Finite Group Databasebeta

← C2×#11721034 #13150323

Subgroups — C2×#11721034 #13150323

Order 1728.

1410 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D3×C9C211yesmaximal, characteristic, complemented
2C22×#6195C211yesmaximal, characteristic, complemented
3C22×#6194C211yesmaximal, characteristic, complemented
4C22×#6192C211yesmaximal, characteristic, complemented
5C2×#88536C211yesmaximal, characteristic, complemented
6C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
7C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
8#11721034C218yesmaximal, direct product factor
9C2×#63906—31nomaximal
10C2×#63492—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C23×D3×C9C2211yescharacteristic
13C23×C3×D9C2211yescharacteristic
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×C4×D3×C9—24no—
19C2×C4×D3×C9C2214yescomplemented
20C2×#6195—24no—
21C2×#6195C2214yescomplemented
22C2×#6194—24no—
23C2×#6194C2214yescomplemented
24C2×#6192—24no—
25C2×#6192C2214yescomplemented
26#88536C2214yescomplemented
27C3×(C2×D9)⋊C4C2214yescomplemented
28Dic3⋊Dic9C2214yescomplemented
29C22×C3×C4×D3D311yescharacteristic
30C22×#626—31no—
31C22×#626—31no—
32C22×#625—31no—
33C2×#3752—31no—
34C2×C3×(C2×D3)⋊C4—31no—
35C2×Dic3⋊Dic3—31no—
36#63906—38no—
37C23×C4×C9—31no—
38C22×#611—31no—
39C22×#611—31no—
40C22×D36—31no—
41#63492—38no—
42C2×(C2×D9)⋊C4—31no—
43C2×(C2×D9)⋊C4—61no—
44C2×(C2×D9)⋊C4D311yescharacteristic, complemented
45C2×C4⋊Dic9—31no—
46C23×C3×C9C2311yescharacteristic
47C22×C3⋊D9—22no—
48C22×C3⋊D9—22no—
49C22×C3⋊D9—22no—
50C22×C3⋊D9C2312yes—
51C22×D3×C9—22no—
52C22×D3×C9—22no—
53C22×D3×C9D414yescomplemented
54C22×D3×C9C2312yes—
55C22×C3×D9—22no—
56C22×C3×D9—22no—
57C22×C3×D9—22no—
58C22×C3×D9C2312yes—
59C2×C3×C4×C9—22no—
60C2×C3×C4×C9C2312yes—
61C2×C3⋊Dic9—22no—
62C2×C3⋊Dic9C2312yes—
63C2×C9×Dic3D414yescomplemented
64C2×C9×Dic3C2312yes—
65C2×C3×Dic9—22no—
66C2×C3×Dic9C2312yes—
67C4×D3×C9—28no—
68#6195—28no—
69#6194—28no—
70#6192—28no—
71C2×#930—91no—
72C2×#884—91no—
73C23×C3⋊D3—31no—
74C23×C3×D3—31no—
75C23×C3×D3C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C22×C3⋊Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3C2×D311yescharacteristic
80C2×C3×C4×D3—24no—
81C2×C3×C4×D3C2×D314yes—
82C2×#626—34no—
83C2×#626—34no—
84C2×#626—64no—
85C2×#626—64no—
86C2×#625—34no—
87C2×#625—64no—
88C24×C9—31no—
89C23×D9—31no—
90C23×D9—61no—
91C23×D9C2×D311yescharacteristic
92#3752—34no—
93C3×(C2×D3)⋊C4—34no—
94Dic3⋊Dic3—34no—
95C22×C4×C9—21no—
96C22×C4×C9—31no—
97C22×C4×C9—34no—
98C22×C4×C9—64no—
99C22×C4×C9C2×D311yescharacteristic
100C2×#611—34no—
101C2×#611—34no—
102C2×#611—64no—
103C2×#611—64no—
104C22×Dic9—31no—
105C22×Dic9—61no—
106C22×Dic9C2×D311yescharacteristic
107C2×D36—34no—
108C2×D36—64no—
109(C2×D9)⋊C4—34no—
110(C2×D9)⋊C4—64no—
111(C2×D9)⋊C4C2×D314yescomplemented
112C4⋊Dic9—34no—
113C22×C3×C9C2×D412yes—
114C22×C3×C9C2×D412yes—
115C22×C3×C9#4112yes—
116C22×C3×C9C2411yescharacteristic
117C2×C3⋊D9—22no—
118C2×C3⋊D9—22no—
119C2×C3⋊D9—22no—
120C2×C3⋊D9—44no—
121C2×D3×C9—28no—
122C2×D3×C9—22no—
