Finite Group Databasebeta

← #13167993

Subgroups — #13167993

Order 1728.

1590 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D32C211yesmaximal, characteristic
2#11723689C211yesmaximal, characteristic, complemented
3C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#11723424C212yesmaximal, complemented
5C3×#63841C212yesmaximal, complemented
6#11722913C214yesmaximal, complemented
7#11722902C212yesmaximal
8Dic3×#624C212yesmaximal, complemented
9D3×#926—32nomaximal
10#10594602—31nomaximal
11C22×C3×D32C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×C4×D3C2212yes—
14C2×#6294C2212yes—
15C2×Dic3×C3⋊D3C2212yes—
16C2×C3×#624C2211yescharacteristic
17C2×C3×D3×Dic3C2212yes—
18C3×C4×D32C418yescomplemented
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
20C3×C4⋊(C3⋊Dic3)C2212yescomplemented
21C32×C4⋊Dic3C2211yescharacteristic, complemented
22#88815C2212yescomplemented
23#88807C2212yescomplemented
24(C2×D3)⋊(C3⋊Dic3)C2214yes—
25Dic3×C3⋊Dic3C2214yescomplemented
26C3×(C2×D3)⋊Dic3C2214yes—
27C3×Dic32C2214yescomplemented
28C22×C3×C4×D3—32no—
29C2×C4×D32D311yescharacteristic
30#64135—32no—
31C4⋊C4×C3⋊D3—31no—
32C3×#926—32no—
33C3×#892—32no—
34D3×#217—34no—
35#63955—34no—
36#63921—32no—
37#63900—31no—
38D3×C4⋊Dic3—32no—
39D3×C4⋊Dic3—32no—
40D3×C4⋊Dic3—61no—
41C4×#624—32no—
42#63877—34no—
43C4×D3×Dic3—34no—
44#63855—34no—
45#63841—32no—
46#63841—32no—
47#63841—62no—
48#63841D312yescomplemented
49C22×C3×C3⋊D3C2311yescharacteristic
50C22×C32×D3C2312yes—
51C2×C3×D32—22no—
52C2×C3×D32—22no—
53C2×C3×D32C2×C414yes—
54C2×C33×C4C2311yescharacteristic
55C2×C3⋊(C3⋊Dic3)C2312yes—
56C2×C3×C3⋊Dic3C2314yes—
57C2×C3×C3⋊Dic3C2311yescharacteristic
58C3×C4×C3⋊D3C2×C414yes—
59C2×C32×Dic3C2312yes—
60C2×C32×Dic3C2312yes—
61C32×C4×D3C2×C418yescomplemented
62#6294—24no—
63Dic3×C3⋊D3—24no—
64C3×#624C2×C414yes—
65C3×D3×Dic3C2×C418yescomplemented
66C2×#926—91no—
67D3×#117—92no—
68C23×C3×D3—32no—
69C22×D32C2×D311yescharacteristic
70C22×C32×C4—32no—
71C2×C4×C3⋊D3—32no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C22×C3×Dic3—32no—
74C2×C3×C4×D3—22no—
75C2×C3×C4×D3—32no—
76C2×C3×C4×D3—32no—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—316no—
79C2×C3×C4×D3—61no—
80C2×C3×C4×D3C2×D312yes—
81C2×C3×C4×D3C2×D312yes—
82C2×#624—32no—
83C2×#624C2×D311yescharacteristic
84C2×D3×Dic3—34no—
85C2×D3×Dic3—34no—
86C2×D3×Dic3—62no—
87C2×D3×Dic3C2×D312yes—
88C4×D32Dic318yescomplemented
89C32×C4⋊C4—31no—
90#3757—34no—
