Finite Group Databasebeta

← C4×#88639 #13147591

Subgroups — C4×#88639 #13147591

Order 1728.

1038 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88639C211yesmaximal, characteristic
2C42×C3⋊D9C211yesmaximal, characteristic, complemented
3C4×C9×D12C211yesmaximal, characteristic, complemented
4C3×C4×Dic18C211yesmaximal, characteristic, complemented
5C4×#6195C212yesmaximal, complemented
6#11721025C211yesmaximal, characteristic, complemented
7#11721014C212yesmaximal, complemented
8#11721011C212yesmaximal, complemented
9C4×D3×Dic9C212yesmaximal, complemented
10#11720922C211yesmaximal, characteristic, complemented
11#11720910C211yesmaximal, characteristic, complemented
12C4×#3796—31nomaximal
13C4×#3699—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×D12C2211yescharacteristic
18C2×C4×D3×C9C2212yes—
19C2×C3×Dic18C2211yescharacteristic
20C2×#6195C2212yes—
21C2×D3×Dic9C2212yes—
22C2×D3×Dic9C2212yes—
23#88639C418yesdirect product factor
24C3×C42×C9C2211yescharacteristic
25C4×C3⋊Dic9C2211yescharacteristic
26C4×C3⋊Dic9C2211yescharacteristic
27C4×C3⋊Dic9C2211yescharacteristic
28C9×(C2×D3)⋊C4C2212yes—
29C9×C4⋊Dic3C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31C3×Dic9⋊C4C2212yes—
32C3×C4×Dic9C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34#88445C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37Dic3×Dic9C2212yes—
38C2×#3796—31no—
39C42×C3⋊D3—31no—
40C3×C4×D12D311yescharacteristic
41C3×C4×Dic6—31no—
42C4×#626—32no—
43#63898—31no—
44#63881—32no—
45#63878—32no—
46C4×D3×Dic3—32no—
47#63849—31no—
48#63843—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×D9—31no—
59C42×D9—61no—
60C4×Dic18D311yescharacteristic, complemented
61C22×C3⋊D9C2311yescharacteristic
62C22×D3×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C4×C3⋊D9—22no—
70C4×C3⋊D9—22no—
71C4×C3⋊D9C2×C414yescomplemented
72C2×C9×Dic3C2312yes—
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2312yes—
75C9×D12C2×C414yescomplemented
76C4×D3×C9—24no—
77C3×Dic18C2×C414yescomplemented
78#6195C2×C418yescomplemented
79D3×Dic9—24no—
80D3×Dic9C2×C418yescomplemented
81C4×#239—91no—
82C4×#237—91no—
83C2×C4×C3⋊D3—31no—
84C2×C4×C3⋊D3—31no—
85C2×C4×C3⋊D3—31no—
86C2×C3×D12C2×D311yescharacteristic
87C2×C3×C4×D3C2×D312yes—
88C2×C3×Dic6—31no—
89C2×#626—32no—
90C2×D3×Dic3—32no—
91C2×D3×Dic3—32no—
92#3796—38no—
93C32×C42C2×D311yescharacteristic
94C4×C3⋊Dic3—31no—
95C4×C3⋊Dic3—31no—
96C4×C3⋊Dic3—31no—
97C3×(C2×D3)⋊C4C2×D312yes—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3C2×D311yescharacteristic
100C3×Dic3⋊C4—32no—
101C3×C4×Dic3—32no—
102Dic3⋊Dic3—32no—
103#3722—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×D9—31no—
114C2×C4×D9—31no—
115C2×C4×D9—31no—
116C2×C4×D9—61no—
117C2×C4×D9—61no—
118C2×C4×D9—61no—
119C2×Dic18C2×D311yescharacteristic
120C9×C4⋊C4—31no—
121C9×C22⋊C4—32no—
122C42×C9—21no—
123C42×C9C2×D311yescharacteristic
124#3676—32no—
125(C2×D9)⋊C4—32no—
126C4⋊Dic9C2×D311yescharacteristic
