Finite Group Databasebeta

← D9×#887 #13148801

Subgroups — D9×#887 #13148801

Order 1728.

964 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D9×Dic6C211yesmaximal, characteristic
2#11721743C211yesmaximal, characteristic, complemented
3C3×C4⋊C4×D9C211yesmaximal, characteristic
4C9×#887C211yesmaximal, characteristic, complemented
5D9×Dic3⋊C4C212yesmaximal
6C4×Dic3×D9C211yesmaximal, characteristic
7C4×Dic3×D9C212yesmaximal
8#11720912C211yesmaximal, characteristic, complemented
9#11720911C211yesmaximal, characteristic, complemented
10#11720902C212yesmaximal, complemented
11#11720900C212yesmaximal, complemented
12D3×#887—31nomaximal
13C4×Q8×D9—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C3×C4×D9C2212yes—
16C2×C3×C4×D9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×Dic3×D9C2211yescharacteristic
19C2×Dic3×D9C2212yes—
20C2×Dic3×D9C2211yescharacteristic
21C2×#6189C2212yes—
22D9×Dic6C418yescomplemented
23C3×C9×C4⋊C4C2211yescharacteristic
24#88534C2212yes—
25C4×C3⋊Dic9C2212yes—
26C4×C3⋊Dic9C2211yescharacteristic
27C9×Dic3⋊C4C2212yes—
28C4×C9×Dic3C2212yes—
29C4×C9×Dic3C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31C3×Dic9⋊C4C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34Dic3×Dic9C2211yescharacteristic
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2211yescharacteristic
37Dic3×Dic9C2212yes—
38C2×D3×Dic6—31no—
39#64096—31no—
40C3×D3×C4⋊C4—31no—
41C3×#887D311yescharacteristic
42D3×Dic3⋊C4—32no—
43C4×D3×Dic3—31no—
44C4×D3×Dic3—32no—
45#63844—31no—
46#63843—31no—
47#63838—32no—
48#63837—32no—
49C2×Q8×D9—31no—
50C4×Q8×C9—31no—
51Q8×Dic9—31no—
52C4⋊C4×D9—32no—
53C4⋊C4×D9D311yescharacteristic
54#63449—31no—
55#63449—32no—
56#63449—61no—
57C42×D9—31no—
58C42×D9—32no—
59C4×Dic18—31no—
60C4×Dic18—32no—
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2311yescharacteristic
67#6226C2×C414yescomplemented
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C3×C4×D9—24no—
74C3×C4×D9C2×C414yes—
75C9×Dic6C2×C414yescomplemented
76Dic3×D9—22no—
77Dic3×D9C2×C418yes—
78Dic3×D9Q814yescomplemented
79#6189C2×C418yescomplemented
80C4×D3×Q8—91no—
81C2×#887—91no—
82C2×#634—31no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—32no—
85C2×C3×Dic6C2×D311yescharacteristic
86C2×#627—32no—
87C2×D3×Dic3—32no—
88C2×D3×Dic3—31no—
89C2×D3×Dic3—31no—
90D3×Dic6—38no—
91C32×C4⋊C4C2×D311yescharacteristic
92#3750—32no—
93C4×C3⋊Dic3—32no—
94C4×C3⋊Dic3—31no—
95C3×C4⋊Dic3—31no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4C2×D312yes—
98C3×C4×Dic3C2×D312yes—
99C3×C4×Dic3C2×D311yescharacteristic
100#3724—32no—
101Dic3⋊Dic3—32no—
102Dic32—31no—
103Dic32—32no—
104Dic32—31no—
105Dic32—32no—
106C2×Q8×C9—31no—
107Q8×D9—38no—
108C2×C4×D9—31no—
109C2×C4×D9—32no—
110C2×C4×D9—31no—
111C2×C4×D9C2×D312yes—
