Finite Group Databasebeta

← #13171734

Subgroups — #13171734

Order 1728.

1638 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#624C211yesmaximal, characteristic
2#11723723C211yesmaximal, characteristic
3C3⋊D3×#217C211yesmaximal, characteristic
4#11723594C212yesmaximal, complemented
5C3×#63955C212yesmaximal, complemented
6#11722915C214yesmaximal, complemented
7#11722908C212yesmaximal, complemented
8Dic3×#624C212yesmaximal, complemented
9D3×#926—32nomaximal
10#10594052—31nomaximal
11C23×C3×C3⋊D3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C2×#6294C2212yes—
14C2×Dic3×C3⋊D3C2212yes—
15C2×C3×#624C418yescomplemented
16C2×C3×#624C2211yescharacteristic
17C2×C3×#624C2211yescharacteristic
18C2×C3×#624C2212yes—
19#88864C2211yescharacteristic
20C3×#3757C2212yes—
21C32×#217C2211yescharacteristic
22#88814C2212yes—
23Dic3⋊(C3⋊Dic3)C2214yescomplemented
24#88803C2212yes—
25Dic3×C3⋊Dic3C2214yescomplemented
26C3×Dic3⋊Dic3C2214yescomplemented
27C3×Dic32C2214yescomplemented
28C22×C3×C4×D3—32no—
29C22×#624D311yescharacteristic
30#64135—32no—
31C22⋊C4×C3⋊D3—31no—
32C3×#926—32no—
33C3×#877—32no—
34#63974—31no—
35D3×#217—32no—
36D3×#217—32no—
37D3×#217—61no—
38#63955—32no—
39#63955—32no—
40#63955—62no—
41#63955D312yescomplemented
42D3×C4⋊Dic3—34no—
43#63883—32no—
44C4×#624—32no—
45C4×D3×Dic3—34no—
46#63855—34no—
47#63841—34no—
48#63836—34no—
49C23×C33C2311yescharacteristic
50C22×C3×C3⋊D3—24no—
51C22×C3×C3⋊D3C2×C414yes—
52C22×C3×C3⋊D3C2311yescharacteristic
53C22×C3×C3⋊D3C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)C2312yes—
55C2×C3×C3⋊Dic3C2314yes—
56C2×C32×Dic3C2×C418yescomplemented
57C2×C32×Dic3C2312yes—
58C2×C32×Dic3C2312yes—
59C2×C32×Dic3C2312yes—
60#6294—24no—
61Dic3×C3⋊D3—24no—
62C3×#624C2×C414yes—
63C3×#624C2×C414yes—
64C3×#624C2×C418yescomplemented
65C2×#926—91no—
66D3×#117—92no—
67C23×C3⋊D3C2×D311yescharacteristic
68C23×C3×D3—32no—
69C23×C3×D3—61no—
70C22×C32×C4—32no—
71C2×C4×C3⋊D3—32no—
72C22×C3×Dic3—22no—
73C22×C3×Dic3C2×D312yes—
74C22×C3×Dic3C2×D312yes—
75C2×C3×C4×D3—316no—
76C2×C3×C4×D3—32no—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—32no—
80C2×#624—32no—
81C2×#624Dic318yescomplemented
82C2×#624C2×D311yescharacteristic
83C2×#624C2×D311yescharacteristic
84C2×#624C2×D312yes—
85C2×D3×Dic3—34no—
86C2×D3×Dic3—34no—
87C2×D3×Dic3—62no—
88C32×C22⋊C4—31no—
89#3757—31no—
90#3757—32no—
91#3757—62no—
92#3757—61no—
93#3757—121no—
94#3757C2×D312yes—
95#3752—32no—
96C4⋊(C3⋊Dic3)—34no—
97C4×C3⋊Dic3—34no—
98C3×#217—21no—
99C3×#217—32no—
100C3×#217—32no—
101C3×#217—62no—
102C3×#217C2×D312yes—
103C3×C4⋊Dic3—34no—
104C3×Dic3⋊C4—34no—
105C3×C4×Dic3—34no—
106C3×C4×Dic3—34no—
107Dic3⋊Dic3—34no—
108Dic3⋊Dic3—34no—
109Dic3⋊Dic3—64no—
110Dic3⋊Dic3C2×D314yescomplemented
111#3722—32no—
112(C2×D3)⋊Dic3—34no—
113(C2×D3)⋊Dic3—34no—
114(C2×D3)⋊Dic3—62no—
115Dic32—34no—
116Dic32—34no—
117Dic32—64no—
118Dic32C2×D314yescomplemented
119C22×C33—22no—