127Dic9⋊C4—32no—
128Dic9⋊C4C2×D312yes—
129C4×Dic9—31no—
130C4×Dic9—31no—
131C4×Dic9—31no—
132C4×Dic9—32no—
133C4×Dic9—32no—
134C4×Dic9—61no—
135C4×Dic9—61no—
136C4×Dic9—61no—
137C4×Dic9C2×D312yes—
138C22×C3×C9C2411yescharacteristic
139C2×C3⋊D9—21no—
140C2×C3⋊D9—21no—
141C2×C3⋊D9C22×C412yes—
142C2×D3×C9—22no—
143C2×D3×C9—22no—
144C2×D3×C9C22×C414yes—
145C3×C4×C9—21no—
146C3×C4×C9C22×C412yes—
147C3×C4×C9#4112yescomplemented
148C3⋊Dic9—21no—
149C3⋊Dic9C22×C412yes—
150C3⋊Dic9#4112yescomplemented
151C9×Dic3—22no—
152C3×Dic9—22no—
153C3×Dic9C22×C414yes—
154C2×#239—91no—
155C2×#237—91no—
156C3×C4×Q8—91no—
157C3×C4×D4—31no—
158Q8×Dic3—91no—
159D4×Dic3—91no—
160C4×#67—92no—
161C2×C4×Dic3—92no—
162#893—91no—
163#893—92no—
164#892—91no—
165#892—92no—
166#892—91no—
167#887—91no—
168#67⋊C4—92no—
169#877—92no—
170C4×D12—92no—
171C4×D12D911yescharacteristic, complemented
172C42×D3—92no—
173C42×D3—91no—
174C42×D3—91no—
175C42×D3—181no—
176C4×Dic6—31no—
177C22×C3⋊D3—31no—
178C22×C3×D3C22×D312yes—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C3⋊Dic3—31no—
183C2×C3⋊Dic3—31no—
184C2×C3⋊Dic3—31no—
185C4×C3⋊D3—34no—
186C4×C3⋊D3—62no—
187C4×C3⋊D3—62no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3C22×D312yes—
191C3×D12C4×D314yes—
192C3×C4×D3—24no—
193C3×Dic6—34no—
194#626—38no—
195D3×Dic3—38no—
196D3×Dic3—64no—
197C23×C9—32no—
198C22×D9—31no—
199C22×D9—61no—
200D4×C9—34no—
201C2×C4×C9—21no—
202C2×C4×C9—21no—
203C2×C4×C9—21no—
204C2×C4×C9—32no—
205C2×C4×C9—64no—
206C2×C4×C9C22×D311yescharacteristic
207C2×C4×C9C22×D311yescharacteristic
208C2×C4×C9C22×D311yescharacteristic
209#611—38no—
210C2×Dic9—31no—
211C2×Dic9—31no—
212C2×Dic9—38no—
213C2×Dic9—31no—
214C2×Dic9—61no—
215C2×Dic9—61no—
216C2×Dic9—61no—
217C2×Dic9—64no—
218C2×Dic9C22×D312yes—
219C2×Dic9C22×D312yes—
220C4×D9—34no—
221C4×D9—64no—
222C4×D9—62no—
223C4×D9—64no—
224C4×D9—64no—
225C4×D9—62no—
226Dic18C4×D314yescomplemented
227C4×#41—271noSylow
228C2×C3×C9C23×C411yescharacteristic
229C2×C3×C9C2×#4111yescharacteristic
230C2×C3×C9C2×#4111yescharacteristic
231C3⋊D9—22no—
232D3×C9—24no—
233C2×C3×Q8—91no—
234C2×C3×D4—31no—
235C22×C3×C4—32no—
236C2×#67—92no—
237C22×Dic3—92no—
238C22×Dic3—92no—
239#239—98no—
240#237—98no—
241C2×D12—92no—
242C2×D12C2×D911yescharacteristic
243C2×C4×D3—92no—
244C2×C4×D3—91no—
245C2×C4×D3—91no—
246C2×C4×D3—92no—
247C2×C4×D3—91no—
248C2×C4×D3—91no—
249C2×C4×D3—91no—
250C2×C4×D3—91no—
251C2×C4×D3—181no—
252C2×C4×D3—181no—
253C2×C4×D3—181no—
254C2×C4×D3C2×D912yes—
255C2×Dic6—31no—
256C3×C4⋊C4—31no—
257C3×C4⋊C4—92no—
258C3×C4⋊C4—91no—
259C3×C22⋊C4—32no—
260C3×C42—21no—
261C3×C42—92no—
262C3×C42C2×D911yescharacteristic
263C3×C42D3211yescharacteristic
264#217—92no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—92no—
267(C2×D3)⋊C4—92no—
268(C2×D3)⋊C4C2×D912yes—
269C4⋊Dic3—31no—
270C4⋊Dic3—92no—
271C4⋊Dic3C2×D911yescharacteristic
272Dic3⋊C4—32no—
273Dic3⋊C4—92no—
274C4×Dic3—32no—
275C4×Dic3—92no—
276C4×Dic3—92no—
