Finite Group Databasebeta

← C2×#67×Dic9 #13152013

Subgroups — C2×#67×Dic9 #13152013

Order 1728.

1410 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C9×#67C211yesmaximal, characteristic
2C23×C3×Dic9C211yesmaximal, characteristic, complemented
3C22×D3×Dic9C211yesmaximal, characteristic, complemented
4C2×#88551C211yesmaximal, characteristic, complemented
5#67×Dic9C218yesmaximal, direct product factor
6C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic, complemented
7C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
8C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
9C2×Dic3×#67—31nomaximal
10C2×D4×Dic9—31nomaximal
11C24×C3×C9C2211yescharacteristic
12C23×D3×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C2×C9×#67C418yescomplemented
16C2×C9×#67C2214yes—
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9—24no—
19C22×C3×Dic9C2211yescharacteristic
20C22×C3×Dic9C2211yescharacteristic
21C22×C3×Dic9C2214yescomplemented
22C2×D3×Dic9—24no—
23C2×D3×Dic9C2214yescomplemented
24#88551C2214yescomplemented
25(C2×D3)⋊Dic9C2214yescomplemented
26Dic3⋊Dic9C2214yescomplemented
27Dic3×Dic9C2214yescomplemented
28C22×C3×#67D311yescharacteristic
29C23×C3×Dic3—31no—
30C22×D3×Dic3—31no—
31C2×#3757—31no—
32Dic3×#67—38no—
33C2×Dic3⋊Dic3—31no—
34C2×(C2×D3)⋊Dic3—31no—
35C2×Dic32—31no—
36C22×D4×C9—31no—
37C23×Dic9—31no—
38C23×Dic9D311yescharacteristic, complemented
39C2×#3676—31no—
40C2×#3676—31no—
41C2×#3676—61no—
42D4×Dic9—38no—
43C2×C4⋊Dic9—31no—
44C2×C4×Dic9—31no—
45C23×C3×C9—22no—
46C23×C3×C9—22no—
47C23×C3×C9C2×C414yescomplemented
48C23×C3×C9C2312yes—
49C23×C3×C9C2311yescharacteristic
50C22×D3×C9—22no—
51C22×D3×C9—22no—
52C22×D3×C9C2×C414yescomplemented
53C22×D3×C9C2312yes—
54C2×C3⋊Dic9—22no—
55C2×C3⋊Dic9—22no—
56C2×C3⋊Dic9C2312yes—
57C2×C3⋊Dic9C2312yes—
58C9×#67C2×C4116yescomplemented
59C2×C9×Dic3C2×C414yescomplemented
60C2×C9×Dic3C2312yes—
61C2×C3×Dic9—22no—
62C2×C3×Dic9—28no—
63C2×C3×Dic9D414yescomplemented
64C2×C3×Dic9C2312yes—
65C2×C3×Dic9C2312yes—
66D3×Dic9—28no—
67C2×D4×Dic3—91no—
68C2×C4×#67—91no—
69C24×C32C2×D311yescharacteristic
70C23×C3×D3C2×D311yescharacteristic
71C22×C3⋊Dic3—31no—
72C22×C3⋊Dic3—31no—
73C2×C3×#67Dic318yes—
74C2×C3×#67C2×D314yes—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—34no—
78C22×C3×Dic3—64no—
79C22×C3×Dic3C2×D311yescharacteristic
80C2×D3×Dic3—34no—
81C2×D3×Dic3—64no—
82C24×C9—21no—
83C24×C9—31no—
84C24×C9C2×D311yescharacteristic
85#3757—34no—
86Dic3⋊Dic3—34no—
87(C2×D3)⋊Dic3—34no—
88Dic32—34no—
89C2×D4×C9—38no—
90C2×D4×C9—34no—
91C22×C4×C9—31no—
92C22×Dic9—24no—
93C22×Dic9—31no—
94C22×Dic9—31no—
95C22×Dic9—34no—
96C22×Dic9—64no—
97C22×Dic9—61no—
98C22×Dic9—61no—
99C22×Dic9C2×D311yescharacteristic
100C22×Dic9C2×D311yescharacteristic
101C22×Dic9C2×D314yescomplemented
102#3676—34no—
103#3676—34no—
104#3676—64no—
105C4⋊Dic9—34no—
106C4×Dic9—34no—
107C22×C3×C9—28no—
108C22×C3×C9—22no—
109C22×C3×C9—22no—
110C22×C3×C9C22×C412yes—
111C22×C3×C9C22×C414yescomplemented
112C22×C3×C9C2×D412yes—
113C22×C3×C9#4112yes—
114C22×C3×C9C2411yescharacteristic
115C2×D3×C9—28no—
116C2×D3×C9—22no—
