Finite Group Databasebeta

← #13181010

Subgroups — #13181010

Order 1728.

1256 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×#67C211yesmaximal, characteristic
2C2×#88994C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊#157C211yesmaximal, characteristic, complemented
4#67⋊#157C214yesmaximal, complemented
5#67×#157C214yesmaximal, complemented
6D3×C22⋊#157C212yesmaximal, complemented
7#11724261C212yesmaximal, complemented
8#10596642—31nomaximal
9C23×C3×C3⋊D3C2211yescharacteristic
10C22×D3×C3⋊D3C2211yescharacteristic
11C22×C3⋊#157C2212yescomplemented
12C22×C3×#157C2212yescomplemented
13C2×D3×#157C2214yescomplemented
14C2×#6312C411yescharacteristic, complemented
15C2×C32×#67C411yescharacteristic, complemented
16C3⋊D3×#67C412yescomplemented
17C3⋊D3×#67C412yescomplemented
18C3⋊D3×#67C2212yes—
19C3⋊D3×#67C2212yes—
20C2×#6292C411yescharacteristic, complemented
21C2×#6290C411yescharacteristic, complemented
22C2×Dic3×C3⋊D3C2211yescharacteristic
23#88994C2212yes—
24#88994C2212yescomplemented
25C3×C22⋊#157C2212yes—
26C3×C22⋊#157C2212yescomplemented
27Dic3⋊#157C2214yescomplemented
28(C2×D3)⋊#157C2214yescomplemented
29Dic3×#157C2214yescomplemented
30C2×D4×C3⋊D3—31no—
31C2×D3×#67—62no—
32C2×C22⋊#157—32no—
33C2×C22⋊#157—31no—
34C2×C22⋊#157D311yescharacteristic, complemented
35D4×#157—34no—
36D4×#157—34no—
37#64286—32no—
38C23×C33C2×C411yescharacteristic, complemented
39C22×C3⋊(C3⋊D3)C2×C411yescharacteristic, complemented
40C22×C3×C3⋊D3—42no—
41C22×C3×C3⋊D3C2×C411yescharacteristic, complemented
42C22×C3×C3⋊D3C2×C411yescharacteristic, complemented
43C22×C3×C3⋊D3C2×C411yescharacteristic, complemented
44C22×C3×C3⋊D3C2311yescharacteristic
45C22×C3×C3⋊D3C2311yescharacteristic
46C22×C3×C3⋊D3C2311yescharacteristic
47C22×C32×D3C2×C411yescharacteristic, complemented
48C2×D3×C3⋊D3—42no—
49C2×D3×C3⋊D3C2×C412yescomplemented
50C2×D3×C3⋊D3C2312yes—
51C2×C3⋊#157—42no—
52C2×C3⋊#157C2314yes—
53C2×C3×#157—42no—
54C2×C3×#157C2314yes—
55D3×#157—44no—
56#6312C2×C412yescomplemented
57#6312C2×C412yescomplemented
58C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic, complemented
59C32×#67C2×C412yescomplemented
60C32×#67C2×C412yescomplemented
61C2×C32×Dic3C2×C411yescharacteristic, complemented
62#6292C2×C412yescomplemented
63#6292C2×C412yescomplemented
64#6290C2×C412yescomplemented
65#6290C2×C412yescomplemented
66Dic3×C3⋊D3C2×C412yescomplemented
67Dic3×C3⋊D3C2312yes—
68#5486—91nomaximal
69C23×C3⋊D3—31no—
70C23×C3⋊D3C2×D311yescharacteristic
71C23×C3×D3—62no—
72C22×D32—62no—
73C22×#157—32no—
74C22×#157—34no—
75C22×#157C2×D312yescomplemented
76C2×C32×D4—31no—
77C2×#638—31no—
78C2×#638—31no—
79C2×#638—62no—
80C2×#638—122no—
