Finite Group Databasebeta

← D9×#875 #13147708

Subgroups — D9×#875 #13147708

Order 1728.

1066 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D9×D12C211yesmaximal, characteristic
2C2×D9×Dic6C211yesmaximal, characteristic, complemented
3#11721712C211yesmaximal, characteristic, complemented
4C9×#875C211yesmaximal, characteristic, complemented
5C3×C42×D9C211yesmaximal, characteristic, complemented
6D9×(C2×D3)⋊C4C214yesmaximal, complemented
7#11720964C211yesmaximal, characteristic, complemented
8#11720956C214yesmaximal, complemented
9#11720953C211yesmaximal, characteristic, complemented
10D3×#875—31nomaximal
11D9×#124—31nomaximal
12C22×D3×D9C2212yes—
13C2×C4⋊(C3⋊D9)C2211yescharacteristic
14C2×#6226C2211yescharacteristic, complemented
15C2×C9×D12C2211yescharacteristic
16C2×C3×C4×D9C2212yes—
17C2×C3×C4×D9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic, complemented
19C2×#6195C2212yes—
20C2×Dic3×D9C2212yes—
21C2×#6189C2212yescomplemented
22D9×D12—24no—
23D9×Dic6—24no—
24C3×C42×C9C2211yescharacteristic, complemented
25#88536C2214yescomplemented
26C9×(C2×D3)⋊C4C2214yescomplemented
27C3×C4×Dic9C2211yescharacteristic, complemented
28C3×C4×Dic9C2212yescomplemented
29(C2×D3)⋊Dic9C2214yescomplemented
30#88445C2214yescomplemented
31C2×D3×D12—31no—
32C2×D3×Dic6—31no—
33#64084—31no—
34C3×#875D311yescharacteristic
35C3×C42×D3—31no—
36D3×(C2×D3)⋊C4—34no—
37#63864—31no—
38#63860—31no—
39#63852—34no—
40C2×Q8×D9—31no—
41C2×D4×D9—31no—
42C9×#124—31no—
43#63509—31no—
44#63498—31no—
45#63447—34no—
46C22⋊C4×D9—34no—
47#63437—31no—
48#63437—61no—
49C42×D9D311yescharacteristic, complemented
50C22×C3⋊D9C2312yes—
51C22×D3×C9C2312yes—
52C22×C3×D9C2311yescharacteristic
53C2×D3×D9—24no—
54C2×D3×D9—28no—
55C2×C3×C4×C9C2312yes—
56C2×C3×C4×C9C2311yescharacteristic
57C2×C3⋊Dic9C2312yes—
58C4⋊(C3⋊D9)—22no—
59#6226—22no—
60C2×C9×Dic3C2312yes—
61C2×C3×Dic9C2312yes—
62C2×C3×Dic9C2311yescharacteristic
63C9×D12—22no—
64C3×C4×D9—24no—
65C3×C4×D9D414yes—
66C9×Dic6—22no—
67#6195—24no—
68Dic3×D9—24no—
69#6189—24no—
70D3×#124—91no—
71C2×#875—91no—
72C22×D32—32no—
73C2×C4⋊(C3⋊D3)—31no—
74C2×#634—31no—
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×C4×D3—31no—
77C2×C3×C4×D3—32no—
78C2×C3×Dic6C2×D311yescharacteristic
79C2×#627—32no—
80C2×#626—32no—
81C2×D3×Dic3—32no—
82D3×D12—64no—
83D3×Dic6—64no—
84C23×D9—32no—
85C32×C42C2×D311yescharacteristic
86#3752—34no—
87C3×(C2×D3)⋊C4C2×D314yes—
88C3×C4×Dic3—32no—
89C3×C4×Dic3—31no—
90#3722—34no—
91(C2×D3)⋊Dic3—34no—
92C2×Q8×C9—31no—
93C2×D4×C9—31no—
94C2×#611—32no—
95Q8×D9—64no—
96D4×D9—64no—
97C2×D36—31no—
98C2×D36—61no—
99C2×C4×D9—32no—
100C2×C4×D9C2×D312yes—
