Finite Group Databasebeta

← #13151090

Subgroups — #13151090

Order 1728.

910 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721837C211yesmaximal, characteristic, complemented
2C3×#63499C211yesmaximal, characteristic, complemented
3C9×#936C211yesmaximal, characteristic, complemented
4#11721278C212yesmaximal, complemented
5#11721276C212yesmaximal, complemented
6#11721257C212yesmaximal, complemented
7#11721242C212yesmaximal, complemented
8#11721003C212yesmaximal, complemented
9#11720940C212yesmaximal, complemented
10#10595262—31nomaximal
11#10589785—31nomaximal
12C23×C3⋊D9C2211yescharacteristic, complemented
13C23×C3×D9C2211yescharacteristic, complemented
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C3×#611C2212yes—
17C2×C9×Dic6C2211yescharacteristic, complemented
18C2×#6192C2212yes—
19C2×#6192C2212yes—
20C2×#6189C2212yescomplemented
21#88551C2211yescharacteristic, complemented
22#88536C2212yescomplemented
23C9×#217C2212yescomplemented
24C9×#217C2212yescomplemented
25C3×#3676C2211yescharacteristic, complemented
26C3×(C2×D9)⋊C4C2212yescomplemented
27C4×C9×Dic3C2211yescharacteristic, complemented
28#88445C2212yescomplemented
29#88445C2212yescomplemented
30(C2×D9)⋊Dic3C2212yescomplemented
31(C2×D9)⋊Dic3C2212yescomplemented
32Dic3×Dic9C2212yescomplemented
33#64146—31no—
34C3×#937—31no—
35C3×#936D311yescharacteristic
36#63981—32no—
37#63977—32no—
38#63959—32no—
39#63953—32no—
40#63907—32no—
41#63852—32no—
42C9×#124—31no—
43#63509—32no—
44#63499—31no—
45#63499—61no—
46#63499D311yescharacteristic, complemented
47#63447—32no—
48#63447—32no—
49#63444—32no—
50#63444—32no—
51#63437—32no—
52C23×C3×C9C2311yescharacteristic
53C23×C3×C9C2311yescharacteristic
54C22×C3⋊D9—42no—
55C22×C3⋊D9—41no—
56C22×C3⋊D9C2312yes—
57C22×C3×D9—41no—
58C22×C3×D9—42no—
59C22×C3×D9C2312yes—
60C3×D4×C9—41no—
61C2×C3×C4×C9C2311yescharacteristic
62#6233—42no—
63C2×C3⋊Dic9C2312yes—
64C3×#611—42no—
65C2×C9×Dic3C2312yes—
66C2×C9×Dic3C2312yes—
67C2×C3×Dic9C2312yes—
68C9×Dic6—41no—
69#6192—42no—
70#6192—42no—
71#6189—42no—
72#5683—91no—
73#5594—91no—
74C23×C3⋊D3—31no—
75C23×C3×D3—31no—
76C2×C32×D4C2×D311yescharacteristic
77C2×#638—32no—
78C2×C3×#67—32no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×#627—32no—
81C2×#626—32no—
82C2×#626—32no—
83C23×D9—31no—
84C23×D9—61no—
85C23×D9C2×D311yescharacteristic, complemented
86#3757—31no—
87#3752—32no—
88C3×#217—31no—
89C3×#217C2×D312yes—
90C3×#217C2×D312yes—
91C3×(C2×D3)⋊C4—32no—
92C3×C4×Dic3C2×D311yescharacteristic
93#3722—32no—
94#3722—32no—
95(C2×D3)⋊Dic3—32no—
96(C2×D3)⋊Dic3—32no—
97Dic32—32no—
98C2×Q8×C9—31no—
99C2×D4×C9—21no—
100C2×D4×C9C2×D311yescharacteristic
101C2×#611—32no—
102C2×#611—62no—
103C2×#611C2×D312yes—
104C2×D36—32no—
105C2×D36—32no—
106C2×Dic18—32no—
107C9×C22⋊C4—32no—
108C9×C22⋊C4—32no—
109C42×C9—31no—
110#3676—31no—
