Finite Group Databasebeta

← C3×#10590651 #13157813

Subgroups — C3×#10590651 #13157813

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×#63916C211yesmaximal, characteristic
2C3×#63849C211yesmaximal, characteristic, complemented
3C3×#63647C211yesmaximal, characteristic, complemented
4C32×#821C211yesmaximal, characteristic, complemented
5C32×#818C211yesmaximal, characteristic, complemented
6C3×#63574C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8C3×#4885—31nomaximal
9C3×#4784—31nomaximal
10#10590651C311yesmaximal, characteristic, direct product factor
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×#626—21no—
14C2×C33×C8C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C3×#3722—21no—
20C3×(C2×D3)⋊Dic3—21no—
21C3×Dic32—21no—
22C3×#938—31no—
23C3×#895—31no—
24C3×#892—31no—
25C3×#892—31no—
26#63916C2×C311yescharacteristic
27#63849C2×C311yescharacteristic, complemented
28C32×#107—31no—
29C32×#102—31no—
30#63647C2×C311yescharacteristic, complemented
31C3×#832—31no—
32C3×#821—21no—
33C3×#821D311yescharacteristic, complemented
34C3×#821C2×C311yescharacteristic, complemented
35C3×(C2×D3)⋊C8—31no—
36C3×(C2×D3)⋊C8—31no—
37C3×(C2×D3)⋊C8—61no—
38C3×#818—21no—
39C3×#818D311yescharacteristic, complemented
40C3×#818C2×C311yescharacteristic, complemented
41C3×#809—31no—
42C3×#808—31no—
43C3×#808—31no—
44#63574C2×C311yescharacteristic, complemented
45#63562C2×C311yescharacteristic, complemented
46C22×C3×C3⋊D3D411yescharacteristic
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4×C3⋊D3—21no—
51C3×C4×C3⋊D3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×D12—21no—
56C32×D12—21no—
57C3×#626—42no—
58C33×C8—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C3×#482—91no—
61#4885—31no—
62#4784—31no—
63C2×C32×D4—31no—
64C2×C4×C3⋊D3C22×C311yescharacteristic
65C2×C3×#67—61no—
66C2×C3×D12—21no—
67C2×C3×D12—61no—
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×D12C22×C311yescharacteristic
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—61no—
73C2×#626—21no—
74C2×C32×C8—21no—
75C2×C32×C8—21no—
76C2×C32×C8—21no—
77C2×C32×C8—41no—
78C2×C32×C8C2×D311yescharacteristic
79C2×C32×C8C2×D311yescharacteristic
80C2×C32×C8C22×C311yescharacteristic
81C32×C4⋊C4—31no—
82C32×C4⋊C4—31no—
83C32×C4⋊C4—31no—
84C2×C3⋊(C3⋊C8)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—21no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C22×C311yescharacteristic
96C3×C4⋊Dic3C22×C311yescharacteristic
97C3×C4⋊Dic3C22×C311yescharacteristic
98C3×C4×Dic3—61no—
99C3×C4×Dic3—61no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103#3722—21no—
104(C2×D3)⋊Dic3—21no—
105Dic32—21no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—21no—
108C2×C3×C3⋊D3—21no—
109C2×C3×C3⋊D3—21no—
110C2×C32×D3—41no—
111C2×C32×D3—41no—
112C2×C32×D3—41no—
113C33×C4#4111yescharacteristic
114C33×C4#4111yescharacteristic
115C3×C3⋊Dic3—21no—
