Finite Group Databasebeta

← C4×D4×C3⋊D9 #13153915

Subgroups — C4×D4×C3⋊D9 #13153915

Order 1728.

2130 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊D9C211yesmaximal, characteristic
2C22×C4×C3⋊D9C212yesmaximal, complemented
3C3×C4×D4×C9C211yesmaximal, characteristic, complemented
4D4×C3⋊Dic9C211yesmaximal, characteristic, complemented
5C4×#6233C212yesmaximal, complemented
6#11721749C211yesmaximal, characteristic, complemented
7C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
8#11721727C212yesmaximal, complemented
9C22⋊C4×C3⋊D9C212yesmaximal, complemented
10C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
11C42×C3⋊D9C211yesmaximal, characteristic, complemented
12C4×D4×C3⋊D3—31nomaximal
13C4×D4×D9—33nomaximal
14C23×C3⋊D9C2212yes—
15C2×C3×D4×C9C2211yescharacteristic
16C22×C3×C4×C9C2212yescomplemented
17C2×#6233C2212yes—
18C22×C3⋊Dic9C2212yescomplemented
19D4×C3⋊D9C418yesdirect product factor
20C2×C4⋊(C3⋊D9)C2211yescharacteristic
21C2×C4×C3⋊D9—28no—
22C2×C4×C3⋊D9C2212yes—
23C2×C4×C3⋊D9C2211yescharacteristic
24C2×C4×C3⋊D9C2211yescharacteristic
25C2×C4×C3⋊D9C2211yescharacteristic
26C2×C4×C3⋊D9C2212yescomplemented
27C2×C4×C3⋊D9C2212yescomplemented
28C3×C9×C4⋊C4C2211yescharacteristic, complemented
29C3×C9×C22⋊C4C2212yescomplemented
30C3×C42×C9C2211yescharacteristic, complemented
31#88551C2212yescomplemented
32#88536C2212yescomplemented
33#88536C2212yescomplemented
34C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
35#88534C2212yescomplemented
36C4×C3⋊Dic9C2211yescharacteristic, complemented
37C4×C3⋊Dic9C2211yescharacteristic, complemented
38C4×C3⋊Dic9C2211yescharacteristic, complemented
39C2×D4×C3⋊D3—31no—
40C22×C4×C3⋊D3—32no—
41C32×C4×D4D311yescharacteristic
42D4×C3⋊Dic3—31no—
43C4×#638—32no—
44#64102—31no—
45C4⋊C4×C3⋊D3—31no—
46#64090—32no—
47C22⋊C4×C3⋊D3—32no—
48C4×C4⋊(C3⋊D3)—31no—
49C42×C3⋊D3—31no—
50C2×D4×D9—33no—
51C22×C4×D9—36no—
52C4×D4×C9D313yescomplemented
53D4×Dic9—33no—
54C4×#611—36no—
55#63455—33no—
56C4⋊C4×D9—33no—
57#611⋊C4—36no—
58C22⋊C4×D9—36no—
59C4×D36—33no—
60C42×D9—33no—
61C23×C3×C9C2312yes—
62C22×C3⋊D9—24no—
63C22×C3⋊D9—24no—
64C22×C3⋊D9C2×C418yescomplemented
65C22×C3⋊D9C2312yes—
66C22×C3⋊D9C2311yescharacteristic
67C3×D4×C9C2×C414yescomplemented
68C2×C3×C4×C9—24no—
69C2×C3×C4×C9C2312yes—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72C2×C3×C4×C9C2311yescharacteristic
