Finite Group Databasebeta

← #10588493⋊C3 #13141956

Subgroups — #10588493⋊C3 #13141956

Order 1728.

363 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 94 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
1#63508⋊C3C211yesmaximal, characteristic
2C2×#3674⋊C3C211yesmaximal, characteristic, complemented
3C2×C8×AGL(1,ℤ/9)C211yesmaximal, characteristic
4#63459⋊C3C211yesmaximal, characteristic
53−1+2×#103C211yesmaximal, characteristic, complemented
6#63376⋊C3C211yesmaximal, characteristic, complemented
7#63366⋊C3C211yesmaximal, characteristic, complemented
8C3×#4813—31nomaximal
9#10588493C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
12C2×3−1+2⋊Q8C2211yescharacteristic
13C2×C8×3−1+2C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15#3674⋊C3—22no—
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17(C2×AGL(1,ℤ/9))⋊C4—21no—
18C4⋊(3−1+2⋊C4)—21no—
19C4⋊(3−1+2⋊C4)C2211yescharacteristic
20(3−1+2⋊C4)⋊C4—21no—
21C2×3−1+2⋊C8C2211yescharacteristic
22C8×AGL(1,ℤ/9)—22no—
23C3×#946—31no—
24C2×C3×#216—31no—
25C2×C3×D3×C8—31no—
26C3×#897—31no—
27C32×#103D311yescharacteristic
28C3×#816—31no—
29C3×#807—31no—
30#63508C2×C311yescharacteristic
31C2×#3674C2×C311yescharacteristic, complemented
32C2×C8×D9C2×C311yescharacteristic
33#63459C2×C311yescharacteristic
34C9×#103—21no—
35C9×#103C2×C311yescharacteristic, complemented
36#63376C2×C311yescharacteristic, complemented
37#63366C2×C311yescharacteristic, complemented
38C22×AGL(1,ℤ/9)D411yescharacteristic
39Q8×3−1+2—21no—
40Q8×3−1+2—21no—
41C2×C4×3−1+2—21no—
42C2×C4×3−1+2—21no—
43C2×C4×3−1+2C2311yescharacteristic
44C2×3−1+2⋊C4—21no—
45C2×3−1+2⋊C4—21no—
46C2×3−1+2⋊C4D411yescharacteristic
47C4×AGL(1,ℤ/9)—21no—
48C4×AGL(1,ℤ/9)—21no—
493−1+2⋊Q8—21no—
503−1+2⋊Q8—21no—
51C8×3−1+2—21no—
523−1+2⋊C8D412yes—
53C3×#467—91no—
54#4813—31no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C3×C4×D3—31no—
57C2×C3×Dic6—31no—
58C2×C32×C8C2×D311yescharacteristic
59C32×C4⋊C4C2×D311yescharacteristic
60C3×#216—62no—
61C3×(C2×D3)⋊C4—61no—
62C3×(C2×D3)⋊C4—61no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—61no—
65C3×Dic3⋊C4—61no—
66C2×C3×C3⋊C8—31no—
67C3×D3×C8—62no—
68C2×Q8×C9—21no—
69C2×Q8×C9C22×C311yescharacteristic
70C2×C4×D9C22×C311yescharacteristic
71C2×Dic18C22×C311yescharacteristic
72C2×C8×C9—21no—
73C2×C8×C9C22×C311yescharacteristic
74C9×C4⋊C4—21no—
75C9×C4⋊C4C22×C311yescharacteristic
76#3674—22no—
77(C2×D9)⋊C4—21no—
78(C2×D9)⋊C4—21no—
79C4⋊Dic9—21no—
80C4⋊Dic9C22×C311yescharacteristic
81Dic9⋊C4—21no—
82C2×C9⋊C8C22×C311yescharacteristic
83C8×D9—22no—
84C22×3−1+2C2×D411yescharacteristic
85C2×AGL(1,ℤ/9)—21no—
86C2×AGL(1,ℤ/9)—21no—
87C2×AGL(1,ℤ/9)Q1612yescomplemented
88C4×3−1+2—41no—
89C4×3−1+2—41no—
90C4×3−1+2C2×D411yescharacteristic