117C2×D3×C9—22no—
118C2×D3×C9C22×C414yescomplemented
119C3⋊Dic9—22no—
120C3⋊Dic9—22no—
121C9×Dic3C22×C414yescomplemented
122C3×Dic9—22no—
123C3×Dic9C2×D414yescomplemented
124C22×C3×D4—31no—
125C23×C3×C4—91no—
126C22×#67D911yescharacteristic
127C23×Dic3—31no—
128C23×Dic3—91no—
129C22×C4×D3—91no—
130C2×#217—91no—
131C2×#217—91no—
132C2×#217—91no—
133C2×#217—181no—
134D4×Dic3—98no—
135C4×#67—98no—
136C2×(C2×D3)⋊C4—91no—
137C2×C4⋊Dic3—91no—
138C2×Dic3⋊C4—91no—
139C2×C4×Dic3—91no—
140C2×C4×Dic3—91no—
141C23×C32—22no—
142C23×C32—22no—
143C23×C32C2×Dic314yes—
144C23×C32C22×D312yes—
145C23×C32C22×D311yescharacteristic
146C22×C3×D3—22no—
147C22×C3×D3—22no—
148C22×C3×D3C2×Dic314yes—
149C22×C3×D3C22×D312yes—
150C2×C3⋊Dic3—32no—
151C2×C3⋊Dic3—32no—
152C2×C3⋊Dic3—62no—
153C2×C3⋊Dic3—62no—
154C3×#67C2×Dic3116yes—
155C2×C3×Dic3—34no—
156C2×C3×Dic3—32no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—68no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3C2×Dic314yes—
161C2×C3×Dic3C22×D312yes—
162D3×Dic3—68no—
163C23×C9—22no—
164C23×C9—22no—
165C23×C9—24no—
166C23×C9—21no—
167C23×C9—22no—
168C23×C9—34no—
169C23×C9—32no—
170C23×C9—42no—
171C23×C9—42no—
172C23×C9—62no—
173C23×C9—62no—
174C23×C9C4×D314yescomplemented
175C23×C9C22×D312yes—
176C23×C9C22×D311yescharacteristic
177D4×C9—316no—
178C2×C4×C9—34no—
179C2×C4×C9—32no—
180C2×Dic9—22no—
181C2×Dic9—28no—
182C2×Dic9—32no—
183C2×Dic9—32no—
184C2×Dic9—62no—
185C2×Dic9—62no—
186C2×Dic9—62no—
187C2×Dic9—62no—
188C2×Dic9—68no—
189C2×Dic9—122no—
190C2×Dic9—122no—
191C2×Dic9#6714yesdirect product factor
192C2×Dic9C22×D312yes—
193C2×Dic9C22×D312yes—
194C2×C4×D4—271noSylow
195C2×C3×C9—24no—
196C2×C3×C9C4×D414yescomplemented
197C2×C3×C9C23×C411yescharacteristic
198C2×C3×C9C22×D411yescharacteristic
199C2×C3×C9C2×#4111yescharacteristic
200D3×C9—24no—
201C24×C3—21no—
202C24×C3—31no—
203C24×C3C2×D911yescharacteristic
204C24×C3D3211yescharacteristic
205C23×D3C2×D911yescharacteristic
206C2×C3×D4—38no—
207C2×C3×D4—34no—
208C22×C3×C4—31no—
209C22×C3×C4—91no—
210C22×C3×C4—91no—
211C22×C3×C4—94no—
212C22×C3×C4—184no—
213C2×#67Dic918yesdirect product factor
214C2×#67C2×D914yes—
215C22×Dic3—31no—
216C22×Dic3—31no—
217C22×Dic3—34no—
218C22×Dic3—64no—
219C22×Dic3—94no—
220C22×Dic3—91no—
221C22×Dic3—91no—
222C22×Dic3—91no—
223C22×Dic3—91no—
224C22×Dic3—181no—
225C22×Dic3—181no—
226C22×Dic3—184no—
227C22×Dic3C2×D911yescharacteristic
228C2×C4×D3—94no—
229C2×C4×D3—184no—
230#217—94no—
231#217—94no—
232#217—94no—
233#217—184no—
234(C2×D3)⋊C4—94no—
235C4⋊Dic3—94no—
236Dic3⋊C4—94no—
237C4×Dic3—94no—
238C4×Dic3—94no—
239C22×C32—28no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32D3×D412yes—
243C22×C32#23712yes—
244C22×C32C22×Dic312yes—
245C22×C32C22×Dic314yes—
246C22×C32C23×D311yescharacteristic
247C2×C3×D3—28no—
248C2×C3×D3—22no—
249C2×C3×D3—22no—
250C2×C3×D3C22×Dic314yes—
251C3⋊Dic3—62no—
252C3⋊Dic3—62no—
253C3×Dic3—34no—
254C3×Dic3—62no—
255C3×Dic3C22×Dic314yes—
256C22×C9—28no—
257C22×C9—22no—
258C22×C9—22no—
259C22×C9—21no—
260C22×C9—22no—
261C22×C9—24no—
262C22×C9—22no—
263C22×C9—22no—
264C22×C9—34no—