123C2×D3×C9—22no—
124C2×D3×C9C2×D414yescomplemented
125C2×C3×D9—22no—
126C2×C3×D9—22no—
127C2×C3×D9—22no—
128C2×C3×D9—44no—
129C3×C4×C9—22no—
130C3⋊Dic9—22no—
131C9×Dic3C2×D414yescomplemented
132C3×Dic9—22no—
133C23×C3×C4—31no—
134C22×#67—91no—
135C22×#67—91no—
136C22×#67—91no—
137C22×#67—91no—
138C22×D12—91no—
139C22×D12—91no—
140C22×C4×D3D911yescharacteristic, complemented
141C2×C3×C22⋊C4—91no—
142#930—98no—
143C2×(C2×D3)⋊C4—31no—
144C2×(C2×D3)⋊C4—91no—
145C2×(C2×D3)⋊C4—91no—
146C2×(C2×D3)⋊C4—181no—
147C2×C4⋊Dic3—91no—
148C2×Dic3⋊C4—91no—
149#884—98no—
150C23×C32C22×D311yescharacteristic
151C22×C3⋊D3—32no—
152C22×C3⋊D3—62no—
153C22×C3⋊D3—62no—
154C22×C3⋊D3—62no—
155C22×C3×D3—22no—
156C22×C3×D3—22no—
157C22×C3×D3—32no—
158C22×C3×D3—62no—
159C22×C3×D3—62no—
160C22×C3×D3—62no—
161C22×C3×D3D1214yes—
162C22×C3×D3C22×D312yes—
163C2×C32×C4—22no—
164C2×C32×C4C22×D312yes—
165C2×C3⋊Dic3—32no—
166C2×C3⋊Dic3—62no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3#6714yes—
170C2×C3×Dic3C22×D312yes—
171C3×C4×D3—28no—
172#626—68no—
173#626—68no—
174#625—68no—
175C23×C9—21no—
176C23×C9—34no—
177C23×C9—32no—
178C23×C9—62no—
179C23×C9—62no—
180C23×C9C22×D311yescharacteristic
181C22×D9—22no—
182C22×D9—22no—
183C22×D9—22no—
184C22×D9—32no—
185C22×D9—62no—
186C22×D9—64no—
187C22×D9—62no—
188C22×D9—62no—
189C22×D9—62no—
190C22×D9—122no—
191C22×D9—122no—
192C22×D9C22×D312yes—
193C2×C4×C9—22no—
194C2×C4×C9—22no—
195C2×C4×C9—34no—
196C2×C4×C9—32no—
197C2×C4×C9—42no—
198C2×C4×C9—68no—
199C2×C4×C9C22×D312yes—
200#611—68no—
201#611—68no—
202C2×Dic9—22no—
203C2×Dic9—32no—
204C2×Dic9—62no—
205C2×Dic9—62no—
206C2×Dic9—122no—
207C2×Dic9C22×D312yes—
208D36—68no—
209C2×#121—271noSylow
210C2×C3×C9#12114yescomplemented
211C2×C3×C9C22×D411yescharacteristic
212C2×C3×C9C22×D411yescharacteristic
213C2×C3×C9C2×#4111yescharacteristic
214C3⋊D9—42no—
215D3×C9—24no—
216C3×D9—42no—
217C24×C3—31no—
218C24×C3—91no—
219C23×D3—31no—
220C23×D3—91no—
221C23×D3—91no—
222C23×D3—181no—
223C23×D3C2×D911yescharacteristic
224C22×C3×C4—21no—
225C22×C3×C4—31no—
226C22×C3×C4—34no—
227C22×C3×C4—64no—
228C22×C3×C4—91no—
229C22×C3×C4C2×D911yescharacteristic
230C22×C3×C4D3211yescharacteristic
231C2×#67—94no—
232C2×#67—94no—
233C2×#67—94no—
234C2×#67—94no—
235C2×#67—184no—
236C2×#67—184no—
237C2×#67—184no—
238C2×#67—184no—
239C22×Dic3—31no—
240C22×Dic3—91no—
241C22×Dic3—91no—
242C22×Dic3—181no—
243C22×Dic3C2×D911yescharacteristic
244C2×D12—94no—
245C2×D12—94no—
246C2×D12—184no—
247C2×D12—184no—
248C2×C4×D3—24no—
249C2×C4×D3C2×D914yescomplemented
250C3×C22⋊C4—94no—
251(C2×D3)⋊C4—34no—
252(C2×D3)⋊C4—94no—
253(C2×D3)⋊C4—94no—
254(C2×D3)⋊C4—184no—
255C4⋊Dic3—94no—
256Dic3⋊C4—94no—
257C22×C32C2×D1212yes—
258C22×C32#23512yes—
259C22×C32C2×#6712yes—
260C22×C32C23×D311yescharacteristic
261C2×C3⋊D3—62no—
262C2×C3⋊D3—62no—
263C2×C3⋊D3—62no—
264C2×C3⋊D3—124no—
265C2×C3×D3—28no—
266C2×C3×D3—22no—
267C2×C3×D3—22no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—124no—
272C2×C3×D3C2×D1214yes—
273C32×C4—22no—
274C3⋊Dic3—62no—
275C3×Dic3—62no—
276C3×Dic3C2×#6714yes—