101C2×C4×D9C2×D311yescharacteristic
102C2×Dic18—31no—
103C2×Dic18—32no—
104C2×Dic18—61no—
105C9×C22⋊C4—34no—
106C42×C9—21no—
107C42×C9C2×D311yescharacteristic, complemented
108#3676—34no—
109(C2×D9)⋊C4—34no—
110(C2×D9)⋊C4—34no—
111(C2×D9)⋊C4—64no—
112C4×Dic9C2×D311yescharacteristic, complemented
113C4×Dic9C2×D312yescomplemented
114C22×C3×C9C2411yescharacteristic
115C2×C3⋊D9—22no—
116C2×C3⋊D9—24no—
117C2×D3×C9—22no—
118C2×D3×C9—24no—
119C2×C3×D9C2×D412yes—
120C2×C3×D9#4114yes—
121D3×D9—48no—
122C3×C4×C9—22no—
123C3×C4×C9C2×D412yes—
124C3⋊Dic9—22no—
125C9×Dic3—22no—
126C3×Dic9—22no—
127C3×Dic9C2×D412yes—
128C2×D3×Q8—91no—
129C2×D3×D4—91no—
130C22×D12—91no—
131C22×Dic6—91no—
132C3×#124—31no—
133C2×C3×C42—91no—
134#947—91no—
135#936—91no—
136C2×(C2×D3)⋊C4—94no—
137#885—94no—
138D3×C22⋊C4—94no—
139#875—91no—
140#875—91no—
141#875—91no—
142#875—91no—
143#875—94no—
144#875—181no—
145#875D911yescharacteristic, direct product factor
146C42×D3—31no—
147C22×C3⋊D3—32no—
148C22×C3×D3—31no—
149C22×C3×D3C22×D312yes—
150C2×D32—68no—
151C2×D32—64no—
152C2×C32×C4C22×D312yes—
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—32no—
155C4⋊(C3⋊D3)—62no—
156#634—62no—
157C2×C3×Dic3—31no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3C22×D312yes—
160C3×D12—22no—
161C3×C4×D3—34no—
162C3×C4×D3—64no—
163C3×Dic6—22no—
164#627—64no—
165#626—64no—
166D3×Dic3—64no—
167C23×C9—32no—
168C22×D9—32no—
169C22×D9—62no—
170C22×D9—64no—
171C22×D9—68no—
172C22×D9C22×D311yescharacteristic
173Q8×C9—62no—
174D4×C9—62no—
175C2×C4×C9—21no—
176C2×C4×C9—22no—
177C2×C4×C9—32no—
178C2×C4×C9C22×D312yes—
179C2×C4×C9C22×D311yescharacteristic
180#611—64no—
181C2×Dic9—32no—
182C2×Dic9—62no—
183C2×Dic9C22×D312yes—
184C2×Dic9C22×D311yescharacteristic
185D36—62no—
186D36—122no—
187C4×D9—24no—
188C4×D9—64no—
189C4×D9D1214yes—
190Dic18—62no—
191Dic18—64no—
192Dic18—122no—
193C2×#124—271noSylow
194C2×C3×C9C22×D411yescharacteristic
195C2×C3×C9C2×#4112yes—
196C3⋊D9—42no—
197D3×C9—42no—
198C3×D9#12414yescomplemented
199C23×D3—92no—
200C23×D3—92no—
201C2×C3×Q8—31no—
202C2×C3×D4—31no—
203C22×C3×C4—91no—
204C22×C3×C4—92no—
205C2×#67—92no—
206C22×Dic3—92no—
207D3×Q8—184no—
208D3×D4—184no—
209C2×D12—92no—
210C2×D12—91no—
211C2×D12—91no—
212C2×D12—181no—
213C2×D12—184no—
214C2×D12C2×D911yescharacteristic
215C2×C4×D3—31no—
216C2×C4×D3—32no—
217C2×C4×D3—92no—
218C2×Dic6—92no—
219C2×Dic6—91no—
220C2×Dic6—91no—
221C2×Dic6—92no—
222C2×Dic6—181no—
223C2×Dic6—184no—
224C2×Dic6C2×D911yescharacteristic, complemented
225C3×C22⋊C4—34no—
226C3×C42—21no—
227C3×C42—92no—
228C3×C42—91no—
229C3×C42C2×D911yescharacteristic, complemented
230C3×C42D3211yescharacteristic