112C2×C4×D9C2×D311yescharacteristic
113C2×Dic18—31no—
114C2×Dic18—32no—
115C2×Dic18—61no—
116C9×C4⋊C4—21no—
117C9×C4⋊C4—32no—
118C9×C4⋊C4C2×D311yescharacteristic
119C42×C9—32no—
120C42×C9—31no—
121C4⋊Dic9—32no—
122C4⋊Dic9C2×D311yescharacteristic
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4—62no—
126Dic9⋊C4C2×D312yes—
127C4×Dic9—31no—
128C4×Dic9—32no—
129C4×Dic9—32no—
130C4×Dic9—32no—
131C4×Dic9—31no—
132C4×Dic9—31no—
133C4×Dic9—62no—
134C4×Dic9—61no—
135C22×C3×C9C2411yescharacteristic
136C2×C3×D9C22×C412yes—
137C2×C3×D9C2×Q812yes—
138C2×C3×D9#4112yes—
139C3×C4×C9—22no—
140C3×C4×C9C22×C412yes—
141C3⋊Dic9—21no—
142C3⋊Dic9C22×C414yes—
143C3⋊Dic9C2×Q812yescomplemented
144C9×Dic3—21no—
145C9×Dic3C22×C414yes—
146C9×Dic3C2×Q812yescomplemented
147C3×Dic9—22no—
148C3×Dic9C22×C412yes—
149C2×D3×Q8—91no—
150C22×Dic6—91no—
151C3×C4×Q8—31no—
152C2×C3×C4⋊C4—91no—
153Q8×Dic3—91no—
154C2×Dic3⋊C4—92no—
155C2×C4×Dic3—92no—
156C2×C4×Dic3—91no—
157D3×C4⋊C4—31no—
158D3×C4⋊C4—92no—
159#887—92no—
160#887—91no—
161#887—91no—
162#887—91no—
163#887—92no—
164#887—92no—
165#887—91no—
166#887—181no—
167#887D911yescharacteristic, direct product factor
168C42×D3—91no—
169C42×D3—92no—
170C4×Dic6—92no—
171C4×Dic6—91no—
172C22×C3×D3—31no—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—32no—
177C2×C3⋊Dic3—31no—
178#634—34no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D312yes—
183C2×C3×Dic3C22×D311yescharacteristic
184C3×C4×D3—34no—
185C3×C4×D3—64no—
186C3×Dic6C4×D314yes—
187#627—38no—
188D3×Dic3—38no—
189D3×Dic3—34no—
190D3×Dic3—62no—
191C22×D9C22×D311yescharacteristic
192Q8×C9—34no—
193C2×C4×C9—21no—
194C2×C4×C9—22no—
195C2×C4×C9—31no—
196C2×C4×C9—32no—
197C2×C4×C9—31no—
198C2×C4×C9C22×D312yes—
199C2×C4×C9C22×D311yescharacteristic
200C2×Dic9—31no—
201C2×Dic9—32no—
202C2×Dic9—31no—
203C2×Dic9—61no—
204C2×Dic9—62no—
205C2×Dic9—61no—
206C2×Dic9C22×D312yes—
207C2×Dic9C22×D311yescharacteristic
208C4×D9—24no—
209C4×D9—38no—
210C4×D9—34no—
211C4×D9—62no—
212C4×D9C4×D314yes—
213Dic18—34no—
214Dic18—38no—
215Dic18—64no—
216C2×C4×Q8—271noSylow
217C2×C3×C9C23×C411yescharacteristic
218C2×C3×C9C22×Q811yescharacteristic
219C2×C3×C9C2×#4111yescharacteristic
220C3×D9C4×Q814yescomplemented
221C2×C3×Q8—31no—
222C22×C3×C4—91no—
223C22×C3×C4—92no—
224C22×Dic3—91no—
225C22×Dic3—92no—
226C22×Dic3—91no—
227D3×Q8—98no—
228C2×C4×D3—31no—
229C2×C4×D3—32no—
230C2×C4×D3—91no—
231C2×C4×D3—92no—
232C2×C4×D3—91no—
233C2×Dic6—92no—
234C2×Dic6—91no—
235C2×Dic6—91no—
236C2×Dic6—92no—
237C2×Dic6—98no—
238C2×Dic6—181no—
239C2×Dic6C2×D911yescharacteristic
240C3×C4⋊C4—21no—
241C3×C4⋊C4—32no—
242C3×C4⋊C4—92no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4C2×D911yescharacteristic
245C3×C4⋊C4D3211yescharacteristic
246C3×C42—32no—
247C3×C42—31no—
