Finite Group Databasebeta

← C2×D3×(C2×D9)⋊C4 #13150520

Subgroups — C2×D3×(C2×D9)⋊C4 #13150520

Order 1728.

2154 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D3×D9C211yesmaximal, characteristic
2C22×C4×D3×C9C211yesmaximal, characteristic, complemented
3C22×D3×Dic9C211yesmaximal, characteristic, complemented
4C2×#88536C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
6C2×#88445C211yesmaximal, characteristic, complemented
7C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
8D3×(C2×D9)⋊C4C218yesmaximal, direct product factor
9C2×D3×(C2×D3)⋊C4—31nomaximal
10C22×(C2×D9)⋊C4—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C23×D3×C9C2211yescharacteristic
13C23×C3×D9C2211yescharacteristic
14C22×D3×D9—24no—
15C22×D3×D9—24no—
16C22×D3×D9C418yescomplemented
17C22×D3×D9C2214yes—
18C22×C3×C4×C9C2211yescharacteristic, complemented
19C22×C3⋊Dic9C2211yescharacteristic, complemented
20C22×C9×Dic3C2211yescharacteristic, complemented
21C22×C3×Dic9C2211yescharacteristic, complemented
22C2×C4×D3×C9—24no—
23C2×C4×D3×C9C2214yescomplemented
24C2×D3×Dic9—24no—
25C2×D3×Dic9C2214yescomplemented
26#88536C2214yescomplemented
27C3×(C2×D9)⋊C4C2214yescomplemented
28#88445C2214yescomplemented
29(C2×D9)⋊Dic3C2214yescomplemented
30C23×D32—31no—
31C22×C3×C4×D3D311yescharacteristic
32C22×D3×Dic3—31no—
33C24×D9—31no—
34C2×#3752—31no—
35C2×C3×(C2×D3)⋊C4—31no—
36C2×#3722—31no—
37C2×(C2×D3)⋊Dic3—31no—
38D3×(C2×D3)⋊C4—38no—
39C23×C4×C9—31no—
40C23×Dic9—31no—
41C2×(C2×D9)⋊C4—31no—
42C2×(C2×D9)⋊C4—31no—
43C2×(C2×D9)⋊C4—31no—
44C2×(C2×D9)⋊C4—38no—
45C2×(C2×D9)⋊C4—61no—
46C2×(C2×D9)⋊C4D311yescharacteristic, direct product factor
47C23×C3×C9C2311yescharacteristic
48C22×C3⋊D9—22no—
49C22×C3⋊D9—22no—
50C22×C3⋊D9C2×C414yescomplemented
51C22×C3⋊D9C2312yes—
52C22×D3×C9C2×C414yes—
53C22×D3×C9D414yes—
54C22×D3×C9D414yes—
55C22×D3×C9C2312yes—
56C22×C3×D9—22no—
57C22×C3×D9—22no—
58C22×C3×D9C2×C414yescomplemented
59C22×C3×D9C2312yes—
60C2×D3×D9—232no—
61C2×D3×D9—28no—
62C2×D3×D9—28no—
63C2×D3×D9C2×C4116yescomplemented
64C2×C3×C4×C9—22no—
65C2×C3×C4×C9C2312yescomplemented
66C2×C3⋊Dic9—22no—
67C2×C3⋊Dic9C2312yescomplemented
68C2×C9×Dic3—22no—
69C2×C9×Dic3C2312yescomplemented
70C2×C3×Dic9—22no—
71C2×C3×Dic9C2312yescomplemented
72C4×D3×C9—28no—
73D3×Dic9—28no—
74C22×(C2×D3)⋊C4—91no—
75C2×D3×C22⋊C4—91no—
76C23×C3⋊D3—31no—
77C23×C3×D3—31no—
78C23×C3×D3C2×D311yescharacteristic
79C22×D32—38no—
80C22×D32—34no—
81C22×D32—64no—
82C22×D32—64no—
83C22×C32×C4C2×D311yescharacteristic
84C22×C3⋊Dic3—31no—
85C22×C3×Dic3—31no—
86C22×C3×Dic3C2×D311yescharacteristic
87C2×C3×C4×D3—24no—
88C2×C3×C4×D3C2×D314yes—
89C2×D3×Dic3—34no—
90C2×D3×Dic3—64no—
91C24×C9—31no—
92C23×D9—31no—
93C23×D9—38no—
94C23×D9—34no—
95C23×D9—64no—
96C23×D9—64no—
97C23×D9—61no—
98C23×D9C2×D311yescharacteristic
99#3752—34no—
100C3×(C2×D3)⋊C4—34no—