91C4⋊(C3⋊Dic3)—31no—
92C4⋊(C3⋊Dic3)—32no—
93C4⋊(C3⋊Dic3)—62no—
94C4⋊(C3⋊Dic3)—61no—
95C4⋊(C3⋊Dic3)—121no—
96C4⋊(C3⋊Dic3)C2×D312yescomplemented
97#3750—32no—
98C4×C3⋊Dic3—34no—
99C3×#217—34no—
100C3×(C2×D3)⋊C4—34no—
101C3×C4⋊Dic3—21no—
102C3×C4⋊Dic3—32no—
103C3×C4⋊Dic3—32no—
104C3×C4⋊Dic3—62no—
105C3×C4⋊Dic3C2×D312yescomplemented
106C3×C4×Dic3—34no—
107C3×C4×Dic3—34no—
108#3724—32no—
109Dic3⋊Dic3—34no—
110Dic3⋊Dic3—34no—
111Dic3⋊Dic3—62no—
112(C2×D3)⋊Dic3—34no—
113(C2×D3)⋊Dic3—34no—
114(C2×D3)⋊Dic3—64no—
115(C2×D3)⋊Dic3C2×D314yes—
116Dic32—34no—
117Dic32—34no—
118Dic32—64no—
119Dic32C2×D314yescomplemented
120C22×C33C2411yescharacteristic
121C2×C3×C3⋊D3C22×C412yes—
122C2×C3×C3⋊D3#4112yes—
123C2×C3×C3⋊D3#4112yes—
124C2×C32×D3—22no—
125C2×C32×D3—22no—
126C2×C32×D3C22×C414yes—
127C3×D32—28no—
128C33×C4C22×C412yes—
129C3⋊(C3⋊Dic3)—22no—
130C3×C3⋊Dic3—24no—
131C3×C3⋊Dic3C22×C412yes—
132C32×Dic3—22no—
133C32×Dic3C22×C414yes—
134C23×C3×C4—91no—
135C22×C4×D3—32no—
136C3×#117—92no—
137C2×#217—92no—
138#926—32no—
139#926—94no—
140#926—92no—
141#926—92no—
142#926—182no—
143C2×C4⋊Dic3—91no—
144C2×C4⋊Dic3—91no—
145C2×C4×Dic3—92no—
146#892—32no—
147#892—92no—
148D3×C4⋊C4—92no—
149D3×C4⋊C4—92no—
150D3×C4⋊C4—181no—
151D3×C22⋊C4—94no—
152#877—94no—
153#872—94no—
154C42×D3—94no—
155C23×C32—32no—
156C22×C3⋊D3C22×D311yescharacteristic
157C22×C3×D3—22no—
158C22×C3×D3—32no—
159C22×C3×D3—38no—
160C22×C3×D3—61no—
161C22×C3×D3—64no—
162C22×C3×D3—64no—
163C22×C3×D3C22×D312yes—
164C22×C3×D3C22×D312yes—
165C2×D32—22no—
166C2×D32—22no—
167C2×D32C2×Dic314yes—
168C2×C32×C4—21no—
169C2×C32×C4—22no—
170C2×C32×C4—32no—
171C2×C32×C4—38no—
172C2×C32×C4—32no—
173C2×C32×C4—41no—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—32no—
177C2×C3⋊Dic3—34no—
178C2×C3⋊Dic3—64no—
179C2×C3⋊Dic3—62no—
180C2×C3⋊Dic3—122no—
181C2×C3⋊Dic3C22×D314yes—
182C2×C3⋊Dic3C22×D311yescharacteristic
183C4×C3⋊D3—64no—
184C4×C3⋊D3C2×Dic314yes—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—38no—
189C2×C3×Dic3—34no—
190C2×C3×Dic3—34no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—64no—
193C2×C3×Dic3C22×D314yes—
194C2×C3×Dic3C22×D312yes—
195C2×C3×Dic3C22×D312yes—
196C3×C4×D3—28no—
197C3×C4×D3—38no—
198C3×C4×D3—38no—
199C3×C4×D3—38no—
200C3×C4×D3—64no—
201C3×C4×D3C4×D318yescomplemented
202C3×C4×D3C2×Dic318yescomplemented
203#624—64no—
204#624C2×Dic314yes—
205D3×Dic3—68no—
206D3×Dic3—68no—
207D3×Dic3—124no—
208D3×Dic3C2×Dic318yescomplemented
