
215C2×#67—61no—
216C2×D12—31no—
217C2×D12—31no—
218C2×D12—61no—
219C2×D12—61no—
220C2×C4×D3C2×C3×D311yescharacteristic
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—41no—
227C2×C3×C8—61no—
228C2×C3×C8D3211yescharacteristic
229C2×C3×C8C2×C3×D311yescharacteristic
230C2×C3×C8C2×C3×D311yescharacteristic
231C3×C4⋊C4—31no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—61no—
237C3×C4⋊C4—91no—
238C3×C22⋊C4—181no—
239C3×C22⋊C4—181no—
240C3×C42—181no—
241#217—61no—
242(C2×D3)⋊C4—61no—
243(C2×D3)⋊C4—61no—
244(C2×D3)⋊C4—61no—
245C4⋊Dic3—31no—
246C4⋊Dic3—31no—
247C4⋊Dic3—61no—
248C4⋊Dic3C2×C3×D311yescharacteristic
249C4⋊Dic3C2×C3×D311yescharacteristic
250C4×Dic3—61no—
251C4×Dic3—61no—
252C2×C3⋊C8C2×C3×D311yescharacteristic
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—41no—
257C22×C32—61no—
258C22×C32—61no—
259C22×C32—61no—
260C22×C32C2×D1211yescharacteristic
261C22×C32D3×D411yescharacteristic
262C22×C32C2×C3×D411yescharacteristic
263C2×C3⋊D3—41no—
264C2×C3⋊D3—41no—
265C2×C3⋊D3—41no—
266C2×C3×D3—22no—
267C2×C3×D3—21no—
268C2×C3×D3—21no—
269C2×C3×D3—21no—
270C2×C3×D3—21no—
271C2×C3×D3—21no—
272C2×C3×D3—21no—
273C2×C3×D3—41no—
274C2×C3×D3—41no—
275C2×C3×D3—122no—
276C2×C3×D3—121no—
277C2×C3×D3—121no—
278C2×C3×D3—121no—
279C2×C3×D3—121no—
280C2×C3×D3—121no—
281C2×C3×D3—121no—
282C2×C3×D3—241no—
283C2×C3×D3—241no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—41no—
291C32×C4—41no—
292C32×C4—61no—
293C32×C4—121no—
294C32×C4—121no—
295C32×C4#23511yescharacteristic
296C32×C4#23711yescharacteristic
297C32×C4#23711yescharacteristic
298C32×C4#23911yescharacteristic
299C32×C4C3×#4111yescharacteristic
300C32×C4C3×#4111yescharacteristic
301C3⋊Dic3—41no—
302C3×Dic3—21no—
303C3×Dic3—21no—
304C3×Dic3—41no—
305C3×Dic3—41no—
306C3×Dic3—41no—
307C3×Dic3—41no—
308C3×Dic3—41no—
309C3×Dic3—81no—
310C3×Dic3—81no—
311C3×Dic3—121no—
312C3×Dic3—121no—
313C3×Dic3—241no—
314#121—91no—
315#117—91no—
316#107—91no—
317#106—31no—
318#102—91no—
319#102—91no—
320C22⋊C8—31no—
321C33#48211yescharacteristic, complemented, Sylow
322C23×C3—31no—
323C23×C3—31no—
324C23×C3—61no—
325C23×C3—181no—
326C22×D3—61no—
327C22×D3—61no—
328C22×D3—121no—
329C22×D3C3×D1211yescharacteristic
330C3×D4—181no—
331C3×D4—181no—
332C3×D4—362no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—41no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—121no—
349C2×C3×C4—121no—
350C2×C3×C4—121no—
351C2×C3×C4—121no—
352C2×C3×C4—181no—
353C2×C3×C4C2×D3211yescharacteristic
354C2×C3×C4C22×C3×D311yescharacteristic
355C2×C3×C4C22×C3×D311yescharacteristic
356#67—122no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×Dic3—121no—
362C2×Dic3C3×D1211yescharacteristic
363D12—62no—
364D12—61no—
365D12—61no—
366D12—61no—
367D12—61no—
368D12—121no—
369D12—122no—
370C4×D3—21no—
371C4×D3—21no—
372C3×C8—22no—
373C3×C8—21no—
374C3×C8—21no—
375C3×C8—21no—
376C3×C8—41no—
377C3×C8—41no—
378C3×C8—42no—
379C3×C8—61no—
380C3×C8—61no—
381C3×C8—121no—
382C3⋊C8—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—41no—
393C2×C32—41no—
394C2×C32—41no—
395C2×C32—121no—
396C2×C32#88611yescharacteristic
397C2×C32#89411yescharacteristic
398C2×C32#90411yescharacteristic
399C2×C32#90811yescharacteristic
400C2×C32#91411yescharacteristic
401C2×C32#91611yescharacteristic
402C2×C32C3×#12311yescharacteristic
403C2×C32C3×#13511yescharacteristic
404C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
405C3⋊D3—81no—
406C3×D3—42no—
407C3×D3—41no—
408C3×D3—41no—
409C3×D3—242no—
410C3×D3—241no—
411C3×D3—241no—
412C2×D4—91no—
413C2×D4—181no—
414C22×C4—31no—
415C2×C8—31no—
416C2×C8C3×D3211yescharacteristic
417C4⋊C4—31no—
