Finite Group Databasebeta

← C3×#10590435 #13157528

Subgroups — C3×#10590435 #13157528

Order 1728.

600 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64102C211yesmaximal, characteristic
2C3×#63872C211yesmaximal, characteristic, complemented
3C32×#808C211yesmaximal, characteristic, complemented
4C32×#807C211yesmaximal, characteristic, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic
7C3×#63564C211yesmaximal, characteristic
8C3×#4891—31nomaximal
9C3×#4871—31nomaximal
10#10590435C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3—21no—
13C33×C4⋊C4C2211yescharacteristic
14C3×#3752—21no—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22#64102C2×C311yescharacteristic
23C3×#893—31no—
24C3×#893—31no—
25C3×#893—61no—
26C3×#890—31no—
27C3×#890—31no—
28#63872C2×C311yescharacteristic, complemented
29C32×#107—31no—
30C32×#106—31no—
31C3×#821—31no—
32C3×#821—31no—
33C3×Dic3⋊C8—31no—
34C3×Dic3⋊C8—31no—
35C3×#809—31no—
36C3×#809—31no—
37C3×#808—21no—
38C3×#808D311yescharacteristic, complemented
39C3×#808C2×C311yescharacteristic, complemented
40C3×#807—21no—
41C3×#807D311yescharacteristic, complemented
42C3×#807C2×C311yescharacteristic, complemented
43#63572C2×C311yescharacteristic, complemented
44#63564C2×C311yescharacteristic
45#63564C2×C311yescharacteristic
46C22×C3×C3⋊D3—21no—
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3D411yescharacteristic
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)Q812yes—
53C3×C4×C3⋊D3—42no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C32×C3⋊C8—21no—
57C32×C3⋊C8—21no—
58C3×#478—91no—
59#4891—31no—
60#4871—31no—
61C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
62C2×C4×C3⋊D3—21no—
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×C3×D12—61no—
66C2×C3×C4×D3—61no—
67C2×C3×C4×D3—61no—
68C2×C3×C4×D3—62no—
69C2×C32×C8—31no—
70C2×C32×C8—31no—
71C32×C4⋊C4—21no—
72C32×C4⋊C4—21no—
73C32×C4⋊C4—21no—
74C32×C4⋊C4—31no—
75C32×C4⋊C4—31no—
76C32×C4⋊C4—41no—
77C32×C4⋊C4C2×D311yescharacteristic
78C32×C4⋊C4C2×D311yescharacteristic
79C32×C4⋊C4C22×C311yescharacteristic
80#3752—21no—
81C4×C3⋊Dic3C22×C311yescharacteristic
82C3×(C2×D3)⋊C4—61no—
83C3×(C2×D3)⋊C4—61no—
84C3×(C2×D3)⋊C4—62no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C22×C311yescharacteristic
92C3×C4⋊Dic3C22×C311yescharacteristic
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—31no—
97C3×C4×Dic3—61no—
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8C2×D311yescharacteristic
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C22×C311yescharacteristic
103C2×C3×C3⋊C8C22×C311yescharacteristic
104Dic3⋊Dic3—21no—
105Dic3⋊Dic3—21no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3—41no—
109C2×C3×C3⋊D3—41no—
110C33×C4—41no—
111C33×C4C2×Q811yescharacteristic
112C33×C4#4111yescharacteristic
113C3×C3⋊Dic3—21no—
114C3×C3⋊Dic3—21no—
115C32×Dic3—41no—
116C32×Dic3—41no—
117C3×#125—91no—
118C3×C4×D4—91no—
119#893—31no—
120#893—31no—
121#893—61no—
122#890—31no—
123#890—31no—
124C3×#107—31no—
125C3×#107—31no—
126C3×#107—61no—
127C3×#106—31no—
128C3×#106—31no—
129C3×#106—61no—
130C3×C4⋊C8—91no—
131C3×#102—91no—
132C3×#102—91no—
133#821—31no—
134#821—31no—
135Dic3⋊C8—31no—
136Dic3⋊C8—31no—
137#809—31no—
138#809—31no—
139#808C3×D311yescharacteristic, complemented
140#807C3×D311yescharacteristic, complemented
141C22×C3⋊D3—21no—
142C22×C3×D3—62no—
143C22×C3×D3—61no—
144C22×C3×D3—61no—
145C2×C32×C4—21no—
146C2×C32×C4—22no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—41no—
153C2×C32×C4—41no—
154C2×C32×C4—41no—
155C2×C32×C4—42no—
156C2×C32×C4—61no—
157C2×C32×C4—61no—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C23×C311yescharacteristic