265C22×C9—42no—
266C22×C9—42no—
267C22×C9—48no—
268C22×C9—68no—
269C22×C9—62no—
270C22×C9—62no—
271C22×C9C2×C4×D312yes—
272C22×C9C2×C4×D314yescomplemented
273C22×C9#23512yes—
274C22×C9C2×#6712yes—
275C22×C9C23×D311yescharacteristic
276C4×C9—34no—
277Dic9—22no—
278Dic9—62no—
279Dic9—62no—
280Dic9—122no—
281Dic9—122no—
282Dic9C2×#6714yesdirect product factor
283C22×D4—31no—
284C23×C4—91no—
285C23×C4—271no—
286C4×D4—278no—
287C2×C4⋊C4—271no—
288C2×C22⋊C4—271no—
289C2×C22⋊C4—271no—
290C2×C42—271no—
291C3×C9C2×C4×D411yescharacteristic, complemented, Sylow
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—24no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3—32no—
300C23×C3—34no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—62no—
304C23×C3—62no—
305C23×C3C2×Dic914yescomplemented
306C23×C3C22×D912yes—
307C23×C3C22×D911yescharacteristic
308C23×C3D3×Dic314yes—
309C23×C3C2×D3212yes—
310C23×C3C2×D3211yescharacteristic
311C22×D3—22no—
312C22×D3—22no—
313C22×D3C2×Dic914yescomplemented
314C22×D3C22×D912yes—
315C3×D4—316no—
316C2×C3×C4—32no—
317C2×C3×C4—34no—
318C2×C3×C4—94no—
319C2×C3×C4—92no—
320C2×C3×C4—92no—
321C2×C3×C4—188no—
322C2×C3×C4—182no—
323#67C2×Dic9116yesdirect product factor
324C2×Dic3—32no—
325C2×Dic3—34no—
326C2×Dic3—32no—
327C2×Dic3—68no—
328C2×Dic3—62no—
329C2×Dic3—92no—
330C2×Dic3—92no—
331C2×Dic3—92no—
332C2×Dic3—92no—
333C2×Dic3—182no—
334C2×Dic3—182no—
335C2×Dic3—188no—
336C2×Dic3—182no—
337C2×Dic3—182no—
338C2×Dic3—182no—
339C2×Dic3—182no—
340C2×Dic3—362no—
341C2×Dic3—362no—
342C2×Dic3C2×Dic914yescomplemented
343C2×Dic3C22×D912yes—
344C4×D3—188no—
345C2×C32—24no—
346C2×C32D4×Dic314yes—
347C2×C32C2×D3×D411yescharacteristic
348C2×C32C2×#23711yescharacteristic
349C2×C32C23×Dic311yescharacteristic
350C3×D3—24no—
351C2×C9—24no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9—24no—
356C2×C9—44no—
357C2×C9—64no—
358C2×C9C4×#6714yescomplemented
359C2×C9C22×C4×D311yescharacteristic
360C2×C9C2×#23511yescharacteristic
361C2×C9C22×#6711yescharacteristic
362C24—31no—
363C24D3×D911yescharacteristic
364C2×D4—38no—
365C2×D4—34no—
366C22×C4—31no—
367C22×C4—91no—
368C22×C4—94no—
369C22×C4—91no—
370C22×C4—184no—
371C22×C4—274no—
372C22×C4—271no—
373C22×C4—271no—
374C22×C4—544no—
375C4⋊C4—274no—
376C22⋊C4—274no—
377C22⋊C4—274no—
378C42—274no—
379C22×C3—28no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—28no—
385C22×C3—21no—
386C22×C3—22no—
387C22×C3—24no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—34no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—48no—
394C22×C3—62no—
395C22×C3—62no—
396C22×C3—68no—
397C22×C3D4×D912yes—
398C22×C3#369912yes—
399C22×C3C22×Dic912yes—
400C22×C3C22×Dic914yescomplemented
401C22×C3C23×D911yescharacteristic
402C22×C3C2×D3×Dic312yes—
403C22×C3C2×D3×Dic314yes—
404C22×C3#380412yes—
405C22×C3D3×#6712yes—
406C22×C3C22×D3211yescharacteristic
407C2×D3—28no—
408C2×D3—22no—
409C2×D3—22no—
410C2×D3C22×Dic914yescomplemented
411C3×C4—34no—
412C3×C4—94no—
413C3×C4—182no—
414Dic3—34no—
415Dic3—62no—
416Dic3—182no—
417Dic3—182no—
418Dic3—182no—
419Dic3—182no—
420Dic3—362no—