101#3722—34no—
102(C2×D3)⋊Dic3—34no—
103C22×C4×C9—21no—
104C22×C4×C9—31no—
105C22×C4×C9—34no—
106C22×C4×C9—64no—
107C22×C4×C9C2×D311yescharacteristic, complemented
108C22×Dic9—31no—
109C22×Dic9—34no—
110C22×Dic9—64no—
111C22×Dic9—61no—
112C22×Dic9C2×D311yescharacteristic, complemented
113(C2×D9)⋊C4—34no—
114(C2×D9)⋊C4—34no—
115(C2×D9)⋊C4—34no—
116(C2×D9)⋊C4—64no—
117(C2×D9)⋊C4C2×D314yesdirect product factor
118C22×C3×C9C22×C412yes—
119C22×C3×C9C2×D412yes—
120C22×C3×C9C2×D412yes—
121C22×C3×C9C2411yescharacteristic
122C2×C3⋊D9—28no—
123C2×C3⋊D9—22no—
124C2×C3⋊D9—22no—
125C2×C3⋊D9C22×C414yescomplemented
126C2×D3×C9C22⋊C4116yescomplemented
127C2×D3×C9C22×C414yes—
128C2×D3×C9C2×D414yes—
129C2×D3×C9C2×D414yes—
130C2×C3×D9—28no—
131C2×C3×D9—22no—
132C2×C3×D9—22no—
133C2×C3×D9C22×C414yescomplemented
134D3×D9—232no—
135C3×C4×C9—22no—
136C3⋊Dic9—22no—
137C9×Dic3—22no—
138C3×Dic9—22no—
139C24×D3—91no—
140C24×D3—91no—
141C23×C3×C4—31no—
142C23×Dic3—91no—
143C22×C4×D3—91no—
144C22×C4×D3D911yescharacteristic, complemented
145C2×C3×C22⋊C4—91no—
146C2×#217—91no—
147C2×(C2×D3)⋊C4—31no—
148C2×(C2×D3)⋊C4—98no—
149C2×(C2×D3)⋊C4—91no—
150C2×(C2×D3)⋊C4—91no—
151C2×(C2×D3)⋊C4—91no—
152C2×(C2×D3)⋊C4—91no—
153C2×(C2×D3)⋊C4—91no—
154C2×(C2×D3)⋊C4—181no—
155D3×C22⋊C4—98no—
156C23×C32C22×D311yescharacteristic
157C22×C3⋊D3—34no—
158C22×C3⋊D3—32no—
159C22×C3⋊D3—62no—
160C22×C3⋊D3—62no—
161C22×C3×D3—34no—
162C22×C3×D3—32no—
163C22×C3×D3—62no—
164C22×C3×D3—62no—
165C22×C3×D3C4×D314yes—
166C22×C3×D3D1214yes—
167C22×C3×D3#6714yes—
168C22×C3×D3C22×D312yes—
169C2×D32—316no—
170C2×D32—632no—
171C2×D32—68no—
172C2×D32—68no—
173C2×C32×C4—22no—
174C2×C32×C4C22×D312yes—
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—62no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3C22×D312yes—
181C3×C4×D3—28no—
182D3×Dic3—68no—
183C23×C9—21no—
184C23×C9—34no—
185C23×C9—34no—
186C23×C9—34no—
187C23×C9—32no—
188C23×C9C22×D311yescharacteristic
189C22×D9—22no—
190C22×D9—22no—
191C22×D9—34no—
192C22×D9—32no—
193C22×D9—316no—
194C22×D9—62no—
195C22×D9—64no—
196C22×D9—632no—
197C22×D9—68no—
198C22×D9—68no—
199C22×D9—62no—
200C22×D9—62no—
201C22×D9—122no—
202C22×D9—122no—
203C22×D9C4×D314yescomplemented
204C22×D9C22×D312yes—
205C2×C4×C9—22no—
206C2×C4×C9—22no—
207C2×C4×C9—32no—
208C2×C4×C9—42no—
209C2×C4×C9—68no—
210C2×C4×C9—62no—
211C2×C4×C9C22×D312yescomplemented
212C2×Dic9—22no—
213C2×Dic9—32no—
214C2×Dic9—62no—
215C2×Dic9—68no—
216C2×Dic9—62no—
217C2×Dic9—122no—
218C2×Dic9C22×D312yescomplemented
219C22×C22⋊C4—271noSylow
220C2×C3×C9C2×C22⋊C414yescomplemented
221C2×C3×C9C23×C411yescharacteristic
222C2×C3×C9C22×D411yescharacteristic
223C2×C3×C9C22×D411yescharacteristic
224C3⋊D9—24no—
225D3×C9C2×C22⋊C418yescomplemented
226C3×D9—24no—
227C24×C3—31no—
228C24×C3—91no—
229C23×D3—31no—
230C23×D3—94no—
