Finite Group Databasebeta

← C4×#88640 #13147642

Subgroups — C4×#88640 #13147642

Order 1728.

1038 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88640C211yesmaximal, characteristic
2C42×C3⋊D9C211yesmaximal, characteristic, complemented
3C42×D3×C9C211yesmaximal, characteristic, complemented
4C3×C42×D9C211yesmaximal, characteristic, complemented
5C4×#6195C212yesmaximal, complemented
6C4×#6194C212yesmaximal, complemented
7C4×#6192C212yesmaximal, complemented
8C4×#6189C212yesmaximal, complemented
9#11720959C211yesmaximal, characteristic, complemented
10#11720943C211yesmaximal, characteristic, complemented
11#11720931C211yesmaximal, characteristic, complemented
12C4×#3798—31nomaximal
13C4×#3697—31nomaximal
14C2×C4×C3⋊D9C2212yes—
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C4×D3×C9C2212yes—
17C2×C4×D3×C9C2211yescharacteristic
18C2×C3×C4×D9C2212yes—
19C2×C3×C4×D9C2211yescharacteristic
20C2×#6195C2211yescharacteristic
21C2×#6194C2211yescharacteristic
22C2×#6192C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24#88640C418yesdirect product factor
25C3×C42×C9C2211yescharacteristic
26C4×C3⋊Dic9C2212yes—
27C4×C3⋊Dic9C2211yescharacteristic
28C4×C9×Dic3C2212yes—
29C4×C9×Dic3C2211yescharacteristic
30C3×C4×Dic9C2212yes—
31C3×C4×Dic9C2211yescharacteristic
32(C2×D3)⋊Dic9C2212yes—
33#88445C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35Dic3⋊Dic9C2212yes—
36#88442C2212yes—
37Dic9⋊Dic3C2212yes—
38C2×#3798—31no—
39C42×C3⋊D3—31no—
40C3×C42×D3—31no—
41C3×C42×D3D311yescharacteristic
42C4×#627—32no—
43C4×#626—32no—
44C4×#626—32no—
45C4×#625—32no—
46#63874—31no—
47#63855—31no—
48#63855—31no—
49C2×#3697—31no—
50C2×C42×C9—31no—
51C4×#611—32no—
52C4×#611—32no—
53#63488—31no—
54C4×D36—32no—
55#63434—31no—
56#63434—31no—
57C42×D9—31no—
58C42×D9—61no—
59C42×D9D311yescharacteristic, complemented
60C4×Dic18—32no—
61C22×C3⋊D9C2311yescharacteristic
62C22×D3×C9C2311yescharacteristic
63C22×C3×D9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2312yes—
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2311yescharacteristic
68C4×C3⋊D9—24no—
69C4×C3⋊D9C2×C414yescomplemented
70C2×C9×Dic3C2312yes—
71C2×C9×Dic3C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2311yescharacteristic
74C4×D3×C9—24no—
75C4×D3×C9C2×C414yescomplemented
76C3×C4×D9—24no—
77C3×C4×D9C2×C414yescomplemented
78#6195C2×C414yescomplemented
79#6194C2×C414yescomplemented
80#6192C2×C414yescomplemented
81#6189C2×C414yescomplemented
82C4×#235—91no—
83C4×#235—91no—
84C2×C4×C3⋊D3—31no—
85C2×C4×C3⋊D3—32no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3C2×D312yes—
89C2×C3×C4×D3C2×D311yescharacteristic
90C2×#627—31no—
91C2×#626—31no—
92C2×#626—31no—
93C2×#625—31no—
94#3798—38no—
95C32×C42C2×D311yescharacteristic
96C4×C3⋊Dic3—31no—
97C4×C3⋊Dic3—32no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3C2×D311yescharacteristic
101C3×C4×Dic3C2×D312yes—
102#3724—32no—
103Dic3⋊Dic3—32no—
104Dic3⋊Dic3—32no—
105#3722—32no—