248C4⋊Dic3—31no—
249C4⋊Dic3—92no—
250Dic3⋊C4—32no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—182no—
257Dic3⋊C4C2×D912yes—
258C4×Dic3—91no—
259C4×Dic3—92no—
260C4×Dic3—91no—
261C4×Dic3—92no—
262C4×Dic3—92no—
263C4×Dic3—91no—
264C4×Dic3—92no—
265C4×Dic3—91no—
266C4×Dic3—91no—
267C4×Dic3—92no—
268C4×Dic3—92no—
269C4×Dic3—91no—
270C4×Dic3—182no—
271C4×Dic3—181no—
272C4×Dic3C2×D911yescharacteristic
273C4×Dic3C2×D912yes—
274C22×C32C23×D311yescharacteristic
275C2×C3×D3—32no—
276C2×C3×D3—32no—
277C2×C3×D3—32no—
278C32×C4—22no—
279C32×C4C2×C4×D312yes—
280C3⋊Dic3—34no—
281C3⋊Dic3—32no—
282C3⋊Dic3—61no—
283C3×Dic3—21no—
284C3×Dic3—32no—
285C3×Dic3—62no—
286C3×Dic3C2×C4×D314yes—
287C3×Dic3D3×Q812yes—
288C22×C9—21no—
289C22×C9C23×D311yescharacteristic
290C2×D9C2×C4×D312yes—
291C2×D9#23712yes—
292C2×D9D3×Q812yes—
293C4×C9—22no—
294C4×C9—22no—
295C4×C9—34no—
296C4×C9—32no—
297C4×C9—42no—
298C4×C9—61no—
299C4×C9C2×C4×D312yes—
300Dic9—22no—
301Dic9—34no—
302Dic9—32no—
303Dic9—64no—
304Dic9—62no—
305Dic9—61no—
306Dic9—121no—
307Dic9C2×C4×D312yes—
308C22×Q8—271no—
309C4×Q8—31no—
310C4×Q8—272no—
311C4×Q8—271no—
312C4×Q8—272no—
313C4×Q8—271no—
314C4×Q8—271no—
315C2×C4⋊C4—91no—
316C2×C4⋊C4—272no—
317C2×C42—271no—
318C2×C42—272no—
319C3×C9C2×C4×Q811yescharacteristic, complemented, Sylow
320C23×C3—91no—
321C22×D3—31no—
322C3×Q8—34no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—31no—
326C2×C3×C4—32no—
327C2×C3×C4—31no—
328C2×C3×C4—91no—
329C2×C3×C4—94no—
330C2×C3×C4—92no—
331C2×C3×C4—184no—
332C2×C3×C4C22×D912yes—
333C2×C3×C4C22×D911yescharacteristic
334C2×C3×C4C2×D3212yes—
335C2×C3×C4C2×D3211yescharacteristic
336C2×Dic3—31no—
337C2×Dic3—32no—
338C2×Dic3—91no—
339C2×Dic3—92no—
340C2×Dic3—91no—
341C2×Dic3—92no—
342C2×Dic3—98no—
343C2×Dic3—94no—
344C2×Dic3—91no—
345C2×Dic3—91no—
346C2×Dic3—181no—
347C2×Dic3—182no—
348C2×Dic3—181no—
349C2×Dic3—182no—
350C2×Dic3C22×D911yescharacteristic
351C2×Dic3C22×D912yes—
352C2×Dic3C22×D911yescharacteristic
353C4×D3—34no—
354C4×D3—64no—
355C4×D3—94no—
356C4×D3—98no—
357C4×D3—182no—
358Dic6—98no—
359Dic6—94no—
360Dic6—94no—
361Dic6—98no—
362Dic6—184no—
363Dic6C4×D914yescomplemented
364C2×C32C22×C4×D311yescharacteristic
365C2×C32C2×D3×Q811yescharacteristic
366C2×C32D3×#4111yescharacteristic
367C3×D3—34no—
368C2×C9—21no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9C22×C4×D311yescharacteristic
372C2×C9C2×#23711yescharacteristic
373C2×C9C2×D3×Q811yescharacteristic
374D9#88714yesdirect product factor
375C2×Q8—31no—
376C2×Q8—272no—
377C2×Q8—278no—
378C2×Q8—271no—
379C22×C4—91no—
380C22×C4—92no—
381C22×C4—271no—
382C22×C4—272no—
383C22×C4—271no—
384C4⋊C4—32no—
385C4⋊C4—92no—
386C4⋊C4—91no—
387C4⋊C4—272no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C4⋊C4D3×D911yescharacteristic
