Finite Group Databasebeta

← D9×#919 #13149984

Subgroups — D9×#919 #13149984

Order 1728.

860 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D9×#239C212yesmaximal, complemented
2#11721805C211yesmaximal, characteristic, complemented
3C9×#919C211yesmaximal, characteristic, complemented
4C3×Q16×D9C211yesmaximal, characteristic, complemented
5#11721216C212yesmaximal, complemented
6D9×#215C212yesmaximal, complemented
7#11721209C212yesmaximal, complemented
8#11720824C211yesmaximal, characteristic, complemented
9D9×D24C211yesmaximal, characteristic, complemented
10#11720812C211yesmaximal, characteristic, complemented
11D3×C8×D9C211yesmaximal, characteristic, complemented
12D3×#919—31nomaximal
13D9×#135—31nomaximal
14#88746C2212yescomplemented
15C9×#239C2212yescomplemented
16C3×Q8×D9C2212yes—
17D9×D12—22no—
18D9×D12C2212yes—
19C4×D3×D9—22no—
20C4×D3×D9C2211yescharacteristic
21#88642—22no—
22#88642C2212yescomplemented
23#88639—22no—
24#88639C2212yescomplemented
25C3×C9×Q16C2211yescharacteristic, complemented
26#88548C2212yescomplemented
27#88525C2211yescharacteristic, complemented
28C8×C3⋊D9C2211yescharacteristic, complemented
29C9×#215C2212yescomplemented
30C3×#3674C2212yescomplemented
31C9×D24C2211yescharacteristic, complemented
32D3×C8×C9C2211yescharacteristic, complemented
33C3×C8×D9C2211yescharacteristic
34C3×Dic36C2211yescharacteristic, complemented
35#88423C2212yescomplemented
36#88422C2212yescomplemented
37D3×C9⋊C8C2211yescharacteristic, complemented
38#88416C2212yescomplemented
39(C3⋊D9)⋊C8C2211yescharacteristic, complemented
40D9×C3⋊C8C2211yescharacteristic
41D3×#239—32no—
42#64128—31no—
43C3×#919D311yescharacteristic
44C3×D3×Q16—31no—
45#63950—32no—
46#63949—32no—
47D3×#215—32no—
48#63810—31no—
49#63808—31no—
50D3×D24—31no—
51D32×C8—31no—
52#41×D9—32no—
53C9×#135—31no—
54#63513—32no—
55#63481—31no—
56#63481—61no—
57Q16×D9D311yescharacteristic, complemented
58#63478—32no—
59SD16×D9—32no—
60#63474—31no—
61D8×D9—31no—
62#63467—31no—
63C2×C8×D9—31no—
64C2×D3×D9—22no—
65C2×D3×D9D411yescharacteristic
66C3×Q8×C9C2312yes—
67C4⋊(C3⋊D9)—22no—
68C4⋊(C3⋊D9)C2312yes—
69C4×C3⋊D9—22no—
70C4×C3⋊D9C2311yescharacteristic
71C9×D12—22no—
72C9×D12C2312yes—
73C4×D3×C9—22no—
74C4×D3×C9C2311yescharacteristic
75C3×C4×D9—22no—
76C3×C4×D9C2311yescharacteristic
77C3×Dic18—22no—
78C3×Dic18C2312yes—
79#6195—22no—
80#6195—22no—
81#6195—22no—
82D3×Dic9—22no—
83D3×Dic9D411yescharacteristic
84(C3⋊D9)⋊C4—22no—
85(C3⋊D9)⋊C4D411yescharacteristic
86Dic3×D9D411yescharacteristic
87C3×C8×C9C2311yescharacteristic
