Finite Group Databasebeta

← #13147643

Subgroups — #13147643

Order 1728.

1006 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88640C211yesmaximal, characteristic
2#11721709C211yesmaximal, characteristic, complemented
3C9×#872C211yesmaximal, characteristic, complemented
4C3×C42×D9C211yesmaximal, characteristic, complemented
5#11721013C212yesmaximal, complemented
6#11721012C212yesmaximal, complemented
7C4×D3×Dic9C211yesmaximal, characteristic, complemented
8#11721002C212yesmaximal, complemented
9C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
10#11720943C211yesmaximal, characteristic, complemented
11#11720901C212yesmaximal, complemented
12#10593965—31nomaximal
13#10589624—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C4×D3×C9C2211yescharacteristic
16C2×C3×C4×D9C2212yes—
17C2×C3×C4×D9C2211yescharacteristic
18C2×#6195C2211yescharacteristic
19C2×#6194C2211yescharacteristic
20C2×D3×Dic9C2212yes—
21C2×#6192C2211yescharacteristic
22C2×(C3⋊D9)⋊C4C2212yes—
23C2×#6189C2211yescharacteristic
24#88640C418yescomplemented
25C3×C42×C9C2211yescharacteristic
26#88536C2212yes—
27#88534C2212yes—
28C4×C3⋊Dic9C2211yescharacteristic
29C9×(C2×D3)⋊C4C2212yes—
30C9×Dic3⋊C4C2212yes—
31C4×C9×Dic3C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33C3×C4×Dic9C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37Dic3×Dic9C2212yes—
38C2×#3798—31no—
39#64081—31no—
40C3×#872D311yescharacteristic
41C3×C42×D3—31no—
42#63904—32no—
43C4×#624—31no—
44#63880—32no—
45#63879—32no—
46C4×D3×Dic3—31no—
47#63855—31no—
48#63837—32no—
49C2×#3697—31no—
50C9×#117—31no—
51C2×C4×Dic9—31no—
52#63455—32no—
53#63449—32no—
54#611⋊C4—32no—
55#611⋊C4—32no—
56#63434—31no—
57#63434—31no—
58#63434—61no—
59C42×D9—31no—
60C42×D9D311yescharacteristic, complemented
61C22×C3⋊D9C2311yescharacteristic
62C22×D3×C9C2311yescharacteristic
63C22×C3×D9C2311yescharacteristic
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2311yescharacteristic
68C4×C3⋊D9C2×C414yescomplemented
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73C4×D3×C9C2×C414yescomplemented
74C3×C4×D9—24no—
75C3×C4×D9C2×C414yescomplemented
76#6195C2×C414yescomplemented
77#6194C2×C414yescomplemented
78D3×Dic9—24no—
79#6192C2×C414yescomplemented
80(C3⋊D9)⋊C4—24no—
81#6189C2×C414yescomplemented
82#5522—91no—
83C4×#235—91no—
84C2×C4×C3⋊D3—31no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—32no—
87C2×C3×C4×D3C2×D311yescharacteristic
88C2×#627—31no—
89C2×#626—31no—
90C2×#626—31no—
91C2×#625—31no—
92C2×#624—32no—
93C2×D3×Dic3—32no—
94#3798—38no—
95C32×C42C2×D311yescharacteristic
96#3752—32no—
97#3750—32no—
98C4×C3⋊Dic3—31no—
99C3×(C2×D3)⋊C4C2×D312yes—
100C3×Dic3⋊C4C2×D312yes—
101C3×C4×Dic3—32no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3C2×D311yescharacteristic
104Dic3⋊Dic3—32no—
105(C2×D3)⋊Dic3—32no—
106Dic32—32no—
