Finite Group Databasebeta

← #13148805

Subgroups — #13148805

Order 1728.

1006 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88634C211yesmaximal, characteristic
2#11721748C211yesmaximal, characteristic, complemented
3C3×#63455C211yesmaximal, characteristic, complemented
4C9×#887C211yesmaximal, characteristic, complemented
5#11721002C212yesmaximal, complemented
6Dic3×D36C211yesmaximal, characteristic, complemented
7C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
8#11720977C212yesmaximal, complemented
9#11720975C212yesmaximal, complemented
10C4×Dic3×D9C212yesmaximal, complemented
11Dic6×Dic9C211yesmaximal, characteristic, complemented
12#10594360—31nomaximal
13C4×#3701—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9C2212yes—
17C2×C9×Dic6C2211yescharacteristic
18C2×#6192C2212yes—
19C2×(C3⋊D9)⋊C4C2211yescharacteristic
20C2×(C3⋊D9)⋊C4C2211yescharacteristic
21C2×Dic3×D9C2212yes—
22C2×Dic3×D9C2212yes—
23#88634C418yescomplemented
24C3×C9×C4⋊C4C2211yescharacteristic
25#88536C2212yes—
26C4⋊(C3⋊Dic9)C2211yescharacteristic
27C4×C3⋊Dic9C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C9×Dic3⋊C4C2212yes—
30C4×C9×Dic3C2212yes—
31C4×C9×Dic3C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2212yes—
37Dic3×Dic9C2211yescharacteristic
38C2×#3796—31no—
39#64101—31no—
40C3×#893—31no—
41C3×#887D311yescharacteristic
42#63904—32no—
43Dic3×D12—31no—
44C4×#624—31no—
45#63880—32no—
46#63878—32no—
47C4×D3×Dic3—32no—
48Dic3×Dic6—31no—
49C2×#3701—31no—
50C4×Q8×C9—31no—
51Q8×Dic9—31no—
52#63455—32no—
53#63455D311yescharacteristic, complemented
54#63454—31no—
55#63454—32no—
56#63454—61no—
57C4×D36—31no—
58C4×D36—32no—
59C42×D9—32no—
60C42×D9—31no—
61C22×C3⋊D9C2311yescharacteristic
62C22×C3×D9C2312yes—
63C2×C3×C4×C9C2312yes—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2311yescharacteristic
67C4×C3⋊D9C2×C414yescomplemented
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C3×D36C2×C414yescomplemented
74C3×C4×D9—24no—
75C9×Dic6C2×C414yescomplemented
76#6192C2×C418yescomplemented
77(C3⋊D9)⋊C4—22no—
78(C3⋊D9)⋊C4—22no—
79Dic3×D9—24no—
80Dic3×D9C2×C418yescomplemented
81C4×#239—91no—
82C4×#237—91no—
83C2×C4×C3⋊D3—31no—
84C2×C3×D12—31no—
85C2×C3×C4×D3—32no—
86C2×C3×Dic6C2×D311yescharacteristic
87C2×#626—32no—
88C2×#624—31no—
89C2×#624—31no—
90C2×D3×Dic3—32no—
91C2×D3×Dic3—32no—
92#3796—38no—
93C32×C4⋊C4C2×D311yescharacteristic
94#3752—32no—
95C4⋊(C3⋊Dic3)—31no—
96C4×C3⋊Dic3—32no—
97C3×(C2×D3)⋊C4—32no—
98C3×Dic3⋊C4C2×D312yes—
99C3×C4×Dic3—31no—
100C3×C4×Dic3C2×D312yes—
101C3×C4×Dic3C2×D311yescharacteristic
102Dic3⋊Dic3—32no—
103(C2×D3)⋊Dic3—32no—
104Dic32—32no—
105Dic32—31no—
106Dic32—31no—
107C2×Q8×C9—31no—
108#3701—38no—
109C2×D36—32no—
110C2×D36C2×D311yescharacteristic
111C2×C4×D9—32no—
112C2×C4×D9—31no—
113C2×C4×D9—32no—
114C2×C4×D9—31no—
115C2×C4×D9—31no—
116C2×C4×D9—61no—
117C2×C4×D9C2×D312yes—
118C9×C4⋊C4—21no—
