Finite Group Databasebeta

← #13140400

Subgroups — #13140400

Order 1728.

620 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×D4×C9C211yesmaximal, characteristic
2#11720906C211yesmaximal, characteristic, complemented
3C2×#88420C211yesmaximal, characteristic, complemented
4#11720576C211yesmaximal, characteristic, complemented
5C3×#63392C211yesmaximal, characteristic, complemented
6#11720295C211yesmaximal, characteristic, complemented
7#11720294C211yesmaximal, characteristic, complemented
8#10591421—31nomaximal
9#10588896—31nomaximal
10C23×D3×C9—21no—
11C2×C3×D4×C9C2211yescharacteristic
12C2×C9×#67—21no—
13D3×D4×C9—22no—
14D3×D4×C9C414yescomplemented
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9C2211yescharacteristic
17C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
18C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
19C3×C4⋊Dic9C2211yescharacteristic, complemented
20C2×C3×C9⋊C8C2211yescharacteristic, complemented
21(C2×D3)⋊Dic9—21no—
22Dic3×Dic9—21no—
23#88420—22no—
24C2×C3×D3×D4D311yescharacteristic
25#63841—31no—
26C2×#3711—31no—
27C22×D4×C9—31no—
28#63659—31no—
29C3×#832—31no—
30#63563—31no—
31#63562—31no—
32#63488—31no—
33C2×#3667—31no—
34#63392—31no—
35#63392—31no—
36#63392—31no—
37#63392—61no—
38#63392D311yescharacteristic, complemented
39C23×C3×C9—21no—
40C22×D3×C9—21no—
41C22×D3×C9—24no—
42C22×D3×C9—42no—
43C22×D3×C9—42no—
44C22×D3×C9D411yescharacteristic
45C3×D4×C9—21no—
46C3×D4×C9C2×C412yes—
47C2×C3×C4×C9C2311yescharacteristic
48C2×C3⋊Dic9—21no—
49C9×#67—24no—
50C2×C9×Dic3D411yescharacteristic
51C2×C3×Dic9—21no—
52C9×D12—21no—
53C9×D12C2×C412yes—
54C4×D3×C9C2×C412yes—
55C4×D3×C9D412yes—
56C3⋊(C9⋊C8)—21no—
57C3×C9⋊C8—21no—
58#5215—91no—
59#4770—91no—
60C23×C3×D3—21no—
61C2×C32×D4C2×D311yescharacteristic
62C2×C3×#67—21no—
63C3×D3×D4—22no—
64C3×D3×D4Dic314yes—
65C2×C3×D12C2×D311yescharacteristic
66C2×C3×C4×D3C2×D311yescharacteristic
67C24×C9—61no—
68C4⋊(C3⋊Dic3)—31no—
69C2×C3⋊(C3⋊C8)—31no—
70C3×C4⋊Dic3—31no—
71C2×C3×C3⋊C8—31no—
72(C2×D3)⋊Dic3—61no—
73Dic32—61no—
74#3711—62no—
75C2×D4×C9—21no—
76C2×D4×C9—34no—
77C2×D4×C9—31no—
78C2×D4×C9—62no—
79C2×D4×C9—61no—
80C2×D4×C9C2×D311yescharacteristic
81C22×C4×C9—31no—
82#3676—61no—
83C4⋊Dic9—31no—
84C4⋊Dic9—61no—
85C4⋊Dic9C2×D311yescharacteristic, complemented
86C4×Dic9—61no—
87#3667—62no—
88C2×C9⋊C8—31no—
89C2×C9⋊C8—61no—
90C2×C9⋊C8C2×D311yescharacteristic, complemented
91C22×C3×C9—22no—
92C22×C3×C9—41no—
93C22×C3×C9—41no—
94C22×C3×C9C2×D411yescharacteristic
95C2×D3×C9—21no—
96C2×D3×C9—21no—
97C2×D3×C9—22no—
98C2×D3×C9—48no—
99C2×D3×C9—41no—
100C2×D3×C9—41no—
101C2×D3×C9C22⋊C412yes—
102C3×C4×C9C22×C411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C3⋊Dic9—41no—
105C9×Dic3C22⋊C412yes—
106C3×Dic9—41no—
107C22×C3×D4—31no—
108C2×D3×D4D911yescharacteristic
109#926—91no—
110C2×#208—91no—
111C2×ΓL(1,𝔽25)—91no—
112#892—91no—
113C3×#102—91no—
114#832—31no—
115#832—91no—
116#832—91no—
117#832—91no—
118#832—91no—
119#832—181no—
120#821—91no—
121#809—91no—
122C23×C32—21no—
123C22×C3×D3—21no—
124C22×C3×D3—24no—
125C22×C3×D3—42no—
126C22×C3×D3—42no—
127C22×C3×D3#6711yescharacteristic
128C32×D4—21no—
129C32×D4C2×Dic312yes—
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—61no—
132C3×#67—24no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3#6711yescharacteristic
