Finite Group Databasebeta

← #13140397

Subgroups — #13140397

Order 1728.

524 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#237C211yesmaximal, characteristic
2#11720906C211yesmaximal, characteristic, complemented
3C2×D3×C9⋊C8C211yesmaximal, characteristic, complemented
4#11720576C211yesmaximal, characteristic, complemented
5C3×#63392C211yesmaximal, characteristic, complemented
6#11720298C211yesmaximal, characteristic, complemented
7#11720297C211yesmaximal, characteristic, complemented
8#10591420—31nomaximal
9#10588917—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×C9×#67—21no—
12C22×C9×Dic3—21no—
13C9×#237—22no—
14C9×#237C414yescomplemented
15C2×C4×D3×C9C2211yescharacteristic
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—
23D3×C9⋊C8—22no—
24C2×C3×#237D311yescharacteristic
25#63841—31no—
26C2×D3×C3⋊C8—31no—
27C2×C9×#41—31no—
28#63659—31no—
29C3×#832—31no—
30#63566—31no—
31#63565—31no—
32#63488—31no—
33C22×C9⋊C8—31no—
34#63395—31no—
35#63395—31no—
36#63392—31no—
37#63392—61no—
38#63392D311yescharacteristic, complemented
39C23×C3×C9—21no—
40C22×D3×C9D411yescharacteristic
41C3×D4×C9—21no—
42C3×D4×C9C2×C412yes—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C9×#67—24no—
46C2×C9×Dic3—21no—
47C2×C9×Dic3—24no—
48C2×C9×Dic3D411yescharacteristic
49C2×C3×Dic9—21no—
50C4×D3×C9C2×C412yes—
51C4×D3×C9D412yes—
52C9×Dic6—21no—
53C9×Dic6C2×C412yes—
54C3⋊(C9⋊C8)—21no—
55C3×C9⋊C8—21no—
56#5233—91no—
57#4772—91no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C3×#67—21no—
60C22×C3×Dic3—21no—
61C3×#237—22no—
62C3×#237Dic314yes—
63C2×C3×C4×D3C2×D311yescharacteristic
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—
71D3×C3⋊C8—62no—
72C9×#41—34no—
73C9×#41—62no—
74C2×Q8×C9—31no—
75C2×D4×C9—21no—
76C2×D4×C9—61no—
77C2×D4×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—
85C2×C9⋊C8—31no—
86C2×C9⋊C8—62no—
87C2×C9⋊C8—61no—
88C2×C9⋊C8C2×D311yescharacteristic
89C22×C3×C9—22no—
90C22×C3×C9—41no—
91C22×C3×C9—41no—
92C22×C3×C9C2×D411yescharacteristic
93C2×D3×C9—21no—
94C2×D3×C9—21no—
95C2×D3×C9C22⋊C412yes—
96C3×C4×C9C22×C411yescharacteristic
97C3×C4×C9C2×D411yescharacteristic
98C3⋊Dic9—41no—
99C9×Dic3—22no—
100C9×Dic3C22⋊C412yes—
101C3×Dic9—41no—
102C2×C3×#41—31no—
103C2×#237D911yescharacteristic
104#926—91no—
105C22×C3⋊C8—91no—
106C2×D3×C8—91no—
107#892—91no—
108C3×#102—91no—
109#835—91no—
110#835—91no—
111#832—31no—
112#832—91no—
113#832—91no—
114#832—181no—
115#816—91no—
116#810—91no—
117C23×C32—21no—
118C22×C3×D3#6711yescharacteristic
119C32×D4—21no—
120C32×D4C2×Dic312yes—
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—61no—
123C3×#67—24no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3#6711yescharacteristic
128C3×C4×D3C2×Dic312yes—
129C3×C4×D3#6712yes—
130C3×Dic6—21no—
131C3×Dic6C2×Dic312yes—
132C23×C9—21no—
133C23×C9—31no—
134C23×C9—41no—
135C3⋊(C3⋊C8)—61no—
136C3×C3⋊C8—61no—
