Finite Group Databasebeta

← #13172182

Subgroups — #13172182

Order 1728.

1986 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6291C211yesmaximal, characteristic
2#11723715C211yesmaximal, characteristic, complemented
3#11723627C212yesmaximal, complemented
4C3×D3×#217C212yesmaximal, complemented
5C3×#63957C211yesmaximal, characteristic, complemented
6#11722935C214yesmaximal, complemented
7#11722932C214yesmaximal, complemented
8#10595510—31nomaximal
9#10594182—32nomaximal
10C23×C3⋊(C3⋊D3)C2211yescharacteristic
11C23×C32×D3C2211yescharacteristic
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C2×#6291C418yescomplemented
14C2×#6291C2212yes—
15C2×#6291C2212yes—
16C2×#6289C2214yescomplemented
17C2×C3×D3×Dic3C2214yescomplemented
18C3×#3757C2212yes—
19C32×#217C2212yes—
20#88815C2212yescomplemented
21#88813C2212yescomplemented
22#88804C2214yescomplemented
23C3×#3724C2212yescomplemented
24C3×(C2×D3)⋊Dic3C2214yescomplemented
25C3×Dic32C2212yescomplemented
26C22×#638—31no—
27C22×#626—32no—
28C22×#626—61no—
29C22⋊C4×C3⋊D3—32no—
30C2×C3×#217—32no—
31C3×D3×C22⋊C4—32no—
32C3×#877—32no—
33#63974—32no—
34#63974—32no—
35#63974—62no—
36D3×#217D312yescomplemented
37#63957—31no—
38#63957D311yescharacteristic, complemented
39#63955—32no—
40#63952—34no—
41#63952—34no—
42#63904—34no—
43C4×#626—34no—
44#63881—34no—
45#63880—34no—
46C23×C33C2311yescharacteristic
47C22×C3⋊(C3⋊D3)—42no—
48C22×C3⋊(C3⋊D3)C2×C414yescomplemented
49C22×C3⋊(C3⋊D3)C2312yes—
50C22×C32×D3—42no—
51C22×C32×D3C2×C414yescomplemented
52C22×C32×D3C2312yes—
53C2×C3×C3⋊Dic3C2×C414yescomplemented
54C2×C3×C3⋊Dic3C2314yes—
55C2×C3×C3⋊Dic3C2311yescharacteristic
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C2×C32×Dic3C2314yes—
58#6291C2×C418yescomplemented
59#6291C2×C418yescomplemented
60#6289—44no—
61C3×D3×Dic3—44no—
62#5521—91no—
63#5486—92no—
64C24×C32—31no—
65C23×C3⋊D3—32no—
66C23×C3⋊D3—31no—
67C23×C3⋊D3—61no—
68C23×C3⋊D3—62no—
69C23×C3⋊D3—121no—
70C23×C3×D3—21no—
71C23×C3×D3C2×D312yes—
72C2×#638—38no—
73C2×#638—32no—
74C2×#638—32no—
75C22×C3⋊Dic3C2×D311yescharacteristic
76C2×C4×C3⋊D3—34no—
77C22×C3×Dic3—32no—
78C22×C3×Dic3—34no—
79C22×C3×Dic3—61no—
80C2×C3×C4×D3—34no—
81C2×#626—316no—
82C2×#626—32no—
83C2×#626—32no—
84C2×#626—32no—
85C2×#626—32no—
86C2×#626—68no—
87C2×#626—62no—
88C2×#626—62no—
89C2×#624—34no—
90C2×#624—34no—
91C2×#624—64no—
92C2×D3×Dic3C2×D314yescomplemented
93C32×C22⋊C4—32no—
94#3757C2×D312yes—
95#3752—34no—
96C3×#217—22no—
97C3×#217—32no—
