Finite Group Databasebeta

← #13169151

Subgroups — #13169151

Order 1728.

1642 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊#1113C211yesmaximal, characteristic
2Dic3×C4⋊(C3⋊D3)C212yesmaximal, complemented
3#11723404C211yesmaximal, characteristic, complemented
4C3×#63894C212yesmaximal, complemented
5C3×#63874C211yesmaximal, characteristic, complemented
6#11722938C212yesmaximal, complemented
7#11722931C212yesmaximal, complemented
8#11722905C214yesmaximal, complemented
9#10594606—31nomaximal
10#10593994—32nomaximal
11C22×#1113C2212yes—
12C2×C3×C4⋊(C3⋊D3)C2212yes—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×#6295C2212yes—
15C4⋊#1113C418yescomplemented
16C2×Dic3×C3⋊D3C2214yescomplemented
17C2×C3×#624C2214yescomplemented
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
19C32×C4⋊Dic3C2211yescharacteristic, complemented
20C32×C4×Dic3C2212yescomplemented
21#88803C2212yes—
22#88803C2214yescomplemented
23Dic3×C3⋊Dic3C2212yescomplemented
24C3×#3724C2211yescharacteristic, complemented
25C3×#3724C2211yescharacteristic, complemented
26C3×#3722C2212yes—
27C3×#3722C2214yescomplemented
28C2×#3802—31no—
29C2×D3×D12—32no—
30#64101—31no—
31C4×C4⋊(C3⋊D3)—32no—
32C3×#893—32no—
33C3×C4×D12—32no—
34C3×#872—32no—
35#63921—34no—
36D3×(C2×D3)⋊C4—34no—
37D3×(C2×D3)⋊C4—34no—
38#63904—34no—
39#63902—32no—
40C4×#625—32no—
41#63894D312yescomplemented
42Dic3×D12—32no—
43Dic3×D12—32no—
44Dic3×D12—62no—
45#63881—34no—
46#63874D311yescharacteristic, complemented
47#63841—32no—
48#63841—61no—
49C22×C3×C3⋊D3C2311yescharacteristic
50C22×C3×C3⋊D3C2314yes—
51C2×#1113—42no—
52C2×#1113—42no—
53C2×#1113C2×C418yescomplemented
54C2×C33×C4C2311yescharacteristic
55C2×C3⋊(C3⋊Dic3)C2312yes—
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C3×C4⋊(C3⋊D3)C2×C418yescomplemented
58C3×C4×C3⋊D3C2×C414yescomplemented
59C2×C32×Dic3C2312yes—
60C2×C32×Dic3C2312yes—
61C2×C32×Dic3C2312yes—
62#6295C2×C418yescomplemented
63Dic3×C3⋊D3—44no—
64C3×#624—44no—
65#5545—92no—
66#5521—91no—
67C22×D32—34no—
68C22×D32—32no—
69C2×C4⋊(C3⋊D3)C2×D312yes—
70C2×C4×C3⋊D3—34no—
71C2×C4×C3⋊D3C2×D311yescharacteristic
72C2×C3×D12—32no—
73C2×C3×D12—32no—
74C2×C3×D12—62no—
75C2×C3×C4×D3—32no—
76C2×C3×C4×D3—34no—
77C2×C3×C4×D3—34no—
78C2×C3×C4×D3—61no—
79C2×#626—34no—
80C2×#625—32no—
81C2×#624C2×D314yescomplemented
82C2×D3×Dic3—34no—
83C2×D3×Dic3—34no—
84C2×D3×Dic3—64no—
85#3802—38no—
86D3×D12—316no—
87C32×C4⋊C4—31no—
88C32×C42—32no—
89#3752—34no—
90#3752—32no—
