Finite Group Databasebeta

← #10588489⋊C3 #13141953

Subgroups — #10588489⋊C3 #13141953

Order 1728.

465 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×AGL(1,ℤ/9)C211yesmaximal, characteristic
2C2×#3674⋊C3C211yesmaximal, characteristic, complemented
3C2×Dic36⋊C3C211yesmaximal, characteristic, complemented
4#63459⋊C3C211yesmaximal, characteristic
53−1+2×#103C211yesmaximal, characteristic, complemented
6(C2×AGL(1,ℤ/9))⋊C8C211yesmaximal, characteristic
7#63368⋊C3C211yesmaximal, characteristic, complemented
8C3×#4809—31nomaximal
9#10588489C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11Q8×AGL(1,ℤ/9)—22no—
12Q8×AGL(1,ℤ/9)—22no—
13C2×C4×AGL(1,ℤ/9)—21no—
14C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
15C2×3−1+2⋊Q8—21no—
16C2×3−1+2⋊Q8C2211yescharacteristic
17C2×3−1+2⋊Q8C2211yescharacteristic
18C2×C8×3−1+2C2211yescharacteristic
19C4⋊C4×3−1+2C2211yescharacteristic
20#3674⋊C3—22no—
21(C2×AGL(1,ℤ/9))⋊C4—21no—
22C4⋊(3−1+2⋊C4)—21no—
23C2×3−1+2⋊C8C2211yescharacteristic
24Dic36⋊C3—22no—
25C2×C3×D3×Q8—31no—
26C2×C3×#216—31no—
27C2×C3×Dic12—31no—
28C3×#897—31no—
29C2×Q8×D9C2×C311yescharacteristic
30C32×#103D311yescharacteristic
31C3×(C2×D3)⋊C8—31no—
32C3×#810—31no—
33C2×#3674C2×C311yescharacteristic, complemented
34C2×Dic36C2×C311yescharacteristic, complemented
35#63459C2×C311yescharacteristic
36C9×#103—21no—
37C9×#103C2×C311yescharacteristic, complemented
38(C2×D9)⋊C8C2×C311yescharacteristic
39#63368C2×C311yescharacteristic, complemented
40C22×AGL(1,ℤ/9)D411yescharacteristic
41Q8×3−1+2—21no—
42Q8×3−1+2D412yes—
43C2×C4×3−1+2—21no—
44C2×C4×3−1+2—21no—
45C2×C4×3−1+2C2311yescharacteristic
46C2×3−1+2⋊C4—21no—
47C2×3−1+2⋊C4—21no—
48C2×3−1+2⋊C4D411yescharacteristic
49C4×AGL(1,ℤ/9)—24no—
50C4×AGL(1,ℤ/9)—21no—
51C4×AGL(1,ℤ/9)—21no—
523−1+2⋊Q8—24no—
533−1+2⋊Q8—21no—
543−1+2⋊Q8—21no—
553−1+2⋊Q8—21no—
563−1+2⋊Q8D412yes—
57C8×3−1+2—21no—
583−1+2⋊C8—21no—
59C3×#451—91no—
60#4809—31no—
61C2×C32×Q8C2×D311yescharacteristic
62C3×D3×Q8—62no—
63C3×D3×Q8—62no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—61no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—61no—
69C2×C32×C8C2×D311yescharacteristic
70C32×C4⋊C4C2×D311yescharacteristic
71C3×#216—62no—
72C3×(C2×D3)⋊C4—61no—
73C3×C4⋊Dic3—61no—
74C2×C3×C3⋊C8—31no—
75C3×Dic12—62no—
76C2×Q8×C9—21no—
77C2×Q8×C9C22×C311yescharacteristic
78Q8×D9—22no—
79Q8×D9—22no—
80C2×C4×D9—21no—
81C2×C4×D9C22×C311yescharacteristic
82C2×Dic18—21no—
83C2×Dic18C22×C311yescharacteristic
84C2×Dic18C22×C311yescharacteristic
85C2×C8×C9—21no—
86C2×C8×C9C22×C311yescharacteristic
87C9×C4⋊C4—21no—
88C9×C4⋊C4C22×C311yescharacteristic
89#3674—22no—
90(C2×D9)⋊C4—21no—
91C4⋊Dic9—21no—
92C2×C9⋊C8C22×C311yescharacteristic
93Dic36—22no—
94C22×3−1+2C2×D411yescharacteristic
95C2×AGL(1,ℤ/9)—21no—
96C2×AGL(1,ℤ/9)—21no—
97C2×AGL(1,ℤ/9)Q1612yescomplemented
98C4×3−1+2—22no—
99C4×3−1+2—41no—
100C4×3−1+2C2×D411yescharacteristic
101C4×3−1+2C2×D411yescharacteristic
1023−1+2⋊C4—21no—
1033−1+2⋊C4—22no—
1043−1+2⋊C4—41no—
105C22×C3×Q8—91no—