91C4×3−1+2#4111yescharacteristic
923−1+2⋊C4—21no—
933−1+2⋊C4—41no—
943−1+2⋊C4—41no—
95C2×C3×Q16—91no—
96C22×C3×C8—91no—
97C3×C22⋊Q8—91no—
98C3×C22⋊Q8—91no—
99#946—31no—
100C2×#216—31no—
101C2×D3×C8—31no—
102#897—31no—
103C3×#107—91no—
104C3×#103—31no—
105C3×#103—91no—
106C3×#103C3×D311yescharacteristic
107#816—31no—
108#807—31no—
109C22×C3×D3—31no—
110C32×Q8—21no—
111C32×Q8—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3×Dic3—31no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C3×C4×D3—61no—
119C3×C4×D3—61no—
120C3×Dic6—61no—
121C3×Dic6—61no—
122C22×D9C3×D411yescharacteristic
123C32×C8—21no—
124C3×C3⋊C8—32no—
125Q8×C9—22no—
126Q8×C9—21no—
127Q8×C9—21no—
128Q8×C9—41no—
129C2×C4×C9—21no—
130C2×C4×C9—21no—
131C2×C4×C9—21no—
132C2×C4×C9—41no—
133C2×C4×C9—41no—
134C2×C4×C9C23×C311yescharacteristic
135C2×Dic9—21no—
136C2×Dic9—21no—
137C2×Dic9C3×D411yescharacteristic
138C4×D9—21no—
139C4×D9—21no—
140Dic18—21no—
141Dic18—21no—
142C8×C9—21no—
143C8×C9—41no—
144C9⋊C8C3×D412yes—
145#467—91noSylow
146C2×3−1+2#12111yescharacteristic
147C2×3−1+2C2×Q1611yescharacteristic
148C2×3−1+2#13511yescharacteristic
149AGL(1,ℤ/9)—22no—
150C2×C3×Q8—31no—
151C2×C3×Q8—91no—
152C2×C3×Q8C2×C3×D311yescharacteristic
153C22×C3×C4—91no—
154C2×C4×D3—31no—
155C2×Dic6—31no—
156C3×Q16—182no—
157C2×C3×C8—31no—
158C2×C3×C8—91no—
159C2×C3×C8—182no—
160C2×C3×C8C2×C3×D311yescharacteristic
161C3×C4⋊C4—31no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—181no—
164C3×C4⋊C4—181no—
165C3×C4⋊C4C2×C3×D311yescharacteristic
166C3×C22⋊C4—181no—
167C3×C22⋊C4—181no—
168#216—62no—
169(C2×D3)⋊C4—61no—
170(C2×D3)⋊C4—61no—
171C4⋊Dic3—31no—
172C4⋊Dic3—61no—
173Dic3⋊C4—61no—
174C2×C3⋊C8—31no—
175D3×C8—62no—
176C22×C32D3×D411yescharacteristic
177C2×C3×D3—32no—
178C2×C3×D3—61no—
179C2×C3×D3—61no—
180C32×C4—41no—
181C32×C4—41no—
182C32×C4#23511yescharacteristic
183C32×C4D3×D411yescharacteristic
184C3×Dic3—61no—
185C3×Dic3—121no—
186C3×Dic3—121no—
187C22×C9—21no—
188C22×C9C2×C3×D411yescharacteristic
189C2×D9—21no—
190C2×D9—21no—
191C2×D9C3×Q1612yescomplemented
192C4×C9—21no—
193C4×C9—21no—
194C4×C9—42no—
195C4×C9—41no—
196C4×C9—41no—
197C4×C9—81no—
198C4×C9C2×C3×D411yescharacteristic
199C4×C9C3×#4111yescharacteristic
200Dic9—21no—
201Dic9—41no—
202Dic9—41no—
203C2×Q16—91no—
204C22×C8—91no—
205C22⋊Q8—91no—
206C22⋊Q8—91no—
207#107—91no—
208#103—91no—
209#103AGL(1,ℤ/9)11yescharacteristic, complemented
2103−1+2#46711yescharacteristic, complemented, Sylow
211C23×C3—91no—
212C22×D3—31no—
213C3×Q8—21no—
214C3×Q8—21no—
215C3×Q8—61no—
216C3×Q8—61no—
217C3×Q8—181no—
218C3×Q8—181no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—31no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4C22×C3×D311yescharacteristic
