Finite Group Databasebeta

← C4×#88638 #13147589

Subgroups — C4×#88638 #13147589

Order 1728.

1000 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88638C211yesmaximal, characteristic
2C4×#6226C211yesmaximal, characteristic, complemented
3C4×C9×D12C211yesmaximal, characteristic, complemented
4C3×C42×D9C211yesmaximal, characteristic, complemented
5C4×#6194C212yesmaximal, complemented
6D12×Dic9C211yesmaximal, characteristic, complemented
7#11721012C212yesmaximal, complemented
8#11721010C212yesmaximal, complemented
9C4×D3×Dic9C212yesmaximal, complemented
10#11720913C211yesmaximal, characteristic, complemented
11#11720907C211yesmaximal, characteristic, complemented
12C4×#3795—31nomaximal
13C4×#3699—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9C2212yes—
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×C3×C4×D9C2211yescharacteristic
20C2×#6194C2212yes—
21C2×D3×Dic9C2212yes—
22C2×D3×Dic9C2212yes—
23#88638C418yesdirect product factor
24C3×C42×C9C2211yescharacteristic
25C4⋊(C3⋊Dic9)C2211yescharacteristic
26#88534C2212yes—
27C4×C3⋊Dic9C2212yes—
28C9×(C2×D3)⋊C4C2212yes—
29C9×C4⋊Dic3C2211yescharacteristic
30C3×C4×Dic9C2211yescharacteristic
31C3×C4×Dic9C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33(C2×D3)⋊Dic9C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37Dic3×Dic9C2212yes—
38C2×#3795—31no—
39C4×#634—31no—
40C3×C4×D12D311yescharacteristic
41C3×C42×D3—31no—
42C4×#625—32no—
43Dic3×D12—31no—
44#63879—32no—
45#63877—32no—
46C4×D3×Dic3—32no—
47#63844—31no—
48#63841—31no—
49C2×#3699—31no—
50C4×D4×C9—31no—
51D4×Dic9—31no—
52C4×#611—32no—
53C2×C4×Dic9—32no—
54#63454—31no—
55#63449—31no—
56#611⋊C4—32no—
57#63439—32no—
58C42×D9D311yescharacteristic, complemented
59C4×Dic18—31no—
60C4×Dic18—61no—
61C22×D3×C9C2312yes—
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2312yes—
68#6226C2×C414yescomplemented
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C9×D12C2×C414yescomplemented
74C4×D3×C9—24no—
75C3×C4×D9—22no—
76C3×C4×D9—22no—
77C3×C4×D9C2×C414yescomplemented
78#6194C2×C418yescomplemented
79D3×Dic9—24no—
80D3×Dic9C2×C418yescomplemented
81C4×#237—91no—
82C4×#235—91no—
83C2×#634—31no—
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3C2×D312yes—
89C2×#625—32no—
90C2×D3×Dic3—32no—
91C2×D3×Dic3—32no—
92#3795—38no—
93C32×C42C2×D311yescharacteristic
94C4⋊(C3⋊Dic3)—31no—
95#3750—32no—
96C4×C3⋊Dic3—32no—
97C3×(C2×D3)⋊C4C2×D312yes—
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×C4×Dic3—31no—
100C3×C4×Dic3—31no—
101C3×C4×Dic3—31no—
102#3724—32no—
103(C2×D3)⋊Dic3—32no—
104(C2×D3)⋊Dic3—32no—
105Dic32—32no—
106Dic32—32no—
107C2×D4×C9—31no—
108C22×C4×C9—32no—
109C2×#611—32no—
110C22×Dic9—32no—
111C22×Dic9—32no—
112#3699—38no—
113C2×C4×D9C2×D311yescharacteristic
114C2×C4×D9C2×D311yescharacteristic
115C2×C4×D9C2×D311yescharacteristic
116C2×Dic18—31no—
117C2×Dic18—61no—
118C9×C4⋊C4—31no—