88C3⋊(C9⋊C8)C2311yescharacteristic
89C9×C3⋊C8C2311yescharacteristic
90C3×C9⋊C8C2311yescharacteristic
91D3×#135—91no—
92C2×#919—91no—
93#3832—32no—
94C3×#239C2×D312yes—
95C3×D3×Q8—32no—
96D3×D12—32no—
97D3×D12—62no—
98C4×D32—31no—
99C4×D32—62no—
100#3799—32no—
101#3799—62no—
102#3796—32no—
103#3796—62no—
104C32×Q16C2×D311yescharacteristic
105#3755—32no—
106#3745—31no—
107C8×C3⋊D3—31no—
108C3×#216—32no—
109C3×#215C2×D312yes—
110C3×Dic12—31no—
111C3×D24C2×D311yescharacteristic
112C3×D3×C8—31no—
113C3×D3×C8C2×D311yescharacteristic
114#3717—32no—
115#3715—32no—
116#3714—32no—
117#3710—31no—
118D3×C3⋊C8—31no—
119D3×C3⋊C8—31no—
120C9×#41—32no—
121#3701—32no—
122#3701—62no—
123Q8×D9C2×D312yes—
124#3699—32no—
125#3699—62no—
126D4×D9—32no—
127D4×D9—62no—
128#3697—32no—
129#3697—62no—
130C2×C4×D9—31no—
131C2×C4×D9—62no—
132C9×Q16—21no—
133C9×Q16C2×D311yescharacteristic, complemented
134C9×SD16—32no—
135C9×D8—31no—
136C2×C8×C9—31no—
137#3675—32no—
138#3675—62no—
139#3674C2×D312yescomplemented
140#3673—32no—
141#3672—32no—
142C2×C9⋊C8—31no—
143D72—31no—
144D72—61no—
145#3664—32no—
146C8×D9—31no—
147C8×D9—31no—
148C8×D9—31no—
149C8×D9—61no—
150C8×D9C2×D311yescharacteristic
151Dic36C2×D311yescharacteristic, complemented
152C2×C3⋊D9—22no—
153C2×C3⋊D9C2×D411yescharacteristic
154C2×D3×C9—22no—
155C2×D3×C9C2×D411yescharacteristic
156C2×C3×D9C2×D411yescharacteristic
157D3×D9—22no—
158D3×D9—44no—
159C3×C4×C9—22no—
160C3×C4×C9C2411yescharacteristic
161C3⋊Dic9C2×D411yescharacteristic
162C9×Dic3C2×D411yescharacteristic
163C3×Dic9—22no—
164C3×Dic9C2×D411yescharacteristic
165D3×#41—92no—
166C2×#239—92no—
167C3×#135—31no—
168C2×C3×Q16—91no—
169#950—92no—
170C2×#215—92no—
171#919—92no—
172#919—91no—
173#919—91no—
174#919—92no—
175#919—91no—
176#919—91no—
177#919—181no—
178#919D911yescharacteristic, direct product factor
179D3×Q16—31no—
180#916—92no—
181D3×SD16—92no—
182#912—91no—
183D3×D8—91no—
184#904—91no—
185C2×D24—91no—
186C2×D3×C8—91no—
187C2×D3×C8—91no—
188C2×D32—31no—
189C2×D32—62no—
190C32×Q8C22×D312yes—
191C4⋊(C3⋊D3)—32no—
192C4⋊(C3⋊D3)—62no—
193C4×C3⋊D3—31no—
194C4×C3⋊D3—62no—
195C3×D12—22no—
196C3×D12C22×D312yes—
197C3×C4×D3—22no—
198C3×C4×D3—31no—
199C3×C4×D3—62no—
200C3×C4×D3C22×D311yescharacteristic
201C3×Dic6—32no—
202C3×Dic6—62no—
203#626—62no—
204#626—62no—
205#626—62no—
206#624—31no—
207#624—62no—
208D3×Dic3—31no—
209D3×Dic3—31no—
210D3×Dic3—62no—
211C22×D9—31no—
212C22×D9—62no—
213C32×C8C22×D311yescharacteristic
214C3⋊(C3⋊C8)—31no—
215C3×C3⋊C8—31no—
216C3×C3⋊C8C22×D311yescharacteristic