120C22×C33C22×C412yes—
121C22×C33C2411yescharacteristic
122C2×C3×C3⋊D3—22no—
123C2×C3×C3⋊D3—28no—
124C2×C3×C3⋊D3C22×C412yes—
125C2×C3×C3⋊D3C22×C412yes—
126C2×C3×C3⋊D3#4114yes—
127C3⋊(C3⋊Dic3)—22no—
128C3×C3⋊Dic3—24no—
129C32×Dic3—22no—
130C32×Dic3C22×C414yes—
131C32×Dic3C22×C414yes—
132C23×C3×C4—91no—
133C22×C4×D3—32no—
134C3×#117—92no—
135C2×#217—91no—
136C2×#217—91no—
137#926—32no—
138#926—94no—
139#926—92no—
140#926—92no—
141#926—182no—
142C2×C4⋊Dic3—92no—
143C2×C4×Dic3—92no—
144#892—94no—
145D3×C4⋊C4—94no—
146D3×C22⋊C4—92no—
147D3×C22⋊C4—92no—
148D3×C22⋊C4—181no—
149#877—32no—
150#877—92no—
151#872—94no—
152C42×D3—94no—
153C23×C32—21no—
154C23×C32—22no—
155C23×C32—41no—
156C23×C32C22×D312yes—
157C23×C32C22×D311yescharacteristic
158C22×C3⋊D3—24no—
159C22×C3⋊D3C2×Dic314yes—
160C22×C3⋊D3C22×D311yescharacteristic
161C22×C3⋊D3C22×D311yescharacteristic
162C22×C3×D3—32no—
163C22×C3×D3—38no—
164C22×C3×D3—32no—
165C22×C3×D3—61no—
166C22×C3×D3—61no—
167C22×C3×D3—64no—
168C22×C3×D3—68no—
169C22×C3×D3—124no—
170C2×C32×C4—38no—
171C2×C32×C4—32no—
172C2×C32×C4—32no—
173C2×C32×C4—32no—
174C2×C3⋊Dic3—32no—
175C2×C3⋊Dic3—34no—
176C2×C3⋊Dic3—64no—
177C2×C3⋊Dic3—62no—
178C2×C3⋊Dic3—122no—
179C2×C3⋊Dic3C22×D314yes—
180C4×C3⋊D3—64no—
181C2×C3×Dic3—28no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—34no—
186C2×C3×Dic3—34no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3C4×D318yescomplemented
189C2×C3×Dic3C2×Dic318yescomplemented
190C2×C3×Dic3C22×D314yes—
191C2×C3×Dic3C22×D312yes—
192C2×C3×Dic3C22×D312yes—
193C2×C3×Dic3C22×D312yes—
194C2×C3×Dic3C22×D312yes—
195C3×C4×D3—38no—
196C3×C4×D3—38no—
197C3×C4×D3—38no—
198C3×C4×D3—38no—
199#624—64no—
200#624C2×Dic314yes—
201#624C2×Dic314yes—
202#624C2×Dic318yescomplemented
203D3×Dic3—68no—
204D3×Dic3—68no—
205D3×Dic3—124no—
206C2×#117—271noSylow
207C2×C33—22no—
208C2×C33C23×C411yescharacteristic
209C2×C33C2×#4112yes—
210C3×C3⋊D3—22no—
211C3×C3⋊D3#11714yescomplemented
212C24×C3—91no—
213C23×D3—32no—
214C23×D3—61no—
215C22×C3×C4—32no—
216C22×C3×C4—32no—
217C22×C3×C4—62no—
218C22×C3×C4—91no—
219C22×C3×C4—91no—
220C22×C3×C4—92no—
221C22×C3×C4—98no—
222C22×Dic3—92no—
223C22×Dic3—92no—
224C22×Dic3D3212yes—
225C2×C4×D3—316no—
226C2×C4×D3—32no—
227C2×C4×D3—32no—
228C2×C4×D3—32no—
229C2×C4×D3—32no—
230C2×C4×D3—94no—
231C2×C4×D3—94no—
232C2×C4×D3—182no—
233C3×C4⋊C4—94no—
234C3×C22⋊C4—32no—
235C3×C22⋊C4—61no—
236C3×C22⋊C4—92no—
237C3×C42—94no—
238#217—32no—
239#217—32no—
240#217—62no—
241#217—92no—
242#217—92no—
243#217—91no—
244#217—92no—
245#217—181no—
246#217—182no—
247#217—361no—
248#217D3211yescharacteristic
249(C2×D3)⋊C4—94no—
250(C2×D3)⋊C4—94no—
251(C2×D3)⋊C4—182no—
252C4⋊Dic3—34no—
253C4⋊Dic3—94no—
254C4⋊Dic3—94no—
255C4⋊Dic3—184no—
256Dic3⋊C4—34no—
257Dic3⋊C4—94no—
258C4×Dic3—34no—
259C4×Dic3—34no—
260C4×Dic3—94no—
261C4×Dic3—94no—
262C4×Dic3—94no—
263C4×Dic3—184no—
264C22×C32—22no—
265C22×C32—24no—