81D4×C3⋊D3—32no—
82D4×C3⋊D3—32no—
83D4×C3⋊D3—32no—
84D4×C3⋊D3—32no—
85C2×C4⋊(C3⋊D3)—31no—
86C2×C4×C3⋊D3—31no—
87C2×C3×#67—22no—
88D3×#67—64no—
89D3×#67—64no—
90D3×#67—64no—
91D3×#67—64no—
92C2×#626—62no—
93C2×#625—62no—
94C2×D3×Dic3—62no—
95C22⋊#157—32no—
96C22⋊#157—32no—
97C22⋊#157—34no—
98C22⋊#157C2×D312yes—
99C22⋊#157C2×D312yescomplemented
100C4⋊#157—34no—
101C4×#157—34no—
102C22×C33—41no—
103C22×C33C22×C411yescharacteristic
104C22×C33C22×C411yescharacteristic
105C22×C33C22×C411yescharacteristic
106C2×C3⋊(C3⋊D3)—41no—
107C2×C3⋊(C3⋊D3)C22×C412yes—
108C2×C3×C3⋊D3—22no—
109C2×C3×C3⋊D3—22no—
110C2×C3×C3⋊D3—22no—
111C2×C3×C3⋊D3—41no—
112C2×C3×C3⋊D3—41no—
113C2×C3×C3⋊D3—41no—
114C2×C3×C3⋊D3C22×C411yescharacteristic
115C2×C3×C3⋊D3C22×C411yescharacteristic
116C2×C3×C3⋊D3C22×C411yescharacteristic
117C2×C3×C3⋊D3C2411yescharacteristic
118C2×C32×D3—41no—
119C2×C32×D3C22×C412yes—
120D3×C3⋊D3—24no—
121D3×C3⋊D3—42no—
122C3⋊#157—24no—
123C3×#157—24no—
124C3⋊(C3⋊Dic3)C22×C412yes—
125C32×Dic3C22×C412yes—
126C22×#67—91no—
127C2×D3×D4—182no—
128C2×C3×C22⋊C4—91no—
129C2×#217—91no—
130C4×#67—94no—
131#67⋊C4—94no—
132D3×C22⋊C4—92no—
133#877—92no—
134C23×C32—22no—
135C23×C32—31no—
136C23×C32—42no—
137C23×C32C4×D311yescharacteristic, complemented
138C22×C3⋊D3—31no—
139C22×C3⋊D3—32no—
140C22×C3⋊D3—32no—
141C22×C3⋊D3—42no—
142C22×C3⋊D3—62no—
143C22×C3⋊D3—122no—
144C22×C3⋊D3—122no—
145C22×C3⋊D3C4×D311yescharacteristic, complemented
146C22×C3⋊D3C4×D311yescharacteristic, complemented
147C22×C3⋊D3C4×D311yescharacteristic, complemented
148C22×C3⋊D3C22×D311yescharacteristic
149C22×C3⋊D3C22×D311yescharacteristic
150C22×C3⋊D3C22×D311yescharacteristic
151C22×C3×D3—22no—
152C22×C3×D3—62no—
153C22×C3×D3—62no—
154C22×C3×D3—62no—
155C22×C3×D3—62no—
156C22×C3×D3—62no—
157C22×C3×D3—62no—
158C22×C3×D3—128no—
159C2×D32—64no—
160C2×D32—64no—
161C2×D32—128no—
162C2×#157—34no—
163C2×#157—42no—
164C2×#157—124no—
165C2×#157—122no—
166C2×#157C22×D314yes—
167C32×D4—32no—
168C32×D4—32no—
169C2×C32×C4—31no—
170#638—32no—
171#638—32no—
172#638—32no—
173#638—32no—
174#638—64no—
175#638—64no—
176#638—124no—
177#638—124no—
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—62no—
180C2×C3⋊Dic3—122no—
181C4⋊(C3⋊D3)—32no—
182C4⋊(C3⋊D3)—32no—
183C4×C3⋊D3—32no—
184C4×C3⋊D3—32no—
185C3×#67—24no—
186C3×#67—24no—
187C2×C3×Dic3—22no—
188#626—64no—
189#626—64no—
190#625—64no—
191#625—64no—
192D3×Dic3—64no—
193D3×Dic3—64no—
194#518—271noSylow
195C2×C33—21no—
196C2×C33—21no—
197C2×C33—21no—
198C2×C33C23×C411yescharacteristic
199C3⋊(C3⋊D3)—22no—
200C3×C3⋊D3—21no—
201C3×C3⋊D3—21no—
202C3×C3⋊D3—21no—
203C3×C3⋊D3H(5,𝔽2)12yes—
204C32×D3—22no—
205C24×C3—91no—