119C9×C4⋊C4—32no—
120C9×C4⋊C4C2×D311yescharacteristic
121C42×C9—32no—
122C42×C9—31no—
123(C2×D9)⋊C4—32no—
124(C2×D9)⋊C4—32no—
125(C2×D9)⋊C4—62no—
126(C2×D9)⋊C4C2×D312yes—
127C4⋊Dic9—31no—
128C4⋊Dic9—32no—
129C4⋊Dic9—61no—
130C4×Dic9—32no—
131C4×Dic9—32no—
132C4×Dic9—31no—
133C4×Dic9—31no—
134C4×Dic9—62no—
135C4×Dic9C2×D311yescharacteristic
136C22×C3×C9C2411yescharacteristic
137C2×C3⋊D9—21no—
138C2×C3⋊D9—21no—
139C2×C3⋊D9C22×C412yes—
140C2×C3×D9—22no—
141C2×C3×D9—22no—
142C2×C3×D9C22×C414yes—
143C3×C4×C9—22no—
144C3×C4×C9C22×C412yes—
145C3⋊Dic9—22no—
146C3⋊Dic9C22×C412yes—
147C9×Dic3—21no—
148C9×Dic3C22×C414yes—
149C9×Dic3#4112yescomplemented
150C3×Dic9—21no—
151C3×Dic9#4112yescomplemented
152C2×#239—91no—
153C2×#237—91no—
154C3×C4×Q8—31no—
155C3×C4×D4—91no—
156Q8×Dic3—91no—
157D4×Dic3—91no—
158C4×#67—92no—
159C2×C4×Dic3—92no—
160#893—31no—
161#893—92no—
162#892—92no—
163#892—91no—
164#892—91no—
165#892—181no—
166#887D911yescharacteristic, complemented
167#67⋊C4—92no—
168#877—92no—
169C4×D12—92no—
170C4×D12—91no—
171C42×D3—91no—
172C42×D3—92no—
173C42×D3—91no—
174C4×Dic6—91no—
175C22×C3⋊D3—31no—
176C22×C3×D3—32no—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—32no—
181C4×C3⋊D3—34no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3C22×D311yescharacteristic
185C2×C3×Dic3C22×D312yes—
186C2×C3×Dic3C22×D311yescharacteristic
187C3×D12—34no—
188C3×C4×D3—64no—
189C3×Dic6C4×D314yes—
190#626—38no—
191#624—62no—
192#624—62no—
193D3×Dic3—38no—
194D3×Dic3—64no—
195C22×D9—31no—
196C22×D9—61no—
197C22×D9C22×D312yes—
198Q8×C9—34no—
199C2×C4×C9—21no—
200C2×C4×C9—22no—
201C2×C4×C9—32no—
202C2×C4×C9—31no—
203C2×C4×C9—31no—
204C2×C4×C9C22×D312yes—
205C2×C4×C9C22×D311yescharacteristic
206C2×Dic9—31no—
207C2×Dic9—32no—
208C2×Dic9—61no—
209C2×Dic9—62no—
210C2×Dic9C22×D311yescharacteristic
211C2×Dic9C22×D311yescharacteristic
212D36—38no—
213D36C4×D314yescomplemented
214C4×D9—24no—
215C4×D9—38no—
216C4×D9—34no—
217C4×D9—64no—
218C4×D9—62no—
219C4×D9—64no—
220C4×D9—62no—
221C4×#41—271noSylow
222C2×C3×C9C23×C411yescharacteristic
223C2×C3×C9C2×#4111yescharacteristic
224C2×C3×C9C2×#4111yescharacteristic
225C3⋊D9—22no—
226C3×D9—24no—
227C2×C3×Q8—31no—
228C2×C3×D4—91no—
229C22×C3×C4—92no—
230C2×#67—92no—
231C22×Dic3—92no—
232C22×Dic3—92no—
233#239—98no—
234#237—98no—
235C2×D12—31no—
236C2×D12—92no—
237C2×C4×D3—32no—
238C2×C4×D3—92no—
239C2×C4×D3—91no—
240C2×C4×D3—91no—
241C2×C4×D3—91no—
242C2×C4×D3—91no—
243C2×C4×D3—92no—
244C2×C4×D3—91no—
245C2×C4×D3—91no—
246C2×C4×D3—181no—
247C2×Dic6C2×D911yescharacteristic
248C3×C4⋊C4—21no—
249C3×C4⋊C4—32no—
250C3×C4⋊C4C2×D911yescharacteristic
251C3×C4⋊C4D3211yescharacteristic
252C3×C22⋊C4—92no—
253C3×C42—32no—
254C3×C42—31no—
255C3×C42—91no—
256#217—92no—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—182no—
262C4⋊Dic3—92no—
263C4⋊Dic3—91no—
264C4⋊Dic3—91no—
265C4⋊Dic3—181no—
266Dic3⋊C4—92no—
267Dic3⋊C4C2×D912yes—