106C2×D3×Q8—31no—
107C2×C3×Q16—91no—
108C2×C3×Q16—91no—
109C3×C22⋊Q8—91no—
110C2×#216—31no—
111C2×Dic12—31no—
112#897—31no—
113C3×#103—31no—
114C3×#103—91no—
115C3×#103C3×D311yescharacteristic
116C3×C22⋊C8—91no—
117(C2×D3)⋊C8—31no—
118#810—31no—
119C22×C3×D3—31no—
120C32×Q8—21no—
121C32×Q8D1212yes—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3×Dic3—31no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C3×C4×D3—64no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C3×Dic6—32no—
132C3×Dic6—64no—
133C3×Dic6—61no—
134C3×Dic6—61no—
135C3×Dic6—61no—
136C22×D9C3×D411yescharacteristic
137C32×C8—21no—
138C3×C3⋊C8—61no—
139Q8×C9—21no—
140Q8×C9—22no—
141Q8×C9—41no—
142Q8×C9C3×D412yes—
143C2×C4×C9—21no—
144C2×C4×C9—21no—
145C2×C4×C9—21no—
146C2×C4×C9—41no—
147C2×C4×C9—41no—
148C2×C4×C9C23×C311yescharacteristic
149C2×Dic9—21no—
150C2×Dic9—21no—
151C2×Dic9C3×D411yescharacteristic
152C4×D9—24no—
153C4×D9—21no—
154C4×D9—21no—
155Dic18—24no—
156Dic18—21no—
157Dic18—21no—
158Dic18—21no—
159Dic18C3×D412yes—
160C8×C9—21no—
161C8×C9—41no—
162C9⋊C8—21no—
163#451—91noSylow
164C2×3−1+2#12011yescharacteristic
165C2×3−1+2C2×Q1611yescharacteristic
166C2×3−1+2#13711yescharacteristic
167AGL(1,ℤ/9)—22no—
168C2×C3×Q8—31no—
169C2×C3×Q8—91no—
170C2×C3×Q8—91no—
171C2×C3×Q8—181no—
172C2×C3×Q8—182no—
173C2×C3×Q8—182no—
174C2×C3×Q8C2×C3×D311yescharacteristic
175C22×C3×C4—91no—
176C22×C3×C4—181no—
177D3×Q8—62no—
178D3×Q8—62no—
179C2×C4×D3—31no—
180C2×C4×D3—61no—
181C2×Dic6—31no—
182C2×Dic6—31no—
183C2×Dic6—61no—
184C3×Q16—182no—
185C3×Q16—182no—
186C2×C3×C8—31no—
187C2×C3×C8—91no—
188C2×C3×C8C2×C3×D311yescharacteristic
189C3×C4⋊C4—31no—
190C3×C4⋊C4—181no—
191C3×C4⋊C4C2×C3×D311yescharacteristic
192C3×C22⋊C4—181no—
193#216—62no—
194(C2×D3)⋊C4—61no—
195C4⋊Dic3—61no—
196C2×C3⋊C8—31no—
197Dic12—62no—
198C22×C32D3×D411yescharacteristic
199C2×C3×D3—32no—
200C2×C3×D3—61no—
201C2×C3×D3—61no—
202C32×C4—22no—
203C32×C4—41no—
204C32×C4C2×D1211yescharacteristic
205C32×C4D3×D411yescharacteristic
206C3×Dic3—62no—
207C3×Dic3—61no—
208C3×Dic3—121no—
209C22×C9—21no—
210C22×C9C2×C3×D411yescharacteristic
211C2×D9—21no—
212C2×D9—21no—
213C2×D9C3×Q1612yescomplemented
214C4×C9—22no—
215C4×C9—21no—
216C4×C9—21no—
217C4×C9—42no—
218C4×C9—41no—
219C4×C9—81no—
220C4×C9C2×C3×D411yescharacteristic
221C4×C9C2×C3×D411yescharacteristic
222Dic9—21no—
223Dic9—22no—
224Dic9—41no—
225C22×Q8—91no—
226C2×Q16—91no—
227C2×Q16—91no—
228C22⋊Q8—91no—
229#103—91no—
230#103AGL(1,ℤ/9)11yescharacteristic, complemented
231C22⋊C8—91no—
2323−1+2#45111yescharacteristic, complemented, Sylow
233C23×C3—91no—
234C22×D3—31no—
235C3×Q8—21no—
236C3×Q8—32no—
237C3×Q8—61no—
238C3×Q8—92no—
239C3×Q8—181no—
240C3×Q8—181no—
241C3×Q8—181no—
242C3×Q8—184no—
243C3×Q8C3×D1212yes—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—31no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—91no—
250C2×C3×C4—181no—
251C2×C3×C4—181no—
252C2×C3×C4—184no—
253C2×C3×C4—181no—
254C2×C3×C4—181no—
255C2×C3×C4C22×C3×D311yescharacteristic