231#217—94no—
232(C2×D3)⋊C4—94no—
233(C2×D3)⋊C4—94no—
234(C2×D3)⋊C4—94no—
235(C2×D3)⋊C4—94no—
236(C2×D3)⋊C4—94no—
237(C2×D3)⋊C4—184no—
238(C2×D3)⋊C4C2×D914yescomplemented
239C4×Dic3—32no—
240C4×Dic3—31no—
241C22×C32C23×D311yescharacteristic
242C2×C3⋊D3—62no—
243C2×C3⋊D3—64no—
244C2×C3×D3—22no—
245C2×C3×D3—24no—
246C2×C3×D3—32no—
247C2×C3×D3—34no—
248D32—128no—
249C32×C4—22no—
250C32×C4D3×D412yes—
251C3⋊Dic3—62no—
252C3×Dic3—22no—
253C3×Dic3—32no—
254C3×Dic3—62no—
255C22×C9—21no—
256C22×C9—62no—
257C22×C9—64no—
258C22×C9C23×D311yescharacteristic
259C2×D9—62no—
260C2×D9—64no—
261C2×D9—122no—
262C2×D9—124no—
263C2×D9—128no—
264C2×D9C2×D1212yes—
265C2×D9#23514yes—
266C4×C9—22no—
267C4×C9—22no—
268C4×C9—42no—
269C4×C9—62no—
270C4×C9C2×D1212yes—
271Dic9—22no—
272Dic9—62no—
273Dic9—122no—
274Dic9C2×D1212yes—
275C22×Q8—271no—
276C22×D4—271no—
277#124—31no—
278#124—274no—
279#124—271no—
280#124—271no—
281#124—271no—
282C2×C22⋊C4—274no—
283C2×C42—91no—
284C3×C9C2×#12411yescharacteristic, complemented, Sylow
285C23×C3—32no—
286C23×C3—91no—
287C22×D3—31no—
288C22×D3—92no—
289C22×D3—92no—
290C22×D3—182no—
291C22×D3—184no—
292C22×D3—188no—
293C22×D3—184no—
294C22×D3—188no—
295C22×D3C22×D912yes—
296C3×Q8—62no—
297C3×D4—62no—
298C2×C3×C4—21no—
299C2×C3×C4—22no—
300C2×C3×C4—32no—
301C2×C3×C4—91no—
302C2×C3×C4—94no—
303C2×C3×C4—92no—
304C2×C3×C4—184no—
305C2×C3×C4C22×D912yes—
306C2×C3×C4C22×D911yescharacteristic
307C2×C3×C4C2×D3212yes—
308C2×C3×C4C2×D3211yescharacteristic
309#67—184no—
310C2×Dic3—31no—
311C2×Dic3—32no—
312C2×Dic3—92no—
313C2×Dic3—92no—
314C2×Dic3—182no—
315C2×Dic3—184no—
316C2×Dic3C22×D912yes—
317D12—22no—
318D12—184no—
319D12—182no—
320D12—182no—
321D12—362no—
322C4×D3—34no—
323C4×D3—64no—
324C4×D3—184no—
325Dic6—22no—
326Dic6—184no—
327Dic6—184no—
328Dic6—182no—
329Dic6—182no—
330Dic6—362no—
331C2×C32C2×D3×D411yescharacteristic
332C2×C32D3×#4112yes—
333C3⋊D3—122no—
334C3×D3—34no—
335C3×D3—42no—
336C2×C9—21no—
337C2×C9—22no—
338C2×C9—122no—
339C2×C9C22×D1211yescharacteristic
340C2×C9C2×#23512yes—
341D9—122no—
342D9—242no—
343D9#87514yesdirect product factor
344C24—272no—
345C2×Q8—31no—
346C2×Q8—272no—
347C2×Q8—271no—
348C2×Q8—544no—
349C2×D4—31no—
350C2×D4—272no—
351C2×D4—271no—
352C2×D4—544no—
353C22×C4—91no—
354C22×C4—92no—
355C22×C4—272no—
356C22⋊C4—34no—
357C22⋊C4—274no—
358C22⋊C4—274no—
359C22⋊C4—274no—
360C42—92no—
361C42—91no—
362C42D3×D911yescharacteristic, complemented
363C22×C3—21no—
364C22×C3—62no—
365C22×C3—64no—
366C22×C3—92no—
367C22×C3—94no—
368C22×C3C23×D911yescharacteristic
369C22×C3C22×D3211yescharacteristic
370C2×D3—22no—
371C2×D3—24no—
372C2×D3—32no—