418C4⋊C4—31no—
419C4⋊C4—91no—
420C22⋊C4—181no—
421C22⋊C4—181no—
422C42—181no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—41no—
427C22×C3—62no—
428C22×C3—61no—
429C22×C3—61no—
430C22×C3—61no—
431C22×C3—61no—
432C22×C3—61no—
433C22×C3—61no—
434C22×C3—121no—
435C22×C3—121no—
436C22×C3—361no—
437C22×C3—361no—
438C22×C3—361no—
439C22×C3D3×D1211yescharacteristic
440C22×C3C2×C3×D1211yescharacteristic
441C22×C3C3×D3×D411yescharacteristic
442C2×D3—21no—
443C2×D3—21no—
444C2×D3—21no—
445C2×D3—122no—
446C2×D3—121no—
447C2×D3—121no—
448C2×D3—121no—
449C2×D3—121no—
450C2×D3—121no—
451C2×D3—121no—
452C2×D3—241no—
453C2×D3—241no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—41no—
461C3×C4—41no—
462C3×C4—61no—
463C3×C4—61no—
464C3×C4—121no—
465C3×C4—121no—
466C3×C4—121no—
467C3×C4—121no—
468C3×C4—121no—
469C3×C4—241no—
470C3×C4—241no—
471C3×C4—361no—
472C3×C4#379611yescharacteristic
473C3×C4#380411yescharacteristic
474C3×C4C3×#23511yescharacteristic
475C3×C4C3×#23711yescharacteristic
476C3×C4C3×#23711yescharacteristic
477C3×C4C3×#23911yescharacteristic
478Dic3—21no—
479Dic3—41no—
480Dic3—41no—
481Dic3—121no—
482Dic3—121no—
483Dic3—241no—
484C32—21no—
485C32—21no—
486C32—21no—
487C32—41no—
488C32#473711yescharacteristic, complemented
489C32#486611yescharacteristic, complemented
490C32C3×#48211yescharacteristic, complemented
491C23—31no—
492C23—181no—
493D4—181no—
494D4—181no—
495D4—362no—
496C2×C4—31no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—181no—
502C2×C4C2×C3×D3211yescharacteristic
503C8—21no—
504C8—61no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—41no—
515C2×C3—41no—
516C2×C3—41no—
517C2×C3—122no—
518C2×C3—121no—
519C2×C3—121no—
520C2×C3—721no—
521C2×C3#6379411yescharacteristic
522C2×C3#6380611yescharacteristic
523C2×C3#6389011yescharacteristic
524C2×C3C3×#88611yescharacteristic
525C2×C3C3×#89411yescharacteristic
526C2×C3C3×#90411yescharacteristic
527C2×C3C3×#90811yescharacteristic
528C2×C3C3×#91411yescharacteristic
529C2×C3C3×#91611yescharacteristic
530D3—41no—
531D3—242no—
532D3—241no—
533D3—241no—
534C22—61no—
535C22—61no—
536C22—61no—
537C22—361no—
538C22—361no—
539C22—361no—
540C22C3×D3×D1211yescharacteristic
541C4—61no—
542C4—121no—
543C4—121no—
544C4—361no—
545C4C3×#379611yescharacteristic
546C4C3×#380411yescharacteristic
547C3—21no—
548C3—21no—
549C3—21no—
550C3—41no—
551C3#1059070411yescharacteristic, direct product factor
552C3C3×#473711yescharacteristic, complemented
553C3C3×#486611yescharacteristic, complemented
554C2—121no—
555C2—721no—
556C2C3×#6379411yescharacteristic
557C2C3×#6380611yescharacteristic
558C2C3×#6389011yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC3×#63909C2
hypercenterC2×C3×C8D32
hypocenterC32C3×#482
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×#63841C2semidirect
3. C3×#63648C2semidirect
4. C32×(C2×D3)⋊C8C2semidirect
5. C32×#817C2semidirect
6. C3×#63575C2semidirect
7. C3×#63564C2semidirect
10. #10590704C3direct (N × H)
27. #63841C2×C3semidirect
30. #63648C2×C3semidirect
36. C3×(C2×D3)⋊C8D3semidirect
37. C3×(C2×D3)⋊C8C2×C3semidirect
40. C3×#817D3semidirect
41. C3×#817C2×C3semidirect
44. #63575C2×C3semidirect
45. #63564C2×C3semidirect
138. (C2×D3)⋊C8C3×D3semidirect
140. #817C3×D3semidirect
321. C33#482semidirect
488. C32#4737semidirect
489. C32#4866semidirect
490. C32C3×#482semidirect
551. C3#10590704direct (N × H)
552. C3C3×#4737semidirect
553. C3C3×#4866semidirect

Maximal subgroups

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

subgroupindex
C3×#639092
C3×#638412
C3×#636482
C32×(C2×D3)⋊C82
C32×#8172
C3×#635752
C3×#635642
C3×#48663
C3×#47373
#105907043