Finite Group Databasebeta

← #13149989

Subgroups — #13149989

Order 1728.

858 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722514C212yesmaximal, complemented
2Q16×C3⋊D9C211yesmaximal, characteristic, complemented
3C9×#919C211yesmaximal, characteristic, complemented
4C3×#63481C211yesmaximal, characteristic, complemented
5#11721215C212yesmaximal, complemented
6#11721212C212yesmaximal, complemented
7#11721207C212yesmaximal, complemented
8#11720822C211yesmaximal, characteristic, complemented
9#11720818C211yesmaximal, characteristic, complemented
10#11720811C211yesmaximal, characteristic, complemented
11D3×C8×D9C211yesmaximal, characteristic, complemented
12#10594764—31nomaximal
13D9×#135—31nomaximal
14Q8×C3⋊D9C2212yes—
15C9×#239C2212yescomplemented
16C3×#3701C2212yescomplemented
17#88646—22no—
18#88646C2212yes—
19C4×D3×D9—22no—
20C4×D3×D9C2211yescharacteristic
21#88641—22no—
22#88641C2212yescomplemented
23#88638—22no—
24#88638C2212yescomplemented
25C3×C9×Q16C2211yescharacteristic, complemented
26#88547C2212yescomplemented
27C8×C3⋊D9C2211yescharacteristic
28#88521C2211yescharacteristic, complemented
29C9×#215C2212yescomplemented
30C3×#3675C2212yescomplemented
31C9×D24C2211yescharacteristic, complemented
32D3×C8×C9C2211yescharacteristic, complemented
33C3×D72C2211yescharacteristic, complemented
34C3×C8×D9C2211yescharacteristic, complemented
35#88424C2212yescomplemented
36#88421C2212yescomplemented
37D3×C9⋊C8C2211yescharacteristic, complemented
38#88415C2212yescomplemented
39(C3⋊D9)⋊C8C2211yescharacteristic
40D9×C3⋊C8C2211yescharacteristic, complemented
41#64319—32no—
42Q16×C3⋊D3—31no—
43C3×#919—31no—
44C3×#919D311yescharacteristic
45#63947—32no—
46#63947—32no—
47#63940—32no—
48#63807—31no—
49#63807—31no—
50#63797—31no—
51D32×C8—31no—
52#41×D9—32no—
53C9×#135—31no—
54#63513—32no—
55#63481D311yescharacteristic, complemented
56Q16×D9—31no—
57Q16×D9—61no—
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⋊D9C2311yescharacteristic
69#6226—22no—
70#6226C2312yes—
71C9×D12—22no—
72C9×D12C2312yes—
73C4×D3×C9—22no—
74C4×D3×C9C2311yescharacteristic
75C3×D36—22no—
76C3×D36C2312yes—
77C3×C4×D9—22no—
78C3×C4×D9C2311yescharacteristic
79#6194—22no—
80#6194—22no—
81#6194—22no—
82D3×Dic9—22no—
83D3×Dic9D411yescharacteristic
84(C3⋊D9)⋊C4D411yescharacteristic
85Dic3×D9—22no—
86Dic3×D9D411yescharacteristic
87C3×C8×C9C2311yescharacteristic
88C3⋊(C9⋊C8)C2311yescharacteristic
89C9×C3⋊C8C2311yescharacteristic
90C3×C9⋊C8C2311yescharacteristic
91D3×#135—91no—
92D3×#135—91no—
93Q8×C3⋊D3—32no—
94C3×#239—32no—
95C3×#239C2×D312yes—
96#3802—32no—
97#3802—62no—
98C4×D32—31no—
99C4×D32—62no—
100#3795—32no—
101#3795—32no—
102#3795—62no—
103#3795—62no—
104C32×Q16C2×D311yescharacteristic
105#3756—32no—
106#3746—31no—
107C8×C3⋊D3—31no—
108C3×#215—32no—
