Finite Group Databasebeta

← #13150285

Subgroups — #13150285

Order 1728.

800 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721813C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
3C9×#924C211yesmaximal, characteristic, complemented
4D9×#217C212yesmaximal
5#11721253C212yesmaximal, complemented
6#11720999C212yesmaximal, complemented
7#11720996C212yesmaximal, complemented
8#11720991C212yesmaximal, complemented
9#11720984C212yesmaximal, complemented
10#10594971—31nomaximal
11#10589736—31nomaximal
12C23×C3×D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C2×#6226C2211yescharacteristic, complemented
15C22×C3×Dic9C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic, complemented
17C2×Dic3×D9C2212yes—
18C2×Dic3×D9C2212yes—
19C2×#6189C2212yescomplemented
20#88551C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
22#88534C2212yescomplemented
23C9×#217C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic, complemented
27C9×Dic3⋊C4C2212yescomplemented
28(C2×D9)⋊Dic3C2212yes—
29(C2×D9)⋊Dic3C2212yes—
30Dic3⋊Dic9C2212yescomplemented
31#88442C2212yescomplemented
32Dic9⋊Dic3C2212yescomplemented
33#64133—31no—
34C2×C3×(C2×D3)⋊C4—31no—
35C3×#924D311yescharacteristic
36D3×#217—32no—
37#63964—32no—
38#63899—32no—
39#63896—32no—
40#63893—32no—
41#63887—32no—
42C9×C22⋊Q8—31no—
43#63508—32no—
44C2×(C2×D9)⋊C4D311yescharacteristic
45#63486—31no—
46#63486—61no—
47#63459—32no—
48#63458—32no—
49#63458—32no—
50C22⋊C4×D9—32no—
51#63440—32no—
52C23×C3×C9C2311yescharacteristic
53C22×C3×D9—41no—
54C22×C3×D9—41no—
55C22×C3×D9Q814yescomplemented
56C22×C3×D9C2312yes—
57C2×C3×C4×C9—41no—
58C2×C3×C4×C9C2311yescharacteristic
59C2×C3×C4×C9C2311yescharacteristic
60C2×C3⋊Dic9C2312yes—
61C2×C3⋊Dic9C2312yes—
62#6226—41no—
63C2×C9×Dic3C2312yes—
64C2×C9×Dic3C2312yes—
65C2×C3×Dic9—41no—
66C2×C3×Dic9C2312yes—
67C9×Dic6—41no—
68Dic3×D9—42no—
69Dic3×D9—42no—
70#6189—42no—
71#5634—91no—
72#5483—91no—
73C23×C3×D3—31no—
74C22×C32×C4C2×D311yescharacteristic
75C2×#634—31no—
76C22×C3×Dic3—31no—
77C2×C3×Dic6C2×D311yescharacteristic
78C2×#627—32no—
79C2×D3×Dic3—32no—
80C2×D3×Dic3—32no—
81C23×D9C2×D311yescharacteristic
82#3757—32no—
83C4⋊(C3⋊Dic3)—31no—
84#3750—32no—
85C3×#217C2×D312yes—
86C3×(C2×D3)⋊C4—32no—
87C3×(C2×D3)⋊C4—32no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×Dic3⋊C4C2×D312yes—
90#3724—32no—
91Dic3⋊Dic3—32no—
92Dic3⋊Dic3—32no—
93(C2×D3)⋊Dic3—32no—
94(C2×D3)⋊Dic3—32no—
95C2×Q8×C9—31no—
96C22×C4×C9—21no—
97C22×C4×C9C2×D311yescharacteristic
98C22×Dic9C2×D311yescharacteristic
99C2×C4×D9—32no—
100C2×C4×D9—32no—
101C2×Dic18—31no—
102C2×Dic18—32no—
103C2×Dic18—61no—
104C9×C4⋊C4—32no—
105C9×C4⋊C4—31no—
106C9×C22⋊C4—32no—
107#3676—32no—
108#3676—62no—
109(C2×D9)⋊C4—32no—
