Finite Group Databasebeta

← #13148802

Subgroups — #13148802

Order 1728.

868 conjugacy classes of subgroups.

The subgroup lattice has 470 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×#63454C211yesmaximal, characteristic
4C9×#887C211yesmaximal, characteristic, complemented
5#11720975C212yesmaximal
6#11720974C212yesmaximal
7C4×#6189C212yesmaximal, complemented
8#11720943C211yesmaximal, characteristic
9#11720912C211yesmaximal, characteristic, complemented
10#11720911C211yesmaximal, characteristic, complemented
11#11720901C212yesmaximal, complemented
12#10594357—31nomaximal
13#10589674—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C3×C4×D9C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic
17C2×Dic3×D9C2212yes—
18C2×#6189C2212yes—
19D9×Dic6C418yescomplemented
20C3×C9×C4⋊C4C2211yescharacteristic
21#88534C2212yes—
22C4×C3⋊Dic9C2212yes—
23C4×C3⋊Dic9C2211yescharacteristic
24C3×(C2×D9)⋊C4C2212yes—
25C9×Dic3⋊C4C2212yes—
26C4×C9×Dic3C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4×Dic9C2212yes—
30(C2×D9)⋊Dic3C2211yescharacteristic
31(C2×D9)⋊Dic3C2211yescharacteristic
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2211yescharacteristic
35Dic9⋊Dic3C2211yescharacteristic
36Dic3×Dic9C2212yes—
37Dic3×Dic9C2212yes—
38C2×D3×Dic6—31no—
39#64096—31no—
40C3×#892—31no—
41C3×#887D311yescharacteristic
42C4×#627—32no—
43#63878—32no—
44#63877—32no—
45#63855—31no—
46#63844—31no—
47#63843—31no—
48#63837—32no—
49C2×Q8×D9—31no—
50C4×Q8×C9—31no—
51Q8×Dic9—31no—
52#63454—32no—
53#63454D311yescharacteristic
54#63449—31no—
55#63449—32no—
56#63449—61no—
57#63434—31no—
58#63434—32no—
59C4×Dic18—31no—
60C4×Dic18—32no—
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67#6226C2×C414yes—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C3×C4×D9C2×C414yes—
74C9×Dic6C2×C414yes—
75Dic3×D9C2×C418yes—
76#6189C2×C418yes—
77#5572—91no—
78#5516—91no—
79C2×#634—31no—
80C2×C3×C4×D3—31no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×#627—32no—
83C2×D3×Dic3—32no—
84D3×Dic6—38no—
85C32×C4⋊C4C2×D311yescharacteristic
86#3750—32no—
87C4×C3⋊Dic3—31no—
88C4×C3⋊Dic3—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×C4⋊Dic3—31no—
91C3×Dic3⋊C4C2×D312yes—
92C3×C4×Dic3—32no—
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C2×D312yes—
95#3724—32no—
96Dic3⋊Dic3—31no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—31no—
99(C2×D3)⋊Dic3—31no—
100(C2×D3)⋊Dic3—31no—
101Dic32—32no—
102Dic32—32no—
103C2×Q8×C9—31no—
104Q8×D9—38no—
105C2×C4×D9—32no—
106C2×C4×D9C2×D311yescharacteristic
107C2×Dic18—31no—
108C2×Dic18—32no—
109C2×Dic18—61no—
110C9×C4⋊C4—21no—
111C9×C4⋊C4—32no—
112C9×C4⋊C4C2×D311yescharacteristic
113C42×C9—32no—
114C42×C9—31no—
115(C2×D9)⋊C4—31no—
116(C2×D9)⋊C4—31no—
117(C2×D9)⋊C4C2×D312yes—
118C4⋊Dic9—32no—
119C4⋊Dic9C2×D311yescharacteristic
120Dic9⋊C4—32no—
121Dic9⋊C4—31no—
122Dic9⋊C4—31no—
123Dic9⋊C4—32no—
124Dic9⋊C4—62no—
125C4×Dic9—31no—
126C4×Dic9—32no—
127C4×Dic9—32no—
128C4×Dic9—32no—
129C4×Dic9—62no—
130C4×Dic9—61no—
131C4×Dic9C2×D312yes—
132C22×C3×C9C2411yescharacteristic
133C2×C3×D9—21no—
134C2×C3×D9—21no—
135C2×C3×D9C22×C412yes—
136C3×C4×C9—22no—
137C3×C4×C9C22×C412yes—
138C3⋊Dic9—21no—
139C3⋊Dic9—21no—
140C3⋊Dic9C22×C414yes—
141C9×Dic3—21no—
142C9×Dic3—21no—
143C9×Dic3C22×C414yes—
144C3×Dic9—22no—
145C3×Dic9C22×C412yes—
146C2×D3×Q8—91no—
147C22×Dic6—91no—