217Q8×C9—22no—
218Q8×C9C22×D312yes—
219D4×C9—32no—
220D4×C9—62no—
221C2×C4×C9—31no—
222C2×C4×C9—62no—
223#611—62no—
224#611—62no—
225#611—62no—
226C2×Dic9—31no—
227C2×Dic9—62no—
228D36—32no—
229D36—62no—
230D36—62no—
231D36—122no—
232C4×D9—22no—
233C4×D9—31no—
234C4×D9—31no—
235C4×D9—31no—
236C4×D9—61no—
237C4×D9—62no—
238C4×D9—62no—
239C4×D9—122no—
240C4×D9C22×D311yescharacteristic
241Dic18—22no—
242Dic18C22×D312yes—
243C8×C9—21no—
244C8×C9—31no—
245C8×C9C22×D311yescharacteristic
246C9⋊C8—31no—
247C9⋊C8—61no—
248C9⋊C8C22×D311yescharacteristic
249C2×#135—271noSylow
250C2×C3×C9C22×D411yescharacteristic
251C3⋊D9—21no—
252C3⋊D9—42no—
253D3×C9—21no—
254D3×C9—42no—
255C3×D9#13512yescomplemented
256C3×#41—32no—
257C2×C3×Q8—92no—
258#239—92no—
259#239—92no—
260#239—92no—
261#239—92no—
262#239—182no—
263#239—182no—
264#239—182no—
265#239C2×D912yescomplemented
266D3×Q8—32no—
267#237—92no—
268#237—182no—
269D3×D4—92no—
270D3×D4—182no—
271#235—92no—
272#235—182no—
273C2×D12—92no—
274C2×D12—182no—
275C2×C4×D3—91no—
276C2×C4×D3—91no—
277C2×C4×D3—182no—
278C2×C4×D3—182no—
279C3×Q16—21no—
280C3×Q16—92no—
281C3×Q16—91no—
282C3×Q16C2×D911yescharacteristic, complemented
283C3×Q16D3211yescharacteristic
284C3×SD16—32no—
285C3×D8—31no—
286C2×C3×C8—31no—
287C2×C3×C8—91no—
288#216—32no—
289#215—92no—
290#215—92no—
291#215—92no—
292#215—92no—
293#215—182no—
294#215C2×D912yescomplemented
295#214—92no—
296#213—92no—
297C2×C3⋊C8—91no—
298C2×C3⋊C8—91no—
299Dic12—31no—
300D24—92no—
301D24—91no—
302D24—91no—
303D24—181no—
304D24C2×D911yescharacteristic, complemented
305#204—92no—
306D3×C8—31no—
307D3×C8—91no—
308D3×C8—91no—
309D3×C8—91no—
310D3×C8—91no—
311D3×C8—91no—
312D3×C8—91no—
313D3×C8—181no—
314D3×C8C2×D911yescharacteristic, complemented
315C2×C3⋊D3—31no—
316C2×C3⋊D3—62no—
317C2×C3×D3—22no—
318C2×C3×D3—31no—
319C2×C3×D3D3×D411yescharacteristic
320D32—62no—
321D32—124no—
322C32×C4—22no—
323C32×C4C23×D311yescharacteristic
324C3⋊Dic3—31no—
325C3×Dic3—31no—
326C3×Dic3—62no—
327C3×Dic3D3×D411yescharacteristic
328C22×C9—31no—
329C22×C9—62no—
330C2×D9—31no—
331C2×D9—61no—
332C2×D9—62no—
333C2×D9—62no—
334C2×D9—122no—
335C2×D9—124no—
336C2×D9D3×D411yescharacteristic
337C4×C9—22no—
338C4×C9—21no—
339C4×C9—31no—
340C4×C9—42no—
341C4×C9C23×D311yescharacteristic
342Dic9—22no—
343Dic9—31no—
344Dic9—61no—
345Dic9D3×D411yescharacteristic
346C2×#41—272no—
347#135—31no—
348#135—271no—
349#135—271no—
350#135—272no—
351#135—272no—
352#135—271no—
353C2×Q16—91no—
354C2×SD16—272no—
355C2×D8—271no—
356C22×C8—271no—