206C23×D3—62no—
207C23×D3—91no—
208C23×D3—182no—
209C2×C3×D4—62no—
210C22×C3×C4—92no—
211C2×#67—91no—
212C2×#67—91no—
213C2×#67—92no—
214C2×#67—92no—
215C2×#67—92no—
216C2×#67—92no—
217C2×#67—91no—
218C2×#67—182no—
219C2×#67—182no—
220C2×#67—362no—
221C2×#67#15711yescharacteristic, complemented
222C22×Dic3—91no—
223C22×Dic3—92no—
224D3×D4—184no—
225D3×D4—184no—
226D3×D4—184no—
227D3×D4—184no—
228C2×D12—182no—
229C2×C4×D3—94no—
230C2×C4×D3—182no—
231C3×C22⋊C4—92no—
232C3×C22⋊C4—92no—
233#217—92no—
234#217—92no—
235(C2×D3)⋊C4—94no—
236Dic3⋊C4—94no—
237C4×Dic3—94no—
238C22×C32—22no—
239C22×C32—22no—
240C22×C32—22no—
241C22×C32—32no—
242C22×C32—42no—
243C22×C32—42no—
244C22×C32—42no—
245C22×C32—41no—
246C22×C32—44no—
247C22×C32—84no—
248C22×C32—121no—
249C22×C32C2×C4×D311yescharacteristic
250C22×C32C2×C4×D311yescharacteristic
251C22×C32C2×C4×D311yescharacteristic
252C2×C3⋊D3—22no—
253C2×C3⋊D3—22no—
254C2×C3⋊D3—22no—
255C2×C3⋊D3—32no—
256C2×C3⋊D3—41no—
257C2×C3⋊D3—41no—
258C2×C3⋊D3—41no—
259C2×C3⋊D3—64no—
260C2×C3⋊D3—64no—
261C2×C3⋊D3—124no—
262C2×C3⋊D3—122no—
263C2×C3⋊D3—121no—
264C2×C3⋊D3—124no—
265C2×C3⋊D3—244no—
266C2×C3⋊D3C2×C4×D311yescharacteristic
267C2×C3⋊D3C2×C4×D311yescharacteristic
268C2×C3⋊D3C2×C4×D311yescharacteristic
269C2×C3⋊D3C23×D311yescharacteristic
270C2×C3×D3—24no—
271C2×C3×D3—44no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—68no—
277C2×C3×D3—124no—
278C2×C3×D3—124no—
279C2×C3×D3—124no—
280C2×C3×D3—124no—
281C2×C3×D3—124no—
282D32—128no—
283D32—128no—
284#157—24no—
285#157—64no—
286C32×C4—32no—
287C3⋊Dic3—32no—
288C3⋊Dic3—64no—
289C3⋊Dic3—124no—
290C3×Dic3—24no—
291C22×D4—271no—
292C4×D4—274no—
293C4×D4—274no—
294#117—272no—
295C2×C22⋊C4—91no—
296C2×C22⋊C4—272no—
297C2×C22⋊C4—271no—
298C33#51811yescharacteristic, complemented, Sylow
299C23×C3—22no—
300C23×C3—42no—
301C23×C3—62no—
302C23×C3—91no—
303C23×C3—91no—
304C23×C3—91no—
305C23×C3—91no—
306C23×C3—91no—
307C23×C3—91no—
308C23×C3—362no—
309C23×C3C2×#15711yescharacteristic, complemented
310C22×D3—62no—
311C22×D3—62no—
312C22×D3—62no—
313C22×D3—62no—
314C22×D3—62no—
315C22×D3—62no—
316C22×D3—91no—
317C22×D3—92no—
318C22×D3—92no—
319C22×D3—128no—
320C22×D3—184no—
321C22×D3—184no—
322C22×D3—182no—
323C22×D3—362no—
324C22×D3—362no—
325C22×D3—368no—
326C22×D3C2×#15711yescharacteristic, complemented
327C3×D4—64no—
328C3×D4—64no—
329C2×C3×C4—62no—
330C2×C3×C4—94no—
331C2×C3×C4—362no—
332#67—92no—
333#67—92no—
334#67—92no—
335#67—92no—
336#67—92no—
337#67—92no—
338#67—184no—
339#67—184no—
340#67—184no—
341#67—184no—
342#67—364no—
343#67—364no—
344#67C2×#15712yescomplemented
345#67C2×#15712yescomplemented
346C2×Dic3—91no—
