Finite Group Databasebeta

← C3×#10590429 #13157500

Subgroups — C3×#10590429 #13157500

Order 1728.

564 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#892C211yesmaximal, characteristic
2C3×#63857C211yesmaximal, characteristic
3C3×#63637C211yesmaximal, characteristic, complemented
4C32×#807C211yesmaximal, characteristic, complemented
5C3×#63574C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8C3×#4996—31nomaximal
9C3×#4888—31nomaximal
10#10590429C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×(C2×D3)⋊C4—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3—21no—
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×Dic3⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C32×#117—31no—
23C3×#924—31no—
24C3×#897—31no—
25C3×#892—21no—
26C3×#892D311yescharacteristic
27C3×#892C2×C311yescharacteristic
28#63857C2×C311yescharacteristic
29C32×#107—31no—
30#63637C2×C311yescharacteristic, complemented
31C3×#830—31no—
32C3×(C2×D3)⋊C8—31no—
33C3×#816—31no—
34C3×#810—31no—
35C3×#810—31no—
36C3×#810—31no—
37C3×#810—61no—
38C3×#808—31no—
39C3×#808—31no—
40C3×#807—21no—
41C3×#807D311yescharacteristic, complemented
42C3×#807C2×C311yescharacteristic, complemented
43#63574C2×C311yescharacteristic, complemented
44#63567C2×C311yescharacteristic, complemented
45(C2×D3)⋊(C3⋊C8)C2×C311yescharacteristic
46C22×C32×D3D411yescharacteristic
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C3×#634—21no—
51C3×#634—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C32×C4×D3—21no—
56C32×C4×D3—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59C3×#485—91no—
60#4996—31no—
61#4888—31no—
62C22×C32×C4—31no—
63C2×#634C22×C311yescharacteristic
64C2×C3×C4×D3—21no—
65C2×C3×C4×D3C2×D311yescharacteristic
66C2×C3×C4×D3C22×C311yescharacteristic
67C2×C3×Dic6—31no—
68C2×C3×Dic6—31no—
69C2×C3×Dic6—61no—
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
80C32×C22⋊C4—61no—
81C32×C42—61no—
82C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
83C3×#217—61no—
84C3×(C2×D3)⋊C4—22no—
85C3×(C2×D3)⋊C4—21no—
86C3×(C2×D3)⋊C4—21no—
87C3×(C2×D3)⋊C4—61no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C22×C311yescharacteristic
94C3×C4⋊Dic3C22×C311yescharacteristic
95C3×Dic3⋊C4—61no—
96C3×C4×Dic3—22no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3—21no—
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C22×C311yescharacteristic
105Dic3⋊Dic3—21no—
106(C2×D3)⋊Dic3—21no—
107C22×C33C2×D411yescharacteristic
108C2×C32×D3—21no—
109C2×C32×D3—21no—
110C2×C32×D3—21no—
111C33×C4—41no—
112C33×C4#4111yescharacteristic
113C33×C4#4111yescharacteristic
114C3×C3⋊Dic3—41no—
115C32×Dic3—21no—
116C32×Dic3—41no—
117C32×Dic3—41no—
118C3×C22⋊Q8—91no—
119C3×#117—31no—
120C3×#117—31no—
121C3×#117—61no—
122#924—31no—
123#897—31no—
124#892C3×D311yescharacteristic
125C3×#107—31no—
126C3×#107—31no—
127C3×#107—61no—
128C3×#106—91no—
129C3×#103—91no—
130C3×#103—91no—
131C3×C22⋊C8—91no—
132#830—31no—
133(C2×D3)⋊C8—31no—
134#816—31no—
135#810—31no—
136#810—31no—
137#810—31no—
138#810—61no—
139#808—31no—
140#808—31no—
141#807C3×D311yescharacteristic, complemented
142C23×C32—31no—
143C22×C3×D3—21no—
144C22×C3×D3#6711yescharacteristic
145C22×C3×D3C3×D411yescharacteristic
146C2×C32×C4—21no—
147C2×C32×C4—22no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—31no—
154C2×C32×C4—41no—
155C2×C32×C4—41no—
156C2×C32×C4—41no—
157C2×C32×C4—42no—
158C2×C32×C4—61no—
159C2×C32×C4—61no—
160C2×C32×C4—61no—
161C2×C32×C4—61no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C23×C311yescharacteristic
165C2×C3⋊Dic3—21no—
166#634—21no—
167#634—21no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—41no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—121no—