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108C22×C4×C9—31no—
109C22×C4×C9—32no—
110C2×#611—31no—
111C2×#611—31no—
112#3697—38no—
113C2×D36—31no—
114C2×C4×D9—31no—
115C2×C4×D9—32no—
116C2×C4×D9—61no—
117C2×C4×D9—62no—
118C2×C4×D9C2×D312yes—
119C2×C4×D9C2×D311yescharacteristic
120C2×Dic18—31no—
121C42×C9—21no—
122C42×C9—32no—
123C42×C9—31no—
124C42×C9C2×D311yescharacteristic
125#3676—32no—
126(C2×D9)⋊C4—32no—
127(C2×D9)⋊C4—32no—
128C4⋊Dic9—32no—
129Dic9⋊C4—32no—
130Dic9⋊C4—32no—
131C4×Dic9—31no—
132C4×Dic9—32no—
133C4×Dic9—62no—
134C4×Dic9—61no—
135C4×Dic9C2×D311yescharacteristic
136C4×Dic9C2×D312yes—
137C22×C3×C9C2411yescharacteristic
138C2×C3⋊D9—22no—
139C2×C3⋊D9C22×C412yes—
140C2×D3×C9—22no—
141C2×D3×C9C22×C412yes—
142C2×C3×D9—22no—
143C2×C3×D9C22×C412yes—
144C3×C4×C9C22×C412yes—
145C3×C4×C9#4114yescomplemented
146C3⋊Dic9—22no—
147C3⋊Dic9C22×C412yes—
148C9×Dic3—22no—
149C9×Dic3C22×C412yes—
150C3×Dic9—22no—
151C3×Dic9C22×C412yes—
152C2×#235—91no—
153C2×#235—91no—
154C2×C3×C42—31no—
155C2×C3×C42—91no—
156C4×#67—92no—
157C4×#67—92no—
158C4×#67—92no—
159C4×#67—92no—
160#926—91no—
161#926—91no—
162C4×D12—92no—
163C4×D12—92no—
164#872—91no—
165#872—91no—
166#872—91no—
167#872—91no—
168C42×D3—31no—
169C42×D3—91no—
170C42×D3—91no—
171C42×D3—181no—
172C42×D3D911yescharacteristic, complemented
173C4×Dic6—92no—
174C4×Dic6—92no—
175C22×C3⋊D3—31no—
176C22×C3×D3—31no—
177C22×C3×D3C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D312yes—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—32no—
182C4×C3⋊D3—34no—
183C4×C3⋊D3—64no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3C22×D312yes—
187C2×C3×Dic3C22×D311yescharacteristic
188C3×C4×D3—24no—
189C3×C4×D3—34no—
190C3×C4×D3—64no—
191C3×C4×D3C4×D314yes—
192#627—34no—
193#626—34no—
194#626—34no—
195#625—34no—
196C23×C9—31no—
197C22×D9—31no—
198C22×D9—61no—
199C22×D9C22×D311yescharacteristic
200C2×C4×C9—22no—
201C2×C4×C9—21no—
202C2×C4×C9—31no—
203C2×C4×C9—34no—
204C2×C4×C9—32no—
205C2×C4×C9—64no—
206C2×C4×C9C22×D311yescharacteristic
207C2×C4×C9C22×D312yes—
208#611—34no—
209#611—34no—
210C2×Dic9—31no—
211C2×Dic9—32no—
212C2×Dic9—61no—
213C2×Dic9—62no—
214C2×Dic9C22×D312yes—
215C2×Dic9C22×D311yescharacteristic
216D36—34no—
217C4×D9—24no—
218C4×D9—34no—
219C4×D9—64no—
220C4×D9—68no—
221C4×D9—64no—
222C4×D9C4×D314yescomplemented
223Dic18—34no—
224C4×#41—271noSylow
225C2×C3×C9C23×C411yescharacteristic
226C2×C3×C9C2×#4112yes—
227C3⋊D9—22no—
228D3×C9—22no—
229C3×D9—22no—
230C22×C3×C4—31no—
231C22×C3×C4—32no—
232C22×C3×C4—91no—
233C22×C3×C4—92no—
234C2×#67—91no—
235C2×#67—91no—
236C2×#67—91no—
237C2×#67—91no—
238#235—98no—
239#235—98no—
240C2×D12—91no—
241C2×D12—91no—
242C2×C4×D3—31no—
243C2×C4×D3—32no—
244C2×C4×D3—91no—
245C2×C4×D3—91no—
246C2×C4×D3—92no—
247C2×C4×D3—92no—
248C2×C4×D3—181no—
249C2×C4×D3—182no—
250C2×C4×D3C2×D912yes—
251C2×C4×D3C2×D911yescharacteristic
252C2×Dic6—91no—
253C2×Dic6—91no—
254C3×C42—21no—
255C3×C42—32no—
256C3×C42—31no—