109C3×#215C2×D312yes—
110C3×D24—31no—
111C3×D24C2×D311yescharacteristic
112C3×D3×C8—31no—
113C3×D3×C8C2×D311yescharacteristic
114#3716—32no—
115#3716—32no—
116#3713—32no—
117#3710—31no—
118D3×C3⋊C8—31no—
119D3×C3⋊C8—31no—
120C9×#41—32no—
121#3701C2×D312yescomplemented
122Q8×D9—32no—
123Q8×D9—62no—
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#3675C2×D312yescomplemented
138#3674—32no—
139#3674—62no—
140#3673—32no—
141#3672—32no—
142C2×C9⋊C8—31no—
143D72C2×D311yescharacteristic, complemented
144#3664—32no—
145C8×D9—31no—
146C8×D9—31no—
147C8×D9—31no—
148C8×D9—61no—
149C8×D9C2×D311yescharacteristic, complemented
150Dic36—31no—
151Dic36—61no—
152C2×C3⋊D9C2×D411yescharacteristic
153C2×D3×C9—22no—
154C2×D3×C9C2×D411yescharacteristic
155C2×C3×D9—22no—
156C2×C3×D9C2×D411yescharacteristic
157D3×D9—22no—
158D3×D9—44no—
159C3×C4×C9—22no—
160C3×C4×C9C2411yescharacteristic
161C3⋊Dic9—22no—
162C3⋊Dic9C2×D411yescharacteristic
163C9×Dic3C2×D411yescharacteristic
164C3×Dic9C2×D411yescharacteristic
165D3×#41—92no—
166D3×#41—92no—
167C3×#135—31no—
168C3×#135—91no—
169#950—92no—
170#950—92no—
171#919—31no—
172#919D911yescharacteristic, complemented
173D3×Q16—91no—
174D3×Q16—91no—
175D3×Q16—181no—
176#916—92no—
177#916—92no—
178D3×SD16—92no—
179D3×SD16—92no—
180#912—91no—
181#912—91no—
182D3×D8—91no—
183D3×D8—91no—
184#904—91no—
185#904—91no—
186C2×D3×C8—91no—
187C2×D3×C8—91no—
188C2×D32—31no—
189C2×D32—62no—
190C32×Q8C22×D312yes—
191C4×C3⋊D3—31no—
192C4×C3⋊D3—62no—
193#634—32no—
194#634—62no—
195C3×D12—22no—
196C3×D12—32no—
197C3×D12—62no—
198C3×D12C22×D312yes—
199C3×C4×D3—22no—
200C3×C4×D3—31no—
201C3×C4×D3—62no—
202C3×C4×D3C22×D311yescharacteristic
203#625—62no—
204#625—62no—
205#625—62no—
206#624—31no—
207D3×Dic3—31no—
208D3×Dic3—31no—
209D3×Dic3—62no—
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—22no—
229D36C22×D312yes—
230C4×D9—22no—
231C4×D9—31no—
232C4×D9—31no—
233C4×D9—31no—
234C4×D9—61no—
235C4×D9—62no—
236C4×D9—62no—
237C4×D9—122no—
238C4×D9C22×D311yescharacteristic
239Dic18—32no—
240Dic18—62no—
241Dic18—62no—
242Dic18—122no—
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#13512yescomplemented
252D3×C9—21no—
253D3×C9—42no—
254C3×D9—21no—
255C3×D9—42no—
256C3×#41—32no—
257C3×#41—92no—
258#239—32no—
259#239C2×D912yescomplemented
260D3×Q8—92no—
261D3×Q8—92no—
262D3×Q8—182no—
263#237—92no—
264#237—92no—
265#237—182no—
266#237—182no—
267D3×D4—92no—
268D3×D4—92no—
269D3×D4—182no—
270D3×D4—182no—
271#235—92no—
272#235—92no—
273#235—182no—
274#235—182no—
275C2×C4×D3—91no—
276C2×C4×D3—91no—
277C2×C4×D3—182no—
278C2×C4×D3—182no—
279C3×Q16—21no—
280C3×Q16C2×D911yescharacteristic, complemented
281C3×Q16D3211yescharacteristic
282C3×SD16—32no—
283C3×SD16—92no—
284C3×D8—31no—
285C3×D8—91no—
286C2×C3×C8—31no—
