Finite Group Databasebeta

← C4×#88635 #13147568

Subgroups — C4×#88635 #13147568

Order 1728.

1024 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88635C211yesmaximal, characteristic
2C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
3C3×C42×D9C211yesmaximal, characteristic, complemented
4C4×C9×Dic6C211yesmaximal, characteristic, complemented
5#11721002C212yesmaximal, complemented
6C4×#6192C212yesmaximal, complemented
7#11720992C211yesmaximal, characteristic, complemented
8C4×(C3⋊D9)⋊C4C212yesmaximal, complemented
9Dic6×Dic9C211yesmaximal, characteristic, complemented
10#11720907C211yesmaximal, characteristic, complemented
11#11720899C212yesmaximal, complemented
12C4×#3799—31nomaximal
13C4×#3701—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C3×C4×D9C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×C9×Dic6C2211yescharacteristic
20C2×#6192C2212yes—
21C2×(C3⋊D9)⋊C4C2212yes—
22C2×(C3⋊D9)⋊C4C2212yes—
23#88635C418yesdirect product factor
24C3×C42×C9C2211yescharacteristic
25#88536C2212yes—
26C4⋊(C3⋊Dic9)C2211yescharacteristic
27C9×C4⋊Dic3C2211yescharacteristic
28C9×Dic3⋊C4C2212yes—
29C4×C9×Dic3C2212yes—
30C3×C4×Dic9C2211yescharacteristic
31C3×C4×Dic9C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33#88445C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35Dic3⋊Dic9C2212yes—
36Dic3×Dic9C2212yes—
37Dic3×Dic9C2212yes—
38C2×#3799—31no—
39C4×C4⋊(C3⋊D3)—31no—
40C3×C42×D3—31no—
41C3×C4×Dic6D311yescharacteristic
42#63904—32no—
43C4×#626—32no—
44#63894—31no—
45C4×#624—32no—
46Dic3×Dic6—31no—
47#63841—31no—
48#63836—32no—
49C2×#3701—31no—
50C4×Q8×C9—31no—
51Q8×Dic9—31no—
52#63455—32no—
53#63455—31no—
54#63454—31no—
55#63454—32no—
56C4×D36—32no—
57C4×D36—31no—
58C4×D36—61no—
59C42×D9—32no—
60C42×D9D311yescharacteristic, complemented
61C22×C3⋊D9C2312yes—
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C4⋊(C3⋊D9)C2×C414yescomplemented
68C4×C3⋊D9—24no—
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C3×C4×D9—22no—
75C3×C4×D9—22no—
76C3×C4×D9C2×C414yescomplemented
77C9×Dic6C2×C414yescomplemented
78#6192C2×C418yescomplemented
79(C3⋊D9)⋊C4—24no—
80(C3⋊D9)⋊C4C2×C418yescomplemented
81C4×#239—91no—
82C4×#235—91no—
83C2×C4⋊(C3⋊D3)—31no—
84C2×C4×C3⋊D3—32no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—31no—
88C2×C3×Dic6C2×D311yescharacteristic
89C2×#626—32no—
90C2×#624—32no—
91C2×#624—32no—
92#3799—38no—
93C32×C42C2×D311yescharacteristic
94#3752—32no—
95C4⋊(C3⋊Dic3)—31no—
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×Dic3⋊C4C2×D312yes—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—31no—
100C3×C4×Dic3—31no—
101C3×C4×Dic3C2×D312yes—
102Dic3⋊Dic3—32no—
103#3722—32no—
104(C2×D3)⋊Dic3—32no—
105Dic32—32no—
106Dic32—32no—
107C2×Q8×C9—31no—
108#3701—38no—
109C2×D36—32no—
110C2×D36—31no—
111C2×D36—61no—
112C2×C4×D9—32no—
113C2×C4×D9—32no—
114C2×C4×D9—32no—
115C2×C4×D9—62no—
116C2×C4×D9C2×D311yescharacteristic
117C2×C4×D9C2×D311yescharacteristic
118C2×C4×D9C2×D311yescharacteristic