73#6233C2×C418yescomplemented
74C2×C3⋊Dic9—24no—
75C2×C3⋊Dic9C2312yes—
76C2×C3⋊Dic9C2311yescharacteristic
77C2×C3⋊Dic9C2311yescharacteristic
78C2×C3⋊Dic9C2311yescharacteristic
79C4⋊(C3⋊D9)C2×C414yescomplemented
80C4×C3⋊D9—24no—
81C4×C3⋊D9—22no—
82C4×C3⋊D9—24no—
83C4×C3⋊D9—24no—
84C4×C3⋊D9C2×C414yescomplemented
85C4×C3⋊D9D414yesdirect product factor
86C4×D3×D4—93no—
87C4×D3×D4—91no—
88C23×C3⋊D3—32no—
89C2×C32×D4C2×D311yescharacteristic
90C22×C32×C4C2×D312yes—
91C2×#638—32no—
92C22×C3⋊Dic3—32no—
93D4×C3⋊D3—38no—
94C2×C4⋊(C3⋊D3)—31no—
95C2×C4×C3⋊D3—31no—
96C2×C4×C3⋊D3—31no—
97C2×C4×C3⋊D3—31no—
98C2×C4×C3⋊D3—32no—
99C2×C4×C3⋊D3—32no—
100C2×C4×C3⋊D3—32no—
101C2×C4×C3⋊D3—68no—
102C23×D9—36no—
103C32×C4⋊C4C2×D311yescharacteristic
104C32×C22⋊C4C2×D312yes—
105C32×C42C2×D311yescharacteristic
106#3757—32no—
107#3752—32no—
108#3752—32no—
109C4⋊(C3⋊Dic3)—31no—
110#3750—32no—
111C4×C3⋊Dic3—31no—
112C4×C3⋊Dic3—31no—
113C4×C3⋊Dic3—31no—
114C2×D4×C9C2×D313yes—
115C22×C4×C9C2×D316yescomplemented
116C2×#611—36no—
117C22×Dic9—36no—
118D4×D9—324no—
119C2×D36—33no—
120C2×C4×D9—33no—
121C2×C4×D9—33no—
122C2×C4×D9—36no—
123C2×C4×D9—33no—
124C2×C4×D9—36no—
125C2×C4×D9—36no—
126C2×C4×D9—624no—
127C9×C4⋊C4C2×D313yescomplemented
128C9×C22⋊C4C2×D316yescomplemented
129C42×C9C2×D313yescomplemented
130#3676—36no—
131(C2×D9)⋊C4—36no—
132(C2×D9)⋊C4—36no—
133C4⋊Dic9—33no—
134Dic9⋊C4—36no—
135C4×Dic9—33no—
136C4×Dic9—33no—
137C4×Dic9—33no—
138C22×C3×C9—22no—
139C22×C3×C9—22no—
140C22×C3×C9C22×C414yescomplemented
141C22×C3×C9C2411yescharacteristic
142C2×C3⋊D9—22no—
143C2×C3⋊D9—22no—
144C2×C3⋊D9—216no—
145C2×C3⋊D9C22×C412yes—
146C2×C3⋊D9C22×C414yescomplemented
147C2×C3⋊D9C2×D412yes—
148C2×C3⋊D9#4112yes—
149C3×C4×C9—22no—
150C3×C4×C9—21no—
151C3×C4×C9C22×C412yescomplemented
152C3×C4×C9C2×D412yescomplemented
153C3⋊Dic9—22no—
154C3⋊Dic9—21no—
155C3⋊Dic9C22×C412yescomplemented
156C3⋊Dic9C2×D412yescomplemented
157C2×D3×D4—93no—
158C2×D3×D4—91no—
159C22×C4×D3—96no—
160C22×C4×D3—92no—
161C3×C4×D4D913yescomplemented
162C3×C4×D4C3⋊D311yescharacteristic
163D4×Dic3—93no—
164D4×Dic3—91no—
165C4×#67—96no—
166C4×#67—92no—
167#893—93no—
168#893—91no—
169D3×C4⋊C4—93no—
170D3×C4⋊C4—91no—
171#67⋊C4—96no—
172#67⋊C4—92no—
173D3×C22⋊C4—96no—
174D3×C22⋊C4—92no—
175C4×D12—93no—
176C4×D12—91no—
177C42×D3—93no—
178C42×D3—91no—
179C23×C32C22×D312yes—
180C22×C3⋊D3—31no—