116C32×Dic3—41no—
117C32×Dic3—41no—
118C32×Dic3—41no—
119C3×#121—91no—
120C3×#117—91no—
121#938—31no—
122#895—31no—
123#892—31no—
124#892—31no—
125C3×#107—31no—
126C3×#107—31no—
127C3×#107—61no—
128C3×#106—91no—
129C3×#102—31no—
130C3×#102—31no—
131C3×#102—61no—
132C3×#102—91no—
133C3×C22⋊C8—91no—
134#832—31no—
135#821C3×D311yescharacteristic, complemented
136(C2×D3)⋊C8—31no—
137(C2×D3)⋊C8—31no—
138(C2×D3)⋊C8—61no—
139#818C3×D311yescharacteristic, complemented
140#809—31no—
141#808—31no—
142#808—31no—
143C23×C32—61no—
144C22×C3⋊D3C3×D411yescharacteristic
145C22×C3×D3—21no—
146C22×C3×D3—21no—
147C22×C3×D3—31no—
148C22×C3×D3—31no—
149C22×C3×D3—41no—
150C22×C3×D3—61no—
151C32×D4—61no—
152C32×D4—61no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—41no—
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⋊Dic3C3×D411yescharacteristic
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—21no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—41no—
176C2×C3×Dic3—41no—
177C2×C3×Dic3—41no—
178C2×C3×Dic3—61no—
179C3×D12—22no—
180C3×D12—21no—
181C3×D12—21no—
182C3×D12—21no—
183C3×D12—21no—
184C3×D12—41no—
185C3×D12—122no—
186C3×C4×D3—61no—
187C3×C4×D3—61no—
188C3×C4×D3—61no—
189C3×C4×D3—61no—
190C3×C4×D3—121no—
191C3×C4×D3—121no—
192#626—42no—
193C32×C8—21no—
194C32×C8—22no—
195C32×C8—21no—
196C32×C8—21no—
197C32×C8—41no—
198C32×C8—41no—
199C32×C8—42no—
200C3⋊(C3⋊C8)—21no—
201C3×C3⋊C8—61no—
202C3×C3⋊C8—61no—
203C3×C3⋊C8—121no—
204#482—91noSylow
205C2×C33#12311yescharacteristic
206C2×C33#13511yescharacteristic
207C2×C33AGL(1,ℤ/8)11yescharacteristic
208C3×C3⋊D3—41no—
209C32×D3—81no—
210C2×C3×D4—31no—
211C2×C3×D4—31no—
212C2×C3×D4—61no—
213C2×C3×D4—181no—
214C22×C3×C4—91no—
215C2×#67—61no—
216C2×D12—61no—
217C2×D12C2×C3×D311yescharacteristic
218C2×C4×D3—31no—
219C2×C4×D3—31no—
220C2×C4×D3—61no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—41no—
225C2×C3×C8—91no—
226C2×C3×C8D3211yescharacteristic
227C2×C3×C8C2×C3×D311yescharacteristic
228C2×C3×C8C2×C3×D311yescharacteristic
229C3×C4⋊C4—31no—
230C3×C4⋊C4—31no—
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—61no—
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⋊Dic3C2×C3×D311yescharacteristic
246C4⋊Dic3C2×C3×D311yescharacteristic
247C4⋊Dic3C2×C3×D311yescharacteristic
248C4×Dic3—61no—
249C4×Dic3—61no—
250C2×C3⋊C8—31no—
251C2×C3⋊C8—31no—
252C2×C3⋊C8—61no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—41no—
257C22×C32—121no—
258C22×C32—121no—
259C22×C32—121no—
260C22×C32D3×D411yescharacteristic
261C22×C32D3×D411yescharacteristic
262C22×C32C2×C3×D411yescharacteristic
263C2×C3⋊D3—21no—
264C2×C3⋊D3—21no—
265C2×C3⋊D3—21no—
266C2×C3×D3—42no—
267C2×C3×D3—41no—
268C2×C3×D3—41no—
269C2×C3×D3—41no—
270C2×C3×D3—41no—
271C2×C3×D3—41no—
272C2×C3×D3—41no—
273C2×C3×D3—62no—
274C2×C3×D3—61no—
275C2×C3×D3—61no—
276C2×C3×D3—61no—
277C2×C3×D3—61no—
278C2×C3×D3—61no—