231C23×D3—98no—
232C23×D3—91no—
233C23×D3—91no—
234C23×D3—98no—
235C23×D3—94no—
236C23×D3—181no—
237C23×D3—184no—
238C23×D3—184no—
239C23×D3—184no—
240C23×D3—184no—
241C23×D3C2×D911yescharacteristic
242C22×C3×C4—21no—
243C22×C3×C4—31no—
244C22×C3×C4—34no—
245C22×C3×C4—64no—
246C22×C3×C4—91no—
247C22×C3×C4C2×D911yescharacteristic, complemented
248C22×C3×C4D3211yescharacteristic
249C22×Dic3—31no—
250C22×Dic3—94no—
251C22×Dic3—91no—
252C22×Dic3—91no—
253C22×Dic3—181no—
254C22×Dic3—184no—
255C22×Dic3C2×D911yescharacteristic, complemented
256C2×C4×D3—24no—
257C2×C4×D3—94no—
258C2×C4×D3—184no—
259C2×C4×D3C2×D914yescomplemented
260C3×C22⋊C4—94no—
261#217—94no—
262(C2×D3)⋊C4—34no—
263(C2×D3)⋊C4—94no—
264(C2×D3)⋊C4—94no—
265(C2×D3)⋊C4—94no—
266(C2×D3)⋊C4—94no—
267(C2×D3)⋊C4—94no—
268(C2×D3)⋊C4—184no—
269C22×C32C2×C4×D312yes—
270C22×C32C2×D1212yes—
271C22×C32C2×#6712yes—
272C22×C32C23×D311yescharacteristic
273C2×C3⋊D3—34no—
274C2×C3⋊D3—68no—
275C2×C3⋊D3—62no—
276C2×C3⋊D3—62no—
277C2×C3×D3—34no—
278C2×C3×D3—68no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3(C2×D3)⋊C4116yes—
282C2×C3×D3C2×C4×D314yes—
283C2×C3×D3C2×D1214yes—
284C2×C3×D3C2×#6714yes—
285D32—632no—
286C32×C4—22no—
287C3⋊Dic3—62no—
288C3×Dic3—22no—
289C3×Dic3—62no—
290C22×C9—21no—
291C22×C9—22no—
292C22×C9—22no—
293C22×C9—22no—
294C22×C9—316no—
295C22×C9—34no—
296C22×C9—34no—
297C22×C9—34no—
298C22×C9C2×C4×D312yes—
299C22×C9D3×D412yes—
300C22×C9D3×D412yes—
301C22×C9C23×D311yescharacteristic
302C2×D9—28no—
303C2×D9—22no—
304C2×D9—22no—
305C2×D9—34no—
306C2×D9—64no—
307C2×D9—632no—
308C2×D9—68no—
309C2×D9—62no—
310C2×D9—62no—
311C2×D9—122no—
312C2×D9—122no—
313C2×D9—128no—
314C2×D9C2×C4×D314yescomplemented
315C4×C9—22no—
316C4×C9—42no—
317C4×C9—62no—
318Dic9—22no—
319Dic9—62no—
320Dic9—122no—
321C25—271no—
322C23×C4—31no—
323C23×C4—271no—
324C2×C22⋊C4—91no—
325C2×C22⋊C4—278no—
326C2×C22⋊C4—271no—
327C2×C22⋊C4—271no—
328C2×C22⋊C4—271no—
329C3×C9C22×C22⋊C411yescharacteristic, complemented, Sylow
330C23×C3—21no—
331C23×C3—32no—
332C23×C3—34no—
333C23×C3—34no—
334C23×C3—34no—
335C23×C3—94no—
336C23×C3—92no—
337C23×C3—182no—
338C23×C3—182no—
339C23×C3C22×D911yescharacteristic
340C23×C3C2×D3211yescharacteristic
341C22×D3—32no—
342C22×D3—34no—
343C22×D3—62no—
344C22×D3—62no—
345C22×D3—916no—
346C22×D3—92no—
347C22×D3—94no—
348C22×D3—92no—
349C22×D3—94no—
350C22×D3—916no—
351C22×D3—182no—
352C22×D3—184no—
353C22×D3—188no—
354C22×D3—188no—
355C22×D3—1832no—
356C22×D3—182no—
357C22×D3—182no—
358C22×D3—182no—
359C22×D3—182no—
360C22×D3—1832no—
361C22×D3—188no—
362C22×D3—188no—
363C22×D3—362no—
364C22×D3—362no—
365C22×D3C4×D914yes—
366C22×D3D3614yes—
367C22×D3#61114yes—
368C22×D3C22×D912yes—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—32no—
373C2×C3×C4—42no—
374C2×C3×C4—68no—
375C2×C3×C4—62no—