181C22×C3⋊D3—38no—
182C22×C3⋊D3—32no—
183C22×C3⋊D3—64no—
184C22×C3⋊D3—64no—
185C32×D4C4×D314yes—
186C2×C32×C4—24no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D312yes—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191#638—38no—
192C2×C3⋊Dic3—31no—
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—32no—
196C2×C3⋊Dic3—64no—
197C4⋊(C3⋊D3)—34no—
198C4×C3⋊D3—34no—
199C4×C3⋊D3—34no—
200C4×C3⋊D3—64no—
201C4×C3⋊D3—62no—
202C4×C3⋊D3—64no—
203C4×C3⋊D3—64no—
204C23×C9C22×D316yes—
205C22×D9—33no—
206C22×D9—324no—
207C22×D9—36no—
208C22×D9—612no—
209C22×D9—612no—
210D4×C9C4×D3112yescomplemented
211C2×C4×C9—212no—
212C2×C4×C9C22×D313yes—
213C2×C4×C9C22×D316yes—
214C2×C4×C9C22×D313yes—
215C2×C4×C9C22×D313yes—
216#611—324no—
217C2×Dic9—33no—
218C2×Dic9—33no—
219C2×Dic9—33no—
220C2×Dic9—36no—
221C2×Dic9—612no—
222D36—312no—
223C4×D9—312no—
224C4×D9—312no—
225C4×D9—612no—
226C4×D9—66no—
227C4×D9—612no—
228C4×D9—612no—
229C2×C4×D4—271noSylow
230C2×C3×C9—24no—
231C2×C3×C9C23×C411yescharacteristic
232C2×C3×C9C22×D411yescharacteristic
233C2×C3×C9C2×#4111yescharacteristic
234C3⋊D9—24no—
235C3⋊D9C4×D414yesdirect product factor
236C23×D3—96no—
237C23×D3—92no—
238C2×C3×D4C2×D913yes—
239C2×C3×D4C2×C3⋊D311yescharacteristic
240C22×C3×C4C2×D916yescomplemented
241C22×C3×C4C2×C3⋊D312yes—
242C2×#67—96no—
243C2×#67—92no—
244C22×Dic3—96no—
245C22×Dic3—92no—
246D3×D4—924no—
247D3×D4—98no—
248C2×D12—93no—
249C2×D12—91no—
250C2×C4×D3—93no—
251C2×C4×D3—93no—
252C2×C4×D3—91no—
253C2×C4×D3—91no—
254C2×C4×D3—93no—
255C2×C4×D3—96no—
256C2×C4×D3—96no—
257C2×C4×D3—96no—
258C2×C4×D3—91no—
259C2×C4×D3—92no—
260C2×C4×D3—92no—
261C2×C4×D3—92no—
262C2×C4×D3—188no—
263C2×C4×D3—1824no—
264C3×C4⋊C4C2×D913yescomplemented
265C3×C4⋊C4C2×C3⋊D311yescharacteristic
266C3×C22⋊C4C2×D916yescomplemented
267C3×C22⋊C4C2×C3⋊D312yes—
268C3×C42C2×D913yescomplemented
269C3×C42C2×C3⋊D311yescharacteristic
270#217—96no—
271#217—92no—
272(C2×D3)⋊C4—96no—
273(C2×D3)⋊C4—96no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276C4⋊Dic3—93no—
277C4⋊Dic3—91no—
278Dic3⋊C4—96no—
279Dic3⋊C4—92no—
280C4×Dic3—93no—
281C4×Dic3—93no—
282C4×Dic3—93no—
283C4×Dic3—91no—
284C4×Dic3—91no—
285C4×Dic3—91no—
286C22×C32—22no—
287C22×C32—22no—
288C22×C32C2×C4×D314yes—
289C22×C32C23×D311yescharacteristic
290C2×C3⋊D3—32no—
291C2×C3⋊D3—32no—
292C2×C3⋊D3—32no—
293C2×C3⋊D3—34no—
294C2×C3⋊D3—616no—