391C42—32no—
392C42—31no—
393C42—271no—
394C42—272no—
395C42—272no—
396C42—271no—
397C42—272no—
398C42—271no—
399C22×C3—21no—
400C22×C3—92no—
401C22×C3—92no—
402C22×C3—92no—
403C22×C3C23×D911yescharacteristic
404C22×C3C22×D3211yescharacteristic
405C2×D3—32no—
406C2×D3—32no—
407C2×D3—32no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—34no—
412C3×C4—32no—
413C3×C4—42no—
414C3×C4—61no—
415C3×C4—92no—
416C3×C4—182no—
417C3×C4C2×C4×D912yes—
418C3×C4C4×D3212yes—
419Dic3—21no—
420Dic3—32no—
421Dic3—62no—
422Dic3—94no—
423Dic3—92no—
424Dic3—94no—
425Dic3—92no—
426Dic3—184no—
427Dic3—182no—
428Dic3—181no—
429Dic3—181no—
430Dic3—361no—
431Dic3C2×C4×D914yes—
432Dic3Q8×D912yescomplemented
433C32C4×D3×Q811yescharacteristic
434C9—21no—
435C9C2×#88711yescharacteristic, complemented
436C23—91no—
437Q8—34no—
438Q8—278no—
439Q8—274no—
440C2×C4—31no—
441C2×C4—32no—
442C2×C4—31no—
443C2×C4—91no—
444C2×C4—94no—
445C2×C4—92no—
446C2×C4—184no—
447C2×C4—278no—
448C2×C4—274no—
449C2×C4—271no—
450C2×C4—272no—
451C2×C4—271no—
452C2×C4—542no—
453C2×C4C2×D3×D912yes—
454C2×C4C2×D3×D911yescharacteristic
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—94no—
459C2×C3C22×C4×D911yescharacteristic
460C2×C3C2×Q8×D911yescharacteristic
461C2×C3#41×D911yescharacteristic
462C2×C3C2×C4×D3211yescharacteristic
463C2×C3D3×#23711yescharacteristic
464C2×C3D32×Q811yescharacteristic
465D3—34no—
466C22—92no—
467C22—92no—
468C22—92no—
469C22C22×D3×D911yescharacteristic
470C4—22no—
471C4—32no—
472C4—34no—
473C4—61no—
474C4—92no—
475C4—182no—
476C4—272no—
477C4—274no—
478C4—541no—
479C4C4×D3×D912yes—
480C3—21no—
481C3C4×Q8×D911yescharacteristic, complemented
482C3D3×#88711yescharacteristic
483C2—94no—
484C2C2×C4×D3×D911yescharacteristic
485C2D9×#23711yescharacteristic
486C2D3×Q8×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3×D9C23
cosocleC22×C3×C9C24
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C2×C4×Q8
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. #11721743C2semidirect
4. C9×#887C2semidirect
8. #11720912C2semidirect
9. #11720911C2semidirect
10. #11720902C2semidirect
11. #11720900C2semidirect
22. D9×Dic6C4semidirect
67. #6226C2×C4semidirect
75. C9×Dic6C2×C4semidirect
78. Dic3×D9Q8semidirect
79. #6189C2×C4semidirect
143. C3⋊Dic9C2×Q8semidirect
146. C9×Dic3C2×Q8semidirect
167. #887D9direct (N × H)
220. C3×D9C4×Q8semidirect
319. C3×C9C2×C4×Q8semidirect
363. Dic6C4×D9semidirect
374. D9#887direct (N × H)
432. Dic3Q8×D9semidirect
435. C9C2×#887semidirect
481. C3C4×Q8×D9semidirect

Maximal subgroups

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

subgroupindex
C2×D9×Dic62
#117217432
C3×C4⋊C4×D92
C9×#8872
D9×Dic3⋊C42
C4×Dic3×D92
#117209122
#117209112
#117209022
#117209002
D3×#8873
C4×Q8×D93