376C2×C3×C4—92no—
377C2×C3×C4—182no—
378C2×C3×C4C22×D912yescomplemented
379C2×C3×C4C2×D3212yes—
380C2×Dic3—22no—
381C2×Dic3—32no—
382C2×Dic3—62no—
383C2×Dic3—92no—
384C2×Dic3—92no—
385C2×Dic3—182no—
386C2×Dic3—188no—
387C2×Dic3—182no—
388C2×Dic3—182no—
389C2×Dic3—362no—
390C2×Dic3C22×D912yescomplemented
391C4×D3—28no—
392C4×D3—188no—
393C2×C32C2×(C2×D3)⋊C414yes—
394C2×C32C22×C4×D311yescharacteristic
395C2×C32C22×D1211yescharacteristic
396C2×C32C22×#6711yescharacteristic
397C3⋊D3—64no—
398C3×D3—64no—
399C3×D3C2×(C2×D3)⋊C418yes—
400C2×C9—21no—
401C2×C9—21no—
402C2×C9—21no—
403C2×C9—24no—
404C2×C9—38no—
405C2×C9D3×C22⋊C414yescomplemented
406C2×C9C22×C4×D311yescharacteristic
407C2×C9C2×D3×D411yescharacteristic
408C2×C9C2×D3×D411yescharacteristic
409D9—24no—
410D9—64no—
411D9—124no—
412C24—31no—
413C24—91no—
414C24—274no—
415C24—278no—
416C24—271no—
417C24—544no—
418C24—544no—
419C22×C4—31no—
420C22×C4—34no—
421C22×C4—64no—
422C22×C4—91no—
423C22×C4—274no—
424C22×C4—271no—
425C22×C4—544no—
426C22×C4D3×D911yescharacteristic, complemented
427C22⋊C4—94no—
428C22⋊C4—274no—
429C22⋊C4—274no—
430C22⋊C4—274no—
431C22×C3—21no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—34no—
436C22×C3—34no—
437C22×C3—34no—
438C22×C3—316no—
439C22×C3—94no—
440C22×C3—188no—
441C22×C3—182no—
442C22×C3—182no—
443C22×C3C2×C4×D912yes—
444C22×C3C2×D3612yes—
445C22×C3C2×#61112yes—
446C22×C3C23×D911yescharacteristic
447C22×C3C4×D3212yes—
448C22×C3D3×D1212yes—
449C22×C3D3×#6712yes—
450C22×C3C22×D3211yescharacteristic
451C2×D3—34no—
452C2×D3—62no—
453C2×D3—62no—
454C2×D3—68no—
455C2×D3—94no—
456C2×D3—94no—
457C2×D3—184no—
458C2×D3—1832no—
459C2×D3—188no—
460C2×D3—182no—
461C2×D3—182no—
462C2×D3—188no—
463C2×D3—182no—
464C2×D3—182no—
465C2×D3—1832no—
466C2×D3—362no—
467C2×D3—368no—
468C2×D3—362no—
469C2×D3(C2×D9)⋊C4116yesdirect product factor
470C2×D3C2×C4×D914yes—
471C2×D3C2×D3614yes—
472C2×D3C2×#61114yes—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4—62no—
477C3×C4—182no—
478Dic3—22no—
479Dic3—62no—
480Dic3—182no—
481Dic3—182no—
482Dic3—362no—
483C32C22×(C2×D3)⋊C411yescharacteristic
484C9—21no—
485C9C2×D3×C22⋊C411yescharacteristic, complemented
486C23—32no—
487C23—34no—
488C23—34no—
489C23—34no—
490C23—92no—
491C23—94no—
492C23—182no—
493C23—182no—
494C23—2716no—
495C23—272no—
496C23—274no—
497C23—548no—
498C23—5432no—
499C23—548no—
500C23—542no—
501C23—542no—
502C23C2×D3×D911yescharacteristic
503C2×C4—22no—
504C2×C4—32no—
505C2×C4—68no—
506C2×C4—62no—
507C2×C4—92no—
508C2×C4—182no—
509C2×C4—272no—
510C2×C4—548no—
511C2×C4—542no—
512C2×C4C2×D3×D912yescomplemented
513C2×C3—21no—
514C2×C3—24no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—38no—
518C2×C3—184no—
519C2×C3C2×(C2×D9)⋊C414yescomplemented
520C2×C3C22×C4×D911yescharacteristic
521C2×C3C22×D3611yescharacteristic