295C2×C3⋊D3—62no—
296C2×C3⋊D3—62no—
297C32×C4—22no—
298C32×C4—21no—
299C32×C4C2×C4×D312yes—
300C32×C4D3×D412yes—
301C3⋊Dic3—32no—
302C3⋊Dic3—32no—
303C3⋊Dic3—62no—
304C3⋊Dic3—61no—
305C22×C9—26no—
306C22×C9—26no—
307C22×C9C2×C4×D3112yescomplemented
308C22×C9C23×D313yes—
309C2×D9—36no—
310C2×D9—36no—
311C2×D9—36no—
312C2×D9—312no—
313C2×D9—648no—
314C2×D9—66no—
315C2×D9—66no—
316C4×C9—26no—
317C4×C9—23no—
318C4×C9C2×C4×D316yescomplemented
319C4×C9D3×D416yescomplemented
320Dic9—36no—
321Dic9—36no—
322Dic9—66no—
323Dic9—63no—
324C22×D4—271no—
325C23×C4—272no—
326C4×D4—271no—
327C4×D4—272no—
328C4×D4—271no—
329C4×D4—272no—
330C4×D4—271no—
331C4×D4C3⋊D911yescharacteristic, direct product factor
332C2×C4⋊C4—271no—
333C2×C22⋊C4—272no—
334C2×C42—271no—
335C3×C9C2×C4×D411yescharacteristic, complemented, Sylow
336C23×C3C22×D916yes—
337C23×C3C22×C3⋊D312yes—
338C22×D3—91no—
339C22×D3—93no—
340C22×D3—92no—
341C22×D3—924no—
342C22×D3—96no—
343C22×D3—98no—
344C22×D3—184no—
345C22×D3—184no—
346C22×D3—1812no—
347C22×D3—1812no—
348C3×D4C4×D9112yescomplemented
349C3×D4C4×C3⋊D314yes—
350C2×C3×C4—212no—
351C2×C3×C4—24no—
352C2×C3×C4C22×D913yes—
353C2×C3×C4C22×D916yes—
354C2×C3×C4C22×D913yes—
355C2×C3×C4C22×D913yes—
356C2×C3×C4C22×C3⋊D311yescharacteristic
357C2×C3×C4C22×C3⋊D312yes—
358C2×C3×C4C22×C3⋊D311yescharacteristic
359C2×C3×C4C22×C3⋊D311yescharacteristic
360#67—924no—
361#67—98no—
362C2×Dic3—91no—
363C2×Dic3—91no—
364C2×Dic3—93no—
365C2×Dic3—93no—
366C2×Dic3—91no—
367C2×Dic3—92no—
368C2×Dic3—93no—
369C2×Dic3—96no—
370C2×Dic3—184no—
371C2×Dic3—1812no—
372D12—912no—
373D12—94no—
374C4×D3—912no—
375C4×D3—912no—
376C4×D3—94no—
377C4×D3—94no—
378C4×D3—184no—
379C4×D3—1812no—
380C4×D3—182no—
381C4×D3—184no—
382C4×D3—186no—
383C4×D3—1812no—
384C4×D3—1812no—
385C4×D3—184no—
386C2×C32—24no—
387C2×C32C22×C4×D311yescharacteristic
388C2×C32C2×D3×D411yescharacteristic
389C2×C32D3×#4111yescharacteristic
390C3⋊D3—34no—
391C3⋊D3—64no—
392C2×C9—212no—
393C2×C9C22×C4×D313yes—
394C2×C9C2×D3×D413yes—
395C2×C9D3×#4113yes—
396D9—312no—
397D9—612no—
398C24—272no—
399C2×D4—272no—
400C2×D4—278no—
401C2×D4—271no—
402C2×D4C2×C3⋊D911yescharacteristic
403C22×C4—271no—
404C22×C4—271no—
405C22×C4—272no—
406C22×C4—271no—
407C22×C4—272no—
408C22×C4—272no—
409C22×C4—272no—
410C22×C4—548no—
411C22×C4C2×C3⋊D912yescomplemented
412C4⋊C4—272no—
413C4⋊C4—271no—
