Finite Group Databasebeta

← C2×#11721041 #13150470

Subgroups — C2×#11721041 #13150470

Order 1728.

1514 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊D9C211yesmaximal, characteristic, complemented
2C22×#6195C211yesmaximal, characteristic, complemented
3C22×#6194C211yesmaximal, characteristic, complemented
4C22×#6192C211yesmaximal, characteristic, complemented
5C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
7C2×#88442C211yesmaximal, characteristic, complemented
8#11721041C218yesmaximal, direct product factor
9C2×#63912—31nomaximal
10C2×#63446—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C23×D3×C9C2211yescharacteristic
13C23×C3×D9C2211yescharacteristic
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C2×C4×C3⋊D9—24no—
17C2×C4×C3⋊D9C2214yescomplemented
18C22×C9×Dic3C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×#6195—24no—
21C2×#6195C2214yescomplemented
22C2×#6194—24no—
23C2×#6194C2214yescomplemented
24C2×#6192—24no—
25C2×#6192C2214yescomplemented
26C9×(C2×D3)⋊C4C2214yescomplemented
27C3×(C2×D9)⋊C4C2214yescomplemented
28#88442C2214yescomplemented
29C22×C4×C3⋊D3—31no—
30C22×#626—31no—
31C22×#626—31no—
32C22×#625—31no—
33C2×C3×(C2×D3)⋊C4—31no—
34C2×C3×(C2×D3)⋊C4D311yescharacteristic
35C2×#3724—31no—
36#63912—38no—
37C22×#611—31no—
38C22×#611—31no—
39C22×D36—31no—
40C22×C4×D9—31no—
41C22×C4×D9—61no—
42C2×C9×C22⋊C4—31no—
43C2×(C2×D9)⋊C4D311yescharacteristic, complemented
44C2×Dic9⋊C4—31no—
45#63446—38no—
46C23×C3×C9C2311yescharacteristic
47C22×C3⋊D9—22no—
48C22×C3⋊D9—22no—
49C22×C3⋊D9D414yescomplemented
50C22×C3⋊D9C2312yes—
51C22×D3×C9—22no—
52C22×D3×C9—22no—
53C22×D3×C9—22no—
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⋊Dic9D414yescomplemented
62C2×C3⋊Dic9C2312yes—
63C4×C3⋊D9—28no—
64C2×C9×Dic3—22no—
65C2×C9×Dic3C2312yes—
66C2×C3×Dic9—22no—
67C2×C3×Dic9C2312yes—
68#6195—28no—
69#6194—28no—
70#6192—28no—
71C2×#884—91no—
72C2×#884—91no—
73C23×C3⋊D3—31no—
74C23×C3×D3—31no—
75C23×C3×D3C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C22×C3⋊Dic3—31no—
78C2×C4×C3⋊D3—34no—
79C2×C4×C3⋊D3—64no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3C2×D311yescharacteristic
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
92C3×(C2×D3)⋊C4—34no—
93C3×(C2×D3)⋊C4C2×D314yes—
94#3724—34no—
95C22×C4×C9—21no—
96C22×C4×C9—31no—
97C22×C4×C9C2×D311yescharacteristic
98C2×#611—34no—
99C2×#611—34no—
100C2×#611—64no—
101C2×#611—64no—
102C22×Dic9—31no—
103C22×Dic9—61no—
104C22×Dic9C2×D311yescharacteristic
105C2×D36—34no—
106C2×D36—64no—
107C2×C4×D9—34no—
108C2×C4×D9—64no—
109C2×C4×D9—64no—
110C2×C4×D9—68no—
111C9×C22⋊C4—34no—
112(C2×D9)⋊C4C2×D314yescomplemented
113Dic9⋊C4—34no—
114C22×C3×C9C2×D412yes—
115C22×C3×C9C2×D412yes—
116C22×C3×C9#4112yes—
117C22×C3×C9C2411yescharacteristic
118C2×C3⋊D9—28no—
119C2×C3⋊D9—22no—
120C2×C3⋊D9—22no—
121C2×C3⋊D9C2×D414yescomplemented