421Dic3—362no—
422Dic3C22×Dic914yescomplemented
423C32C2×D4×Dic311yescharacteristic
424C9—21no—
425C9C2×C4×#6711yescharacteristic, complemented
426C23—22no—
427C23—22no—
428C23—32no—
429C23—34no—
430C23—62no—
431C23—62no—
432C23D3×Dic914yescomplemented
433C23C2×D3×D912yes—
434C23C2×D3×D911yescharacteristic
435D4—316no—
436C2×C4—32no—
437C2×C4—34no—
438C2×C4—92no—
439C2×C4—94no—
440C2×C4—92no—
441C2×C4—188no—
442C2×C4—182no—
443C2×C4—272no—
444C2×C4—272no—
445C2×C4—548no—
446C2×C4—542no—
447C2×C4—542no—
448C2×C3—24no—
449C2×C3—24no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—24no—
454C2×C3—44no—
455C2×C3—64no—
456C2×C3D4×Dic914yescomplemented
457C2×C3C2×D4×D911yescharacteristic
458C2×C3C2×#369911yescharacteristic
459C2×C3C23×Dic911yescharacteristic
460C2×C3Dic3×#6714yes—
461C2×C3C22×D3×Dic311yescharacteristic
462C2×C3C2×#380411yescharacteristic
463C2×C3C2×D3×#6711yescharacteristic
464D3—24no—
465C22—22no—
466C22—22no—
467C22—28no—
468C22—34no—
469C22—62no—
470C22—62no—
471C22—68no—
472C22C2×D3×Dic912yes—
473C22C2×D3×Dic914yescomplemented
474C22#8866212yes—
475C22D9×#6712yes—
476C22C22×D3×D911yescharacteristic
477C4—34no—
478C4—94no—
479C4—182no—
480C4—542no—
481C4—542no—
482C3—21no—
483C3C2×D4×Dic911yescharacteristic, complemented
484C3C2×Dic3×#6711yescharacteristic
485C2—24no—
486C2—64no—
487C2#67×Dic914yesdirect product factor
488C2C22×D3×Dic911yescharacteristic
489C2C2×#8866211yescharacteristic
490C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC22×C3×#67D3
cosocleC22×C3×C9C24
hypercenterC24D3×D9
hypocenterC3×C9C2×C4×D4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×C3×Dic9C2semidirect
3. C22×D3×Dic9C2semidirect
4. C2×#88551C2semidirect
5. #67×Dic9C2direct (N × H)
6. C2×(C2×D3)⋊Dic9C2semidirect
7. C2×Dic3⋊Dic9C2semidirect
8. C2×Dic3×Dic9C2semidirect
15. C2×C9×#67C4semidirect
21. C22×C3×Dic9C22semidirect
23. C2×D3×Dic9C22semidirect
24. #88551C22semidirect
25. (C2×D3)⋊Dic9C22semidirect
26. Dic3⋊Dic9C22semidirect
27. Dic3×Dic9C22semidirect
38. C23×Dic9D3semidirect
47. C23×C3×C9C2×C4semidirect
52. C22×D3×C9C2×C4semidirect
58. C9×#67C2×C4semidirect
59. C2×C9×Dic3C2×C4semidirect
63. C2×C3×Dic9D4semidirect
101. C22×Dic9C2×D3semidirect
111. C22×C3×C9C22×C4semidirect
118. C2×D3×C9C22×C4semidirect
121. C9×Dic3C22×C4semidirect
123. C3×Dic9C2×D4semidirect
174. C23×C9C4×D3semidirect
191. C2×Dic9#67direct (N × H)
196. C2×C3×C9C4×D4semidirect
213. C2×#67Dic9direct (N × H)
272. C22×C9C2×C4×D3semidirect
282. Dic9C2×#67direct (N × H)
291. C3×C9C2×C4×D4semidirect
305. C23×C3C2×Dic9semidirect
313. C22×D3C2×Dic9semidirect
323. #67C2×Dic9direct (N × H)
342. C2×Dic3C2×Dic9semidirect
358. C2×C9C4×#67semidirect
400. C22×C3C22×Dic9semidirect
410. C2×D3C22×Dic9semidirect
422. Dic3C22×Dic9semidirect
425. C9C2×C4×#67semidirect
432. C23D3×Dic9semidirect
456. C2×C3D4×Dic9semidirect
473. C22C2×D3×Dic9semidirect
483. C3C2×D4×Dic9semidirect
487. C2#67×Dic9direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C9×#672
C23×C3×Dic92
C22×D3×Dic92
C2×#885512
#67×Dic92
C2×(C2×D3)⋊Dic92
C2×Dic3⋊Dic92
C2×Dic3×Dic92
C2×Dic3×#673
C2×D4×Dic93