107Dic32—32no—
108C22×C4×C9—31no—
109C2×#611—31no—
110C2×#611—31no—
111C22×Dic9—32no—
112#3697—38no—
113C2×D36—31no—
114C2×C4×D9—31no—
115C2×C4×D9—32no—
116C2×C4×D9—61no—
117C2×C4×D9C2×D312yes—
118C2×C4×D9C2×D311yescharacteristic
119C2×Dic18—31no—
120C9×C4⋊C4—32no—
121C9×C22⋊C4—32no—
122C42×C9—21no—
123C42×C9—31no—
124C42×C9C2×D311yescharacteristic
125(C2×D9)⋊C4—32no—
126(C2×D9)⋊C4—32no—
127(C2×D9)⋊C4—62no—
128Dic9⋊C4—32no—
129Dic9⋊C4—32no—
130Dic9⋊C4—62no—
131C4×Dic9—31no—
132C4×Dic9—32no—
133C4×Dic9—32no—
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×C9—22no—
145C3×C4×C9C22×C412yes—
146C3⋊Dic9—22no—
147C3⋊Dic9C22×C412yes—
148C9×Dic3—22no—
149C9×Dic3C22×C412yes—
150C3×Dic9C22×C412yes—
151C3×Dic9#4114yescomplemented
152C2×#235—91no—
153C2×#235—91no—
154C3×#117—31no—
155C2×C3×C42—91no—
156C4×#67—92no—
157C4×#67—92no—
158#926—91no—
159C2×C4×Dic3—91no—
160#893—92no—
161#887—92no—
162#67⋊C4—92no—
163#67⋊C4—92no—
164C4×D12—92no—
165#872—91no—
166#872—91no—
167#872—91no—
168#872—181no—
169#872D911yescharacteristic, complemented
170C42×D3—31no—
171C42×D3—91no—
172C42×D3—91no—
173C42×D3—91no—
174C4×Dic6—92no—
175C22×C3⋊D3—31no—
176C22×C3×D3—31no—
177C22×C3×D3C22×D311yescharacteristic
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D311yescharacteristic
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—32no—
182C4×C3⋊D3—34no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3C22×D312yes—
186C2×C3×Dic3C22×D311yescharacteristic
187C3×C4×D3—34no—
188C3×C4×D3—64no—
189C3×C4×D3C4×D314yes—
190#627—34no—
191#626—34no—
192#626—34no—
193#625—34no—
194#624—64no—
195D3×Dic3—64no—
196C23×C9—31no—
197C22×D9—31no—
198C22×D9—61no—
199C22×D9C22×D311yescharacteristic
200C2×C4×C9—21no—
201C2×C4×C9—22no—
202C2×C4×C9—31no—
203C2×C4×C9—34no—
204C2×C4×C9—32no—
205C2×C4×C9C22×D312yes—
206C2×C4×C9C22×D311yescharacteristic
207#611—34no—
208#611—34no—
209C2×Dic9—31no—
210C2×Dic9—32no—
211C2×Dic9—61no—
212C2×Dic9—62no—
213C2×Dic9—64no—
214C2×Dic9C22×D311yescharacteristic
215C2×Dic9C22×D312yes—
216D36—34no—
217C4×D9—24no—
218C4×D9—34no—
219C4×D9—64no—
220C4×D9—64no—
221C4×D9C4×D314yescomplemented
222Dic18—34no—
223C4×#41—271noSylow
224C2×C3×C9C23×C411yescharacteristic
225C2×C3×C9C2×#4112yes—
226C3⋊D9—22no—
227D3×C9—22no—
228C3×D9—22no—
229C22×C3×C4—31no—
230C22×C3×C4—91no—
231C22×C3×C4—92no—
232C2×#67—91no—
233C2×#67—91no—
234C2×#67—91no—
235C2×#67—91no—
236C22×Dic3—92no—
237#235—98no—
238#235—98no—
239C2×D12—91no—
240C2×D12—91no—
241C2×C4×D3—31no—
242C2×C4×D3—32no—
243C2×C4×D3—91no—
244C2×C4×D3—91no—
245C2×C4×D3—92no—
246C2×C4×D3—92no—
247C2×C4×D3—92no—
248C2×C4×D3—181no—
249C2×C4×D3C2×D911yescharacteristic
250C2×Dic6—91no—
251C2×Dic6—91no—
252C3×C4⋊C4—32no—
253C3×C22⋊C4—32no—
254C3×C42—21no—
255C3×C42—31no—
256C3×C42—92no—
257C3×C42—91no—
258C3×C42C2×D911yescharacteristic