122C2×D3×C9—22no—
123C2×D3×C9—22no—
124C2×D3×C9—22no—
125C2×D3×C9—44no—
126C2×C3×D9—22no—
127C2×C3×D9—22no—
128C2×C3×D9—22no—
129C2×C3×D9—44no—
130C3×C4×C9—22no—
131C3⋊Dic9C2×D414yescomplemented
132C9×Dic3—22no—
133C3×Dic9—22no—
134C22×#67—91no—
135C22×#67—91no—
136C22×#67—91no—
137C22×#67—91no—
138C22×D12—91no—
139C22×D12—91no—
140C22×C4×D3—91no—
141C22×C4×D3—91no—
142C22×C4×D3—181no—
143C2×C3×C22⋊C4—31no—
144C2×C3×C22⋊C4—91no—
145C2×(C2×D3)⋊C4—31no—
146C2×(C2×D3)⋊C4D911yescharacteristic, complemented
147C2×Dic3⋊C4—91no—
148C2×Dic3⋊C4—91no—
149#884—98no—
150#884—98no—
151C23×C32C22×D311yescharacteristic
152C22×C3⋊D3—34no—
153C22×C3⋊D3—32no—
154C22×C3⋊D3—62no—
155C22×C3⋊D3—62no—
156C22×C3×D3—22no—
157C22×C3×D3—22no—
158C22×C3×D3—22no—
159C22×C3×D3—32no—
160C22×C3×D3—62no—
161C22×C3×D3—62no—
162C22×C3×D3—62no—
163C22×C3×D3C22×D312yes—
164C2×C32×C4—22no—
165C2×C32×C4C22×D312yes—
166C2×C3⋊Dic3—34no—
167C2×C3⋊Dic3—32no—
168C4×C3⋊D3—68no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3C22×D312yes—
173#626—68no—
174#626—68no—
175#625—68no—
176C23×C9—21no—
177C23×C9—32no—
178C23×C9—62no—
179C23×C9—62no—
180C23×C9—62no—
181C23×C9C22×D311yescharacteristic
182C22×D9—22no—
183C22×D9—22no—
184C22×D9—22no—
185C22×D9—34no—
186C22×D9—32no—
187C22×D9—62no—
188C22×D9—64no—
189C22×D9—64no—
190C22×D9—64no—
191C22×D9—62no—
192C22×D9—62no—
193C22×D9C22×D312yes—
194C2×C4×C9—24no—
195C2×C4×C9—22no—
196C2×C4×C9—22no—
197C2×C4×C9—32no—
198C2×C4×C9—62no—
199C2×C4×C9C22×D312yes—
200#611—68no—
201#611—68no—
202C2×Dic9—22no—
203C2×Dic9—34no—
204C2×Dic9—32no—
205C2×Dic9—64no—
206C2×Dic9—62no—
207C2×Dic9C22×D312yes—
208D36—68no—
209C4×D9—68no—
210C4×D9—616no—
211C2×#121—271noSylow
212C2×C3×C9#12114yescomplemented
213C2×C3×C9C22×D411yescharacteristic
214C2×C3×C9C22×D411yescharacteristic
215C2×C3×C9C2×#4111yescharacteristic
216C3⋊D9—24no—
217D3×C9—42no—
218C3×D9—42no—
219C24×C3—31no—
220C24×C3—91no—
221C23×D3—31no—
222C23×D3—91no—
223C23×D3—91no—
224C23×D3—181no—
225C23×D3C2×D911yescharacteristic
226C22×C3×C4—21no—
227C22×C3×C4—31no—
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—94no—
249C2×C4×D3—94no—
250C2×C4×D3—184no—
251C2×C4×D3—184no—
252C2×C4×D3—184no—
253C2×C4×D3—188no—
254C3×C22⋊C4—34no—
255C3×C22⋊C4—94no—
256(C2×D3)⋊C4—34no—
257(C2×D3)⋊C4C2×D914yescomplemented
258Dic3⋊C4—94no—
259Dic3⋊C4—94no—
260C22×C32#23512yes—
261C22×C32D3×D412yes—
262C22×C32D3×D412yes—
263C22×C32C23×D311yescharacteristic
264C2×C3⋊D3—34no—
265C2×C3⋊D3—68no—
266C2×C3⋊D3—62no—
267C2×C3⋊D3—62no—
268C2×C3×D3—22no—
269C2×C3×D3—22no—
270C2×C3×D3—22no—
271C2×C3×D3—44no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—124no—
276C32×C4—22no—
277C3⋊Dic3—34no—
278C3×Dic3—22no—
279C3×Dic3—62no—
280C22×C9—21no—
281C22×C9—22no—
282C22×C9—22no—
283C22×C9—22no—
284C22×C9—62no—
285C22×C9—62no—
286C22×C9—62no—
287C22×C9—124no—
288C22×C9#23512yes—