277C22×C9—21no—
278C22×C9—22no—
279C22×C9—22no—
280C22×C9—22no—
281C22×C9—34no—
282C22×C9—68no—
283C22×C9—62no—
284C22×C9—62no—
285C22×C9#23512yes—
286C22×C9D3×D412yes—
287C22×C9D3×D412yes—
288C22×C9C23×D311yescharacteristic
289C2×D9—22no—
290C2×D9—22no—
291C2×D9—22no—
292C2×D9—44no—
293C2×D9—64no—
294C2×D9—62no—
295C2×D9—62no—
296C2×D9—62no—
297C2×D9—122no—
298C2×D9—122no—
299C2×D9—128no—
300C2×D9—124no—
301C4×C9—22no—
302C4×C9—34no—
303C4×C9—42no—
304Dic9—22no—
305Dic9—62no—
306Dic9—122no—
307C22×D4—271no—
308C22×D4—271no—
309C22×D4—271no—
310C23×C4—31no—
311#121—278no—
312C2×C4⋊C4—271no—
313C2×C22⋊C4—91no—
314C2×C22⋊C4—271no—
315C3×C9C2×#12111yescharacteristic, complemented, Sylow
316C23×C3—21no—
317C23×C3—32no—
318C23×C3—34no—
319C23×C3—62no—
320C23×C3—62no—
321C23×C3—92no—
322C23×C3—182no—
323C23×C3—182no—
324C23×C3—182no—
325C23×C3C22×D911yescharacteristic
326C23×C3C2×D3211yescharacteristic
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—32no—
330C22×D3—62no—
331C22×D3—62no—
332C22×D3—62no—
333C22×D3—92no—
334C22×D3—92no—
335C22×D3—182no—
336C22×D3—184no—
337C22×D3—182no—
338C22×D3—182no—
339C22×D3—182no—
340C22×D3—182no—
341C22×D3—182no—
342C22×D3—182no—
343C22×D3—362no—
344C22×D3—362no—
345C22×D3D3614yescomplemented
346C22×D3C22×D912yes—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—32no—
351C2×C3×C4—34no—
352C2×C3×C4—42no—
353C2×C3×C4—68no—
354C2×C3×C4—92no—
355C2×C3×C4—182no—
356C2×C3×C4C22×D912yes—
357C2×C3×C4C2×D3212yes—
358#67—188no—
359#67—188no—
360#67—188no—
361#67—188no—
362C2×Dic3—32no—
363C2×Dic3—62no—
364C2×Dic3—92no—
365C2×Dic3—92no—
366C2×Dic3—182no—
367C2×Dic3—182no—
368C2×Dic3—182no—
369C2×Dic3—362no—
370C2×Dic3#61114yescomplemented
371C2×Dic3C22×D912yes—
372D12—188no—
373D12—188no—
374C4×D3—28no—
375C2×C32#93014yes—
376C2×C32C22×D1211yescharacteristic
377C2×C32C2×#23511yescharacteristic
378C2×C32C22×#6711yescharacteristic
379C3⋊D3—122no—
380C3×D3—24no—
381C3×D3—122no—
382C2×C9—21no—
383C2×C9—21no—
384C2×C9—21no—
385C2×C9—24no—
386C2×C9—64no—
387C2×C9#88414yescomplemented
388C2×C9C2×#23511yescharacteristic
389C2×C9C2×D3×D411yescharacteristic
390C2×C9C2×D3×D411yescharacteristic
391D9—42no—
392D9—124no—
393D9—122no—
394C24—31no—
395C24—91no—
396C24—271no—
397C2×D4—274no—
398C2×D4—274no—
399C2×D4—274no—
400C2×D4—544no—
401C2×D4—544no—
402C2×D4—544no—
403C22×C4—31no—
404C22×C4—34no—
405C22×C4—64no—
406C22×C4—91no—
407C22×C4—271no—
408C22×C4D3×D911yescharacteristic
409C4⋊C4—274no—
410C22⋊C4—94no—
411C22⋊C4—274no—
412C22×C3—22no—
413C22×C3—21no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—34no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3—68no—
420C22×C3—182no—
421C22×C3—182no—
422C22×C3—182no—
423C22×C3—364no—
424C22×C3C2×D3612yes—
425C22×C3#369712yes—
426C22×C3C2×#61112yes—
427C22×C3C23×D911yescharacteristic
428C22×C3#379812yes—
429C22×C3D3×D1212yes—
430C22×C3D3×#6712yes—
431C22×C3C22×D3211yescharacteristic
432C2×D3—28no—
433C2×D3—22no—
434C2×D3—22no—
435C2×D3—62no—
436C2×D3—62no—
437C2×D3—62no—
438C2×D3—124no—
439C2×D3—184no—