119C9×C4⋊C4—32no—
120C9×C4⋊C4—31no—
121C42×C9—21no—
122C42×C9—32no—
123C42×C9C2×D311yescharacteristic
124(C2×D9)⋊C4—32no—
125(C2×D9)⋊C4—32no—
126(C2×D9)⋊C4—32no—
127(C2×D9)⋊C4—62no—
128C4⋊Dic9—31no—
129C4⋊Dic9—32no—
130C4⋊Dic9—61no—
131C4×Dic9—32no—
132C4×Dic9—32no—
133C4×Dic9C2×D311yescharacteristic
134C4×Dic9C2×D311yescharacteristic
135C4×Dic9C2×D311yescharacteristic
136C22×C3×C9C2411yescharacteristic
137C2×C3⋊D9—22no—
138C2×C3⋊D9—22no—
139C2×C3⋊D9C22×C414yes—
140C2×C3×D9—21no—
141C2×C3×D9—21no—
142C2×C3×D9C22×C412yes—
143C3×C4×C9—21no—
144C3×C4×C9C22×C412yes—
145C3×C4×C9#4112yescomplemented
146C3⋊Dic9—22no—
147C9×Dic3—22no—
148C9×Dic3C22×C414yes—
149C3×Dic9—21no—
150C3×Dic9C22×C412yes—
151C3×Dic9#4112yescomplemented
152C2×#239—91no—
153C2×#235—91no—
154C3×C4×Q8—31no—
155C2×C3×C42—91no—
156Q8×Dic3—91no—
157C4×#67—92no—
158C4×#67—92no—
159#926—91no—
160#893—91no—
161#893—92no—
162#892—92no—
163#892—91no—
164C4×D12—92no—
165C4×D12—91no—
166C4×D12—91no—
167C4×D12—91no—
168C4×D12—181no—
169#872—92no—
170C42×D3—31no—
171C42×D3—92no—
172C42×D3—92no—
173C4×Dic6—91no—
174C4×Dic6D911yescharacteristic, complemented
175C22×C3⋊D3—32no—
176C22×C3×D3—31no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C3⋊Dic3—32no—
181C4⋊(C3⋊D3)—34no—
182C4×C3⋊D3—64no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3C22×D312yes—
187C2×C3×Dic3C22×D312yes—
188C3×C4×D3—34no—
189C3×C4×D3—62no—
190C3×C4×D3—62no—
191C3×Dic6C4×D314yes—
192#626—38no—
193#624—38no—
194#624—64no—
195C22×D9—32no—
196C22×D9—62no—
197C22×D9C22×D311yescharacteristic
198Q8×C9—34no—
199C2×C4×C9—21no—
200C2×C4×C9—21no—
201C2×C4×C9—21no—
202C2×C4×C9—32no—
203C2×C4×C9—32no—
204C2×C4×C9C22×D311yescharacteristic
205C2×C4×C9C22×D311yescharacteristic
206C2×C4×C9C22×D311yescharacteristic
207C2×Dic9—32no—
208C2×Dic9—62no—
209C2×Dic9C22×D311yescharacteristic
210C2×Dic9C22×D311yescharacteristic
211C2×Dic9C22×D311yescharacteristic
212D36—38no—
213D36—34no—
214D36—64no—
215C4×D9—22no—
216C4×D9—22no—
217C4×D9—38no—
218C4×D9—64no—
219C4×D9—64no—
220C4×D9—124no—
221C4×D9C4×D314yescomplemented
222C4×#41—271noSylow
223C2×C3×C9C23×C411yescharacteristic
224C2×C3×C9C2×#4111yescharacteristic
225C2×C3×C9C2×#4111yescharacteristic
226C3⋊D9—24no—
227C3×D9—22no—
228C2×C3×Q8—31no—
229C22×C3×C4—91no—
230C22×C3×C4—91no—
231C22×C3×C4—91no—
232C2×#67—92no—
233#239—98no—
234#235—98no—
235C2×D12—92no—
236C2×D12—91no—
237C2×D12—91no—
238C2×D12—181no—
239C2×C4×D3—31no—
240C2×C4×D3—31no—
241C2×C4×D3—31no—
242C2×C4×D3—92no—
243C2×C4×D3—92no—
244C2×C4×D3—92no—
245C2×C4×D3—92no—
246C2×C4×D3—92no—
247C2×C4×D3—92no—
248C2×C4×D3—182no—
249C2×Dic6C2×D911yescharacteristic
250C3×C4⋊C4—32no—
251C3×C4⋊C4—31no—
252C3×C42—21no—
253C3×C42—32no—
254C3×C42—91no—
255C3×C42—91no—
256C3×C42—91no—
257C3×C42C2×D911yescharacteristic
258C3×C42D3211yescharacteristic
259#217—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—182no—
266C4⋊Dic3—92no—
267C4⋊Dic3—91no—
268C4⋊Dic3—91no—
269C4⋊Dic3—181no—