137Q8×C9—32no—
138Q8×C9—61no—
139D4×C9—21no—
140D4×C9—22no—
141D4×C9—41no—
142D4×C9—64no—
143D4×C9C4×D312yes—
144C2×C4×C9—21no—
145C2×C4×C9—31no—
146C2×C4×C9—32no—
147C2×C4×C9—32no—
148C2×C4×C9—64no—
149C2×C4×C9—61no—
150C2×C4×C9C22×D311yescharacteristic
151C2×Dic9—21no—
152C2×Dic9—61no—
153C2×Dic9—121no—
154C9⋊C8—21no—
155C9⋊C8—61no—
156C9⋊C8—121no—
157#416—271noSylow
158C2×C3×C9—42no—
159C2×C3×C9C2×C22⋊C411yescharacteristic
160C2×C3×C9#13511yescharacteristic
161C2×C3×C9#13511yescharacteristic
162D3×C9—22no—
163C3×#41—34no—
164C3×#41—62no—
165C2×C3×Q8—31no—
166C2×C3×D4—21no—
167C2×C3×D4—61no—
168C2×C3×D4C2×D911yescharacteristic
169C2×C3×D4D3211yescharacteristic
170C22×C3×C4—31no—
171C22×C3×C4—61no—
172C2×#67—21no—
173C22×Dic3—21no—
174#237—22no—
175#237Dic914yescomplemented
176C2×C4×D3C2×D911yescharacteristic
177C2×Dic6C2×D911yescharacteristic
178C2×C3×C8—91no—
179C3×C4⋊C4—91no—
180#217—181no—
181(C2×D3)⋊C4—181no—
182C4⋊Dic3—31no—
183C4⋊Dic3—91no—
184C4⋊Dic3—91no—
185C4⋊Dic3—181no—
186C4×Dic3—181no—
187C4×Dic3—181no—
188C2×C3⋊C8—31no—
189C2×C3⋊C8—91no—
190C2×C3⋊C8—91no—
191C2×C3⋊C8—181no—
192C2×C3⋊C8—182no—
193D3×C8—182no—
194C22×C32—22no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32C2×#6711yescharacteristic
198C2×C3×D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3#21712yes—
201C32×C4C22×Dic311yescharacteristic
202C32×C4C2×#6711yescharacteristic
203C3⋊Dic3—121no—
204C3×Dic3—22no—
205C3×Dic3—121no—
206C3×Dic3#21712yes—
207C22×C9—22no—
208C22×C9—21no—
209C22×C9—32no—
210C22×C9—42no—
211C22×C9—41no—
212C22×C9—41no—
213C22×C9—61no—
214C22×C9—61no—
215C22×C9—81no—
216C22×C9—81no—
217C22×C9D3×D411yescharacteristic
218C4×C9—21no—
219C4×C9—21no—
220C4×C9—32no—
221C4×C9—62no—
222C4×C9C2×C4×D311yescharacteristic
223C4×C9D3×D411yescharacteristic
224Dic9—41no—
225Dic9—121no—
226Dic9—241no—
227C2×#41—31no—
228C22×C8—271no—
229#117—271no—
230#103—271no—
231#103—271no—
232#102—91no—
233#102—271no—
234C3×C9#41611yescharacteristic, complemented, Sylow
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—31no—
238C23×C3—41no—
239C22×D3#61111yescharacteristic
240C3×Q8—32no—
241C3×Q8—61no—
242C3×D4—21no—
243C3×D4—21no—
244C3×D4—22no—
245C3×D4—41no—
246C3×D4—64no—
247C3×D4C2×Dic912yes—
248C3×D4D3×Dic312yes—
249C2×C3×C4—21no—
250C2×C3×C4—31no—
251C2×C3×C4—32no—
252C2×C3×C4—32no—
253C2×C3×C4—64no—
254C2×C3×C4—61no—
255C2×C3×C4—181no—
256C2×C3×C4C22×D911yescharacteristic
257C2×C3×C4C2×D3211yescharacteristic
258#67—24no—
259C2×Dic3—21no—
260C2×Dic3—24no—
261C2×Dic3—61no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264C2×Dic3—361no—
265C2×Dic3#61111yescharacteristic
266C4×D3C2×Dic912yes—
267C4×D3#61112yes—
268Dic6—21no—
269Dic6C2×Dic912yes—
270C3×C8—181no—
271C3⋊C8—61no—
272C3⋊C8—181no—
273C3⋊C8—181no—
274C3⋊C8—361no—
275C2×C32—42no—
276C2×C32#95011yescharacteristic
277C2×C32#95011yescharacteristic
278C2×C32C2×#21711yescharacteristic