266C22×C32—22no—
267C22×C32—21no—
268C22×C32—22no—
269C22×C32—24no—
270C22×C32—41no—
271C22×C32—42no—
272C22×C32—42no—
273C22×C32—44no—
274C22×C32—82no—
275C22×C32C2×C4×D314yes—
276C22×C32C22×Dic312yes—
277C22×C32C23×D312yes—
278C22×C32C23×D311yescharacteristic
279C2×C3⋊D3—22no—
280C2×C3⋊D3—28no—
281C2×C3⋊D3#23514yes—
282C2×C3⋊D3C22×Dic312yes—
283C2×C3⋊D3C22×Dic312yes—
284C2×C3×D3—34no—
285C2×C3×D3—38no—
286C2×C3×D3—34no—
287C2×C3×D3—62no—
288C2×C3×D3—64no—
289C2×C3×D3—62no—
290C2×C3×D3—616no—
291C2×C3×D3—64no—
292C2×C3×D3—128no—
293C2×C3×D3—122no—
294C32×C4—34no—
295C32×C4—34no—
296C32×C4—62no—
297C3⋊Dic3—24no—
298C3⋊Dic3—62no—
299C3⋊Dic3—64no—
300C3⋊Dic3—122no—
301C3⋊Dic3—124no—
302C3⋊Dic3—242no—
303C3×Dic3—24no—
304C3×Dic3—24no—
305C3×Dic3—24no—
306C3×Dic3—42no—
307C3×Dic3—64no—
308C3×Dic3—64no—
309C3×Dic3—124no—
310C3×Dic3C2×C4×D314yes—
311C3×Dic3C2×C4×D314yes—
312C3×Dic3C22×Dic314yes—
313C3×Dic3C22×Dic314yes—
314C23×C4—91no—
315#117—92no—
316#117—274no—
317#117—272no—
318C2×C4⋊C4—272no—
319C2×C22⋊C4—271no—
320C2×C22⋊C4—271no—
321C2×C42—272no—
322C33C2×#11711yescharacteristic, complemented, Sylow
323C23×C3—22no—
324C23×C3—21no—
325C23×C3—41no—
326C23×C3—91no—
327C23×C3—91no—
328C23×C3—94no—
329C23×C3—184no—
330C23×C3C2×D3211yescharacteristic
331C23×C3C2×D3212yes—
332C22×D3—32no—
333C22×D3—32no—
334C22×D3—38no—
335C22×D3—61no—
336C22×D3—64no—
337C22×D3—61no—
338C22×D3—68no—
339C22×D3—124no—
340C2×C3×C4—32no—
341C2×C3×C4—32no—
342C2×C3×C4—32no—
343C2×C3×C4—32no—
344C2×C3×C4—38no—
345C2×C3×C4—38no—
346C2×C3×C4—34no—
347C2×C3×C4—62no—
348C2×C3×C4—62no—
349C2×C3×C4—68no—
350C2×C3×C4—62no—
351C2×C3×C4—94no—
352C2×C3×C4—94no—
353C2×C3×C4—98no—
354C2×C3×C4—94no—
355C2×Dic3—34no—
356C2×Dic3—34no—
357C2×Dic3—64no—
358C2×Dic3—94no—
359C2×Dic3—92no—
360C2×Dic3—184no—
361C2×Dic3—182no—
362C2×Dic3—184no—
363C2×Dic3—184no—
364C2×Dic3—362no—
365C2×Dic3D3×Dic318yescomplemented
366C2×Dic3C2×D3212yes—
367C2×Dic3C2×D3212yes—
368C2×Dic3C2×D3212yes—
369C4×D3—38no—
370C4×D3—38no—
371C4×D3—38no—
372C4×D3—38no—
373C4×D3—188no—
374C4×D3—188no—
375C4×D3—364no—
376C2×C32—24no—
377C2×C32—22no—
378C2×C32—21no—
379C2×C32—24no—
380C2×C32—22no—
381C2×C32—22no—
382C2×C32—41no—
383C2×C32—42no—
384C2×C32—42no—
385C2×C32—44no—
386C2×C32—82no—
387C2×C32C22×C4×D312yes—
388C2×C32C2×#23512yes—
389C2×C32D3×#4114yes—
390C2×C32C23×Dic311yescharacteristic
391C3⋊D3—22no—
392C3⋊D3#92614yescomplemented
393C3×D3—38no—
394C3×D3—64no—
395C3×D3—64no—
396C3×D3—122no—
397C24—91no—
398C22×C4—32no—
399C22×C4—91no—
400C22×C4—91no—
401C22×C4—92no—
402C22×C4—98no—
403C22×C4—272no—
404C22×C4—272no—
405C4⋊C4—94no—
406C4⋊C4—274no—
407C22⋊C4—31no—
408C22⋊C4—92no—
409C22⋊C4—272no—
410C22⋊C4—272no—
411C22⋊C4—271no—
412C42—94no—
413C42—274no—
414C22×C3—24no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—21no—
418C22×C3—24no—
419C22×C3—22no—
420C22×C3—42no—
421C22×C3—44no—
422C22×C3—41no—