259C3×C42D3211yescharacteristic
260#217—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—182no—
265(C2×D3)⋊C4C2×D912yes—
266C4⋊Dic3—92no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—182no—
271Dic3⋊C4C2×D912yes—
272C4×Dic3—31no—
273C4×Dic3—32no—
274C4×Dic3—92no—
275C4×Dic3—92no—
276C4×Dic3—91no—
277C4×Dic3—91no—
278C4×Dic3—92no—
279C4×Dic3—92no—
280C4×Dic3—181no—
281C4×Dic3C2×D911yescharacteristic
282C22×C32C23×D311yescharacteristic
283C2×C3⋊D3—32no—
284C2×C3⋊D3—62no—
285C2×C3×D3—22no—
286C2×C3×D3—32no—
287C2×C3×D3—62no—
288C2×C3×D3C2×C4×D312yes—
289C32×C4—22no—
290C32×C4C2×C4×D312yes—
291C3⋊Dic3—32no—
292C3⋊Dic3—62no—
293C3×Dic3—22no—
294C3×Dic3—32no—
295C3×Dic3—34no—
296C3×Dic3C2×C4×D312yes—
297C22×C9—21no—
298C22×C9—32no—
299C22×C9—62no—
300C22×C9C23×D311yescharacteristic
301C2×D9—22no—
302C2×D9—32no—
303C2×D9—62no—
304C2×D9—62no—
305C2×D9—122no—
306C2×D9C2×C4×D312yes—
307C4×C9—22no—
308C4×C9—22no—
309C4×C9—32no—
310C4×C9—42no—
311C4×C9—62no—
312C4×C9C2×C4×D312yes—
313Dic9—32no—
314Dic9—62no—
315Dic9—62no—
316Dic9—122no—
317Dic9C2×C4×D312yes—
318Dic9#23514yescomplemented
319C2×#41—271no—
320C4×Q8—272no—
321C4×D4—272no—
322C4×D4—272no—
323C4×D4—272no—
324#117—31no—
325#117—271no—
326#117—271no—
327C2×C42—91no—
328C2×C42—271no—
329C2×C42—271no—
330C3×C9C4×#4111yescharacteristic, complemented, Sylow
331C23×C3—31no—
332C23×C3—91no—
333C22×D3—31no—
334C22×D3—91no—
335C22×D3—91no—
336C22×D3—181no—
337C22×D3C22×D911yescharacteristic
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—31no—
341C2×C3×C4—34no—
342C2×C3×C4—32no—
343C2×C3×C4—92no—
344C2×C3×C4—91no—
345C2×C3×C4—94no—
346C2×C3×C4—184no—
347C2×C3×C4C22×D912yes—
348C2×C3×C4C22×D911yescharacteristic
349C2×C3×C4C2×D3212yes—
350C2×C3×C4C2×D3211yescharacteristic
351#67—94no—
352#67—94no—
353#67—94no—
354#67—94no—
355C2×Dic3—32no—
356C2×Dic3—31no—
357C2×Dic3—91no—
358C2×Dic3—91no—
359C2×Dic3—92no—
360C2×Dic3—92no—
361C2×Dic3—181no—
362C2×Dic3—182no—
363C2×Dic3—184no—
364C2×Dic3C22×D912yes—
365C2×Dic3C22×D911yescharacteristic
366D12—94no—
367D12—94no—
368C4×D3—34no—
369C4×D3—64no—
370C4×D3—94no—
371C4×D3—94no—
372C4×D3—184no—
373C4×D3—184no—
374C4×D3—184no—
375C4×D3—184no—
376C4×D3C4×D914yescomplemented
377Dic6—94no—
378Dic6—94no—
379C2×C32C22×C4×D311yescharacteristic
380C2×C32D3×#4112yes—
381C3⋊D3—62no—
382C3×D3—22no—
383C3×D3—62no—
384C2×C9—21no—
385C2×C9—22no—
386C2×C9—62no—
387C2×C9C22×C4×D311yescharacteristic
388C2×C9C2×#23512yes—
389D9—22no—
390D9—62no—
391D9—122no—
392#41—278no—
393C2×Q8—271no—
394C2×D4—271no—
395C2×D4—271no—
396C2×D4—271no—
397C22×C4—31no—
398C22×C4—91no—
399C22×C4—92no—
400C22×C4—272no—
401C22×C4—272no—
402C22×C4—271no—
403C4⋊C4—32no—
404C4⋊C4—272no—
405C4⋊C4—272no—
406C22⋊C4—32no—
407C22⋊C4—272no—
408C22⋊C4—272no—
409C42—31no—
410C42—92no—
411C42—91no—
412C42—272no—