257C3×C42—92no—
258C3×C42—91no—
259C3×C42C2×D911yescharacteristic
260C3×C42D3211yescharacteristic
261#217—92no—
262#217—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—92no—
267C4⋊Dic3—92no—
268C4⋊Dic3—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—92no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273C4×Dic3—32no—
274C4×Dic3—31no—
275C4×Dic3—92no—
276C4×Dic3—91no—
277C4×Dic3—92no—
278C4×Dic3—91no—
279C4×Dic3—181no—
280C4×Dic3—182no—
281C4×Dic3C2×D912yes—
282C4×Dic3C2×D911yescharacteristic
283C22×C32C23×D311yescharacteristic
284C2×C3⋊D3—32no—
285C2×C3⋊D3—62no—
286C2×C3×D3—22no—
287C2×C3×D3—32no—
288C2×C3×D3—62no—
289C2×C3×D3C2×C4×D312yes—
290C32×C4C2×C4×D312yes—
291C32×C4#23514yes—
292C3⋊Dic3—32no—
293C3⋊Dic3—62no—
294C3×Dic3—22no—
295C3×Dic3—32no—
296C3×Dic3—62no—
297C3×Dic3C2×C4×D312yes—
298C22×C9—21no—
299C22×C9—32no—
300C22×C9—62no—
301C22×C9C23×D311yescharacteristic
302C2×D9—22no—
303C2×D9—32no—
304C2×D9—62no—
305C2×D9—64no—
306C2×D9—62no—
307C2×D9C2×C4×D312yes—
308C4×C9—24no—
309C4×C9—22no—
310C4×C9—32no—
311C4×C9—62no—
312C4×C9C2×C4×D312yes—
313C4×C9#23514yescomplemented
314Dic9—22no—
315Dic9—32no—
316Dic9—62no—
317Dic9—64no—
318Dic9—62no—
319Dic9C2×C4×D312yes—
320C2×#41—271no—
321C4×Q8—272no—
322C4×D4—272no—
323C4×D4—272no—
324C4×D4—272no—
325#117—271no—
326#117—271no—
327#117—271no—
328C2×C42—31no—
329C2×C42—91no—
330C2×C42—271no—
331C3×C9C4×#4111yescharacteristic, complemented, Sylow
332C23×C3—31no—
333C23×C3—91no—
334C22×D3—31no—
335C22×D3—91no—
336C22×D3—91no—
337C22×D3—181no—
338C22×D3C22×D911yescharacteristic
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—31no—
342C2×C3×C4—34no—
343C2×C3×C4—32no—
344C2×C3×C4—64no—
345C2×C3×C4—91no—
346C2×C3×C4—94no—
347C2×C3×C4—92no—
348C2×C3×C4—184no—
349C2×C3×C4C22×D911yescharacteristic
350C2×C3×C4C22×D912yes—
351C2×C3×C4C2×D3212yes—
352C2×C3×C4C2×D3211yescharacteristic
353#67—94no—
354#67—94no—
355#67—94no—
356#67—94no—
357C2×Dic3—31no—
358C2×Dic3—32no—
359C2×Dic3—91no—
360C2×Dic3—91no—
361C2×Dic3—92no—
362C2×Dic3—92no—
363C2×Dic3—181no—
364C2×Dic3—182no—
365C2×Dic3C22×D912yes—
366C2×Dic3C22×D911yescharacteristic
367D12—94no—
368D12—94no—
369C4×D3—24no—
370C4×D3—34no—
371C4×D3—64no—
372C4×D3—94no—
373C4×D3—94no—
374C4×D3—184no—
375C4×D3—188no—
376C4×D3—184no—
377C4×D3—184no—
378C4×D3C4×D914yescomplemented
379Dic6—94no—
380Dic6—94no—
381C2×C32C22×C4×D311yescharacteristic
382C2×C32C2×#23512yes—
383C3⋊D3—62no—
384C3×D3—22no—
385C3×D3—62no—
386C2×C9—22no—
387C2×C9—21no—
388C2×C9—62no—
389C2×C9C22×C4×D311yescharacteristic
390C2×C9C2×#23512yes—
391D9—22no—
392D9—64no—
393D9—62no—
394#41—278no—
395C2×Q8—271no—
396C2×D4—271no—
397C2×D4—271no—
398C2×D4—271no—
399C22×C4—31no—
400C22×C4—32no—
401C22×C4—91no—
402C22×C4—92no—
403C22×C4—271no—
404C22×C4—272no—
405C4⋊C4—272no—
406C4⋊C4—272no—
407C4⋊C4—272no—
408C22⋊C4—272no—
409C22⋊C4—272no—
410C22⋊C4—272no—
411C42—31no—
412C42—32no—
413C42—91no—
414C42—92no—
415C42—271no—
416C42—272no—