98C3×#217—32no—
99C3×#217—34no—
100C3×#217—62no—
101C3×#217C2×D312yes—
102C3×#217C2×D312yes—
103C3×(C2×D3)⋊C4—34no—
104C3×Dic3⋊C4—34no—
105C3×C4×Dic3—34no—
106#3724—32no—
107#3724C2×D312yescomplemented
108Dic3⋊Dic3—34no—
109#3722—34no—
110#3722—34no—
111#3722—64no—
112(C2×D3)⋊Dic3C2×D314yescomplemented
113Dic32—34no—
114Dic32—32no—
115Dic32C2×D312yescomplemented
116C22×C33—22no—
117C22×C33C22×C412yes—
118C22×C33C2411yescharacteristic
119C2×C3⋊(C3⋊D3)—24no—
120C2×C3⋊(C3⋊D3)—44no—
121C2×C3⋊(C3⋊D3)C22×C414yes—
122C2×C32×D3—24no—
123C2×C32×D3—44no—
124C2×C32×D3C22×C414yes—
125C3×C3⋊Dic3—24no—
126C3×C3⋊Dic3C22×C412yes—
127C3×C3⋊Dic3C22×C412yes—
128C32×Dic3—24no—
129C22×#67—92no—
130C22×#67—181no—
131C22×D12—91no—
132C3×#117—91no—
133C2×C3×C22⋊C4—92no—
134C2×#217—32no—
135C4×#67—94no—
136C4×#67—94no—
137#926—91no—
138#893—94no—
139#67⋊C4—94no—
140#67⋊C4—94no—
141D3×C22⋊C4—32no—
142D3×C22⋊C4—92no—
143D3×C22⋊C4—92no—
144D3×C22⋊C4—92no—
145D3×C22⋊C4—182no—
146#877—32no—
147#877—92no—
148C4×D12—94no—
149C23×C32—21no—
150C23×C32—22no—
151C23×C32—32no—
152C23×C32—34no—
153C23×C32—41no—
154C23×C32—122no—
155C23×C32C22×D312yes—
156C23×C32C22×D311yescharacteristic
157C22×C3⋊D3—32no—
158C22×C3⋊D3—32no—
159C22×C3⋊D3—32no—
160C22×C3⋊D3—38no—
161C22×C3⋊D3—34no—
162C22×C3⋊D3—62no—
163C22×C3⋊D3—62no—
164C22×C3⋊D3—68no—
165C22×C3⋊D3—64no—
166C22×C3⋊D3—62no—
167C22×C3⋊D3—124no—
168C22×C3⋊D3—122no—
169C22×C3⋊D3—124no—
170C22×C3⋊D3—122no—
171C22×C3⋊D3—128no—
172C22×C3⋊D3—124no—
173C22×C3⋊D3—128no—
174C22×C3×D3—24no—
175C22×C3×D3—22no—
176C22×C3×D3—44no—
177C22×C3×D3—44no—
178C22×C3×D3C4×D318yescomplemented
179C22×C3×D3C22×D312yes—
180C22×C3×D3C22×D312yes—
181C2×C32×C4—34no—
182#638—38no—
183#638—38no—
184C2×C3⋊Dic3C4×D314yescomplemented
185C2×C3⋊Dic3C22×D314yes—
186C2×C3⋊Dic3C22×D311yescharacteristic
187C2×C3⋊Dic3C22×D311yescharacteristic
188C4×C3⋊D3—124no—
189C2×C3×Dic3—24no—
190C2×C3×Dic3—38no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—34no—
194C2×C3×Dic3—34no—
195C2×C3×Dic3—64no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—64no—
199C2×C3×Dic3—124no—
200C2×C3×Dic3C22×D314yes—
201C2×C3×Dic3C22×D314yes—
202C3×C4×D3—124no—
203#626—38no—
204#626—38no—
205#626—38no—
206#626—38no—
207#626—68no—
208#626—68no—
209#624—124no—
210#624—124no—
211#624—128no—
212D3×Dic3—44no—
213#518—271noSylow
214C2×C33—22no—
215C2×C33C23×C411yescharacteristic
216C2×C33H(5,𝔽2)12yes—
217C3⋊(C3⋊D3)—24no—
218C32×D3—24no—
219C24×C3—32no—
220C24×C3—61no—
221C23×D3—91no—