413C42—272no—
414C42—271no—
415C42D3×D911yescharacteristic
416C22×C3—21no—
417C22×C3—32no—
418C22×C3—62no—
419C22×C3—92no—
420C22×C3—182no—
421C22×C3C23×D911yescharacteristic
422C22×C3C22×D3211yescharacteristic
423C2×D3—22no—
424C2×D3—32no—
425C2×D3—62no—
426C2×D3—92no—
427C2×D3—92no—
428C2×D3—182no—
429C2×D3—182no—
430C2×D3—182no—
431C2×D3—362no—
432C2×D3C2×C4×D912yes—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—32no—
437C3×C4—42no—
438C3×C4—62no—
439C3×C4—92no—
440C3×C4—94no—
441C3×C4C2×C4×D912yes—
442C3×C4C4×D3212yes—
443Dic3—22no—
444Dic3—32no—
445Dic3—34no—
446Dic3—92no—
447Dic3—92no—
448Dic3—182no—
449Dic3—182no—
450Dic3—182no—
451Dic3—362no—
452Dic3C2×C4×D912yes—
453C32#552211yescharacteristic
454C9—21no—
455C9C4×#23511yescharacteristic, complemented
456C23—31no—
457C23—91no—
458C23—271no—
459Q8—274no—
460D4—274no—
461D4—274no—
462D4—274no—
463C2×C4—31no—
464C2×C4—34no—
465C2×C4—32no—
466C2×C4—92no—
467C2×C4—91no—
468C2×C4—94no—
469C2×C4—184no—
470C2×C4—271no—
471C2×C4—274no—
472C2×C4—272no—
473C2×C4—544no—
474C2×C4—544no—
475C2×C4C2×D3×D912yes—
476C2×C4C2×D3×D911yescharacteristic
477C2×C3—21no—
478C2×C3—22no—
479C2×C3—62no—
480C2×C3—182no—
481C2×C3C22×C4×D911yescharacteristic
482C2×C3#41×D912yes—
483C2×C3C2×C4×D3211yescharacteristic
484C2×C3D3×#23512yes—
485D3—22no—
486D3—62no—
487D3—182no—
488D3—182no—
489D3—362no—
490C22—32no—
491C22—62no—
492C22—92no—
493C22—182no—
494C22—272no—
495C22—542no—
496C22C22×D3×D911yescharacteristic
497C4—22no—
498C4—32no—
499C4—62no—
500C4—92no—
501C4—94no—
502C4—272no—
503C4—542no—
504C4C4×D3×D912yes—
505C3—21no—
506C3#1058962411yescharacteristic, complemented
507C3#1059396511yescharacteristic
508C2—62no—
509C2—182no—
510C2—542no—
511C2C2×C4×D3×D911yescharacteristic
512C2D9×#23512yes—
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#88640C2
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. #11721709C2semidirect
3. C9×#872C2semidirect
4. C3×C42×D9C2semidirect
5. #11721013C2semidirect
6. #11721012C2semidirect
7. C4×D3×Dic9C2semidirect
8. #11721002C2semidirect
9. C4×(C3⋊D9)⋊C4C2semidirect
10. #11720943C2semidirect
11. #11720901C2semidirect
24. #88640C4semidirect
60. C42×D9D3semidirect
68. C4×C3⋊D9C2×C4semidirect
73. C4×D3×C9C2×C4semidirect
75. C3×C4×D9C2×C4semidirect
76. #6195C2×C4semidirect
77. #6194C2×C4semidirect
79. #6192C2×C4semidirect
81. #6189C2×C4semidirect
151. C3×Dic9#41semidirect
169. #872D9semidirect
221. C4×D9C4×D3semidirect
318. Dic9#235semidirect
330. C3×C9C4×#41semidirect
376. C4×D3C4×D9semidirect
455. C9C4×#235semidirect
506. C3#10589624semidirect

Maximal subgroups

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

subgroupindex
C2×#886402
#117217092
C9×#8722
C3×C42×D92
#117210132
#117210122
C4×D3×Dic92
#117210022
C4×(C3⋊D9)⋊C42
#117209432
#117209012
#105939653
#105896243