Finite Group Databasebeta

← C2×#67⋊#157 #13181009

Subgroups — C2×#67⋊#157 #13181009

Order 1728.

1432 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3⋊#157C211yesmaximal, characteristic, complemented
2C22×D3×#157C211yesmaximal, characteristic, complemented
3C2×C3⋊D3×#67C211yesmaximal, characteristic
4C2×C3×C22⋊#157C211yesmaximal, characteristic, complemented
5#67⋊#157C218yesmaximal, direct product factor
6C2×Dic3⋊#157C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊#157C211yesmaximal, characteristic, complemented
8C2×Dic3×#157C211yesmaximal, characteristic, complemented
9C2×D4×#157—31nomaximal
10C23×C3×C3⋊D3C2211yescharacteristic
11C22×D3×C3⋊D3C2211yescharacteristic
12C22×C3⋊#157—24no—
13C22×C3⋊#157C2211yescharacteristic
14C22×C3⋊#157C2211yescharacteristic
15C22×C3⋊#157C2214yescomplemented
16C22×C3×#157C2211yescharacteristic
17C22×C3×#157C2211yescharacteristic
18C2×D3×#157—24no—
19C2×D3×#157C2214yescomplemented
20C2×#6312C411yescharacteristic, complemented
21C2×C32×#67C411yescharacteristic, complemented
22C3⋊D3×#67C414yescomplemented
23C3⋊D3×#67C2214yes—
24C2×#6292C411yescharacteristic, complemented
25C2×#6290C411yescharacteristic, complemented
26C2×Dic3×C3⋊D3C2211yescharacteristic
27C3×C22⋊#157C2214yescomplemented
28Dic3⋊#157C2214yescomplemented
29(C2×D3)⋊#157C2214yescomplemented
30Dic3×#157C2214yescomplemented
31C23×#157—31no—
32C23×#157—31no—
33C2×D4×C3⋊D3—31no—
34C2×D3×#67—62no—
35C2×C22⋊#157—31no—
36C2×C22⋊#157D311yescharacteristic, complemented
37D4×#157—38no—
38C2×C4⋊#157—31no—
39C2×C4×#157—31no—
40C23×C33C2×C411yescharacteristic, complemented
41C22×C3⋊(C3⋊D3)C2×C411yescharacteristic, complemented
42C22×C3×C3⋊D3—22no—
43C22×C3×C3⋊D3—22no—
44C22×C3×C3⋊D3C2×C412yescomplemented
45C22×C3×C3⋊D3C2×C411yescharacteristic, complemented
46C22×C3×C3⋊D3C2312yes—
47C22×C3×C3⋊D3C2311yescharacteristic
48C22×C32×D3C2×C411yescharacteristic, complemented
49C2×D3×C3⋊D3—22no—
50C2×D3×C3⋊D3—22no—
51C2×D3×C3⋊D3C2×C412yescomplemented
52C2×D3×C3⋊D3C2312yes—
53C2×C3⋊#157—22no—
54C2×C3⋊#157—28no—
55C2×C3⋊#157D414yescomplemented
56C2×C3⋊#157C2312yes—
57C2×C3⋊#157C2312yes—
58C2×C3×#157—22no—
59C2×C3×#157—22no—
60C2×C3×#157C2312yes—
61C2×C3×#157C2312yes—
62D3×#157—28no—
63#6312C2×C414yescomplemented
64C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic, complemented
65C32×#67C2×C414yescomplemented
66C2×C32×Dic3C2×C411yescharacteristic, complemented
67#6292C2×C414yescomplemented
68#6290C2×C414yescomplemented
69Dic3×C3⋊D3C2×C412yescomplemented
70Dic3×C3⋊D3C2312yes—
71C2×#67⋊C4—91nomaximal
72C23×C3⋊D3—31no—
73C23×C3⋊D3C2×D311yescharacteristic
74C23×C3×D3—62no—
75C22×D32—62no—
76C22×#157—31no—
77C22×#157—31no—
78C22×#157—34no—
79C22×#157—34no—
80C22×#157—64no—
81C22×#157—64no—
82C22×#157C2×D311yescharacteristic