440C2×D3—182no—
441C2×D3—182no—
442C2×D3—182no—
443C2×D3—182no—
444C2×D3—182no—
445C2×D3—182no—
446C2×D3—362no—
447C2×D3—362no—
448C2×D3—368no—
449C2×D3—364no—
450C2×D3—364no—
451C2×D3C2×D3614yescomplemented
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—34no—
455C3×C4—42no—
456C3×C4—182no—
457Dic3—62no—
458Dic3—182no—
459Dic3—182no—
460Dic3—362no—
461Dic3C2×#61114yescomplemented
462C32C2×#93011yescharacteristic
463C9—21no—
464C9C2×#88411yescharacteristic, complemented
465C23—32no—
466C23—34no—
467C23—62no—
468C23—62no—
469C23—92no—
470C23—182no—
471C23—182no—
472C23—182no—
473C23—272no—
474C23—542no—
475C23—542no—
476C23—542no—
477C23C2×D3×D911yescharacteristic
478D4—548no—
479D4—548no—
480D4—548no—
481C2×C4—22no—
482C2×C4—32no—
483C2×C4—34no—
484C2×C4—68no—
485C2×C4—92no—
486C2×C4—182no—
487C2×C4—272no—
488C2×C4—542no—
489C2×C4C2×D3×D912yes—
490C2×C3—21no—
491C2×C3—24no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—64no—
495C2×C3—362no—
496C2×C3#6349214yescomplemented
497C2×C3C22×D3611yescharacteristic
498C2×C3C2×#369711yescharacteristic
499C2×C3C22×#61111yescharacteristic
500C2×C3#6390614yes—
501C2×C3C2×#379811yescharacteristic
502C2×C3C2×D3×D1211yescharacteristic
503C2×C3C2×D3×#6711yescharacteristic
504D3—24no—
505D3—122no—
506D3—364no—
507D3—362no—
508D3—362no—
509C22—34no—
510C22—62no—
511C22—68no—
512C22—62no—
513C22—182no—
514C22—182no—
515C22—182no—
516C22—364no—
517C22—542no—
518C22—542no—
519C22—542no—
520C22—1084no—
521C22#8864012yes—
522C22D3×D3612yes—
523C22D3×#61112yes—
524C22C22×D3×D911yescharacteristic
525C4—22no—
526C4—34no—
527C4—182no—
528C4—542no—
529C3—21no—
530C3C2×#6349211yescharacteristic, complemented
531C3C2×#6390611yescharacteristic
532C2—64no—
533C2—362no—
534C2—1082no—
535C2#1172103414yesdirect product factor
536C2C2×#8864011yescharacteristic
537C2C2×D3×D3611yescharacteristic
538C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#121
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×D3×C9C2semidirect
2. C22×#6195C2semidirect
3. C22×#6194C2semidirect
4. C22×#6192C2semidirect
5. C2×#88536C2semidirect
6. C2×C3×(C2×D9)⋊C4C2semidirect
7. C2×Dic3⋊Dic9C2semidirect
8. #11721034C2direct (N × H)
19. C2×C4×D3×C9C22semidirect
21. C2×#6195C22semidirect
23. C2×#6194C22semidirect
25. C2×#6192C22semidirect
26. #88536C22semidirect
27. C3×(C2×D9)⋊C4C22semidirect
28. Dic3⋊Dic9C22semidirect
44. C2×(C2×D9)⋊C4D3semidirect
53. C22×D3×C9D4semidirect
63. C2×C9×Dic3D4semidirect
111. (C2×D9)⋊C4C2×D3semidirect
124. C2×D3×C9C2×D4semidirect
131. C9×Dic3C2×D4semidirect
140. C22×C4×D3D9semidirect
210. C2×C3×C9#121semidirect
249. C2×C4×D3C2×D9semidirect
315. C3×C9C2×#121semidirect
345. C22×D3D36semidirect
370. C2×Dic3#611semidirect
387. C2×C9#884semidirect
451. C2×D3C2×D36semidirect
461. Dic3C2×#611semidirect
464. C9C2×#884semidirect
496. C2×C3#63492semidirect
530. C3C2×#63492semidirect
535. C2#11721034direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D3×C92
C22×#61952
C22×#61942
C22×#61922
C2×#885362
C2×C3×(C2×D9)⋊C42
C2×Dic3⋊Dic92
#117210342
C2×#639063
C2×#634923