Finite Group Databasebeta

← #13140774

Subgroups — #13140774

Order 1728.

492 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×Q8×C9C211yesmaximal, characteristic
2#11720906C211yesmaximal, characteristic
3C2×#88420C211yesmaximal, characteristic
4#11720582C211yesmaximal, characteristic, complemented
5C3×#63395C211yesmaximal, characteristic, complemented
6#11720298C211yesmaximal, characteristic, complemented
7#11720297C211yesmaximal, characteristic, complemented
8#10591557—31nomaximal
9#10588906—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11D3×Q8×C9—22no—
12D3×Q8×C9C414yescomplemented
13C2×C4×D3×C9—21no—
14C2×C4×D3×C9C2211yescharacteristic
15C2×C9×Dic6—21no—
16C2×C9×Dic6C2211yescharacteristic
17C4⋊(C3⋊Dic9)C2211yescharacteristic
18C2×C3⋊(C9⋊C8)C2211yescharacteristic
19C3×C4⋊Dic9C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21(C2×D3)⋊Dic9—21no—
22Dic3×Dic9—21no—
23#88420—22no—
24C2×C3×D3×Q8D311yescharacteristic
25#63841—31no—
26C2×#3711—31no—
27C22×Q8×C9—31no—
28#63662—31no—
29C3×#835—31no—
30#63566—31no—
31#63565—31no—
32#63488—31no—
33C2×#3667—31no—
34#63395—31no—
35#63395—31no—
36#63395—31no—
37#63395—61no—
38#63395D311yescharacteristic, complemented
39C22×D3×C9D411yescharacteristic
40C3×Q8×C9—21no—
41C3×Q8×C9C2×C412yes—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C2×C9×Dic3—21no—
46C2×C9×Dic3D411yescharacteristic
47C2×C3×Dic9—21no—
48C4×D3×C9—24no—
49C4×D3×C9C2×C412yes—
50C4×D3×C9D412yes—
51C9×Dic6—21no—
52C9×Dic6—24no—
53C9×Dic6C2×C412yes—
54C3⋊(C9⋊C8)—21no—
55C3×C9⋊C8—21no—
56#5225—91no—
57#4802—91no—
58C2×C32×Q8C2×D311yescharacteristic
59C3×D3×Q8—22no—
60C3×D3×Q8Dic314yes—
61C2×C3×C4×D3—21no—
62C2×C3×C4×D3C2×D311yescharacteristic
63C2×C3×Dic6—21no—
64C2×C3×Dic6C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)—31no—
66C2×C3⋊(C3⋊C8)—31no—
67C3×C4⋊Dic3—31no—
68C2×C3×C3⋊C8—31no—
69(C2×D3)⋊Dic3—61no—
70Dic32—61no—
71#3711—62no—
72C2×Q8×C9—21no—
73C2×Q8×C9—34no—
74C2×Q8×C9—31no—
75C2×Q8×C9—61no—
76C2×Q8×C9—62no—
77C2×Q8×C9C2×D311yescharacteristic
78C22×C4×C9—31no—
79C22×C4×C9—61no—
80#3676—61no—
81C4⋊Dic9—31no—
82C4⋊Dic9—61no—
83C4⋊Dic9C2×D311yescharacteristic
84C4×Dic9—61no—
85#3667—62no—
86C2×C9⋊C8—31no—
87C2×C9⋊C8—61no—
88C2×C9⋊C8C2×D311yescharacteristic
89C22×C3×C9C2×D411yescharacteristic
90C2×D3×C9—21no—
91C2×D3×C9—21no—
92C2×D3×C9C22⋊C412yes—
93C3×C4×C9—22no—
94C3×C4×C9C22×C411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C3⋊Dic9—41no—
97C9×Dic3—22no—
98C9×Dic3C22⋊C412yes—
99C3×Dic9—41no—
100C22×C3×Q8—31no—
101C2×D3×Q8D911yescharacteristic
102#926—91no—
103C2×#208—91no—
104C2×ΓL(1,𝔽25)—91no—
105#892—91no—
106C3×#103—91no—
107#835—31no—
108#835—91no—
109#835—91no—
110#835—91no—
111#835—91no—
112#835—181no—
113#816—91no—
114#810—91no—
115C22×C3×D3#6711yescharacteristic
116C32×Q8—21no—