222C23×D3—92no—
223C23×D3—182no—
224C23×D3—181no—
225C23×D3—361no—
226C23×D3D3211yescharacteristic
227C22×C3×C4—94no—
228C22×C3×C4—91no—
229C2×#67—92no—
230C2×#67—92no—
231C2×#67—92no—
232C2×#67—92no—
233C2×#67—916no—
234C2×#67—182no—
235C2×#67—182no—
236C2×#67—188no—
237C22×Dic3—32no—
238C22×Dic3—34no—
239C22×Dic3—61no—
240C2×D12—98no—
241C2×D12—92no—
242C2×D12—92no—
243C2×C4×D3—34no—
244C2×C4×D3—94no—
245C2×C4×D3—94no—
246C2×C4×D3—94no—
247C2×C4×D3—184no—
248C3×C4⋊C4—92no—
249C3×C22⋊C4—32no—
250C3×C22⋊C4—32no—
251C3×C22⋊C4—62no—
252C3×C22⋊C4—94no—
253C3×C22⋊C4—92no—
254C3×C42—92no—
255#217—32no—
256#217—32no—
257#217—34no—
258#217—62no—
259#217D3212yes—
260(C2×D3)⋊C4—34no—
261(C2×D3)⋊C4—94no—
262(C2×D3)⋊C4—94no—
263(C2×D3)⋊C4—94no—
264(C2×D3)⋊C4—184no—
265C4⋊Dic3—92no—
266Dic3⋊C4—34no—
267Dic3⋊C4—94no—
268C4×Dic3—34no—
269C4×Dic3—94no—
270C4×Dic3—92no—
271C22×C32—22no—
272C22×C32—24no—
273C22×C32—24no—
274C22×C32—22no—
275C22×C32—21no—
276C22×C32—24no—
277C22×C32—22no—
278C22×C32—34no—
279C22×C32—42no—
280C22×C32—41no—
281C22×C32—44no—
282C22×C32—44no—
283C22×C32—64no—
284C22×C32—124no—
285C22×C32C2×C4×D312yes—
286C22×C32C2×C4×D314yes—
287C22×C32C23×D312yes—
288C22×C32C23×D311yescharacteristic
289C2×C3⋊D3—34no—
290C2×C3⋊D3—34no—
291C2×C3⋊D3—34no—
292C2×C3⋊D3—64no—
293C2×C3⋊D3—64no—
294C2×C3⋊D3—64no—
295C2×C3⋊D3—64no—
296C2×C3⋊D3—68no—
297C2×C3⋊D3—64no—
298C2×C3⋊D3—64no—
299C2×C3⋊D3—124no—
300C2×C3⋊D3—124no—
301C2×C3⋊D3—124no—
302C2×C3⋊D3—128no—
303C2×C3⋊D3—128no—
304C2×C3⋊D3—124no—
305C2×C3⋊D3—1216no—
306C2×C3⋊D3—128no—
307C2×C3⋊D3—1216no—
308C2×C3×D3—24no—
309C2×C3×D3—24no—
310C2×C3×D3—24no—
311C2×C3×D3—44no—
312C2×C3×D3—48no—
313C2×C3×D3—48no—
314C2×C3×D3C2×C4×D314yes—
315C2×C3×D3C2×C4×D314yes—
316C32×C4—64no—
317C3⋊Dic3—24no—
318C3⋊Dic3C2×C4×D312yes—
319C3⋊Dic3C2×C4×D312yes—
320C3×Dic3—24no—
321C3×Dic3—24no—
322C3×Dic3—34no—
323C3×Dic3—34no—
324C3×Dic3—44no—
325C3×Dic3—62no—
326C3×Dic3—62no—
327C3×Dic3—64no—
328C3×Dic3—64no—
329C3×Dic3—124no—
330C22×D4—271no—
331C4×D4—274no—
332C4×D4—274no—
333#117—91no—
334#117—271no—
335C2×C22⋊C4—92no—
336C2×C22⋊C4—272no—
337C33#51811yescharacteristic, complemented, Sylow
338C23×C3—21no—
339C23×C3—22no—
340C23×C3—32no—
341C23×C3—32no—
342C23×C3—38no—
343C23×C3—41no—
344C23×C3—62no—
345C23×C3—64no—
346C23×C3—124no—
347C23×C3—124no—
348C23×C3C2×D3212yes—
349C23×C3C2×D3211yescharacteristic
350C22×D3—42no—
351C22×D3—92no—
352C22×D3—92no—
353C22×D3—92no—
354C22×D3—98no—
355C22×D3—94no—