135C3×D12—21no—
136C3×D12C2×Dic312yes—
137C3×C4×D3C2×Dic312yes—
138C3×C4×D3#6712yes—
139C23×C9—21no—
140C23×C9—31no—
141C23×C9—41no—
142C23×C9—64no—
143C23×C9—61no—
144C23×C9—122no—
145C23×C9—122no—
146C3⋊(C3⋊C8)—61no—
147C3×C3⋊C8—61no—
148D4×C9—21no—
149D4×C9—22no—
150D4×C9—32no—
151D4×C9—41no—
152D4×C9—64no—
153D4×C9—61no—
154D4×C9C4×D312yes—
155C2×C4×C9—21no—
156C2×C4×C9—31no—
157C2×C4×C9—32no—
158C2×C4×C9—32no—
159C2×C4×C9C22×D311yescharacteristic
160C2×Dic9—21no—
161C2×Dic9—61no—
162C2×Dic9—121no—
163C9⋊C8—21no—
164C9⋊C8—61no—
165C9⋊C8—121no—
166#418—271noSylow
167C2×C3×C9—42no—
168C2×C3×C9C2×C22⋊C411yescharacteristic
169C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
170C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
171D3×C9—22no—
172D3×C9—42no—
173C24×C3—61no—
174C23×D3—21no—
175C2×C3×D4—21no—
176C2×C3×D4—34no—
177C2×C3×D4—31no—
178C2×C3×D4—62no—
179C2×C3×D4—61no—
180C2×C3×D4C2×D911yescharacteristic
181C2×C3×D4D3211yescharacteristic
182C22×C3×C4—31no—
183C2×#67—21no—
184D3×D4—22no—
185D3×D4Dic914yescomplemented
186C2×D12C2×D911yescharacteristic
187C2×C4×D3C2×D911yescharacteristic
188C2×C3×C8—91no—
189C3×C4⋊C4—91no—
190#217—181no—
191(C2×D3)⋊C4—181no—
192C4⋊Dic3—31no—
193C4⋊Dic3—91no—
194C4⋊Dic3—91no—
195C4⋊Dic3—181no—
196C4×Dic3—181no—
197C4×Dic3—181no—
198#208—182no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—91no—
202C2×C3⋊C8—181no—
203ΓL(1,𝔽25)—182no—
204C22×C32—22no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32C2×#6711yescharacteristic
208C2×C3×D3—21no—
209C2×C3×D3—21no—
210C2×C3×D3—22no—
211C2×C3×D3—48no—
212C2×C3×D3—41no—
213C2×C3×D3—41no—
214C2×C3×D3#21712yes—
215C32×C4C22×Dic311yescharacteristic
216C32×C4C2×#6711yescharacteristic
217C3⋊Dic3—121no—
218C3×Dic3—121no—
219C3×Dic3#21712yes—
220C22×C9—22no—
221C22×C9—21no—
222C22×C9—32no—
223C22×C9—42no—
224C22×C9—41no—
225C22×C9—41no—
226C22×C9—61no—
227C22×C9—61no—
228C22×C9—62no—
229C22×C9—81no—
230C22×C9—81no—
231C22×C9—128no—
232C22×C9—121no—
233C22×C9—121no—
234C22×C9D3×D411yescharacteristic
235C4×C9—21no—
236C4×C9—21no—
237C4×C9—32no—
238C4×C9C2×C4×D311yescharacteristic
239C4×C9D3×D411yescharacteristic
240Dic9—41no—
241Dic9—121no—
242Dic9—241no—
243C22×D4—31no—
244C2×M16—271no—
245#117—271no—
246#102—91no—
247#102—271no—
248#102—271no—
249#102—271no—
250C3×C9#41811yescharacteristic, complemented, Sylow
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—31no—
254C23×C3—41no—
255C23×C3—64no—
256C23×C3—61no—
257C23×C3—122no—
258C23×C3—122no—
259C22×D3—21no—
260C22×D3—24no—
261C22×D3—42no—
262C22×D3—42no—
263C22×D3#61111yescharacteristic
264C3×D4—21no—
265C3×D4—21no—
266C3×D4—22no—
267C3×D4—32no—
268C3×D4—41no—
269C3×D4—64no—
270C3×D4—61no—
271C3×D4C2×Dic912yes—
272C3×D4D3×Dic312yes—
273C2×C3×C4—21no—
274C2×C3×C4—31no—
275C2×C3×C4—32no—
276C2×C3×C4—32no—
277C2×C3×C4—181no—
278C2×C3×C4C22×D911yescharacteristic
279C2×C3×C4C2×D3211yescharacteristic
280#67—24no—
281C2×Dic3—61no—
282C2×Dic3—181no—
283C2×Dic3—181no—
284C2×Dic3—361no—
285C2×Dic3#61111yescharacteristic
286D12—21no—
287D12C2×Dic912yes—
288C4×D3C2×Dic912yes—
289C4×D3#61112yes—
290C3×C8—181no—
291C3⋊C8—61no—
292C3⋊C8—181no—
293C3⋊C8—181no—
294C3⋊C8—361no—
295C2×C32—42no—
296C2×C32#93211yescharacteristic
297C2×C32#93211yescharacteristic
298C2×C32C2×#21711yescharacteristic