117C32×Q8C2×Dic312yes—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—61no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3#6711yescharacteristic
124C3×C4×D3—24no—
125C3×C4×D3C2×Dic312yes—
126C3×C4×D3#6712yes—
127C3×Dic6—21no—
128C3×Dic6—24no—
129C3×Dic6C2×Dic312yes—
130C23×C9—31no—
131C3⋊(C3⋊C8)—61no—
132C3×C3⋊C8—61no—
133Q8×C9—21no—
134Q8×C9—22no—
135Q8×C9—32no—
136Q8×C9—41no—
137Q8×C9—64no—
138Q8×C9—61no—
139Q8×C9C4×D312yes—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—31no—
143C2×C4×C9—32no—
144C2×C4×C9—32no—
145C2×C4×C9—41no—
146C2×C4×C9—64no—
147C2×C4×C9—61no—
148C2×C4×C9C22×D311yescharacteristic
149C2×Dic9—21no—
150C2×Dic9—61no—
151C2×Dic9—121no—
152C9⋊C8—21no—
153C9⋊C8—61no—
154C9⋊C8—121no—
155#419—271noSylow
156C2×C3×C9C2×C22⋊C411yescharacteristic
157C2×C3×C9#13711yescharacteristic
158C2×C3×C9#13711yescharacteristic
159D3×C9—22no—
160C2×C3×Q8—21no—
161C2×C3×Q8—34no—
162C2×C3×Q8—31no—
163C2×C3×Q8—61no—
164C2×C3×Q8—62no—
165C2×C3×Q8C2×D911yescharacteristic
166C2×C3×Q8D3211yescharacteristic
167C22×C3×C4—31no—
168C22×C3×C4—61no—
169D3×Q8—22no—
170D3×Q8Dic914yescomplemented
171C2×C4×D3—21no—
172C2×C4×D3C2×D911yescharacteristic
173C2×Dic6—21no—
174C2×Dic6C2×D911yescharacteristic
175C2×C3×C8—91no—
176C3×C4⋊C4—91no—
177#217—181no—
178(C2×D3)⋊C4—181no—
179C4⋊Dic3—31no—
180C4⋊Dic3—91no—
181C4⋊Dic3—91no—
182C4⋊Dic3—181no—
183C4×Dic3—181no—
184C4×Dic3—181no—
185#208—182no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—181no—
190ΓL(1,𝔽25)—182no—
191C22×C32C2×#6711yescharacteristic
192C2×C3×D3—21no—
193C2×C3×D3—21no—
194C2×C3×D3#21712yes—
195C32×C4—22no—
196C32×C4C22×Dic311yescharacteristic
197C32×C4C2×#6711yescharacteristic
198C3⋊Dic3—121no—
199C3×Dic3—22no—
200C3×Dic3—121no—
201C3×Dic3#21712yes—
202C22×C9—21no—
203C22×C9—32no—
204C22×C9—61no—
205C22×C9—61no—
206C22×C9D3×D411yescharacteristic
207C4×C9—22no—
208C4×C9—21no—
209C4×C9—21no—
210C4×C9—32no—
211C4×C9—42no—
212C4×C9—62no—
213C4×C9C2×C4×D311yescharacteristic
214C4×C9D3×D411yescharacteristic
215Dic9—41no—
216Dic9—121no—
217Dic9—241no—
218C22×Q8—31no—
219C2×M16—271no—
220#117—271no—
221#103—91no—
222#103—271no—
223#103—271no—
224#103—271no—
225C3×C9#41911yescharacteristic, complemented, Sylow
226C23×C3—31no—
227C22×D3#61111yescharacteristic
228C3×Q8—21no—
229C3×Q8—21no—
230C3×Q8—22no—
231C3×Q8—32no—
232C3×Q8—41no—
233C3×Q8—64no—
234C3×Q8—61no—
235C3×Q8C2×Dic912yes—
236C3×Q8D3×Dic312yes—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—31no—
241C2×C3×C4—32no—
242C2×C3×C4—32no—
243C2×C3×C4—41no—
244C2×C3×C4—64no—
245C2×C3×C4—61no—
246C2×C3×C4—181no—
247C2×C3×C4C22×D911yescharacteristic
248C2×C3×C4C2×D3211yescharacteristic
249C2×Dic3—21no—
250C2×Dic3—61no—
251C2×Dic3—181no—
252C2×Dic3—181no—
253C2×Dic3—361no—
254C2×Dic3#61111yescharacteristic
255C4×D3—24no—
256C4×D3C2×Dic912yes—
257C4×D3#61112yes—
258Dic6—21no—