373C2×D3—34no—
374C2×D3—182no—
375C2×D3—184no—
376C2×D3—182no—
377C2×D3—184no—
378C2×D3—362no—
379C2×D3—364no—
380C2×D3—368no—
381C2×D3—368no—
382C3×C4—22no—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—42no—
386C3×C4—62no—
387C3×C4—92no—
388C3×C4—182no—
389C3×C4D4×D912yes—
390C3×C4D3×D1212yes—
391Dic3—22no—
392Dic3—32no—
393Dic3—62no—
394Dic3—182no—
395Dic3—182no—
396Dic3—362no—
397C32D3×#12411yescharacteristic
398C9—21no—
399C9C2×#87511yescharacteristic, complemented
400C23—32no—
401C23—91no—
402C23—272no—
403C23—544no—
404C23—548no—
405Q8—62no—
406Q8—544no—
407Q8—542no—
408D4—62no—
409D4—544no—
410D4—542no—
411C2×C4—32no—
412C2×C4—91no—
413C2×C4—94no—
414C2×C4—92no—
415C2×C4—184no—
416C2×C4—272no—
417C2×C4—544no—
418C2×C4C2×D3×D912yes—
419C2×C4C2×D3×D911yescharacteristic
420C2×C3—21no—
421C2×C3—22no—
422C2×C3—94no—
423C2×C3—122no—
424C2×C3C2×D4×D911yescharacteristic
425C2×C3#41×D912yes—
426C2×C3C2×D3×D1211yescharacteristic
427C2×C3D3×#23512yes—
428D3—34no—
429D3—42no—
430D3—362no—
431D3—362no—
432D3—722no—
433C22—62no—
434C22—64no—
435C22—92no—
436C22—94no—
437C22—542no—
438C22—544no—
439C22—1088no—
440C22C22×D3×D911yescharacteristic
441C4—22no—
442C4—62no—
443C4—92no—
444C4—182no—
445C4—542no—
446C4D9×D1212yes—
447C3—21no—
448C3D9×#12411yescharacteristic, complemented
449C3D3×#87511yescharacteristic
450C2—94no—
451C2—122no—
452C2—1082no—
453C2C2×D9×D1211yescharacteristic
454C2D9×#23512yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×D9×D12C2
hypercenterC42D3×D9
hypocenterC3×C9C2×#124
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×D9×Dic6C2semidirect
3. #11721712C2semidirect
4. C9×#875C2semidirect
5. C3×C42×D9C2semidirect
6. D9×(C2×D3)⋊C4C2semidirect
7. #11720964C2semidirect
8. #11720956C2semidirect
9. #11720953C2semidirect
14. C2×#6226C22semidirect
18. C2×C9×Dic6C22semidirect
21. C2×#6189C22semidirect
24. C3×C42×C9C22semidirect
25. #88536C22semidirect
26. C9×(C2×D3)⋊C4C22semidirect
27. C3×C4×Dic9C22semidirect
28. C3×C4×Dic9C22semidirect
29. (C2×D3)⋊Dic9C22semidirect
30. #88445C22semidirect
49. C42×D9D3semidirect
107. C42×C9C2×D3semidirect
112. C4×Dic9C2×D3semidirect
113. C4×Dic9C2×D3semidirect
145. #875D9direct (N × H)
198. C3×D9#124semidirect
224. C2×Dic6C2×D9semidirect
229. C3×C42C2×D9semidirect
238. (C2×D3)⋊C4C2×D9semidirect
284. C3×C9C2×#124semidirect
343. D9#875direct (N × H)
362. C42D3×D9semidirect
399. C9C2×#875semidirect
448. C3D9×#124semidirect

Maximal subgroups

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

subgroupindex
C2×D9×D122
C2×D9×Dic62
#117217122
C9×#8752
C3×C42×D92
D9×(C2×D3)⋊C42
#117209642
#117209562
#117209532
D3×#8753
D9×#1243