417C42D3×D911yescharacteristic
418C22×C3—21no—
419C22×C3—32no—
420C22×C3—62no—
421C22×C3—92no—
422C22×C3—182no—
423C22×C3C23×D911yescharacteristic
424C22×C3C22×D3211yescharacteristic
425C2×D3—22no—
426C2×D3—32no—
427C2×D3—62no—
428C2×D3—92no—
429C2×D3—92no—
430C2×D3—182no—
431C2×D3—184no—
432C2×D3—182no—
433C2×D3—182no—
434C2×D3C2×C4×D912yes—
435C3×C4—24no—
436C3×C4—22no—
437C3×C4—32no—
438C3×C4—62no—
439C3×C4—92no—
440C3×C4—182no—
441C3×C4C2×C4×D912yes—
442C3×C4#369714yescomplemented
443C3×C4#379814yes—
444C3×C4C4×D3212yes—
445Dic3—22no—
446Dic3—32no—
447Dic3—62no—
448Dic3—92no—
449Dic3—92no—
450Dic3—182no—
451Dic3—184no—
452Dic3—182no—
453Dic3—182no—
454Dic3C2×C4×D912yes—
455C32C4×#23511yescharacteristic
456C9—21no—
457C9C4×#23511yescharacteristic, complemented
458C23—31no—
459C23—91no—
460C23—271no—
461Q8—274no—
462D4—274no—
463D4—274no—
464D4—274no—
465C2×C4—31no—
466C2×C4—34no—
467C2×C4—32no—
468C2×C4—64no—
469C2×C4—91no—
470C2×C4—94no—
471C2×C4—92no—
472C2×C4—184no—
473C2×C4—271no—
474C2×C4—274no—
475C2×C4—272no—
476C2×C4—544no—
477C2×C4C2×D3×D912yes—
478C2×C4C2×D3×D911yescharacteristic
479C2×C3—22no—
480C2×C3—21no—
481C2×C3—62no—
482C2×C3—182no—
483C2×C3C22×C4×D911yescharacteristic
484C2×C3C2×#369712yes—
485C2×C3C2×#379812yes—
486C2×C3C2×C4×D3211yescharacteristic
487D3—22no—
488D3—62no—
489D3—184no—
490D3—182no—
491D3—182no—
492C22—32no—
493C22—62no—
494C22—92no—
495C22—182no—
496C22—272no—
497C22—542no—
498C22C22×D3×D911yescharacteristic
499C4—32no—
500C4—62no—
501C4—92no—
502C4—182no—
503C4—272no—
504C4—542no—
505C4#8864014yesdirect product factor
506C4C4×D3×D912yes—
507C3—21no—
508C3C4×#369711yescharacteristic, complemented
509C3C4×#379811yescharacteristic
510C2—62no—
511C2—182no—
512C2—542no—
513C2C2×#8864012yes—
514C2C2×C4×D3×D911yescharacteristic
subgroupG/H
center, hypercenterC42D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#88640C2
cosocleC22×C3×C9C24
hypocenterC3×C9C4×#41
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C42×C3⋊D9C2semidirect
3. C42×D3×C9C2semidirect
4. C3×C42×D9C2semidirect
5. C4×#6195C2semidirect
6. C4×#6194C2semidirect
7. C4×#6192C2semidirect
8. C4×#6189C2semidirect
9. #11720959C2semidirect
10. #11720943C2semidirect
11. #11720931C2semidirect
24. #88640C4direct (N × H)
59. C42×D9D3semidirect
69. C4×C3⋊D9C2×C4semidirect
75. C4×D3×C9C2×C4semidirect
77. C3×C4×D9C2×C4semidirect
78. #6195C2×C4semidirect
79. #6194C2×C4semidirect
80. #6192C2×C4semidirect
81. #6189C2×C4semidirect
145. C3×C4×C9#41semidirect
172. C42×D3D9semidirect
222. C4×D9C4×D3semidirect
313. C4×C9#235semidirect
331. C3×C9C4×#41semidirect
378. C4×D3C4×D9semidirect
442. C3×C4#3697semidirect
457. C9C4×#235semidirect
505. C4#88640direct (N × H)
508. C3C4×#3697semidirect

Maximal subgroups

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

subgroupindex
C2×#886402
C42×C3⋊D92
C42×D3×C92
C3×C42×D92
C4×#61952
C4×#61942
C4×#61922
C4×#61892
#117209592
#117209432
#117209312
C4×#37983
C4×#36973