270C4⋊Dic3C2×D911yescharacteristic
271Dic3⋊C4—92no—
272Dic3⋊C4C2×D912yes—
273C4×Dic3—31no—
274C4×Dic3—31no—
275C4×Dic3—31no—
276C4×Dic3—92no—
277C4×Dic3—92no—
278C4×Dic3—92no—
279C4×Dic3—92no—
280C4×Dic3C2×D912yes—
281C22×C32C23×D311yescharacteristic
282C2×C3⋊D3—34no—
283C2×C3⋊D3—62no—
284C2×C3⋊D3—62no—
285C2×C3×D3—32no—
286C2×C3×D3—61no—
287C2×C3×D3—61no—
288C32×C4—21no—
289C32×C4C2×C4×D312yes—
290C32×C4#23912yes—
291C3⋊Dic3—62no—
292C3×Dic3—22no—
293C3×Dic3—32no—
294C3×Dic3—32no—
295C3×Dic3—61no—
296C3×Dic3C2×C4×D314yes—
297C22×C9—21no—
298C22×C9C23×D311yescharacteristic
299C2×D9—21no—
300C2×D9—21no—
301C2×D9—34no—
302C2×D9—64no—
303C2×D9—62no—
304C2×D9—62no—
305C2×D9—122no—
306C2×D9—122no—
307C2×D9C2×C4×D312yes—
308C4×C9—21no—
309C4×C9—22no—
310C4×C9—22no—
311C4×C9—34no—
312C4×C9—41no—
313C4×C9—62no—
314C4×C9C2×C4×D312yes—
315C4×C9#23512yescomplemented
316Dic9—21no—
317Dic9—62no—
318Dic9—122no—
319Dic9C2×C4×D312yes—
320Dic9#23512yescomplemented
321C2×#41—271no—
322C4×Q8—31no—
323C4×Q8—271no—
324C4×D4—272no—
325C4×D4—271no—
326C4×D4—272no—
327C4×D4—271no—
328#117—272no—
329#117—271no—
330C2×C42—91no—
331C2×C42—272no—
332C3×C9C4×#4111yescharacteristic, complemented, Sylow
333C23×C3—91no—
334C22×D3—31no—
335C22×D3—92no—
336C22×D3—92no—
337C22×D3—182no—
338C3×Q8—34no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—32no—
343C2×C3×C4—32no—
344C2×C3×C4—91no—
345C2×C3×C4—91no—
346C2×C3×C4—94no—
347C2×C3×C4—91no—
348C2×C3×C4—182no—
349C2×C3×C4—182no—
350C2×C3×C4C22×D911yescharacteristic
351C2×C3×C4C22×D911yescharacteristic
352C2×C3×C4C22×D911yescharacteristic
353C2×C3×C4C2×D3211yescharacteristic
354C2×C3×C4C2×D3211yescharacteristic
355C2×C3×C4C2×D3211yescharacteristic
356#67—98no—
357C2×Dic3—31no—
358C2×Dic3—31no—
359C2×Dic3—31no—
360C2×Dic3—92no—
361C2×Dic3—92no—
362C2×Dic3—182no—
363C2×Dic3C22×D912yes—
364C2×Dic3C22×D912yes—
365D12—98no—
366D12—94no—
367D12—94no—
368D12—184no—
369C4×D3—34no—
370C4×D3—62no—
371C4×D3—62no—
372C4×D3—98no—
373C4×D3—98no—
374C4×D3—184no—
375C4×D3—184no—
376C4×D3—184no—
377C4×D3—184no—
378C4×D3—364no—
379Dic6C4×D914yescomplemented
380C2×C32C22×C4×D311yescharacteristic
381C2×C32C2×#23911yescharacteristic
382C2×C32D3×#4111yescharacteristic
383C3⋊D3—64no—
384C3×D3—62no—
385C2×C9—21no—
386C2×C9—21no—
387C2×C9—21no—
388C2×C9C22×C4×D311yescharacteristic
389C2×C9C2×#23511yescharacteristic
390C2×C9C2×#23511yescharacteristic
391D9—22no—
392D9—64no—
393D9—124no—
394#41—278no—
395C2×Q8—31no—
396C2×D4—272no—
397C2×D4—271no—
398C22×C4—91no—
399C22×C4—91no—
400C22×C4—91no—
401C22×C4—272no—
402C22×C4—272no—
403C22×C4—272no—
404C4⋊C4—32no—
405C4⋊C4—31no—
406C4⋊C4—272no—
407C4⋊C4—271no—
408C22⋊C4—272no—
409C22⋊C4—272no—
410C22⋊C4—272no—
411C42—32no—
412C42—91no—
413C42—91no—
414C42—91no—
415C42—272no—
416C42—272no—
417C42D3×D911yescharacteristic
418C22×C3—21no—
419C22×C3—92no—
420C22×C3—181no—
421C22×C3—181no—
422C22×C3C23×D911yescharacteristic
423C22×C3C22×D3211yescharacteristic