91C4⋊(C3⋊Dic3)—32no—
92C4⋊(C3⋊Dic3)—31no—
93C4⋊(C3⋊Dic3)—61no—
94C4⋊(C3⋊Dic3)—62no—
95C4⋊(C3⋊Dic3)—121no—
96C4×C3⋊Dic3—32no—
97C3×(C2×D3)⋊C4—34no—
98C3×(C2×D3)⋊C4—34no—
99C3×(C2×D3)⋊C4—32no—
100C3×(C2×D3)⋊C4—32no—
101C3×C4⋊Dic3—21no—
102C3×C4⋊Dic3C2×D312yescomplemented
103C3×Dic3⋊C4—32no—
104C3×Dic3⋊C4—32no—
105C3×C4×Dic3—22no—
106C3×C4×Dic3C2×D312yescomplemented
107C3×C4×Dic3C2×D312yescomplemented
108#3724C2×D311yescharacteristic, complemented
109#3724C2×D311yescharacteristic, complemented
110#3722C2×D312yes—
111#3722C2×D314yescomplemented
112(C2×D3)⋊Dic3—34no—
113(C2×D3)⋊Dic3—34no—
114(C2×D3)⋊Dic3—34no—
115(C2×D3)⋊Dic3—62no—
116(C2×D3)⋊Dic3—64no—
117Dic32—34no—
118Dic32—62no—
119C22×C33C2411yescharacteristic
120C2×C3×C3⋊D3—24no—
121C2×C3×C3⋊D3—24no—
122C2×C3×C3⋊D3—21no—
123C2×C3×C3⋊D3—21no—
124C2×C3×C3⋊D3C22×C418yes—
125C2×C3×C3⋊D3C22×C412yes—
126#1113—48no—
127C33×C4C22×C412yes—
128C3⋊(C3⋊Dic3)—22no—
129C3×C3⋊Dic3C22×C412yes—
130C32×Dic3—22no—
131C32×Dic3—22no—
132C32×Dic3—22no—
133C2×D3×D4—92no—
134C22×D12—91no—
135C3×C4×D4—92no—
136C3×#117—91no—
137D4×Dic3—92no—
138C4×#67—94no—
139C2×(C2×D3)⋊C4—94no—
140#926—91no—
141#893—32no—
142#893—92no—
143#893—92no—
144#892—92no—
145#892—181no—
146#67⋊C4—94no—
147D3×C22⋊C4—94no—
148D3×C22⋊C4—94no—
149C4×D12—32no—
150C4×D12—92no—
151C4×D12—92no—
152C4×D12—182no—
153#872—32no—
154C22×C3⋊D3C22×D311yescharacteristic
155C22×C3⋊D3C22×D314yes—
156C22×C3×D3—32no—
157C22×C3×D3—34no—
158C22×C3×D3—34no—
159C22×C3×D3—61no—
160C22×C3×D3—64no—
161C2×D32—316no—
162C2×D32—38no—
163C2×D32—124no—
164C2×D32—122no—
165C2×D32—124no—
166C2×D32—122no—
167C2×C32×C4—22no—
168C2×C32×C4—21no—
169C2×C32×C4—32no—
170C2×C32×C4—32no—
171C2×C32×C4—32no—
172C2×C32×C4—41no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D312yes—
175C2×C3⋊Dic3—34no—
176C2×C3⋊Dic3—32no—
177C2×C3⋊Dic3—62no—
178C2×C3⋊Dic3—64no—
179C2×C3⋊Dic3—122no—
180C2×C3⋊Dic3C22×D311yescharacteristic
181C4⋊(C3⋊D3)C4×D318yescomplemented
182C4×C3⋊D3—124no—
183C4×C3⋊D3C4×D314yescomplemented
184C2×C3×Dic3—22no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3C22×D314yes—
190C2×C3×Dic3C22×D312yes—
191C2×C3×Dic3C22×D312yes—
192C2×C3×Dic3C22×D312yes—
193C2×C3×Dic3C22×D312yes—
194C3×D12—38no—
195C3×D12—38no—
196C3×D12—68no—
197C3×C4×D3—38no—
198C3×C4×D3—64no—
199C3×C4×D3—124no—
200C3×C4×D3—124no—
201#626—316no—
202#625—38no—
203#624—44no—
204D3×Dic3—124no—
205D3×Dic3—124no—
206D3×Dic3—128no—
207#518—271noSylow