299C3×D3—22no—
300C3×D3—42no—
301C2×C9—21no—
302C2×C9—21no—
303C2×C9—21no—
304C2×C9—42no—
305C2×C9—62no—
306C2×C9—82no—
307C2×C9—122no—
308C2×C9D3×C22⋊C411yescharacteristic
309C2×C9#91111yescharacteristic
310C2×C9#91411yescharacteristic
311C24—61no—
312C2×D4—34no—
313C2×D4—31no—
314C2×D4—62no—
315C2×D4—61no—
316C2×D4D3×D911yescharacteristic
317C22×C4—31no—
318M16—542no—
319C2×C8—91no—
320C2×C8—271no—
321C4⋊C4—91no—
322C4⋊C4—271no—
323C22⋊C4—541no—
324C42—541no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—21no—
328C22×C3—32no—
329C22×C3—42no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—61no—
335C22×C3—61no—
336C22×C3—62no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3—128no—
340C22×C3—121no—
341C22×C3—121no—
342C22×C3C2×#61111yescharacteristic
343C22×C3D3×#6711yescharacteristic
344C2×D3—21no—
345C2×D3—21no—
346C2×D3—22no—
347C2×D3—48no—
348C2×D3—41no—
349C2×D3—41no—
350C2×D3#367612yes—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—32no—
354C3×C4—361no—
355C3×C4C22×Dic911yescharacteristic
356C3×C4C2×#61111yescharacteristic
357C3×C4C2×D3×Dic311yescharacteristic
358C3×C4D3×#6711yescharacteristic
359Dic3—121no—
360Dic3—361no—
361Dic3—361no—
362Dic3—721no—
363Dic3#367612yes—
364C32#521511yescharacteristic
365C9—21no—
366C9#477011yescharacteristic, complemented
367C23—21no—
368C23—31no—
369C23—64no—
370C23—61no—
371C23—122no—
372C23—122no—
373D4—21no—
374D4—32no—
375D4—64no—
376D4—61no—
377D4D3×Dic912yes—
378C2×C4—31no—
379C2×C4—32no—
380C2×C4—32no—
381C2×C4—181no—
382C2×C4—541no—
383C2×C4C2×D3×D911yescharacteristic
384C8—181no—
385C8—541no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—42no—
390C2×C3—42no—
391C2×C3—62no—
392C2×C3—82no—
393C2×C3—122no—
394C2×C3#6349511yescharacteristic
395C2×C3#6349511yescharacteristic
396C2×C3C2×#367611yescharacteristic
397C2×C3#6393211yescharacteristic
398C2×C3#6393411yescharacteristic
399C2×C3D3×#21711yescharacteristic
400D3—22no—
401D3—42no—
402C22—22no—
403C22—32no—
404C22—41no—
405C22—41no—
406C22—61no—
407C22—61no—
408C22—62no—
409C22—128no—
410C22—121no—
411C22—121no—
412C22D3×#61111yescharacteristic
413C4—32no—
414C4—361no—
415C4—1081no—
416C4C2×D3×Dic911yescharacteristic
417C4D3×#61111yescharacteristic
418C3—21no—
419C3#1058889611yescharacteristic, complemented
420C3#1059142111yescharacteristic
421C2—42no—
422C2—62no—
423C2—122no—
424C2#1172116811yescharacteristic
425C2#1172117011yescharacteristic
426C2D3×#367611yescharacteristic
subgroupG/H
centerC22D3×#611
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32C2×#67
prime radicalC2×C3×D3×D4D3
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9#418
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11720906C2semidirect
3. C2×#88420C2semidirect
4. #11720576C2semidirect
5. C3×#63392C2semidirect
6. #11720295C2semidirect
7. #11720294C2semidirect
14. D3×D4×C9C4semidirect
17. C4⋊(C3⋊Dic9)C22semidirect
18. C2×C3⋊(C9⋊C8)C22semidirect
19. C3×C4⋊Dic9C22semidirect
20. C2×C3×C9⋊C8C22semidirect
38. #63392D3semidirect
85. C4⋊Dic9C2×D3semidirect
90. C2×C9⋊C8C2×D3semidirect
185. D3×D4Dic9semidirect
250. C3×C9#418semidirect
366. C9#4770semidirect
419. C3#10588896semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D4×C92
#117209062
C2×#884202
#117205762
C3×#633922
#117202952
#117202942
#105914213
#105888963