356C22×D3—182no—
357C22×D3—182no—
358C22×D3—182no—
359C22×D3—184no—
360C22×D3—188no—
361C22×D3—362no—
362C22×D3—364no—
363C22×D3—364no—
364C22×D3—362no—
365C22×D3—368no—
366C22×D3—364no—
367C22×D3—368no—
368C22×D3#62414yescomplemented
369C22×D3C2×D3212yes—
370C2×C3×C4—34no—
371C2×C3×C4—34no—
372C2×C3×C4—64no—
373C2×C3×C4—91no—
374C2×C3×C4—91no—
375C2×C3×C4—94no—
376C2×C3×C4—94no—
377C2×C3×C4—364no—
378#67—98no—
379#67—98no—
380#67—98no—
381#67—98no—
382#67—188no—
383#67—188no—
384C2×Dic3—32no—
385C2×Dic3—32no—
386C2×Dic3—38no—
387C2×Dic3—34no—
388C2×Dic3—34no—
389C2×Dic3—64no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—64no—
393C2×Dic3—124no—
394C2×Dic3C2×D3214yes—
395D12—98no—
396D12—98no—
397C4×D3—124no—
398C4×D3—364no—
399C4×D3—364no—
400C4×D3—364no—
401C4×D3—368no—
402C2×C32—22no—
403C2×C32—24no—
404C2×C32—24no—
405C2×C32—22no—
406C2×C32—24no—
407C2×C32—21no—
408C2×C32—22no—
409C2×C32—41no—
410C2×C32—42no—
411C2×C32—44no—
412C2×C32—44no—
413C2×C32—64no—
414C2×C32C22×C4×D311yescharacteristic
415C2×C32C22×C4×D312yes—
416C2×C32#100414yes—
417C2×C32#100912yes—
418C3⋊D3—64no—
419C3⋊D3—68no—
420C3⋊D3—64no—
421C3⋊D3—64no—
422C3⋊D3—124no—
423C3⋊D3—124no—
424C3⋊D3—128no—
425C3×D3—24no—
426C3×D3—24no—
427C3×D3—44no—
428C24—31no—
429C24—271no—
430C2×D4—272no—
431C2×D4—272no—
432C2×D4—278no—
433C22×C4—94no—
434C22×C4—91no—
435C22×C4—274no—
436C4⋊C4—92no—
437C4⋊C4—272no—
438C22⋊C4—32no—
439C22⋊C4—94no—
440C22⋊C4—92no—
441C22⋊C4—274no—
442C42—92no—
443C42—272no—
444C22×C3—24no—
445C22×C3—24no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—21no—
449C22×C3—22no—
450C22×C3—24no—
451C22×C3—34no—
452C22×C3—34no—
453C22×C3—44no—
454C22×C3—44no—
455C22×C3—41no—
456C22×C3—42no—
457C22×C3—64no—
458C22×C3—64no—
459C22×C3—64no—
460C22×C3—124no—
461C22×C3—128no—
462C22×C3—128no—
463C22×C3C4×D3214yes—
464C22×C3C2×#62412yes—
465C22×C3C22×D3212yes—
466C22×C3C22×D3211yescharacteristic
467C2×D3—24no—
468C2×D3—44no—
469C2×D3—94no—
470C2×D3—94no—
471C2×D3—94no—
472C2×D3—184no—
473C2×D3—184no—
474C2×D3—184no—
475C2×D3—188no—
476C2×D3—184no—
477C2×D3—184no—
478C2×D3—184no—
479C2×D3—364no—
480C2×D3—364no—
481C2×D3—364no—
482C2×D3—368no—
483C2×D3—368no—
484C2×D3—364no—
485C2×D3—3616no—
486C2×D3—368no—
487C2×D3—3616no—
488C2×D3C2×#62414yes—
489C3×C4—64no—
490C3×C4—64no—
491C3×C4—92no—
492C3×C4—92no—
493C3×C4—124no—
494C3×C4—184no—
495Dic3—24no—
496Dic3—34no—
497Dic3—34no—
498Dic3—62no—
499Dic3—62no—
500Dic3—64no—
501Dic3—64no—
502Dic3—124no—