208C2×C33C23×C411yescharacteristic
209C2×C33H(5,𝔽2)11yescharacteristic
210C2×C33H(5,𝔽2)11yescharacteristic
211C3×C3⋊D3—22no—
212C3×C3⋊D3—28no—
213C23×D3—92no—
214C23×D3—94no—
215C2×C3×D4—92no—
216C22×C3×C4—94no—
217C22×C3×C4—91no—
218C2×#67—94no—
219C22×Dic3—94no—
220D3×D4—916no—
221C2×D12—32no—
222C2×D12—32no—
223C2×D12—62no—
224C2×D12—98no—
225C2×D12—92no—
226C2×C4×D3—32no—
227C2×C4×D3—34no—
228C2×C4×D3—34no—
229C2×C4×D3—61no—
230C2×C4×D3—94no—
231C2×C4×D3—94no—
232C2×C4×D3—184no—
233C3×C4⋊C4—32no—
234C3×C4⋊C4—61no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—91no—
237C3×C22⋊C4—92no—
238C3×C22⋊C4—94no—
239C3×C42—32no—
240C3×C42—32no—
241C3×C42—62no—
242#217—92no—
243#217—94no—
244(C2×D3)⋊C4—32no—
245(C2×D3)⋊C4—32no—
246(C2×D3)⋊C4—34no—
247(C2×D3)⋊C4—34no—
248(C2×D3)⋊C4—94no—
249(C2×D3)⋊C4—94no—
250(C2×D3)⋊C4—94no—
251(C2×D3)⋊C4—184no—
252(C2×D3)⋊C4—182no—
253C4⋊Dic3—91no—
254C4⋊Dic3—92no—
255C4⋊Dic3—182no—
256C4⋊Dic3—181no—
257C4⋊Dic3—361no—
258C4⋊Dic3D3211yescharacteristic, complemented
259Dic3⋊C4—32no—
260Dic3⋊C4—32no—
261C4×Dic3—94no—
262C4×Dic3—92no—
263C4×Dic3—182no—
264C4×Dic3D3212yescomplemented
265C22×C32—21no—
266C22×C32—22no—
267C22×C32—41no—
268C22×C32C23×D312yes—
269C22×C32C23×D311yescharacteristic
270C2×C3⋊D3—24no—
271C2×C3⋊D3—24no—
272C2×C3⋊D3—21no—
273C2×C3⋊D3—21no—
274C2×C3⋊D3C2×C4×D318yes—
275C2×C3⋊D3C2×C4×D312yes—
276C2×C3×D3—38no—
277C2×C3×D3—38no—
278C2×C3×D3—34no—
279C2×C3×D3—62no—
280C2×C3×D3—68no—
281C2×C3×D3—64no—
282C2×C3×D3—62no—
283C2×C3×D3—64no—
284C2×C3×D3—64no—
285C2×C3×D3—64no—
286C2×C3×D3—62no—
287C2×C3×D3—124no—
288C2×C3×D3—124no—
289C2×C3×D3—121no—
290C2×C3×D3—121no—
291D32—128no—
292D32—1216no—
293C32×C4—22no—
294C32×C4—24no—
295C32×C4—42no—
296C32×C4—62no—
297C32×C4—62no—
298C32×C4—62no—
299C32×C4C2×C4×D314yes—
300C32×C4C2×C4×D312yes—
301C3⋊Dic3—62no—
302C3⋊Dic3—64no—
303C3⋊Dic3—124no—
304C3⋊Dic3—122no—
305C3⋊Dic3—242no—
306C3⋊Dic3C2×C4×D312yes—
307C3×Dic3—24no—
308C3×Dic3—22no—
309C3×Dic3—24no—
310C3×Dic3—22no—
311C3×Dic3—22no—
312C3×Dic3—22no—
313C3×Dic3—34no—
314C3×Dic3—42no—
315C3×Dic3—42no—
316C3×Dic3—62no—
317C22×D4—271no—
318C4×D4—92no—
319C4×D4—272no—
320C4×D4—272no—
321C4×D4—272no—
322#117—91no—
323#117—271no—
324C2×C22⋊C4—274no—
325C33#51811yescharacteristic, complemented, Sylow
326C23×C3—91no—
327C23×C3—94no—
328C22×D3—32no—
329C22×D3—34no—
330C22×D3—34no—
331C22×D3—61no—
332C22×D3—64no—
333C22×D3—98no—
334C22×D3—916no—
335C22×D3—364no—
336C22×D3—364no—