230C2×Dic3—31no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C4×D3—61no—
234C4×D3—61no—
235Dic6—61no—
236Dic6—61no—
237C3×C8—21no—
238C3×C8—61no—
239C3×C8—92no—
240C3⋊C8—32no—
241C2×C32#88411yescharacteristic
242C2×C32#91611yescharacteristic
243C2×C32D3×Q1611yescharacteristic
244C3×D3—62no—
245C2×C9—21no—
246C2×C9—21no—
247C2×C9—21no—
248C2×C9C3×#12111yescharacteristic
249C2×C9C2×C3×Q1611yescharacteristic
250C2×C9C3×#13511yescharacteristic
251D9—22no—
252C2×Q8—91no—
253C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
254C22×C4—91no—
255Q16—182no—
256C2×C8—91no—
257C2×C8—182no—
258C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
259C4⋊C4—91no—
260C4⋊C4—181no—
261C4⋊C4—181no—
262C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
263C22⋊C4—181no—
264C22⋊C4—181no—
265C22×C3—31no—
266C22×C3—92no—
267C22×C3—181no—
268C22×C3—181no—
269C22×C3C3×D3×D411yescharacteristic
270C2×D3—32no—
271C2×D3—61no—
272C2×D3—61no—
273C3×C4—31no—
274C3×C4—31no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—121no—
278C3×C4—121no—
279C3×C4—181no—
280C3×C4—361no—
281C3×C4—361no—
282C3×C4C3×#23511yescharacteristic
283C3×C4C3×D3×D411yescharacteristic
284Dic3—61no—
285Dic3—121no—
286Dic3—121no—
287C32#481311yescharacteristic
288C9—21no—
289C9C3×#46711yescharacteristic, complemented
290C23—91no—
291Q8—21no—
292Q8—21no—
293Q8—181no—
294Q8—181no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—91no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4—181no—
301C2×C4—181no—
302C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
303C8—21no—
304C8—92no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—182no—
309C2×C3C3×#88411yescharacteristic
310C2×C3C3×#91611yescharacteristic
311C2×C3C3×D3×Q1611yescharacteristic
312D3—62no—
313C22—92no—
314C22—181no—
315C22—181no—
316C22D4×AGL(1,ℤ/9)11yescharacteristic
317C4—41no—
318C4—41no—
319C4—181no—
320C4—361no—
321C4—361no—
322C4#3697⋊C311yescharacteristic
323C4D4×AGL(1,ℤ/9)11yescharacteristic
324C3—31no—
325C3C3×#481311yescharacteristic
326C2—182no—
327C2#63446⋊C311yescharacteristic
328C2#63478⋊C311yescharacteristic
329C2Q16×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×#467
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
5. 3−1+2×#103C2semidirect
6. #63376⋊C3C2semidirect
7. #63366⋊C3C2semidirect
9. #10588493C3semidirect
31. C2×#3674C2×C3semidirect
35. C9×#103C2×C3semidirect
36. #63376C2×C3semidirect
37. #63366C2×C3semidirect
87. C2×AGL(1,ℤ/9)Q16semidirect
191. C2×D9C3×Q16semidirect
209. #103AGL(1,ℤ/9)semidirect
210. 3−1+2#467semidirect
289. C9C3×#467semidirect

Maximal subgroups

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

subgroupindex
#63508⋊C32
C2×#3674⋊C32
C2×C8×AGL(1,ℤ/9)2
#63459⋊C32
3−1+2×#1032
#63376⋊C32
#63366⋊C32
C3×#48133
#105884933