256C2×Dic3—31no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C4×D3—64no—
260C4×D3—61no—
261C4×D3—61no—
262Dic6—32no—
263Dic6—64no—
264Dic6—61no—
265Dic6—61no—
266Dic6—61no—
267C3×C8—21no—
268C3×C8—61no—
269C3×C8—181no—
270C3⋊C8—61no—
271C2×C32#88211yescharacteristic
272C2×C32#91511yescharacteristic
273C2×C32D3×Q1611yescharacteristic
274C3×D3—62no—
275C2×C9—21no—
276C2×C9—21no—
277C2×C9—21no—
278C2×C9C3×#12011yescharacteristic
279C2×C9C2×C3×Q1611yescharacteristic
280C2×C9C3×#13711yescharacteristic
281D9—22no—
282C2×Q8—91no—
283C2×Q8—91no—
284C2×Q8—181no—
285C2×Q8—182no—
286C2×Q8—182no—
287C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
288C22×C4—91no—
289C22×C4—181no—
290Q16—182no—
291Q16—182no—
292C2×C8—91no—
293C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
294C4⋊C4—181no—
295C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
296C22⋊C4—181no—
297C22×C3—31no—
298C22×C3—92no—
299C22×C3—181no—
300C22×C3—181no—
301C22×C3C3×D3×D411yescharacteristic
302C2×D3—32no—
303C2×D3—61no—
304C2×D3—61no—
305C3×C4—22no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—41no—
309C3×C4—62no—
310C3×C4—121no—
311C3×C4—181no—
312C3×C4—182no—
313C3×C4—361no—
314C3×C4C2×C3×D1211yescharacteristic
315C3×C4C3×D3×D411yescharacteristic
316Dic3—62no—
317Dic3—61no—
318Dic3—121no—
319C32#480911yescharacteristic
320C9—21no—
321C9C3×#45111yescharacteristic, complemented
322C23—91no—
323Q8—21no—
324Q8—92no—
325Q8—184no—
326Q8—181no—
327Q8—181no—
328Q8—181no—
329Q8C4⋊AGL(1,ℤ/9)12yes—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—91no—
333C2×C4—184no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
339C8—21no—
340C8—181no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—31no—
344C2×C3—182no—
345C2×C3C3×#88211yescharacteristic
346C2×C3C3×#91511yescharacteristic
347C2×C3C3×D3×Q1611yescharacteristic
348D3—62no—
349C22—92no—
350C22—181no—
351C22—181no—
352C22D4×AGL(1,ℤ/9)11yescharacteristic
353C4—22no—
354C4—41no—
355C4—181no—
356C4—182no—
357C4—361no—
358C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
359C4D4×AGL(1,ℤ/9)11yescharacteristic
360C3—31no—
361C3C3×#480911yescharacteristic
362C2—182no—
363C2#63444⋊C311yescharacteristic
364C2#63477⋊C311yescharacteristic
365C2Q16×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#103C2
socleC22×C3C3×D3×D4
prime radicalC22×AGL(1,ℤ/9)D4
hypercenter#103AGL(1,ℤ/9)
hypocenterC9C3×#451
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. C2×#3674⋊C3C2semidirect
3. C2×Dic36⋊C3C2semidirect
5. 3−1+2×#103C2semidirect
7. #63368⋊C3C2semidirect
9. #10588489C3semidirect
33. C2×#3674C2×C3semidirect
34. C2×Dic36C2×C3semidirect
37. C9×#103C2×C3semidirect
39. #63368C2×C3semidirect
97. C2×AGL(1,ℤ/9)Q16semidirect
213. C2×D9C3×Q16semidirect
230. #103AGL(1,ℤ/9)semidirect
232. 3−1+2#451semidirect
321. C9C3×#451semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×AGL(1,ℤ/9)2
C2×#3674⋊C32
C2×Dic36⋊C32
#63459⋊C32
3−1+2×#1032
(C2×AGL(1,ℤ/9))⋊C82
#63368⋊C32
C3×#48093
#105884893