111#3676—61no—
112#3676C2×D311yescharacteristic, complemented
113(C2×D9)⋊C4—32no—
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—62no—
119(C2×D9)⋊C4C2×D312yescomplemented
120C4×Dic9—32no—
121C22×C3×C9—21no—
122C22×C3×C9—22no—
123C22×C3×C9—22no—
124C22×C3×C9—21no—
125C22×C3×C9C2411yescharacteristic
126C2×C3⋊D9—22no—
127C2×C3⋊D9—22no—
128C2×C3⋊D9—22no—
129C2×C3⋊D9—44no—
130C2×C3×D9—22no—
131C2×C3×D9—22no—
132C2×C3×D9—22no—
133C2×C3×D9—44no—
134C3×C4×C9—21no—
135C3⋊Dic9—22no—
136C9×Dic3—22no—
137C9×Dic3—22no—
138C3×Dic9—22no—
139C3×#124—31no—
140C3×#120—91no—
141#953—92no—
142#947—92no—
143#937—31no—
144#937—91no—
145#937—91no—
146#937—92no—
147#937—181no—
148#936—92no—
149#936D911yescharacteristic, complemented
150#885—92no—
151#885—92no—
152#885—92no—
153#885—92no—
154#885—92no—
155#882—92no—
156#882—92no—
157#875—92no—
158C23×C32C22×D311yescharacteristic
159C23×C32C22×D311yescharacteristic
160C22×C3⋊D3—32no—
161C22×C3⋊D3—122no—
162C22×C3⋊D3—121no—
163C22×C3×D3—32no—
164C22×C3×D3—121no—
165C22×C3×D3—122no—
166C32×D4—41no—
167C2×C32×C4C22×D311yescharacteristic
168#638—122no—
169C2×C3⋊Dic3—32no—
170C3×#67—122no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3C22×D312yes—
173C2×C3×Dic3C22×D312yes—
174C3×Dic6—41no—
175#627—122no—
176#626—122no—
177#626—122no—
178C23×C9—21no—
179C23×C9—21no—
180C23×C9C22×D311yescharacteristic
181C23×C9C22×D311yescharacteristic
182C22×D9—32no—
183C22×D9—41no—
184C22×D9—42no—
185C22×D9—62no—
186C22×D9—121no—
187C22×D9—122no—
188C22×D9—124no—
189C22×D9—122no—
190C22×D9C22×D312yes—
191Q8×C9—121no—
192D4×C9—41no—
193D4×C9—42no—
194C2×C4×C9—21no—
195C2×C4×C9—32no—
196C2×C4×C9—32no—
197C2×C4×C9C22×D311yescharacteristic
198#611—42no—
199#611—122no—
200#611—124no—
201C2×Dic9—32no—
202C2×Dic9—62no—
203C2×Dic9C22×D312yes—
204D36—122no—
205D36—122no—
206Dic18—122no—
207H(3,𝔽4)—271noSylow
208C2×C3×C9—41no—
209C2×C3×C9—41no—
210C2×C3×C9H(5,𝔽2)11yescharacteristic
211C2×C3×C9H(5,𝔽2)12yes—
212C3⋊D9—42no—
213C3×D9—42no—
214C24×C3—91no—
215C23×D3—31no—
216C23×D3—91no—
217C23×D3—91no—
218C23×D3—181no—
219C2×C3×Q8—31no—
220C2×C3×D4—21no—
221C2×C3×D4—92no—
222C2×C3×D4C2×D911yescharacteristic
223C2×C3×D4D3211yescharacteristic
224C2×#67—32no—
225C2×#67—92no—
226C2×#67—92no—
227C2×#67—92no—
228C2×#67—92no—
229C2×#67—182no—
230C2×D12—92no—
231C2×D12—92no—
232C2×Dic6—92no—
233C2×Dic6—92no—
234C2×Dic6C2×D911yescharacteristic, complemented
235C3×C22⋊C4—32no—
236C3×C22⋊C4—32no—
237C3×C22⋊C4—91no—
238C3×C22⋊C4—92no—
239C3×C42—31no—
240#217—31no—
241#217—91no—
242#217—91no—
243#217—92no—
244#217—92no—
245#217—181no—
246#217C2×D912yescomplemented
247#217C2×D912yescomplemented
248(C2×D3)⋊C4—32no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—182no—
258C4×Dic3—92no—
259C4×Dic3—92no—
260C4×Dic3C2×D911yescharacteristic, complemented