209C2×#117—271noSylow
210C2×C33C23×C411yescharacteristic
211C2×C33C2×#4111yescharacteristic
212C2×C33C2×#4111yescharacteristic
213C3×C3⋊D3#11714yescomplemented
214C32×D3—24no—
215C24×C3—91no—
216C23×D3—32no—
217C22×C3×C4—32no—
218C22×C3×C4—32no—
219C22×C3×C4—62no—
220C22×C3×C4—91no—
221C22×C3×C4—98no—
222C22×C3×C4—92no—
223C22×C3×C4—91no—
224C22×Dic3—32no—
225C22×Dic3—92no—
226C22×Dic3—92no—
227C2×C4×D3—32no—
228C2×C4×D3—32no—
229C2×C4×D3—32no—
230C2×C4×D3—316no—
231C2×C4×D3—61no—
232C2×C4×D3—94no—
233C2×C4×D3—94no—
234C2×C4×D3—182no—
235C2×C4×D3D3212yes—
236C3×C4⋊C4—32no—
237C3×C4⋊C4—61no—
238C3×C4⋊C4—92no—
239C3×C22⋊C4—94no—
240C3×C42—94no—
241#217—34no—
242#217—94no—
243#217—94no—
244#217—184no—
245(C2×D3)⋊C4—34no—
246(C2×D3)⋊C4—94no—
247C4⋊Dic3—32no—
248C4⋊Dic3—32no—
249C4⋊Dic3—62no—
250C4⋊Dic3—92no—
251C4⋊Dic3—92no—
252C4⋊Dic3—91no—
253C4⋊Dic3—92no—
254C4⋊Dic3—181no—
255C4⋊Dic3—182no—
256C4⋊Dic3—361no—
257C4⋊Dic3D3211yescharacteristic, complemented
258Dic3⋊C4—94no—
259Dic3⋊C4—94no—
260Dic3⋊C4—182no—
261C4×Dic3—34no—
262C4×Dic3—34no—
263C4×Dic3—94no—
264C4×Dic3—94no—
265C4×Dic3—94no—
266C4×Dic3—184no—
267C22×C32—21no—
268C22×C32—22no—
269C22×C32—34no—
270C22×C32—41no—
271C22×C32—62no—
272C22×C32—62no—
273C22×C32C23×D312yes—
274C22×C32C23×D311yescharacteristic
275C2×C3⋊D3#23512yes—
276C2×C3⋊D3#23512yes—
277C2×C3⋊D3C22×Dic312yes—
278C2×C3×D3—22no—
279C2×C3×D3—22no—
280C2×C3×D3—22no—
281C2×C3×D3—22no—
282C2×C3×D3—24no—
283C2×C3×D3—34no—
284C2×C3×D3—34no—
285C2×C3×D3—34no—
286C2×C3×D3—42no—
287C2×C3×D3—42no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C2×C3×D3—616no—
292C2×C3×D3C2×C4×D314yes—
293C2×C3×D3C22×Dic314yes—
294D32—28no—
295C32×C4—22no—
296C32×C4—24no—
297C32×C4—34no—
298C32×C4—42no—
299C32×C4—62no—
300C32×C4C2×C4×D314yes—
301C32×C4C22×Dic312yes—
302C3⋊Dic3—24no—
303C3⋊Dic3—62no—
304C3⋊Dic3—64no—
305C3⋊Dic3—122no—
306C3⋊Dic3—124no—
307C3⋊Dic3—242no—
308C3⋊Dic3C22×Dic312yes—
309C3×Dic3—24no—
310C3×Dic3—24no—
311C3×Dic3—34no—
312C3×Dic3—42no—
313C3×Dic3—62no—
314C3×Dic3—64no—
315C3×Dic3—64no—
316C3×Dic3—124no—
317C3×Dic3C2×C4×D314yes—
318C3×Dic3C22×Dic314yes—
319C23×C4—91no—
320#117—92no—
321#117—274no—
322#117—272no—
323C2×C4⋊C4—271no—
324C2×C4⋊C4—271no—
325C2×C22⋊C4—272no—
326C2×C42—272no—
327C33C2×#11711yescharacteristic, complemented, Sylow
328C23×C3—32no—
329C23×C3—32no—
330C23×C3—62no—
331C23×C3—94no—
332C23×C3—91no—
333C23×C3—182no—
334C23×C3—182no—
335C22×D3—32no—
336C22×D3—38no—