83C22×#157C2×D311yescharacteristic
84C2×C32×D4—31no—
85C2×#638—31no—
86C2×#638—31no—
87C2×#638—62no—
88C2×#638—122no—
89D4×C3⋊D3—34no—
90D4×C3⋊D3—34no—
91C2×C4⋊(C3⋊D3)—31no—
92C2×C4×C3⋊D3—31no—
93C2×C3×#67—22no—
94D3×#67—68no—
95D3×#67—68no—
96C2×#626—62no—
97C2×#625—62no—
98C2×D3×Dic3—62no—
99C22⋊#157—34no—
100C22⋊#157C2×D314yescomplemented
101C4⋊#157—34no—
102C4×#157—34no—
103C22×C33—22no—
104C22×C33C22×C411yescharacteristic
105C22×C33C22×C412yescomplemented
106C2×C3⋊(C3⋊D3)—22no—
107C2×C3⋊(C3⋊D3)C22×C412yescomplemented
108C2×C3×C3⋊D3—22no—
109C2×C3×C3⋊D3—24no—
110C2×C3×C3⋊D3—22no—
111C2×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3C22×C411yescharacteristic
113C2×C3×C3⋊D3C22×C412yescomplemented
114C2×C3×C3⋊D3C2×D412yes—
115C2×C3×C3⋊D3#4112yes—
116C2×C3×C3⋊D3C2411yescharacteristic
117C2×C32×D3—22no—
118C2×C32×D3C22×C412yescomplemented
119D3×C3⋊D3—24no—
120D3×C3⋊D3—22no—
121D3×C3⋊D3—22no—
122C3⋊#157—22no—
123C3⋊#157C2×D414yescomplemented
124C3×#157—22no—
125C3×#157—22no—
126C3⋊(C3⋊Dic3)C22×C412yescomplemented
127C32×Dic3C22×C412yescomplemented
128C22×#67—91no—
129C23×Dic3—91no—
130C2×D3×D4—182no—
131C22×C4×D3—91no—
132C2×C3×C22⋊C4—91no—
133C2×(C2×D3)⋊C4—91no—
134C2×Dic3⋊C4—91no—
135C2×C4×Dic3—91no—
136#67⋊C4—98no—
137C23×C32—22no—
138C23×C32—31no—
139C23×C32—42no—
140C23×C32C4×D311yescharacteristic, complemented
141C22×C3⋊D3—22no—
142C22×C3⋊D3—22no—
143C22×C3⋊D3—31no—
144C22×C3⋊D3—32no—
145C22×C3⋊D3—32no—
146C22×C3⋊D3—62no—
147C22×C3⋊D3—62no—
148C22×C3⋊D3—62no—
149C22×C3⋊D3—122no—
150C22×C3⋊D3C4×D312yescomplemented
151C22×C3⋊D3C4×D311yescharacteristic, complemented
152C22×C3⋊D3C22×D312yes—
153C22×C3⋊D3C22×D311yescharacteristic
154C22×C3×D3—22no—
155C22×C3×D3—62no—
156C22×C3×D3—62no—
157C22×C3×D3—64no—
158C22×C3×D3—64no—
159C22×C3×D3—124no—
160C22×C3×D3—124no—
161C2×D32—64no—
162C2×D32—64no—
163C2×D32—124no—
164C2×D32—124no—
165C2×#157—22no—
166C2×#157—22no—
167C2×#157—34no—
168C2×#157—32no—
169C2×#157—32no—
170C2×#157—68no—
171C2×#157—68no—
172C2×#157—62no—
173C2×#157C22×D312yes—
174C2×#157C22×D312yes—
175C32×D4—34no—
176C2×C32×C4—31no—
177#638—34no—
178#638—34no—
179#638—68no—
180#638—128no—
181C2×C3⋊Dic3—31no—
182C2×C3⋊Dic3—62no—
183C2×C3⋊Dic3—122no—
184C4⋊(C3⋊D3)—34no—
185C4×C3⋊D3—32no—
186C4×C3⋊D3—32no—
187C3×#67—28no—
188C2×C3×Dic3—22no—
189#626—68no—
190#625—68no—
191D3×Dic3—64no—
192D3×Dic3—64no—
193C2×C4×D4—271noSylow
194C2×C33—22no—
195C2×C33C4×D412yescomplemented
196C2×C33C23×C411yescharacteristic
197C3⋊(C3⋊D3)—22no—
198C3×C3⋊D3—22no—
199C3×C3⋊D3C4×D412yescomplemented
200C3×C3⋊D3C22×D411yescharacteristic
201C3×C3⋊D3C2×#4111yescharacteristic
202C32×D3—22no—
203C24×C3—91no—
204C23×D3—62no—
205C23×D3—91no—
206C23×D3—182no—
207C2×C3×D4—62no—
208C22×C3×C4—91no—