287C2×C3×C8—91no—
288#216—92no—
289#216—92no—
290#216—182no—
291#215—32no—
292#215C2×D912yescomplemented
293#214—92no—
294#214—92no—
295#213—92no—
296#213—92no—
297C2×C3⋊C8—91no—
298C2×C3⋊C8—91no—
299Dic12—91no—
300Dic12—91no—
301Dic12—181no—
302D24—31no—
303D24C2×D911yescharacteristic, complemented
304#204—92no—
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—22no—
317C2×C3×D3—31no—
318C2×C3×D3—62no—
319C2×C3×D3D3×D411yescharacteristic
320D32—62no—
321D32—124no—
322C32×C4—22no—
323C32×C4C23×D311yescharacteristic
324C3⋊Dic3—31no—
325C3⋊Dic3—62no—
326C3×Dic3—31no—
327C3×Dic3D3×D411yescharacteristic
328C22×C9—31no—
329C22×C9—62no—
330C2×D9—22no—
331C2×D9—31no—
332C2×D9—61no—
333C2×D9—62no—
334C2×D9—124no—
335C2×D9D3×D411yescharacteristic
336C4×C9—22no—
337C4×C9—21no—
338C4×C9—31no—
339C4×C9—42no—
340C4×C9C23×D311yescharacteristic
341Dic9—31no—
342Dic9—61no—
343Dic9—62no—
344Dic9—122no—
345Dic9D3×D411yescharacteristic
346C2×#41—272no—
347#135—31no—
348#135—91no—
349#135—271no—
350#135—271no—
351#135—272no—
352#135—272no—
353C2×Q16—271no—
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×Q8C22×D912yes—
364C3×Q8C2×D3212yes—
365C3×D4—32no—
366C3×D4—62no—
367C3×D4—92no—
368C3×D4—182no—
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—
376#67—182no—
377#67—182no—
378#67—182no—
379C2×Dic3—91no—
380C2×Dic3—91no—
381C2×Dic3—182no—
382C2×Dic3—182no—
383D12—22no—
384D12—32no—
385D12—62no—
386D12C22×D912yes—
387C4×D3—22no—
388C4×D3—31no—
389C4×D3—62no—
390C4×D3—91no—
391C4×D3—91no—
392C4×D3—91no—
393C4×D3—91no—
394C4×D3—91no—
395C4×D3—91no—
396C4×D3—181no—
397C4×D3—182no—
398C4×D3—182no—
399C4×D3—182no—
400C4×D3—182no—
401C4×D3—362no—
402C4×D3C22×D911yescharacteristic
403Dic6—92no—
404Dic6—92no—
405Dic6—182no—
406Dic6—182no—
407Dic6—182no—
408Dic6—362no—
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—32no—
421C3×D3—21no—
422C3×D3—42no—
423C3×D3—61no—
424C3×D3—122no—
425C2×C9—21no—
426C2×C9—61no—
427C2×C9—122no—
428C2×C9C2×D3×D411yescharacteristic
429D9—21no—
430D9—32no—
431D9—42no—
432D9—62no—
433#41—32no—
434#41—92no—
435#41—272no—
436#41—272no—
437#41—542no—
438#41—542no—
439C2×Q8—272no—
440C2×D4—272no—
441C2×D4—542no—
442C22×C4—271no—
443C22×C4—542no—
444Q16—272no—
445Q16—271no—
446Q16D3×D911yescharacteristic, complemented
447SD16—32no—
448SD16—92no—
449SD16—272no—
450SD16—272no—
451D8—31no—
452D8—91no—
453D8—272no—
454C2×C8—31no—
455C2×C8—91no—
456C2×C8—271no—
457C2×C8—271no—
458C2×C8—271no—
459C2×C8—271no—
460C22×C3—31no—
461C22×C3—62no—
462C22×C3—91no—
463C22×C3—182no—
464C2×D3—22no—
465C2×D3—31no—
466C2×D3—62no—
467C2×D3—91no—
468C2×D3—91no—
469C2×D3—181no—
470C2×D3—182no—
471C2×D3—182no—
472C2×D3—364no—
473C2×D3—364no—
474C2×D3D4×D911yescharacteristic
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—21no—