119C9×C22⋊C4—32no—
120C42×C9—21no—
121C42×C9C2×D311yescharacteristic
122#3676—32no—
123(C2×D9)⋊C4—32no—
124C4⋊Dic9—31no—
125C4⋊Dic9—61no—
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4—62no—
129C4×Dic9—32no—
130C4×Dic9—32no—
131C4×Dic9—32no—
132C4×Dic9—62no—
133C4×Dic9C2×D311yescharacteristic
134C4×Dic9C2×D311yescharacteristic
135C4×Dic9C2×D311yescharacteristic
136C22×C3×C9C2411yescharacteristic
137C2×D3×C9—22no—
138C2×D3×C9—22no—
139C2×D3×C9C22×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—
147C3⋊Dic9C22×C414yes—
148C9×Dic3—22no—
149C3×Dic9—21no—
150C3×Dic9C22×C412yes—
151C3×Dic9#4112yescomplemented
152C2×#237—91no—
153C2×#235—91no—
154C3×C4×D4—31no—
155C2×C3×C42—91no—
156D4×Dic3—91no—
157C4×#67—92no—
158C4×#67—92no—
159C4×#67—92no—
160#926—91no—
161C2×C4×Dic3—92no—
162#892—91no—
163#887—91no—
164#67⋊C4—92no—
165#877—92no—
166C4×D12—91no—
167C4×D12D911yescharacteristic, complemented
168#872—92no—
169C42×D3—31no—
170C42×D3—92no—
171C4×Dic6—91no—
172C4×Dic6—91no—
173C4×Dic6—91no—
174C4×Dic6—181no—
175C22×C3×D3—31no—
176C22×C3×D3C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C3⋊Dic3—32no—
181C2×C3⋊Dic3—32no—
182#634—34no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3C22×D312yes—
187C3×D12C4×D314yes—
188C3×C4×D3—24no—
189C3×C4×D3—34no—
190C3×C4×D3—62no—
191C3×C4×D3—62no—
192#625—38no—
193D3×Dic3—38no—
194D3×Dic3—64no—
195C23×C9—32no—
196C22×D9C22×D311yescharacteristic
197D4×C9—34no—
198C2×C4×C9—21no—
199C2×C4×C9—21no—
200C2×C4×C9—21no—
201C2×C4×C9—32no—
202C2×C4×C9—64no—
203C2×C4×C9C22×D311yescharacteristic
204C2×C4×C9C22×D311yescharacteristic
205C2×C4×C9C22×D311yescharacteristic
206#611—38no—
207C2×Dic9—32no—
208C2×Dic9—38no—
209C2×Dic9—32no—
210C2×Dic9—62no—
211C2×Dic9—62no—
212C2×Dic9—64no—
213C2×Dic9C22×D311yescharacteristic
214C2×Dic9C22×D311yescharacteristic
215C2×Dic9C22×D311yescharacteristic
216C4×D9—22no—
217C4×D9—22no—
218C4×D9C4×D314yescomplemented
219Dic18—34no—
220Dic18—64no—
221C4×#41—271noSylow
222C2×C3×C9C23×C411yescharacteristic
223C2×C3×C9C2×#4111yescharacteristic
224C2×C3×C9C2×#4111yescharacteristic
225D3×C9—24no—
226C3×D9—22no—
227C2×C3×D4—31no—
228C22×C3×C4—32no—
229C22×C3×C4—91no—
230C22×C3×C4—91no—
231C22×C3×C4—91no—
232C2×#67—92no—
233C2×#67—92no—
234C22×Dic3—92no—
235C22×Dic3—92no—
236#237—98no—
237#235—98no—
238C2×D12C2×D911yescharacteristic
239C2×C4×D3—31no—
240C2×C4×D3—31no—
241C2×C4×D3—31no—
242C2×C4×D3—92no—
243C2×C4×D3—92no—
244C2×C4×D3C2×D912yes—
245C2×Dic6—91no—
246C2×Dic6—91no—
247C2×Dic6—181no—
248C3×C4⋊C4—31no—
249C3×C22⋊C4—32no—
250C3×C42—21no—
251C3×C42—91no—
252C3×C42—91no—
253C3×C42—91no—
254C3×C42C2×D911yescharacteristic
255C3×C42D3211yescharacteristic
256#217—92no—
257#217—92no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4C2×D912yes—
261C4⋊Dic3—91no—
262C4⋊Dic3—91no—
263C4⋊Dic3—181no—
264C4⋊Dic3C2×D911yescharacteristic