279C3×D3—22no—
280C2×C9—21no—
281C2×C9—21no—
282C2×C9—21no—
283C2×C9—42no—
284C2×C9—62no—
285C2×C9—82no—
286C2×C9D3×C22⋊C411yescharacteristic
287C2×C9#91211yescharacteristic
288C2×C9#91611yescharacteristic
289#41—34no—
290#41—62no—
291C2×Q8—31no—
292C2×D4—61no—
293C2×D4D3×D911yescharacteristic
294C22×C4—31no—
295C22×C4—61no—
296C2×C8—91no—
297C2×C8—271no—
298C2×C8—542no—
299C4⋊C4—91no—
300C4⋊C4—271no—
301C22⋊C4—541no—
302C42—541no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—21no—
306C22×C3—32no—
307C22×C3—42no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—61no—
313C22×C3—61no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3C2×#61111yescharacteristic
317C22×C3D3×#6711yescharacteristic
318C2×D3—21no—
319C2×D3—21no—
320C2×D3#367612yes—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—32no—
324C3×C4—62no—
325C3×C4—361no—
326C3×C4C22×Dic911yescharacteristic
327C3×C4C2×#61111yescharacteristic
328C3×C4C2×D3×Dic311yescharacteristic
329C3×C4D3×#6711yescharacteristic
330Dic3—22no—
331Dic3—121no—
332Dic3—361no—
333Dic3—361no—
334Dic3—721no—
335Dic3#367612yes—
336C32#523311yescharacteristic
337C9—21no—
338C9#477211yescharacteristic, complemented
339C23—21no—
340C23—31no—
341Q8—32no—
342Q8—61no—
343D4—21no—
344D4—64no—
345D4D3×Dic912yes—
346C2×C4—31no—
347C2×C4—32no—
348C2×C4—32no—
349C2×C4—64no—
350C2×C4—61no—
351C2×C4—181no—
352C2×C4—541no—
353C2×C4C2×D3×D911yescharacteristic
354C8—181no—
355C8—541no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—42no—
360C2×C3—42no—
361C2×C3—62no—
362C2×C3—82no—
363C2×C3#6351311yescharacteristic
364C2×C3#6351311yescharacteristic
365C2×C3C2×#367611yescharacteristic
366C2×C3#6393311yescharacteristic
367C2×C3#6393511yescharacteristic
368C2×C3D3×#21711yescharacteristic
369D3—22no—
370C22—22no—
371C22—32no—
372C22—41no—
373C22—41no—
374C22—61no—
375C22—61no—
376C22D3×#61111yescharacteristic
377C4—32no—
378C4—62no—
379C4—361no—
380C4—1081no—
381C4C2×D3×Dic911yescharacteristic
382C4D3×#61111yescharacteristic
383C3—21no—
384C3#1058891711yescharacteristic, complemented
385C3#1059142011yescharacteristic
386C2—42no—
387C2—62no—
388C2#1172116911yescharacteristic
389C2#1172117111yescharacteristic
390C2D3×#367611yescharacteristic
subgroupG/H
centerC22D3×#611
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32C2×#67
prime radicalC2×C3×#237D3
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9#416
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×D3×C9⋊C8C2semidirect
4. #11720576C2semidirect
5. C3×#63392C2semidirect
6. #11720298C2semidirect
7. #11720297C2semidirect
14. C9×#237C4semidirect
38. #63392D3semidirect
175. #237Dic9semidirect
234. C3×C9#416semidirect
338. C9#4772semidirect
384. C3#10588917semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2372
#117209062
C2×D3×C9⋊C82
#117205762
C3×#633922
#117202982
#117202972
#105914203
#105889173