337C22×D3—61no—
338C22×D3—64no—
339C22×D3—64no—
340C22×D3C2×D3212yes—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—32no—
344C2×C3×C4—32no—
345C2×C3×C4—38no—
346C2×C3×C4—38no—
347C2×C3×C4—34no—
348C2×C3×C4—41no—
349C2×C3×C4—68no—
350C2×C3×C4—62no—
351C2×C3×C4—62no—
352C2×C3×C4—94no—
353C2×C3×C4—98no—
354C2×C3×C4—91no—
355C2×C3×C4—94no—
356C2×C3×C4—94no—
357C2×C3×C4C2×D3211yescharacteristic
358C2×C3×C4C2×D3212yes—
359C2×Dic3—32no—
360C2×Dic3—38no—
361C2×Dic3—34no—
362C2×Dic3—34no—
363C2×Dic3—61no—
364C2×Dic3—64no—
365C2×Dic3—94no—
366C2×Dic3—92no—
367C2×Dic3—184no—
368C2×Dic3—182no—
369C2×Dic3—184no—
370C2×Dic3—184no—
371C2×Dic3—362no—
372C2×Dic3C2×D3212yes—
373C2×Dic3C2×D3212yes—
374C4×D3—38no—
375C4×D3—38no—
376C4×D3—38no—
377C4×D3—64no—
378C4×D3—188no—
379C4×D3—188no—
380C4×D3—364no—
381C4×D3D3×Dic318yescomplemented
382C2×C32—21no—
383C2×C32—22no—
384C2×C32—22no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—22no—
388C2×C32—41no—
389C2×C32—41no—
390C2×C32—41no—
391C2×C32—64no—
392C2×C32C22×C4×D312yes—
393C2×C32C2×#23511yescharacteristic
394C2×C32C2×#23511yescharacteristic
395C2×C32D3×#4112yes—
396C2×C32D3×#4112yes—
397C2×C32C23×Dic311yescharacteristic
398C3⋊D3#92614yescomplemented
399C3×D3—24no—
400C3×D3—24no—
401C3×D3—38no—
402C3×D3—44no—
403C3×D3—64no—
404C24—91no—
405C22×C4—32no—
406C22×C4—91no—
407C22×C4—98no—
408C22×C4—92no—
409C22×C4—91no—
410C22×C4—272no—
411C22×C4—272no—
412C4⋊C4—31no—
413C4⋊C4—92no—
414C4⋊C4—272no—
415C4⋊C4—272no—
416C4⋊C4—271no—
417C22⋊C4—94no—
418C22⋊C4—274no—
419C42—94no—
420C42—274no—
421C22×C3—22no—
422C22×C3—21no—
423C22×C3—34no—
424C22×C3—34no—
425C22×C3—41no—
426C22×C3—64no—
427C22×C3—62no—
428C22×C3—62no—
429C22×C3—62no—
430C22×C3—62no—
431C22×C3—92no—
432C22×C3—92no—
433C22×C3—92no—
434C22×C3—122no—
435C22×C3—122no—
436C22×C3—188no—
437C22×C3C22×D3211yescharacteristic
438C22×C3C22×D3212yes—
439C2×D3—22no—
440C2×D3—22no—
441C2×D3—34no—
442C2×D3—34no—
443C2×D3—34no—
444C2×D3—62no—
445C2×D3—62no—
446C2×D3—62no—
447C2×D3—616no—
448C2×D3C2×D3×Dic314yes—
449C3×C4—22no—
450C3×C4—24no—
451C3×C4—34no—
452C3×C4—34no—
453C3×C4—42no—
454C3×C4—64no—
455C3×C4—64no—
456C3×C4—92no—
457C3×C4—122no—
458C3×C4—184no—
459C3×C4C4×D3212yes—
460C3×C4C2×D3×Dic314yes—
461Dic3—22no—
462Dic3—34no—
463Dic3—62no—
464Dic3—64no—
465Dic3—64no—
466Dic3—124no—
467Dic3—184no—
468Dic3—182no—
469Dic3—362no—
470Dic3—364no—
471Dic3—722no—
472Dic3C2×D3×Dic314yes—
473C32—21no—
474C32—22no—