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—182no—
270C4×Dic3—31no—
271C4×Dic3—31no—
272C4×Dic3—31no—
273C4×Dic3—92no—
274C4×Dic3—92no—
275C4×Dic3—92no—
276C4×Dic3—92no—
277C4×Dic3—92no—
278C4×Dic3—92no—
279C4×Dic3—182no—
280C22×C32C23×D311yescharacteristic
281C2×C3×D3—22no—
282C2×C3×D3—22no—
283C2×C3×D3—32no—
284C2×C3×D3—61no—
285C2×C3×D3—61no—
286C2×C3×D3C2×C4×D314yes—
287C32×C4—21no—
288C32×C4C2×C4×D312yes—
289C32×C4#23712yes—
290C3⋊Dic3—34no—
291C3⋊Dic3—62no—
292C3×Dic3—22no—
293C3×Dic3—32no—
294C3×Dic3—32no—
295C3×Dic3—61no—
296C22×C9—21no—
297C22×C9—34no—
298C22×C9—62no—
299C22×C9—62no—
300C22×C9C23×D311yescharacteristic
301C2×D9—21no—
302C2×D9—21no—
303C2×D9C2×C4×D312yes—
304C4×C9—21no—
305C4×C9—22no—
306C4×C9—22no—
307C4×C9—41no—
308C4×C9—62no—
309C4×C9C2×C4×D312yes—
310C4×C9#23512yescomplemented
311Dic9—21no—
312Dic9—34no—
313Dic9—64no—
314Dic9—62no—
315Dic9—122no—
316Dic9C2×C4×D312yes—
317Dic9#23512yescomplemented
318C2×#41—271no—
319C4×Q8—271no—
320C4×Q8—271no—
321C4×D4—31no—
322C4×D4—271no—
323C4×D4—272no—
324C4×D4—272no—
325#117—272no—
326#117—271no—
327C2×C42—91no—
328C2×C42—272no—
329C3×C9C4×#4111yescharacteristic, complemented, Sylow
330C23×C3—32no—
331C23×C3—91no—
332C22×D3—31no—
333C22×D3C22×D912yes—
334C3×D4—34no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—32no—
339C2×C3×C4—64no—
340C2×C3×C4—91no—
341C2×C3×C4—91no—
342C2×C3×C4—94no—
343C2×C3×C4—91no—
344C2×C3×C4—182no—
345C2×C3×C4—182no—
346C2×C3×C4C22×D911yescharacteristic
347C2×C3×C4C22×D911yescharacteristic
348C2×C3×C4C22×D911yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C2×D3211yescharacteristic
352#67—98no—
353#67—98no—
354C2×Dic3—31no—
355C2×Dic3—31no—
356C2×Dic3—31no—
357C2×Dic3—98no—
358C2×Dic3—92no—
359C2×Dic3—92no—
360C2×Dic3—92no—
361C2×Dic3—92no—
362C2×Dic3—182no—
363C2×Dic3—182no—
364C2×Dic3—184no—
365C2×Dic3C22×D912yes—
366D12C4×D914yescomplemented
367C4×D3—24no—
368C4×D3—34no—
369C4×D3—62no—
370C4×D3—62no—
371C4×D3—98no—
372C4×D3—184no—
373Dic6—94no—
374Dic6—94no—
375Dic6—184no—
376C2×C32C22×C4×D311yescharacteristic
377C2×C32C2×#23711yescharacteristic
378C2×C32D3×#4111yescharacteristic
379C3×D3—24no—
380C3×D3—62no—
381C2×C9—21no—
382C2×C9—21no—
383C2×C9—21no—
384C2×C9—64no—
385C2×C9C22×C4×D311yescharacteristic
386C2×C9C2×#23511yescharacteristic
387C2×C9C2×#23511yescharacteristic
388D9—22no—
389#41—278no—
390C2×Q8—271no—
391C2×D4—31no—
392C2×D4—272no—
393C22×C4—32no—
394C22×C4—91no—
395C22×C4—91no—
396C22×C4—91no—
397C22×C4—272no—
398C22×C4—272no—
399C4⋊C4—31no—
400C4⋊C4—272no—
401C4⋊C4—272no—
402C4⋊C4—271no—
403C22⋊C4—32no—
404C22⋊C4—272no—
405C22⋊C4—272no—
406C42—91no—
407C42—91no—
408C42—91no—
409C42—272no—
410C42—272no—
411C42—272no—
412C42D3×D911yescharacteristic
413C22×C3—21no—
414C22×C3—34no—
415C22×C3—62no—
416C22×C3—62no—
417C22×C3—92no—
418C22×C3—181no—
419C22×C3—181no—