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32C23×D311yescharacteristic
266C2×C3⋊D3—62no—
267C2×C3⋊D3—62no—
268C2×C3⋊D3—62no—
269C2×C3⋊D3—124no—
270C2×C3×D3—62no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—124no—
274C32×C4—21no—
275C3⋊Dic3—62no—
276C3×Dic3—22no—
277C3×Dic3—22no—
278C3×Dic3—62no—
279C22×C9—22no—
280C22×C9—22no—
281C22×C9—22no—
282C22×C9—21no—
283C22×C9—21no—
284C22×C9—21no—
285C22×C9—42no—
286C22×C9—42no—
287C22×C9—41no—
288C22×C9C23×D311yescharacteristic
289C2×D9—22no—
290C2×D9—22no—
291C2×D9—22no—
292C2×D9—44no—
293C2×D9—64no—
294C2×D9—62no—
295C2×D9—62no—
296C2×D9—62no—
297C2×D9—122no—
298C2×D9—122no—
299C2×D9—128no—
300C2×D9—124no—
301C4×C9—21no—
302C4×C9—41no—
303C4×C9—62no—
304C4×C9—62no—
305Dic9—22no—
306Dic9—62no—
307Dic9—122no—
308#124—31no—
309#124—272no—
310#124—272no—
311#124—272no—
312#124—272no—
313#120—91no—
314#120—272no—
315#120—272no—
316#120—271no—
317C3×C9H(3,𝔽4)11yescharacteristic, complemented, Sylow
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—92no—
321C23×C3—361no—
322C23×C3—362no—
323C23×C3C22×D911yescharacteristic
324C23×C3C22×D911yescharacteristic
325C23×C3C2×D3211yescharacteristic
326C23×C3C2×D3211yescharacteristic
327C22×D3—32no—
328C22×D3—92no—
329C22×D3—92no—
330C22×D3—122no—
331C22×D3—121no—
332C22×D3—182no—
333C22×D3—362no—
334C22×D3—361no—
335C22×D3—361no—
336C22×D3—362no—
337C22×D3—364no—
338C22×D3—362no—
339C3×Q8—121no—
340C3×D4—41no—
341C3×D4—41no—
342C3×D4—42no—
343C3×D4—362no—
344C2×C3×C4—21no—
345C2×C3×C4—32no—
346C2×C3×C4—32no—
347C2×C3×C4—92no—
348C2×C3×C4C22×D911yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350#67—122no—
351#67—362no—
352#67—362no—
353#67—362no—
354#67—362no—
355#67—364no—
356C2×Dic3—32no—
357C2×Dic3—92no—
358C2×Dic3—92no—
359C2×Dic3—182no—
360C2×Dic3C22×D912yes—
361C2×Dic3C22×D912yes—
362D12—362no—
363D12—362no—
364Dic6—41no—
365Dic6—362no—
366Dic6—362no—
367C2×C32—41no—
368C2×C32—41no—
369C2×C32#100411yescharacteristic
370C2×C32#100912yes—
371C3⋊D3—122no—
372C3×D3—122no—
373C2×C9—22no—
374C2×C9—21no—
375C2×C9—42no—
376C2×C9—41no—
377C2×C9—41no—
378C2×C9—81no—
379C2×C9#100412yes—
380C2×C9#100411yescharacteristic
381D9—42no—
382D9—124no—
383D9—122no—
384C24—91no—
385C24—271no—
386C2×Q8—31no—
387C2×Q8—272no—
388C2×D4—92no—
389C2×D4—272no—
390C2×D4—272no—
391C2×D4—272no—
392C2×D4D3×D911yescharacteristic
393C22⋊C4—32no—
394C22⋊C4—32no—
395C22⋊C4—91no—
396C22⋊C4—92no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—271no—
402C22⋊C4—272no—
403C42—31no—
404C42—272no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—21no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—41no—
418C22×C3—182no—
419C22×C3—182no—
420C22×C3—182no—
421C22×C3—364no—
422C22×C3C23×D911yescharacteristic
423C22×C3C22×D3211yescharacteristic
424C2×D3—62no—
425C2×D3—62no—
426C2×D3—62no—
427C2×D3—124no—