347C2×Dic3—92no—
348C2×Dic3—92no—
349C2×Dic3—94no—
350C2×Dic3—182no—
351C2×Dic3—362no—
352C2×Dic3—362no—
353C2×Dic3C2×#15711yescharacteristic, complemented
354D12—184no—
355D12—184no—
356C4×D3—184no—
357C4×D3—184no—
358C4×D3—364no—
359C2×C32—21no—
360C2×C32—24no—
361C2×C32—22no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—44no—
365C2×C32—42no—
366C2×C32—42no—
367C2×C32—42no—
368C2×C32—62no—
369C2×C32—82no—
370C2×C32—82no—
371C2×C32C22×C4×D311yescharacteristic
372C3⋊D3—21no—
373C3⋊D3—21no—
374C3⋊D3—21no—
375C3⋊D3—62no—
376C3⋊D3—124no—
377C3⋊D3—244no—
378C3⋊D3#100412yes—
379C3×D3—44no—
380C3×D3—64no—
381C3×D3—64no—
382C3×D3—122no—
383C3×D3—122no—
384C24—91no—
385C24—271no—
386C2×D4—31no—
387C2×D4—271no—
388C2×D4—272no—
389C2×D4—272no—
390C2×D4—272no—
391C2×D4—272no—
392C2×D4—271no—
393C2×D4—271no—
394C22×C4—92no—
395C22×C4—271no—
396C22×C4—274no—
397C22×C4—272no—
398C4⋊C4—274no—
399C22⋊C4—92no—
400C22⋊C4—92no—
401C22⋊C4—274no—
402C22⋊C4—272no—
403C22⋊C4—272no—
404C42—274no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—41no—
409C22×C3—44no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—64no—
414C22×C3—84no—
415C22×C3—91no—
416C22×C3—91no—
417C22×C3—91no—
418C22×C3—91no—
419C22×C3—124no—
420C22×C3—182no—
421C22×C3—182no—
422C22×C3—182no—
423C22×C3—361no—
424C22×C3—361no—
425C22×C3—361no—
426C22×C3C22×#15711yescharacteristic
427C22×C3C22×#15711yescharacteristic
428C22×C3C22×#15711yescharacteristic
429C2×D3—41no—
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—68no—
435C2×D3—92no—
436C2×D3—124no—
437C2×D3—124no—
438C2×D3—124no—
439C2×D3—124no—
440C2×D3—124no—
441C2×D3—184no—
442C2×D3—184no—
443C2×D3—364no—
444C2×D3—368no—
445C2×D3—362no—
446C2×D3—368no—
447C2×D3—361no—
448C2×D3—364no—
449C2×D3—724no—
450C2×D3C22×#15712yes—
451C3×C4—64no—
452C3×C4—184no—
453Dic3—92no—
454Dic3—184no—
455Dic3—184no—
456Dic3—364no—
457Dic3C22×#15712yes—
458C32—22no—
459C32—42no—
460C32#548611yescharacteristic, complemented
461C23—31no—
462C23—91no—
463C23—91no—
464C23—91no—
465C23—91no—
466C23—91no—
467C23—91no—
468C23—272no—
469C23—272no—
470C23—271no—
471C23—362no—
472C23—1082no—
473C23D3×#15711yescharacteristic, complemented
474D4—32no—
475D4—32no—
476D4—272no—
477D4—272no—
478D4—272no—
479D4—272no—
480D4—272no—
481D4—272no—
482C2×C4—31no—
483C2×C4—94no—
484C2×C4—272no—
485C2×C4—272no—
486C2×C4—271no—
487C2×C4—274no—
488C2×C4—362no—
489C2×C4—1084no—
490C2×C4—1082no—
491C2×C3—24no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—22no—
496C2×C3—42no—
497C2×C3—42no—
498C2×C3—44no—
499C2×C3—42no—
500C2×C3—82no—
501C2×C3—82no—
502C2×C3—92no—
503C2×C3—124no—
504C2×C3—181no—
505C2×C3—181no—