268C4×Dic3—31no—
269C4×Dic3—91no—
270C4×Dic3—91no—
271C4×Dic3—92no—
272C4×Dic3—92no—
273C4×Dic3—92no—
274C4×Dic3—92no—
275C4×Dic3—91no—
276C4×Dic3—91no—
277C4×Dic3—182no—
278C4×Dic3C2×D911yescharacteristic
279C4×Dic3C2×D912yes—
280C22×C32C23×D311yescharacteristic
281C2×C3⋊D3—32no—
282C2×C3⋊D3—61no—
283C2×C3⋊D3—61no—
284C2×C3×D3—34no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C32×C4—22no—
288C32×C4C2×C4×D312yes—
289C3⋊Dic3—32no—
290C3⋊Dic3—62no—
291C3×Dic3—21no—
292C3×Dic3—32no—
293C3×Dic3—61no—
294C3×Dic3C2×C4×D314yes—
295C3×Dic3#23912yes—
296C22×C9—21no—
297C22×C9C23×D311yescharacteristic
298C2×D9—22no—
299C2×D9—22no—
300C2×D9—32no—
301C2×D9—62no—
302C2×D9—61no—
303C2×D9—61no—
304C2×D9—121no—
305C2×D9—121no—
306C2×D9C2×C4×D314yes—
307C4×C9—22no—
308C4×C9—22no—
309C4×C9—34no—
310C4×C9—32no—
311C4×C9—42no—
312C4×C9—61no—
313C4×C9C2×C4×D312yes—
314Dic9—21no—
315Dic9—32no—
316Dic9—62no—
317Dic9—62no—
318Dic9—122no—
319Dic9#23712yescomplemented
320C2×#41—271no—
321C4×Q8—31no—
322C4×Q8—271no—
323C4×D4—91no—
324C4×D4—272no—
325C4×D4—271no—
326C4×D4—272no—
327#117—272no—
328#117—271no—
329C2×C42—271no—
330C2×C42—272no—
331C3×C9C4×#4111yescharacteristic, complemented, Sylow
332C23×C3—92no—
333C22×D3—32no—
334C22×D3—91no—
335C22×D3—91no—
336C22×D3—181no—
337C3×Q8—34no—
338C3×D4—94no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—32no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—91no—
345C2×C3×C4—91no—
346C2×C3×C4—184no—
347C2×C3×C4C22×D912yes—
348C2×C3×C4C22×D911yescharacteristic
349C2×C3×C4C2×D3212yes—
350C2×C3×C4C2×D3211yescharacteristic
351#67—98no—
352C2×Dic3—31no—
353C2×Dic3—31no—
354C2×Dic3—91no—
355C2×Dic3—91no—
356C2×Dic3—98no—
357C2×Dic3—92no—
358C2×Dic3—92no—
359C2×Dic3—181no—
360C2×Dic3—182no—
361C2×Dic3—184no—
362C2×Dic3C22×D911yescharacteristic
363C2×Dic3C22×D912yes—
364C2×Dic3C22×D911yescharacteristic
365D12—34no—
366D12—98no—
367C4×D3—64no—
368C4×D3—98no—
369C4×D3—94no—
370C4×D3—94no—
371C4×D3—184no—
372C4×D3—182no—
373C4×D3—182no—
374C4×D3—182no—
375C4×D3—184no—
376C4×D3—182no—
377Dic6C4×D914yescomplemented
378C2×C32C22×C4×D311yescharacteristic
379C2×C32C2×#23911yescharacteristic
380C2×C32D3×#4111yescharacteristic
381C3⋊D3—62no—
382C3×D3—64no—
383C2×C9—21no—
384C2×C9—21no—
385C2×C9—21no—
386C2×C9C22×C4×D311yescharacteristic
387C2×C9C2×#23711yescharacteristic
388C2×C9D3×#4111yescharacteristic
389D9—24no—
390D9—62no—
391D9—122no—
392#41—278no—
393C2×Q8—31no—
394C2×D4—91no—
395C2×D4—272no—
396C22×C4—92no—
397C22×C4—271no—
398C22×C4—271no—
399C22×C4—272no—
400C22×C4—272no—
401C22×C4—271no—
402C4⋊C4—32no—
403C4⋊C4—272no—
404C4⋊C4—271no—
405C4⋊C4D3×D911yescharacteristic
406C22⋊C4—92no—
407C22⋊C4—272no—
408C22⋊C4—272no—
409C42—32no—
410C42—31no—
411C42—91no—
412C42—271no—
413C42—271no—
414C42—272no—
415C42—272no—
416C22×C3—21no—
417C22×C3—94no—
418C22×C3—182no—
419C22×C3—182no—
420C22×C3C23×D911yescharacteristic