420C22×C3C23×D911yescharacteristic
421C22×C3C22×D3211yescharacteristic
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—32no—
425C2×D3—61no—
426C2×D3—61no—
427C2×D3C2×C4×D914yes—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—41no—
433C3×C4—62no—
434C3×C4—92no—
435C3×C4—92no—
436C3×C4—181no—
437C3×C4C2×C4×D912yes—
438C3×C4#369912yescomplemented
439C3×C4#379512yes—
440C3×C4C4×D3212yes—
441Dic3—22no—
442Dic3—32no—
443Dic3—32no—
444Dic3—61no—
445Dic3—94no—
446Dic3—94no—
447Dic3—184no—
448Dic3—182no—
449Dic3—182no—
450Dic3—362no—
451C32C4×#23711yescharacteristic
452C9—21no—
453C9C4×#23511yescharacteristic, complemented
454C23—32no—
455C23—91no—
456Q8—274no—
457D4—34no—
458D4—278no—
459C2×C4—32no—
460C2×C4—64no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—94no—
464C2×C4—91no—
465C2×C4—182no—
466C2×C4—182no—
467C2×C4—278no—
468C2×C4—272no—
469C2×C4—272no—
470C2×C4—544no—
471C2×C4C2×D3×D911yescharacteristic
472C2×C4C2×D3×D911yescharacteristic
473C2×C4C2×D3×D911yescharacteristic
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—64no—
478C2×C3—182no—
479C2×C3C22×C4×D911yescharacteristic
480C2×C3C2×#369911yescharacteristic
481C2×C3#41×D911yescharacteristic
482C2×C3C2×#379511yescharacteristic
483C2×C3C2×C4×D3211yescharacteristic
484C2×C3D3×#23511yescharacteristic
485D3—24no—
486D3—62no—
487C22—34no—
488C22—62no—
489C22—62no—
490C22—92no—
491C22—181no—
492C22—181no—
493C22C22×D3×D911yescharacteristic
494C4—21no—
495C4—62no—
496C4—92no—
497C4—92no—
498C4—181no—
499C4—274no—
500C4—542no—
501C4#8863812yesdirect product factor
502C4C4×D3×D912yes—
503C3—21no—
504C3C4×#369911yescharacteristic, complemented
505C3C4×#379511yescharacteristic
506C2—64no—
507C2—182no—
508C2C2×#8863811yescharacteristic
509C2C2×C4×D3×D911yescharacteristic
510C2D9×#23511yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#88638C2
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×#6226C2semidirect
3. C4×C9×D12C2semidirect
4. C3×C42×D9C2semidirect
5. C4×#6194C2semidirect
6. D12×Dic9C2semidirect
7. #11721012C2semidirect
8. #11721010C2semidirect
9. C4×D3×Dic9C2semidirect
10. #11720913C2semidirect
11. #11720907C2semidirect
23. #88638C4direct (N × H)
58. C42×D9D3semidirect
68. #6226C2×C4semidirect
73. C9×D12C2×C4semidirect
77. C3×C4×D9C2×C4semidirect
78. #6194C2×C4semidirect
80. D3×Dic9C2×C4semidirect
145. C3×C4×C9#41semidirect
151. C3×Dic9#41semidirect
167. C4×D12D9semidirect
218. C4×D9C4×D3semidirect
310. C4×C9#235semidirect
317. Dic9#235semidirect
329. C3×C9C4×#41semidirect
366. D12C4×D9semidirect
438. C3×C4#3699semidirect
453. C9C4×#235semidirect
501. C4#88638direct (N × H)
504. C3C4×#3699semidirect

Maximal subgroups

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

subgroupindex
C2×#886382
C4×#62262
C4×C9×D122
C3×C42×D92
C4×#61942
D12×Dic92
#117210122
#117210102
C4×D3×Dic92
#117209132
#117209072
C4×#37953
C4×#36993