428C2×D3—184no—
429C2×D3—182no—
430C2×D3—182no—
431C2×D3—182no—
432C2×D3—182no—
433C2×D3—182no—
434C2×D3—182no—
435C2×D3—362no—
436C2×D3—362no—
437C2×D3—364no—
438C2×D3—368no—
439C2×D3—364no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—41no—
443C3×C4—62no—
444C3×C4—62no—
445C3×C4—182no—
446Dic3—22no—
447Dic3—22no—
448Dic3—62no—
449Dic3—182no—
450Dic3—182no—
451Dic3—362no—
452C32#568311yescharacteristic
453C9—21no—
454C9#559411yescharacteristic, complemented
455C23—92no—
456C23—272no—
457C23—362no—
458C23—361no—
459C23—1082no—
460C23—1081no—
461C23C2×D3×D911yescharacteristic
462C23C2×D3×D911yescharacteristic
463Q8—121no—
464Q8—1082no—
465D4—41no—
466D4—362no—
467D4—1082no—
468D4—1082no—
469D4—1082no—
470C2×C4—32no—
471C2×C4—32no—
472C2×C4—92no—
473C2×C4—272no—
474C2×C4C2×D3×D911yescharacteristic
475C2×C3—22no—
476C2×C3—21no—
477C2×C3—42no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—81no—
483C2×C3—362no—
484C2×C3#6371011yescharacteristic
485C2×C3#6371512yes—
486C2×C3#6431412yes—
487C2×C3#6433011yescharacteristic
488D3—122no—
489D3—364no—
490D3—362no—
491D3—362no—
492C22—22no—
493C22—22no—
494C22—21no—
495C22—21no—
496C22—182no—
497C22—182no—
498C22—182no—
499C22—364no—
500C22—542no—
501C22—542no—
502C22—542no—
503C22—1084no—
504C22C22×D3×D911yescharacteristic
505C4—21no—
506C4—62no—
507C4—62no—
508C4—182no—
509C4—542no—
510C3—21no—
511C3#1058978511yescharacteristic, complemented
512C3#1059526211yescharacteristic
513C2—41no—
514C2—41no—
515C2—362no—
516C2—1082no—
517C2#1172250112yes—
518C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9H(3,𝔽4)
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11721837C2semidirect
2. C3×#63499C2semidirect
3. C9×#936C2semidirect
4. #11721278C2semidirect
5. #11721276C2semidirect
6. #11721257C2semidirect
7. #11721242C2semidirect
8. #11721003C2semidirect
9. #11720940C2semidirect
12. C23×C3⋊D9C22semidirect
13. C23×C3×D9C22semidirect
17. C2×C9×Dic6C22semidirect
20. C2×#6189C22semidirect
21. #88551C22semidirect
22. #88536C22semidirect
23. C9×#217C22semidirect
24. C9×#217C22semidirect
25. C3×#3676C22semidirect
26. C3×(C2×D9)⋊C4C22semidirect
27. C4×C9×Dic3C22semidirect
28. #88445C22semidirect
29. #88445C22semidirect
30. (C2×D9)⋊Dic3C22semidirect
31. (C2×D9)⋊Dic3C22semidirect
32. Dic3×Dic9C22semidirect
46. #63499D3semidirect
85. C23×D9C2×D3semidirect
112. #3676C2×D3semidirect
119. (C2×D9)⋊C4C2×D3semidirect
149. #936D9semidirect
234. C2×Dic6C2×D9semidirect
246. #217C2×D9semidirect
247. #217C2×D9semidirect
260. C4×Dic3C2×D9semidirect
317. C3×C9H(3,𝔽4)semidirect
454. C9#5594semidirect
511. C3#10589785semidirect

Maximal subgroups

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

subgroupindex
#117218372
C3×#634992
C9×#9362
#117212782
#117212762
#117212572
#117212422
#117210032
#117209402
#105952623
#105897853