421C22×C3C22×D3211yescharacteristic
422C2×D3—34no—
423C2×D3—62no—
424C2×D3—62no—
425C2×D3—92no—
426C2×D3—92no—
427C2×D3—182no—
428C2×D3—181no—
429C2×D3—181no—
430C2×D3—181no—
431C2×D3—181no—
432C2×D3—361no—
433C2×D3—361no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—34no—
438C3×C4—32no—
439C3×C4—42no—
440C3×C4—61no—
441C3×C4—92no—
442C3×C4—181no—
443C3×C4C2×C4×D912yes—
444C3×C4C4×D3212yes—
445Dic3—21no—
446Dic3—32no—
447Dic3—61no—
448Dic3—92no—
449Dic3—92no—
450Dic3—182no—
451Dic3—182no—
452Dic3—182no—
453Dic3—362no—
454Dic3C2×C4×D914yes—
455Dic3#370112yescomplemented
456C32C4×#23911yescharacteristic
457C9—21no—
458C9C4×#23711yescharacteristic, complemented
459C23—92no—
460C23—271no—
461Q8—34no—
462D4—94no—
463D4—278no—
464C2×C4—31no—
465C2×C4—32no—
466C2×C4—31no—
467C2×C4—91no—
468C2×C4—91no—
469C2×C4—184no—
470C2×C4—278no—
471C2×C4—271no—
472C2×C4—274no—
473C2×C4—272no—
474C2×C4—542no—
475C2×C4—544no—
476C2×C4—542no—
477C2×C4C2×D3×D912yes—
478C2×C4C2×D3×D911yescharacteristic
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—184no—
483C2×C3C22×C4×D911yescharacteristic
484C2×C3C2×#370111yescharacteristic
485C2×C3#41×D911yescharacteristic
486C2×C3C2×C4×D3211yescharacteristic
487C2×C3D3×#23711yescharacteristic
488C2×C3D3×#23911yescharacteristic
489D3—64no—
490D3—182no—
491D3—182no—
492D3—362no—
493C22—94no—
494C22—182no—
495C22—182no—
496C22—272no—
497C22—541no—
498C22—541no—
499C22C22×D3×D911yescharacteristic
500C4—22no—
501C4—32no—
502C4—34no—
503C4—61no—
504C4—92no—
505C4—181no—
506C4—272no—
507C4—542no—
508C4C4×D3×D912yes—
509C3—21no—
510C3C4×#370111yescharacteristic, complemented
511C3#1059436011yescharacteristic
512C2—184no—
513C2—542no—
514C2C2×C4×D3×D911yescharacteristic
515C2D9×#23711yescharacteristic
516C2D3×#370111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#88634C2
cosocleC22×C3×C9C24
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C4×#41
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721748C2semidirect
3. C3×#63455C2semidirect
4. C9×#887C2semidirect
5. #11721002C2semidirect
6. Dic3×D36C2semidirect
7. C4×(C3⋊D9)⋊C4C2semidirect
8. #11720977C2semidirect
9. #11720975C2semidirect
10. C4×Dic3×D9C2semidirect
11. Dic6×Dic9C2semidirect
23. #88634C4semidirect
53. #63455D3semidirect
67. C4×C3⋊D9C2×C4semidirect
73. C3×D36C2×C4semidirect
75. C9×Dic6C2×C4semidirect
76. #6192C2×C4semidirect
80. Dic3×D9C2×C4semidirect
149. C9×Dic3#41semidirect
151. C3×Dic9#41semidirect
166. #887D9semidirect
213. D36C4×D3semidirect
319. Dic9#237semidirect
331. C3×C9C4×#41semidirect
377. Dic6C4×D9semidirect
455. Dic3#3701semidirect
458. C9C4×#237semidirect
510. C3C4×#3701semidirect

Maximal subgroups

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

subgroupindex
C2×#886342
#117217482
C3×#634552
C9×#8872
#117210022
Dic3×D362
C4×(C3⋊D9)⋊C42
#117209772
#117209752
C4×Dic3×D92
Dic6×Dic92
#105943603
C4×#37013