209C22×C3×C4—91no—
210C2×#67—91no—
211C2×#67—91no—
212C2×#67—94no—
213C2×#67—94no—
214C2×#67—91no—
215C2×#67—182no—
216C2×#67—182no—
217C2×#67—362no—
218C2×#67#15711yescharacteristic, complemented
219C22×Dic3—91no—
220C22×Dic3—91no—
221C22×Dic3—91no—
222C22×Dic3—94no—
223C22×Dic3—184no—
224D3×D4—188no—
225D3×D4—188no—
226C2×D12—182no—
227C2×C4×D3—94no—
228C2×C4×D3—182no—
229C2×C4×D3—184no—
230C3×C22⋊C4—94no—
231(C2×D3)⋊C4—94no—
232Dic3⋊C4—94no—
233C4×Dic3—94no—
234C22×C32—22no—
235C22×C32—24no—
236C22×C32—22no—
237C22×C32—32no—
238C22×C32—42no—
239C22×C32—44no—
240C22×C32—44no—
241C22×C32—62no—
242C22×C32—84no—
243C22×C32C2×C4×D311yescharacteristic
244C22×C32C2×C4×D312yescomplemented
245C2×C3⋊D3—22no—
246C2×C3⋊D3—24no—
247C2×C3⋊D3—22no—
248C2×C3⋊D3—22no—
249C2×C3⋊D3—32no—
250C2×C3⋊D3—64no—
251C2×C3⋊D3—62no—
252C2×C3⋊D3—64no—
253C2×C3⋊D3—62no—
254C2×C3⋊D3—62no—
255C2×C3⋊D3—124no—
256C2×C3⋊D3—124no—
257C2×C3⋊D3—244no—
258C2×C3⋊D3C2×C4×D311yescharacteristic
259C2×C3⋊D3C2×C4×D312yescomplemented
260C2×C3⋊D3D3×D412yes—
261C2×C3⋊D3#23712yes—
262C2×C3⋊D3C23×D311yescharacteristic
263C2×C3×D3—24no—
264C2×C3×D3—44no—
265C2×C3×D3—62no—
266C2×C3×D3—64no—
267C2×C3×D3—64no—
268C2×C3×D3—62no—
269C2×C3×D3—64no—
270C2×C3×D3—124no—
271C2×C3×D3—128no—
272C2×C3×D3—124no—
273C2×C3×D3—124no—
274D32—124no—
275D32—128no—
276D32—124no—
277#157—22no—
278#157—22no—
279#157—34no—
280#157—62no—
281C32×C4—32no—
282C3⋊Dic3—32no—
283C3⋊Dic3—64no—
284C3⋊Dic3—124no—
285C3×Dic3—24no—
286C22×D4—271no—
287C23×C4—271no—
288C23×C4—271no—
289C4×D4—278no—
290C2×C4⋊C4—271no—
291C2×C22⋊C4—91no—
292C2×C22⋊C4—271no—
293C2×C42—271no—
294C33C2×C4×D411yescharacteristic, complemented, Sylow
295C23×C3—22no—
296C23×C3—42no—
297C23×C3—62no—
298C23×C3—91no—
299C23×C3—91no—
300C23×C3—92no—
301C23×C3—92no—
302C23×C3—182no—
303C23×C3—182no—
304C23×C3C2×#15711yescharacteristic, complemented
305C22×D3—62no—
306C22×D3—62no—
307C22×D3—64no—
308C22×D3—64no—
309C22×D3—91no—
310C22×D3—92no—
311C22×D3—92no—
312C22×D3—124no—
313C22×D3—124no—
314C22×D3—184no—
315C22×D3—184no—
316C22×D3—182no—
317C22×D3—182no—
318C22×D3—182no—
319C22×D3—362no—
320C22×D3—364no—
321C22×D3—364no—
322C22×D3C2×#15711yescharacteristic, complemented
323C3×D4—68no—
324C2×C3×C4—62no—
325C2×C3×C4—92no—
326C2×C3×C4—92no—
327C2×C3×C4—182no—
328C2×C3×C4—182no—
329#67—94no—
330#67—94no—
331#67—94no—
332#67—188no—
333#67—188no—
334#67—368no—
335#67C2×#15714yescomplemented
336C2×Dic3—94no—
337C2×Dic3—92no—
338C2×Dic3—91no—
339C2×Dic3—92no—
340C2×Dic3—92no—
341C2×Dic3—92no—
342C2×Dic3—182no—
343C2×Dic3—188no—
344C2×Dic3—182no—
345C2×Dic3—362no—
346C2×Dic3C2×#15711yescharacteristic, complemented
347D12—188no—
348C4×D3—184no—
349C4×D3—184no—
350C4×D3—188no—
351C2×C32—24no—
352C2×C32—22no—