289C22×C9D3×D412yes—
290C22×C9D3×D412yes—
291C22×C9C23×D311yescharacteristic
292C2×D9—22no—
293C2×D9—22no—
294C2×D9—22no—
295C2×D9—34no—
296C2×D9—44no—
297C2×D9—64no—
298C2×D9—64no—
299C2×D9—64no—
300C2×D9—616no—
301C2×D9—68no—
302C2×D9—62no—
303C2×D9—62no—
304C4×C9—24no—
305C4×C9—22no—
306C4×C9—62no—
307Dic9—22no—
308Dic9—34no—
309Dic9—64no—
310C22×D4—271no—
311C22×D4—271no—
312C22×D4—271no—
313C23×C4—271no—
314#121—278no—
315C2×C4⋊C4—271no—
316C2×C22⋊C4—31no—
317C2×C22⋊C4—91no—
318C3×C9C2×#12111yescharacteristic, complemented, Sylow
319C23×C3—21no—
320C23×C3—32no—
321C23×C3—62no—
322C23×C3—62no—
323C23×C3—62no—
324C23×C3—92no—
325C23×C3—182no—
326C23×C3—182no—
327C23×C3—182no—
328C23×C3C22×D911yescharacteristic
329C23×C3C2×D3211yescharacteristic
330C22×D3—22no—
331C22×D3—22no—
332C22×D3—22no—
333C22×D3—32no—
334C22×D3—62no—
335C22×D3—62no—
336C22×D3—62no—
337C22×D3—92no—
338C22×D3—94no—
339C22×D3—92no—
340C22×D3—94no—
341C22×D3—182no—
342C22×D3—184no—
343C22×D3—184no—
344C22×D3—184no—
345C22×D3—182no—
346C22×D3—182no—
347C22×D3—182no—
348C22×D3—182no—
349C22×D3C22×D912yes—
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—32no—
355C2×C3×C4—62no—
356C2×C3×C4—92no—
357C2×C3×C4—182no—
358C2×C3×C4C22×D912yes—
359C2×C3×C4C2×D3212yes—
360#67—188no—
361#67—188no—
362#67—188no—
363#67—188no—
364C2×Dic3—22no—
365C2×Dic3—32no—
366C2×Dic3—62no—
367C2×Dic3—92no—
368C2×Dic3—94no—
369C2×Dic3—92no—
370C2×Dic3—94no—
371C2×Dic3—182no—
372C2×Dic3—184no—
373C2×Dic3C22×D912yes—
374D12—188no—
375D12—188no—
376C4×D3—188no—
377C4×D3—188no—
378C4×D3—1816no—
379C2×C32#88414yes—
380C2×C32C2×#23511yescharacteristic
381C2×C32C2×D3×D411yescharacteristic
382C2×C32C2×D3×D411yescharacteristic
383C3⋊D3—64no—
384C3×D3—42no—
385C3×D3—122no—
386C2×C9—21no—
387C2×C9—21no—
388C2×C9—21no—
389C2×C9—24no—
390C2×C9—122no—
391C2×C9#88414yescomplemented
392C2×C9C2×#23511yescharacteristic
393C2×C9C2×D3×D411yescharacteristic
394C2×C9C2×D3×D411yescharacteristic
395D9—42no—
396D9—68no—
397D9—64no—
398C24—31no—
399C24—91no—
400C24—271no—
401C2×D4—274no—
402C2×D4—274no—
403C2×D4—274no—
404C2×D4—544no—
405C2×D4—544no—
406C2×D4—544no—
407C22×C4—31no—
408C22×C4—91no—
409C22×C4—271no—
410C22×C4—274no—
411C22×C4—544no—
412C22×C4D3×D911yescharacteristic
413C4⋊C4—274no—
414C22⋊C4—34no—
415C22⋊C4—94no—
416C22×C3—22no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—62no—
421C22×C3—62no—
422C22×C3—62no—
423C22×C3—124no—
424C22×C3—182no—
425C22×C3—182no—
426C22×C3—182no—
427C22×C3—364no—
428C22×C3#369712yes—
429C22×C3D4×D912yes—
430C22×C3D4×D912yes—
431C22×C3C23×D911yescharacteristic
432C22×C3#379812yes—
433C22×C3#380212yes—
434C22×C3#381112yes—
435C22×C3C22×D3211yescharacteristic
436C2×D3—22no—
437C2×D3—22no—
438C2×D3—22no—
439C2×D3—44no—
440C2×D3—62no—
441C2×D3—62no—
442C2×D3—62no—
443C2×D3—94no—
444C2×D3—94no—
445C2×D3—124no—
446C2×D3—184no—