423C22×C3—42no—
424C22×C3—82no—
425C22×C3—92no—
426C22×C3—94no—
427C22×C3—92no—
428C22×C3—188no—
429C22×C3—182no—
430C22×C3C4×D3212yes—
431C22×C3C2×D3×Dic314yes—
432C22×C3C22×D3211yescharacteristic
433C22×C3C22×D3212yes—
434C2×D3—38no—
435C2×D3—34no—
436C2×D3—34no—
437C2×D3—64no—
438C2×D3—62no—
439C2×D3—62no—
440C2×D3—616no—
441C2×D3—64no—
442C2×D3—128no—
443C2×D3—122no—
444C3×C4—34no—
445C3×C4—34no—
446C3×C4—34no—
447C3×C4—34no—
448C3×C4—64no—
449C3×C4—64no—
450C3×C4—64no—
451C3×C4—122no—
452C3×C4—184no—
453Dic3—22no—
454Dic3—64no—
455Dic3—64no—
456Dic3—124no—
457Dic3—184no—
458Dic3—182no—
459Dic3—362no—
460Dic3—364no—
461Dic3—722no—
462Dic3C2×D3×Dic314yes—
463Dic3C2×D3×Dic314yes—
464C32—21no—
465C32—22no—
466C32—41no—
467C32D3×#11712yescomplemented
468C32C2×#92611yescharacteristic, complemented
469C23—91no—
470C23—91no—
471C23—94no—
472C23—184no—
473C23D3311yescharacteristic
474C2×C4—32no—
475C2×C4—32no—
476C2×C4—38no—
477C2×C4—32no—
478C2×C4—94no—
479C2×C4—94no—
480C2×C4—98no—
481C2×C4—94no—
482C2×C4—272no—
483C2×C4—544no—
484C2×C4—544no—
485C2×C3—22no—
486C2×C3—24no—
487C2×C3—22no—
488C2×C3—21no—
489C2×C3—24no—
490C2×C3—22no—
491C2×C3—42no—
492C2×C3—44no—
493C2×C3—41no—
494C2×C3—42no—
495C2×C3—82no—
496C2×C3—94no—
497C2×C3—182no—
498C2×C3C2×C4×D3211yescharacteristic
499C2×C3D3×#23514yes—
500C2×C3#6431212yes—
501C2×C3C22×D3×Dic312yes—
502D3—38no—
503D3—64no—
504D3—64no—
505D3—122no—
506C22—22no—
507C22—92no—
508C22—94no—
509C22—92no—
510C22—188no—
511C22—182no—
512C22D32×Dic312yes—
513C22C2×D3311yescharacteristic
514C4—34no—
515C4—34no—
516C4—62no—
517C4—184no—
518C4—542no—
519C3—21no—
520C3—22no—
521C3—41no—
522C3#1059405211yescharacteristic, complemented
523C3D3×#92612yescomplemented
524C2—22no—
525C2—94no—
526C2—182no—
527C2C2×D32×Dic311yescharacteristic
528C2#1172441012yes—
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC23×C33C23
socle, cosocleC22×C33C24
prime radicalC23×C3×C3⋊D3C22
hypercenterC23D33
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11723594C2semidirect
5. C3×#63955C2semidirect
6. #11722915C2semidirect
7. #11722908C2semidirect
8. Dic3×#624C2semidirect
15. C2×C3×#624C4semidirect
23. Dic3⋊(C3⋊Dic3)C22semidirect
25. Dic3×C3⋊Dic3C22semidirect
26. C3×Dic3⋊Dic3C22semidirect
27. C3×Dic32C22semidirect
41. #63955D3semidirect
56. C2×C32×Dic3C2×C4semidirect
64. C3×#624C2×C4semidirect
81. C2×#624Dic3semidirect
110. Dic3⋊Dic3C2×D3semidirect
118. Dic32C2×D3semidirect
188. C2×C3×Dic3C4×D3semidirect
189. C2×C3×Dic3C2×Dic3semidirect
202. #624C2×Dic3semidirect
211. C3×C3⋊D3#117semidirect
322. C33C2×#117semidirect
365. C2×Dic3D3×Dic3semidirect
392. C3⋊D3#926semidirect
467. C32D3×#117semidirect
468. C32C2×#926semidirect
522. C3#10594052semidirect
523. C3D3×#926semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6242
#117237232
C3⋊D3×#2172
#117235942
C3×#639552
#117229152
#117229082
Dic3×#6242
D3×#9263
#105940523