337C22×D3—362no—
338C22×D3—362no—
339C3×D4—98no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—34no—
343C2×C3×C4—32no—
344C2×C3×C4—32no—
345C2×C3×C4—32no—
346C2×C3×C4—32no—
347C2×C3×C4—41no—
348C2×C3×C4—62no—
349C2×C3×C4—62no—
350C2×C3×C4—62no—
351C2×C3×C4—91no—
352C2×C3×C4—94no—
353C2×C3×C4—364no—
354C2×C3×C4C2×D3212yes—
355C2×C3×C4C2×D3211yescharacteristic
356#67—916no—
357C2×Dic3—32no—
358C2×Dic3—61no—
359C2×Dic3—92no—
360C2×Dic3—94no—
361C2×Dic3—182no—
362C2×Dic3—184no—
363C2×Dic3—362no—
364C2×Dic3—364no—
365C2×Dic3C2×D3212yes—
366C2×Dic3C2×D3212yes—
367C2×Dic3C2×D3212yes—
368D12—38no—
369D12—38no—
370D12—68no—
371D12—98no—
372C4×D3—38no—
373C4×D3—64no—
374C4×D3—124no—
375C4×D3—124no—
376C4×D3—364no—
377C4×D3—364no—
378C4×D3—368no—
379C2×C32—21no—
380C2×C32—22no—
381C2×C32—22no—
382C2×C32—22no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—41no—
386C2×C32—41no—
387C2×C32—41no—
388C2×C32C22×C4×D311yescharacteristic
389C2×C32C22×C4×D312yes—
390C2×C32#100412yes—
391C2×C32#100911yescharacteristic
392C2×C32#100911yescharacteristic
393C2×C32#100912yes—
394C3⋊D3—22no—
395C3⋊D3—28no—
396C3×D3—68no—
397C3×D3—64no—
398C3×D3—68no—
399C3×D3—122no—
400C3×D3—128no—
401C24—272no—
402C2×D4—92no—
403C2×D4—278no—
404C2×D4—272no—
405C22×C4—94no—
406C22×C4—91no—
407C22×C4—274no—
408C4⋊C4—31no—
409C4⋊C4—91no—
410C4⋊C4—91no—
411C4⋊C4—271no—
412C22⋊C4—92no—
413C22⋊C4—94no—
414C22⋊C4—272no—
415C22⋊C4—274no—
416C42—32no—
417C42—272no—
418C22×C3—21no—
419C22×C3—22no—
420C22×C3—41no—
421C22×C3—98no—
422C22×C3—92no—
423C22×C3—184no—
424C22×C3—184no—
425C22×C3—181no—
426C22×C3—181no—
427C22×C3C22×D3212yes—
428C22×C3C22×D3211yescharacteristic
429C2×D3—34no—
430C2×D3—38no—
431C2×D3—38no—
432C2×D3—62no—
433C2×D3—68no—
434C2×D3—64no—
435C2×D3—62no—
436C2×D3—64no—
437C2×D3—62no—
438C2×D3—64no—
439C2×D3—64no—
440C2×D3—124no—
441C2×D3—124no—
442C2×D3—121no—
443C2×D3—121no—
444C2×D3—368no—
445C2×D3—3616no—
446C3×C4—22no—
447C3×C4—24no—
448C3×C4—42no—
449C3×C4—64no—
450C3×C4—64no—
451C3×C4—62no—
452C3×C4—62no—
453C3×C4—62no—
454C3×C4—62no—
455C3×C4—92no—
456C3×C4—122no—
457C3×C4—122no—
458C3×C4C4×D3214yes—
459C3×C4C4×D3212yes—
460Dic3—22no—
461Dic3—22no—
462Dic3—22no—
463Dic3—34no—
464Dic3—62no—
465Dic3—184no—
466Dic3—182no—
467Dic3—364no—
468Dic3—362no—
469Dic3—722no—
470C32—21no—
471C32—22no—
472C32—41no—
473C32#552111yescharacteristic, complemented
474C32#554512yescomplemented
475C23—91no—
476C23—94no—
477C23—278no—
478C23—1082no—