148C3×C4×Q8—31no—
149C3×#117—91no—
150Q8×Dic3—91no—
151#926—91no—
152#892—31no—
153#892—92no—
154#887—92no—
155#887—91no—
156#887—91no—
157#887—91no—
158#887—91no—
159#887—181no—
160#887D911yescharacteristic, complemented
161#877—92no—
162#877—92no—
163#872—92no—
164#872—91no—
165C4×Dic6—92no—
166C4×Dic6—91no—
167C4×Dic6—92no—
168C4×Dic6—92no—
169C22×C3×D3—31no—
170C2×C32×C4C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—31no—
174C2×C3⋊Dic3—31no—
175#634—34no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3C22×D311yescharacteristic
179C2×C3×Dic3C22×D311yescharacteristic
180C2×C3×Dic3C22×D312yes—
181C3×C4×D3—34no—
182C3×Dic6C4×D314yes—
183#627—38no—
184D3×Dic3—38no—
185C22×D9C22×D311yescharacteristic
186Q8×C9—34no—
187C2×C4×C9—21no—
188C2×C4×C9—22no—
189C2×C4×C9—32no—
190C2×C4×C9—31no—
191C2×C4×C9—31no—
192C2×C4×C9C22×D312yes—
193C2×C4×C9C22×D311yescharacteristic
194C2×Dic9—32no—
195C2×Dic9—31no—
196C2×Dic9—31no—
197C2×Dic9—61no—
198C2×Dic9—62no—
199C2×Dic9—61no—
200C2×Dic9C22×D312yes—
201C2×Dic9C22×D311yescharacteristic
202C4×D9—38no—
203C4×D9C4×D314yes—
204Dic18—34no—
205Dic18—38no—
206Dic18—64no—
207#519—271noSylow
208C2×C3×C9C23×C411yescharacteristic
209C2×C3×C92−1+411yescharacteristic
210C2×C3×C92−1+411yescharacteristic
211C3×D9—22no—
212C2×C3×Q8—31no—
213C22×C3×C4—91no—
214C22×Dic3—92no—
215D3×Q8—98no—
216C2×C4×D3—31no—
217C2×C4×D3—92no—
218C2×Dic6—92no—
219C2×Dic6—91no—
220C2×Dic6—91no—
221C2×Dic6—92no—
222C2×Dic6—98no—
223C2×Dic6—181no—
224C2×Dic6C2×D911yescharacteristic
225C3×C4⋊C4—21no—
226C3×C4⋊C4—32no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4C2×D911yescharacteristic
229C3×C4⋊C4D3211yescharacteristic
230C3×C22⋊C4—92no—
231C3×C42—32no—
232C3×C42—31no—
233C3×C42—92no—
234#217—91no—
235#217—91no—
236(C2×D3)⋊C4—32no—
237(C2×D3)⋊C4—91no—
238(C2×D3)⋊C4—91no—
239C4⋊Dic3—31no—
240C4⋊Dic3—92no—
241C4⋊Dic3—91no—
242C4⋊Dic3—91no—
243Dic3⋊C4—91no—
244Dic3⋊C4—92no—
245Dic3⋊C4—91no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—182no—
251Dic3⋊C4C2×D912yes—
252C4×Dic3—32no—
253C4×Dic3—92no—
254C4×Dic3—92no—
255C4×Dic3—92no—
256C4×Dic3—91no—
257C4×Dic3—92no—
258C4×Dic3—91no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—182no—
262C4×Dic3—181no—
263C4×Dic3C2×D912yes—
264C4×Dic3C2×D911yescharacteristic
265C22×C32C23×D311yescharacteristic
266C2×C3×D3—32no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C32×C4—22no—
270C32×C4C2×C4×D312yes—
271C3⋊Dic3—34no—
272C3⋊Dic3—61no—
273C3⋊Dic3—61no—
274C3×Dic3—21no—
275C3×Dic3—21no—
276C3×Dic3—32no—
277C3×Dic3—62no—
278C3×Dic3C2×C4×D314yes—
279C22×C9—21no—
280C22×C9C23×D311yescharacteristic
281C2×D9—21no—
282C2×D9—21no—
283C2×D9C2×C4×D312yes—
284C4×C9—22no—
285C4×C9—22no—
286C4×C9—34no—
287C4×C9—42no—
288C4×C9—61no—
289C4×C9—61no—
290C4×C9C2×C4×D312yes—
291Dic9—22no—
292Dic9—34no—
293Dic9—64no—
294Dic9—62no—
295Dic9—61no—
296Dic9—61no—
297Dic9—121no—
298Dic9C2×C4×D312yes—
299C22×Q8—271no—
300C4×Q8—31no—
301C4×Q8—272no—
302C4×Q8—271no—
303C4×Q8—272no—
304C4×Q8—271no—
305C4×Q8—271no—
306#117—91no—
307#117—272no—
308#117—271no—
309#117—272no—
310C3×C9#51911yescharacteristic, complemented, Sylow
311C23×C3—91no—
312C22×D3—31no—
313C3×Q8—34no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—32no—
317C2×C3×C4—31no—
318C2×C3×C4—31no—
319C2×C3×C4—91no—
320C2×C3×C4—94no—
321C2×C3×C4—92no—
322C2×C3×C4C22×D912yes—