259Dic6—24no—
260Dic6C2×Dic912yes—
261C3×C8—181no—
262C3⋊C8—61no—
263C3⋊C8—181no—
264C3⋊C8—181no—
265C3⋊C8—361no—
266C2×C32#94211yescharacteristic
267C2×C32#94211yescharacteristic
268C2×C32C2×#21711yescharacteristic
269C3×D3—22no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9—62no—
274C2×C9D3×C22⋊C411yescharacteristic
275C2×C9#91511yescharacteristic
276C2×C9#91811yescharacteristic
277C2×Q8—34no—
278C2×Q8—31no—
279C2×Q8—61no—
280C2×Q8—62no—
281C2×Q8D3×D911yescharacteristic
282C22×C4—31no—
283C22×C4—61no—
284M16—542no—
285C2×C8—91no—
286C2×C8—271no—
287C4⋊C4—91no—
288C4⋊C4—271no—
289C22⋊C4—541no—
290C42—541no—
291C22×C3—21no—
292C22×C3—32no—
293C22×C3—61no—
294C22×C3—61no—
295C22×C3C2×#61111yescharacteristic
296C22×C3D3×#6711yescharacteristic
297C2×D3—21no—
298C2×D3—21no—
299C2×D3#367612yes—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—32no—
305C3×C4—42no—
306C3×C4—62no—
307C3×C4—361no—
308C3×C4C22×Dic911yescharacteristic
309C3×C4C2×#61111yescharacteristic
310C3×C4C2×D3×Dic311yescharacteristic
311C3×C4D3×#6711yescharacteristic
312Dic3—22no—
313Dic3—121no—
314Dic3—361no—
315Dic3—361no—
316Dic3—721no—
317Dic3#367612yes—
318C32#522511yescharacteristic
319C9—21no—
320C9#480211yescharacteristic, complemented
321C23—31no—
322Q8—21no—
323Q8—32no—
324Q8—64no—
325Q8—61no—
326Q8D3×Dic912yes—
327C2×C4—21no—
328C2×C4—31no—
329C2×C4—32no—
330C2×C4—32no—
331C2×C4—64no—
332C2×C4—61no—
333C2×C4—181no—
334C2×C4—541no—
335C2×C4C2×D3×D911yescharacteristic
336C8—181no—
337C8—541no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—62no—
342C2×C3#6350511yescharacteristic
343C2×C3#6350511yescharacteristic
344C2×C3C2×#367611yescharacteristic
345C2×C3#6394611yescharacteristic
346C2×C3#6394811yescharacteristic
347C2×C3D3×#21711yescharacteristic
348D3—22no—
349C22—32no—
350C22—61no—
351C22—61no—
352C22D3×#61111yescharacteristic
353C4—22no—
354C4—32no—
355C4—62no—
356C4—361no—
357C4—1081no—
358C4C2×D3×Dic911yescharacteristic
359C4D3×#61111yescharacteristic
360C3—21no—
361C3#1058890611yescharacteristic, complemented
362C3#1059155711yescharacteristic
363C2—62no—
364C2#1172120611yescharacteristic
365C2#1172120811yescharacteristic
366C2D3×#367611yescharacteristic
subgroupG/H
centerC22D3×#611
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32C2×#67
prime radicalC22×C3×D3#67
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9#419
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. #11720582C2semidirect
5. C3×#63395C2semidirect
6. #11720298C2semidirect
7. #11720297C2semidirect
12. D3×Q8×C9C4semidirect
38. #63395D3semidirect
170. D3×Q8Dic9semidirect
225. C3×C9#419semidirect
320. C9#4802semidirect
361. C3#10588906semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Q8×C92
#117209062
C2×#884202
#117205822
C3×#633952
#117202982
#117202972
#105915573
#105889063