503C32—22no—
504C32—21no—
505C32—41no—
506C32#548612yescomplemented
507C32#552111yescharacteristic, complemented
508C23—32no—
509C23—34no—
510C23—122no—
511C23—272no—
512C23—274no—
513C23—1082no—
514C23D3311yescharacteristic
515D4—278no—
516D4—278no—
517C2×C4—34no—
518C2×C4—91no—
519C2×C4—91no—
520C2×C4—94no—
521C2×C4—94no—
522C2×C4—364no—
523C2×C4—1084no—
524C2×C3—24no—
525C2×C3—24no—
526C2×C3—22no—
527C2×C3—21no—
528C2×C3—22no—
529C2×C3—24no—
530C2×C3—22no—
531C2×C3—44no—
532C2×C3—44no—
533C2×C3—41no—
534C2×C3—42no—
535C2×C3—64no—
536C2×C3—64no—
537C2×C3—124no—
538C2×C3C2×C4×D3212yes—
539C2×C3#6431414yes—
540C2×C3C22×#62411yescharacteristic
541C2×C3#6433012yes—
542D3—24no—
543D3—188no—
544D3—184no—
545D3—184no—
546D3—184no—
547D3—364no—
548D3—364no—
549D3—368no—
550C22—22no—
551C22—34no—
552C22—64no—
553C22—124no—
554C22—274no—
555C22—544no—
556C22—1084no—
557C22D3×#62412yes—
558C22C2×D3311yescharacteristic
559C4—64no—
560C4—92no—
561C4—92no—
562C4—184no—
563C3—22no—
564C3—21no—
565C3—41no—
566C3#1059418212yescomplemented
567C3#1059551011yescharacteristic, complemented
568C2—22no—
569C2—64no—
570C2—544no—
571C2C2×D3×#62411yescharacteristic
572C2#1172441912yes—
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC23×C33C23
socle, cosocleC22×C33C24
prime radicalC22×#6291C2
hypercenterC23D33
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #11723715C2semidirect
3. #11723627C2semidirect
4. C3×D3×#217C2semidirect
5. C3×#63957C2semidirect
6. #11722935C2semidirect
7. #11722932C2semidirect
13. C2×#6291C4semidirect
16. C2×#6289C22semidirect
17. C2×C3×D3×Dic3C22semidirect
20. #88815C22semidirect
21. #88813C22semidirect
22. #88804C22semidirect
23. C3×#3724C22semidirect
24. C3×(C2×D3)⋊Dic3C22semidirect
25. C3×Dic32C22semidirect
36. D3×#217D3semidirect
38. #63957D3semidirect
48. C22×C3⋊(C3⋊D3)C2×C4semidirect
51. C22×C32×D3C2×C4semidirect
53. C2×C3×C3⋊Dic3C2×C4semidirect
58. #6291C2×C4semidirect
59. #6291C2×C4semidirect
92. C2×D3×Dic3C2×D3semidirect
107. #3724C2×D3semidirect
112. (C2×D3)⋊Dic3C2×D3semidirect
115. Dic32C2×D3semidirect
178. C22×C3×D3C4×D3semidirect
184. C2×C3⋊Dic3C4×D3semidirect
337. C33#518semidirect
368. C22×D3#624semidirect
506. C32#5486semidirect
507. C32#5521semidirect
566. C3#10594182semidirect
567. C3#10595510semidirect

Maximal subgroups

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

subgroupindex
C22×#62912
#117237152
#117236272
C3×D3×#2172
C3×#639572
#117229352
#117229322
#105955103
#105941823