323C2×C3×C4C22×D911yescharacteristic
324C2×C3×C4C2×D3212yes—
325C2×C3×C4C2×D3211yescharacteristic
326C2×Dic3—31no—
327C2×Dic3—32no—
328C2×Dic3—92no—
329C2×Dic3—92no—
330C2×Dic3—98no—
331C2×Dic3—91no—
332C2×Dic3—91no—
333C2×Dic3—91no—
334C2×Dic3—91no—
335C2×Dic3—181no—
336C2×Dic3—182no—
337C2×Dic3—181no—
338C2×Dic3C22×D911yescharacteristic
339C2×Dic3C22×D911yescharacteristic
340C2×Dic3C22×D912yes—
341C4×D3—34no—
342C4×D3—98no—
343Dic6—98no—
344Dic6—94no—
345Dic6—94no—
346Dic6—98no—
347Dic6—184no—
348Dic6C4×D914yes—
349C2×C32C22×C4×D311yescharacteristic
350C2×C32#100711yescharacteristic
351C2×C32#101011yescharacteristic
352C3×D3—62no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9—21no—
356C2×C9C22×C4×D311yescharacteristic
357C2×C9#101011yescharacteristic
358C2×C9#101011yescharacteristic
359D9—22no—
360C2×Q8—31no—
361C2×Q8—272no—
362C2×Q8—278no—
363C2×Q8—271no—
364C22×C4—91no—
365C22×C4—272no—
366C4⋊C4—32no—
367C4⋊C4—91no—
368C4⋊C4—272no—
369C4⋊C4—272no—
370C4⋊C4—271no—
371C4⋊C4—271no—
372C4⋊C4—272no—
373C4⋊C4D3×D911yescharacteristic
374C22⋊C4—92no—
375C22⋊C4—271no—
376C22⋊C4—271no—
377C42—32no—
378C42—31no—
379C42—92no—
380C42—272no—
381C42—272no—
382C42—272no—
383C42—271no—
384C22×C3—21no—
385C22×C3—92no—
386C22×C3—181no—
387C22×C3—181no—
388C22×C3C23×D911yescharacteristic
389C22×C3C22×D3211yescharacteristic
390C2×D3—32no—
391C2×D3—61no—
392C2×D3—61no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—34no—
397C3×C4—42no—
398C3×C4—61no—
399C3×C4—61no—
400C3×C4—92no—
401C3×C4—182no—
402C3×C4C2×C4×D912yes—
403C3×C4C4×D3212yes—
404Dic3—21no—
405Dic3—21no—
406Dic3—32no—
407Dic3—62no—
408Dic3—94no—
409Dic3—94no—
410Dic3—184no—
411Dic3—182no—
412Dic3—181no—
413Dic3—181no—
414Dic3—181no—
415Dic3—181no—
416Dic3—361no—
417Dic3C2×C4×D914yes—
418C32#557211yescharacteristic
419C9—21no—
420C9#551611yescharacteristic, complemented
421C23—91no—
422Q8—34no—
423Q8—278no—
424Q8—274no—
425C2×C4—32no—
426C2×C4—31no—
427C2×C4—31no—
428C2×C4—91no—
429C2×C4—94no—
430C2×C4—92no—
431C2×C4—278no—
432C2×C4—272no—
433C2×C4—271no—
434C2×C4—271no—
435C2×C4C2×D3×D912yes—
436C2×C4C2×D3×D911yescharacteristic
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—182no—
441C2×C3C22×C4×D911yescharacteristic
442C2×C3#6371311yescharacteristic
443C2×C3#6371611yescharacteristic
444C2×C3C2×C4×D3211yescharacteristic
445C2×C3#6430911yescharacteristic
446C2×C3#6431511yescharacteristic
447D3—62no—
448C22—92no—
449C22—181no—
450C22—181no—
451C22C22×D3×D911yescharacteristic
452C4—22no—
453C4—34no—
454C4—61no—
455C4—61no—
456C4—92no—
457C4—182no—
458C4—274no—
459C4—541no—
460C4—541no—
461C4C4×D3×D912yes—
462C3—21no—
463C3#1058967411yescharacteristic, complemented
464C3#1059435711yescharacteristic
465C2—182no—
466C2C2×C4×D3×D911yescharacteristic
467C2#1172249411yescharacteristic
468C2#1172251011yescharacteristic
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×C9#519
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
7. C4×#6189C2semidirect
9. #11720912C2semidirect
10. #11720911C2semidirect
11. #11720901C2semidirect
19. D9×Dic6C4semidirect
160. #887D9semidirect
310. C3×C9#519semidirect
420. C9#5516semidirect
463. C3#10589674semidirect

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×#634542
C9×#8872
#117209752
#117209742
C4×#61892
#117209432
#117209122
#117209112
#117209012
#105943573
#105896743