353C2×C32—22no—
354C2×C32—44no—
355C2×C32—42no—
356C2×C32—44no—
357C2×C32—62no—
358C2×C32—84no—
359C2×C32#67⋊C412yescomplemented
360C2×C32C22×C4×D311yescharacteristic
361C3⋊D3—22no—
362C3⋊D3—62no—
363C3⋊D3—124no—
364C3⋊D3—244no—
365C3⋊D3#67⋊C412yescomplemented
366C3⋊D3C2×D3×D411yescharacteristic
367C3⋊D3C2×#23711yescharacteristic
368C3×D3—44no—
369C3×D3—64no—
370C3×D3—62no—
371C3×D3—62no—
372C3×D3—124no—
373C24—91no—
374C24—271no—
375C2×D4—31no—
376C2×D4—271no—
377C2×D4—274no—
378C2×D4—274no—
379C2×D4—271no—
380C2×D4—271no—
381C22×C4—91no—
382C22×C4—91no—
383C22×C4—271no—
384C22×C4—274no—
385C22×C4—271no—
386C22×C4—271no—
387C22×C4—274no—
388C22×C4—544no—
389C22×C4—544no—
390C4⋊C4—274no—
391C22⋊C4—94no—
392C22⋊C4—274no—
393C42—274no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—24no—
397C22×C3—44no—
398C22×C3—42no—
399C22×C3—44no—
400C22×C3—64no—
401C22×C3—84no—
402C22×C3—92no—
403C22×C3—92no—
404C22×C3—91no—
405C22×C3—91no—
406C22×C3—92no—
407C22×C3—124no—
408C22×C3—182no—
409C22×C3—182no—
410C22×C3—184no—
411C22×C3—182no—
412C22×C3C22×#15711yescharacteristic
413C22×C3C22×#15712yescomplemented
414C2×D3—22no—
415C2×D3—62no—
416C2×D3—62no—
417C2×D3—64no—
418C2×D3—64no—
419C2×D3—64no—
420C2×D3—92no—
421C2×D3—124no—
422C2×D3—128no—
423C2×D3—124no—
424C2×D3—124no—
425C2×D3—184no—
426C2×D3—182no—
427C2×D3—182no—
428C2×D3—184no—
429C2×D3—182no—
430C2×D3—364no—
431C2×D3—364no—
432C2×D3—368no—
433C2×D3—364no—
434C2×D3—364no—
435C2×D3—724no—
436C2×D3C22×#15712yescomplemented
437C3×C4—64no—
438C3×C4—182no—
439C3×C4—182no—
440Dic3—94no—
441Dic3—92no—
442Dic3—184no—
443Dic3—182no—
444Dic3—364no—
445Dic3C22×#15712yescomplemented
446C32—22no—
447C32—42no—
448C32C2×#67⋊C411yescharacteristic, complemented
449C23—31no—
450C23—91no—
451C23—91no—
452C23—92no—
453C23—92no—
454C23—182no—
455C23—182no—
456C23—272no—
457C23—272no—
458C23—271no—
459C23—542no—
460C23—542no—
461C23D3×#15711yescharacteristic, complemented
462D4—34no—
463D4—274no—
464D4—274no—
465D4—274no—
466C2×C4—31no—
467C2×C4—92no—
468C2×C4—92no—
469C2×C4—182no—
470C2×C4—182no—
471C2×C4—272no—
472C2×C4—272no—
473C2×C4—272no—
474C2×C4—274no—
475C2×C4—271no—
476C2×C4—272no—
477C2×C4—548no—
478C2×C4—548no—
479C2×C4—542no—
480C2×C3—22no—
481C2×C3—22no—
482C2×C3—24no—
483C2×C3—44no—
484C2×C3—42no—
485C2×C3—44no—
486C2×C3—84no—
487C2×C3—91no—
488C2×C3—91no—
489C2×C3—92no—
490C2×C3—124no—
491C2×C3—182no—
492C2×C3D4×#15712yescomplemented
493C2×C3C23×#15711yescharacteristic
494D3—22no—
495D3—64no—
496D3—62no—
497D3—62no—
498D3—124no—
499D3—182no—
500D3—364no—
501D3—724no—
502C22—22no—
503C22—32no—
504C22—62no—
505C22—91no—
506C22—91no—
507C22—92no—
508C22—92no—
509C22—92no—
510C22—182no—
511C22—182no—
512C22—184no—
513C22—182no—
514C22—272no—
515C22—542no—