110(C2×D9)⋊C4—32no—
111(C2×D9)⋊C4C2×D312yes—
112(C2×D9)⋊C4C2×D312yes—
113C4⋊Dic9—31no—
114C4⋊Dic9—32no—
115C4⋊Dic9—61no—
116Dic9⋊C4—32no—
117Dic9⋊C4—32no—
118Dic9⋊C4—32no—
119Dic9⋊C4—62no—
120C22×C3×C9—21no—
121C22×C3×C9—21no—
122C22×C3×C9C2×Q812yes—
123C22×C3×C9C2411yescharacteristic
124C2×C3×D9—22no—
125C2×C3×D9—22no—
126C2×C3×D9—44no—
127C2×C3×D9C2×Q814yes—
128C3×C4×C9—21no—
129C3×C4×C9—21no—
130C3⋊Dic9—22no—
131C3⋊Dic9—22no—
132C9×Dic3—22no—
133C9×Dic3—22no—
134C3×Dic9—22no—
135C3×C22⋊Q8—31no—
136C2×C3×C22⋊C4—91no—
137C2×#217—92no—
138#946—92no—
139C2×(C2×D3)⋊C4—31no—
140#924—91no—
141#924—91no—
142#924—92no—
143#924—181no—
144#924D911yescharacteristic, complemented
145#897—92no—
146#896—92no—
147#896—92no—
148D3×C22⋊C4—92no—
149#878—92no—
150#878—92no—
151#878—92no—
152#878—92no—
153#878—92no—
154C23×C32C22×D311yescharacteristic
155C22×C3×D3—32no—
156C22×C3×D3—34no—
157C22×C3×D3—121no—
158C22×C3×D3—121no—
159C2×C32×C4—41no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C3⋊Dic3—32no—
163C2×C3⋊Dic3—32no—
164#634—121no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—121no—
167C2×C3×Dic3C22×D312yes—
168C2×C3×Dic3C22×D312yes—
169C3×Dic6—41no—
170#627—122no—
171D3×Dic3—122no—
172D3×Dic3—122no—
173C23×C9—21no—
174C23×C9C22×D311yescharacteristic
175C22×D9—41no—
176C22×D9—41no—
177C22×D9Dic614yescomplemented
178C22×D9C22×D312yes—
179Q8×C9—121no—
180C2×C4×C9—21no—
181C2×C4×C9—21no—
182C2×C4×C9—32no—
183C2×C4×C9—32no—
184C2×C4×C9—41no—
185C2×C4×C9—42no—
186C2×C4×C9C22×D311yescharacteristic
187C2×C4×C9C22×D311yescharacteristic
188C2×Dic9—32no—
189C2×Dic9—32no—
190C2×Dic9—41no—
191C2×Dic9—62no—
192C2×Dic9—62no—
193C2×Dic9C22×D312yes—
194C4×D9—122no—
195C4×D9—122no—
196Dic18—122no—
197Dic18—121no—
198Dic18—122no—
199#543—271noSylow
200C2×C3×C9—22no—
201C2×C3×C9C22×Q811yescharacteristic
202C2×C3×C9H(5,𝔽2)11yescharacteristic
203C2×C3×C9H(5,𝔽2)11yescharacteristic
204C3×D9—24no—
205C24×C3—91no—
206C23×D3—31no—
207C2×C3×Q8—31no—
208C22×C3×C4—21no—
209C22×C3×C4—91no—
210C22×C3×C4C2×D911yescharacteristic
211C22×C3×C4D3211yescharacteristic
212C22×Dic3—31no—
213C22×Dic3—92no—
214C22×Dic3—92no—
215C2×C4×D3—92no—
216C2×C4×D3—92no—
217C2×Dic6—92no—
218C2×Dic6—91no—
219C2×Dic6—91no—
220C2×Dic6—92no—
221C2×Dic6—181no—
222C2×Dic6C2×D911yescharacteristic, complemented
223C3×C4⋊C4—31no—
224C3×C4⋊C4—32no—
225C3×C22⋊C4—32no—
226C3×C22⋊C4—92no—
227C3×C22⋊C4—92no—
228#217—92no—
229#217—92no—
230#217—92no—
231#217—92no—
232#217—182no—
233#217C2×D912yes—
234(C2×D3)⋊C4—32no—
235(C2×D3)⋊C4—32no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—92no—
238C4⋊Dic3—92no—
239C4⋊Dic3—91no—
240C4⋊Dic3—91no—
241C4⋊Dic3—92no—
242C4⋊Dic3—181no—
243C4⋊Dic3C2×D911yescharacteristic, complemented