357C3×C9C2×#13511yescharacteristic, complemented, Sylow
358C22×D3—91no—
359C22×D3—91no—
360C22×D3—182no—
361C22×D3—182no—
362C3×Q8—22no—
363C3×Q8—92no—
364C3×Q8—182no—
365C3×Q8C22×D912yes—
366C3×Q8C2×D3212yes—
367C3×D4—32no—
368C3×D4—62no—
369C2×C3×C4—31no—
370C2×C3×C4—62no—
371C2×C3×C4—91no—
372C2×C3×C4—182no—
373#67—182no—
374#67—182no—
375#67—182no—
376C2×Dic3—91no—
377C2×Dic3—91no—
378C2×Dic3—182no—
379D12—22no—
380D12—92no—
381D12—92no—
382D12—182no—
383D12—182no—
384D12—182no—
385D12—182no—
386D12—182no—
387D12—182no—
388D12—362no—
389D12C22×D912yes—
390C4×D3—22no—
391C4×D3—31no—
392C4×D3—62no—
393C4×D3—91no—
394C4×D3—91no—
395C4×D3—91no—
396C4×D3—91no—
397C4×D3—91no—
398C4×D3—91no—
399C4×D3—181no—
400C4×D3—182no—
401C4×D3—182no—
402C4×D3—182no—
403C4×D3—182no—
404C4×D3—182no—
405C4×D3—362no—
406C4×D3C22×D911yescharacteristic
407Dic6—32no—
408Dic6—62no—
409C3×C8—21no—
410C3×C8—31no—
411C3×C8—91no—
412C3×C8C22×D911yescharacteristic
413C3×C8C2×D3211yescharacteristic
414C3⋊C8—31no—
415C3⋊C8—91no—
416C3⋊C8—91no—
417C3⋊C8—181no—
418C3⋊C8C22×D911yescharacteristic
419C2×C32C2×D3×D411yescharacteristic
420C3⋊D3—61no—
421C3⋊D3—122no—
422C3×D3—21no—
423C3×D3—32no—
424C3×D3—42no—
425C2×C9—21no—
426C2×C9—61no—
427C2×C9—122no—
428C2×C9C2×D3×D411yescharacteristic
429D9—61no—
430D9—121no—
431D9—122no—
432D9—242no—
433D9#91912yesdirect product factor
434#41—32no—
435#41—272no—
436#41—272no—
437#41—272no—
438#41—542no—
439#41—542no—
440C2×Q8—92no—
441C2×D4—272no—
442C2×D4—542no—
443C22×C4—271no—
444C22×C4—542no—
445Q16—92no—
446Q16—91no—
447Q16D3×D911yescharacteristic, complemented
448SD16—32no—
449SD16—272no—
450SD16—272no—
451SD16—272no—
452D8—31no—
453D8—272no—
454D8—271no—
455C2×C8—31no—
456C2×C8—91no—
457C2×C8—271no—
458C2×C8—271no—
459C2×C8—271no—
460C2×C8—271no—
461C22×C3—31no—
462C22×C3—62no—
463C22×C3—91no—
464C2×D3—22no—
465C2×D3—31no—
466C2×D3—91no—
467C2×D3—91no—
468C2×D3—181no—
469C2×D3—182no—
470C2×D3—182no—
471C2×D3—182no—
472C2×D3—182no—
473C2×D3—362no—
474C2×D3—364no—
475C2×D3—364no—
476C2×D3D4×D911yescharacteristic
477C3×C4—22no—
478C3×C4—22no—
479C3×C4—21no—
480C3×C4—31no—
481C3×C4—42no—
482C3×C4—91no—
483C3×C4—182no—
484C3×C4C23×D911yescharacteristic
485C3×C4C22×D3211yescharacteristic
486Dic3—31no—
487Dic3—62no—
488Dic3—91no—
489Dic3—91no—
490Dic3—181no—
491Dic3D4×D911yescharacteristic
492C32D3×#13511yescharacteristic
493C9—21no—
494C9C2×#91911yescharacteristic, complemented
495C23—271no—
496C23—542no—
497Q8—92no—
498Q8—182no—
499Q8C2×D3×D912yes—
500D4—32no—
501D4—62no—
502D4—272no—