414C4⋊C4C2×C3⋊D911yescharacteristic, complemented
415C22⋊C4—272no—
416C22⋊C4—272no—
417C22⋊C4—272no—
418C22⋊C4C2×C3⋊D912yescomplemented
419C42—271no—
420C42—271no—
421C42—271no—
422C42C2×C3⋊D911yescharacteristic, complemented
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—26no—
426C22×C3—26no—
427C22×C3C2×C4×D9112yescomplemented
428C22×C3C23×D913yes—
429C22×C3C2×C4×C3⋊D314yes—
430C22×C3C23×C3⋊D311yescharacteristic
431C2×D3—92no—
432C2×D3—92no—
433C2×D3—92no—
434C2×D3—96no—
435C2×D3—96no—
436C2×D3—96no—
437C2×D3—94no—
438C2×D3—912no—
439C2×D3—1816no—
440C2×D3—1848no—
441C2×D3—182no—
442C2×D3—182no—
443C2×D3—186no—
444C2×D3—186no—
445C3×C4—22no—
446C3×C4—21no—
447C3×C4—26no—
448C3×C4—23no—
449C3×C4C2×C4×D916yescomplemented
450C3×C4D4×D916yescomplemented
451C3×C4C2×C4×C3⋊D312yes—
452C3×C4D4×C3⋊D312yes—
453Dic3—92no—
454Dic3—92no—
455Dic3—96no—
456Dic3—96no—
457Dic3—182no—
458Dic3—181no—
459Dic3—186no—
460Dic3—183no—
461C32C4×D3×D411yescharacteristic
462C9C4×D3×D413yescomplemented
463C23—271no—
464C23—272no—
465C23—278no—
466C23—544no—
467C23—544no—
468C23C22×C3⋊D912yes—
469D4—278no—
470D4—274no—
471D4C4×C3⋊D914yesdirect product factor
472C2×C4—24no—
473C2×C4—271no—
474C2×C4—271no—
475C2×C4—274no—
476C2×C4—274no—
477C2×C4—271no—
478C2×C4—272no—
479C2×C4—544no—
480C2×C4—544no—
481C2×C4—542no—
482C2×C4—544no—
483C2×C4—544no—
484C2×C4C22×C3⋊D911yescharacteristic
485C2×C4C22×C3⋊D912yes—
486C2×C4C22×C3⋊D911yescharacteristic
487C2×C4C22×C3⋊D911yescharacteristic
488C2×C3—24no—
489C2×C3—212no—
490C2×C3C22×C4×D913yes—
491C2×C3C2×D4×D913yes—
492C2×C3#41×D913yes—
493C2×C3C22×C4×C3⋊D311yescharacteristic
494C2×C3C2×D4×C3⋊D311yescharacteristic
495C2×C3#41×C3⋊D311yescharacteristic
496D3—94no—
497D3—912no—
498D3—184no—
499D3—1812no—
500C22—22no—
501C22—22no—
502C22—272no—
503C22—272no—
504C22—272no—
505C22—274no—
506C22—5416no—
507C22—542no—
508C22—542no—
509C22C2×C4×C3⋊D914yescomplemented
510C22C23×C3⋊D911yescharacteristic
511C4—22no—
512C4—21no—
513C4—272no—
514C4—272no—
515C4—542no—
516C4—541no—
517C4C2×C4×C3⋊D912yescomplemented
518C4D4×C3⋊D912yesdirect product factor
519C3C4×D4×D913yescomplemented
520C3C4×D4×C3⋊D311yescharacteristic
521C2—24no—
522C2—274no—
523C2—544no—
524C2C22×C4×C3⋊D911yescharacteristic
525C2C2×D4×C3⋊D911yescharacteristic
526C2#41×C3⋊D911yescharacteristic
subgroupG/H