277C4×Dic3—91no—
278C4×Dic3—91no—
279C4×Dic3—91no—
280C4×Dic3—91no—
281C4×Dic3—91no—
282C4×Dic3—91no—
283C4×Dic3—92no—
284C4×Dic3—92no—
285C4×Dic3—181no—
286C4×Dic3—181no—
287C4×Dic3—181no—
288C22×C32C23×D311yescharacteristic
289C2×C3⋊D3—32no—
290C2×C3⋊D3—61no—
291C2×C3⋊D3—61no—
292C2×C3×D3—22no—
293C2×C3×D3—22no—
294C2×C3×D3C2×C4×D314yes—
295C32×C4—21no—
296C32×C4C2×C4×D312yes—
297C32×C4#23712yes—
298C3⋊Dic3—32no—
299C3⋊Dic3—32no—
300C3⋊Dic3—61no—
301C3×Dic3—22no—
302C3×Dic3—34no—
303C3×Dic3—62no—
304C22×C9—21no—
305C22×C9—34no—
306C22×C9—62no—
307C22×C9—62no—
308C22×C9C23×D311yescharacteristic
309C2×D9—32no—
310C2×D9—62no—
311C2×D9—62no—
312C2×D9—62no—
313C2×D9—61no—
314C2×D9—61no—
315C4×C9—22no—
316C4×C9—21no—
317C4×C9—22no—
318C4×C9—22no—
319C4×C9—62no—
320C4×C9C2×C4×D312yes—
321C4×C9#23912yescomplemented
322Dic9—22no—
323Dic9—32no—
324Dic9—32no—
325Dic9—62no—
326Dic9—62no—
327Dic9—62no—
328Dic9—61no—
329Dic9C2×C4×D314yes—
330C2×#41—271no—
331C4×Q8—91no—
332C4×Q8—271no—
333C4×D4—31no—
334C4×D4—272no—
335C4×D4—272no—
336C4×D4—271no—
337#117—272no—
338#117—271no—
339C2×C42—272no—
340C2×C42—271no—
341C3×C9C4×#4111yescharacteristic, complemented, Sylow
342C23×C3—32no—
343C22×D3—91no—
344C22×D3—91no—
345C22×D3—181no—
346C22×D3C22×D912yes—
347C3×Q8—94no—
348C3×D4—34no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—32no—
353C2×C3×C4—64no—
354C2×C3×C4—92no—
355C2×C3×C4—92no—
356C2×C3×C4C22×D911yescharacteristic
357C2×C3×C4C22×D911yescharacteristic
358C2×C3×C4C22×D911yescharacteristic
359C2×C3×C4C2×D3211yescharacteristic
360C2×C3×C4C2×D3211yescharacteristic
361C2×C3×C4C2×D3211yescharacteristic
362#67—98no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—98no—
366C2×Dic3—91no—
367C2×Dic3—91no—
368C2×Dic3—91no—
369C2×Dic3—91no—
370C2×Dic3—91no—
371C2×Dic3—91no—
372C2×Dic3—181no—
373C2×Dic3—181no—
374C2×Dic3—181no—
375C2×Dic3—184no—
376C2×Dic3C22×D912yes—
377D12—98no—
378D12C4×D914yescomplemented
379C4×D3—24no—
380C4×D3—98no—
381C4×D3—94no—
382C4×D3—94no—
383C4×D3—184no—
384C4×D3—182no—
385C4×D3—182no—
386C4×D3—184no—
387C4×D3—184no—
388C4×D3—184no—
389C4×D3—182no—
390C4×D3—182no—
391Dic6—34no—
392C2×C32C22×C4×D311yescharacteristic
393C2×C32C2×#23711yescharacteristic
394C2×C32D3×#4111yescharacteristic
395C3⋊D3—62no—
396C3×D3—24no—
397C2×C9—21no—
398C2×C9—21no—
399C2×C9—21no—
400C2×C9—64no—
401C2×C9C22×C4×D311yescharacteristic
402C2×C9C2×#23911yescharacteristic
403C2×C9D3×#4111yescharacteristic
404D9—64no—
405D9—62no—
406#41—278no—
407C2×Q8—91no—
408C2×D4—31no—
409C2×D4—272no—
410C22×C4—32no—
411C22×C4—272no—
412C22×C4—272no—
413C22×C4—271no—
414C22×C4—271no—
415C22×C4—271no—
416C4⋊C4—31no—
417C4⋊C4—92no—
418C4⋊C4—91no—
419C4⋊C4—272no—
420C22⋊C4—32no—
421C22⋊C4—272no—
422C22⋊C4—272no—
423C42—92no—
424C42—272no—
425C42—272no—
426C42—271no—
427C42—271no—
428C42—271no—
429C42D3×D911yescharacteristic
430C22×C3—21no—
431C22×C3—34no—
432C22×C3—62no—
433C22×C3—62no—
434C22×C3C23×D911yescharacteristic