178C2×C3×Dic3#6711yescharacteristic
179C2×C3×Dic3C3×D411yescharacteristic
180C3×C4×D3—22no—
181C3×C4×D3—22no—
182C3×C4×D3—21no—
183C3×C4×D3—21no—
184C3×C4×D3—21no—
185C3×C4×D3—21no—
186C3×Dic6—62no—
187C3×Dic6—61no—
188C3×Dic6—61no—
189C3×Dic6—61no—
190C3×Dic6—61no—
191C3×Dic6—121no—
192C32×C8—61no—
193C3⋊(C3⋊C8)—21no—
194C3×C3⋊C8—22no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—61no—
198C3×C3⋊C8—61no—
199C3×C3⋊C8—121no—
200#485—91noSylow
201C2×C33#12311yescharacteristic
202C2×C33#13511yescharacteristic
203C2×C33#13711yescharacteristic
204C32×D3—41no—
205C2×C3×Q8—91no—
206C22×C3×C4—31no—
207C22×C3×C4—31no—
208C22×C3×C4—61no—
209C2×C4×D3C2×C3×D311yescharacteristic
210C2×Dic6—31no—
211C2×Dic6—31no—
212C2×Dic6—61no—
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—61no—
216C2×C3×C8—91no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—41no—
225C3×C4⋊C4—61no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4—181no—
228C3×C4⋊C4D3211yescharacteristic
229C3×C4⋊C4C2×C3×D311yescharacteristic
230C3×C4⋊C4C2×C3×D311yescharacteristic
231C3×C22⋊C4—61no—
232C3×C22⋊C4—61no—
233C3×C22⋊C4—62no—
234C3×C22⋊C4—181no—
235C3×C42—61no—
236C3×C42—61no—
237C3×C42—62no—
238#217—61no—
239(C2×D3)⋊C4—21no—
240(C2×D3)⋊C4—61no—
241C4⋊Dic3—61no—
242C4⋊Dic3C2×C3×D311yescharacteristic
243C4⋊Dic3C2×C3×D311yescharacteristic
244Dic3⋊C4—61no—
245C4×Dic3—21no—
246C2×C3⋊C8—31no—
247C2×C3⋊C8—31no—
248C2×C3⋊C8—61no—
249C2×C3⋊C8C2×C3×D311yescharacteristic
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32—61no—
255C22×C32—61no—
256C22×C32—61no—
257C22×C32D3×D411yescharacteristic
258C22×C32C2×#6711yescharacteristic
259C22×C32C2×C3×D411yescharacteristic
260C2×C3×D3—22no—
261C2×C3×D3—21no—
262C2×C3×D3—21no—
263C2×C3×D3—21no—
264C2×C3×D3—21no—
265C2×C3×D3—21no—
266C2×C3×D3—21no—
267C2×C3×D3—41no—
268C2×C3×D3—41no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—41no—
276C32×C4—41no—
277C32×C4—41no—
278C32×C4—42no—
279C32×C4—41no—
280C32×C4—41no—
281C32×C4—61no—
282C32×C4—81no—
283C32×C4—81no—
284C32×C4—82no—
285C32×C4—121no—
286C32×C4—121no—
287C32×C4#23511yescharacteristic
288C32×C4#23511yescharacteristic
289C32×C4#23511yescharacteristic
290C32×C4#23911yescharacteristic
291C32×C4C3×#4111yescharacteristic
292C32×C4C3×#4111yescharacteristic
293C3⋊Dic3—41no—
294C3×Dic3—22no—
295C3×Dic3—21no—
296C3×Dic3—21no—
297C3×Dic3—42no—
298C3×Dic3—41no—
299C3×Dic3—41no—
300C3×Dic3—41no—
301C3×Dic3—41no—
302C3×Dic3—81no—
303C3×Dic3—122no—
304C3×Dic3—121no—
305C3×Dic3—121no—
306C22⋊Q8—91no—
307#117—31no—
308#107—31no—
309#106—91no—
310#103—91no—
311#103—91no—
312C22⋊C8—91no—
313C33#48511yescharacteristic, complemented, Sylow
314C23×C3—31no—
315C23×C3—31no—
316C23×C3—61no—
317C22×D3C3×#6711yescharacteristic
318C3×Q8—181no—
319C3×Q8—181no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4—61no—
334C2×C3×C4—62no—
335C2×C3×C4—62no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—62no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—121no—
346C2×C3×C4—181no—
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C22×C3×D311yescharacteristic
349C2×C3×C4C22×C3×D311yescharacteristic
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—121no—
355C2×Dic3C3×#6711yescharacteristic
356C4×D3—21no—
357C4×D3—21no—
358Dic6—62no—
359Dic6—61no—
360Dic6—61no—
361Dic6—61no—
362Dic6—61no—
363Dic6—121no—
364C3×C8—62no—
365C3×C8—61no—
366C3×C8—61no—
367C3×C8—181no—
368C3⋊C8—21no—
369C3⋊C8—61no—
370C3⋊C8—61no—
371C3⋊C8—121no—
372C2×C32—21no—
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—41no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32—121no—