centerC2×C4C22×C3⋊D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C23×C3⋊D3
FittingC3×C4×D4×C9C2
socleC22×C32C23×D3
prime radicalC2×D4×C3⋊D9C2
cosocleC22×C3×C9C24
hypercenterC4×D4C3⋊D9
hypocenterC3×C9C2×C4×D4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C22×C4×C3⋊D9C2semidirect
3. C3×C4×D4×C9C2semidirect
4. D4×C3⋊Dic9C2semidirect
5. C4×#6233C2semidirect
6. #11721749C2semidirect
7. C4⋊C4×C3⋊D9C2semidirect
8. #11721727C2semidirect
9. C22⋊C4×C3⋊D9C2semidirect
10. C4×C4⋊(C3⋊D9)C2semidirect
11. C42×C3⋊D9C2semidirect
16. C22×C3×C4×C9C22semidirect
18. C22×C3⋊Dic9C22semidirect
19. D4×C3⋊D9C4direct (N × H)
26. C2×C4×C3⋊D9C22semidirect
27. C2×C4×C3⋊D9C22semidirect
28. C3×C9×C4⋊C4C22semidirect
29. C3×C9×C22⋊C4C22semidirect
30. C3×C42×C9C22semidirect
31. #88551C22semidirect
32. #88536C22semidirect
33. #88536C22semidirect
34. C4⋊(C3⋊Dic9)C22semidirect
35. #88534C22semidirect
36. C4×C3⋊Dic9C22semidirect
37. C4×C3⋊Dic9C22semidirect
38. C4×C3⋊Dic9C22semidirect
52. C4×D4×C9D3semidirect
64. C22×C3⋊D9C2×C4semidirect
67. C3×D4×C9C2×C4semidirect
73. #6233C2×C4semidirect
79. C4⋊(C3⋊D9)C2×C4semidirect
84. C4×C3⋊D9C2×C4semidirect
85. C4×C3⋊D9D4direct (N × H)
115. C22×C4×C9C2×D3semidirect
127. C9×C4⋊C4C2×D3semidirect
128. C9×C22⋊C4C2×D3semidirect
129. C42×C9C2×D3semidirect
140. C22×C3×C9C22×C4semidirect
146. C2×C3⋊D9C22×C4semidirect
151. C3×C4×C9C22×C4semidirect
152. C3×C4×C9C2×D4semidirect
155. C3⋊Dic9C22×C4semidirect
156. C3⋊Dic9C2×D4semidirect
161. C3×C4×D4D9semidirect
210. D4×C9C4×D3semidirect
235. C3⋊D9C4×D4direct (N × H)
240. C22×C3×C4C2×D9semidirect
264. C3×C4⋊C4C2×D9semidirect
266. C3×C22⋊C4C2×D9semidirect
268. C3×C42C2×D9semidirect
307. C22×C9C2×C4×D3semidirect
318. C4×C9C2×C4×D3semidirect
319. C4×C9D3×D4semidirect
331. C4×D4C3⋊D9direct (N × H)
335. C3×C9C2×C4×D4semidirect
348. C3×D4C4×D9semidirect
411. C22×C4C2×C3⋊D9semidirect
414. C4⋊C4C2×C3⋊D9semidirect
418. C22⋊C4C2×C3⋊D9semidirect
422. C42C2×C3⋊D9semidirect
427. C22×C3C2×C4×D9semidirect
449. C3×C4C2×C4×D9semidirect
450. C3×C4D4×D9semidirect
462. C9C4×D3×D4semidirect
471. D4C4×C3⋊D9direct (N × H)
509. C22C2×C4×C3⋊D9semidirect
517. C4C2×C4×C3⋊D9semidirect
518. C4D4×C3⋊D9direct (N × H)
519. C3C4×D4×D9semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D92
C22×C4×C3⋊D92
C3×C4×D4×C92
D4×C3⋊Dic92
C4×#62332
#117217492
C4⋊C4×C3⋊D92
#117217272
C22⋊C4×C3⋊D92
C4×C4⋊(C3⋊D9)2
C42×C3⋊D92
C4×D4×C3⋊D33
C4×D4×D93