161C2×C3⋊Dic3C3×D411yescharacteristic
162C2×C3⋊Dic3C3×D411yescharacteristic
163C4⋊(C3⋊D3)—21no—
164C4⋊(C3⋊D3)C3×Q812yes—
165C4×C3⋊D3—42no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—41no—
175C2×C3×Dic3—41no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C3×D12—32no—
179C3×D12—32no—
180C3×D12—62no—
181C3×D12—62no—
182C3×D12—61no—
183C3×D12—61no—
184C3×C4×D3—124no—
185C3×C4×D3—122no—
186C3×C4×D3—122no—
187C32×C8—61no—
188C32×C8—61no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8—22no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8—21no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8—21no—
195#478—91noSylow
196C2×C33C22⋊Q811yescharacteristic
197C2×C33#13511yescharacteristic
198C2×C33AGL(1,ℤ/8)11yescharacteristic
199C3×C3⋊D3—42no—
200C2×C3×D4—91no—
201C22×C3×C4—181no—
202C2×D12—31no—
203C2×D12—31no—
204C2×D12—61no—
205C2×C4×D3—61no—
206C2×C4×D3—61no—
207C2×C4×D3—62no—
208C2×C3×C8—31no—
209C2×C3×C8—31no—
210C2×C3×C8—31no—
211C2×C3×C8—31no—
212C2×C3×C8—61no—
213C2×C3×C8—61no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—31no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—41no—
222C3×C4⋊C4—61no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4—181no—
225C3×C4⋊C4—181no—
226C3×C4⋊C4D3211yescharacteristic
227C3×C4⋊C4C2×C3×D311yescharacteristic
228C3×C4⋊C4C2×C3×D311yescharacteristic
229C3×C22⋊C4—181no—
230C3×C42—91no—
231(C2×D3)⋊C4—61no—
232(C2×D3)⋊C4—61no—
233(C2×D3)⋊C4—62no—
234C4⋊Dic3—61no—
235C4⋊Dic3—61no—
236C4⋊Dic3C2×C3×D311yescharacteristic
237C4⋊Dic3C2×C3×D311yescharacteristic
238Dic3⋊C4—61no—
239Dic3⋊C4—61no—
240C4×Dic3—31no—
241C4×Dic3—31no—
242C4×Dic3—61no—
243C2×C3⋊C8C2×C3×D311yescharacteristic
244C2×C3⋊C8C2×C3×D311yescharacteristic
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—41no—
249C22×C32D3×D411yescharacteristic
250C22×C32D3×D411yescharacteristic
251C22×C32C2×C3×D411yescharacteristic
252C2×C3⋊D3—22no—
253C2×C3⋊D3—41no—
254C2×C3⋊D3—41no—
255C2×C3×D3—64no—
256C2×C3×D3—62no—
257C2×C3×D3—62no—
258C2×C3×D3—122no—
259C2×C3×D3—122no—
260C2×C3×D3—121no—
261C2×C3×D3—121no—
262C2×C3×D3—121no—
263C2×C3×D3—121no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—41no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4—42no—
274C32×C4—41no—
275C32×C4—41no—
276C32×C4—81no—
277C32×C4—81no—
278C32×C4—82no—
279C32×C4—121no—
280C32×C4—121no—
281C32×C4#23511yescharacteristic
282C32×C4#23511yescharacteristic
283C32×C4D3×Q811yescharacteristic
284C32×C4D3×Q811yescharacteristic
285C32×C4C2×C3×Q811yescharacteristic
286C32×C4C3×#4111yescharacteristic
287C3⋊Dic3—21no—
288C3⋊Dic3—21no—
289C3×Dic3—41no—
290C3×Dic3—41no—
291C3×Dic3—41no—
292C3×Dic3—41no—
293C3×Dic3—62no—
294C3×Dic3—61no—
295C3×Dic3—61no—
296C3×Dic3—61no—
297C3×Dic3—61no—
298C3×Dic3—81no—
299C3×Dic3—81no—
300C3×Dic3—121no—
301#125—91no—
302C4×D4—91no—
303#107—31no—
304#106—31no—
305C4⋊C8—91no—
306#102—91no—
307#102—91no—
308C33#47811yescharacteristic, complemented, Sylow
309C23×C3—181no—
310C22×D3—62no—
311C22×D3—61no—
312C22×D3—61no—
313C3×D4—92no—
314C3×D4—181no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—121no—
333C2×C3×C4—121no—
334C2×C3×C4—362no—
335C2×C3×C4C2×D3211yescharacteristic
336C2×C3×C4C22×C3×D311yescharacteristic
337C2×C3×C4C22×C3×D311yescharacteristic
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—31no—
341C2×Dic3—31no—
342C2×Dic3—31no—
343C2×Dic3—31no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346D12—32no—
347D12—32no—
348D12—62no—
349D12—62no—
350D12—61no—
351D12—61no—
352C4×D3—124no—
353C4×D3—122no—
354C4×D3—122no—
355C3×C8—62no—
356C3×C8—62no—
357C3×C8—61no—
358C3×C8—61no—
359C3×C8—61no—
360C3×C8—61no—
361C3⋊C8—21no—