385C2×C32#89411yescharacteristic
386C2×C32#91211yescharacteristic
387C2×C32#91811yescharacteristic
388C2×C32#92911yescharacteristic
389C2×C32#95011yescharacteristic
390C2×C32#95111yescharacteristic
391C2×C32C3×#12311yescharacteristic
392C2×C32C3×#13511yescharacteristic
393C2×C32C3×#13711yescharacteristic
394C3×D3—42no—
395C3×D3—41no—
396C3×D3—41no—
397C2×Q8—91no—
398C22×C4—31no—
399C2×C8—31no—
400C2×C8—91no—
401C4⋊C4—31no—
402C4⋊C4—31no—
403C4⋊C4—181no—
404C4⋊C4C3×D3211yescharacteristic
405C22⋊C4—61no—
406C22⋊C4—181no—
407C42—61no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—41no—
412C22×C3—62no—
413C22×C3—61no—
414C22×C3—61no—
415C22×C3—61no—
416C22×C3—61no—
417C22×C3—61no—
418C22×C3—61no—
419C22×C3—121no—
420C22×C3—121no—
421C22×C3D3×#6711yescharacteristic
422C22×C3C3×D3×D411yescharacteristic
423C22×C3C2×C3×#6711yescharacteristic
424C2×D3—21no—
425C2×D3—21no—
426C2×D3—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—42no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—62no—
440C3×C4—61no—
441C3×C4—61no—
442C3×C4—81no—
443C3×C4—81no—
444C3×C4—82no—
445C3×C4—122no—
446C3×C4—121no—
447C3×C4—121no—
448C3×C4—121no—
449C3×C4—121no—
450C3×C4—241no—
451C3×C4—361no—
452C3×C4#379811yescharacteristic
453C3×C4#379911yescharacteristic
454C3×C4C3×#23511yescharacteristic
455C3×C4C3×#23511yescharacteristic
456C3×C4C3×#23511yescharacteristic
457C3×C4C3×#23911yescharacteristic
458Dic3—21no—
459Dic3—41no—
460Dic3—41no—
461Dic3—122no—
462Dic3—121no—
463Dic3—121no—
464C32—21no—
465C32—21no—
466C32—21no—
467C32—41no—
468C32#488811yescharacteristic, complemented
469C32#499611yescharacteristic, complemented
470C32C3×#48511yescharacteristic, complemented
471C23—31no—
472Q8—181no—
473Q8—181no—
474C2×C4—21no—
475C2×C4—31no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—181no—
481C2×C4C2×C3×D3211yescharacteristic
482C8—61no—
483C8—181no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3—41no—
496C2×C3—122no—
497C2×C3—121no—
498C2×C3—121no—
499C2×C3#6385011yescharacteristic
500C2×C3#6393311yescharacteristic
501C2×C3#6394311yescharacteristic
502C2×C3C3×#89411yescharacteristic
503C2×C3C3×#91211yescharacteristic
504C2×C3C3×#91811yescharacteristic
505C2×C3C3×#92911yescharacteristic
506C2×C3C3×#95011yescharacteristic
507C2×C3C3×#95111yescharacteristic
508D3—41no—
509C22—61no—
510C22—61no—
511C22—61no—
512C22C3×D3×#6711yescharacteristic
513C4—41no—
514C4—61no—
515C4—121no—
516C4—121no—
517C4—361no—
518C4C3×#379811yescharacteristic
519C4C3×#379911yescharacteristic
520C3—21no—
521C3—21no—
522C3—21no—
523C3—41no—
524C3#1059042911yescharacteristic, direct product factor
525C3C3×#488811yescharacteristic, complemented
526C3C3×#499611yescharacteristic, complemented
527C2—121no—
528C2C3×#6385011yescharacteristic
529C2C3×#6393311yescharacteristic
530C2C3×#6394311yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#485
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
3. C3×#63637C2semidirect
4. C32×#807C2semidirect
5. C3×#63574C2semidirect
6. C3×#63567C2semidirect
10. #10590429C3direct (N × H)
30. #63637C2×C3semidirect
41. C3×#807D3semidirect
42. C3×#807C2×C3semidirect
43. #63574C2×C3semidirect
44. #63567C2×C3semidirect
141. #807C3×D3semidirect
313. C33#485semidirect
468. C32#4888semidirect
469. C32#4996semidirect
470. C32C3×#485semidirect
524. C3#10590429direct (N × H)
525. C3C3×#4888semidirect
526. C3C3×#4996semidirect

Maximal subgroups

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

subgroupindex
C32×#8922
C3×#638572
C3×#636372
C32×#8072
C3×#635742
C3×#635672
C3×(C2×D3)⋊(C3⋊C8)2
C3×#49963
C3×#48883
#105904293