478C3×C4—31no—
479C3×C4—42no—
480C3×C4—91no—
481C3×C4C23×D911yescharacteristic
482C3×C4C22×D3211yescharacteristic
483Dic3—31no—
484Dic3—91no—
485Dic3—91no—
486Dic3—181no—
487Dic3—182no—
488Dic3—182no—
489Dic3—362no—
490Dic3D4×D911yescharacteristic
491C32D3×#13511yescharacteristic
492C9—21no—
493C9D3×#13511yescharacteristic, complemented
494C23—271no—
495C23—542no—
496Q8—272no—
497Q8—542no—
498Q8C2×D3×D912yes—
499D4—32no—
500D4—62no—
501D4—92no—
502D4—182no—
503D4—542no—
504D4—542no—
505D4—542no—
506C2×C4—31no—
507C2×C4—62no—
508C2×C4—91no—
509C2×C4—182no—
510C2×C4—271no—
511C2×C4—271no—
512C2×C4—271no—
513C2×C4—271no—
514C2×C4—542no—
515C2×C4—542no—
516C2×C4—542no—
517C8—31no—
518C8—91no—
519C8—271no—
520C8C2×D3×D911yescharacteristic
521C2×C3—21no—
522C2×C3—61no—
523C2×C3—122no—
524C2×C3—181no—
525C2×C3—362no—
526C2×C3C2×D4×D911yescharacteristic
527C2×C3D32×D411yescharacteristic
528D3—21no—
529D3—42no—
530D3—61no—
531D3—92no—
532D3—92no—
533D3—122no—
534D3—182no—
535C22—31no—
536C22—62no—
537C22—91no—
538C22—182no—
539C22—271no—
540C22—542no—
541C22—1084no—
542C4—22no—
543C4—31no—
544C4—91no—
545C4—271no—
546C4—542no—
547C4C22×D3×D911yescharacteristic
548C3—21no—
549C3D9×#13511yescharacteristic, complemented
550C3#1059476411yescharacteristic
551C2—61no—
552C2—122no—
553C2—181no—
554C2—272no—
555C2—362no—
556C2D3×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. #11722514C2semidirect
2. Q16×C3⋊D9C2semidirect
3. C9×#919C2semidirect
4. C3×#63481C2semidirect
5. #11721215C2semidirect
6. #11721212C2semidirect
7. #11721207C2semidirect
8. #11720822C2semidirect
9. #11720818C2semidirect
10. #11720811C2semidirect
11. D3×C8×D9C2semidirect
15. C9×#239C22semidirect
16. C3×#3701C22semidirect
22. #88641C22semidirect
24. #88638C22semidirect
25. C3×C9×Q16C22semidirect
26. #88547C22semidirect
28. #88521C22semidirect
29. C9×#215C22semidirect
30. C3×#3675C22semidirect
31. C9×D24C22semidirect
32. D3×C8×C9C22semidirect
33. C3×D72C22semidirect
34. C3×C8×D9C22semidirect
35. #88424C22semidirect
36. #88421C22semidirect
37. D3×C9⋊C8C22semidirect
38. #88415C22semidirect
40. D9×C3⋊C8C22semidirect
55. #63481D3semidirect
121. #3701C2×D3semidirect
133. C9×Q16C2×D3semidirect
137. #3675C2×D3semidirect
143. D72C2×D3semidirect
149. C8×D9C2×D3semidirect
172. #919D9semidirect
251. C3⋊D9#135semidirect
259. #239C2×D9semidirect
280. C3×Q16C2×D9semidirect
292. #215C2×D9semidirect
303. D24C2×D9semidirect
314. D3×C8C2×D9semidirect
357. C3×C9C2×#135semidirect
446. Q16D3×D9semidirect
493. C9D3×#135semidirect
549. C3D9×#135semidirect

Maximal subgroups

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

subgroupindex
#117225142
Q16×C3⋊D92
C9×#9192
C3×#634812
#117212152
#117212122
#117212072
#117208222
#117208182
#117208112
D3×C8×D92
#105947643
D9×#1353