279C2×C3×D3—61no—
280C2×C3×D3—81no—
281C2×C3×D3—81no—
282C2×C3×D3—121no—
283C2×C3×D3—121no—
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—121no—
293C32×C4—121no—
294C32×C4—121no—
295C32×C4#23511yescharacteristic
296C32×C4#23511yescharacteristic
297C32×C4#23711yescharacteristic
298C32×C4#23911yescharacteristic
299C32×C4C3×#4111yescharacteristic
300C32×C4C3×#4111yescharacteristic
301C3⋊Dic3—21no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3—41no—
305C3×Dic3—41no—
306C3×Dic3—41no—
307C3×Dic3—41no—
308C3×Dic3—61no—
309C3×Dic3—61no—
310C3×Dic3—81no—
311C3×Dic3—81no—
312C3×Dic3—81no—
313C3×Dic3—121no—
314#121—91no—
315#117—91no—
316#107—31no—
317#106—91no—
318#102—31no—
319#102—91no—
320C22⋊C8—91no—
321C33#48211yescharacteristic, complemented, Sylow
322C23×C3—61no—
323C23×C3—61no—
324C23×C3—91no—
325C23×C3—121no—
326C22×D3—21no—
327C22×D3—31no—
328C22×D3—31no—
329C22×D3—61no—
330C3×D4—62no—
331C3×D4—61no—
332C3×D4—61no—
333C3×D4—61no—
334C3×D4—61no—
335C3×D4—121no—
336C3×D4—362no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—41no—
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—91no—
348C2×C3×C4—121no—
349C2×C3×C4—121no—
350C2×C3×C4—121no—
351C2×C3×C4—181no—
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—21no—
360C2×Dic3—31no—
361C2×Dic3—31no—
362C2×Dic3—61no—
363D12—21no—
364D12—21no—
365D12—122no—
366C4×D3—61no—
367C4×D3—61no—
368C4×D3—61no—
369C4×D3—61no—
370C4×D3—121no—
371C4×D3—121no—
372C3×C8—22no—
373C3×C8—21no—
374C3×C8—21no—
375C3×C8—21no—
376C3×C8—41no—
377C3×C8—41no—
378C3×C8—42no—
379C3×C8—181no—
380C3⋊C8—61no—
381C3⋊C8—61no—
382C3⋊C8—121no—
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—241no—
396C2×C32#88311yescharacteristic
397C2×C32#89411yescharacteristic
398C2×C32#91111yescharacteristic
399C2×C32#91211yescharacteristic
400C2×C32#91411yescharacteristic
401C2×C32#91911yescharacteristic
402C2×C32C3×#12311yescharacteristic
403C2×C32C3×#13511yescharacteristic
404C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
405C3⋊D3—41no—
406C3×D3—82no—
407C3×D3—81no—
408C3×D3—81no—
409C3×D3—122no—
410C3×D3—121no—
411C3×D3—121no—
412C2×D4—31no—
413C2×D4—181no—
414C22×C4—91no—
415C2×C8—91no—
416C2×C8C3×D3211yescharacteristic
417C4⋊C4—31no—
418C4⋊C4—31no—
419C4⋊C4—31no—
420C22⋊C4—181no—
421C22⋊C4—181no—
422C42—181no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—41no—
427C22×C3—122no—
428C22×C3—121no—
429C22×C3—121no—
430C22×C3—121no—
431C22×C3—121no—
432C22×C3—121no—
433C22×C3—121no—
434C22×C3—181no—
435C22×C3—181no—
436C22×C3—181no—
437C22×C3—241no—
438C22×C3—241no—
439C22×C3#380211yescharacteristic
440C22×C3C3×D3×D411yescharacteristic
441C22×C3C3×D3×D411yescharacteristic
442C2×D3—41no—
443C2×D3—41no—
444C2×D3—41no—
445C2×D3—62no—
446C2×D3—61no—
447C2×D3—61no—
448C2×D3—61no—
449C2×D3—61no—
450C2×D3—61no—
451C2×D3—61no—
452C2×D3—121no—
453C2×D3—121no—