435C22×C3C22×D3211yescharacteristic
436C2×D3—22no—
437C2×D3—22no—
438C2×D3—92no—
439C2×D3—92no—
440C2×D3—182no—
441C2×D3—182no—
442C2×D3—182no—
443C2×D3—181no—
444C2×D3—181no—
445C2×D3—181no—
446C2×D3—181no—
447C2×D3C2×C4×D914yes—
448C3×C4—22no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—62no—
454C3×C4—94no—
455C3×C4—182no—
456C3×C4C2×C4×D912yes—
457C3×C4#369912yescomplemented
458C3×C4#379612yes—
459C3×C4C4×D3212yes—
460Dic3—22no—
461Dic3—34no—
462Dic3—62no—
463Dic3—92no—
464Dic3—92no—
465Dic3—92no—
466Dic3—92no—
467Dic3—182no—
468Dic3—182no—
469Dic3—182no—
470Dic3—181no—
471Dic3—181no—
472C32C4×#23711yescharacteristic
473C9—21no—
474C9C4×#23911yescharacteristic, complemented
475C23—32no—
476C23—271no—
477Q8—94no—
478D4—34no—
479D4—278no—
480C2×C4—32no—
481C2×C4—64no—
482C2×C4—92no—
483C2×C4—92no—
484C2×C4—278no—
485C2×C4—271no—
486C2×C4—271no—
487C2×C4—274no—
488C2×C4—271no—
489C2×C4—542no—
490C2×C4—544no—
491C2×C4—542no—
492C2×C4C2×D3×D911yescharacteristic
493C2×C4C2×D3×D911yescharacteristic
494C2×C4C2×D3×D911yescharacteristic
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—64no—
499C2×C3C22×C4×D911yescharacteristic
500C2×C3C2×#369911yescharacteristic
501C2×C3#41×D911yescharacteristic
502C2×C3C2×#379611yescharacteristic
503C2×C3C2×C4×D3211yescharacteristic
504C2×C3#6430611yescharacteristic
505D3—24no—
506D3—184no—
507D3—182no—
508D3—182no—
509C22—34no—
510C22—62no—
511C22—62no—
512C22—272no—
513C22—541no—
514C22—541no—
515C22C22×D3×D911yescharacteristic
516C4—21no—
517C4—62no—
518C4—94no—
519C4—182no—
520C4—272no—
521C4—272no—
522C4—541no—
523C4#8863912yesdirect product factor
524C4C4×D3×D912yes—
525C3—21no—
526C3C4×#369911yescharacteristic, complemented
527C3C4×#379611yescharacteristic
528C2—64no—
529C2—542no—
530C2C2×#8863911yescharacteristic
531C2C2×C4×D3×D911yescharacteristic
532C2#1172247511yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#88639C2
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. C42×C3⋊D9C2semidirect
3. C4×C9×D12C2semidirect
4. C3×C4×Dic18C2semidirect
5. C4×#6195C2semidirect
6. #11721025C2semidirect
7. #11721014C2semidirect
8. #11721011C2semidirect
9. C4×D3×Dic9C2semidirect
10. #11720922C2semidirect
11. #11720910C2semidirect
23. #88639C4direct (N × H)
60. C4×Dic18D3semidirect
71. C4×C3⋊D9C2×C4semidirect
75. C9×D12C2×C4semidirect
77. C3×Dic18C2×C4semidirect
78. #6195C2×C4semidirect
80. D3×Dic9C2×C4semidirect
147. C3×C4×C9#41semidirect
150. C3⋊Dic9#41semidirect
171. C4×D12D9semidirect
226. Dic18C4×D3semidirect
321. C4×C9#239semidirect
341. C3×C9C4×#41semidirect
378. D12C4×D9semidirect
457. C3×C4#3699semidirect
474. C9C4×#239semidirect
523. C4#88639direct (N × H)
526. C3C4×#3699semidirect

Maximal subgroups

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

subgroupindex
C2×#886392
C42×C3⋊D92
C4×C9×D122
C3×C4×Dic182
C4×#61952
#117210252
#117210142
#117210112
C4×D3×Dic92
#117209222
#117209102
C4×#37963
C4×#36993