503D4—542no—
504D4—542no—
505D4—542no—
506D4—542no—
507C2×C4—31no—
508C2×C4—62no—
509C2×C4—91no—
510C2×C4—182no—
511C2×C4—271no—
512C2×C4—271no—
513C2×C4—271no—
514C2×C4—271no—
515C2×C4—542no—
516C2×C4—542no—
517C2×C4—542no—
518C8—31no—
519C8—91no—
520C8—271no—
521C8C2×D3×D911yescharacteristic
522C2×C3—21no—
523C2×C3—61no—
524C2×C3—92no—
525C2×C3—122no—
526C2×C3C2×D4×D911yescharacteristic
527C2×C3D32×D411yescharacteristic
528D3—21no—
529D3—32no—
530D3—42no—
531D3—181no—
532D3—181no—
533D3—361no—
534D3—362no—
535D3—362no—
536D3—722no—
537C22—31no—
538C22—62no—
539C22—91no—
540C22—271no—
541C22—542no—
542C22—542no—
543C22—1084no—
544C4—22no—
545C4—31no—
546C4—91no—
547C4—182no—
548C4—271no—
549C4C22×D3×D911yescharacteristic
550C3—21no—
551C3D9×#13511yescharacteristic, complemented
552C3D3×#91911yescharacteristic
553C2—61no—
554C2—92no—
555C2—122no—
556C2—541no—
557C2—1082no—
558C2D3×D4×D911yescharacteristic
subgroupG/H
centerC2D3×D4×D9
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC3×C9×Q16C22
socleC2×C32C2×D3×D4
hypercenterQ16D3×D9
hypocenterC3×C9C2×#135
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C4 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. D9×#239C2semidirect
2. #11721805C2semidirect
3. C9×#919C2semidirect
4. C3×Q16×D9C2semidirect
5. #11721216C2semidirect
6. D9×#215C2semidirect
7. #11721209C2semidirect
8. #11720824C2semidirect
9. D9×D24C2semidirect
10. #11720812C2semidirect
11. D3×C8×D9C2semidirect
14. #88746C22semidirect
15. C9×#239C22semidirect
22. #88642C22semidirect
24. #88639C22semidirect
25. C3×C9×Q16C22semidirect
26. #88548C22semidirect
27. #88525C22semidirect
28. C8×C3⋊D9C22semidirect
29. C9×#215C22semidirect
30. C3×#3674C22semidirect
31. C9×D24C22semidirect
32. D3×C8×C9C22semidirect
34. C3×Dic36C22semidirect
35. #88423C22semidirect
36. #88422C22semidirect
37. D3×C9⋊C8C22semidirect
38. #88416C22semidirect
39. (C3⋊D9)⋊C8C22semidirect
57. Q16×D9D3semidirect
133. C9×Q16C2×D3semidirect
139. #3674C2×D3semidirect
151. Dic36C2×D3semidirect
178. #919D9direct (N × H)
255. C3×D9#135semidirect
265. #239C2×D9semidirect
282. C3×Q16C2×D9semidirect
294. #215C2×D9semidirect
304. D24C2×D9semidirect
314. D3×C8C2×D9semidirect
357. C3×C9C2×#135semidirect
433. D9#919direct (N × H)
447. Q16D3×D9semidirect
494. C9C2×#919semidirect
551. C3D9×#135semidirect

Maximal subgroups

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

subgroupindex
D9×#2392
#117218052
C9×#9192
C3×Q16×D92
#117212162
D9×#2152
#117212092
#117208242
D9×D242
#117208122
D3×C8×D92
D3×#9193
D9×#1353