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—182no—
251Dic3⋊C4C2×D912yescomplemented
252C22×C32—21no—
253C22×C32—21no—
254C22×C32D3×Q812yes—
255C22×C32C23×D311yescharacteristic
256C2×C3×D3—34no—
257C2×C3×D3—62no—
258C2×C3×D3—62no—
259C2×C3×D3—124no—
260C32×C4—21no—
261C32×C4—21no—
262C3⋊Dic3—62no—
263C3⋊Dic3—62no—
264C3×Dic3—22no—
265C3×Dic3—22no—
266C3×Dic3—62no—
267C22×C9—21no—
268C22×C9—21no—
269C22×C9—21no—
270C22×C9—22no—
271C22×C9—41no—
272C22×C9—41no—
273C22×C9C2×Dic612yes—
274C22×C9C23×D311yescharacteristic
275C2×D9—22no—
276C2×D9—22no—
277C2×D9—44no—
278C2×D9C2×Dic614yes—
279C4×C9—22no—
280C4×C9—21no—
281C4×C9—21no—
282C4×C9—41no—
283C4×C9—62no—
284C4×C9—62no—
285Dic9—22no—
286Dic9—62no—
287Dic9—62no—
288Dic9—122no—
289Dic9—122no—
290C22⋊Q8—31no—
291C22⋊Q8—272no—
292C22⋊Q8—272no—
293C22⋊Q8—272no—
294C22⋊Q8—272no—
295C22⋊Q8—272no—
296C22⋊Q8—271no—
297C2×C22⋊C4—91no—
298C2×C22⋊C4—272no—
299C3×C9#54311yescharacteristic, complemented, Sylow
300C23×C3—21no—
301C23×C3—92no—
302C23×C3—94no—
303C23×C3—361no—
304C23×C3—361no—
305C23×C3C22×D911yescharacteristic
306C23×C3C2×D3211yescharacteristic
307C22×D3—32no—
308C22×D3—34no—
309C22×D3—121no—
310C22×D3—121no—
311C3×Q8—121no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—32no—
315C2×C3×C4—32no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—42no—
319C2×C3×C4—92no—
320C2×C3×C4—361no—
321C2×C3×C4C22×D911yescharacteristic
322C2×C3×C4C22×D911yescharacteristic
323C2×C3×C4C2×D3211yescharacteristic
324C2×C3×C4C2×D3211yescharacteristic
325C2×Dic3—32no—
326C2×Dic3—92no—
327C2×Dic3—92no—
328C2×Dic3—92no—
329C2×Dic3—92no—
330C2×Dic3—121no—
331C2×Dic3—182no—
332C2×Dic3—182no—
333C2×Dic3—362no—
334C2×Dic3—362no—
335C2×Dic3C22×D912yes—
336C2×Dic3C22×D912yes—
337C4×D3—362no—
338C4×D3—362no—
339Dic6—41no—
340Dic6—362no—
341Dic6—362no—
342Dic6—361no—
343Dic6—361no—
344Dic6—362no—
345C2×C32—22no—
346C2×C32#100411yescharacteristic
347C2×C32C2×D3×Q811yescharacteristic
348C2×C32#100911yescharacteristic
349C3×D3—64no—
350C2×C9—22no—
351C2×C9—21no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—42no—
355C2×C9C22×Dic611yescharacteristic
356C2×C9#100411yescharacteristic
357C2×C9#100411yescharacteristic
358D9—24no—
359C24—91no—
360C2×Q8—31no—
361C2×Q8—272no—
362C2×Q8—271no—
363C22×C4—91no—
364C22×C4—272no—
365C22×C4—272no—
366C22×C4D3×D911yescharacteristic
367C4⋊C4—32no—
368C4⋊C4—31no—
369C4⋊C4—272no—
370C4⋊C4—272no—
371C4⋊C4—272no—
372C4⋊C4—272no—
373C4⋊C4—271no—
374C22⋊C4—32no—
375C22⋊C4—92no—
376C22⋊C4—92no—
377C22⋊C4—272no—
378C22⋊C4—272no—
379C22⋊C4—272no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—22no—
385C22×C3—21no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—94no—