516C22—542no—
517C22—544no—
518C22—542no—
519C22C2×D3×#15711yescharacteristic
520C22C2×D3×#15712yescomplemented
521C4—32no—
522C4—182no—
523C4—182no—
524C4—274no—
525C4—272no—
526C4—542no—
527C3—22no—
528C3—42no—
529C3C2×D4×#15711yescharacteristic, complemented
530C2—22no—
531C2—62no—
532C2—91no—
533C2—91no—
534C2—92no—
535C2—182no—
536C2—542no—
537C2#67⋊#15712yesdirect product factor
538C2C22×D3×#15711yescharacteristic
subgroupG/H
centerC22C2×D3×#157
commutatorC2×C33C23×C4
FrattiniC2C22×D3×#157
FittingC23×C33C2×C4
socleC22×C33C22×C4
prime radicalC2×C3⋊D3×#67C2
cosocleC2×C3×C3⋊D3C24
hypercenterC23D3×#157
hypocenterC33C2×C4×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊#157C2semidirect
2. C22×D3×#157C2semidirect
4. C2×C3×C22⋊#157C2semidirect
5. #67⋊#157C2direct (N × H)
6. C2×Dic3⋊#157C2semidirect
7. C2×(C2×D3)⋊#157C2semidirect
8. C2×Dic3×#157C2semidirect
15. C22×C3⋊#157C22semidirect
19. C2×D3×#157C22semidirect
20. C2×#6312C4semidirect
21. C2×C32×#67C4semidirect
22. C3⋊D3×#67C4semidirect
24. C2×#6292C4semidirect
25. C2×#6290C4semidirect
27. C3×C22⋊#157C22semidirect
28. Dic3⋊#157C22semidirect
29. (C2×D3)⋊#157C22semidirect
30. Dic3×#157C22semidirect
36. C2×C22⋊#157D3semidirect
40. C23×C33C2×C4semidirect
41. C22×C3⋊(C3⋊D3)C2×C4semidirect
44. C22×C3×C3⋊D3C2×C4semidirect
45. C22×C3×C3⋊D3C2×C4semidirect
48. C22×C32×D3C2×C4semidirect
51. C2×D3×C3⋊D3C2×C4semidirect
55. C2×C3⋊#157D4semidirect
63. #6312C2×C4semidirect
64. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
65. C32×#67C2×C4semidirect
66. C2×C32×Dic3C2×C4semidirect
67. #6292C2×C4semidirect
68. #6290C2×C4semidirect
69. Dic3×C3⋊D3C2×C4semidirect
100. C22⋊#157C2×D3semidirect
105. C22×C33C22×C4semidirect
107. C2×C3⋊(C3⋊D3)C22×C4semidirect
113. C2×C3×C3⋊D3C22×C4semidirect
118. C2×C32×D3C22×C4semidirect
123. C3⋊#157C2×D4semidirect
126. C3⋊(C3⋊Dic3)C22×C4semidirect
127. C32×Dic3C22×C4semidirect
140. C23×C32C4×D3semidirect
150. C22×C3⋊D3C4×D3semidirect
151. C22×C3⋊D3C4×D3semidirect
195. C2×C33C4×D4semidirect
199. C3×C3⋊D3C4×D4semidirect
218. C2×#67#157semidirect
244. C22×C32C2×C4×D3semidirect
259. C2×C3⋊D3C2×C4×D3semidirect
294. C33C2×C4×D4semidirect
304. C23×C3C2×#157semidirect
322. C22×D3C2×#157semidirect
335. #67C2×#157semidirect
346. C2×Dic3C2×#157semidirect
359. C2×C32#67⋊C4semidirect
365. C3⋊D3#67⋊C4semidirect
413. C22×C3C22×#157semidirect
436. C2×D3C22×#157semidirect
445. Dic3C22×#157semidirect
448. C32C2×#67⋊C4semidirect
461. C23D3×#157semidirect
492. C2×C3D4×#157semidirect
520. C22C2×D3×#157semidirect
529. C3C2×D4×#157semidirect
537. C2#67⋊#157direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3⋊#1572
C22×D3×#1572
C2×C3⋊D3×#672
C2×C3×C22⋊#1572
#67⋊#1572
C2×Dic3⋊#1572
C2×(C2×D3)⋊#1572
C2×Dic3×#1572
C2×D4×#1573
C2×#67⋊C49