447C2×D3—184no—
448C2×D3—1816no—
449C2×D3—184no—
450C2×D3—182no—
451C2×D3—182no—
452C2×D3—188no—
453C2×D3—188no—
454C2×D3—182no—
455C2×D3—182no—
456C3×C4—24no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—62no—
460C3×C4—182no—
461Dic3—22no—
462Dic3—62no—
463Dic3—94no—
464Dic3—94no—
465Dic3—184no—
466C32C2×#88411yescharacteristic
467C9—21no—
468C9C2×#88411yescharacteristic, complemented
469C23—32no—
470C23—62no—
471C23—62no—
472C23—62no—
473C23—92no—
474C23—182no—
475C23—182no—
476C23—182no—
477C23—272no—
478C23—274no—
479C23—542no—
480C23—542no—
481C23C2×D3×D911yescharacteristic
482D4—548no—
483D4—548no—
484D4—548no—
485C2×C4—22no—
486C2×C4—32no—
487C2×C4—62no—
488C2×C4—92no—
489C2×C4—182no—
490C2×C4—272no—
491C2×C4—274no—
492C2×C4—548no—
493C2×C4C2×D3×D912yes—
494C2×C3—21no—
495C2×C3—24no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—122no—
499C2×C3—362no—
500C2×C3#6344614yescomplemented
501C2×C3C2×#369711yescharacteristic
502C2×C3C2×D4×D911yescharacteristic
503C2×C3C2×D4×D911yescharacteristic
504C2×C3#6391214yes—
505C2×C3C2×#379811yescharacteristic
506C2×C3C2×#380211yescharacteristic
507C2×C3C2×#381111yescharacteristic
508D3—42no—
509D3—122no—
510D3—188no—
511D3—184no—
512D3—184no—
513C22—62no—
514C22—62no—
515C22—62no—
516C22—124no—
517C22—182no—
518C22—182no—
519C22—182no—
520C22—274no—
521C22—364no—
522C22—542no—
523C22—548no—
524C22—542no—
525C22#8864012yes—
526C22#8864612yes—
527C22#8866812yes—
528C22C22×D3×D911yescharacteristic
529C4—22no—
530C4—62no—
531C4—182no—
532C4—274no—
533C3—21no—
534C3C2×#6344611yescharacteristic, complemented
535C3C2×#6391211yescharacteristic
536C2—122no—
537C2—362no—
538C2—544no—
539C2#1172104114yesdirect product factor
540C2C2×#8864011yescharacteristic
541C2C2×#8864611yescharacteristic
542C2C2×#8866811yescharacteristic
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×C3⋊D9C2semidirect
2. C22×#6195C2semidirect
3. C22×#6194C2semidirect
4. C22×#6192C2semidirect
5. C2×C9×(C2×D3)⋊C4C2semidirect
6. C2×C3×(C2×D9)⋊C4C2semidirect
7. C2×#88442C2semidirect
8. #11721041C2direct (N × H)
17. C2×C4×C3⋊D9C22semidirect
21. C2×#6195C22semidirect
23. C2×#6194C22semidirect
25. C2×#6192C22semidirect
26. C9×(C2×D3)⋊C4C22semidirect
27. C3×(C2×D9)⋊C4C22semidirect
28. #88442C22semidirect
43. C2×(C2×D9)⋊C4D3semidirect
49. C22×C3⋊D9D4semidirect
61. C2×C3⋊Dic9D4semidirect
112. (C2×D9)⋊C4C2×D3semidirect
121. C2×C3⋊D9C2×D4semidirect
131. C3⋊Dic9C2×D4semidirect
146. C2×(C2×D3)⋊C4D9semidirect
212. C2×C3×C9#121semidirect
257. (C2×D3)⋊C4C2×D9semidirect
318. C3×C9C2×#121semidirect
391. C2×C9#884semidirect
468. C9C2×#884semidirect
500. C2×C3#63446semidirect
534. C3C2×#63446semidirect
539. C2#11721041direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D92
C22×#61952
C22×#61942
C22×#61922
C2×C9×(C2×D3)⋊C42
C2×C3×(C2×D9)⋊C42
C2×#884422
#117210412
C2×#639123
C2×#634463