522C2×C3C22×#61111yescharacteristic
523C2×C3D3×(C2×D3)⋊C414yes—
524C2×C3C2×C4×D3211yescharacteristic
525C2×C3C2×D3×D1211yescharacteristic
526C2×C3C2×D3×#6711yescharacteristic
527D3—64no—
528D3—184no—
529D3—184no—
530D3—364no—
531D3C2×(C2×D9)⋊C418yesdirect product factor
532C22—34no—
533C22—316no—
534C22—34no—
535C22—34no—
536C22—94no—
537C22—182no—
538C22—188no—
539C22—182no—
540C22—274no—
541C22—5432no—
542C22—542no—
543C22—548no—
544C22—542no—
545C22C4×D3×D912yes—
546C22D3×D3612yes—
547C22D3×#61112yes—
548C22C22×D3×D911yescharacteristic
549C4—22no—
550C4—62no—
551C4—182no—
552C4—542no—
553C3—21no—
554C3C22×(C2×D9)⋊C411yescharacteristic, complemented
555C3C2×D3×(C2×D3)⋊C411yescharacteristic
556C2—38no—
557C2—184no—
558C2—544no—
559C2D3×(C2×D9)⋊C414yesdirect product factor
560C2C2×C4×D3×D911yescharacteristic
561C2C2×D3×D3611yescharacteristic
562C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×D3×D9C2
cosocleC22×C3×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9C22×C22⋊C4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×D3×C9C2semidirect
3. C22×D3×Dic9C2semidirect
4. C2×#88536C2semidirect
5. C2×C3×(C2×D9)⋊C4C2semidirect
6. C2×#88445C2semidirect
7. C2×(C2×D9)⋊Dic3C2semidirect
8. D3×(C2×D9)⋊C4C2direct (N × H)
16. C22×D3×D9C4semidirect
18. C22×C3×C4×C9C22semidirect
19. C22×C3⋊Dic9C22semidirect
20. C22×C9×Dic3C22semidirect
21. C22×C3×Dic9C22semidirect
23. C2×C4×D3×C9C22semidirect
25. C2×D3×Dic9C22semidirect
26. #88536C22semidirect
27. C3×(C2×D9)⋊C4C22semidirect
28. #88445C22semidirect
29. (C2×D9)⋊Dic3C22semidirect
46. C2×(C2×D9)⋊C4D3direct (N × H)
50. C22×C3⋊D9C2×C4semidirect
58. C22×C3×D9C2×C4semidirect
63. C2×D3×D9C2×C4semidirect
65. C2×C3×C4×C9C23semidirect
67. C2×C3⋊Dic9C23semidirect
69. C2×C9×Dic3C23semidirect
71. C2×C3×Dic9C23semidirect
107. C22×C4×C9C2×D3semidirect
112. C22×Dic9C2×D3semidirect
117. (C2×D9)⋊C4C2×D3direct (N × H)
125. C2×C3⋊D9C22×C4semidirect
126. C2×D3×C9C22⋊C4semidirect
133. C2×C3×D9C22×C4semidirect
144. C22×C4×D3D9semidirect
203. C22×D9C4×D3semidirect
211. C2×C4×C9C22×D3semidirect
218. C2×Dic9C22×D3semidirect
220. C2×C3×C9C2×C22⋊C4semidirect
225. D3×C9C2×C22⋊C4semidirect
247. C22×C3×C4C2×D9semidirect
255. C22×Dic3C2×D9semidirect
259. C2×C4×D3C2×D9semidirect
314. C2×D9C2×C4×D3semidirect
329. C3×C9C22×C22⋊C4semidirect
378. C2×C3×C4C22×D9semidirect
390. C2×Dic3C22×D9semidirect
405. C2×C9D3×C22⋊C4semidirect
426. C22×C4D3×D9semidirect
469. C2×D3(C2×D9)⋊C4direct (N × H)
485. C9C2×D3×C22⋊C4semidirect
512. C2×C4C2×D3×D9semidirect
519. C2×C3C2×(C2×D9)⋊C4semidirect
531. D3C2×(C2×D9)⋊C4direct (N × H)
554. C3C22×(C2×D9)⋊C4semidirect
559. C2D3×(C2×D9)⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D3×D92
C22×C4×D3×C92
C22×D3×Dic92
C2×#885362
C2×C3×(C2×D9)⋊C42
C2×#884452
C2×(C2×D9)⋊Dic32
D3×(C2×D9)⋊C42
C2×D3×(C2×D3)⋊C43
C22×(C2×D9)⋊C43