479C23—1082no—
480D4—98no—
481D4—278no—
482C2×C4—32no—
483C2×C4—32no—
484C2×C4—32no—
485C2×C4—91no—
486C2×C4—94no—
487C2×C4—272no—
488C2×C4—364no—
489C2×C4—1084no—
490C2×C4D3311yescharacteristic
491C2×C3—21no—
492C2×C3—22no—
493C2×C3—22no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—22no—
497C2×C3—41no—
498C2×C3—41no—
499C2×C3—41no—
500C2×C3—182no—
501C2×C3—188no—
502C2×C3C2×C4×D3211yescharacteristic
503C2×C3C2×C4×D3212yes—
504C2×C3#6430712yes—
505C2×C3#6430812yes—
506C2×C3#6431311yescharacteristic
507C2×C3#6432011yescharacteristic
508D3—68no—
509D3—64no—
510D3—68no—
511D3—122no—
512D3—128no—
513C22—92no—
514C22—98no—
515C22—184no—
516C22—181no—
517C22—184no—
518C22—181no—
519C22—1088no—
520C22C2×D3311yescharacteristic
521C4—62no—
522C4—62no—
523C4—62no—
524C4—92no—
525C4—542no—
526C4#8894912yes—
527C3—22no—
528C3—21no—
529C3—41no—
530C3#1059399412yescomplemented
531C3#1059460611yescharacteristic, complemented
532C2—182no—
533C2—188no—
534C2#1172438011yescharacteristic
535C2#1172439011yescharacteristic
536C2C2×#8894911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC2×C4⋊#1113C2
hypercenterC2×C4D33
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. Dic3×C4⋊(C3⋊D3)C2semidirect
3. #11723404C2semidirect
4. C3×#63894C2semidirect
5. C3×#63874C2semidirect
6. #11722938C2semidirect
7. #11722931C2semidirect
8. #11722905C2semidirect
15. C4⋊#1113C4semidirect
16. C2×Dic3×C3⋊D3C22semidirect
17. C2×C3×#624C22semidirect
18. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
19. C32×C4⋊Dic3C22semidirect
20. C32×C4×Dic3C22semidirect
22. #88803C22semidirect
23. Dic3×C3⋊Dic3C22semidirect
24. C3×#3724C22semidirect
25. C3×#3724C22semidirect
27. C3×#3722C22semidirect
41. #63894D3semidirect
46. #63874D3semidirect
53. C2×#1113C2×C4semidirect
57. C3×C4⋊(C3⋊D3)C2×C4semidirect
58. C3×C4×C3⋊D3C2×C4semidirect
62. #6295C2×C4semidirect
81. C2×#624C2×D3semidirect
102. C3×C4⋊Dic3C2×D3semidirect
106. C3×C4×Dic3C2×D3semidirect
107. C3×C4×Dic3C2×D3semidirect
108. #3724C2×D3semidirect
109. #3724C2×D3semidirect
111. #3722C2×D3semidirect
181. C4⋊(C3⋊D3)C4×D3semidirect
183. C4×C3⋊D3C4×D3semidirect
258. C4⋊Dic3D32semidirect
264. C4×Dic3D32semidirect
325. C33#518semidirect
473. C32#5521semidirect
474. C32#5545semidirect
530. C3#10593994semidirect
531. C3#10594606semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
Dic3×C4⋊(C3⋊D3)2
#117234042
C3×#638942
C3×#638742
#117229382
#117229312
#117229052
#105946063
#105939943