389C22×C3—182no—
390C22×C3—182no—
391C22×C3—364no—
392C22×C3Q8×D912yes—
393C22×C3C23×D911yescharacteristic
394C22×C3D3×Dic612yes—
395C22×C3C22×D3211yescharacteristic
396C2×D3—34no—
397C2×D3—62no—
398C2×D3—62no—
399C2×D3—124no—
400C3×C4—22no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—41no—
406C3×C4—62no—
407C3×C4—62no—
408C3×C4—182no—
409Dic3—22no—
410Dic3—22no—
411Dic3—62no—
412Dic3—182no—
413Dic3—182no—
414Dic3—182no—
415Dic3—182no—
416Dic3—362no—
417Dic3—362no—
418C32#563411yescharacteristic
419C9—21no—
420C9#548311yescharacteristic, complemented
421C23—92no—
422C23—94no—
423C23—361no—
424C23—361no—
425C23C2×D3×D911yescharacteristic
426Q8—121no—
427Q8—1081no—
428Q8—1082no—
429C2×C4—32no—
430C2×C4—32no—
431C2×C4—41no—
432C2×C4—92no—
433C2×C4—272no—
434C2×C4—272no—
435C2×C4—361no—
436C2×C4—1082no—
437C2×C4—1082no—
438C2×C4C2×D3×D911yescharacteristic
439C2×C4C2×D3×D911yescharacteristic
440C2×C3—22no—
441C2×C3—22no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—42no—
446C2×C3—184no—
447C2×C3#6371011yescharacteristic
448C2×C3C2×Q8×D911yescharacteristic
449C2×C3#6371511yescharacteristic
450C2×C3C2×D3×Dic611yescharacteristic
451C2×C3#6430711yescharacteristic
452C2×C3#6433011yescharacteristic
453D3—64no—
454C22—21no—
455C22—21no—
456C22—94no—
457C22—182no—
458C22—182no—
459C22—364no—
460C22D9×Dic612yes—
461C22C22×D3×D911yescharacteristic
462C4—21no—
463C4—21no—
464C4—62no—
465C4—62no—
466C4—182no—
467C4—542no—
468C4—542no—
469C3—21no—
470C3#1058973611yescharacteristic, complemented
471C3#1059497111yescharacteristic
472C2—22no—
473C2—184no—
474C2C2×D9×Dic611yescharacteristic
475C2#1172247711yescharacteristic
476C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC23×C3×D9C22
hypercenterC22×C4D3×D9
hypocenterC3×C9#543
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11721813C2semidirect
3. C9×#924C2semidirect
5. #11721253C2semidirect
6. #11720999C2semidirect
7. #11720996C2semidirect
8. #11720991C2semidirect
9. #11720984C2semidirect
14. C2×#6226C22semidirect
16. C2×C9×Dic6C22semidirect
19. C2×#6189C22semidirect
21. C4⋊(C3⋊Dic9)C22semidirect
22. #88534C22semidirect
26. C9×C4⋊Dic3C22semidirect
27. C9×Dic3⋊C4C22semidirect
30. Dic3⋊Dic9C22semidirect
31. #88442C22semidirect
32. Dic9⋊Dic3C22semidirect
55. C22×C3×D9Q8semidirect
144. #924D9semidirect
177. C22×D9Dic6semidirect
222. C2×Dic6C2×D9semidirect
243. C4⋊Dic3C2×D9semidirect
251. Dic3⋊C4C2×D9semidirect
299. C3×C9#543semidirect
420. C9#5483semidirect
470. C3#10589736semidirect

Maximal subgroups

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

subgroupindex
#117218132
C2×C3×(C2×D9)⋊C42
C9×#9242
D9×#2172
#117212532
#117209992
#117209962
#117209912
#117209842
#105949713
#105897363