362C3⋊C8—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—41no—
373C2×C32—41no—
374C2×C32—41no—
375C2×C32#89611yescharacteristic
376C2×C32#89611yescharacteristic
377C2×C32#91111yescharacteristic
378C2×C32#91411yescharacteristic
379C2×C32#91611yescharacteristic
380C2×C32#91911yescharacteristic
381C2×C32C3×C22⋊Q811yescharacteristic
382C2×C32C3×#13511yescharacteristic
383C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
384C3⋊D3—42no—
385C3×D3—124no—
386C3×D3—122no—
387C3×D3—122no—
388C2×D4—91no—
389C22×C4—181no—
390C2×C8—31no—
391C2×C8—31no—
392C4⋊C4—31no—
393C4⋊C4—31no—
394C4⋊C4—181no—
395C4⋊C4—181no—
396C4⋊C4C3×D3211yescharacteristic
397C22⋊C4—181no—
398C42—91no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—41no—
403C22×C3—182no—
404C22×C3—361no—
405C22×C3—361no—
406C22×C3#381111yescharacteristic
407C22×C3C3×D3×D411yescharacteristic
408C22×C3C3×D3×D411yescharacteristic
409C2×D3—64no—
410C2×D3—62no—
411C2×D3—62no—
412C2×D3—122no—
413C2×D3—122no—
414C2×D3—121no—
415C2×D3—121no—
416C2×D3—121no—
417C2×D3—121no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—42no—
425C3×C4—41no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—81no—
431C3×C4—81no—
432C3×C4—82no—
433C3×C4—121no—
434C3×C4—121no—
435C3×C4—121no—
436C3×C4—121no—
437C3×C4—181no—
438C3×C4—181no—
439C3×C4—241no—
440C3×C4—241no—
441C3×C4#379711yescharacteristic
442C3×C4#379811yescharacteristic
443C3×C4C3×#23511yescharacteristic
444C3×C4C3×#23511yescharacteristic
445C3×C4C3×D3×Q811yescharacteristic
446C3×C4C3×D3×Q811yescharacteristic
447Dic3—41no—
448Dic3—41no—
449Dic3—62no—
450Dic3—61no—
451Dic3—61no—
452Dic3—61no—
453Dic3—61no—
454Dic3—121no—
455C32—21no—
456C32—21no—
457C32—21no—
458C32—41no—
459C32#487111yescharacteristic, complemented
460C32#489111yescharacteristic, complemented
461C32C3×#47811yescharacteristic, complemented
462C23—181no—
463D4—92no—
464D4—181no—
465C2×C4—21no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—91no—
469C2×C4—91no—
470C2×C4—362no—
471C2×C4C2×C3×D3211yescharacteristic
472C8—61no—
473C8—61no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—362no—
487C2×C3#6386511yescharacteristic
488C2×C3#6393711yescharacteristic
489C2×C3#6395011yescharacteristic
490C2×C3C3×#89611yescharacteristic
491C2×C3C3×#89611yescharacteristic
492C2×C3C3×#91111yescharacteristic
493C2×C3C3×#91411yescharacteristic
494C2×C3C3×#91611yescharacteristic
495C2×C3C3×#91911yescharacteristic
496D3—124no—
497D3—122no—
498D3—122no—
499C22—182no—
500C22—361no—
501C22—361no—
502C22C3×#381111yescharacteristic
503C4—41no—
504C4—121no—
505C4—121no—
506C4—181no—
507C4—181no—
508C4C3×#379711yescharacteristic
509C4C3×#379811yescharacteristic
510C3—21no—
511C3—21no—
512C3—21no—
513C3—41no—
514C3#1059043511yescharacteristic, direct product factor
515C3C3×#487111yescharacteristic, complemented
516C3C3×#489111yescharacteristic, complemented
517C2—362no—
518C2C3×#6386511yescharacteristic
519C2C3×#6393711yescharacteristic
520C2C3×#6395011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#63872C2semidirect
3. C32×#808C2semidirect
4. C32×#807C2semidirect
5. C3×#63572C2semidirect
10. #10590435C3direct (N × H)
28. #63872C2×C3semidirect
38. C3×#808D3semidirect
39. C3×#808C2×C3semidirect
41. C3×#807D3semidirect
42. C3×#807C2×C3semidirect
43. #63572C2×C3semidirect
139. #808C3×D3semidirect
140. #807C3×D3semidirect
308. C33#478semidirect
459. C32#4871semidirect
460. C32#4891semidirect
461. C32C3×#478semidirect
514. C3#10590435direct (N × H)
515. C3C3×#4871semidirect
516. C3C3×#4891semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
C3×#638722
C32×#8082
C32×#8072
C3×#635722
C3×#635642
C3×#48913
C3×#48713
#105904353