424C2×D3—32no—
425C2×D3—61no—
426C2×D3—61no—
427C2×D3—94no—
428C2×D3—94no—
429C2×D3—184no—
430C2×D3—182no—
431C2×D3—182no—
432C2×D3—182no—
433C2×D3—182no—
434C2×D3—362no—
435C2×D3—362no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—34no—
441C3×C4—41no—
442C3×C4—62no—
443C3×C4—92no—
444C3×C4—92no—
445C3×C4—181no—
446C3×C4C2×C4×D912yes—
447C3×C4#370112yescomplemented
448C3×C4#379912yes—
449C3×C4C4×D3212yes—
450Dic3—22no—
451Dic3—32no—
452Dic3—32no—
453Dic3—61no—
454Dic3—182no—
455Dic3—182no—
456Dic3—362no—
457Dic3C2×C4×D914yes—
458C32C4×#23911yescharacteristic
459C9—21no—
460C9C4×#23511yescharacteristic, complemented
461C23—91no—
462C23—272no—
463Q8—34no—
464D4—278no—
465D4—274no—
466C2×C4—32no—
467C2×C4—32no—
468C2×C4—91no—
469C2×C4—91no—
470C2×C4—94no—
471C2×C4—91no—
472C2×C4—182no—
473C2×C4—182no—
474C2×C4—278no—
475C2×C4—272no—
476C2×C4—544no—
477C2×C4—544no—
478C2×C4C2×D3×D911yescharacteristic
479C2×C4C2×D3×D911yescharacteristic
480C2×C4C2×D3×D911yescharacteristic
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—182no—
485C2×C3C22×C4×D911yescharacteristic
486C2×C3C2×#370111yescharacteristic
487C2×C3#41×D911yescharacteristic
488C2×C3C2×#379911yescharacteristic
489C2×C3C2×C4×D3211yescharacteristic
490C2×C3D3×#23511yescharacteristic
491D3—62no—
492D3—184no—
493D3—184no—
494D3—364no—
495C22—92no—
496C22—181no—
497C22—181no—
498C22—274no—
499C22—542no—
500C22—542no—
501C22C22×D3×D911yescharacteristic
502C4—21no—
503C4—34no—
504C4—62no—
505C4—92no—
506C4—92no—
507C4—181no—
508C4—542no—
509C4#8863512yesdirect product factor
510C4C4×D3×D912yes—
511C3—21no—
512C3C4×#370111yescharacteristic, complemented
513C3C4×#379911yescharacteristic
514C2—182no—
515C2—544no—
516C2C2×#8863511yescharacteristic
517C2C2×C4×D3×D911yescharacteristic
518C2D9×#23511yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#88635C2
cosocleC22×C3×C9C24
hypercenterC42D3×D9
hypocenterC3×C9C4×#41
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊(C3⋊D9)C2semidirect
3. C3×C42×D9C2semidirect
4. C4×C9×Dic6C2semidirect
5. #11721002C2semidirect
6. C4×#6192C2semidirect
7. #11720992C2semidirect
8. C4×(C3⋊D9)⋊C4C2semidirect
9. Dic6×Dic9C2semidirect
10. #11720907C2semidirect
11. #11720899C2semidirect
23. #88635C4direct (N × H)
60. C42×D9D3semidirect
67. C4⋊(C3⋊D9)C2×C4semidirect
76. C3×C4×D9C2×C4semidirect
77. C9×Dic6C2×C4semidirect
78. #6192C2×C4semidirect
80. (C3⋊D9)⋊C4C2×C4semidirect
145. C3×C4×C9#41semidirect
151. C3×Dic9#41semidirect
174. C4×Dic6D9semidirect
221. C4×D9C4×D3semidirect
315. C4×C9#235semidirect
320. Dic9#235semidirect
332. C3×C9C4×#41semidirect
379. Dic6C4×D9semidirect
447. C3×C4#3701semidirect
460. C9C4×#235semidirect
509. C4#88635direct (N × H)
512. C3C4×#3701semidirect

Maximal subgroups

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

subgroupindex
C2×#886352
C4×C4⋊(C3⋊D9)2
C3×C42×D92
C4×C9×Dic62
#117210022
C4×#61922
#117209922
C4×(C3⋊D9)⋊C42
Dic6×Dic92
#117209072
#117208992
C4×#37993
C4×#37013