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—121no—
463C3×C4—121no—
464C3×C4—121no—
465C3×C4—121no—
466C3×C4—121no—
467C3×C4—121no—
468C3×C4—181no—
469C3×C4—241no—
470C3×C4—241no—
471C3×C4—241no—
472C3×C4#379911yescharacteristic
473C3×C4#380411yescharacteristic
474C3×C4C3×#23511yescharacteristic
475C3×C4C3×#23511yescharacteristic
476C3×C4C3×#23711yescharacteristic
477C3×C4C3×#23911yescharacteristic
478Dic3—41no—
479Dic3—41no—
480Dic3—41no—
481Dic3—61no—
482Dic3—61no—
483Dic3—121no—
484C32—21no—
485C32—21no—
486C32—21no—
487C32—41no—
488C32#478411yescharacteristic, complemented
489C32#488511yescharacteristic, complemented
490C32C3×#48211yescharacteristic, complemented
491C23—61no—
492C23—91no—
493D4—61no—
494D4—61no—
495D4—362no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—91no—
500C2×C4—181no—
501C2×C4—181no—
502C2×C4C2×C3×D3211yescharacteristic
503C8—21no—
504C8—181no—
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—242no—
518C2×C3—241no—
519C2×C3—241no—
520C2×C3—361no—
521C2×C3#6379811yescharacteristic
522C2×C3#6381011yescharacteristic
523C2×C3#6385411yescharacteristic
524C2×C3C3×#88311yescharacteristic
525C2×C3C3×#89411yescharacteristic
526C2×C3C3×#91111yescharacteristic
527C2×C3C3×#91211yescharacteristic
528C2×C3C3×#91411yescharacteristic
529C2×C3C3×#91911yescharacteristic
530D3—81no—
531D3—122no—
532D3—121no—
533D3—121no—
534C22—121no—
535C22—121no—
536C22—121no—
537C22—181no—
538C22—181no—
539C22—181no—
540C22C3×#380211yescharacteristic
541C4—121no—
542C4—121no—
543C4—121no—
544C4—181no—
545C4C3×#379911yescharacteristic
546C4C3×#380411yescharacteristic
547C3—21no—
548C3—21no—
549C3—21no—
550C3—41no—
551C3#1059065111yescharacteristic, direct product factor
552C3C3×#478411yescharacteristic, complemented
553C3C3×#488511yescharacteristic, complemented
554C2—241no—
555C2—361no—
556C2C3×#6379811yescharacteristic
557C2C3×#6381011yescharacteristic
558C2C3×#6385411yescharacteristic
subgroupG/H
centerC22×C3#3802
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC3×#63916C2
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×#63849C2semidirect
3. C3×#63647C2semidirect
4. C32×#821C2semidirect
5. C32×#818C2semidirect
6. C3×#63574C2semidirect
7. C3×#63562C2semidirect
10. #10590651C3direct (N × H)
27. #63849C2×C3semidirect
30. #63647C2×C3semidirect
33. C3×#821D3semidirect
34. C3×#821C2×C3semidirect
39. C3×#818D3semidirect
40. C3×#818C2×C3semidirect
44. #63574C2×C3semidirect
45. #63562C2×C3semidirect
135. #821C3×D3semidirect
139. #818C3×D3semidirect
321. C33#482semidirect
488. C32#4784semidirect
489. C32#4885semidirect
490. C32C3×#482semidirect
551. C3#10590651direct (N × H)
552. C3C3×#4784semidirect
553. C3C3×#4885semidirect

Maximal subgroups

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

subgroupindex
C3×#639162
C3×#638492
C3×#636472
C32×#8212
C32×#8182
C3×#635742
C3×#635622
C3×#48853
C3×#47843
#105906513