475C32—41no—
476C32D3×#11712yescomplemented
477C32C2×#92611yescharacteristic, complemented
478C23—32no—
479C23—94no—
480C23—91no—
481C23—182no—
482C23—182no—
483C2×C4—32no—
484C2×C4—38no—
485C2×C4—32no—
486C2×C4—94no—
487C2×C4—98no—
488C2×C4—91no—
489C2×C4—94no—
490C2×C4—94no—
491C2×C4—272no—
492C2×C4—544no—
493C2×C4—544no—
494C2×C4D3311yescharacteristic
495C2×C3—21no—
496C2×C3—22no—
497C2×C3—22no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—22no—
501C2×C3—41no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—64no—
505C2×C3—64no—
506C2×C3—94no—
507C2×C3—124no—
508C2×C3C2×C4×D3211yescharacteristic
509C2×C3D3×#23512yes—
510C2×C3D3×#23512yes—
511C2×C3#6431211yescharacteristic
512C2×C3#6431911yescharacteristic
513C2×C3C22×D3×Dic312yes—
514D3—24no—
515D3—38no—
516D3—64no—
517C22—34no—
518C22—62no—
519C22—62no—
520C22—92no—
521C22—92no—
522C22—92no—
523C22—188no—
524C22C2×D3311yescharacteristic
525C4—34no—
526C4—62no—
527C4—92no—
528C4—184no—
529C4—542no—
530C4D32×Dic312yes—
531C3—21no—
532C3—22no—
533C3—41no—
534C3#1059460211yescharacteristic, complemented
535C3D3×#92612yescomplemented
536C2—64no—
537C2—94no—
538C2#1172438811yescharacteristic
539C2#1172438911yescharacteristic
540C2C2×D32×Dic311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC22×C3×D32C22
hypercenterC2×C4D33
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723689C2semidirect
3. C3⋊D3×C4⋊Dic3C2semidirect
4. #11723424C2semidirect
5. C3×#63841C2semidirect
6. #11722913C2semidirect
8. Dic3×#624C2semidirect
18. C3×C4×D32C4semidirect
19. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
20. C3×C4⋊(C3⋊Dic3)C22semidirect
21. C32×C4⋊Dic3C22semidirect
22. #88815C22semidirect
23. #88807C22semidirect
25. Dic3×C3⋊Dic3C22semidirect
27. C3×Dic32C22semidirect
48. #63841D3semidirect
61. C32×C4×D3C2×C4semidirect
65. C3×D3×Dic3C2×C4semidirect
88. C4×D32Dic3semidirect
96. C4⋊(C3⋊Dic3)C2×D3semidirect
105. C3×C4⋊Dic3C2×D3semidirect
119. Dic32C2×D3semidirect
201. C3×C4×D3C4×D3semidirect
202. C3×C4×D3C2×Dic3semidirect
208. D3×Dic3C2×Dic3semidirect
213. C3×C3⋊D3#117semidirect
257. C4⋊Dic3D32semidirect
327. C33C2×#117semidirect
381. C4×D3D3×Dic3semidirect
398. C3⋊D3#926semidirect
476. C32D3×#117semidirect
477. C32C2×#926semidirect
534. C3#10594602semidirect
535. C3D3×#926semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D322
#117236892
C3⋊D3×C4⋊Dic32
#117234242
C3×#638412
#117229132
#117229022
Dic3×#6242
D3×#9263
#105946023