506C2×C3—181no—
507C2×C3C23×#15711yescharacteristic
508D3—22no—
509D3—64no—
510D3—64no—
511D3—122no—
512D3—122no—
513D3—182no—
514D3—364no—
515D3—724no—
516C22—32no—
517C22—41no—
518C22—91no—
519C22—91no—
520C22—91no—
521C22—91no—
522C22—121no—
523C22—182no—
524C22—182no—
525C22—182no—
526C22—272no—
527C22—361no—
528C22—361no—
529C22—361no—
530C22—544no—
531C22—1081no—
532C22—1082no—
533C22C2×D3×#15711yescharacteristic
534C22C2×D3×#15711yescharacteristic
535C22C2×D3×#15711yescharacteristic
536C4—32no—
537C4—184no—
538C4—272no—
539C4—544no—
540C3—22no—
541C3—42no—
542C3#1059664211yescharacteristic, complemented
543C2—21no—
544C2—21no—
545C2—21no—
546C2—62no—
547C2—92no—
548C2—181no—
549C2—181no—
550C2—181no—
551C2—542no—
552C2C22×D3×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×#157
commutator, socleC2×C33C23×C4
FittingC23×C33C2×C4
prime radicalC2×C3⋊D3×#67C2
cosocleC2×C3×C3⋊D3C24
hypercenterC23D3×#157
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88994C2semidirect
3. C2×C3×C22⋊#157C2semidirect
4. #67⋊#157C2semidirect
5. #67×#157C2semidirect
6. D3×C22⋊#157C2semidirect
7. #11724261C2semidirect
11. C22×C3⋊#157C22semidirect
12. C22×C3×#157C22semidirect
13. C2×D3×#157C22semidirect
14. C2×#6312C4semidirect
15. C2×C32×#67C4semidirect
16. C3⋊D3×#67C4semidirect
17. C3⋊D3×#67C4semidirect
20. C2×#6292C4semidirect
21. C2×#6290C4semidirect
24. #88994C22semidirect
26. C3×C22⋊#157C22semidirect
27. Dic3⋊#157C22semidirect
28. (C2×D3)⋊#157C22semidirect
29. Dic3×#157C22semidirect
34. C2×C22⋊#157D3semidirect
38. C23×C33C2×C4semidirect
39. C22×C3⋊(C3⋊D3)C2×C4semidirect
41. C22×C3×C3⋊D3C2×C4semidirect
42. C22×C3×C3⋊D3C2×C4semidirect
43. C22×C3×C3⋊D3C2×C4semidirect
47. C22×C32×D3C2×C4semidirect
49. C2×D3×C3⋊D3C2×C4semidirect
56. #6312C2×C4semidirect
57. #6312C2×C4semidirect
58. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
59. C32×#67C2×C4semidirect
60. C32×#67C2×C4semidirect
61. C2×C32×Dic3C2×C4semidirect
62. #6292C2×C4semidirect
63. #6292C2×C4semidirect
64. #6290C2×C4semidirect
65. #6290C2×C4semidirect
66. Dic3×C3⋊D3C2×C4semidirect
75. C22×#157C2×D3semidirect
99. C22⋊#157C2×D3semidirect
137. C23×C32C4×D3semidirect
145. C22×C3⋊D3C4×D3semidirect
146. C22×C3⋊D3C4×D3semidirect
147. C22×C3⋊D3C4×D3semidirect
221. C2×#67#157semidirect
298. C33#518semidirect
309. C23×C3C2×#157semidirect
326. C22×D3C2×#157semidirect
344. #67C2×#157semidirect
345. #67C2×#157semidirect
353. C2×Dic3C2×#157semidirect
460. C32#5486semidirect
473. C23D3×#157semidirect
542. C3#10596642semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×#672
C2×#889942
C2×C3×C22⋊#1572
#67⋊#1572
#67×#1572
D3×C22⋊#1572
#117242612
#105966423
#54869